Boundary Value Problems
|
|
- Christal Boyd
- 6 years ago
- Views:
Transcription
1 Volume 29 Boundary Value Problems Editor-in-Chief: Ravi P. Agarwal Special Issue Singular Boundary Value Problems for Ordinary Differential Equations Guest Editors Juan J. Nieto and Donal O Regan Hindawi Publishing Corporation
2 Singular Boundary Value Problems for Ordinary Differential Equations
3
4 Boundary Value Problems Singular Boundary Value Problems for Ordinary Differential Equations Guest Editors: Juan J. Nieto and Donal O Regan
5 Copyright 29 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in volume 29 of Boundary Value Problems. All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
6 Editor-in-Chief Ravi P. Agarwal, Florida Institute of Technology, USA Associate Editors Ugur G. Abdulla, USA Peter Bates, USA Michel C. Chipot, Switzerland Emmanuele Dibenedetto, USA Pavel Drábek, Czech Republic Lawrence C. Evans, USA Robert Finn, USA Avner Friedman, USA Robert Bob Gilbert, USA Nobuyuki Kenmochi, Japan Ivan T. Kiguradze, Georgia V. Lakshmikantham, USA Gary M. Lieberman, USA Raul F. Manásevich, Chile Jean Mawhin, Belgium Patrick Joseph McKenna, USA Salim A. Messaoudi, Saudi Arabia Donal O Regan, Ireland Kanishka Perera, USA Irena Rachůnkova, Czech Republic Vicentiu D. Radulescu, Romania Colin Rogers, Australia Sandro Salsa, Italy Martin D. Schechter, USA Veli B. Shakhmurov, Turkey RogerM.Temam,USA Roberto Triggiani, USA Zhitao Zhang, China Wenming Zou, China
7
8 Contents Singular Boundary Value Problems for Ordinary Differential Equations, Juan J. Nieto and Donal O Regan Volume 29, Article ID 89529, 2 pages A Viral Infection Model with a Nonlinear Infection Rate,YumeiYu,JuanJ.Nieto,AngelaTorres, and Kaifa Wang Volume 29, Article ID 95816, 19 pages An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces, Mouffak Benchohra, Alberto Cabada, and Djamila Seba Volume 29, Article ID , 11 pages Antiperiodic Boundary Value Problems for Finite Dimensional Differential Systems,Y.Q.Chen, D. O Regan, F. L. Wang, and S. L. Zhou Volume 29, Article ID , 11 pages Constant Sign and Nodal Solutions for Problems with the p-laplacian and a Nonsmooth Potential Using Variational Techniques, Ravi P. Agarwal, Michael E. Filippakis, Donal O Regan, and Nikolaos S. Papageorgiou Volume 29, Article ID 82237, 32 pages Electroelastic Wave Scattering in a Cracked Dielectric Polymer under a Uniform Electric Field, Yasuhide Shindo and Fumio Narita Volume 29, Article ID , 27 pages Existence and Exponential Stability of Positive Almost Periodic Solutions for a Model of Hematopoiesis, J. O. Alzabut, J. J. Nieto, and G. Tr. Stamov Volume 29, Article ID 12751, 1 pages Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems, J. Caballero Mena, J. Harjani, and K. Sadarangani Volume 29, Article ID 42131, 1 pages Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular m-point Boundary Value Problems, Xinsheng Du and Zengqin Zhao Volume 29, Article ID , 13 pages Existence of Periodic Solution for a Nonlinear Fractional Differential Equation, Mohammed Belmekki, Juan J. Nieto, and Rosana Rodríguez-López Volume 29, Article ID , 18 pages Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses, Jianli Li and Juan J. Nieto Volume 29, Article ID , 12 pages Existence of Solutions for Fractional Differential Inclusions with Antiperiodic Boundary Conditions, Bashir Ahmad and Victoria Otero-Espinar Volume 29, Article ID , 11 pages
9 Existence Results for Nonlinear Boundary Value Problems of Fractional Integrodifferential Equations with Integral Boundary Conditions, Bashir Ahmad and Juan J. Nieto Volume 29, Article ID 78576, 11 pages First-Order Singular and Discontinuous Differential Equations, Daniel C. Biles and Rodrigo López Pouso Volume 29, Article ID 57671, 25 pages Homoclinic Solutions of Singular Nonautonomous Second-Order Differential Equations, Irena RachůnkováandJanTomeček Volume 29, Article ID , 21 pages Limit Properties of Solutions of Singular Second-Order Differential Equations,IrenaRachůnková, SvatoslavStaněk, Ewa Weinmüller,and Michael Zenz Volume 29, Article ID 95769, 28 pages Multiplicity Results Using Bifurcation Techniques for a Class of Fourth-Order m-point Boundary Value Problems, Yansheng Liu and Donal O Regan Volume 29, Article ID 97135, 2 pages Multipoint Singular Boundary-Value Problem for Systems of Nonlinear Differential Equations, Jaromír Baštinec,Josef Diblík, and Zdeněk Šmarda Volume 29, Article ID , 2 pages New Results on Multiple Solutions for Nth-Order Fuzzy Differential Equations under Generalized Differentiability, A. Khastan, F. Bahrami, and K. Ivaz Volume 29, Article ID , 13 pages On Some Generalizations Bellman-Bihari Result for Integro-Functional Inequalities for Discontinuous Functions and Their Applications, Angela Gallo and Anna Maria Piccirillo Volume 29, Article ID 88124, 14 pages On Step-Like Contrast Structure of Singularly Perturbed Systems, Mingkang Ni and Zhiming Wang Volume 29, Article ID , 17 pages Positive Solutions for a Class of Coupled System of Singular Three-Point Boundary Value Problems, Naseer Ahmad Asif and Rahmat Ali Khan Volume 29, Article ID 27363, 18 pages Positive Solutions of Singular Multipoint Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations on Infinite Intervals in Banach Spaces, Xingqiu Zhang Volume 29, Article ID 97865, 22 pages Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales, Liang-Gen Hu, Ti-Jun Xiao, and Jin Liang Volume 29, Article ID 93764, 19 pages
10 Contents Recent Existence Results for Second-Order Singular Periodic Differential Equations,JifengChuand Juan J. Nieto Volume 29, Article ID 54863, 2 pages Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems,IgorBoglaev Volume 29, Article ID 3266, 17 pages
11 Hindawi Publishing Corporation Boundary Value Problems Volume 29, Article ID 89529, 2 pages doi:1.1155/29/89529 Editorial Singular Boundary Value Problems for Ordinary Differential Equations Juan J. Nieto 1 and Donal O Regan 2 1 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, Spain 2 Department of Mathematics, National University of Ireland, Galway, Ireland Correspondence should be addressed to Juan J. Nieto, juanjose.nieto.roig@usc.es Received 31 December 29; Accepted 31 December 29 Copyright q 29 J. J. Nieto and D. O Regan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Singular boundary value problems for ordinary differential equations model many real world phenomena ranging from different physics equations to biological, physiological, and medical processes 1 3. This special issue places its emphasis on the study, theory, and applications of boundary value problems involving singularities. It includes some review articles such as 4, equations with discontinuous nonlinearities 5, boundary value problems with uncertainty 6, fractional differential equations 7, periodic or antiperiodic solutions 8, and biological 9 or medical applications 1. Different methods and techniques are used ranging from variational methods 11 to bifurcation techniques 12. The editors aimed at a volume that may serve as a reference in the topic of the special issue and collect twenty five original and cutting-edge research articles by some of the top researchers in boundary value problems for ordinary differential equations worldwide and from many different countries Algeria, Austria, Bulgaria, China, Czech Republic, Greece, Iran, Ireland, Italy, Japan, New Zealand, Pakistan, Saudi Arabia, South Korea, Spain, USA. We would like to thank the authors for their contributions, the Editor-in-Chief of the journal, Professor Ravi P. Agarwal, and the Editorial Office of the journal for their support. References 1 R. P. Agarwal and D. O Regan, Singular Differential and Integral Equations with Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, A. Cabada, E. Liz, and J. J. Nieto, Mathematical Models in Engineering, Biology and Medicine, American Institute of Physics, Melville, NY, USA, D. O Regan, Theory of Singular Boundary Value Problems, World Scientific Publishing, River Edge, NJ, USA, 1994.
12 2 Boundary Value Problems 4 J. Chu and J. J. Nieto, Recent existence results for second-order singular periodic differential equations, Boundary Value Problems, vol. 29, Article ID 54863, 2 pages, D. C. Biles and R. López-Pouso, First-order singular and discontinuous differential equations, Boundary Value Problems, vol. 29, Article ID 57671, 25 pages, A. Khastan, F. Bahrami, and K. Ivaz, New results on multiple solutions for Nth-order fuzzy differential equations under generalized differentiability, Boundary Value Problems, vol. 29, Article ID , 13 pages, M. Belmekki, J. J. Nieto, and R. Rodríguez-López, Existence of periodic solution for a nonlinear fractional differential equation, Boundary Value Problems, vol. 29, Article ID , 18 pages, Y. Q. Chen, D. O Regan, F. L. Wang, and S. L. Zhou, Antiperiodic boundary value problems for finite dimensional differential systems, Boundary Value Problems, vol. 29, Article ID , 11 pages, Y. Yu, J. J. Nieto, A. Torres, and K. Wang, A viral infection model with a nonlinear infection rate, Boundary Value Problems, vol. 29, Article ID 95816, 19 pages, J. O. Alzabut, J. J. Nieto, and G. Tr. Stamov, Existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis, Boundary Value Problems, vol. 29, Article ID 12751, 1 pages, R. P. Agarwal, M. E. Filippakis, D. O Regan, and N. S. Papageorgiou, Constant sign and nodal solutions for problems with the p-laplacian and a nonsmooth potential using variational techniques, Boundary Value Problems, vol. 29, Article ID 82237, 32 pages, Y. Liu and D. O Regan, Multiplicity results using bifurcation techniques for a class of fourth-order m-point boundary value problems, Boundary Value Problems, vol. 29, Article ID 97135, 2 pages, 29.
