Solutions For Several Kinds Of Nonlinear Differential-integral Equations

Nonlinear functional analysis is an important branch of mathematical analysis. It has formed a complete theoretical system in 1950s. In recent years, physics, aerospace, biotechnology and other practical problems have emerged, nonlinear functional analysis has become an important tool to solve these nonlinear problems.Existence and multiplicity of solutions for nonlinear differential-integral equations is an important research topic, it can describe the specific problems in physics, chemistry, economics and other applied disciplines. In this paper, we use several methods of nonlinear functional analysis such as cone theory, monotone iterative method, fixed point theorem to study several types of nonlinear differential-integral equations and get the existence and multiplicity of solutions. We improve and promotion some main theorems which appeared in previous literatures, and give examples to illustrate the effectiveness of the resulting conclusions. This paper is divided into three chapters, its main contents are as follows:In the first chapter, we introduces the significance, background and research status about nonlinear differential-integral equations, give the main theorem of this paper.In the scend chapter, we first establish the existence and uniqueness of fixed point with three arguments, give a concise proof by using normal cone, and mixed monotonic operator. Then we use the theory to get the existence of positive almost automorphic solutions for neutral integral equations where are almost automorphic functions. Finally, we give an example to illustrate the correctness of the result.In the third chapter, we use properties of Green function and the cone expansion-compression fixed point theorem to study a nonlinear singular impulsive Sturm-Liouville boundary value problemwhere J=[0,+∞),J+=(0,+∞),Parameters α1,α2,β1,β2,γi,δi∈J,ηi,tk∈J+,i=1,2, …m-2,t1<t2<…<tn,tk≠ηi,f∈C[J+×J+,J+],p∈C[J,J+]∩C1[J+,J+], IK∈C[J,J],(?)+∞01/p(v)dv<+∞,ρ=α2β1+α1β2+α1α2 (?)+∞0 1/p(v)dv>0,△x’(tk)=x’(tk+)-x(tk-),x’(tk-),x’(tk+)are the left and right limits of x’(t)at point k.