Donal O'Regan

Positive solutions of differential, difference and integral equations

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.

Generalized contractions in partially ordered metric spaces

We present some fixed point results for monotone operators in a metric space endowed with a partial order using a weak generalized contraction-type assumption.

Dynamic equations on time scales: a survey

The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. In this paper we give an introduction to the time scales calculus. We also present various properties of the exponential function on an arbitrary time scale, and use it to solve linear dynamic equations of first order. Several examples and applications, among them an insect population model, are considered. We then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients. Finally, we consider self-adjoint equations and, more generally, so-called symplectic systems, and present several results on the positivity of quadratic functionals.

Infinite Interval Problems for Differential, Difference and Integral Equations

Preface. 1. Second Order Boundary Value Problems. 2. Higher Order Boundary Value Problems. 3. Continuous Systems. 4. Integral Equations. 5. Discrete Systems. 6. Equations in Banach Spaces. 7. Multivalued Equations. 8. Equations on Time Scales. Subject Index.

Oscillation Theory for Difference and Functional Differential Equations

Preface. 1. Oscillation of Difference Equations. 1.1. Introduction. 1.2. Oscillation of Scalar Difference Equations. 1.3. Oscillation of Orthogonal Polynomials. 1.4. Oscillation of Functions Recurrence Equations. 1.5. Oscillation in Ordered Sets. 1.6. Oscillation in Linear Spaces. 1.7. Oscillation in Archimedean Spaces. 1.8. Oscillation of Partial Recurrence Equations. 1.9. Oscillation of System of Equations. 1.10. Oscillation Between Sets. 1.11. Oscillation of Continuous-Discrete Recurrence Equations. 1.12. Second Order Quasilinear Difference Equations. 1.13. Oscillation of Even Order Difference Equations. 1.14. Oscillation of Odd Order Difference Equations. 1.15. Oscillation of Neutral Difference Equations. 1.16. Oscillation of Mixed Difference Equations. 1.17. Difference Equations Involving Quasi-differences. 1.18. Difference Equations with Distributed Deviating Arguments. 1.19. Oscillation of Systems of Higher Order Difference Equations. 1.20. Partial Difference Equations with Continuous Variables. 2. Oscillation of Functional Differential Equations. 2.1. Introduction. 2.2. Definitions, Notations and Preliminaries. 2.3. Ordinary Difference Equations. 2.4. Functional Difference Equations. 2.5. Comparison of Equations of the Same Form. 2.6. Comparison of Equations with Others of Lower Order. 2.7. Further Comparison Results. 2.8. Equations with Middle Term of Order (n - 1). 2.9. Forced Differential Equations. 2.10.Forced Equations with Middle Term of Order (n - 1). 2.11. Superlinear Forced Equations. 2.12. Sublinear Forced Equations. 2.13. Perturbed Functional Equations. 2.14. Comparison of Neutral Equations with Nonneutral Equations. 2.15 Comparison of Neutral Equations with Equations of the Same Form. 2.16. Neutral Differential Equations of Mixed Type. 2.17. Functional Differential Equations Involving Quasi-derivatives. 2.18. Neutral and Damped Functional Differential Equations Involving Quasi-derivatives. 2.19. Forced Functional Differential Equations Involving Quasi-derivatives. 2.20. Systems of Higher Order Functional Differential Equations. References. Subject Index.

Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations

Preface. 1. Preliminaries. 2. Oscillation and Nonoscillation of Linear Ordinary Differential Equations. 3. Oscillation and Nonoscillation of Half-Linear Differential Equations. 4. Oscillation Theory for Superlinear Differential Equations. 5. Oscillation Theory for Sublinear Differential Equations. 6. Further Results on the Oscillation of Differential Equations. 7. Oscillation Results for Differential Systems. 8. Asymptotic Behavior of Solutions of Certain Differential Equations. 9. Miscellaneous Topics. 10. Nonoscillation Theory for Multivalued Differential Equations. Subject Index.

Existence theory for nonlinear integral and integrodifferential equations

Preface. 1. Introduction and Preliminaries. 2. Existence Theory for Nonlinear Fredholm and Volterra Integrodifferential Equations. 3. Solution Sets of Abstract Volterra Equations. 4. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Compact Intervals. 5. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Half-Open Intervals. 6. Existence Theory for Nonlinear Nonresonant Operator and Integral Equations. 7. Existence Theory for Nonlinear Resonant Operator and Integral Equations. 8. Integral Inclusions. 9. Approximation of Solutions of Operator Equations on the Half Line. 10. Operator Equations in Banach Spaces Relative to the Weak Topology. 11. Stochastic Integral Equations. 12. Periodic Solutions for Operator Equations. Index.

Fixed Point Theory for Lipschitzian-type Mappings with Applications

Fundamentals.- Convexity, Smoothness, and Duality Mappings.- Geometric Coefficients of Banach Spaces.- Existence Theorems in Metric Spaces.- Existence Theorems in Banach Spaces.- Approximation of Fixed Points.- Strong Convergence Theorems.- Applications of Fixed Point Theorems.

Discrete Oscillation Theory

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Multiple positive solutions of singular discrete p-Laplacian problems via variational methods

We obtain multiple positive solutions of singular discrete p-Laplacian problems using variational methods.