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

Abstract

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 $ \mathcal{(P)}:\left\{ \begin{array}{ccl} D^{\lambda}x(t) & = & f(t,x(t)) = Fx(t) \;{\rm{ if }}\; t\in I_0 = (0,T] \\ x(0) & = & x(T) = r \\ \end{array} \right\} . $ Moreover, we present other applications to systems of linear equations and Fredholm type integral equation.

Keywords

• fixed point
• j-metric space
• fractional differential equation
• system of linear equations