Estimates of the gap between extreme solutions of nonlinear difference equations

We study bounded solutions of nonlinear difference equations on arbitrary index sets within a unified fixed-point framework. Working in the Banach lattice l^∞ (X,R) l ∞ ( X , R ) , we consider equations of the form M(u) = λ u + Df(u) + w, M ( u ) = λ u + D f ( u ) + w , where M is an increasing linear operator, Df D f is the diagonal operator induced by a scalar super-linear nonlinearity f∈ C¹(R,R) f ∈ C 1 ( R , R ) , and w∈ l^∞ (X,R) w ∈ l ∞ ( X , R ) . Our abstract results apply to a broad class of discrete models, including discrete Laplace-type operators on Rⁿ R n , Banach limits and difference equations with spatial structure.