Solvability of infinite system of implicit fractional integral equations of two variables in tempered sequence spaces via generalized Darbo’s fixed point theorem

In this article, we study the existence of solutions for an infinite system of implicit fractional integral equations of two variables in tempered sequence spaces \(c_{0}^{\alpha }\) $c_0^{\alpha }$ and \(\ell _{p}^{\alpha }\) ${\ell }_p^{\alpha }$ \((p\ge 1)\) $(p\ge 1)$ . To achieve our main objective, we establish a new fixed point theorem using the measure of noncompactness and a new contraction operator, which generalizes Darbo’s fixed point theorem and several other theorems from the literature. The advantage of the proposed fixed point theorem is that the requirement of the compactness of the domain is relaxed, which is essential in some fixed point theorems. We apply this generalized Darbo’s fixed point theorem to the infinite system of implicit fractional integral equations of two variables in tempered sequence spaces \(c_{0}^{\alpha }\) $c_0^{\alpha }$ and \(\ell _{p}^{\alpha }\) ${\ell }_p^{\alpha }$ \((p\ge 1)\) $(p\ge 1)$ . We have justified the finding results of these sequence spaces with the help of numerical examples.