Research Article Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form
|
|
- Spencer Reynolds
- 5 years ago
- Views:
Transcription
1 Difference Equations Article ID pages Research Article Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form Vu Van Khuong 12 and Tran Hong Thai 1 1 Department of Mathematics Hung Yen University of Technology and Education Khoai Chau Hung Yen Vietnam 2 Department of Mathematical Analysis University of Transport and Communications Dong Da Hanoi Vietnam Correspondence should be addressed to Vu Van Khuong; vuvankhuong@gmail.com Received 31 May 2014; Revised 10 September 2014; Accepted 24 September 2014; Published 13 October 2014 Academic Editor: Honglei Xu Copyright 2014 V. Van Khuong and T. Hong Thai. 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. The goal of this paper is to study the boundedness the persistence and the asymptotic behavior of the positive solutions of the system of two difference equations of exponential form: x n1 (abe y n /(dhy n y n1 (abe x n /( n where a b c dandh are positive constants and the initial values x 0 y 0 are positive real values. Also we determine the rate of convergence of a solution that converges to the equilibrium E(x y of this system. 1. Introduction In [1] the authors studied the boundedness the asymptotic behavior the periodicity and the stability of the positive solutions of the difference equation: y n1 αβey n (1 γy n1 where α β γ are positive constants and the initial values y 1 y 0 are positive numbers. Motivated by the above paper we will extend the above difference equation to a system of difference equations; our goal will be to investigate the boundedness the persistence and the asymptotic behavior of the positive solutions of the following system of exponential form: x n1 abey n dhy n y n1 abex n n where a b c d h are positive constants and the initial values x 0 y 0 are positive real values. (2 Difference equations and systems of difference equations of exponential form can be found in [2 6]. Moreover as difference equations have many applications in applied sciences there are many papers and books that can be found concerning the theory and applications of difference equations; see [7 9] and the references cited therein. 2. Global Behavior of Solutions of System (2 In the first lemma we study the boundedness and persistence ofthepositivesolutionsof(2. Lemma 1. Every positive solution of (2 is bounded and persists. Proof. Let (x n y n be an arbitrary solution of (2. From (2 we can see that x n abc d y n abc n12... (3 d
2 2 Difference Equations In addition from (2 and(3 we get x n abe(abc/d ce (abc/d dh ((abc /d y n abe(abc/d ce (abc/d dh ((abc /d n23... Therefore from (3 and (4 the proof of the lemma is complete. In order to prove the main result of this section we recall the next theorem without its proof. See [10 11]. Theorem 2. Let R [a 1 b 1 ] [c 1 d 1 ] and (4 f:r [a 1 b 1 ] g : R [c 1 d 1 ] (5 be a continuous functions such that the following hold: (a f(x y is decreasing in both variables and g(x y is decreasing in both variables for each (x y R; (b if (m 1 M 1 m 2 M 2 R 2 is a solution of M 1 f(m 1 m 2 m 1 f(m 1 M 2 M 2 g(m 1 m 2 m 2 g(m 1 M 2 then m 1 M 1 and m 2 M 2. Then the following system of difference equations (6 x n1 f(x n y n y n1 g(x n y n (7 has a unique equilibrium (x y and every solution (x n y n of the system (7 with (x 0 y 0 R converges to the unique equilibrium (x y. In addition the equilibrium (x y is globally asymptotically stable. Now we state the main theorem of this section. Theorem 3. Consider system (2. Suppose that the following relation holds true: bc<d. (8 Then system (2 has a unique positive equilibrium (x y and every positive solution of (2 tends to the unique positive equilibrium (x y as n. In addition the equilibrium (x y is globally asymptotically stable. Proof. We consider the functions f (u V abev ce u dhv (9 g (u V abeu ce V dhu where a(bc e(abc/d u V I[ ch ((abc /d abc ]. (10 d It is easy to see that f(u V g(u V are decreasing in both variables for each (u V I I. In addition from (9 and (10 wehavef(u V I g(u V Ias (u V I Iand so f:i I I g:i I I. Now let m 1 M 1 m 2 M 2 be positive real numbers such that M 1 abem 2 ce m 1 dhm 2 M 2 abem 1 ce m 2 dhm 1 m 1 abem 2 ce M 1 dhm 2 m 2 abem 1 ce M 2 dhm 1. (11 Moreover arguing as in the proof of Theorem 2itsufficesto assume that From (11we get which imply that m 1 M 1 m 2 M 2. (12 be m 2 ce m 1 (dhm 2 M 1 a be M 2 ce M 1 (dhm 2 m 1 a be m 1 ce m 2 (dhm 1 M 2 a be M 1 ce M 2 (dhm 1 m 2 a d(m 1 m 1 h(m 1 m 2 M 2 m 1 b(e m 2 e M 2 c(e m 1 e M 1 be m 2M 2 (e M 2 e m 2 ce m 1M 1 (e M 1 e m 1 d(m 2 m 2 h(m 2 m 1 M 2 m 2 b(e m 1 e M 1 c(e m 2 e M 2 (13 be m 1M 1 (e M 1 e m 1 ce m 2M 2 (e M 2 e m 2. (14 Moreover we get e M 1 e m 1 e α (M 1 m 1 m 1 α M 1 e M 2 e m 2 e β (M 2 m 2 m 2 β M 2. Then by adding the two relations (14weobtain d(m 1 m 1 d(m 2 m 2 (bc (15 [e m 1M 1 α (M 1 m 1 e m 2M 2 β (M 2 m 2 ]. (16
