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

‎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.