13 Hindawi Publishing Corporation Boundary Value Problems Volume 29, Article ID 95816, 19 pages doi:1.1155/29/95816 Research Article A Viral Infection Model with a Nonlinear Infection Rate Yumei Yu, 1 Juan J. Nieto, 2 Angela Torres, 3 and Kaifa Wang 4 1 School of Science, Dalian Jiaotong University, Dalian 11628, China 2 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santioga de compostela, Spain 3 Departamento de Psiquiatría, Radiología ysaludpública, Facultad de Medicina, Universidad de Santiago de Compostela, Santioga de compostela, Spain 4 Department of Computers Science, Third Military Medical University, Chongqing 438, China Correspondence should be addressed to Kaifa Wang, kaifawang@yahoo.com.cn Received 28 February 29; Revised 23 April 29; Accepted 27 May 29 Recommended by Donal O Regan A viral infection model with a nonlinear infection rate is constructed based on empirical evidences. Qualitative analysis shows that there is a degenerate singular infection equilibrium. Furthermore, bifurcation of cusp-type with codimension two i.e., Bogdanov-Takens bifurcation is confirmed under appropriate conditions. As a result, the rich dynamical behaviors indicate that the model can display an Allee effect and fluctuation effect, which are important for making strategies for controlling the invasion of virus. Copyright q 29 Yumei Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Mathematical models can provide insights into the dynamics of viral load in vivo. A basic viral infection model 1 has been widely used for studying the dynamics of infectious agents such as hepatitis B virus HBV, hepatitis C virus HCV, and human immunodeficiency virus HIV, which has the following forms: dx λ dx βxv, dt dy βxv ay, dt dv ky uv, dt 1.1
14 2 Boundary Value Problems where susceptible cells x t are produced at a constant rate λ, die at a density-dependent rate dx, and become infected with a rate βuv; infected cells y t are produced at rate βuv and die at a density-dependent rate ay; free virus particles v t are released from infected cells at the rate ky and die at a rate uv. Recently, there have been many papers on virus dynamics within-host in different aspects based on the 1.1. For example, the influences of spatial structures on virus dynamics have been considered, and the existence of traveling waves is established via the geometric singular perturbation method 2. For more literature, we list 3, 4 and references cited therein. Usually, there is a plausible assumption that the amount of free virus is simply proportional to the number of infected cells because the dynamics of the virus is substantially faster than that of the infected cells, u a, k λ. Thus, the number of infected cells y t can also be considered as a measure of virus load v t e.g., see 5 7. As a result, the model 1.1 is reduced to dx λ dx βxy, dt dy βxy ay. dt 1.2 As for this model, it is easy to see that the basic reproduction number of virus is given by R βλ/ad, which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process. Furthermore, we know that the infection-free equilibrium E λ/d, is globally asymptotically stable if R < 1, and so is the infection equilibrium E 1 a/β, βλ ad /aβ if R > 1. Note that both infection terms in 1.1 and 1.2 are based on the mass-action principle Perelson and Nelson 8 ; that is, the infection rate per susceptible cell and per virus is a constant β. However, infection experiments of Ebert et al. 9 and McLean and Bostock 1 suggest that the infection rate of microparasitic infections is an increasing function of the parasite dose and is usually sigmoidal in shape. Thus, as Regoes et al. 11, we take the nonlinear infection rate into account by relaxing the mass-action assumption that is made in 1.2 and obtain dx dt λ dx β y ) x, dy dt β y ) x ay, 1.3 where the infection rate per susceptible cell, β y, is a sigmoidal function of the virus parasite concentration because the number of infected cells y t can also be considered as a measure of virus load e.g., see 5 7, which is represented in the following form: β y ) y/id 5 κ κ, κ > y/id 5 Here, ID 5 denotes the infectious dose at which 5% of the susceptible cells are infected, κ measures the slope of the sigmoidal curve at ID 5 and approximates the average number
15 Boundary Value Problems 3 of virus that enters a single host cell at the begin stage of invasion, y/id 5 κ measures the infection force of the virus, and 1/ 1 y/id 5 κ measures the inhibition effect from the behavioral change of the susceptible cells when their number increases or from the production of immune response which depends on the infected cells. In fact, many investigators have introduced different functional responses into related equations for epidemiological modeling, of which we list and references cited therein. However, a few studies have considered the influences of nonlinear infection rate on virus dynamics. When the parameter κ 1, 18, 19 considered a viral mathematical model with the nonlinear infection rate and time delay. Furthermore, some different types of nonlinear functional responses, in particular of the form βx q y or Holling-type functional response, were investigated in Note that κ>1in 1.4. To simplify the study, we fix the slope κ 2 in the present paper, and system 1.3 becomes dx dt λ dx y 2 x, y2 ID 2 5 dy dt y 2 x ay. ID 2 5 y2 1.5 To be concise in notations, rescale 1.5 by X x/id 5,Y y/id 5. For simplicity, we still use variables x, y instead of X, Y and obtain dx dt m dx y2 1 y 2 x, dy dt y2 1 y x ay, where m λ/id 5.Notethat1/d is the average life time of susceptible cells and 1/a is the average life-time of infected cells. Thus, a d is always valid by means of biological detection. If a d, the virus does not kill infected cells. Therefore, the virus is non cytopathic in vivo. However, when a > d, which means that the virus kills infected cells before its average life time, the virus is cytopathic in vivo. The main purpose of this paper is to study the effect of the nonlinear infection rate on the dynamics of 1.6. We will perform a qualitative analysis and derive the Allee-type dynamics which result from the appearance of bistable states or saddle-node state in 1.6. The bifurcation analysis indicates that 1.6 undergoes a Bogdanov-Takens bifurcation at the degenerate singular infection equilibrium which includes a saddle-node bifurcation, a Hopf bifurcation, and a homoclinic bifurcation. Thus, the nonlinear infection rate can induce the complex dynamic behaviors in the viral infection model. The organization of the paper is as follows. In Section 2, the qualitative analysis of system 1.6 is performed, and the stability of the equilibria is obtained. The results indicate that 1.6 can display an Allee effect. Section 3 gives the bifurcation analysis, which indicates that the dynamics of 1.6 is more complex than that of 1.1 and 1.2. Finally, a brief discussion on the direct biological implications of the results is given in Section 4.
16 4 Boundary Value Problems 2. Qualitative Analysis Since we are interested in virus pathogenesis and not initial processes of infection, we assume that the initial data for the system 1.6 are such that x >, y >. 2.1 The objective of this section is to perform a qualitative analysis of system 1.6 and derive the Allee-type dynamics. Clearly, the solutions of system 1.6 with positive initial values are positive and bounded. Let g y y/ 1 y 2, and note that 1.6 has one and only one infection-free equilibrium E m/d,. Then by using the formula of a basic reproduction number for the compartmental models in van den Driessche and Watmough 24, weknow that the basic reproduction number of virus of 1.6 is R 1 a m d g, 2.2 which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process as zero. Although it is zero, we will show that the virus can still persist in host. We start by studying the equilibria of 1.6. Obviously, the infection-free equilibrium E m/d, always exists and is a stable hyperbolic node because the corresponding characteristic equation is ω d ω a. In order to find the positive infection equilibria, set y2 m dx x, 1 y2 y x a, 1 y2 2.3 then we have the equation a 1 d y 2 my ad. 2.4 Based on 2.4, we can obtain that i there is no infection equilibria if m 2 < 4a 2 d 1 d ; ii there is a unique infection equilibrium E 1 x,y if m 2 4a 2 d 1 d ; iii there are two infection equilibria E 11 x 1, y 1 and E 12 x 2, y 2 if m 2 > 4a 2 d 1 d.