3 Difference Equations 3 Therefore from (16we have (M 1 m 1 [d(bc e m 1M 1 α ] (M 2 m 2 [d(bc e m 2M 2 β ]0. (17 Then using (8 (12 and (17 givesusm 1 M 1 and m 2 M 2.HencefromTheorem 2 system (2hasauniquepositive equilibrium (x y and every positive solution of (2 tends totheuniquepositiveequilibrium(x y as n.in addition the equilibrium (xy is globally asymptotically stable. This completes the proof of the theorem. 3. Rate of Convergence In this section we give the rate of convergence of a solution that converges to the equilibrium E (x y of the system (2 for all values of parameters. The rate of convergence of solutions that converge to an equilibrium has been obtained for some two-dimensional systems in [12 13]. The following results give the rate of convergence of solutions of a system of difference equations: x n1 [AB(n] x n (18 where x n is a k-dimensional vector A C k k is a constant matrix and B:Z C k k is a matrix function satisfying B (n 0 when n (19 where denotes any matrix norm which is associated with the vector norm; also denotes the Euclidean norm in R 2 given by x (x y x 2 y 2. (20 Theorem 4 (see [14]. Assume that condition (19 holds. If x n isasolutionofsystem(18 then either x n 0for all large n or ρ lim n n x n (21 exists and is equal to the modulus of one of the eigenvalues of matrix A. Theorem 5 (see [14]. Assume that condition (19 holds. If x n isasolutionofsystem(18 then either x n 0for all large n or ρ lim x n1 n x (22 n exists and is equal to the modulus of one of the eigenvalues of matrix A. The equilibrium point of the system (2 satisfies the following system of equations: x abey ce x dh y (23 y abex ce y. dh x If bc < dwecaneasilyseethatthesystem(23 hasan unique equilibrium E(x x. The map T associated with the system (2is abe y ce x T(xy( f(xy g(xy ( dhy. (24 abe x ce y The Jacobian matrix of T is dhy J T ( be x ( h (abe x ce y ( 2 be y (d hy h (a be y ce x (d hy 2 ce y. (25 By using the system (23 value of the Jacobian matrix of T at the equilibrium point E(x y (x x is J T ( be x ( h [a(bc e x ] ( 2 be x ( h [a(bc e x ] ( 2. (26
4 4 Difference Equations Our goal in this section is to determine the rate of convergence of every solution of the system (2in the regions where the parameters a b c d h (0 (b c < d and initial conditions x 0 and y 0 are arbitrary nonnegative numbers. Theorem 6. The error vector e n ( e1 n ( x nx e 2 y n n y of every solution x n 0 of (2 satisfies both of the following asymptotic relations: lim n n e n λ i (J T (E for some i 1 2 lim e (27 n1 n e n λ i (J T (E for some i 1 2 where λ i (J T (E is equal to the modulus of one of the eigenvalues of the Jacobian matrix evaluated at the equilibrium J T (E. Proof. First we will find a system satisfied by the error terms. The error terms are given as x n1 x abey n dhy n abey ce x dhy ((abe y n (dhy (abe y ce x (dhy n ((d hy n (dhy 1 (bd(e y n e y cd(e x n e x ah(yy n bh(e y n ye y y n ch(e x n ye x y n ((d hy n (dhy 1 bd (e y n e y e yny (d hy n (dhy cd (e x n e x e xnx (d hy n (dhy bh (d hy n (dhy (e y n ye y n y n e y n y n e y y n ch (d hy n (dhy (e x n ye x n y n e x n y n e y y n ah (d hy n (dhy (y n y b e y ny (d hy (ey n e y c e x nx (d hy (ex n e x h abe y n (d hy n (dhy (y n y b e y n (d hy (e y ny 1 c e x n (d hy (e x nx 1 h abe y n (d hy n (dhy (y n y b e y n (d hy [(y n y O 2 ((y n y 2 ] c e x n (d hy [(x n x O 1 ((x n x 2 ] h abe y n (d hy n (dhy (y n y c e x n (d hy (x n x bey n (d hy n h(abe y n (d hy n (dhy (y n y O 1 ((x n x 2 O 2 ((y n y 2. By calculating similarly we get y n1 y bex n (d hx n h(abe x n (d hx n ( c e y n ( (y n y O 3 ((x n x 2 O 4 ((y n y 2. From (28and(29wehave x n1 x c e x n (d hy (x n x (x n x bey n (d hy n h(abe y n (d hy n (dhy (y n y (28 (29
5 Difference Equations 5 Set y n1 y bex n (d hx n h(abe x n (d hx n ( (x n x c e y n ( (y n y. (30 e 1 n x n x e 2 n y n y. (31 Then system (30 can be represented as where e 1 n1 a ne 1 n b ne 2 n (32 e 2 n1 c ne 1 n d ne 2 n a n c e x n (d hy b n bey n (d hy n h(abe y n (d hy n (dhy c n bex n (d hx n h(abe x n (d hx n ( d n c e y n (. (33 Taking the limits of a n b n c n andd n as n weobtain lim a c n n e x ( that is lim b n n bex ( h [a (bc e x ] ( 2 lim c n n bex ( h [a (bc e x ] ( 2 lim d c n n e x ( ; a n c e x ( α n b n bex ( h [a (bc e x ] ( 2 β n c n bex ( h [a (bc e x ] ( 2 γ n d n c e x ( δ n (34 (35 where α n 0 β n 0 γ n 0andδ n 0as n. Now we have system of the form (18: where e n1 (AB(n e n (36 A( be x ( h [a(bc e x ] ( 2 be x ( h [a(bc e x ] ( 2 (37 β n α n B (n ( δ n γ n B (n 0 as n. Thus the limiting system of error terms can be written as ( e 1 n1 e 2 n1 A( e 1 n e 2 n. (38 The system is exactly linearized system of (2 evaluated at the equilibrium E(x y (x x. Then Theorems 4 and 5 imply the result. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] I. Ozturk F. Bozkurt and S. Ozen On the difference equation y n1 (αbe yn /(γy n1 Applied Mathematics and Computationvol.181no.2pp [2] D. C. Zhang and B. Shi Oscillation and global asymptotic stability in a discrete epidemic model Mathematical Analysis and Applicationsvol.278no.1pp
