Published by European Centre for Research Training and Development UK ( GRONWALL BELLMAN OU-IANG TYPE INEQUALITIES
|
|
- Randolph Bell
- 5 years ago
- Views:
Transcription
1 GRONWALL BELLMAN OU-IANG TYPE INEQUALITIES Jaashree V. Patil Department of Mathematics Vasantrao Naik Mahavidalaa,Aurangabad (M.S.) INDIA ABSTRACT: In this paper we prove some retarded nonlinear inequalities which provide eplicit bounds on unknown function. Applications are also given to illustrate the usefulness of one of our results. Kewords: Integral Inequalities, Retarded Equation, Eplicit Bounds, Boundedness. INTRODUCTION In the development of the theor of differential and integral equations integral inequalities which provide eplicit bounds on unknown function take ver important place. The literature on such inequalities is vast see [6, 8] and the references there in. One of the most useful inequalities in the development of theor of differential equation is given in following Lemma b Ou-Iang see []. Lemma. Let u and f are non-negative functions on [, ) satisfing u (t) < k + t f (s)u(s)ds for all, t[, ), where k > is a constant then u(t) < k + for all t[, ). t f(s)ds In literature, the Ou-Iang tpe inequalities and their generalizations occurs which used effectivel in the stud of qualitative as well as quantitative properties of solutions of differential equations see [-5, 7,9 ].In this paper we prove some new retarded inequalities which is generalization of above lemma and used to stud the boundedness properties of solutions of integral and differential equations. 43
2 MAIN RESULTS Theorem. :If u(, ), m(, ) are real valued nonnegative continuous functions defined for >, >, f(,, s, t) be continuous non-decreasing in and for each t, s. < () <, < () <, ' (), '() > are real valued continuous functions defined for >, > satisfies u (, ) < m(, ) + f (,, s, t) u( s, t) dt ds ---- (.) then u(, ) < m(, ) f (,, s, t)dtds ---- (.) Proof. Let us assume that m(, ) >. Fied an numbers and with and Define a function z(, ) f (,,s, t)u(s, t)dtds ---- (.3) From (.) we get u (, ) < m, z, ---- (.4) and z(, ) = z (, ) =, clearl z (, ) is positive non-decreasing function in each variable. Hence for [, ] and [, ] b direct calculation, we get 44
3 z = f(,, (), ()) u( (), ()) '() '() '(),, (), t u (), t dt + + '(),,s, () u, () ds f (,,s, t) u(s, t) dt ds f,, (), () m(, ) z(, ) '() '() ( ) '() f,, (), t m (), t z (), t () '() f,, s, () m s, () z s, () ds () ( ) m s, t z s, t f,,s, t { dt ds () m, z, f,, (), () '() '() '() f,, (), tdt '() f,,s,, (ds z ( ) f,, s, t dt ds} Z, z, m, f,,s, t dt ds on the other hand z, f,,s, tdt ds m, z, ---- (.5) 45
4 Keeping fied in (.5), integrating with respect to from to, and using fact that z (, ) =, we deduce. Z, Z, m, f,, s, t dt ds ---- (.6) Now keeping fied in (.6) and integrating with respect to from to we get m, Z, m, f,,s, tdt ds for (, ) [, ] [, ] let = and = in above inequalit we get m, z, m, f,,s, tdt ds As and are arbitraril and u(, ) < m, z, we have obtain inequalit in (.). Corollar... Let m, n, g C (R+ R+, R+),, C (R+, R+) and let α, β is non-decreasing with () <, () < if u C (R+ R+, R+) Satisfies u(, ) < m, f,,s, tdt ds >, > Theorem.. Let m, n, g C (R+ R+, R+),, C (R+, R+) and let α, β is non-decreasing with () <, () < if u C (R+ R+, R+) Satisfies 46
