Existence Of Positive Solutions For Nonlinear Differential System

In last few years, more and more nonlinear problems have resulted from mathematics , physics, chemistry, biology, medicine, economics, engineering, sybernetics and so on.In solving these problems, many important methods and theory such as partial ordering method, topological degree method, the theory of cone and the variational method have been developed gradually. They become very effective theoretical tool to solve many nonlinear problems in the fields of the science and technology. And what is more, they are important approaches to study nonlinear integral equations.This paper mainly investigates the existence of positive solutions for boundary value problems of nonlinear differential systems, including the singular boundary value problems by using topological degree method. The existence and uniqueness of positive solutions for differential equations have been considered extensively for last twenty years ([4]-[31]). This paper discusses the problems of differential systems more generally on the basis of above references.Chapter 1 investigates the existence of positive solutions for three-point boundary value problem of differential system on by using fixed point index theoremwheref,g∈C[(0,1)×R⁺×R⁺, (0,+∞)],λ∈R⁺,η∈(0,1),α> 0, 0 <αη< 1, R⁺ = [0,+∞). The main result is given in follow: Theorem 1.2.1 Suppose (H_1.1)-(H_1.3) hold. Then for any r > 0, there existsλ(r) > 0 such that BVP(1.1.1) has at least two positive solutions (x₁,y₁) and (x₂,y₂) satisfying 0 <‖(x₁,y₁)‖< r <‖(x₂, y₂)‖asλ∈(0,λ(r)).By using cone expansion and conpression theorem, Chapter 2 investigates the existence of positive solutions for boundary value systems with p-Laplacianwhere (?)₁ and (?) : R→R are the increasing homomorphisms and positive homomorphisms , (?)₁(0) = 0, (?)₂(0) = 0,f,g∈C[R⁺×R⁺, (0, +∞)], R⁺ = [0, +∞). a,b∈C[(0,1), R⁺]. The main results are as follows:Theorem 2.2.1 Suppose (H_2.1)-(H_2.6) hold. Then BVP(2.1.1) has at least two positive solutions (x₁,y₁) and (x₂,y₂) satisfying 0 <‖(x₁,y₁)‖< H₀ <‖(x₂,y₂)‖.Theorem 2.2.2 Suppose (H_2.1)-(H_2.3) and (H_2.7)-(H_2.9) hold. Then BVP(2.1.1) has at least two positive solutions (x₃,y₃) and (x₄,y₄) satisfying 0 <‖(x₃, y₃)‖< (?)<‖(x₄,y₄)‖.Chapter 3 investigates the existence of multiple positive solutions for singular boundary value problem with p-Laplacianwhereφ_p(u) =|u|^p-2u, p > 1,α,β,γ,δ> 0,f∈C[(0,1)×(0, +∞), R⁺], R⁺ = [0, +∞). The main result is given in follow:Theorem 3.2.1 Suppose (H_3.1) and (H_3.2) hold, then for r > 0. there existsλ(r) > 0, such thatλ∈(0,λ(r)) SBVP(3.1.1)(3.1.2) has at least two positive solutions u(t) and v(t) satisfying 0 <‖u(t)‖< r <‖v(t)‖.