Study For The Existence Of Positive Solution Of Several Boundary Value Problems Of Differential Equation

In later years, all sorts of nonlinear problems have resulted from mathematics,physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. During the development of solving such problems, nonlinear functionalanalysis has been bing one of the most important research fields in modern mathematics. It mainly includes partial ordering method, topological degree method and the variational method. Also it provides a much effect theoretical tool for solving many nonlinear problems in the fields of the science and technology.And what is more, it is an important approach for studying nonlinear differential equations arising from many applied mathematics. L. E. J. Brouwer had established the conception of topological degree for finite dimensional space in 1912. J. Leray and J. Schauder had extend the conception to completely continuousfield of Banach space in 1934, afterward E. Rothe, M. A. Krasnosel'skii, P. H. Rabinowitz, H. Amann, K. Deimling had carried on embedded research on topological degree and cone theory. Many well known mathematicians in China, say Zhang Gongqing, Guo Dajun, Chen Wenyuan, Ding Guanggui, Sun Jingxianetc., had proud works in various fields of nonlinear functional analysis. (See [1-12]).The singular ordinary differential equation is an important aspect of differentialequation, it arise in the fields of gas dynamics, newtonian fluid mechanics, nuclear physics, the theory of boundary layer, nonlinear optics and so on. Theorefore,it has been considered extensively(See [16,46-49] and reference therein).The present paper mainly investigates existence of positive solutions, multiplicityfor several classes of ordinary differential equations as well as singular differential equation boundary value problem by using topological degree, cone theory and lower and upper solution method. The main contents are as follows:Chapter 1 gives serval lemmas on existence of fixed point and computation of fixed point index, which play an important role in next chapters.Chapter 2 considers the positive solution of fourth-order differential equation boundary value problem withe two varible parameterwhere A(t),B(t)∈C[0,1],ξ_i∈(0,1),a_i,b_i∈[0,+∞),i=1,2,…,m-2are given constants. Using the fixed point index theory, we obtain the sufficient condition for the above BVP has at least one and two positive solution.Chapter 3 considers the positive solution of singular semipositone (n,p) eigenvalue problem.In section 1, using the fixed point index theorem on cones, we investigates the existence of positive solution for the following singular (n, p) boundary value problem.where n≥2,1≤p≤n-1 are fixed.λ>0 is a constant, q∈L¹(0,1),q≥0,a．e．f:[0,1]×R⁺→R is continuous.In section 2, we replace the suppose of continuity on f by relative weaker Caratheodory condition, and removed the strict restriction of lower bounded on f. we still obtained the existence of positive solution and multiplicity for the above BVP.Chapter 4 investigates singular semipositone (k, n - k) conjugate m-point boundary value problemwhere a_i∈[0,∞),i=1,2,…,m-2,∑_i=1^m-2a_i>0,0<ξ₁<ξ₂<…<ξ_m-2<1 are constants,m≥3,f:(0,1)×[0,∞)→[0,+∞)is continuous,p:(0,1)→(-∞,+∞) Lebesgue integrable. By using the fixed point indextheory, we obtain the sufficient condition of positive solution for the above BVP. Chapter 5 investigates the following sigular higher-order m-point boundary value problemwhere a_i∈[0,∞),i=1,2,…,m-2,∑_i=1^m-2 a_i>0,0<ξ₁<ξ₂<…<ξ_m-2<1 are constants,m≥3,f:(0,1)×[0,∞)→[0,+∞)is continuous. we givea simpler expression of Green's function for the above problem, and discuss it's properties. then by using the fixed point index theory, we obtained the existence of positive solution for the above BVP.Chapter 6 investigateswhereψ_p(t)=|t|^p-2t,p>1,0<ξ<η<1,are constants,αandβare rightcontinuous on [ξ,η), left continuous at t =η, and nondecreasing on [ξ,η], withα(ξ)=β(ξ)=0;∫_ξ^ηx(Ï„)dα(Ï„)and∫_ξ^ηψ_p(x"(Ï„))dβ(Ï„)denote the Riemann-Stieltjes integrals of x andψ_p(x") with respect toαandβ, respectively, and0<∫_ξ^ηdα(Ï„)<1 and 0<∫_ξ^ηdβ(Ï„)<1.By using the lower and upper solutionmethod, we obtain the sufficient and necessary condition of pseudoC³[0,1] positive solution and C²[0,1]positive solution.