We prove a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order, using a weak contractivity type of assumption. Besides including several recent developments, our theorem can be used to investigate a large class of problems. As an application, we discuss the existence and uniqueness of solution for a periodic boundary value problem.

Fixed point theorems in partially ordered metric spaces and applications

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 …

Impulsive Differential Equations and Inclusions

We introduce the concept of a mixed g -monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. Presented theorems are generalizations of the recent fixed point theorems due to Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393] and include several recent developments.

Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces

We develop a versatile general equilibrium framework to determine the spatial distribution of economic activity on (nearly) any surface with (nearly) any geography. Combining the gravity structure of trade with labor mobility, we provide conditions for the existence and uniqueness of a spatial economic equilibrium and derive a simple set of dierential equations which govern the relationship between economic activity and the geography of the surface. We then use the framework to identify the underlying topography of productivities and amenities both in the United States and across the world. We nd that geographic location is an important determinant of the observed spatial distribution of income. Finally, we show how changing the underlying geography would aect the equilibrium distribution of economic activity.

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Trade and the Topography of the Spatial Economy

We generalize the fixed-point theorem of Leggett-Williams, which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a Banach space. We then show how to apply our theorem to prove the existence of three positive solutions to a second-order discrete boundary value problem.

Three positive fixed points of nonlinear operators on ordered banach spaces

Nonlinear problems in abstract cones

Preface. 1. Preliminaries. 2. Nonlinear Integral Equations in Banach Spaces. 3. Nonlinear Integro-Differential Equations in Banach Spaces. 4. Nonlinear Impulsive Integral Equations in Banach Spaces. References.

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Dajun Guo

Papers

Nonlinear problems in abstract cones

Nonlinear Integral Equations in Abstract Spaces

Extremal Solutions of Nonlinear Impulsive Integrodifferential Equations in Banach Spaces

Periodic Boundary Value Problems for a Class of Second-Order Impulsive Integro-Differential Equations in Banach Spaces

Multiple positive solutions of boundary-value problems for impulsive differential equations