Pseudo antiperiodic solutions to Volterra difference equations (Q2112043)

The authors study the existence of pseudo \(\omega\)-antiperiodic solutions for the following Volterra difference equation of convolution type: \[ u(n+1) = \lambda \sum \limits ^n_{j=-\infty} a(n-j) u(j) + p(n, u(\gamma(n))) , \qquad n \in \mathbb{Z}, \] where \( \lambda \in \mathbb{C}\), \( a(\cdot)\) is a \(\mathbb{C}\)-valued summable sequence, \( p : \mathbb{Z}\times X \to X\) is a bounded function on bounded sets of the Banach space \(X\), \(\gamma : \mathbb{Z} \to \mathbb{Z}\). The notion of pseudo \(\omega\)-antiperiodic sequence can be seen as the discrete version of a pseudo \(\omega\)-antiperiodic function. Some basic results, such as completeness of the space, convolution and superposition theorems for pseudo \(\omega\)-antiperiodic sequences are derived. Such general results and notions are then used to investigate the existence of pseudo \(\omega\)-antiperiodic solutions to the above Volterra difference equation. In particular, the existence and uniqueness are established by means of the global Lipschitz growth condition on the second variable of the nonlinear term with different Lipschitz coefficients, namely a constant, a summable function or a \(q\)-th summable function respectively. Some existence results are obtained under non-Lipschitz growth conditions on the second variable of the nonlinear term.