‎In this work‎, ‎we first introduce the concept of weighted sequence space $m_\omega(\Delta_{\mathfrak{v}}^{\varsigma}‎, ‎\psi,q)$‎. ‎Then‎, ‎we construct a Hausdorff measure of noncompactness on this sequence space‎. ‎Further
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‎In this work‎, ‎we first introduce the concept of weighted sequence space $m_\omega(\Delta_{\mathfrak{v}}^{\varsigma}‎, ‎\psi,q)$‎. ‎Then‎, ‎we construct a Hausdorff measure of noncompactness on this sequence space‎. ‎Furthermore‎, ‎by employing this measure of noncompactness we discuss the solvability of an infinite system of nonlinear third-order differential equations with initial conditions in the weighted sequence space $m_\omega(\Delta_{\mathfrak{v}}^{\varsigma}‎, ‎\psi,q)$‎. ‎Eventually‎, ‎we demonstrate an example to show the usefulness of the obtained result‎.
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