The existence theorem for a solution of a system of integral equations along with cyclical common fixed point technique

In this paper, we introduce the concept of generalized cyclic contraction pairs in b-metric spaces. We establish some fixed point results on b-metric spaces and also give some examples to illustrate the main results. As applications, we show the existence of a common solution for the following system of integral equations: $$\begin{aligned} {\left\{ \begin{array}{ll} x(t) = \int ^b_a K_1(t,r,x(r))\mathrm{d}r,\\ x(t) = \int ^b_a K_2(t,r,x(r))\mathrm{d}r, \end{array}\right. } \end{aligned}$$ where \(a, b \in \mathbb {R}\) with \(areal value functions defined on \([a,b] \subseteq \mathbb {R}\)) and \(K_1, K_2 : [a,b] \times [a,b] \times \mathbb {R} \rightarrow \mathbb {R}\) are mappings.