Existence results by Mönch's fixed point theorem for a tripled system of sequential fractional differential equations

In this paper, we study the existence of the solutions for a tripled system of Caputo sequential fractional differential equations. The main results are established with the aid of Mönch's fixed point theorem. The stability of the tripled system is also investigated via the Ulam-Hyer technique. In addition, an applied example with graphs of the behaviour of the system solutions with different fractional orders are provided to support the theoretical results obtained in this study

Symmetry analysis for nonlinear fractional terminal system under w-Hilfer fractional derivative in different weighted Banach spaces

Our objective in this study is to investigate the behavior of a nonlinear terminal fractional system under $w$-Hilfer fractional derivative in different weighted Banach spaces. We examine the system's dynamics and understand the effects of different weighted Banach spaces on the properties of solutions, including existence, uniqueness, stability, and symmetry. We derive the equivalent integral equations and employ the Schauder and Banach fixed point theorems. Additionally, we discuss three symmetric cases of the system to show how the choice of the weighted function $w(\iota)$ impacts the solutions and their symmetry properties. We study the stability of the solutions in the Ulam sense to assess the robustness and reliability of these solutions under various conditions. Finally, to understand the system's behavior, we present an illustrative example with graphs of the symmetric cases

Existence of Solutions and Relative Controllability of a Stochastic System with Nonpermutable Matrix Coefficients

In this study, time-delayed stochastic dynamical systems of linear and nonlinear equations are discussed. The existence and uniqueness of the stochastic semilinear time-delay system in finite dimensional space is investigated. Introducing the delay Gramian matrix, we establish some sufficient and necessary conditions for the relative approximate controllability of time-delayed linear stochastic dynamical systems. In addition, by applying the Banach fixed point theorem, we establish some sufficient relative approximate controllability conditions for semilinear time-delayed stochastic differential systems. Finally, concrete examples are given to illustrate the main results

Relative Controllability and Ulam–Hyers Stability of the Second-Order Linear Time-Delay Systems

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and Ω. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam–Hyers stability of a second-order semi-linear time-delayed system

Existence and Ulam–Hyers Stability Results for a System of Coupled Generalized Liouville–Caputo Fractional Langevin Equations with Multipoint Boundary Conditions

We study the existence and uniqueness of solutions for coupled Langevin differential equations of fractional order with multipoint boundary conditions involving generalized Liouville–Caputo fractional derivatives. Furthermore, we discuss Ulam–Hyers stability in the context of the problem at hand. The results are shown with examples. Results are asymmetric when a generalized Liouville–Caputo fractional derivative (ρ) parameter is changed

Fractional Stochastic Integro-Differential Equations with Nonintantaneous Impulses: Existence, Approximate Controllability and Stochastic Iterative Learning Control

In this paper, existence/uniqueness of solutions and approximate controllability concept for Caputo type stochastic fractional integro-differential equations (SFIDE) in a Hilbert space with a noninstantaneous impulsive effect are studied. In addition, we study different types of stochastic iterative learning control for SFIDEs with noninstantaneous impulses in Hilbert spaces. Finally, examples are given to support the obtained results

Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay

Various scholars have lately employed a wide range of strategies to resolve two specific types of symmetrical fractional differential equations. The evolution of a number of real-world systems in the physical and biological sciences exhibits impulsive dynamical features that can be represented via impulsive differential equations. In this paper, we explore some existence and controllability theories for the Caputo order q∈(1,2) of delay- and random-effect-affected fractional functional integroevolution equations (FFIEEs). In order to prove that random solutions exist, we must prove a random fixed point theorem using a stochastic domain and the mild solution. Then we demonstrate that our solutions are controllable. At the end, applications and example is illustrated which indicates the applicability of this manuscript

On a System of <i>ψ</i>-Caputo Hybrid Fractional Differential Equations with Dirichlet Boundary Conditions

In this article, we investigate sufficient conditions for the existence and stability of solutions to a coupled system of ψ-Caputo hybrid fractional derivatives of order 1υ≤2 subjected to Dirichlet boundary conditions. We discuss the existence and uniqueness of solutions with the assistance of the Leray–Schauder alternative theorem and Banach’s contraction principle. In addition, by using some mathematical techniques, we examine the stability results of Ulam–Hyers. Finally, we provide one example in order to show the validity of our results

Relative Controllability and Ulam&ndash;Hyers Stability of the Second-Order Linear Time-Delay Systems

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and &Omega;. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam&ndash;Hyers stability of a second-order semi-linear time-delayed system

Applicability of Mönch’s Fixed Point Theorem on a System of (<i>k</i>, <i>ψ</i>)-Hilfer Type Fractional Differential Equations

In this article, we study a system of Hilfer (k,ψ)-fractional differential equations, subject to nonlocal boundary conditions involving Hilfer (k,ψ)-derivatives and (k,ψ)-integrals. The results for the mentioned system are established by using Mönch’s fixed point theorem, then the Ulam–Hyers technique is used to verify the stability of the solution for the proposed system. In general, symmetry and fractional differential equations are related to each other. When a generalized Hilfer fractional derivative is modified, asymmetric results are obtained. This study concludes with an applied example illustrating the existence results obtained by Mönch’s theorem