6 6 Difference Equations [3] E. El-Metwally E. A. Grove G. Ladas R. Levins and M. Radin On the difference equation x n 1 (aβ xn 1e x n Nonlinear Analysis: Theory Methods & Applicationsvol.47no. 7 pp [4] G. Papaschinopoulos M. A. Radin and C. J. Schinas On the system of two difference equations of exponential form: x n1 abx n1 e y ny n1 cdy n1 e x n Mathematical and Computer Modellingvol.54no.11-12pp [5] G. Stefanidou G. Papaschinopoulos and C. J. Schinas On a system of two exponential type difference equations Communications on Applied Nonlinear Analysis vol.17no.2pp [6] S. Stević On a discrete epidemic model Discrete Dynamics in Nature and Societyvol.2007ArticleID pages2007. [7] E.A.GroveandG.LadasPeriodicities in Nonlinear Difference Equations Chapman & Hall/CRC [8] R. P. Agarwal Difference Equations and Inequalities Marcel Dekker New York NY USA 2nd edition [9]V.L.KocicandG.LadasGlobal Behavior of Nonlinear Difference Equations of Higher Order with Applications Kluwer Academic Dordrecht The Netherlands [10] M. R. S. Kulenović andg. Ladas Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures Chapman and Hall/CRC Boca Raton Fla USA [11] D. BurgićandZ.Nurkanović An example of a globally asymptotically stable anti-monotonic system of rational difference equations in the plane Sarajevo Mathematics vol. 5no.18pp [12] M. R. S. Kulenović and M. Nurkanović Asymptotic behavior of a competitive system of linear fractional difference equations Advances in Difference Equations vol.2006articleid [13] M.R.S.Kulenović and Z.Nurkanović The rate of convergence of solution of a three dimensional linear fractional systems of difference equations Zbornik radova PMF Tuzla Svezak Matematikavol.2pp [14] M. Pituk More on Poincare s and Peron s theorems for difference equations Difference Equations and Applicationsvol.8pp
7 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization
GLOBAL DYNAMICS OF THE SYSTEM OF TWO EXPONENTIAL DIFFERENCE EQUATIONS
Electronic Journal of Mathematical Analysis an Applications Vol. 7(2) July 209, pp. 256-266 ISSN: 2090-729X(online) http://math-frac.org/journals/ejmaa/ GLOBAL DYNAMICS OF THE SYSTEM OF TWO EXPONENTIAL
More informationResearch Article Global Attractivity of a Higher-Order Difference Equation
Discrete Dynamics in Nature and Society Volume 2012, Article ID 930410, 11 pages doi:10.1155/2012/930410 Research Article Global Attractivity of a Higher-Order Difference Equation R. Abo-Zeid Department
More informationGlobal dynamics of two systems of exponential difference equations by Lyapunov function
Khan Advances in Difference Equations 2014 2014:297 R E S E A R C H Open Access Global dynamics of two systems of exponential difference equations by Lyapunov function Abdul Qadeer Khan * * Correspondence:
More informationReview Article Solution and Attractivity for a Rational Recursive Sequence
Discrete Dynamics in Nature and Society Volume 2011, Article ID 982309, 17 pages doi:10.1155/2011/982309 Review Article Solution and Attractivity for a Rational Recursive Sequence E. M. Elsayed 1, 2 1
More informationResearch Article On the Difference Equation x n 1 x n x n k / x n k 1 a bx n x n k
Abstract and Applied Analysis Volume 2012, Article ID 108047, 9 pages doi:10.1155/2012/108047 Research Article On the Difference Equation x n 1 x n x n k / x n k 1 a bx n x n k Stevo Stević, 1 Josef Diblík,
More informationON THE STABILITY OF SOME SYSTEMS OF EXPONENTIAL DIFFERENCE EQUATIONS. N. Psarros, G. Papaschinopoulos, and C.J. Schinas
Opuscula Math. 38, no. 1 2018, 95 115 https://doi.org/10.7494/opmath.2018.38.1.95 Opuscula Mathematica ON THE STABILITY OF SOME SYSTEMS OF EXPONENTIAL DIFFERENCE EQUATIONS N. Psarros, G. Papaschinopoulos,
More informationTwo Modifications of the Beverton Holt Equation
International Journal of Difference Equations ISSN 0973-5321 Volume 4 Number 1 pp. 115 136 2009 http://campus.mst.edu/ijde Two Modifications of the Beverton Holt Equation G. Papaschinopoulos. J. Schinas
More informationResearch Article Global Attractivity and Periodic Character of Difference Equation of Order Four
