Uniqueness of positive solutions for several classes of sum operator equations and applications

作者：Yang Chen; Zhai Chengbo^*; Hao Mengru

来源：JOURNAL OF INEQUALITIES AND APPLICATIONS, 2014, 2014(1): 58.

DOI：10.1186/1029-242X-2014-58

• positive solution
• operator equation
• normal cone
• integral equation
• Lane-Emden-Fowler equation

In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation
u(t) = lambda integral(b)(a) G(t, s)f(s, u(s)) ds, where f and G are both nonnegative, lambda > 0 is a parameter; the other is the elliptic boundary value problem for the Lane-Emden-Fowler equation -Delta u = lambda f(x, u), u(x) > 0 in Omega, u(x) = 0 on partial derivative Omega, where Omega is a bounded domain with smooth boundary in R-N (N >= 1), lambda > 0 and f (x, u) is allowed to be singular on partial derivative Omega. The new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.

• 出版日期2014-2-10
• 单位山西大学