Nonlinear Singular Differential And Integral Equations Boundary Value Problems And Applications

Along with development of science and technology, various non-linear problem hasaroused widespread interest day by day, and so the nonlinear analysis has become oneimportant research direction in modern mathematics. The nonlinear functional analysisis an important branch in nonlinear analysis, because it can explain well various the nat-ural phenomenon. The boundary value problem of nonlinear differential equation stemsfrom the applied mathematics, the physics, the cybernetics and each kind of applicationdiscipline. It is one of most active domains of functional analysis studiesin at present.The singular nonlinear differential equation boundary value problem is also the hot spotwhich has been discussed in recent years. So it become a very important domain of dif-ferential equation research at present. In this paper, we use the cone theory, the fixedpoint theory, the topological degree theory as well as the fixed point index theory andcombined with a iterative technique, to study several kinds of boundary value problemsfor nonlinear singular differential equation and we apply the main results to the boundaryvalue problem for the singular integral differential equation.The thesis is divided into four chapters according to contents.In chapter 1, we use the cone theory and M(o|¨)nch fixed point theorem combined witha monotone iterative technique to investigate the positive solutions of a class of boundaryproblems for nth-order nonlinear impulsive singular integro-differential equations of mixedtype on an infinite interval in Banach spaces.We not only obtain the existence of a positive solution for singular boundary value problem(1.1.1) but also develop an iterative sequence for the solution. This paper generalize andimprove the results in [5, 14], and apply the main results to the infinite system of scalarsecond order impulsive singular integro-differential equations.In chapter 2, the fixed point theory and monotone iterative technique are used toinvestigate the unique positive solution of boundary problems (1.1.1). In addition, anexplicit iterative approximation of the solution for the boundary value problem are de-rived. In this paper, we generalize and improve the results in [7, 15, 18, 20, 21], andapply the main results to the infinite system of scalar second order impulsive singularintegro-differential equations. In chapter 3, we study the existence and multiplicity of nontrivial solutions for thefourth order m-point singular boundary value problems:u⁴(t)=f(t,u(t),-u″(t)),t∈J′,Making use of the theory of the fixed point index in a cone and the Leray-Schauder degree,we prove that there exist at least ten different nontrivial solutions for the fourth orderm-point singular boundary value problems. We obtain the main results in this paperwhich is the important improvements and the supplement to the main theorems 1, 2 andthe corollaries 1-3 in paper[28].In chapter 4, we obtain the existence of multiple positive solutions of m-point bound-ary value problem for 2nth-order singular nonlinear integro differential equations in aBanach space by means of fixed point index theory of completely continuous operators.For the special case f∈C(J′×[0,∞)^N, [0,∞)), paper [50] has investigated BVP(4.1.1)andobtained some necessary and sufficient conditions for the existence of a positive solutionby means of the fixed point theorems. When u²ⁱ(1)=0 or n = 2 in (4.1.1) a sameresult was gotten in paper [30, 51] by constructing lower and upper solutions. For thegeneral case f∈C(J×Rⁿ,R) in (4.1.1) i.e., f is continuous, BVP(4.1.1)is nonsingular,some sufficient conditions for the one or more solutions of BVP(4.1.1)have been gottenby paper [52, 52] applying Leggett-Williams fixed point theorem. But this paper is toestablish the existence of multiple positive solutions to the BVP(4.1.1) by using the thefixed point index theory of completely continuous operators, which is different from thatof papers [30, 50-53].