A new type of three dimensional metric spaces with applications to fractional differential equations

Abstract

<abstract><p>In this manuscript, we introduce a three dimension metric type spaces so called $ J $-metric spaces. We prove the existence and uniqueness of a fixed point for self mappings in such spaces with different types of contractions. We use our result to prove the existence and uniqueness of a solution of the following fractional differential equations such as</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathcal{(P)}:\left\{ \begin{array}{ccl} D^{\lambda}x(t) &amp; = &amp; f(t,x(t)) = Fx(t) \;{\rm{ if }}\; t\in I_0 = (0,T] \\ x(0) &amp; = &amp; x(T) = r \\ \end{array} \right\} . $\end{document} </tex-math></disp-formula></p> <p>Moreover, we present other applications to systems of linear equations and Fredholm type integral equation.</p></abstract>