Multiple positive periodic solutions for nonlinear first order functional difference equations (Q2912438)

The authors obtain sufficient conditions for the existence of three positive solutions for each of the finite difference equations, NEWLINE\[NEWLINE \Delta x(n) = \mp a(n) x(n) \pm \lambda b(n) f\big ( n, x(h(n)) \big ). NEWLINE\]NEWLINE Throughout they assume that \(a(n), b(n),\) and \(h(n)\) are \(T\)-periodic positive sequences with \(T \geq 1\), and that \(f (n, x)\) is \(T\)-periodic in \(n\) and continuous in \(x\) for each \(n \in \mathbb{Z}\). They also assume that \(0 < a(n) < 1\) and hence the problems are invertible. After they invert the problem, they determine bounds on the kernel \(G(n, s)\). Using the well-known cone-theoretic Leggett-Williams fixed point theorem, they establish sufficient conditions on the nonlinear growth term \(f\) for the existence of three solutions of the difference equations.