Constant-Sign Lp Solutions for a System of Integral Equations

We consider the following system of integral equations $$u_i(t)=\int^1_0g_i(t,s)f(s,u_1(s),u_2(s),\cdots,u_n(s))ds,\quad t\in \lbrack 0,1\rbrack,1\leq i\leq n.$$ Our aim is to establish criteria such that the above system has a constant-sign solution (u₁, u₂, …, u _n) ∈ (L^p0, 1])ⁿ, where the integer 1 ≤ p < ∞ is fixed. We shall tackle the case when f is ‘nonnegative’ as well as the case when f is ‘semipositone’. The above problem is also extended to that on the half-line 0, ∞) $$u_i(t)=\int^1_0g_i(t,s)f(s,u_1(s),u_2(s),\cdots,u_n(s))ds,\quad t\in \lbrack 0,\infty ),1\leq i\leq n.$$