$$(\psi ,\phi )$$ ( ψ , ϕ ) -Wardowski contraction pairs and some applications

In this paper, we propose the notion of $$(\psi ,\phi )$$ -Wardowski contraction pairs and achieve a common fixed point theorem consistent with such kind of contractions in complete metric spaces. One of the main motivations of this article is to define a contractive condition which does not compel the mappings to be continuous at their common fixed points, an interesting question posed by Rhoades (Contemporary Math 72:233–245, 1988). Besides, we provide some applications of our obtained results in certain type of operator equations, fractional differential equation boundary value problems, functional equations arising in dynamic programmings, and a particular type of non-linear quadratic integral equations which are relevant to events arising in physics, economics, engineering, operations research, and many other relevant disciplines.