Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

Abstract

A modified implicit relation and $\varpi$-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided.
As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation $\mathcal{X}=\mathcal{B} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*}\mathcal{T} \mathcal{(X)}\mathcal{A}_{i}.$
An example is given, using matrices that are randomly generated,
as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.