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The main focus of this manuscript is to study an impulsive fractional integro‐differential equation with delay and Caputo fractional derivative. The existence solution of such a class of fractional differential equations is discussed for linear and nonlinear case with the help of direct integral method. Moreover, Banach's fixed point theorem and Schaefer's fixed point theorem are use to discuss the uniqueness and at least one solution of the said fractional differential equations, respectively. Some hypothesis and inequalities are utilize to present four different types of Hyers–Ulam stability of the mentioned impulsive integro‐differential equation. Example is provide for the illustration of main results. [ABSTRACT FROM AUTHOR]

In this dissertation, we study the coupled system of q$$ q $$‐fractional Langevin differential equations involving q$$ q $$‐Caputo derivative having q$$ q $$‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this article, with the help of Laplace transform, the existence of solution was established in a finite dimensional setting for nonlinear ψ-Hilfer fractional stochastic equation with both retarded and advanced arguments driven by multiplicative and fractional noises, with Hurst index H∈(1/2,1). At first, we obtained the existence and uniqueness results by using the Banach fixed point theorem (FPT). Second, the existence result was also obtained by applying Schaefer's fixed point theorem with less conservative conditions. Furthermore, we investigated the Hyers Ulam Rasisas stability for the aforementioned system. At the end, an example was illustrated to validate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder's alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study. [ABSTRACT FROM AUTHOR]

In this research study, we formulate two generalized nonlinear multi-order fractional boundary value problems with the help of quantum diffeerence operators. To investigate the exis- tence property for possible solutions of these q-di_erence FBVPs, we apply two separate methods motivated by some notions in relation to the measure of noncompactness and end-point technique. The condensing functions and multifunctions having the (AE)-property play an important role in our study. As well as, two examples corresponding to both techniques are provided to ensure the compatibility of the findings numerically. [ABSTRACT FROM AUTHOR]

In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

In this paper, existence and uniqueness of solution for a coupled impulsive Hilfer–Hadamard type fractional differential system are obtained by using Kransnoselskii’s fixed point theorem. Different types of Hyers–Ulam stability are also discussed.We provide an example demonstrating consistency to the theoretical findings.