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This article outlines the necessary requirements for a coupled system of fractional order boundary value involving the Caputo fractional derivative, including its existence, uniqueness, and various forms of Ulam stability. We demonstrate the existence and uniqueness of the proposed coupled system by using the cone-type Leray–Schauder result and the Banach contraction principle. Based on the traditional method of nonlinear functional analysis, the stability is examined. An example is used to provide a clear illustration of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation ∂tu - div a(x, t,∇u) + λ (|u| p(x,t)-2u) = 0 in ΩT, where λ≥0 and ∂tu denote the partial derivative of u with respect to the time variable t, while∇u denotes the one with respect to the space variable x. Moreover, the vector-field a(x, t, ·) satisfies certain nonstandard p(x, t)-growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values. [ABSTRACT FROM AUTHOR]

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient conditions for the existence, uniqueness and Ulam–Hyers stability of the solutions of a coupled system of nonlinear fractional impulsive differential equations. The existence and uniqueness results are carried out via Banach contraction principle and Schauder's fixed point theorem. The main theoretical results are well illustrated with the help of an example. [ABSTRACT FROM AUTHOR]

This manuscript deals with the existence theory, uniqueness, and various kinds of Ulam–Hyers stability of solutions for a class and coupled system of fractional order differential equations involving Caputo derivatives. Applying Schaefer and Banach's fixed point approaches, existence and uniqueness results are obtained for the proposed problems. Stability results are investigated by using the classical technique of nonlinear functional analysis. Examples are given with each problem to illustrate the main results. [ABSTRACT FROM AUTHOR]