A fixed point method for nonlinear equations involving a duality mapping defined on product spaces

The aim of this paper is to obtain solutions for the equation $$ J_{q,p} (u_1,u_2) =N_{f,g}(u_1,u_2), $$ where $J_{q,p}$ is the duality mapping on a product of two real, reflexive and smooth Banach spaces $X_1, X_2$, corresponding to the gauge functions $varphi_1(t)=t^{q-1}$, $varphi_2(t)=t^{p-1}$, $1