Several Types Of Nonlinear Differential Equations Boundary Value Problems

The nonlinear analysis is an important branch in nonlinear analysis. The nonlinear functional analysisit is a subject of profound theories and broad ap-plications, bases on nonlinear problems of math and science, constructs general theories and methods, and plays an important role in dealing with all kinds of nonlinear integral equations,especial in nonlinear differential equations and par-tial differential equations. Because it can commendably explain all kinds of nat-ural phenomenal, with the widely application in the realistic production and life. With the development of modern physics and applied mathematics, vaious non-linear problem has aroused people's widespread interest day and day. Especial in the boundary value problem of nonlinear stems,it arising in the physical sciences, applied mathematics the cybernetics and each kind of application discipline,it is an important kind of question in the differential equations, and it is one of most active domains of functional analysis studiesin at present. In this paper, we use the monotone iterative technique, the method of upper and lower solutions ,Leray-Schauder principle, to study the existence of positive solutions for several kinds of boundary value problems for nonlinear singular differential equation.The thesis is divided into three chapters according to contents.In Chapter 1, we discuss the existence of positive solutions to the following nonlinear singular integral boundary value problem: where q∈L1[0,1] is nonnegative,symmetric on [0,1] and q(t)(?)0 on any subinter-val of [0,1] and may be singular at t= 0 and/or 1,f:[0,1] x R+x R x R-â†'> R+ is continuous,symmetric in which R-= (-∞,0], R+= [0,+∞). By using mono-tone iterative technique and combining with the relevent knowledge of cone the-orey,we prove that the above boundary value problem has symmetric positive solutions. The problem we investigated is more general than that is considered in [15], and our results generalize and extend previous results in the field. In Chapter 2, we using of the method of upper and lower solutions and Schauder fixed point theorem, consider the following fourth-order boundary value problem with integral boundary conditions: where nonlinear f is depending on all lower-order derivatives of x,∫:[0,1]×R4â†'R is continuous; gi:Râ†'Ris integral and 0< gi(s)< 1(i= 1,2,3,4); ki(i= 1,2,3,4) are constants with 1