Estimates of the gap between extreme solutions of nonlinear difference equations

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