Existence of solutions for a class of nonlinear Volterra integro-dynamic equations on time scales

Purpose This paper encompasses recent developments on a class of nonlinear Volterra integro-dynamic equations on arbitrary time scales. More precisely, we provide some conditions under which the considered equations have at least one, at least two and at least three solutions. Volterra studied the hereditary influences when he was examining a population growth model. The research work resulted in a specific topic, where both differential and integral operators appeared together in the same equation. This new type of equations was termed as Volterra integro-differential equations. Any Volterra integro-differential equation is characterized by the existence of one or more derivatives of the unknown outside the integral sign. The Volterra integro-differential equations may be observed when we convert an initial value problem to an integral equation by using Leibnitz rule. Design/methodology/approach Firstly, we find a suitable integral representation of the solutions of the considered class of equations. Then, we construct two operators so that any fixed point of their sum is a solution of the considered nonlinear Volterra integro-dynamic equations on arbitrary time scales. Findings For the proof of our main results, we propose a new approach based upon recent fixed point theorems. The proposed approach gives complimentary results to the existing results obtained by using the classical fixed point theorems. Moreover, we prove that our considered equations have at least one, at least two and at least three solutions. Originality/value All results are novel and generalize other existing results. The proposed approach gives complimentary results to the existing literature using the classical fixed point theorems.