Existence And Uniqueness Of Two Classes Of Integral Equations On Λ-bounded Variation Solutions

Integral equations are an important branch of modern mathematics. The development of integral equations is closely related to the problems of mathematics and physics, moreover, it is widely used in engineering science and mechanics. In fact, the solutions of many integral equations often describe some physical phenomena in real life, and these solutions are often A-bounded variation functions. So the research on A-bounded varation solutions of the integral equations is an important direction of the integral equations theory. Taking into account a lot of the facts that the fractional derivatives exist widely in nature and in many technological sciences, thus this article mainly studies the nonlinear integral equations and the fractional order derivative of the nonlinear integral equations about the existence and uniqueness of solutions. This article is organized as follows:The first chapter introduces the research on A-bounded varation solutions of the integral equations and the solutions of the fractional order derivative of integral equations, as well as some of the historical background of the concept of A-bounded variation and the fractional derivative. We state A-bounded varation functions and the fractional order derivatives which are widely applied in various subject areas, and they are aboundant of profound physical background and theoretical connotation.The second chapter begins from the practical significance of A-bounded varation solu-tions of a concrete integral equation in physical field, thus we obtain a class of nonlinear integral equations, then we promote the nonlinear integral equations and define the operator G which is a contraction mapping. Taking advantage of the Banach contraction principle to prove the existence of the solutions in the space of ABV(I). Meanwhile, we define the map-ping P that satisfies the Lipschitz condition, Furthermore, F’ is a superposition operator of higher order, thus we get the uniqueness of A-bounded varation solutions by the Lovelady fixed point theorem. We obtain the main conclusions as follows:Theorem 2.2.1 In the equation (1.3.1), assume that:(1) g:I�'R is a ABV function;(2) q(s) is a bounded function,|q(s)|≤K, K is a constant;(3) p(s) a bounded function,|ρ(s)|≤L, L is a constant;(4) F:I×I�'R is a function such that VΛ(K(·,s):I)≤M(s) for a.e.s∈I, where M:I�'R+、F(·, s) is integrable in the Lebesgue sense;(5) Let F’(x)(t)=f(t,x(t)),f(0,0)=0,f(t,x(t)) exists continuous partial derivatives with regard to x(t),(?)f/((?)x(t))|t=0=0;Under the above assumptions there exists λ,ξ,η>0, such that ω2<λ,g∈BΛ(0,η), equation (2.1) has a unique ΛBV solution x(t)∈BΛ(0,ξ).The third chapter is based on the last chapter, we achieve a class of the fractional order derivative of nonlinear integral equations by introducing the concept of Caputo fractional derivatives. We define the space E of continuous functions and the operator A under the con-dition of A-bounded varation functions, taking advantage of the Schauder fixed point theory to prove the existence of the solutions in the space of ABV(I). Meanwhile, H(x)(t),f(t, x(t)) satisfy the Lipschitz condition, so we get the uniqueness of A-bounded varation solutions. We obtain the main conclusions as follows:Theorem 3.2.1 Suppose conditions (2), (3), (4) of Theorem 2.2.1 are satified, f(t,x(t)) is a bounded function. Then the equation (3.1) exists a ΛBV solution:x(t)∈C[0,h]∩ L1[0,h], Dqx(t)∈C(I)∩L1(I).Theorem 3.2.2 Suppose conditions of Theorem 3.2.1 are satified, H(x)(t),f(t,x(t)) fulfill a Lipschitz condition: |H(x)(t)-H(x)(t)|≤L’|x(t)-x(t)|, |f(t,x(t))-f(t,x(t))|≤L"|x(t)-x(t)|. Then the equation (3.1) exists at most a ΛBV solution:x(t)∈C[0,h]∩L1[0,h],Dqx(t)∈ C(I)∩L1(I).