Discrete Dynamics in Nature and Society Volume 2012, Article ID 746738, 20 pages doi:10.1155/2012/746738 Research Article Global Attractivity and Periodic Character of Difference Equation of Order Four
More informationResearch Article On Boundedness of Solutions of the Difference Equation x n 1 px n qx n 1 / 1 x n for q>1 p>1
Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 463169, 11 pages doi:10.1155/2009/463169 Research Article On Boundedness of Solutions of the Difference Equation
More informationGlobal Attractivity of a Higher-Order Nonlinear Difference Equation
International Journal of Difference Equations ISSN 0973-6069, Volume 5, Number 1, pp. 95 101 (010) http://campus.mst.edu/ijde Global Attractivity of a Higher-Order Nonlinear Difference Equation Xiu-Mei
More informationDynamics of a Rational Recursive Sequence
International Journal of Difference Equations ISSN 0973-6069, Volume 4, Number 2, pp 185 200 (2009) http://campusmstedu/ijde Dynamics of a Rational Recursive Sequence E M Elsayed Mansoura University Department
More informationGlobal Attractivity of a Rational Difference Equation
Math. Sci. Lett. 2, No. 3, 161-165 (2013) 161 Mathematical Sciences Letters An International Journal http://dx.doi.org/10.12785/msl/020302 Global Attractivity of a Rational Difference Equation Nouressadat
More informationON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE
IJMMS 2004:55, 295 2926 PII. S067204402270 http://ijmms.hindawi.com Hindawi Publishing Corp. ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE GEORGE L. KARAKOSTAS and STEVO STEVIĆ Received 25 February 2004
More informationResearch Article Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear Damping
Mathematical Problems in Engineering Volume 15, Article ID 194, 5 pages http://dx.doi.org/1.1155/15/194 Research Article Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear
More informationResearch Article Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space
Journal of Applied Mathematics Volume 2012, Article ID 435676, 15 pages doi:10.1155/2012/435676 Research Article Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space Bin-Chao Deng,
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 An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem
Abstract and Applied Analysis, Article ID 60813, 5 pages http://dx.doi.org/10.1155/014/60813 Research Article An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem Luoyi
More informationOn the Dynamics of a Rational Difference Equation, Part 1
International Journal of Difference Equations (IJDE). ISSN 0973-6069 Volume 3 Number 1 (2008), pp. 1 35 Research India Publications http://www.ripublication.com/ijde.htm On the Dynamics of a Rational Difference
More informationGlobal Attractivity in a Nonlinear Difference Equation and Applications to a Biological Model
International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 2, pp. 233 242 (204) http://campus.mst.edu/ijde Global Attractivity in a Nonlinear Difference Equation and Applications to
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 informationResearch Article A Note about the General Meromorphic Solutions of the Fisher Equation
Mathematical Problems in Engineering, Article ID 793834, 4 pages http://dx.doi.org/0.55/204/793834 Research Article A Note about the General Meromorphic Solutions of the Fisher Equation Jian-ming Qi, Qiu-hui
More informationResearch Article Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications
Abstract and Applied Analysis Volume 2012, Article ID 479438, 13 pages doi:10.1155/2012/479438 Research Article Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and
More informationResearch Article Strong Convergence of Parallel Iterative Algorithm with Mean Errors for Two Finite Families of Ćirić Quasi-Contractive Operators
Abstract and Applied Analysis Volume 01, Article ID 66547, 10 pages doi:10.1155/01/66547 Research Article Strong Convergence of Parallel Iterative Algorithm with Mean Errors for Two Finite Families of
More informationResearch Article Localization and Perturbations of Roots to Systems of Polynomial Equations
International Mathematics and Mathematical Sciences Volume 2012, Article ID 653914, 9 pages doi:10.1155/2012/653914 Research Article Localization and Perturbations of Roots to Systems of Polynomial Equations
More informationResearch Article On the Difference Equation x n a n x n k / b n c n x n 1 x n k
Abstract and Applied Analysis Volume 2012, Article ID 409237, 20 pages doi:10.1155/2012/409237 Research Article On the Difference Equation x n a n x n k / b n c n x n 1 x n k Stevo Stević, 1 Josef Diblík,
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 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 informationA System of Difference Equations with Solutions Associated to Fibonacci Numbers
