Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions

Abstract

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of "bang---bang" principle for control systems is given as well by utilizing the relaxation theorem.