17 Boundary Value Problems 5 Here, x a ) 1 y 2 y, y m 2a 1 d, y 1 m m 2 4a 2 d 1 d, x 1 2a 1 d y 2 m m 2 4a 2 d 1 d, x 2 2a 1 d ) a 1 y 2 1, y 1 a 1 y 2 2 ) y Thus, the surface SN { } m, d, a : m 2 4a 2 d 1 d 2.6 is a Saddle-Node bifurcation surface, that is, on one side of the surface SN system 1.6 has not any positive equilibria; on the surface SN system 1.6 has only one positive equilibrium; on the other side of the surface SN system 1.6 has two positive equilibria. The detailed results will follow. Next, we determine the stability of E 11 and E 12. The Jacobian matrix at E 11 is J E11 d y2 1 1 y 2 1 y y 2 1 2x 1y 1 1 y 2 1 ) 2 a 2x 1y 1 1 y 2 1 ) After some calculations, we have )) det a 1 d 4a ) 2 d 1 d m m 2 4a 2 d 1 d m J E11 ) a 2 1 d m m m 2 4a 2 d 1 d Since m 2 > 4a 2 d 1 d in this case, 4a 2 d 1 d m m 2 4a 2 d 1 d m > is valid. Thus, det J E11 < and the equilibrium E 11 is a saddle. The Jacobian matrix at E 12 is J E12 d y2 2 1 y 2 2 y y 2 2 2x 2y 2 1 y 2 2 ) 2 a 2x 2y 2 1 y 2 2 )
18 6 Boundary Value Problems By a similar argument as above, we can obtain that det J E12 >. Thus, the equilibrium E 12 is a node, or a focus, or a center. For the sake of simplicity, we denote m m ε 2a d 1 d, a 2 1 2d a d 1 a d, if a>2d 1 d. 2.1 We have the following results on the stability of E 12. Theorem 2.1. Suppose that equilibrium E 12 exists; that is, m>m ε.thene 12 is always stable if d a 2d 1 d.whena>2d 1 d, we have i E 12 is stable if m>m ; ii E 12 is unstable if m<m ; iii E 12 is a linear center if m m. Proof. After some calculations, the matrix trace of J E12 is tr ) 2a 3 1 d 1 2d m 1 a d m ) m 2 4a 2 d 1 d J E12 2a 2 1 d m m ), 2.11 m 2 4a 2 d 1 d and its sign is determined by ) F m 2a 3 1 d 1 2d m 1 a d m m 2 4a 2 d 1 d Note that F m 1 a d 2m m 2 4a 2 d 1 d ) m 2 <, 2.13 m2 4a 2 d 1 d which means that F m is a monotone decreasing function of variable m. Clearly, >, if a>2d 1 d, F m ε 2a 2 1 d a 2d 1 d, if a 2d 1 d Note that F m implies that 2a 3 1 d 1 2d m 1 a d m m 2 4a 2 d 1 d. 2.15
19 Boundary Value Problems 7 Squaring 2.15 we find that 4a 6 1 d 2 1 2d 2 m 2 1 a d 2 4a3 1 d 1 2d 1 a d m 2 m 2 4a 2 d 1 d Thus, a 4 1 d 1 2d 2 m 2 1 a d 2 m a 1 2d 1 a d a 2 1 2d a d 1 a d. a d 1 d d, 1 a d 2.17 This means that F m. Thus, under the condition of m > m ε and the sign of F m, tr J E12 < is always valid if a 2d 1 d. When a>2d 1 d, tr J E12 < ifm>m, tr J E12 > ifm<m,andtr J E12 ifm m. For 1.6, its asymptotic behavior is determined by the stability of E 12 if it does not have a limit cycle. Next, we begin to consider the nonexistence of limit cycle in 1.6. Note that E 11 is a saddle and E 12 is a node, a focus, or a center. A limit cycle of 1.6 must include E 12 and does not include E 11. Since the flow of 1.6 moves toward down on the line where y y 1 and x<x 1 and moves towards up on the line where y y 1 and x>x 1, it is easy to see that any potential limit cycle of 1.6 must lie in the region where y>y 1. Take a Dulac function D 1 y 2 /y 2, and denote the right-hand sides of 1.6 by P 1 and P 2, respectively. We have DP 1 x DP 2 y 1 a d y2 a d y 2, 2.18 which is negative if y 2 > a d / 1 a d. Hence, we can obtain the following result. Theorem 2.2. There is no limit cycle in 1.6 if y 2 1 > a d 1 a d Note that y 1 > as long as it exists. Thus, inequality 2.19 is always valid if a d. When a > d, using the expression of y 1 in 2.5, we have that inequality 2.19 that is equivalent to 2a 3 1 d 1 2d 1 a d <m 2 < a 4 1 2d 2 a d 1 a d. 2.2
20 8 Boundary Value Problems Indeed, since m 2 2a 2 1 d 2 y 2 1 m 2 2a 2 1 d d 2 1 d m m2 4a 2 d 1 d, 2a 2 1 d 2 d 1 d a d 1 a d m 2 2a 2 1 d 2 a 1 2d 1 d 1 a d, 2.21 we have 2.19 that is equivalent to m 2 2a 2 1 d a 1 2d 2 1 d 1 a d > m m2 4a 2 d 1 d, a 2 1 d 2 that is, m 2 2a3 1 d 2 1 2d 1 d 1 a d >m m 2 4a 2 d 1 d Thus, m 2 > 2a3 1 d 2 1 2d 1 d 1 a d On the other hand, squaring 2.23 we find that m 4 4a3 1 d 2 1 2d 1 d 1 a d m2 4a6 1 d 4 1 2d 2 1 d 2 1 a d 2 >m4 4a 2 d 1 d m 2, 2.25 which is equivalent to m 2 < a 4 1 2d 2 a d 1 a d The combination of 2.24 and 2.26 yields 2.2. Furthermore, 4a 2 d 1 d < a 4 1 2d 2 a d 1 a d 2.27
21 Boundary Value Problems 9 is equivalent to a / 2d 1 d,both 2a 3 1 d 1 2d 1 a d < 2a 3 1 d 1 2d 1 a d a 4 1 2d 2 a d 1 a d, < 4a 2 d 1 d 2.28 are equivalent to a<2d 1 d. Consequently, we have the following. Corollary 2.3. There is no limit cycle in 1.6 if either of the following conditions hold: i a d and m 2 > 4a 2 d 1 d ; ii d<a<2d 1 d and 4a 2 d 1 d <m 2 <a 4 1 2d 2 / a d 1 a d. When m 2 4a 2 d 1 d, system 1.6 has a unique infection equilibrium E 1.The Jacobian matrix at E 1 is J E1 d y 2 2x y 1 y 2 ) 1 y 2 2 y 2 1 y 2 a x y ) 1 y 2 2 The determinant of J E1 is det J E1 ) a 1 d 4a 2 d 1 d m 2) m 2 4a 2 1 d 2, 2.3 and the trace of J E1 is tr J E1 ) 4a 2 1 d a 2d 1 d m 2 4a 2 1 d Thus, E 1 is a degenerate singular point. Since its singularity, complex dynamic behaviors may occur, which will be studied in the next section. 3. Bifurcation Analysis In this section, the Bogdanov-Takens bifurcation for short, BT bifurcation of system 1.6 is studied when there is a unique degenerate infection equilibrium E 1.
22 1 Boundary Value Problems For simplicity of computation, we introduce the new time τ by dt 1 y 2 dτ, rewrite τ as t,andobtain dx dt m dx my2 1 d xy 2, dy dt ay xy2 ay Note that 3.1 and 1.6 are C -equivalent; both systems have the same dynamics only the time changes. As the above mentioned, assume that H1 m 2 4a 2 d 1 d. Then 3.1 admits a unique positive equilibrium E 1 x,y, where x 2a2 1 2d, y m m 2a 1 d. 3.2 In order to translate the positive equilibrium E 1 to origin, we set X x x,y y y and obtain dx dt 2dX 2aY 2a2 1 d Y 2 m m a XY 1 d XY2, dy dt d 1 d X 2dY m a 1 d XY 2a2 1 d Y 2 XY 2 ay 3. m 3.3 Since we are interested in codimension 2 bifurcation, we assume further that H2 a 2d 1 d. Then, after some transformations, we have the following result. Theorem 3.1. The equilibrium E 1 of 1.6 is a cusp of codimension 2 if H1) and H2) hold; that is, it is a Bogdanov-Takens singularity. Proof. Under assumptions H1 and H2, it is clear that the linearized matrix of 3.3 2d 2a M d 1 d 2d 3.4 has two zero eigenvalues. Let x X, y 2dX 2aY. Since the parameters m, a, d satisfy the assumptions H1 and H2, after some algebraic calculations, 3.3 is transformed into dx dt y md 2a 2 x2 1 d 2m y2 f 1 x, y ), dy dt md2 2d 1 x 2 2md2 m 2d 1 xy y 2 ) f a 2 a 2 4a 2 2 x, y, 3.5
23 Boundary Value Problems 11 where f i x, y, i 1, 2, are smooth functions in variables x, y at least of the third order. Using an affine translation u x y/2d, v y to 3.5,weobtain du dt v m 2a u2 m a 2 uv f 1 u, v, dv dt md2 2d 1 u 2 md a 2 a uv f 2 2 u, v, 3.6 where f i u, v, i 1, 2, are smooth functions in variables u, v at least of order three. To obtain the canonical normal forms, we perform the transformation of variables by Then, 3.6 becomes x u m 2a 2 u2, y v m 2a u dy dt md2 2d 1 x 2 a 2 dx dt y F ) 1 x, y, md 2d 1 a 2 xy F 2 x, y ), 3.8 where F i x, y, i 1, 2, are smooth functions in x, y at least of the third order. Obviously, md 2 2d 1 >, a 2 md 2d 1 >. a This implies that the origin of 3.3, thatis,e 1 of 1.6, is a cusp of codimension 2 by in 25, Theorem 3, Section In the following we will investigate the approximating BT bifurcation curves. The parameters m and a are chosen as bifurcation parameters. Consider the following perturbed system: dx dt m λ 1 dx xy2 1 y, 2 dy dt xy2 1 y a 2 λ 2 y, 3.1 where m,a and d are positive constants while H1 and H2 are satisfied. That is to say, m 2 4a2 d 1 d, a 2d 1 d. 3.11
24 12 Boundary Value Problems λ 1 and λ 2 are in the small neighborhood of, ; x and y are in the small neighborhood of x,y, where x 2a2 1 2d, y m m 2a 1 d Clearly, if λ 1 λ 2, x,y is the degenerate equilibrium E 1 of 1.6. Substituting X x x,y y y into 3.1 and using Taylor expansion, we obtain dx dt 1 y 2) λ 1 d 1 d y 2) X 2 a 1 2d m λ 1 y ) Y m d 1 x λ 1 Y 2 m XY f 1 X, Y, λ, a dy dt y 1 y 2) λ 2 y 2 X 2x y a 1 3y 2) 1 3y 2) ) λ 2 Y 2y XY x 3a y 3y ) λ 2 Y 2 f 2 X, Y, λ, 3.13 where λ λ 1,λ 2, f i X, Y, λ, i 1, 2, are smooth functions of X, Y and λ at least of order three in variables X, Y. Making the change of variables x X, y 2dX 2 a y λ 1 Y to 3.13 and noting the conditions in 3.11 and expressions in 3.12, we have dx dt 1 y 2) m d λ 1 y 2a 2 2 ) d2 λ a 2 1 x a 2 2 λ 1 m 2d dy dt β β 1 x β 2 y β 3 x 2 β 4 xy β 5 y 2 f 2 x, y, λ ), ) y 2 f 1 x, y, λ ), 3.14 where a 2 a y λ 1, β 2d 1 y 2) λ 1 2a 2 y 1 y 2) λ 2, β 1 2d 1 d y λ 1 2d 1 3y 2) λ 2, β 2 1 3y 2) λ 2, β 3 m d 2 2d 1 a a 2 4m d 2 a 2 1 d λ 1 6d2 y a 2 λ 2, 3.15 β 4 2m d 2 a a 2 2m d a 2 1 d λ 1 6dy a 2 λ 2, β 5 m 2d 1 4a 2 a 3y 2a 2 λ 2.