International Journal of Difference Equations ISSN 0973-6069 Volume Number pp 6 77 06) http://campusmstedu/ijde A System of Difference Equations with Solutions Associated to Fibonacci Numbers Yacine Halim
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 informationResearch Article Applying GG-Convex Function to Hermite-Hadamard Inequalities Involving Hadamard Fractional Integrals
International Journal of Mathematics and Mathematical Sciences Volume 4, Article ID 3635, pages http://dx.doi.org/.55/4/3635 Research Article Applying GG-Convex Function to Hermite-Hadamard Inequalities
More informationResearch Article Some New Explicit Values of Quotients of Ramanujan s Theta Functions and Continued Fractions
International Mathematics and Mathematical Sciences Article ID 534376 7 pages http://dx.doi.org/0.55/04/534376 Research Article Some New Explicit Values of Quotients of Ramanujan s Theta Functions and
More informationResearch Article Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent
International Differential Equations Volume 2015, Article ID 494907, 4 pages http://dx.doi.org/10.1155/2015/494907 Research Article Existence for Elliptic Equation Involving Decaying Cylindrical Potentials
More informationCorrespondence should be addressed to Abasalt Bodaghi;
Function Spaces, Article ID 532463, 5 pages http://dx.doi.org/0.55/204/532463 Research Article Approximation on the Quadratic Reciprocal Functional Equation Abasalt Bodaghi and Sang Og Kim 2 Department
More informationResearch Article Advanced Discrete Halanay-Type Inequalities: Stability of Difference Equations
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 29, Article ID 535849, 11 pages doi:1.1155/29/535849 Research Article Advanced Discrete Halanay-Type Inequalities: Stability
More informationResearch Article The Dirichlet Problem on the Upper Half-Space
Abstract and Applied Analysis Volume 2012, Article ID 203096, 5 pages doi:10.1155/2012/203096 Research Article The Dirichlet Problem on the Upper Half-Space Jinjin Huang 1 and Lei Qiao 2 1 Department of
More informationResearch Article On Prime Near-Rings with Generalized Derivation
International Mathematics and Mathematical Sciences Volume 2008, Article ID 490316, 5 pages doi:10.1155/2008/490316 Research Article On Prime Near-Rings with Generalized Derivation Howard E. Bell Department
More informationResearch Article An Inverse Eigenvalue Problem for Jacobi Matrices
Mathematical Problems in Engineering Volume 2011 Article ID 571781 11 pages doi:10.1155/2011/571781 Research Article An Inverse Eigenvalue Problem for Jacobi Matrices Zhengsheng Wang 1 and Baoiang Zhong
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 informationResearch Article Translative Packing of Unit Squares into Squares
International Mathematics and Mathematical Sciences Volume 01, Article ID 61301, 7 pages doi:10.1155/01/61301 Research Article Translative Packing of Unit Squares into Squares Janusz Januszewski Institute
More informationON THE GLOBAL ATTRACTIVITY AND THE PERIODIC CHARACTER OF A RECURSIVE SEQUENCE. E.M. Elsayed
Opuscula Mathematica Vol. 30 No. 4 2010 http://dx.doi.org/10.7494/opmath.2010.30.4.431 ON THE GLOBAL ATTRACTIVITY AND THE PERIODIC CHARACTER OF A RECURSIVE SEQUENCE E.M. Elsayed Abstract. In this paper
More informationResearch Article Fourier Series of the Periodic Bernoulli and Euler Functions
Abstract and Applied Analysis, Article ID 85649, 4 pages http://dx.doi.org/.55/24/85649 Research Article Fourier Series of the Periodic Bernoulli and Euler Functions Cheon Seoung Ryoo, Hyuck In Kwon, 2
More informationResearch Article Stability Switches and Hopf Bifurcations in a Second-Order Complex Delay Equation
Hindawi Mathematical Problems in Engineering Volume 017, Article ID 679879, 4 pages https://doi.org/10.1155/017/679879 Research Article Stability Switches and Hopf Bifurcations in a Second-Order Complex
More informationResearch Article Coupled Fixed Point Theorems for a Pair of Weakly Compatible Maps along with CLRg Property in Fuzzy Metric Spaces
Applied Mathematics Volume 2012, Article ID 961210, 13 pages doi:10.1155/2012/961210 Research Article Coupled Fixed Point Theorems for a Pair of Weakly Compatible Maps along with CLRg Property in Fuzzy
More informationResearch Article Global Existence and Boundedness of Solutions to a Second-Order Nonlinear Differential System
Applied Mathematics Volume 212, Article ID 63783, 12 pages doi:1.1155/212/63783 Research Article Global Existence and Boundedness of Solutions to a Second-Order Nonlinear Differential System Changjian
More informationON THE RATIONAL RECURSIVE SEQUENCE X N+1 = γx N K + (AX N + BX N K ) / (CX N DX N K ) Communicated by Mohammad Asadzadeh. 1.