5 (.7) International Journal of Mathematics and Statistics Studies u (, ) m(, ) n(, ) g(s, t) u(s, t)dt ds ---- then when >, > u(, ) < m(, ) n(, ) g s, t dt ds ---- (.8) Proof. Let us assume that m(, ) >. Fied an number and with < < and < < z(, ) = m (, ) + n (, ) g(s, t) u(s, t) dt ds Then z(, ) >, z(, ) = z (, ) = m (, ) and from (.7) we get u (, ) < z (, ) i.e. u (, ) < (, ) z ---- (.9) Clearl z(, ) is positive non-decreasing function in each variable. Hence we get z = n (, ) g (), (). u (), (). '(). '() n(, ) g( (), () ) z (), () '(). '() z n(, ) g (), () '() '() z, < i.e. 47
6 z International Journal of Mathematics and Statistics Studies z, n(, ) g (), () '() '() Keeping fied, integrating above inequalit with respect to from to and using fact that z (, ) =, we get z, z, n(, ) g (), (t) '() '(t) d t Now keeping fied in above inequalit and integrating with respect to from to, we deduce n, m, z, g (s), (t) '(s) '(t) dsdt B making change of variable on right-hand side of above inequalit, we get z, m, n, g s, tdt ds Taking =, = in the inequalit (.), since and are arbitrar and as u(, ) < z, we get the required inequalit in (.8). Theorem.3. Assume m, f, α, β be as in Theorem. ω(u) is positive continuous non-decreasing for u > with ω () = and ds s if u C(R+ R+, R+) satisfies u (, ) < m (, ) + f,,s, t u(s, t) dt ds,, ---- (.) then 48
7 u(, ) < International Journal of Mathematics and Statistics Studies G G m, f (,,s, t) dt ds ---- (.) where G(r) = r ds s, r > Proof : Assume, be fied and let z(, ) = f (,,s, t) u(s, t) dt ds Clearl z (, ) is non-decreasing for and hence for [, ], [, ], we have z = f,, (), () u( (), () '() '() + '() f,,s, ( ) u(s, ( )) ds + + '() f,, (), t u( (), t) dt f,,s, t u(s, t) dt ds < f,, (), () m (), z(, ). '() '() + '() f,, (), t m (), t z (), t dt 49
8 '(),,s, () ds + f m s, () z () + f,,s, t m(s, t) z(s, t)dt ds m, z, f,, () (). '(). '() '() f,,,(), t dt '() f,,z, () ds f (,,s, t)dtds i.e. Z z, m, f (,,s, t)dt ds We have f (,,s, t) dt ds m, z, Z Integrating both sides of above inequailt with respect to from to we deduce Z z, m, f (,,s, t)dt ds Integrating above inequalit with respect to from to and using definition of G, we get 5
9 (.3) G International Journal of Mathematics and Statistics Studies G m, z, m, f (,,s, t) dt ds ---- for,,,. As ds s from (.3) we have m, z, G G m, f (,,s, t) dt ds let =, = in above inequalit, we get () m, z, G G m, f,,s, t dt ds As, are arbitraril and u(, ) < m(, ) z(, ) we obtain inequalit in (.). Theorem.4 : Suppose m, n, g C(R+ R+, R+),,C (R+, R+) and m, n, g, are nondecreasing function with () <, () < for >, > assume C(R+, R) be a nondecreasing function such that (t) > for t > and uc(r+ R+, R+) satisfies ds s if u(, ) < m (, ) + n (, ) g(t, s) (.4) u(s, t) dt ds >, >
10 u (, ) < (.5) International Journal of Mathematics and Statistics Studies G G m(, ) n(, ) g(s, t) dt ds, >, > ---- where G(r) = r ds,r s Proof : Let us assume that m (, ) >, Fied an number and with < <, < <. Define a function z(, ) as z(, ) = m, + n, g s, t u(s, t) dt ds Then z (, ) >, z (, ) = z (, ) = m, and from (.4) we have u (, ) < z (, ) implies u (, ) < z, clearl z (, ) is positive non-decreasing function in each variable. Hence we get z = n, g (), () (u (), () '() '() n, g (), () z (), () '() '() z z(, ) n, g (), () '() '() i.e. 5
11 z n, g (), () '() '() z, Keeping fied, integrating above inequalit with respect to from to and using fact that Z (, ) = we get z, n, g (), () '() '(t) dt z, Keeping fied in above inequalit and integrating with respect to from to we get G G z(, ) m(, ) n, g (s), (t) '(s) '(t) dt ds B making change of variable on right-hand side of above inequalit, we get z(, ) G G m(, ) n, g (s, t), dtds for,,, let =, = in (.6) we have z, G G m, n, g s, t dt ds As, are arbitraril and u (, ) < z, we get required inequalit in (.5). 53
