Fixed point approach to solve fractional differential equations in $S^{JS}$-metric spaces

This study aims to establish a new fixed point theorem in the framework of $S^{JS}$-metric spaces, recently introduced by Beg et al. We propose different principles of contraction using various techniques. The theorems obtained represent a new framework for other future work in the considered space. Also, we provide two applications of our results to linear system of equations and the following fractional differential equation

$$\mathcal{(}\mathcal{P}\mathcal{)}:\left\{\begin{array}{ccl}{D}^{\lambda }x(t)& =& f(t,x(t))=Fx(t)\mathrm{i}\mathrm{f}\mathrm{t}\in {\mathrm{I}}_0=(0,\mathrm{T}]\\ x(0)& =& x(T)=r\end{array}\right\}.$$

These applications show the effectiveness of our approach as a powerful tool for solving several types of differential equations.