Existence of positive solutions of boundary value problem for a discrete difference system
The authors study the discrete system \(\Delta^2u_i(k)+f_i(k,u_1(k),u_2(k))=0\) with the Dirichlet boundary conditions \(u_i(0)=u_i(T+2)=0\), where \(i=1,2\), \(T>1\) is a fixed integer, \(k\in\{0,1,2,\dots,T\}\), \(\Delta u(k)=u(k+1)-u(k)\), \(\Delta^2 u(k)=\Delta(\Delta u(k))\). In their recent paper, the authors [ibid. 143, No. 2--3, 213--221 (2003; Zbl 1030.39015)] obtained some sufficient conditions for the existence of three positive (that is, nonnegative and nontrivial) solutions of the above system by using the Leggett-Williams fixed point theorem. In the paper under review, they continue their investigation of this system and prove some sufficient conditions for the existence of one or two positive solutions. In this case the proof is based on a nonlinear alternative of Leray-Schauder type and the Krasnoselskii fixed point theorem on a cone.
nonlinear alternative of Leray-Schauder type
scientific article; zbMATH DE number 2118014