12 APPLICATION In this section we will show that our one of the result is used in proving the boundedness of solutions of nonlinear integral equation. Consider the nonlinear partial integral equation of the form u, h, G,,s, t, u(s, t) dt ds ---- (3.) where all the functions are continuous on their respective domains of their definitions and m, h, ---- (3.) F,, t,s, u f,, t,s u (3.3) for >, > where m, f,, be as in Theorem (.). Using (3.), (3.3) in (3.) and using Theorem. we obtain the bound on the solution u(, ) of (3.). REFERENCES ) L Ou-Iang, The boundedness of solutions of linear differential equatioins, Ann. Mat. Pura. Appl., 68 (995) ) C.M. Dafermos, The second law of thermodnamic and stabilit, Arch. Ration, Mech. Anal.,7(979) ) O. Lipovan, A retarded integral inequalit and its applications, J. math. Anal. Appl., 85 (3) ) O. Lipovan, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl., 3 (6) ) Lianzhong Li, Fanvei Meng, Leliang He, Some generalized integral inequalities and their applications, J. Math.Anal.Appl., 37 ()
13 6) B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic press, New York, ) B.G. Pachpatte, on some new inequalities related to certain inequalities in the theor of differential equations, J. Math. Anal.Appl.,89 (995) ) B.G. pachpatte, Integral and Finite Difference Inequalities and Applications, Elsevier, New York, 6. 9) H. Zhang, F. Meng, Integral inequalities in two independent variables for retarded Volterra equations, Appl.Math.Comp.,99 (8)
Journal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 2, Issue 2, Article 15, 21 ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES B.G. PACHPATTE DEPARTMENT
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics SOME NEW DISCRETE NONLINEAR DELAY INEQUALITIES AND APPLICATION TO DISCRETE DELAY EQUATIONS WING-SUM CHEUNG AND SHIOJENN TSENG Department of Mathematics
More informationGENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS
Communications in Applied Analysis 19 (215), 679 688 GENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS TINGXIU WANG Department of Mathematics, Texas A&M
More informationA generalized Gronwall inequality and its application to a fractional differential equation
J. Math. Anal. Appl. 328 27) 75 8 www.elsevier.com/locate/jmaa A generalized Gronwall inequality and its application to a fractional differential equation Haiping Ye a,, Jianming Gao a, Yongsheng Ding
More informationExistence And Uniqueness Of Mild Solutions Of Second Order Volterra Integrodifferential Equations With Nonlocal Conditions
Applied Mathematics E-Notes, 9(29), 11-18 c ISSN 167-251 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Existence And Uniqueness Of Mild Solutions Of Second Order Volterra Integrodifferential
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 informationIntroduction to Differential Equations
Math0 Lecture # Introduction to Differential Equations Basic definitions Definition : (What is a DE?) A differential equation (DE) is an equation that involves some of the derivatives (or differentials)
More informationSome Integral Inequalities with Maximum of the Unknown Functions
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 6, Number 1, pp. 57 69 2011 http://campus.mst.edu/adsa Some Integral Inequalities with Maximum of the Unknown Functions Snezhana G.
More informationON NONLINEAR INTEGRAL INEQUALITIES AND APPLICATIONS
ON NONLINEAR INTEGRAL INEQUALITIES AND APPLICATIONS A Thesis submitted to the University of Pune for the degree of Doctor of Philosophy in Mathematics By Sitaram Gena Latpate Under the guidance of Dr.