Bulletin of the Iranian Mathematical Society Vol. 36 No. 1 (2010), pp 103-115. ON THE RATIONAL RECURSIVE SEQUENCE X N+1 γx N K + (AX N + BX N K ) / (CX N DX N K ) E.M.E. ZAYED AND M.A. EL-MONEAM* Communicated
More informationResearch Article Completing a 2 2Block Matrix of Real Quaternions with a Partial Specified Inverse
Applied Mathematics Volume 0, Article ID 7978, 5 pages http://dx.doi.org/0.55/0/7978 Research Article Completing a Block Matrix of Real Quaternions with a Partial Specified Inverse Yong Lin, and Qing-Wen
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 informationPeriodicity and Solution of Rational Recurrence Relation of Order Six
Applied Mathematics 3 79-733 http://d.doi.org/.436/am..377 Published Online July (http://www.scirp.org/journal/am) Periodicity and Solution of Rational Recurrence Relation of Order Si Tarek F. Ibrahim
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 informationResearch Article A New Class of Meromorphic Functions Associated with Spirallike Functions
Applied Mathematics Volume 202, Article ID 49497, 2 pages doi:0.55/202/49497 Research Article A New Class of Meromorphic Functions Associated with Spirallike Functions Lei Shi, Zhi-Gang Wang, and Jing-Ping
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 informationGlobal Behavior of a Higher Order Rational Difference Equation
International Journal of Difference Euations ISSN 0973-6069, Volume 10, Number 1,. 1 11 (2015) htt://camus.mst.edu/ijde Global Behavior of a Higher Order Rational Difference Euation Raafat Abo-Zeid The
More informationResearch Article On a New Three-Step Class of Methods and Its Acceleration for Nonlinear Equations
e Scientific World Journal, Article ID 34673, 9 pages http://dx.doi.org/0.55/204/34673 Research Article On a New Three-Step Class of Methods and Its Acceleration for Nonlinear Equations T. Lotfi, K. Mahdiani,
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 informationKingSaudBinAbdulazizUniversityforHealthScience,Riyadh11481,SaudiArabia. Correspondence should be addressed to Raghib Abu-Saris;
Chaos Volume 26, Article ID 49252, 7 pages http://dx.doi.org/.55/26/49252 Research Article On Matrix Projective Synchronization and Inverse Matrix Projective Synchronization for Different and Identical
More informationResearch Article A New Roper-Suffridge Extension Operator on a Reinhardt Domain
Abstract and Applied Analysis Volume 2011, Article ID 865496, 14 pages doi:10.1155/2011/865496 Research Article A New Roper-Suffridge Extension Operator on a Reinhardt Domain Jianfei Wang and Cailing Gao
More informationResearch Article Fixed Point Theorems of Quasicontractions on Cone Metric Spaces with Banach Algebras
Abstract and Applied Analysis Volume 2013, Article ID 187348, 5 pages http://dx.doi.org/10.1155/2013/187348 Research Article Fixed Point Theorems of Quasicontractions on Cone Metric Spaces with Banach
More informationResearch Article Several New Third-Order and Fourth-Order Iterative Methods for Solving Nonlinear Equations
International Engineering Mathematics, Article ID 828409, 11 pages http://dx.doi.org/10.1155/2014/828409 Research Article Several New Third-Order and Fourth-Order Iterative Methods for Solving Nonlinear
More informationDynamics of the equation complex plane
APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Dynamics of the equation in the complex plane Sk. Sarif Hassan 1 and Esha Chatterjee * Received: 0 April 015 Accepted: 08 November 015 Published:
More informationResearch Article Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming
Discrete Dynamics in Nature and Society Volume 20, Article ID 7697, pages doi:0.55/20/7697 Research Article Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming Wei-Xiang
More informationResearch Article Statistical Tests for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Probability and Statistics Volume 2015, Article ID 723924, 5 pages http://dx.doi.org/10.1155/2015/723924 Research Article Statistical Tests for the Reciprocal of a Normal Mean with a Known Coefficient
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 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 A Note on the Central Limit Theorems for Dependent Random Variables
International Scholarly Research Network ISRN Probability and Statistics Volume 22, Article ID 92427, pages doi:.542/22/92427 Research Article A Note on the Central Limit Theorems for Dependent Random
More informationResearch Article Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations
Abstract and Applied Analysis, Article ID 8392, 8 pages http://dxdoiorg/11155/214/8392 Research Article Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential
More informationResearch Article Bessel Equation in the Semiunbounded Interval x [x 0, ]: Solving in the Neighbourhood of an Irregular Singular Point
International Mathematics and Mathematical Sciences Volume 2016, Article ID 6826482, 7 pages http://dx.doi.org/10.1155/2016/6826482 Research Article Bessel Equation in the Semiunbounded Interval x [x 0,
