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The goal of this manuscript is to establish quadruple fixed-point and coincidence-point consequences in the setting of generalized metric spaces equipped with vector-valued metrics and matrix equations. Moreover, some supportive examples and corollaries are presented here to support the theoretical results. Ultimately, the theoretical results are presented here to discuss some applications to support our study.

Quadruple fixed-point techniques for solving integral equations involved with matrices and the Markov process in generalized metric spaces