25 Boundary Value Problems 13 f i u, v, λ, i 1, 2, are smooth functions in variables u, v at least of the third order, and the coefficients depend smoothly on λ 1 and λ 2. Let X x y/2d, Y y.using 3.11 and 3.12, after some algebraic calculations, we obtain dx dt c c 1 X c 2 Y c 3 X 2 c 4 XY F 1 X, Y, λ, dy dt e e 1 X e 2 Y e 3 X 2 e 4 XY F 2 X, Y, λ, 3.16 where F i X, Y, λ, i 1, 2, are smooth functions of X, Y and λ at least of the third order in variables X, Y, c 1 c 1 d a 2y 1 y 2) λ 2, c 3 m d 1 d a a 2 y 1 d λ 1 1 3y 2) λ 2, c 2 1 y a λ 1, 3d 2 1 d λ 2 2a d 1 d λ 1 ), c 4 m 1 2dλ 1, a a 2 e 2d 1 y 2) λ 1 2a 2 y 1 y 2) λ 2, 3.17 e 1 2dc 1, e 2 y 1 d λ 1, e 3 m d 2 2d 1 3 a a 2 e 4 m d a a 2 1 d λ 2 4a 1 d λ 1 1 2a 1 d λ 1 ). ), Let x X, y c c 1 X c 2 Y c 3 X 2 c 4 XY F 1 X, Y, λ. Then 3.16 becomes dx dt y, dy dt b b 1 x b 2 y b 3 x 2 b 4 xy b 5 y 2 G x, y, λ ), 3.18
26 14 Boundary Value Problems where b c 2 e c e 2, b 1 c 2 e 1 c 4 e c 1 e 2 c e 4, c 4 b 2 c 1 c e 2, c 2 b 3 c 2 e 3 c 4 e 1 c 3 e 2 c 1 e 4, 3.19 c 4 c 2 4 b 4 2c 3 c 1 c e c 2 c 2 4, 2 b 5 c 4 c 2. G x, y, λ is smooth function in variables x, y at least of order three, and all the coefficients depend smoothly on λ 1 and λ 2. By setting X x b 2 /b 4,Y y to 3.18, weobtain dx Y, dt dy dt r r 1 X b 3 X 2 b 4 XY b 5 Y 2 G 1 X, Y, λ, 3.2 where G 1 X, Y, λ is smooth function in variables X, Y at least of the third order and r b b 2 4 b 1b 2 b 4 b 3 b 2 2, b r 1 b 1b 4 2b 2 b 3 b 4. Now, introducing a new time variable τ to 3.2, which satisfies dt 1 b 5 X dτ, and still writing τ as t, we have dx dt Y 1 b 5X, dy dt r r 1 X b 3 X 2 b 4 XY b 5 Y 2) 1 b 5 X G 2 X, Y, λ, 3.22
27 Boundary Value Problems 15 where G 2 X, Y, λ is smooth function of X, Y and λ at least of three order in variables X, Y. Setting x X, y Y 1 b 5 X to 3.22, weobtain dx y, dt dy dt r q 1 x q 2 x 2 ) b 4 xy G 3 x, y, λ, 3.23 where G 3 x, y, λ is smooth function of x, y and λ at least of order three in variables x, y and q 1 r 1 2r b 5, q 2 r b 2 5 2r 1b 5 b If λ 1 andλ 2, it is easy to obtain the following results: r, q 1, q 2 m d 2 2d 1 a 2 > 3.25 b 4 m d 2d 1 a 2 >. By setting X b 2 4 /q 2 x q 1 b 2 4 /2q2 2,Y b3 4 /q2 2 and τ q 2/b 4 t, and rewriting X, Y, τ as x, y, t, weobtain dx y, dt dy dt μ 1 μ 2 y x 2 ) xy G 4 x, y, λ, 3.26 where μ 1 r b 4 4 q 3 2 q2 1 b4 4, 4q 4 2 μ 2 q 1b 2 4, 2q and G 4 x, y, λ is smooth function of x, y and λ at least of order three in variables x, y. By the theorem of Bogdanov in 26, 27 and the result of Perko in 25, weobtainthe following local representations of bifurcation curves in a small neighborhood Δ of the origin i.e., E 1 of 1.6.
28 16 Boundary Value Problems H III SN H HL SN III II μ 2 I IV μ 1 II SN IV HL I SN Figure 1: The bifurcation set and the corresponding phase portraits of system 3.26 at origin. Theorem 3.2. Let the assumptions H1) and H2) hold. Then 1.6 admits the following bifurcation behaviors: i there is a saddle-node bifurcation curve SN ± { λ 1,λ 2 : μ 1, μ 2 > or μ 2 < }; ii there is a Hopf bifurcation curve H { λ 1,λ 2 : μ 1 μ 2 2 o λ 2,q 1 < }; iii there is a homoclinic-loop bifurcation curve HL { λ 1,λ 2 : μ 1 49/25 μ 2 2 o λ 2 }. Concretely, as the statement in 28, Chapter 3, when μ 1,μ 2 Δ, the orbital topical structure of the system 3.26 at origin corresponding system 1.6 at E 1 is shown in Figure Discussion Note that most infection experiments suggest that the infection rate of microparasitic infections is an increasing function of the parasite dose, usually sigmoidal in shape. In this paper, we study a viral infection model with a type of nonlinear infection rate, which was introduced by Regoes et al. 11. Qualitative analysis Theorem 2.1 implies that infection equilibrium E 12 is always stable if the virus is noncytopathic, a d, or cytopathic in vivo but its cytopathic effect is less than or equal to an appropriate value, a 2d 1 d. When the cytopathic effect of virus is greater than the threshold value, a > 2d 1 d, the stability of the infection equilibrium E 12 depends on the value of parameter m, which is proportional to the birth rate of susceptible cells λ and is in inverse proportion to the infectious dose ID 5. The infection equilibrium is stable if m>m and becomes unstable if m<m. When m gets to the critical value, m m, the infection equilibrium is a linear center, so the oscillation behaviors may occur. If our model 1.6 does not have a limit cycle see Theorem 2.2 and Corollary 2.3, its asymptotic behavior is determined by the stability of E 12. When E 12 is stable, there is a region outside which positive semiorbits tend to E as t tends to infinity and inside
29 Boundary Value Problems y SM E 12 E 11 UM Persistence SM.8 UM.6 Extinction Extinction x Figure 2: Illustrations of the Allee effect for 1.5. Here, λ 17.6, d 1., a 3., ID 5 2. E 17.6, is stable, E , is a saddle point, E , is stable. Note that SM is the stable manifolds of E 11 solid line, UM is the unstable manifolds of E 11 dash line, and the phase portrait of 1.6 is divided into two domains of extinction and persistence of the virus by SM. E which positive semi-orbits tend to E 12 as t tends to infinity; that is, the virus will persist if the initial position lies in the region and disappear if the initial position lies outside this region. Thus, besides the value of parameters, the initial concentration of the virus can also affect the result of invasion. An invasion threshold may exist in these cases, which is typical for the so-called Allee effect that occurs when the abundance or frequency of a species is positively correlated with its growth rate see 11. Consequently, the unrescaled model 1.5 can display an Allee effect see Figure 2, which is an infrequent phenomenon in current viral infection models though it is reasonable and important in viral infection process. Furthermore, when infection equilibrium becomes a degenerate singular point, we have shown that the dynamics of this model are very rich inside this region see Theorems 3.1 and 3.2 and Figure 1. Static and dynamical bifurcations, including saddle-node bifurcation, Hopf bifurcation, homoclinic bifurcation, and bifurcation of cusp-type with codimension two i.e., Bogdanov-Takens bifurcation, have been exhibited. Thus, besides the Allee effect, our model 1.6 shows that the viral oscillation behaviors can occur in the host based on the appropriate conditions, which was observed in chronic HBV or HCV carriers see These results inform that the viral infection is very complex in the development of a better understanding of diseases. According to the analysis, we find that the cytopathic effect of virus and the birth rate of susceptible cells are both significant to induce the complex and interesting phenomena, which is helpful in the development of various drug therapy strategies against viral infection. Acknowledgments This work is supported by the National Natural Science Fund of China nos and , the Natural Science Foundation Project of CQ CSTC 27BB512, and the Science Fund of Third Military Medical University 6XG1.