More informationASYMPTOTIC BEHAVIOUR OF SECOND-ORDER DIFFERENCE EQUATIONS
ANZIAM J. 462004, 57 70 ASYMPTOTIC BEHAVIOUR OF SECOND-ORDER DIFFERENCE EQUATIONS STEVO STEVIĆ Received 9 December, 200; revised 9 September, 2003 Abstract In this paper we prove several growth theorems
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 New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients
Abstract and Applied Analysis Volume 2010, Article ID 564068, 11 pages doi:10.1155/2010/564068 Research Article New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive
More informationResearch Article A Note on Kantorovich Inequality for Hermite Matrices
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 0, Article ID 5767, 6 pages doi:0.55/0/5767 Research Article A Note on Kantorovich Inequality for Hermite Matrices Zhibing
More informationResearch Article On the Stability of Alternative Additive Equations in Multi-β-Normed Spaces
Function Spaces Volume 206, Article ID 2534597, 7 pages http://dx.doi.org/0.55/206/2534597 Research Article On the Stability of Alternative Additive Equations in Multi-β-Normed Spaces Xiuzhong Yang, Jing
More informationResearch Article Wave Scattering in Inhomogeneous Strings
International Scholarly Research Network ISRN Mathematical Analysis Volume 011, Article ID 64049, 14 pages doi:10.540/011/64049 Research Article Wave Scattering in Inhomogeneous Strings Nezam Iraniparast
More informationResearch Article Approximation of Analytic Functions by Bessel s Functions of Fractional Order
Abstract and Applied Analysis Volume 20, Article ID 923269, 3 pages doi:0.55/20/923269 Research Article Approximation of Analytic Functions by Bessel s Functions of Fractional Order Soon-Mo Jung Mathematics
More informationResearch Article On a Max-Type Difference Equation
Hindawi Publishing Corporation dvances in Difference Equations Volume 2010, rticle ID 584890, 6 pages doi:10.1155/2010/584890 Research rticle On a Max-Type Difference Equation li Gelisken, Cengiz Cinar,
More informationOn the difference equation. x n+1 = ax n l + bx n k + f(x n l, x n k ),
Abdelrahman et al. Advances in Difference Equations (2018) 2018:431 https://doi.org/10.1186/s13662-018-1880-8 R E S E A R C H Open Access On the difference equation x n+1 = ax n l + bx n k + f (x n l,
More informationA Note on the Positive Nonoscillatory Solutions of the Difference Equation
Int. Journal of Math. Analysis, Vol. 4, 1, no. 36, 1787-1798 A Note on the Positive Nonoscillatory Solutions of the Difference Equation x n+1 = α c ix n i + x n k c ix n i ) Vu Van Khuong 1 and Mai Nam
More informationResearch Article Unicity of Entire Functions concerning Shifts and Difference Operators
Abstract and Applied Analysis Volume 204, Article ID 38090, 5 pages http://dx.doi.org/0.55/204/38090 Research Article Unicity of Entire Functions concerning Shifts and Difference Operators Dan Liu, Degui
More informationResearch Article Some New Fixed-Point Theorems for a (ψ, φ)-pair Meir-Keeler-Type Set-Valued Contraction Map in Complete Metric Spaces
Applied Mathematics Volume 2011, Article ID 327941, 11 pages doi:10.1155/2011/327941 Research Article Some New Fixed-Point Theorems for a (ψ, φ)-pair Meir-Keeler-Type Set-Valued Contraction Map in Complete
More informationResearch Article Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the Upper Half-Plane
Abstract and Applied Analysis Volume 2011, Article ID 989625, 10 pages doi:10.1155/2011/989625 Research Article Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the
More informationResearch Article Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler Foundation Using Homotopy Perturbation Method
Mathematical Problems in Engineering Volume 213, Article ID 721294, 9 pages http://dx.doi.org/1.1155/213/721294 Research Article Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler
More informationResearch Article On a Quasi-Neutral Approximation to the Incompressible Euler Equations
Applied Mathematics Volume 2012, Article ID 957185, 8 pages doi:10.1155/2012/957185 Research Article On a Quasi-Neutral Approximation to the Incompressible Euler Equations Jianwei Yang and Zhitao Zhuang
More informationResearch Article Approximation Algorithm for a System of Pantograph Equations
Applied Mathematics Volume 01 Article ID 714681 9 pages doi:101155/01/714681 Research Article Approximation Algorithm for a System of Pantograph Equations Sabir Widatalla 1 and Mohammed Abdulai Koroma
More informationSTABILITY OF THE kth ORDER LYNESS EQUATION WITH A PERIOD-k COEFFICIENT
International Journal of Bifurcation Chaos Vol 7 No 2007 43 52 c World Scientific Publishing Company STABILITY OF THE kth ORDER LYNESS EQUATION WITH A PERIOD-k COEFFICIENT E J JANOWSKI M R S KULENOVIĆ
More informationGlobal Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation
