Integral Boundary Value Problems Of Nonlinear Differential Equations

Along with science's and technology's dcvelopment, various non-linear problem has aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an important branch in nonlinear 'analysis, because it can explain well various the natural phenomenon. The boundary value problem of nonlinear differential equation stems from the applied mathematics, the physics, the cybernetics and each kind of application discipline. It is one of most active domains of functional analysis studiesin at present. The nonlinear differential equation integral boundary value problem is also the hot spot which has been discussed in recent years. So it become a very important domain of differential equation research at present. In this paper, we use the cone theory, the fixed point theory, the topologieal degree theory as well as the fixed point index theory and concerned the first eigenvalue corresponding to the relevant linear operator, to study several kinds of integral boundary value problems for differential equation and we apply the main results to the boundary value problem for the integral differential equation.The thesis is divided into three chapters according to contents.In chapter 1, by combining fixed point index theory and constructing an available integral operator, we establish sufficient conditions for the existence of positive solutions under some conditions concerning the first eigenvalue corre-sponding to the relevant linear operator. In this paper, we are concerned with the existence of positive solutions of the following integral boundary value problemWhere f: [0,1] x [0, +∞)→R is continuous, and there exist M≥0 and b≥0 such that f(t, x)≥-Mx-b. In the case where f can be allowed to change sign,αandβare right continuous on [0,1), left continuous at t = 1, and nondecreasingon [0,1] withα(0) =β(0) = 0; (?)u(r)dα(r) and(?)u(r)dβ(r) denote theRismann - Stieltjes integrals of u with respect toαandβ, respectively. In [1], using the topoiogical degree theory,the author consider the singular two-point boundary value problem,and there exist three constants b > 0, c > 0 andα6 (0,1) such that f(u) >-b - c|u|^αfor all u∈R. We generalize and improve the results in [1], and consider integral boundary value problem under the conditionα= 1, the method is also different from [1].In chapter 2,under the similar condition of chapter 1, the fixed point theory and the topological degree theory are used to investigate the positive solution of Strum - Liouville integral boundary problems (2.1)Where p∈C¹[0, 1], p(t) > 0; q(t)∈C[0,1], q(t) > 0;α,β,δ,γ> 0 are constants such thatβδ+αδ+αγ> 0; f: [0,1] x [0, +∞)→R is continuous;σ₁ andσ₂ are right continuous on [0,1), left continuous at t =1, and nondecreasing on[0,1], withσ₁(0)σ₂(0) = 0; (?)u(s)dσ₁(s) and (?) u(s)dσ₂(s) denote theRiemann -Stieltjes integrals of u with respect toσ₁ andσ₂, respectively. Inthis paper, we generalize and improve the main results in chapter 1, and apply the main results to the integral boundary value problem.In chapter 3, the fixed point theory and the topological degree theory are used to investigate the existence of nontrivial solutions of the following fourthorder Strum - Liouville integral boundary value problemwhereα_iβ_iδ_iγ_i≥0(i = 1, 2) are constants such thatρ_i =β_iδ_i+α_iδ_i +α_iγ_i > 0(i = 1,2); h: (0,1)→[0,+∞) is continuous on (0,1) and may be singular at t = 0 or/and t = 1; f: [0,1]×(-∞, +∞)×(-∞, +∞)→(-∞, +∞) is continuous;φ₁ andφ₂ are right continuous on [0,1). left continuous at t = 1, and non-decreasingon [0,1], withφ₁(0) =φ₂(0) = 0;(?) u(s)dφ₁(s) and (?) u''(s)dφ₂(s) decreasing on [0,1], withφ₁(0) =φ₂(0) = 0; (?)u(s)dφ₁(s) and (?) u''(s)dφ₂(s) denote the Riemann - Stieltjes integral of u and u" with respect to fa andφ₂, respectively. We gained the integral boundary value problem (3.1) at least have a nontrivial solution. By using the Krasonselskii's fixed point theorem in a cone, in [6] ,the author proved the existence of positive solutions of fourth-order Strum-Liouville boundary value problem with changing sign nonlinearitywheref: [0, l]×[0,+∞)×(-∞,0]→[0,+∞) is continuous; b: (0,1)→(-∞,+∞) is Lebesgue integrable. The present paper is motivated by [6], we suppose f: [0,1]×(-∞,+∞)×(-∞,+∞)→(-∞, +∞) is continuous and consider integral boundary conditions. We obtain the existence results of nontrivial solutions,the method and the main results are different from paper [6].The innovation of this paper is as follows. In chapter 1, the nonlinear term has been improved and the integral boundary value problems have been discussed and the methods are also different from those in [1]. At the basis of the chapter 1, the existence of the positive solutions of Strum - Liouville integral boundary problems have been investigated in chapter 2. The discussion is more wide and the results are better. In chapter 3, we obtain the existence results of nontrivial solutions, and the existence results of positive solutions for some cases, the method and the main results are different from paper [6].