More informationExtensions of Gronwall s inequality with logarithmic terms by Je R. L. Webb
Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 2 (23), pp. 149-159. Extensions of Gronwall s inequality with logarithmic terms by Je R. L. Webb Abstract 1. This article is a self-contained
More informationON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS
PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary
More informationA Nonlinear Sturm Picone Comparison Theorem for Dynamic Equations on Time Scales
International Journal of Difference Equations IJDE). ISSN 0973-6069 Volume 2 Number 1 2007), pp. 25 35 Research India Publications http://www.ripublication.com/ijde.htm A Nonlinear Sturm Picone Comparison
More informationSIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM
Electronic Journal of Differential Euations, Vol. 22 (22), No. 26, pp. 9. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SIMULTANEOUS AND NON-SIMULTANEOUS
More informationArchivum Mathematicum
Archivum Mathematicum Deepak B. Pachpatte Some dynamic inequalities applicable to partial integrodifferential equations on time scales Archivum Mathematicum, Vol. 51 2015, No. 3, 143 152 Persistent URL:
More informationPICONE S IDENTITY FOR A SYSTEM OF FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 143, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PICONE S IDENTITY
More informationTHE PERRON PROBLEM FOR C-SEMIGROUPS
Math. J. Okayama Univ. 46 (24), 141 151 THE PERRON PROBLEM FOR C-SEMIGROUPS Petre PREDA, Alin POGAN and Ciprian PREDA Abstract. Characterizations of Perron-type for the exponential stability of exponentially
More informationPOSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF SINGULAR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION
International Journal of Pure and Applied Mathematics Volume 92 No. 2 24, 69-79 ISSN: 3-88 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/.2732/ijpam.v92i2.3
More informationON THE PICARD PROBLEM FOR HYPERBOLIC DIFFERENTIAL EQUATIONS IN BANACH SPACES
Discussiones Mathematicae Differential Inclusions, Control and Optimization 23 23 ) 31 37 ON THE PICARD PROBLEM FOR HYPERBOLIC DIFFERENTIAL EQUATIONS IN BANACH SPACES Antoni Sadowski Institute of Mathematics
More informationMathematical Journal of Okayama University
Mathematical Journal of Okayama University Volume 46, Issue 1 24 Article 29 JANUARY 24 The Perron Problem for C-Semigroups Petre Prada Alin Pogan Ciprian Preda West University of Timisoara University of
More informationEXISTENCE AND NONEXISTENCE OF ENTIRE SOLUTIONS TO THE LOGISTIC DIFFERENTIAL EQUATION
EXISTENCE AND NONEXISTENCE OF ENTIRE SOLUTIONS TO THE LOGISTIC DIFFERENTIAL EQUATION MARIUS GHERGU AND VICENŢIURĂDULESCU Received 26 February 23 We consider the one-dimensional logistic problem r α A u
More informationProperties of some nonlinear partial dynamic equations on time scales
Malaya Journal of Matematik 4)03) 9 Properties of some nonlinear partial dynamic equations on time scales Deepak B. Pachpatte a, a Department of Mathematics, Dr. Babasaheb Ambedekar Marathwada University,
More informationF6 Solving Inequalities
UNIT F6 Solving Inequalities: Tet F6 Solving Inequalities F6. Inequalities on a Number Line An inequalit involves one of the four smbols >,, < or The following statements illustrate the meaning of each
More informationNontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
Advances in Dynamical Systems and Applications ISSN 973-532, Volume 6, Number 2, pp. 24 254 (2 http://campus.mst.edu/adsa Nontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
More informationTHE ANALYSIS OF SOLUTION OF NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS
THE ANALYSIS OF SOLUTION OF NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS FANIRAN TAYE SAMUEL Assistant Lecturer, Department of Computer Science, Lead City University, Ibadan, Nigeria. Email :
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 informationPUBLICATIONS. Bo Zhang
PUBLICATIONS Bo Zhang 1. Bo Zhang, Formulas of Liapunov functions for systems of linear partial differential equations, to appear in Journal of Mathematical Analysis and Applications (available online
More informationApplied Mathematics Letters
Applied Mathematics Letters 24 (211) 219 223 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Laplace transform and fractional differential
More informationStability for solution of Differential Variational Inequalitiy
Stability for solution of Differential Variational Inequalitiy Jian-xun Zhao Department of Mathematics, School of Science, Tianjin University Tianjin 300072, P.R. China Abstract In this paper we study
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationBlow-up collocation solutions of nonlinear homogeneous Volterra integral equations
Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations R. Benítez 1,, V. J. Bolós 2, 1 Dpto. Matemáticas, Centro Universitario de Plasencia, Universidad de Extremadura. Avda.
