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In this article, we introduce and study a boundary value problem for (k,χ¯*)\left(k,{\bar{\chi }}_{* })-Hilfer generalized proportional fractional differential equation of order in an interval (1, 2], equipped with integro-multipoint nonlocal boundary conditions. In the scalar case setting, the existence results are proved via Leray-Schauder nonlinear alternative and Krasnosel’skiĭ’s fixed point theorem, while the existence of a unique solution is established by applying Banach’s contraction mapping principle. In Banach’s space setting, an existence result is proved via Mönch’s fixed point theorem and the measure of noncompactness. Finally, the obtained theoretical results are well illustrated by constructed examples.

In this paper, we study the existence and uniqueness of solutions for a coupled system of Hilfer-Hadamard sequential fractional differential equations with multi-point Riemann-Liouville fractional integral boundary conditions via standard fixed point theorems. The existence of solutions is proved using Krasnoselskii’s fixed point theorem, while the existence and uniqueness of solutions is established using the Banach fixed point theorem. We also discuss the stability of the problem in terms of Ulam-Hyers, Ulam-Hyers-Rassias, generalized Ulam-Hyers, and generalized Ulam-Hyers-Rassias stability. As an application, some examples are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We aim to investigate the stability property for the certain linear and nonlinear fractional q -difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear q -difference equations of the q -Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy q -difference initial value problem, we use the same quantum Laplace transform and the q -Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable. [ABSTRACT FROM AUTHOR]

In this work, we initiate the study of a new class of impulsive boundary value problems consisting of mixed type fractional quantum and Hadamard derivatives. We will establish existence and uniqueness results by using tools from the functional analysis. We prove the uniqueness result via Banach’s contraction mapping principle, while we will use the Leray-Schauder nonlinear alternative to establish an existence result. We also present examples to illustrate the obtained results.

We study a nonlinear system of Riemann-Liouville fractional differential equations equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We make use of the tools of the fixed-point theory to obtain the desired results, which are well-supported with numerical examples.

In studying boundary value problems and coupled systems of fractional order in (1 , 2 ] , involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in (1 , 2 ] , involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach's fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study right quantum calculus on finite intervals with respect to another function. We present new definitions on the right quantum derivative and right quantum integral of a function with respect to another function and study their basic properties. The new definitions generalize the previous existing results in the literature. We provide applications of the newly defined quantum calculus by obtaining new Hermite–Hadamard-type inequalities for convex, h-convex, and modified h-convex functions. [ABSTRACT FROM AUTHOR]

This paper describes a new class of boundary value fractional-order differential equations of the q-Hilfer and q-Caputo types, with separated boundary conditions. The presented problem is converted to an equivalent integral form, and fixed-point theorems are used to prove the existence and uniqueness of solutions. Moreover, several special cases demonstrate how the proposed problems advance beyond the existing literature. Examples are provided to support the analysis presented. [ABSTRACT FROM AUTHOR]

In this paper, we study a quantum difference coupled impulsive system with respect to another function. Some quantum derivative and integral asymmetric graphs with respect to another function are shown to illustrate the behavior of parameters. Existence and uniqueness results are established via Banach contraction mapping principle and Leray–Schauder alternative. Examples illustrating the obtained results are also included. Our results are new and significantly contribute to the literature to this new subject on quantum calculus on finite intervals with respect to another function. [ABSTRACT FROM AUTHOR]

This survey paper complements to our previous review papers on Lyapunov-type inequalities and contains some of the most recent results on these inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. In precise terms, we have included the results related to Riemann-Liouville, Caputo, mixed Riemann-Liouville and Caputo, Riesz-Caputo, ψ -Caputo, Hadamard, Katugampola, Hilfer, ψ - Hilfer, proportional, variable order Hadamard, partial, systems of Riemann-Liouville, bi-ordinal Hilfer-Katugampola, and ψ -Hilfer fractional derivative operators. The Lyapunov-type inequalities for discrete fractional boundary value problems are also presented. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and uniqueness of solutions for a system of Hilfer-Hadamard sequential fractional differential equations using standard fixed-point theorems. We apply the Leray-Schauder alternative and Banach's contraction mapping principle to obtain the existence and uniqueness results for the given problem. Additionally, we discuss illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we studied the existence of solutions for a coupled system of nonlinear sequential proportional ψ-Hilfer fractional differential equations with multi-point boundary conditions. By using a Burton's version of the Krasnosel'skii's fixed-point theorem we established sufficient conditions for the existence result. An example illustrating our main result was also provided. [ABSTRACT FROM AUTHOR]

In this paper, we study boundary value problems of fractional integro-differential equations and inclusions involving Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray-Schauder in the single-valued case, while Martelli’s fixed point theorem, nonlinear alternative for multi-valued maps, and Covitz-Nadler fixed point theorem are used in the inclusion case. Examples illustrating the obtained results are also presented.