Solvability of \(\psi \)-Hilfer Fractional Differential Equations in the Space of Summable Functions

This paper is devoted to study the Cauchy type problem for the nonlinear differential equation with \(\psi \) $\psi$ -Hilfer fractional derivative \(\begin{aligned} \begin{aligned}&{^{H}} \textbf{D}_{a^+}^{\alpha , \nu ; \psi } u(x)= f(x,u(x)) \quad a.e. \; \hbox {on }[a,b] \\&\textbf{I}_{a^+}^{(1-\alpha )(1-\nu );\psi } u(a) =c \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{cc}& _{}^H{\mathbf{D}}_{a^{+}}^{\alpha ,\nu ;\psi }u(x)=f(x,u(x))a.e.\text{on}[a,b]\hfill \\ \hfill & {\mathbf{I}}_{a^{+}}^{(1-\alpha )(1-\nu );\psi }u(a)=c\hfill \end{array}\end{array}$$ in the space of summable functions \(L^{\alpha , \nu }[a,b]\) $L^{\alpha ,\nu }[a,b]$ , where \(\alpha \in (0,1)\) $\alpha \in (0,1)$ , \(\nu \in (0,1)\) $\nu \in (0,1)$ , \(\psi \in C^1[a,b]\) $\psi \in C^1[a,b]$ be a strictly increasing function and \({^{H}} \textbf{D}_{{a}^+}^{\alpha , \nu ; \psi }\) $_{}^H{\mathbf{D}}_{a^{+}}^{\alpha ,\nu ;\psi }$ is the \(\psi \) $\psi$ -Hilfer fractional derivative of u. The equivalence of this problem and the nonlinear Volterra integral equation is established. The existence and uniqueness of solution to the Cauchy type problem is proved using progressive contractions techniques and the Banach fixed point theorem.