Constant-Sign Periodic and Almost Periodic Solutions for a System of Integral Equations

AbstractWe consider the following system of integral equations $${u_{i}(t)=\int\nolimits_{I} g_{i}(t, s)f(s, u_{1}(s), u_{2}(s), \cdots, u_{n}(s))ds, \quad t \in I, \ 1 \leq i\leq n}$$where I is an interval of $$\mathbb{R}$$. Our aim is to establish criteria such that the above system has a constant-sign periodic and almost periodic solution (u1, u2,…,un) when I is an infinite interval of $$\mathbb{R}$$, and a constant-sign periodic solution when I is a finite interval of $$\mathbb{R}$$. The above problem is also extended to that on $$\mathbb{R}$$$$u_{i} {\left( t \right)} = {\int_\mathbb{R} {g_{i} {\left( {t,s} \right)}f_{i} {\left( {s,u_{1} {\left( s \right)},u_{2} {\left( s \right)}, \cdots ,u_{n} {\left( s \right)}} \right)}ds\quad t \in \mathbb{R},\quad 1 \leqslant i \leqslant n.} }$$