Existence of positive solutions to systems of nonlinear integral or differential equations. (English) Zbl 1195.45018

Cheng, Xiyou; Zhang, Zhitao

Topol. Methods Nonlinear Anal. 34, No. 2, 267-277 (2009).

The existence of nontrivial positive solutions \(u,v\) of the system \[ u(x)=\int_{\overline\Omega}k_1(x,y)f_1(y,u(y),v(y))dy\qquad v(x)=\int_{\overline\Omega}k_2(x,y)f_2(y,u(y),v(y))dy \] of two Hammerstein equations is studied. The functions \(f_i\), \(k_i\) are assumed to be nonnegative and continuous for \(u,v\geq0\). Moreover, \(k_i\) are supposed to be symmetric and to satisfy certain hypotheses which imply that the corresponding linear integral operators \(B_i\) with kernel \(k_i\) have positive spectral radius \(r(B_i)\) in \(C(\overline\Omega)\). It is shown that if \(f_1\) grows superlinear with respect to \(u\) at \(0\) and \(\infty\) and if \(f_2\) grows sublinear with respect to \(v\) at \(0\) and \(\infty\) (actually, the growth need only to be faster/slower than that of the linear function \(t\mapsto t/r(B_i)\), and the estimates must be uniform with respect to the other variable), and if \(f_2\) is bounded with respect to \(u\) at \(\infty\) in a certain sense, then the problem has a positive solution \(u,v\neq0\). The proof makes use of the formulas for the Cartesian product of the index on cones and of Krasnoselsk’iĭtype index calculations on cones.
The result is applied to a second order system of two ordinary differential equations with mixed Dirichlet-Neumann boundary conditions.

Reviewer: Martin Väth (Berlin)

MSC:

45G15	Systems of nonlinear integral equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
45M20	Positive solutions of integral equations
47H07	Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H11	Degree theory for nonlinear operators
47H30	Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)

Keywords:

system of Hammerstein equations; positive solution; nontrivial solution; fixed point index on cones