University of Rhode Island DigitalCommons@URI Mathematics Faculty Publications Mathematics 2013 Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation M. Garić-Demirović
More informationResearch Article A Note on the Discrete Spectrum of Gaussian Wells (I): The Ground State Energy in One Dimension
Mathematical Physics Volume 016, Article ID 15769, 4 pages http://dx.doi.org/10.1155/016/15769 Research Article A Note on the Discrete Spectrum of Gaussian Wells (I: The Ground State Energy in One Dimension
More informationResearch Article A Novel Filled Function Method for Nonlinear Equations
Applied Mathematics, Article ID 24759, 7 pages http://dx.doi.org/0.55/204/24759 Research Article A Novel Filled Function Method for Nonlinear Equations Liuyang Yuan and Qiuhua Tang 2 College of Sciences,
More informationResearch Article A Matrix Method Based on the Fibonacci Polynomials to the Generalized Pantograph Equations with Functional Arguments
Advances in Mathematical Physics, Article ID 694580, 5 pages http://dx.doi.org/10.1155/2014/694580 Research Article A Matrix Method Based on the Fibonacci Polynomials to the Generalized Pantograph Equations
More informationResearch Article Another Aspect of Triangle Inequality
International Scholarly Research Network ISRN Mathematical Analysis Volume 2011, Article ID 514184, 5 pages doi:10.5402/2011/514184 Research Article Another Aspect of Triangle Inequality Kichi-Suke Saito,
More informationGlobal Dynamics and Bifurcations of Two Quadratic Fractional Second Order Difference Equations
University of Rhode Island DigitalCommons@URI Mathematics Faculty Publications Mathematics 2016 Global Dynamics and Bifurcations of Two Quadratic Fractional Second Order Difference Equations Senada Kalabušić
More informationResearch Article Exact Solutions of φ 4 Equation Using Lie Symmetry Approach along with the Simplest Equation and Exp-Function Methods
Abstract and Applied Analysis Volume 2012, Article ID 350287, 7 pages doi:10.1155/2012/350287 Research Article Exact Solutions of φ 4 Equation Using Lie Symmetry Approach along with the Simplest Equation
More informationResearch Article Chaos Control on a Duopoly Game with Homogeneous Strategy
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 16, Article ID 74185, 7 pages http://dx.doi.org/1.1155/16/74185 Publication Year 16 Research Article Chaos Control on a Duopoly
More informationResearch Article Circle-Uniqueness of Pythagorean Orthogonality in Normed Linear Spaces
Function Spaces, Article ID 634842, 4 pages http://dx.doi.org/10.1155/2014/634842 Research Article Circle-Uniqueness of Pythagorean Orthogonality in Normed Linear Spaces Senlin Wu, Xinjian Dong, and Dan
More informationResearch Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field
Complexity, Article ID 6235649, 9 pages https://doi.org/10.1155/2018/6235649 Research Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field Jinwang Liu, Dongmei
More informationResearch Article Partial Pole Placement in LMI Region
Control Science and Engineering Article ID 84128 5 pages http://dxdoiorg/11155/214/84128 Research Article Partial Pole Placement in LMI Region Liuli Ou 1 Shaobo Han 2 Yongji Wang 1 Shuai Dong 1 and Lei
More informationResearch Article Domination Conditions for Families of Quasinearly Subharmonic Functions
International Mathematics and Mathematical Sciences Volume 2011, Article ID 729849, 9 pages doi:10.1155/2011/729849 Research Article Domination Conditions for Families of Quasinearly Subharmonic Functions
More informationResearch Article Constrained Solutions of a System of Matrix Equations
Journal of Applied Mathematics Volume 2012, Article ID 471573, 19 pages doi:10.1155/2012/471573 Research Article Constrained Solutions of a System of Matrix Equations Qing-Wen Wang 1 and Juan Yu 1, 2 1
More informationResearch Article Some Results on Characterizations of Matrix Partial Orderings
Applied Mathematics, Article ID 408457, 6 pages http://dx.doi.org/10.1155/2014/408457 Research Article Some Results on Characterizations of Matrix Partial Orderings Hongxing Wang and Jin Xu Department
More informationResearch Article Implicative Ideals of BCK-Algebras Based on the Fuzzy Sets and the Theory of Falling Shadows
International Mathematics and Mathematical Sciences Volume 2010, Article ID 819463, 11 pages doi:10.1155/2010/819463 Research Article Implicative Ideals of BCK-Algebras Based on the Fuzzy Sets and the
More informationResearch Article A Generalization of Ćirić Quasicontractions
Abstract and Applied Analysis Volume 2012, Article ID 518734, 9 pages doi:10.1155/2012/518734 Research Article A Generalization of Ćirić Quasicontractions Erdal Karapınar, 1 Kieu Phuong Chi, 2 and Tran
More informationDepartment of Engineering Sciences, University of Patras, Patras, Greece
International Differential Equations Volume 010, Article ID 436860, 13 pages doi:10.1155/010/436860 Research Article Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of
More information