### OPTIMAL SOLUTIONS AND APPLICATIONS TO NONLINEAR MATRIX AND INTEGRAL EQUATIONS VIA SIMULATION FUNCTIO

#### Abstract

Based on the concepts of $\alpha$-proximal admissible mappings and

simulation function, we establish some best proximity point and coupled best

proximity point results in the context of $b$-complete $b$-metric spaces. We

also provide some concrete examples to illustrate the obtained results.

Moreover, we prove the existence of the solution of nonlinear integral

equation and positive definite solution of nonlinear matrix equation $X=Q +

\sum\limits^{ m}_{i=1}A^{*}_{i}\gamma(X)A_{i}- \sum\limits^{

m}_{i=1}B^{*}_{i}\gamma(X)B_{i}$. The given results not only unify but also generalize a number of existing results on the topic in the corresponding

literature.

simulation function, we establish some best proximity point and coupled best

proximity point results in the context of $b$-complete $b$-metric spaces. We

also provide some concrete examples to illustrate the obtained results.

Moreover, we prove the existence of the solution of nonlinear integral

equation and positive definite solution of nonlinear matrix equation $X=Q +

\sum\limits^{ m}_{i=1}A^{*}_{i}\gamma(X)A_{i}- \sum\limits^{

m}_{i=1}B^{*}_{i}\gamma(X)B_{i}$. The given results not only unify but also generalize a number of existing results on the topic in the corresponding

literature.

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