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Positive Solutions of Differential, Difference and Integral Equations

TL;DR: In this article, the authors present a Coupled System of Boundary Value Problems (CSV) for the first order initial value problems. But they do not address the second order value problems, i.e., the (n,p) boundary value problem.
Abstract: Preface. Ordinary Differential Equations. 1. First Order Initial Value Problems. 2. Second Order Initial Value Problems. 3. Positone Boundary Value Problems. 4. Semi-positone Boundary Value Problems. 5. Semi-Infinite Interval Problems. 6. Mixed Boundary Value Problems. 7. Singular Boundary Value Problems. 8. General Singular and Nonsingular Boundary Value Problems. 9. Quasilinear Boundary Value Problems. 10. Delay Boundary Value Problems. 11. Coupled System of Boundary Value Problems. 12. Higher Order Sturm-Liouville Boundary Value Problems. 13. (n,p) Boundary Value Problems. 14. Focal Boundary Value Problems. 15. General Focal Boundary Value Problems. 16. Conjugate Boundary Value Problems. Difference Equations. 17. Discrete Second Order Boundary Value Problems. 18. Discrete Higher Order Sturm-Liouville Boundary Value Problems. 19. Discrete (n,p) Boundary Value Problems. 20. Discrete Focal Boundary Value Problems. 21. Discrete Conjugate Boundary Value Problems. Integral and Integrodifferential Equations. 22. Volterra Integral Equations. 23. Hammerstein Integral Equations. 24. First Order Integrodifferential Equations. References. Authors Index. Subject Index.

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Book
15 Jun 2001
TL;DR: The Time Scales Calculus as discussed by the authors is a generalization of the time-scales calculus with linear systems and higher-order linear equations, and it can be expressed in terms of linear Symplectic Dynamic Systems.
Abstract: Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

2,581 citations

MonographDOI
01 Jan 2006
TL;DR: Ben-chohra as discussed by the authors dedicates this book to his family members who complete us, and his children, Mohamed, Maroua, and Abdelillah; J. Henderson dedicates to his wife, Darlene and his descendants, Kathy.
Abstract: Dedication We dedicate this book to our family members who complete us. In particular, M. Ben-chohra's dedication is to his wife, Kheira, and his children, Mohamed, Maroua, and Abdelillah; J. Henderson dedicates to his wife, Darlene, and his descendants, Kathy, Contents Preface xi 1. Preliminaries 1 1.1. Definitions and results for multivalued analysis 1 1.2. Fixed point theorems 4 1.3. Semigroups 7 1.4. Some additional lemmas and notions 9 2. Impulsive ordinary differential equations & inclusions 11 2.1. Introduction 11 2.2. Impulsive ordinary differential equations 12 2.3. Impulsive ordinary differential inclusions 24 2.4. Ordinary damped differential inclusions 49 2.5. Notes and remarks 62 3. Impulsive functional differential equations & inclusions 63 3.1. Introduction 63 3.2. Impulsive functional differential equations 63 3.3. Impulsive neutral differential equations 74 3.4. Impulsive functional differential inclusions 80 3.5. Impulsive neutral functional DIs 95 3.6. Impulsive semilinear functional DIs 107 3.7. Notes and remarks 118 4. Impulsive differential inclusions with nonlocal conditions 119 4.1. Introduction 119 4.2. Nonlocal impulsive semilinear differential inclusions 119 4.3. Existence results for impulsive functional semilinear differential inclusions with nonlocal conditions 136 4.4. Notes and remarks 145 5. Positive solutions for impulsive differential equations 147 5.1. Introduction 147 5.2. Positive solutions for impulsive functional differential equations 147 5.3. Positive solutions for impulsive boundary value problems 154 5.4. Double positive solutions for impulsive boundary value problems 159 5.5. Notes and remarks 165 viii Contents 6. Boundary value problems for impulsive differential inclusions 167 6.1. Introduction 167 6.2. First-order impulsive differential inclusions with periodic boundary conditions 167 6.3. Upper-and lower-solutions method for impulsive differential inclusions with nonlinear boundary conditions 184 6.4. Second-order boundary value problems 191 6.5. Notes and remarks 198 7. Nonresonance impulsive differential inclusions 199 7.1. Introduction 199 7.2. Nonresonance first-order impulsive functional differential inclusions with periodic boundary conditions 199 7.3. Nonresonance second-order impulsive functional differential inclusions with periodic boundary conditions 209 7.4. Nonresonance higher-order boundary value problems for impulsive functional differential inclusions 217 7.5. Notes and remarks 227 8. Impulsive differential equations & inclusions with variable times 229 8.1. Introduction 229 8.2. First-order impulsive differential equations with variable times 229 8.3. Higher-order impulsive differential equations with variable times 235 8.4. Boundary value problems for differential inclusions with variable times 241 8.5. Notes and remarks 252 9. Nondensely defined impulsive differential equations & inclusions 253 9.1. Introduction 253 9.2. Nondensely defined impulsive semilinear differential equations with nonlocal conditions 253 9.3. Nondensely defined …

807 citations

Journal ArticleDOI
TL;DR: In this article, the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problem Du + a(t)f(u) = 0, 0 < t < 1,1 < � � 2,
Abstract: In this paper we give sufficient conditions for the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problem Du + a(t)f(u) = 0, 0 < t < 1,1 < � � 2,

295 citations


Cites background from "Positive Solutions of Differential,..."

  • ...[2] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Positive Solutions of Differential, Difference, and Integral Equations,” Kluwer Academic Publishers, Boston, 1999....

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  • ...The texts by Agarwal, O’Regan and Wong [2] and by Guo and Lakshmikantham [13] are excellent resources for the use of fixed point theory in the study of existence of solutions to boundary value problems....

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  • ...Some authors have used fixed point theorems to show the existence of positive solutions to boundary value problems for ordinary differential equations, difference equations, and dynamic equations on time scales, see for example [1, 2, 5, 6, 11, 15, 18, 20, 21, 26, 27, 36] and references therein....

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Journal ArticleDOI
TL;DR: In this paper, the authors employ the critical point theory to establish the existence of multiple solutions of some regular and singular discrete boundary value problems, and show that these solutions can be found in a constant number of rounds.
Abstract: We employ the critical point theory to establish the existence of multiple solutions of some regular as well as singular discrete boundary value problems. ? 2004 Elsevier Ltd. All rights reserved. MSC: 34B16; 39A10

180 citations

Journal ArticleDOI
TL;DR: The existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ (1, 2] are proved by applying some standard fixed point theorems.

175 citations