Several Classes Of Nonlinear Differential Equation Boundary Value Problem For The Existence Of The Solution

Along with the development of modern physics and applied mathematics,various nonlinear problem has aroused people’s widespread attention day by day.The nonlinear functional analysis as an important branch in nonlinear analysishas become an important tool for study mathematics, pysics, chemistry, biology’stechnology. The nonlinear functional analysis ofers efective theoretic meaningfor these problems, and it is a reaserch subjct for profound theories and broadapplication. The nonlinear functional analysis bases on nonlinear problems ofmath and all kinds of scientific field, and constructs general theories and methods.It could solve various the natural phenomenon and problem.The boundary value problem of nonlinear stems from the applied mathe-matics, the physics, the cybernetics and each kind of application discipline. Itis an important kind of question in the diferential equations, it is one of mostactive domains of functional analysis studiesin at present. The singular nonlineardiferential equation boundary value problem is also the hot spot which has beendiscussed in recent years. So it attachs more and more attention.In this paper, we use the cone theory, the fixed point index theory, the lowerand upper solutions as well as the Krasnoselskii’s fixed point theory, to study theexistence of solutions for several kinds of boundary value problems for nonlinearsingular diferential equation.The thesis is divided into three chapters:In Chapter1, we study the following2p-order and2q-order nonlinear singularsemiposition systems:where f, g:(0,1)×(R+)p×(R+)q→R are continuous, f, g may be singular att=0and/or t=1and may take negative values, in which R+=[0,+∞), p, q∈ N+, ai≥0, bi≥0, ci≥0, di≥0, ρi=aici+aiai+bici>0,0≤i≤p1; andαj≥0, βj≥0, γj≥0, δj≥0, ρ j=αjγj+αjδj+βjγj>0,0≤j≤q1. weobtain the existence of positive solution. The results are based upon the fixedpoint index theorem. so this chapter improve the main results of [1],[2].In Chapter2, we study the existence of positive solutions of second-ordersingular boundary value problem with nonlinear boundary conditions on the half-where ai, bi, ci, di≥0, and aidi+aici+bici>0, pi(t)∈C[0,+∞)∩C1(0,+∞),pi(t)>0, t∈(0,+∞), φi0+: R→R+0, fi: R+×R×R×R×R→R arecontinuous functions (i=1,2), Here R+=[0,+∞), R+0=(0,+∞). By applyingthe techniques of unbounded upper and lower solutions, we obtain the existenceof at least one solution.In Chapter3, we considered the existence of positive solutions for the fol-...