More informationA Note on Retarded Ouyang Integral Inequalities
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 3, Number 2, pp. 291 296 (28) http://campus.mst.edu/adsa A Note on Retarded Ouyang Integral Inequalities Fu-Hsiang Wong National Taipei
More informationMath 2 Honors Summer Work
Math Honors Summer Work The purpose of this assignment is to demonstrate our thorough understanding of the topics and concepts presented in Math IH. This work should be of high qualit. Please clearl and
More informationOscillation by Impulses for a Second-Order Delay Differential Equation
PERGAMON Computers and Mathematics with Applications 0 (2006 0 www.elsevier.com/locate/camwa Oscillation by Impulses for a Second-Order Delay Differential Equation L. P. Gimenes and M. Federson Departamento
More informationExistence of Solutions for a Class of Third Order Quasilinear Ordinary Differential Equations with Nonlinear Boundary Value Problems
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 3, Number 2, pp. 35 321 (28) http://campus.mst.edu/adsa Existence of Solutions for a Class of Third Order Quasilinear Ordinary Differential
More informationUpper and lower solution method for fourth-order four-point boundary value problems
Journal of Computational and Applied Mathematics 196 (26) 387 393 www.elsevier.com/locate/cam Upper and lower solution method for fourth-order four-point boundary value problems Qin Zhang a, Shihua Chen
More informationExistence of homoclinic solutions for Duffing type differential equation with deviating argument
2014 9 «28 «3 Sept. 2014 Communication on Applied Mathematics and Computation Vol.28 No.3 DOI 10.3969/j.issn.1006-6330.2014.03.007 Existence of homoclinic solutions for Duffing type differential equation
More informationPólya-Szegö s Principle for Nonlocal Functionals
International Journal of Mathematical Analysis Vol. 12, 218, no. 5, 245-25 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/ijma.218.8327 Pólya-Szegö s Principle for Nonlocal Functionals Tiziano Granucci
More informationBoundedness of solutions to a retarded Liénard equation
Electronic Journal of Qualitative Theory of Differential Equations 21, No. 24, 1-9; http://www.math.u-szeged.hu/ejqtde/ Boundedness of solutions to a retarded Liénard equation Wei Long, Hong-Xia Zhang
More informationABSTRACT AND CONCRETE GRONWALL LEMMAS
Fixed Point Theory, 1(29), No. 2, 221-228 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html ABSTRACT AND CONCRETE GRONWALL LEMMAS CECILIA CRĂCIUN AND NICOLAE LUNGU Colfe s School Horn Park Lane London, SE12
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM
Electronic Journal of Differential Equations, Vol. 211 (211), No. 78, 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 informationPeriodic solutions of a class of nonautonomous second order differential systems with. Daniel Paşca
Periodic solutions of a class of nonautonomous second order differential systems with (q, p) Laplacian Daniel Paşca Abstract Some existence theorems are obtained by the least action principle for periodic
More informationThe complex periodic problem for a Riccati equation
The complex periodic problem for a Riccati equation Rafael Ortega Departamento de Matemática Aplicada Facultad de Ciencias Universidad de Granada, 1871 Granada, Spain rortega@ugr.es Dedicated to Jean Mawhin,
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 informationExistence of Positive Periodic Solutions of Mutualism Systems with Several Delays 1
Advances in Dynamical Systems and Applications. ISSN 973-5321 Volume 1 Number 2 (26), pp. 29 217 c Research India Publications http://www.ripublication.com/adsa.htm Existence of Positive Periodic Solutions
More informationDISSIPATIVE PERIODIC PROCESSES
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 77, Number 6, November 1971 DISSIPATIVE PERIODIC PROCESSES BY J. E. BILLOTTI 1 AND J. P. LASALLE 2 Communicated by Felix Browder, June 17, 1971 1. Introduction.