30 18 Boundary Value Problems References 1 M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, Oxford, UK, 2. 2 K. Wang and W. Wang, Propagation of HBV with spatial dependence, Mathematical Biosciences, vol. 21, no. 1, pp , D.Campos,V.Méndez, and S. Fedotov, The effects of distributed life cycles on the dynamics of viral infections, Journal of Theoretical Biology, vol. 254, no. 2, pp , P. Kr. Srivastava and P. Chandra, Modeling the dynamics of HIV and CD4 T cells during primary infection, Nonlinear Analysis: Real World Applications. In press. 5 C. Bartholdy, J. P. Christensen, D. Wodarz, and A. R. Thomsen, Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in gamma interferon-deficient mice infected with lymphocytic choriomeningitis virus, Journal of Virology, vol. 74, no. 22, pp , 2. 6 S. Bonhoeffer,J.M.Coffin, and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, Journal of Virology, vol. 71, no. 4, pp , D. Wodarz, J. P. Christensen, and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, Trends in Immunology, vol. 23, no. 4, pp , A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, vol. 41, no. 1, pp. 3 44, D. Ebert, C. D. Zschokke-Rohringer, and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, vol. 122, no. 2, pp. 2 29, 2. 1 A. R. McLean and C. J. Bostock, Scrapie infections initiated at varying doses: an analysis of 117 titration experiments, Philosophical Transactions of the Royal Society B, vol. 355, no. 14, pp , R. R. Regoes, D. Ebert, and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proceedings of the Royal Society B, vol. 269, no. 1488, pp , S. Gao, L. Chen, J. J. Nieto, and A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, vol. 24, no , pp , S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, vol. 188, no. 1, pp , S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp , O. Sharomi and A. B. Gumel, Re-infection-induced backward bifurcation in the transmission dynamics of Chlamydia trachomatis, Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp , W. Wang, Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, vol. 3, no. 1, pp , H. Zhang, L. Chen, and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp , D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp , X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp , L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos, Solitons & Fractals, vol. 41, no. 1, pp , W. Wang, J. Shen, and J. J. Nieto, Permanence and periodic solution of predator-prey system with Holling type functional response and impulses, Discrete Dynamics in Nature and Society, vol. 27, Article ID 81756, 15 pages, X. Wang and X. Song, Global stability and periodic solution of a model for HIV infection of CD4 T cells, Applied Mathematics and Computation, vol. 189, no. 2, pp , J. Yang, Dynamics behaviors of a discrete ratio-dependent predator-prey system with Holling type III functional response and feedback controls, Discrete Dynamics in Nature and Society, vol. 28, Article ID , 19 pages, P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, vol. 18, pp , L. Perko, Differential Equations and Dynamical Systems, vol. 7 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 1996.
31 Boundary Value Problems R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta Mathematica Sovietica, vol. 1, pp , R. Bogdanov, Versal deformations of a singular point on the plan in the case of zero eigenvalues, Selecta Mathematica Sovietica, vol. 1, pp , Z. Zhang, C. Li, Z. Zheng, and W. Li, The Base of Bifurcation Theory about Vector Fields, Higher Education Press, Beijing, China, Y. K. Chun, J. Y. Kim, H. J. Woo, et al., No significant correlation exists between core promoter mutations, viral replication, and liver damage in chronic hepatitis B infection, Hepatology, vol. 32, no. 5, pp , 2. 3 G.-H. Deng, Z.-L. Wang, Y.-M. Wang, K.-F. Wang, and Y. Fan, Dynamic determination and analysis of serum virus load in patients with chronic HBV infection, World Chinese Journal of Digestology, vol. 12, no. 4, pp , P. Pontisso, G. Bellati, M. Brunetto, et al., Hepatitis C virus RNA profiles in chronically infected individuals: do they relate to disease activity? Hepatology, vol. 29, no. 2, pp , 1999.
32 Hindawi Publishing Corporation Boundary Value Problems Volume 29, Article ID , 11 pages doi:1.1155/29/ Research Article An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces Mouffak Benchohra, 1 Alberto Cabada, 2 and Djamila Seba 3 1 Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, BP 89, 22 Sidi Bel-Abbès, Algeria 2 Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain 3 Département de Mathématiques, Université de Boumerdès, Avenue de l Indépendance, 35 Boumerdès, Algeria Correspondence should be addressed to Mouffak Benchohra, benchohra@univ-sba.dz Received 3 January 29; Revised 23 March 29; Accepted 15 May 29 Recommended by Juan J. Nieto The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness. Copyright q 29 Mouffak Benchohra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The theory of fractional differential equations has been emerging as an important area of investigation in recent years. Let us mention that this theory has many applications in describing numerous events and problems of the real world. For example, fractional differential equations are often applicable in engineering, physics, chemistry, and biology. See Hilfer 1, Glockle and Nonnenmacher 2, Metzler et al. 3, Podlubny 4, Gaul et al. 5, among others. Fractional differential equations are also often an object of mathematical investigations; see the papers of Agarwal et al. 6, Ahmad and Nieto 7, Ahmad and Otero- Espinar 8, Belarbi et al. 9, Belmekki et al 1, Benchohra et al , Chang and Nieto 14, Daftardar-Gejji and Bhalekar 15, Figueiredo Camargo et al. 16, and the monographs of Kilbas et al. 17 and Podlubny 4. Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain y, y, and so forth. the same requirements of boundary conditions. Caputo s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types, see 18, 19.
33 2 Boundary Value Problems In this paper we investigate the existence of solutions for boundary value problems with fractional order differential equations and nonlinear integral conditions of the form c D r y t f t, y t ), for each t J,T, y y y T y T T T g s, y s ) ds, h s, y s ) ds, 1.1 where c D r, 1 <r 2 is the Caputo fractional derivative, f, g, andh : J E E are given functions satisfying some assumptions that will be specified later, and E is a Banach space with norm. Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics 2 and cellular systems 21. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors such as, for instance, Arara and Benchohra 22, Benchohra et al. 23, 24,Infante 25, Peciulyte et al. 26, and the references therein. In our investigation we apply the method associated with the technique of measures of noncompactness and the fixed point theorem of Mönch type. This technique was mainly initiated in the monograph of Bana and Goebel 27 and subsequently developed and used in many papers; see, for example, Bana and Sadarangoni 28, Guo et al. 29, Lakshmikantham and Leela 3, Mönch 31, and Szufla Preliminaries In this section, we present some definitions and auxiliary results which will be needed in the sequel. Denote by C J, E the Banach space of continuous functions J E, with the usual supremum norm y sup { y t, t J }. 2.1 Let L 1 J, E be the Banach space of measurable functions y : J integrable, equipped with the norm E which are Bochner y L 1 T y s ds. 2.2
34 Boundary Value Problems 3 Let L J, E be the Banach space of measurable functions y : J E which are bounded, equipped with the norm y L inf { c>: y t c, a.e. t J }. 2.3 Let AC 1 J, E be the space of functions y : J E, whose first derivative is absolutely continuous. Moreover, for a given set V of functions v : J E let us denote by V t {v t,v V }, t J, V J {v t : v V }, t J. 2.4 Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness. Definition 2.1 see 27. LetE be a Banach space and Ω E the bounded subsets of E. The Kuratowski measure of noncompactness is the map α : Ω E, defined by { α B inf ɛ>:b } n B i 1 i and diam B i ɛ ; here B Ω E. 2.5 Properties The Kuratowski measure of noncompactness satisfies some properties for more details see 27. a α B B is compact B is relatively compact. b α B α B. c A B α A α B. d α A B α A α B. e α cb c α B ; c R. f α cob α B. Here B and cob denote the closure and the convex hull of the bounded set B, respectively. For completeness we recall the definition of Caputo derivative of fractional order. Definition 2.2 see 17. The fractional order integral of the function h L 1 a, b of order r R ; is defined by Iah r t 1 t Γ r a h s dt, 1 r t s 2.6
35 4 Boundary Value Problems where Γ is the gamma function. When a, we write I r h t h ϕ r t, where ϕ r t tr 1 Γ r for t>, 2.7 ϕ r t fort, and ϕ r δ t as r. Here δ is the delta function. Definition 2.3 see 17. For a function h given on the interval a, b, the Caputo fractionalorder derivative of h, oforderr>, is defined by c Da r h t 1 t Γ n r a h n s ds t s. 1 n r 2.8 Here n r 1and r denotes the integer part of r. Definition 2.4. A map f : J E E is said to be Carathéodory if i t f t, u is measurable for each u E; ii u f t, u is continuous for almost each t J. For our purpose we will only need the following fixed point theorem and the important Lemma. Theorem 2.5 see 31, 33. Let D be a bounded, closed and convex subset of a Banach space such that D, and let N be a continuous mapping of D into itself. If the implication V con V or V N V {} α V 2.9 holds for every subset V of D,thenN has a fixed point. Lemma 2.6 see 32. Let D be a bounded, closed, and convex subset of the Banach space C J, E, G a continuous function on J J, and a function f : J E E satisfies the Carathéodory conditions, and there exists p L 1 J,R such that for each t J and each bounded set B E one has lim k α f J t,k B ) p t α B ; where J t,k t k, t J. 2.1 If V is an equicontinuous subset of D, then { α G s, t f s, y s ) }) ds : y V J J G t, s p s α V s ds. 2.11
36 Boundary Value Problems 5 3. Existence of Solutions Let us start by defining what we mean by a solution of the problem 1.1. Definition 3.1. A function y AC 1 J, E is said to be a solution of 1.1 if it satisfies 1.1. Let σ, ρ 1,ρ 2 : J E be continuous functions and consider the linear boundary value problem c D r y t σ t, t J, y y y T y T T T ρ 1 s ds, ρ 2 s ds. 3.1 Lemma 3.2 see 11. Let 1 <r 2 and let σ, ρ 1,ρ 2 : J E be continuous. A function y is a solution of the fractional integral equation y t P t T G t, s σ s ds 3.2 with T T 1 t t 1 P t ρ 1 s ds ρ 2 s ds, T 2 T 2 t s r 1 1 t T s r 1 1 t T s r 2 G t, s Γ r T 2 Γ r T 2 Γ r 1, s t, 1 t T s r 1 1 t T s r 2 T 2 Γ r T 2 Γ r 1, t s T, T if and only if y is a solution of the fractional boundary value problem 3.1. Remark 3.3. It is clear that the function t T G t, s ds is continuous on J, and hence is bounded. Let { T } G : sup G t, s ds, t J. 3.5
37 6 Boundary Value Problems For the forthcoming analysis, we introduce the following assumptions H1 The functions f, g, h : J E E satisfy the Carathéodory conditions. H2 There exist p f,p g,p h L J,R, such that f t, y ) p f t y for a.e. t J and each y E, g t, y ) p g t y, for a.e. t J and each y E, 3.6 h t, y ) p h t y, for a.e. t J and each y E. H3 For almost each t J and each bounded set B E we have lim k α f J t,k B ) p f t α B, lim k α g J t,k B ) p g t α B, 3.7 lim k α h J t,k B p h t α B. Theorem 3.4. Assume that assumptions H1 H3 hold. If T T 1 T 2 then the boundary value problem 1.1 has at least one solution. [ p g L p h L ] G p f L < 1, 3.8 Proof. We transform the problem 1.1 into a fixed point problem by defining an operator N : C J, E C J, E as ) T Ny t Py t G t, s f s, y s ) ds, 3.9 where P y t T 1 t T 2 T g s, y s ) ds t 1 T 2 T h s, y s ) ds, 3.1 and the function G t, s is given by 3.4. Clearly, the fixed points of the operator N are solution of the problem 1.1. LetR> and consider the set D R { y C J, E : y R } Clearly, the subset D R is closed, bounded, and convex. We will show that N satisfies the assumptions of Theorem 2.5. The proof will be given in three steps.
