We investigate the existence and uniqueness of solutions to a coupled system of the hybrid fractional integro-differential equations involving φ-Caputo fractional operators. To achieve this goal, we make use of a hybrid fixed point theorem for a sum of three operators due to Dhage and also the uniqueness result is obtained by making use of the Banach contraction principle. Moreover, we explore the Ulam–Hyers stability and its generalized version for the given coupled hybrid system. An example is presented to guarantee the validity of our existence results.

In this paper, we study a coupled fully hybrid system of $(\mathbf{k},\mathsf{\Phi })$–Hilfer fractional differential equations equipped with non-symmetric $(\mathbf{k},\mathsf{\Phi })$–Riemann-Liouville ($\mathcal{RL}$) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space ($\mathcal{GBS}$). Moreover, the Ulam–Hyers ($\mathcal{UH}$) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results.

The objective of this paper is to study the existence of extremal solutions for nonlinear boundary value problems of fractional differential equations involving the $\mathsf{\psi }-$Caputo derivative ${}^C{\mathbb{D}}_{a^{+}}^{\sigma ;\mathsf{\psi }}\varrho \left(\mathrm{t}\right)=\mathcal{V}(\mathrm{t},\varrho \left(\mathrm{t}\right))$ under integral boundary conditions $\varrho \left(a\right)=\lambda {\mathbb{I}}^{\nu ;\mathsf{\psi }}\varrho \left(\eta \right)+\delta$. Our main results are obtained by applying the monotone iterative technique combined with the method of upper and lower solutions. Further, we consider three cases for ${\mathsf{\psi }}^{*}\left(\mathrm{t}\right)$ as $\mathrm{t}$, Caputo, $2^{\mathrm{t}}$, $\sqrt{\mathrm{t}}$, and Katugampola (for $\rho =0.5)$ derivatives and examine the validity of the acquired outcomes with the help of two different particular examples.

In this paper, a new class of a neutral functional delay differential equation involving the generalized ψ -Caputo derivative is investigated on a partially ordered Banach space. The existence and uniqueness results to the given boundary value problem are established with the help of the Dhage’s technique and Banach contraction principle. Also, we prove other existence criteria by means of the topological degree method. Finally, Ulam-Hyers type stability and its generalized version are studied. Two illustrative examples are presented to demonstrate the validity of our obtained results.