Multiple Positive Solutions Of Nonlinear Operator Equations And Applications

Various kinds of nonlinear problems arise in many fields of natural science,engineering technology and social science,such as physics,ecology and economics.Most of these problems can be come down to nonlinear differential equations or nonlinear integral equations,which can be abstracted into operator equations to study.Therefore,it is significant to investigate solutions of nonlinear operator equations with nonlinear functional analysis-the powerful tool.This paper includes four chapters,which can be divided into three parts.The first part is Chapter Ⅰ.In Chapter Ⅰ,multiple solutions of some nonlinear operator equations are considered with the cone theory,topological theory and the fixed point index theory.Moreover,the famous Amann's three-solution theorem and Leggett-Williams' three-solution theorem are united.Futhermore,the results are applied to some fourth-order boundary problems and some new conclusions are obtained.The second part are Chapter Ⅱand Chapter Ⅲ.In the two chapters,nonlinear Hammerstein equations are considered respectively from the different angles-the kernel function and the nonlinear term-by using the cone theory and the topological degree theory.The third part is Chpter Ⅳ,which belongs to the concrete application of the nonlinear analysis methods.In Chapter Ⅳ,the existence of the positive solutions of some singular Sturm-Liouville boundary value problems is obtained with the cone theory and the fixed point index theory.The following are these concrete results.Throughout the following of this section,E denotes a real Banach space and P is a cone in E.In Chapter Ⅰ,a new three-solution theorem is obtained.Moreover,the famous Amann's and Leggett-Williams' three-solution theorems in nonlinear functional analysis can be seen as its special cases,namely they are united.So they are improved.The main results can be stated as the following:Let D be a nonempty bounded close convex subset in E,αand βnonnegative continuous functional on D. and α is concave while β is convex.Suppose 01 with Ax = z1 + t(x - z1), then (Ar) < d;(ii) there exists z2 U2 satisfies if there exist x U2, t 1 with Ax = z2 + t(x - z2), then a(.Ax) > a,where dUi denotes the boundary of Ui relative to D, i = 1,2. Then A has at least three fixed points x1, x2 and x3 in D such that (x1) < d, < (13), d < (x3) and a(x3) < a.In Chapter II, we consider the eigenvalues and eigenvectors of the following nonlinear Hammerstein integral equationwhere the sign of the kernel k may be variable. G denotes a bounded closed domain in Euclidean space Rn and the kernel k is defined on G x G. The main conclusion is the following:Theorem 2.2.1 Suppose that (i) k is continuous on G x G and there exists a function h Lp(G) (p > 1) such that G h(x)k(x, y)dx > 0 for all y G; (ii) / is continuous on G R, f(x,0) = 0, f'u(x,u) exists and is continuous for sufficiently small |u|; and (iii) / has the lower bound, lim|u|_+ f(x, u)/|u| = + uniformly on x G1 = G \ {x G : h(x) = 0}. Then(i) for any 0, n, n = 1,2,..., A is an eigenvalue of A, where {An} is the sequence of eigenvalues of the linear integral operator K1 : C(G) - C(G), defined byG(ii) lim -+ \\<|| || = + , where is the eigenvector of A with respect to A;(iii) for any 0, n, n = 1, 2, ..., there exists = ( ) > 0 and R = R( ) > 0 such that, for every C(G) satisfying 0 < || || < , the equation (x) = A (x) + (x) has at least two continuous solutions satisfying 0 and || || < R.Theorem 2.2.1 is an improvement of Theorem 2 in [32], replacing the condition "h(x) is a bounded measurable function" by the weaker one " h Lp(G)(p > 1)". This improvement enables the theorem to be applied to more areas.In Chapter III, we still consider nonlinear Hammerstein integral equation. Being different from Chapter II, we suppose the kernel k is nonnegative...