Some new generalized Volterra–Fredholm type finite difference inequalities involving four iterated sums

In this paper, based on some new generalized Volterra–Fredholm type finite difference inequalities involving four iterated sums, we derive some new explicit bounds for the unknown functions concerned. These inequalities are of more general forms than some existing results in the literature, and are generalization of the main results of Ma [J. Comput. Appl. Math. 233 (2010) 2170–2180]. As application of the established results, we present a Volterra–Fredholm sum-difference equation, and research the qualitative as well as quantitative properties for it.