More informationCopyright MMIX ARACNE editrice S.r.l. via Raffaele Garofalo, 133 A/B Roma (06)
Sergiy Borysenko Gerardo Iovane Differential Models: Dynamical Systems with Perturbations Copyright MMIX ARACNE editrice S.r.l. www.aracneeditrice.it info@aracneeditrice.it via Raffaele Garofalo, 133 A/B
More informationOn Some Qualitiative Properties of Solutions to Certain Third Order Vector Differential Equations with Multiple Constant Deviating Arguments
ISSN 749-3889 (print) 749-3897 (online) International Journal of Nonlinear Science Vol.6(8) No.3pp.8-9 On Some Qualitiative Properties of Solutions to Certain Third Order Vector Differential Equations
More informationOn the Ψ - Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations
DOI: 1.2478/awutm-213-12 Analele Universităţii de Vest, Timişoara Seria Matematică Informatică LI, 2, (213), 7 28 On the Ψ - Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations
More informationFachrichtung 6.1 Mathematik
Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 Mathematik Preprint Nr. A note on splitting-type variational problems with subquadratic growth Dominic reit Saarbrücken
More informationAsymptotic Behavior for Semi-Linear Wave Equation with Weak Damping
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 15, 713-718 HIKARI Ltd, www.m-hikari.com Asymptotic Behavior for Semi-Linear Wave Equation with Weak Damping Ducival Carvalho Pereira State University
More informationOn the Spectral Theory of Operator Pencils in a Hilbert Space
Journal of Nonlinear Mathematical Phsics ISSN: 1402-9251 Print 1776-0852 Online Journal homepage: http://www.tandfonline.com/loi/tnmp20 On the Spectral Theor of Operator Pencils in a Hilbert Space Roman
More informationEXISTENCE THEOREMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVIII, 2(29), pp. 287 32 287 EXISTENCE THEOREMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES A. SGHIR Abstract. This paper concernes with the study of existence
More information610 S.G. HRISTOVA AND D.D. BAINOV 2. Statement of the problem. Consider the initial value problem for impulsive systems of differential-difference equ
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 23, Number 2, Spring 1993 APPLICATION OF THE MONOTONE-ITERATIVE TECHNIQUES OF V. LAKSHMIKANTHAM FOR SOLVING THE INITIAL VALUE PROBLEM FOR IMPULSIVE DIFFERENTIAL-DIFFERENCE
More informationON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay
ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov
More informationBIBO stabilization of feedback control systems with time dependent delays
to appear in Applied Mathematics and Computation BIBO stabilization of feedback control systems with time dependent delays Essam Awwad a,b, István Győri a and Ferenc Hartung a a Department of Mathematics,
More informationLACK OF HÖLDER REGULARITY OF THE FLOW FOR 2D EULER EQUATIONS WITH UNBOUNDED VORTICITY. 1. Introduction
LACK OF HÖLDER REGULARITY OF THE FLOW FOR 2D EULER EQUATIONS WITH UNBOUNDED VORTICITY JAMES P. KELLIHER Abstract. We construct a class of examples of initial vorticities for which the solution to the Euler
More informationIntermediate Math Circles Wednesday November Inequalities and Linear Optimization
WWW.CEMC.UWATERLOO.CA The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Intermediate Math Circles Wednesda November 21 2012 Inequalities and Linear Optimization Review: Our goal is to solve sstems
More informationarxiv: v1 [math.ap] 16 May 2007
arxiv:0705.446v1 [math.ap] 16 May 007 Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity Alexis Vasseur October 3, 018 Abstract In this short note, we give a
More informationStabilization for the Wave Equation with Variable Coefficients and Balakrishnan-Taylor Damping. Tae Gab Ha
TAIWANEE JOURNAL OF MATHEMATIC Vol. xx, No. x, pp., xx 0xx DOI: 0.650/tjm/788 This paper is available online at http://journal.tms.org.tw tabilization for the Wave Equation with Variable Coefficients and
More informationLeighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential Equations
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 5, Number 2, pp. 25 214 (21) http://campus.mst.edu/adsa Leighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential
More informationy = ± x 2 + c. ln y = 2 ln x + 2C sin(x) dx
Worked Solutions Chapter 4: Separable First-Order Equations 43 a Factoring out 2, we get 3 sinx)) 2, which is f x)g), ds with f x) 3 sinx) and g) 2 So the equation is separable 43 c x x )2 x )2 f x)g)
More informationOPTIMAL CONTROL FOR A PARABOLIC SYSTEM MODELLING CHEMOTAXIS
Trends in Mathematics Information Center for Mathematical Sciences Volume 6, Number 1, June, 23, Pages 45 49 OPTIMAL CONTROL FOR A PARABOLIC SYSTEM MODELLING CHEMOTAXIS SANG UK RYU Abstract. We stud the
More informationON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the 3D Navier- Stokes equation belongs to
More informationGRONWALL'S INEQUALITY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 33, Number I, May 1972 GRONWALL'S INEQUALITY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES DONALD R. SNOW Abstract.
More informationASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE VOLTERRA EQUATIONS. Janusz Migda and Małgorzata Migda
Opuscula Math. 36, no. 2 (2016), 265 278 http://dx.doi.org/10.7494/opmath.2016.36.2.265 Opuscula Mathematica ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE VOLTERRA EQUATIONS Janusz Migda and Małgorzata
More informationSection 1.5 Formal definitions of limits
Section.5 Formal definitions of limits (3/908) Overview: The definitions of the various tpes of limits in previous sections involve phrases such as arbitraril close, sufficientl close, arbitraril large,
More informationUpper and lower solutions method and a fractional differential equation boundary value problem.
Electronic Journal of Qualitative Theory of Differential Equations 9, No. 3, -3; http://www.math.u-szeged.hu/ejqtde/ Upper and lower solutions method and a fractional differential equation boundary value
More informationOn Some Estimates of the Remainder in Taylor s Formula
Journal of Mathematical Analysis and Applications 263, 246 263 (2) doi:.6/jmaa.2.7622, available online at http://www.idealibrary.com on On Some Estimates of the Remainder in Taylor s Formula G. A. Anastassiou
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
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 informationGlobal Solutions for a Nonlinear Wave Equation with the p-laplacian Operator
Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática
More informationVANISHING VISCOSITY IN THE PLANE FOR VORTICITY IN BORDERLINE SPACES OF BESOV TYPE
VANISHING VISCOSITY IN THE PLANE FOR VORTICITY IN BORDERLINE SPACES OF BESOV TYPE ELAINE COZZI AND JAMES P. KELLIHER Abstract. The existence and uniqueness of solutions to the Euler equations for initial
More informationStability of solutions to abstract evolution equations with delay
Stability of solutions to abstract evolution equations with delay A.G. Ramm Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu Abstract An equation u = A(t)u+B(t)F
More informationOscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments
Journal of Mathematical Analysis Applications 6, 601 6 001) doi:10.1006/jmaa.001.7571, available online at http://www.idealibrary.com on Oscillation Criteria for Certain nth Order Differential Equations
More informationSecond Proof: Every Positive Integer is a Frobenius Number of Three Generators
International Mathematical Forum, Vol., 5, no. 5, - 7 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/imf.5.54 Second Proof: Ever Positive Integer is a Frobenius Number of Three Generators Firu Kamalov
More informationOscillationofNonlinearFirstOrderNeutral Di erenceequations
AppliedMathematics E-Notes, 1(2001), 5-10 c Availablefreeatmirrorsites ofhttp://math2.math.nthu.edu.tw/»amen/ OscillationofNonlinearFirstOrderNeutral Di erenceequations YingGaoandGuangZhang yz Received1June2000
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 informationPERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian
Electronic Journal of Differential Euations, Vol. 24 (24), No. 64, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PERIODIC SOLUIONS FOR NON-AUONOMOUS
More informationGLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS TO A PETROVSKY EQUATION WITH GENERAL NONLINEAR DISSIPATION AND SOURCE TERM
Georgian Mathematical Journal Volume 3 (26), Number 3, 397 4 GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION TO A PETROVKY EQUATION WITH GENERAL NONLINEAR DIIPATION AND OURCE TERM NOUR-EDDINE AMROUN AND ABBE
More informationA global solution curve for a class of free boundary value problems arising in plasma physics
A global solution curve for a class of free boundary value problems arising in plasma physics Philip Korman epartment of Mathematical Sciences University of Cincinnati Cincinnati Ohio 4522-0025 Abstract
More informationLYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT. Marat Akhmet. Duygu Aruğaslan
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.25.457 DYNAMICAL SYSTEMS Volume 25, Number 2, October 2009 pp. 457 466 LYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
More informationSome fixed point results for dual contractions of rational type
Mathematica Moravica Vol. 21, No. 1 (2017), 139 151 Some fixed point results for dual contractions of rational type Muhammad Nazam, Muhammad Arshad, Stojan Radenović, Duran Turkoglu and Vildan Öztürk Abstract.
