### Fixed point theorems in $C^{*}$-algebra-valued $b_{v}( s)$-metric spaces with application and numerical methods

#### Abstract

We first introduce a novel notion named $C^{*}$-algebra-valued $b_{v}(s)$-metric spaces. Then, we give proofs of the Banach contraction principle, the expansion mapping theorem, and Jungck's theorem in $C^{*}$-algebra-valued $b_{v}(s)$-metric spaces. As an application of our results, we establish a result for an integral equation in a $C^{*}$-algebra-valued $b_{v}(s)$-metric space. Finally, a numerical method is presented to solve the proposed integral equation, and the convergence of this method is also studied. Moreover, a numerical example is given to show applicability and accuracy of the numerical method and guarantee the theoretical results.

#### Keywords

$C^{*}$-algebra; $b_{v}(s)$-metric space; Fixed point theorem; Integral equation; Contractive mapping

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