Boundary Value Problems Of Nonlinear Equations And Their Applications

With the development of modern physics and applied mathematics, the mathematic abilities of analysing and dominating objective phenomenon are demanded highly and precisely. Further, more and more results of nonlinear analysis are accumulated continuously. As a result, an important branch of applied mathematics-nonlinear functional analysis establishes, which is at present one of the most active fields that is studied in analytical mathematics. Among them, the nonlinear boundary value problem are being studied extensively. The present paper employs the cone theory, fixed point index theory, and Krasnoselskii fixed point theorem and so on, to investigate the existence of solutions to boundary value problem of several kinds of nonlinear systems of differential equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into three sections according to contents.In chapter 1, we consider the systems of second-order singular differential equations involving parameterswith boundary conditionswhere f, g : [0,1]×[0,∞)⁴â†'[0,∞) are continuous;φ,Φ: (0,1)â†'[0,∞) are continuous and may be singular with t = 0, or t = 1;α,β,γ,δ≥0. Some theorems for the existence, non-existence, one positive solution and multiplicity of its positive solutions are obtained by the Krasnosel'skii-Guo theorem on cone compression and expansion in a special function space, the upper and lower solutions method. In Chapter 2, by a new maximum principle for operator L₂u = u" -2au'+(a² + b²)u in the boundary condition and a fixed-point theorem in cones, we investigate the existence of positive solutions for the following periodic boundary value problem (PBVP)whereλ≥0; a∈R, b∈(0,Ï€);φ₁,φ₂: (0,1)â†'[0, +∞) are continuous, that isφ₁,φ₂ are probably singular at t = 0 and t = 1; f₁(t, u, v) : [0,1]×(0, +∞)×(0, +∞)â†'(0, +∞) and f₂(t,u,v) : [0,1]×[0,+∞)×(0,+∞)â†'[0,+∞) are two continuous functions, that is f₁(t,u,v) and f₂ (t,u,v) may be singular at u = 0 and v = 0 respectively. We find that whenλlies in some range, (PBVP)(2.1.1) at least has one positive solution.In Chapter 3, by constructing a special cone and applying fixed index theory in cone, we investigate the existence of positive solution for a semi-positone singular boundary value problemwhereλ> 0 is a parameter, f₁ : (0,1)×(0,∞)×[0,∞)â†'R, f₂ : (0,1)×[0,∞)×(0,∞)â†'R are continuous, that is f₁ may be singular at t = 0, t = 1 and u = 0, f₂ may be singular at t = 0, t = 1 and v = 0. We find aλ₀ > 0, such that the system (3.1.1) has at least one positive solution for 0 <λ<λ₀.