Research Article A Viral Infection Model with a Nonlinear Infection Rate
Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 958016, 19 pages doi:10.1155/2009/958016 Research Article A Viral Infection Model with a Nonlinear Infection Rate Yumei Yu,
More informationResearch Article On the Stability Property of the Infection-Free Equilibrium of a Viral Infection Model
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume, Article ID 644, 9 pages doi:.55//644 Research Article On the Stability Property of the Infection-Free Equilibrium of a Viral
More informationResearch Article A New Fractional Integral Inequality with Singularity and Its Application
Abstract and Applied Analysis Volume 212, Article ID 93798, 12 pages doi:1.1155/212/93798 Research Article A New Fractional Integral Inequality with Singularity and Its Application Qiong-Xiang Kong 1 and
More informationMEASURE OF NONCOMPACTNESS AND FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES
Communications in Applied Analysis 2 (28), no. 4, 49 428 MEASURE OF NONCOMPACTNESS AND FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES MOUFFAK BENCHOHRA, JOHNNY HENDERSON, AND DJAMILA SEBA Laboratoire
More informationQualitative Analysis of a Discrete SIR Epidemic Model
ISSN (e): 2250 3005 Volume, 05 Issue, 03 March 2015 International Journal of Computational Engineering Research (IJCER) Qualitative Analysis of a Discrete SIR Epidemic Model A. George Maria Selvam 1, D.
More informationResearch Article On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition
Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 378686, 9 pages doi:10.1155/2011/378686 Research Article On the Existence of Solutions for Dynamic Boundary Value
More informationResearch Article Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations
Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 810453, 13 pages doi:10.1155/2010/810453 Research Article Singular Cauchy Initial Value Problem for Certain Classes
More informationResearch Article Uniqueness of Positive Solutions for a Class of Fourth-Order Boundary Value Problems
Abstract and Applied Analysis Volume 211, Article ID 54335, 13 pages doi:1.1155/211/54335 Research Article Uniqueness of Positive Solutions for a Class of Fourth-Order Boundary Value Problems J. Caballero,
More informationResearch Article Solvability of a Class of Integral Inclusions
Abstract and Applied Analysis Volume 212, Article ID 21327, 12 pages doi:1.1155/212/21327 Research Article Solvability of a Class of Integral Inclusions Ying Chen and Shihuang Hong Institute of Applied
More informationGlobal Analysis of an Epidemic Model with Nonmonotone Incidence Rate
Global Analysis of an Epidemic Model with Nonmonotone Incidence Rate Dongmei Xiao Department of Mathematics, Shanghai Jiaotong University, Shanghai 00030, China E-mail: xiaodm@sjtu.edu.cn and Shigui Ruan
More informationFractional order Pettis integral equations with multiple time delay in Banach spaces
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S. Tomul LXIII, 27, f. Fractional order Pettis integral equations with multiple time delay in Banach spaces Mouffak Benchohra Fatima-Zohra Mostefai Received:
More informationResearch Article Existence and Uniqueness Results for Perturbed Neumann Boundary Value Problems
Hindawi Publishing Corporation Boundary Value Problems Volume 2, Article ID 4942, pages doi:.55/2/4942 Research Article Existence and Uniqueness Results for Perturbed Neumann Boundary Value Problems Jieming
More informationSTUDY OF THE DYNAMICAL MODEL OF HIV
STUDY OF THE DYNAMICAL MODEL OF HIV M.A. Lapshova, E.A. Shchepakina Samara National Research University, Samara, Russia Abstract. The paper is devoted to the study of the dynamical model of HIV. An application
More informationResearch Article The Stability of Gauss Model Having One-Prey and Two-Predators
Abstract and Applied Analysis Volume 2012, Article ID 219640, 9 pages doi:10.1155/2012/219640 Research Article The Stability of Gauss Model Having One-Prey and Two-Predators A. Farajzadeh, 1 M. H. Rahmani
More informationResearch Article The Existence of Countably Many Positive Solutions for Nonlinear nth-order Three-Point Boundary Value Problems
Hindawi Publishing Corporation Boundary Value Problems Volume 9, Article ID 575, 8 pages doi:.55/9/575 Research Article The Existence of Countably Many Positive Solutions for Nonlinear nth-order Three-Point
More informationResearch Article A Fixed Point Theorem for Mappings Satisfying a Contractive Condition of Rational Type on a Partially Ordered Metric Space
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2010, Article ID 190701, 8 pages doi:10.1155/2010/190701 Research Article A Fixed Point Theorem for Mappings Satisfying a Contractive
More informationResearch Article Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular m-point Boundary Value Problems
Hindawi Publishing Corporation Boundary Value Problems Volume 29, Article ID 9627, 3 pages doi:.55/29/9627 Research Article Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular
More informationBifurcation Analysis of a SIRS Epidemic Model with a Generalized Nonmonotone and Saturated Incidence Rate
Bifurcation Analysis of a SIRS Epidemic Model with a Generalized Nonmonotone and Saturated Incidence Rate Min Lu a, Jicai Huang a, Shigui Ruan b and Pei Yu c a School of Mathematics and Statistics, Central
More informationFUZZY SOLUTIONS FOR BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXV 1(26 pp. 119 126 119 FUZZY SOLUTIONS FOR BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS A. ARARA and M. BENCHOHRA Abstract. The Banach fixed point theorem
More informationResearch Article Global Dynamics of a Competitive System of Rational Difference Equations in the Plane
Hindawi Publishing Corporation Advances in Difference Equations Volume 009 Article ID 1380 30 pages doi:101155/009/1380 Research Article Global Dynamics of a Competitive System of Rational Difference Equations
More informationFractional Differential Inclusions with Impulses at Variable Times
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 7, Number 1, pp. 1 15 (212) http://campus.mst.edu/adsa Fractional Differential Inclusions with Impulses at Variable Times Mouffak Benchohra
More informationThe Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
Applied Mathematics, 05, 6, 665-675 Published Online September 05 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/046/am056048 The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
More informationOn boundary value problems for fractional integro-differential equations in Banach spaces
Malaya J. Mat. 3425 54 553 On boundary value problems for fractional integro-differential equations in Banach spaces Sabri T. M. Thabet a, and Machindra B. Dhakne b a,b Department of Mathematics, Dr. Babasaheb
More informationResearch Article Existence and Localization Results for p x -Laplacian via Topological Methods
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 21, Article ID 12646, 7 pages doi:11155/21/12646 Research Article Existence and Localization Results for -Laplacian via Topological
More informationSensitivity and Stability Analysis of Hepatitis B Virus Model with Non-Cytolytic Cure Process and Logistic Hepatocyte Growth
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 3 2016), pp. 2297 2312 Research India Publications http://www.ripublication.com/gjpam.htm Sensitivity and Stability Analysis
More informationResearch Article Positive Solutions for Neumann Boundary Value Problems of Second-Order Impulsive Differential Equations in Banach Spaces
Abstract and Applied Analysis Volume 212, Article ID 41923, 14 pages doi:1.1155/212/41923 Research Article Positive Solutions for Neumann Boundary Value Problems of Second-Order Impulsive Differential
More informationON SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES
Applied Mathematics and Stochastic Analysis 15:1 (2002) 45-52. ON SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES M. BENCHOHRA Université de Sidi Bel Abbés Département de Mathématiques
More informationResearch Article Nonlinear Dynamics and Chaos in a Fractional-Order HIV Model
Mathematical Problems in Engineering Volume 29, Article ID 378614, 12 pages doi:1.1155/29/378614 Research Article Nonlinear Dynamics and Chaos in a Fractional-Order HIV Model Haiping Ye 1, 2 and Yongsheng
More informationResearch Article Almost Periodic Solutions of Prey-Predator Discrete Models with Delay
Hindawi Publishing Corporation Advances in Difference Equations Volume 009, Article ID 976865, 19 pages doi:10.1155/009/976865 Research Article Almost Periodic Solutions of Prey-Predator Discrete Models
More informationBIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL WITH A NONLINEAR INCIDENCE
INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 7, Number 1, Pages 61 72 c 2011 Institute for Scientific Computing and Information BIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL
More informationA Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage
Applied Mathematical Sciences, Vol. 1, 216, no. 43, 2121-213 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.216.63128 A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and
More informationDynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:05 55 Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting K. Saleh Department of Mathematics, King Fahd
More informationResearch Article Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients
Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 606149, 15 pages doi:10.1155/2010/606149 Research Article Frequent Oscillatory Behavior of Delay Partial Difference
More informationGlobal Properties for Virus Dynamics Model with Beddington-DeAngelis Functional Response
Global Properties for Virus Dynamics Model with Beddington-DeAngelis Functional Response Gang Huang 1,2, Wanbiao Ma 2, Yasuhiro Takeuchi 1 1,Graduate School of Science and Technology, Shizuoka University,
More informationExistence Results for Multivalued Semilinear Functional Differential Equations
E extracta mathematicae Vol. 18, Núm. 1, 1 12 (23) Existence Results for Multivalued Semilinear Functional Differential Equations M. Benchohra, S.K. Ntouyas Department of Mathematics, University of Sidi
More informationResearch Article Bounds of Solutions of Integrodifferential Equations
Abstract and Applied Analysis Volume 211, Article ID 571795, 7 pages doi:1.1155/211/571795 Research Article Bounds of Solutions of Integrodifferential Equations Zdeněk Šmarda Department of Mathematics,
More informationExistence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions
Existence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions Mouffak Benchohra a,b 1 and Jamal E. Lazreg a, a Laboratory of Mathematics, University
More informationResearch Article Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 008, Article ID 84607, 9 pages doi:10.1155/008/84607 Research Article Generalized Mann Iterations for Approximating Fixed Points
More informationResearch Article The Mathematical Study of Pest Management Strategy
Discrete Dynamics in Nature and Society Volume 22, Article ID 25942, 9 pages doi:.55/22/25942 Research Article The Mathematical Study of Pest Management Strategy Jinbo Fu and Yanzhen Wang Minnan Science
More informationNONLINEAR DIFFERENTIAL EQUATIONS
Profolio of Researchers and Research Teams NONLINEAR DIFFERENTIAL EQUATIONS (Last update 03/04/2013) Code: GI-1561 Department: Análise Matemática Web: www.usc.es/ednl/ Contact: Nieto Roig, Juan José juanjose.nieto.roig@usc.es
More informationEXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO HIGHER-ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 212 (212), No. 234, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationResearch Article Adaptive Control of Chaos in Chua s Circuit
Mathematical Problems in Engineering Volume 2011, Article ID 620946, 14 pages doi:10.1155/2011/620946 Research Article Adaptive Control of Chaos in Chua s Circuit Weiping Guo and Diantong Liu Institute
More informationGLOBAL STABILITY OF SIR MODELS WITH NONLINEAR INCIDENCE AND DISCONTINUOUS TREATMENT
Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 304, pp. 1 8. SSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GLOBAL STABLTY
More informationResearch Article Nonlinear Systems of Second-Order ODEs
Hindawi Publishing Corporation Boundary Value Problems Volume 28, Article ID 236386, 9 pages doi:1.1155/28/236386 Research Article Nonlinear Systems of Second-Order ODEs Patricio Cerda and Pedro Ubilla
More informationResearch Article Existence and Duality of Generalized ε-vector Equilibrium Problems
Applied Mathematics Volume 2012, Article ID 674512, 13 pages doi:10.1155/2012/674512 Research Article Existence and Duality of Generalized ε-vector Equilibrium Problems Hong-Yong Fu, Bin Dan, and Xiang-Yu
More informationResearch Article Nonlinear Boundary Value Problem of First-Order Impulsive Functional Differential Equations
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 21, Article ID 49741, 14 pages doi:1.1155/21/49741 Research Article Nonlinear Boundary Value Problem of First-Order Impulsive
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1. Yong Zhou. Abstract
EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1 Yong Zhou Abstract In this paper, the initial value problem is discussed for a system of fractional differential
More informationON QUADRATIC INTEGRAL EQUATIONS OF URYSOHN TYPE IN FRÉCHET SPACES. 1. Introduction
ON QUADRATIC INTEGRAL EQUATIONS OF URYSOHN TYPE IN FRÉCHET SPACES M. BENCHOHRA and M. A. DARWISH Abstract. In this paper, we investigate the existence of a unique solution on a semiinfinite interval for
More informationResearch Article Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential Equations
Applied Mathematics Volume 2012, Article ID 615303, 13 pages doi:10.1155/2012/615303 Research Article Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential
More informationResearch Article A Delayed Epidemic Model with Pulse Vaccination
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2008, Article ID 746951, 12 pages doi:10.1155/2008/746951 Research Article A Delayed Epidemic Model with Pulse Vaccination
More informationResearch Article Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces
Abstract and Applied Analysis Volume 2012, Article ID 435790, 6 pages doi:10.1155/2012/435790 Research Article Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly
More informationResearch Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation
International Differential Equations Volume 2010, Article ID 764738, 8 pages doi:10.1155/2010/764738 Research Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation
More informationResearch Article Propagation of Computer Virus under Human Intervention: A Dynamical Model
Discrete Dynamics in Nature and ociety Volume 2012, Article ID 106950, 8 pages doi:10.1155/2012/106950 Research Article Propagation of Computer Virus under Human Intervention: A Dynamical Model Chenquan
More informationInitial value problems for singular and nonsmooth second order differential inclusions
Initial value problems for singular and nonsmooth second order differential inclusions Daniel C. Biles, J. Ángel Cid, and Rodrigo López Pouso Department of Mathematics, Western Kentucky University, Bowling
More informationON A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES
J. Nonlinear Funct. Anal. 28 (28, Article ID 2 https://doi.org/.23952/jnfa.28.2 ON A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES SAÏD ABBAS, MOUFFAK
More informationGlobal Analysis of a HCV Model with CTL, Antibody Responses and Therapy
Applied Mathematical Sciences Vol 9 205 no 8 3997-4008 HIKARI Ltd wwwm-hikaricom http://dxdoiorg/02988/ams20554334 Global Analysis of a HCV Model with CTL Antibody Responses and Therapy Adil Meskaf Department
More informationGlobal Stability of a Computer Virus Model with Cure and Vertical Transmission
International Journal of Research Studies in Computer Science and Engineering (IJRSCSE) Volume 3, Issue 1, January 016, PP 16-4 ISSN 349-4840 (Print) & ISSN 349-4859 (Online) www.arcjournals.org Global
More informationResearch Article Strong Convergence of a Projected Gradient Method
Applied Mathematics Volume 2012, Article ID 410137, 10 pages doi:10.1155/2012/410137 Research Article Strong Convergence of a Projected Gradient Method Shunhou Fan and Yonghong Yao Department of Mathematics,
More informationBoundary value problems for fractional differential equations with three-point fractional integral boundary conditions
Sudsutad and Tariboon Advances in Difference Equations 212, 212:93 http://www.advancesindifferenceequations.com/content/212/1/93 R E S E A R C H Open Access Boundary value problems for fractional differential
More informationResearch Article Solvability for a Coupled System of Fractional Integrodifferential Equations with m-point Boundary Conditions on the Half-Line
Abstract and Applied Analysis Volume 24, Article ID 29734, 7 pages http://dx.doi.org/.55/24/29734 Research Article Solvability for a Coupled System of Fractional Integrodifferential Equations with m-point
More informationEXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY. S. H. Saker
Nonlinear Funct. Anal. & Appl. Vol. 10 No. 005 pp. 311 34 EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY S. H. Saker Abstract. In this paper we derive
More informationResearch Article On Decomposable Measures Induced by Metrics
Applied Mathematics Volume 2012, Article ID 701206, 8 pages doi:10.1155/2012/701206 Research Article On Decomposable Measures Induced by Metrics Dong Qiu 1 and Weiquan Zhang 2 1 College of Mathematics
More informationModelling of the Hand-Foot-Mouth-Disease with the Carrier Population
Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population Ruzhang Zhao, Lijun Yang Department of Mathematical Science, Tsinghua University, China. Corresponding author. Email: lyang@math.tsinghua.edu.cn,
More informationA Comparison of Two Predator-Prey Models with Holling s Type I Functional Response
A Comparison of Two Predator-Prey Models with Holling s Type I Functional Response ** Joint work with Mark Kot at the University of Washington ** Mathematical Biosciences 1 (8) 161-179 Presented by Gunog
More informationMultiplesolutionsofap-Laplacian model involving a fractional derivative
Liu et al. Advances in Difference Equations 213, 213:126 R E S E A R C H Open Access Multiplesolutionsofap-Laplacian model involving a fractional derivative Xiping Liu 1*,MeiJia 1* and Weigao Ge 2 * Correspondence:
More informationResearch Article Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings
Discrete Dynamics in Nature and Society Volume 2011, Article ID 487864, 16 pages doi:10.1155/2011/487864 Research Article Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive
More informationResearch Article Common Fixed Points of Weakly Contractive and Strongly Expansive Mappings in Topological Spaces
