Fixed points of $$(\varphi ,F)$$-weak contractions on metric-like spaces with applications to integral equations on time scales

In this paper, the notion of $$(\varphi , F)$$ -weak contractions in the framework of metric-like spaces is introduced and the corresponding fixed point theorems are established. As an attempt to answer the common question regarding the uniqueness of trivial solutions of homogeneous integral equations on time scales, one of our main results is applied to investigate sufficient criteria for existence and uniqueness of solution of nonlinear Fredholm integral equations of the second kind on time scales. The latter result is obtained through a new Lipschitz condition on the kernel different from the usual ones. Examples are constructed to show that our ideas herein are new and generalize some recently announced results in the literature.