The main purpose of this paper is to investigate the existence and Ulam-Hyers stability (U-Hs) of solutions of a nonlinear neutral stochastic fractional differential system. We prove the existence and uniqueness of solutions to the proposed system by using fixed point theorems and the Banach contraction principle. Also, by using fundamental schemes of fractional calculus, we study the (U-Hs) to the solutions of our suggested system. Besides, we study an example, best describing our main result.

This work investigates the existence and uniqueness of solutions for a coupled system of fractional differential equations with three-point generalized fractional integral boundary conditions within generalized proportional fractional derivatives of the Riemann-Liouville type. By using the Schauder and Banach fixed point theorems, we study the existence and uniqueness of solutions for the aforesaid system. Finally, we present an example to validate our theoretical outcomes.

As we know one of the most important equations which have many applications in various areas of physics, mathematics, and financial markets, is the Sturm–Liouville equation. In this paper, by using the $\alpha$-$\psi$-contraction technique in fixed point theory and employing some functional inequalities, we study the existence of solutions of the partial fractional hybrid case of generalized Sturm–Liouville-Langevin equations under partial boundary value conditions. Towards the end, we present two examples with numerical and graphical simulation to illustrate our main results.

One of the most famous equations that are widely used in various branches of physics, mathematics, financial markets, etc. is the Langevin equation. In this work, we investigate the existence of the solution for two generalized fractional hybrid Langevin equations under different boundary conditions. For this purpose, the problem of the existence of a solution will become the problem of finding a fixed point for an operator defined in the Banach space. To achieve the result, one of the recent fixed point techniques, namely the $ \alpha $-$ \psi $-contraction technique, will be used. We provide sufficient conditions to use this type of contraction in our main theorems. In the calculations of the auxiliary lemmas that we present, the Mittag-Leffler function plays a fundamental role. The fractional derivative operators used are of the Caputo type. Two examples are provided to demonstrate the validity of the obtained theorems. Also, some figures and a table are presented to illustrate the results.