Existence Of Solutions For Several Classes Of Nonlinear Abstract Equations And Integral Equations

Nonlinear problems have gained considerable attention due to their applica-tions in various fields of science and engineering. Nonlinear equations including nonlinear differential equations and integral equations, which are important in the study of nonlinear problems. In this context, the main objective of this paper is to study the existence of solutions to four classes of nonlinear equations and establish a theorem on fixed points of nonlinear contractive mappings.In chapter 3, we are concerned with the existence of mild solutions to the Cauchy problem for fractional evolution equations of neutral type with almost sectorial operators where 0< q< 1, the fractional derivative is understood in the Caputo sense, A is an almost sectorial operator on a complex Banach space, and/, h are given functions. With the help of the theory of measure of noncompactness and a fixed point theorem of Darbo type, we establish a new existence theorem of mild solutions for the Cauchy problem above. By the way, the global attractive property of the solutions is also obtained. Moreover, we give two examples to illustrate our abstract results. Next, we also consider the case q= 1 for a class of nonlinear evolution equations, that is, and establish a new existence theorem.Then in chapter 4, a class of neutral stochastic differential equations with infinite delay is investigated. Under some suitable assumptions, the existence of quadratic mean almost periodic mild solutions for the equation are discussed by means of semigroups of operators and the fixed point method.Thanks to the theory of measure of weak noncompactness and Krasnoselskii’s fixed point theorem, we consider in chapter 5 a class of general nonlinear integral equation and establish a new and general existence theorem for above nonlinear functional integral equation.In chapter 6, based on the works of [3,48,49,50,79], we study the existence of a unique fixed point for the generalized eventual cyclic gross contractive map-pings in metric spaces. Our main result extends some existing fixed point theorems.