Differential Inclusions in a Banach Space

1. Multi-Valued Differential Equation Generated by A Differential Inclusion.- 1. Background of the theory of measurable multi-valued mappings and differential inequalities.- 2. Existence of local solutions of a multi-valued differential equation with conditions of compactness type.- 3. Existence of local solutions of a multi-valued differential equation. Comparison theorems.- 4. Global solutions of a multi-valued differential equation.- 5. Existence theorems of solutions of the multi-valued operator equation.- 6. Notes and Remarks.- 2. Differential Inclusions. Existence of Solutions.- 1. Types of solutions of a differential inclusion.- 2. Semicontinuous multi-functions.- 3. Strongly measurable selectors of multi-functions.- 4. Continuous selectors of multi-functions with decomposable values.- 5. Continuous selectors of a multi-valued mappings generated by differential inclusions with non-convex right hand side.- 6. Multi-valued selectors of a multi-valued mapping generated by a differential inclusion.- 7. Existence of solutions of a differential inclusion with nonconvex right hand side continuous in x.- 8. Existence of the solution of a differential inclusion with right hand side semicontinuous in x.- 9. Notes and Remarks.- 3. Properties of Solutions.- 1. Auxiliary results.- 2. The density theorems.- 3. The co-density theorems.- 4. Compactness of the set of solutions.- 5. Dependence of the set of solutions on initial conditions and parameters.- 6. Connectedness of the set of solutions.- 7. Notes and remarks.- 4. Integral Funnel of the Differential Inclusion.- 1. Auxiliary lemmas.- 2. The equation of the integral funnel.- 3. Solutions of the equation of the integral funnel.- 4. Properties of solutions of the equation of the integral funnel.- 5. Properties of the integralfunnel.- 6. Extreme points of the set of solutions of a linear differential inclusion.- 7. Notes and Remarks.- 5. Inclusions with Non-Compact Right Hand Side.- 1. Continuous selectors of fixed point sets of multi-functions with decomposable values.- 2. Properties of the multi-valued Nemytskii operator.- 3. Continuous selectors of fixed point sets of the multi-valued Nemytskii operator.- 4. Existence and properties of solution sets of differential inclusion.- 5. Notes and Remarks.- Appendices.- A—.- 1. Non-convex problem of calculus of variations.- 2. Existence of optimal control without assumptions of convexity.- 3. Extension in continuity of multi-valued mappings.- 4. Equivalence of differential inclusions and control systems.- References.- Symbols.