More informationHybrid CQ projection algorithm with line-search process for the split feasibility problem
Dang et al. Journal of Inequalities and Applications 016 016:106 DOI 10.1186/s13660-016-1039-7 R E S E A R C H Open Access Hbrid CQ proection algorithm with line-search process for the split feasibilit
More information(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS
(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS Svetlana Janković and Miljana Jovanović Faculty of Science, Department of Mathematics, University
More informationTomasz Człapiński. Communicated by Bolesław Kacewicz
Opuscula Math. 34, no. 2 (214), 327 338 http://dx.doi.org/1.7494/opmath.214.34.2.327 Opuscula Mathematica Dedicated to the Memory of Professor Zdzisław Kamont GLOBAL CONVERGENCE OF SUCCESSIVE APPROXIMATIONS
More informationDIfferential equations of fractional order have been the
Multistage Telescoping Decomposition Method for Solving Fractional Differential Equations Abdelkader Bouhassoun Abstract The application of telescoping decomposition method, developed for ordinary differential
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 informationA Numerical Algorithm for Ambrosetti-Prodi Type Operators
A Numerical Algorithm for Ambrosetti-Prodi Tpe Operators José Teieira Cal Neto and Carlos Tomei March 9, Abstract We consider the numerical solution of the equation u f(u) = g, for the unknown u satisfing
More informationarxiv: v2 [math.ap] 6 Sep 2007
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, arxiv:0708.3067v2 [math.ap] 6 Sep 2007 A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics NOTES ON AN INTEGRAL INEQUALITY QUÔ C ANH NGÔ, DU DUC THANG, TRAN TAT DAT, AND DANG ANH TUAN Department of Mathematics, Mechanics and Informatics,
More informationSecond order Volterra-Fredholm functional integrodifferential equations
Malaya Journal of Matematik )22) 7 Second order Volterra-Fredholm functional integrodifferential equations M. B. Dhakne a and Kishor D. Kucche b, a Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada
More informationDecay in Time of Incompressible Flows
J. math. fluid mech. 5 (23) 231 244 1422-6928/3/3231-14 c 23 Birkhäuser Verlag, Basel DOI 1.17/s21-3-79-1 Journal of Mathematical Fluid Mechanics Decay in Time of Incompressible Flows Heinz-Otto Kreiss,
More informationBounded and periodic solutions of infinite delay evolution equations
J. Math. Anal. Appl. 286 (2003) 705 712 www.elsevier.com/locate/jmaa Bounded and periodic solutions of infinite delay evolution equations James Liu, a,,1 Toshiki Naito, b and Nguyen Van Minh c a Department
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE Variations of Gronwall s Lemma Gronwall s lemma, which solves a certain kind of inequality for a function, is useful in the theory of differential
More informationULAM-HYERS-RASSIAS STABILITY OF SEMILINEAR DIFFERENTIAL EQUATIONS WITH IMPULSES
Electronic Journal of Differential Equations, Vol. 213 (213), No. 172, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ULAM-HYERS-RASSIAS
More informationNonlinear Analysis. A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel Lizorkin spaces
Nonlinear Analysis 74 (11) 5 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A regularity criterion for the 3D magneto-micropolar fluid equations
More informationMATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED
FOM 11 T1 SYSTEMS OF LINEAR INEQUALITIES 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) A SYSTEM OF LINEAR INEQUALITIES = a problem where or more inequalities are graphed on the same grid, the solution
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 informationEngineering Mathematics I
Engineering Mathematics I_ 017 Engineering Mathematics I 1. Introduction to Differential Equations Dr. Rami Zakaria Terminolog Differential Equation Ordinar Differential Equations Partial Differential
More informationExistence of at least two periodic solutions of the forced relativistic pendulum
Existence of at least two periodic solutions of the forced relativistic pendulum Cristian Bereanu Institute of Mathematics Simion Stoilow, Romanian Academy 21, Calea Griviţei, RO-172-Bucharest, Sector
More information