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 746045, 15 pages doi:10.1155/2010/746045 Research Article Common Fixed Points of Weakly Contractive and Strongly
More informationResearch Article The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem
Journal of Applied Mathematics Volume 2012, Article ID 219478, 15 pages doi:10.1155/2012/219478 Research Article The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity
More informationResearch Article Modeling Computer Virus and Its Dynamics
Mathematical Problems in Engineering Volume 213, Article ID 842614, 5 pages http://dx.doi.org/1.1155/213/842614 Research Article Modeling Computer Virus and Its Dynamics Mei Peng, 1 Xing He, 2 Junjian
More informationGLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS
CANADIAN APPIED MATHEMATICS QUARTERY Volume 13, Number 4, Winter 2005 GOBA DYNAMICS OF A MATHEMATICA MODE OF TUBERCUOSIS HONGBIN GUO ABSTRACT. Mathematical analysis is carried out for a mathematical model
More informationCOMMON FIXED POINT THEOREMS FOR A PAIR OF COUNTABLY CONDENSING MAPPINGS IN ORDERED BANACH SPACES
Journal of Applied Mathematics and Stochastic Analysis, 16:3 (2003), 243-248. Printed in the USA c 2003 by North Atlantic Science Publishing Company COMMON FIXED POINT THEOREMS FOR A PAIR OF COUNTABLY
More informationMonotone Iterative Method for a Class of Nonlinear Fractional Differential Equations on Unbounded Domains in Banach Spaces
Filomat 31:5 (217), 1331 1338 DOI 1.2298/FIL175331Z Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Monotone Iterative Method for
More informationResearch Article The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 376852, 7 pages doi:10.1155/2010/376852 Research Article The Solution by Iteration
More informationEXISTENCE OF MILD SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 241, pp. 1 11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF MILD SOLUTIONS TO PARTIAL
More informationSingularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations
Singularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations Irena Rachůnková, Svatoslav Staněk, Department of Mathematics, Palacký University, 779 OLOMOUC, Tomkova
More informationFRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
Opuscula Math. 37, no. 2 27), 265 28 http://dx.doi.org/.7494/opmath.27.37.2.265 Opuscula Mathematica FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
More informationAustralian Journal of Basic and Applied Sciences. Effect of Personal Hygiene Campaign on the Transmission Model of Hepatitis A
Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: 67-73 ISSN:1991-8178 Australian Journal of Basic and Applied Sciences Journal home page: wwwajbaswebcom Effect of Personal Hygiene
More informationEXISTENCE OF SOLUTIONS TO FRACTIONAL-ORDER IMPULSIVE HYPERBOLIC PARTIAL DIFFERENTIAL INCLUSIONS
Electronic Journal of Differential Equations, Vol. 24 (24), No. 96, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF SOLUTIONS TO
More informationResearch Article Quasilinearization Technique for Φ-Laplacian Type Equations
International Mathematics and Mathematical Sciences Volume 0, Article ID 975760, pages doi:0.55/0/975760 Research Article Quasilinearization Technique for Φ-Laplacian Type Equations Inara Yermachenko and
More informationResearch Article Oscillation Criteria of Certain Third-Order Differential Equation with Piecewise Constant Argument
Journal of Applied Mathematics Volume 2012, Article ID 498073, 18 pages doi:10.1155/2012/498073 Research Article Oscillation Criteria of Certain hird-order Differential Equation with Piecewise Constant
More informationResearch Article Generalized α-ψ Contractive Type Mappings and Related Fixed Point Theorems with Applications
Abstract and Applied Analysis Volume 01, Article ID 793486, 17 pages doi:10.1155/01/793486 Research Article Generalized α-ψ Contractive Type Mappings and Related Fixed Point Theorems with Applications
More informationResearch Article Iterative Approximation of a Common Zero of a Countably Infinite Family of m-accretive Operators in Banach Spaces
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 325792, 13 pages doi:10.1155/2008/325792 Research Article Iterative Approximation of a Common Zero of a Countably
More informationTHE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL
THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL HAL L. SMITH* SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES ARIZONA STATE UNIVERSITY TEMPE, AZ, USA 8587 Abstract. This is intended as lecture notes for nd
More informationResearch Article Some Fixed-Point Theorems for Multivalued Monotone Mappings in Ordered Uniform Space
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 186237, 12 pages doi:10.1155/2011/186237 Research Article Some Fixed-Point Theorems for Multivalued Monotone Mappings
More informationResearch Article An Impulse Model for Computer Viruses
Discrete Dynamics in Nature and Society Volume 2012, Article ID 260962, 13 pages doi:10.1155/2012/260962 Research Article An Impulse Model for Computer Viruses Chunming Zhang, Yun Zhao, and Yingjiang Wu
More informationResearch Article Attracting Periodic Cycles for an Optimal Fourth-Order Nonlinear Solver
Abstract and Applied Analysis Volume 01, Article ID 63893, 8 pages doi:10.1155/01/63893 Research Article Attracting Periodic Cycles for an Optimal Fourth-Order Nonlinear Solver Mi Young Lee and Changbum
More informationThe Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 549-560 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.718 The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive
More informationResearch Article Optimality Conditions of Vector Set-Valued Optimization Problem Involving Relative Interior
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 183297, 15 pages doi:10.1155/2011/183297 Research Article Optimality Conditions of Vector Set-Valued Optimization
More informationResearch Article New Exact Solutions for the 2 1 -Dimensional Broer-Kaup-Kupershmidt Equations
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 00, Article ID 549, 9 pages doi:0.55/00/549 Research Article New Exact Solutions for the -Dimensional Broer-Kaup-Kupershmidt Equations
More informationResearch Article Modulus of Convexity, the Coeffcient R 1,X, and Normal Structure in Banach Spaces
Abstract and Applied Analysis Volume 2008, Article ID 135873, 5 pages doi:10.1155/2008/135873 Research Article Modulus of Convexity, the Coeffcient R 1,X, and Normal Structure in Banach Spaces Hongwei
More informationBritish Journal of Applied Science & Technology 10(2): 1-11, 2015, Article no.bjast ISSN:
British Journal of Applied Science & Technology 10(2): 1-11, 2015, Article no.bjast.18590 ISSN: 2231-0843 SCIENCEDOMAIN international www.sciencedomain.org Solutions of Sequential Conformable Fractional
More informationBackward bifurcation underlies rich dynamics in simple disease models
Backward bifurcation underlies rich dynamics in simple disease models Wenjing Zhang Pei Yu, Lindi M. Wahl arxiv:1504.05260v1 [math.ds] 20 Apr 2015 October 29, 2018 Abstract In this paper, dynamical systems
More informationA Stochastic Viral Infection Model with General Functional Response
Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 9, 435-445 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/nade.16.664 A Stochastic Viral Infection Model with General Functional Response
More informationResearch Article Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to F p, q, s Space on the Unit Ball
Abstract and Applied Analysis Volume 2011, Article ID 152635, 9 pages doi:10.1155/2011/152635 Research Article Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to F p,
More informationBull. Math. Soc. Sci. Math. Roumanie Tome 60 (108) No. 1, 2017, 3 18
Bull. Math. Soc. Sci. Math. Roumanie Tome 6 8 No., 27, 3 8 On a coupled system of sequential fractional differential equations with variable coefficients and coupled integral boundary conditions by Bashir
More informationResearch Article Equivalent Extensions to Caristi-Kirk s Fixed Point Theorem, Ekeland s Variational Principle, and Takahashi s Minimization Theorem
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 970579, 20 pages doi:10.1155/2010/970579 Research Article Equivalent Extensions to Caristi-Kirk s Fixed Point
More informationCorrespondence should be addressed to Yagub A. Sharifov,
Abstract and Applied Analysis Volume 212, Article ID 59482, 14 pages doi:1.1155/212/59482 Research Article Existence and Uniqueness of Solutions for the System of Nonlinear Fractional Differential Equations
More informationResearch Article Strong Convergence Bound of the Pareto Index Estimator under Right Censoring
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 200, Article ID 20956, 8 pages doi:0.55/200/20956 Research Article Strong Convergence Bound of the Pareto Index Estimator
More informationVariational iteration method for q-difference equations of second order
Sichuan University From the SelectedWorks of G.C. Wu Summer June 6, 1 Variational iteration method for -difference euations of second order Guo-Cheng Wu Available at: https://works.bepress.com/gcwu/1/
More informationResearch Article Existence of Periodic Positive Solutions for Abstract Difference Equations
Discrete Dynamics in Nature and Society Volume 2011, Article ID 870164, 7 pages doi:10.1155/2011/870164 Research Article Existence of Periodic Positive Solutions for Abstract Difference Equations Shugui
More information