The Basic Theory For Nonlinear Integro-differential Equations

The basic theory of boundary value problem and initial value problem of nonlinear integro-differential equations are considered in this paper.This paper consists of three chapters. In the first chapter, the history and actuality of these problems are summarized. The relevant notes, theories, and definitions are listed. These conclusions are mostly extracted from the results of references books [1] and [2].In the first section of the first chapter, the existence theorems of solutions of nonlinear integro-differential equations of Volterra type in Banach spaces are proved by means of Ascoli-Arzela's fixed-point theorem. The conclusions are as follows:Theorem2.1.1 Let B =(where, Ms areal number),numbersdenotes (Kuratowski) measure of noncompactness. Then the following system has at least one solution on In the second section of the second chapter, the existence theorem of solutions of nonlinear integro-differential equation of Voltrra type with the following boundary value condition:are proved by means of compress mapping principle. The conclusions are as follows:Theorem 2.2.1 Let f be continuous on IxE3 ,and satisfy the following Lipshitz'scondition:whereandThen the boundary value problem has at least one solution x e C2 [I,E].In the third section of the second chapter, the existence theorems of solutions of nonlinear integro-differential equations of mixed type in Banach spaces are proved by means of Darbo fixed-point theorem.whereThe conclusions are as follows:Theorem 2.3.1 suppose that the following assumptions are satisfied.(i)There exists constant N > 0 such thatfor any bounded(ii) There exists constant(iii)k(t,s) andh(t,s) satisfy the conditions in Lemma (2.3.3)and (2.3.4).Thenprovided the boundary valueproblem has at least one solutionIn the first section of the third chapter, the existence of maximum and minimum solutions and uniqueness theorems of solutions of nonlinear integrodifferential equations with impulses are proved.whereis a real Banach spaces,C[E,E] .The conclusions are as follows: Suppose that(HI) There exist y0,z0 PC[I,E] such that:(H2) There exists constantfor anysuch that:(H3) For anyif B beequicontinuouson.thenare constants, andai(H4) There exists constantssuch that:for anyN is normal constant in P.Theorem 3.1.1 Assume that E is a Bnanch space, P is a normal cone in E, and (HI), (H2), (H3) are satisfied . Then the FVP has maximum solution y* and minimum solution z' in [y0,z0] > furthermore , let:uniformlyIn the second section of the third chapter, the existence theorems of solutions of boundary value problem of nonlinear integro-differential equations with impulse are studiedThe conclusions are as follows:Theorem 3.2.1 suppose that the following assumptions are satisfied.:(HI) For anyThen the IBVP has unique solution...