Avery fixed point theorem applied to Hammerstein integral equations

Abstract

We apply a recent Avery et al. fixed point theorem to the Hammerstein integral equation $$ x(t)=\int^{T_2}_{T_1}G(t,s)f(x(s))\,ds, \quad t\in[T_1,T_2]. $$ Under certain conditions on G, we show the existence of positive and positive symmetric solutions. Examples are given where G is a convolution kernel and where G is a Green's function associated with different boundary-value problem.

Keywords

• Hammerstein integral equation
• boundary-value problem
• fractional boundary-value problem