SOLUTIONS OF FRACTIONAL HYBRID DIFFERENTIAL EQUATIONS VIA FIXED POINT THEOREMS AND PICARD APPROXIMATIONS

Abstract

We investigate the following fractional hybrid differential equation:
$$
\left\{\begin{array}{l}
D_{t_0+}^\alpha\left[x(t)-f_1(t, x(t))\right]=f_2(t, x(t)) \text { a.e } t \in J, \\
x\left(t_0\right)=x_0 \in \mathbb{R},
\end{array}\right.
$$
where $D_{t_0+}^\alpha$ is the Riemann-Liouville differential operator order of $\alpha>0, J=\left[t_0, t_0+a\right]$, for some $t_0 \in \mathbb{R}, a>0$, $f_1 \in C(J \times \mathbb{R}, \mathbb{R}), \quad f_2 \in \mathcal{L}_p^\alpha(J \times \mathbb{R}, \mathbb{R}), \quad p \geq 1$ and satisfies certain conditions. We investigate such equations in two cases: $\alpha \in(0,1)$ and $\alpha \geq 1$. In the first case, we prove the existence and uniqueness of a solution which extends the main result of [1]. Moreover, we show that the Picard iteration associated to an operator $T: C(J \times \mathbb{R}) \rightarrow C(J \times \mathbb{R})$ converges to the unique solution of (1.0) for any initial guess $x \in C(J \times \mathbb{R})$. In particular, the rate of convergence is $n^{-1}$. In the second case, we investigate this equation in the space of $k$ times differentiable functions. Naturally, the initial condition $x\left(t_0\right)=x_0$ is replaced by $x^{(k)}\left(t_0\right)=x_0, 0 \leq k \leq n_{\alpha, p}-1$ and the existence and uniqueness of a solution of (1.0) is established. Moreover, the convergence of the Picard iterations to the unique solution of (1.0) is shown. In particular, the rate of convergence is $n^{-1}$. Finally, we provide some examples to show the applicability of the abstract results. These examples cannot be solved by the methods demonstrated in [1].

Received: August 8, 2022 
Revised: September 24, 2022 
Accepted: October 14, 2022