Positive Solutions For Nonlinear Boundary Value Problems And Random Fixed Points Theorems

Presently,many authors were concerned with many kinds of nonlinear second order boundary value problem(BVP) including nonlinear second order two-point BVP, nonlinear second order several-point BVP,et al ([1-42]). In this paper,we are concerned with nonlinear second order singular two-point BVP,nonlinear second order singular several-point BVP and these BVPs with parameters in the boundary condition. We point out that, BVPs considered in this paper has the more generalized formNot only the singularity of q at t = 0,1 is considered ,but also the singularity of 1/p(t) at t = 0,1 is considered in this paper.In Chapter 1 and Chapter 2, we always assume thatThe paper is organized as follows:In Chapter 1,firstly,we consider (1) with the following boundary conditionwhere a, b, c, d ≥ 0, ac+ad+cb > 0,λ1, λ2 are two nonnegative parameters,f ∈ C([0, ∞), [0,∞)). We transform the problem of the existence of solution of (1) and (3) into the problem of the existence of the fixed point of the following integral operatorwherewhere ds.Under suitable conditions,we proved that there exists (1) and (3) has at least two solutions. We combine Leray-Schauder degree theory with the lower solution and upper solution method to obtain the above results. Finally, we obtain the existence of two positive solutions for (1) with boundary condition In Chapter 2,we come to consider (1) with the following boundary conditionwhere .Suppose eitherwhere λ1 is the first eigenvalue of the linear operatorand k(t,s),p, is given by (5) and (6),.Then (1) and (7) has at least one positive solution.Moreover.even if f needs not to be nonnegative.we also prove that (1) and (7) has at least one positive solution under the assumption of that f is bounded below.We point out that the method above is valid in considering (1) with the following boundary conditionsFinally,similarly to the method used in Chapter l,we can consider (1) with the several point boundary condition including parameters.The similar results can be obtain for (1).In Chapter 3,we Studied the existence of random fixed point and random critical point. Firstly,in §3.1, we consider the existence of random fixed points for random generalized inward maps of condensing type.In virtue of the method of [44-46,53-54],we introduced the random fixed point index for random generalized inward maps of condensing type. By using the properties of the random fixed point index,we get some random fixed point theorems for random generalized inward maps of condensing type.As a application,we give the the existence of random positive solution of a random integral equation in L2[0,1].Nextly,in §3.2,we obtained three random fixed points theorem for random weakly inward map by virtue of the ordinary differential equation theory.We have not seen such results in the reference as we known.Finally,in §3.3,by using the Kuratowski-Ryll-Nardzewski measurable selector Theo-rem,we prove that a random continuous and weakly lower continuous function attain the inf on weakly compact set.Even if the domain of the function change with the changeof ω,the result still holds true. Also,we prove that a random C1 function satisfying P.S condition has a random critical point corresponding to any given random critical value of the function.We point out that many of the existence theorems of critical point can be randomized by using this result.We do not discuss it in detail here.