Existence Of Multiple Solutions For Some Boundary Value Problems

In last few years,many sorts of nonlinear problems have resulted from mathe-matics,pysics,chemistry, biology, medicine, economics,engineering,sybernetics and so on.With solving these problems,many important methods and theory such as par-tial ordering method, topological degree method, the theory of cone and the variational method have been developed gradually. They become very effective theoretical tools to solve nonlinear problems in the fields of the science and technology.This paper mainly investigates the existence of multiple solutions for boundary value problems of nonlinear differential system by using the theory of cone,the fixed point theory and the fixed point index theorems.The existence and uniqueness of so-lutions for differential equations have been considered extensively since twenty years ago([17],[23],[24],[30]). This paper discusses the existence of multiple solutions for non-linear differential system on the base of the dissertations.Chapter 1 investigates the existence of multiple solutions for second order three point boundary value problems of nonlinear differential system where f∈C((0,1)×[0,+∞)×(-∞,+∞),(0,+∞)),g∈C((0,1)×[0,+∞)×(-∞,+∞),(-∞,+∞)). 0<η<1,α>0 andαη<1.The papers([1],[15])considered the existence of solutions for second order two point boundary value problems of nonlinear differential system, the paper([24])considered the existence of one solution for second order three point boundary value problems of coupled system. On the base of these papers and the paper[26],this paper considers the above questions.As far as we know,there is no paper to investigate the existence of solutions for nonlinear differential system as this paper does.The main tool used here is the fixed point index theorem of cone.In this paper we suppose that f is nonnegative while g is allowed to change sign.Finally we get the existence of multiple solutions from the Theorem 1.2.1.And we give an example to demonstrate the conditions are suitable.Chapter 2 investigates the existence of three solutions for fourth-order boundary value problems with intergral boundary conditions where(1)f,g∈C([0,1]×[0,+∞),[0,+∞)).For each t∈[0,1], g(t,x) is increasing inx.(2) (?)t∈[0,1],a(t),b(t)≥0;∫01a(t)dt,∫01b(t)dt∈(0,1).The nonlinear problem with intergral boundary condition is investigated by Bitssdze in 1960.Then Samarskili and I Il'in also studied the same work. Recently,more and more attention is paid to this problem,such as papers([3],[5],[27] and references therein).[28] considered the existence of positive and symmetrical solutions for fourth-order boundary value problem with intergral boundary condition.On the base of the papers,this paper considers the existence of three solutions for the above question by using the theory, which is proved in[20].And we give an example to illustrate the conditions are suitable.By using Leggett—Williams fixed point theorem,the last Chapter investigates the existence of three solutions for fourth-order boundary value problems where a≥0,b≥0,c,d>0,ac+bc+ad>0.Also assume thatWe get the Theory 3.2.1 according to papers([6],[13],[14])and we give an example to demonstrate the conditions are suitable.