Constant-sign solutions for systems of singular integral equations of Hammerstein type

Abstract

We consider the system of Hammerstein integral equations u i ( t ) = ∫ 0 T g i ( t , s ) f i ( s , u 1 ( s ) + ρ 1 ( s ) , u 2 ( s ) + ρ 2 ( s ) , … , u n ( s ) + ρ n ( s ) ) d s , t ∈ [ 0 , T ] , 1 ≤ i ≤ n where T > 0 is fixed, ρ i ’s are given functions and the nonlinearities f i ( t , x 1 , x 2 , … , x n ) can be singular at t = 0 and x j = 0 where j ∈ { 1 , 2 , ⋯ , n } . Criteria are offered for the existence of constant-sign solutions, i.e., θ i u i ( t ) ≥ 0 for t ∈ [ 0 , T ] and 1 ≤ i ≤ n , where θ i ∈ { 1 , − 1 } is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.