Implicit Fractional Differential Equations Research Articles

Qualitative Analysis of Implicit Dirichlet Boundary Value Problem for Caputo-Fabrizio Fractional Differential Equations

This article studies a class of implicit fractional differential equations involving a Caputo-Fabrizio fractional derivative under Dirichlet boundary conditions (DBCs). Using classical fixed-point theory techniques due to Banch’s and Krasnoselskii, a qualitative analysis of the concerned problem for the existence of solutions is established. Furthermore, some results about the stability of the Ulam type are also studied for the proposed problem. Some pertinent examples are given to justify the results.

Analysis of (α,β)-order coupled implicit Caputo fractional differential equations using topological degree method

AbstractThis article is devoted to establish the existence of solution of(α,β)$\left(\alpha ,\beta \right)$-order coupled implicit fractional differential equation with initial conditions, using Laplace transform method. The topological degree theory is used to obtain sufficient conditions for uniqueness and at least one solution of the considered system. Beside this, Ulam’s type stabilities are discussed for the proposed system. To support our main results, we present an example.

Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations

In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.

Study of fractional order pantograph type impulsive antiperiodic boundary value problem

In this paper, we study existence and stability results of an anti-periodic boundary value problem of nonlinear delay (pantograph) type implicit fractional differential equations with impulsive conditions. Using Schaefer’s fixed point theorem and Banach’s fixed point theorem, we have established results of at least one solution and uniqueness. Also, using the Hyers–Ulam concept, we have derived various kinds of Ulam stability results for the considered problem. Finally, we have applied our obtained results to a numerical problem.

Existence and uniqueness results for \u03a6-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition

The present paper describes the implicit fractional pantograph differential equation in the context of generalized fractional derivative and anti-periodic conditions. We formulated the Green’s function of the proposed problems. With the aid of a Green’s function, we obtain an analogous integral equation of the proposed problems and demonstrate the existence and uniqueness of solutions using the techniques of the Schaefer and Banach fixed point theorems. Besides, some special cases that show the proposed problems extend the current ones in the literature are presented. Finally, two examples were given as an application to illustrate the results obtained.

Existence and stability results for Hilfer-Katugampola-type fractional implicit differential equations with nonlocal conditions

Existence and stability results for Hilfer-Katugampola-type fractional implicit differential equations with nonlocal conditions

EXISTENCE RESULTS FOR IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL BOUNDARY CONDITIONS

Adv. Math. Sci. J. Vol. 9, No. 4 Open Access EXISTENCE RESULTS FOR IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL BOUNDARY CONDITIONS A. Anguraj, M. Kasthuri and P. Karthikeyan DOI: https://doi.org/10.37418/amsj.9.4.95 Full Text Abstract Abstract In this paper, we examine the existence of solutions for implicit FDE’s with fractional boundary conditions. To prove … page AMSJ9-N4-95 Read More »

Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations

In this paper, we derive the equivalent fractional integral equation to the nonlinear implicit fractional differential equations involving Ψ‐Hilfer fractional derivative subject to nonlocal fractional integral boundary conditions. The existence of a solution, Ulam–Hyers, and Ulam–Hyers–Rassias stability have been acquired by means of an equivalent fractional integral equation. Our investigations depend on the fixed‐point theorem due to Krasnoselskii and the Gronwall inequality involving Ψ‐Riemann–Liouville fractional integral. Finally, examples are provided to show the utilization of primary outcomes.

Integrable Solutions for Caputo-Hadamard Implicit Fractional Order Differential Equations

Integrable Solutions for Caputo-Hadamard Implicit Fractional Order Differential Equations

Periodic boundary value problems for fractional implicit differential equations involving Hilfer fractional derivative

Periodic boundary value problems for fractional implicit differential equations involving Hilfer fractional derivative

Study on Krasnoselskii\u2019s fixed point theorem for Caputo\u2013Fabrizio fractional differential equations

This note is concerned with establishing existence theory of solutions to a class of implicit fractional differential equations (FODEs) involving nonsingular derivative. By using usual classical fixed point theorems of Banach and Krasnoselskii, we develop sufficient conditions for the existence of at least one solution and its uniqueness. Further, some results about Ulam–Hyers stability and its generalization are also discussed. Two suitable examples are given to demonstrate the results.

Analysis of Coupled System of Implicit Fractional Differential Equations Involving Katugampola–Caputo Fractional Derivative

In this paper, we study the existence and uniqueness of solutions to implicit the coupled fractional differential system with the Katugampola–Caputo fractional derivative. Different fixed-point theorems are used to acquire the required results. Moreover, we derive some sufficient conditions to guarantee that the solutions to our considered system are Hyers–Ulam stable. We also provided an example that explains our results.

Study of implicit delay fractional differential equations under anti-periodic boundary conditions

This research work is related to studying a class of special type delay implicit fractional order differential equations under anti-periodic boundary conditions. With the help of classical fixed point theory due to Schauder and Banach, we derive some results about the existence of at least one solution. Further, we also study some results including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam Rassias, and generalized Hyers–Ulam–Rassias stability. We provide some test problems for illustrating our analysis.

Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type

This paper deals with a nonlinear implicit fractional differential equation with the anti-periodic boundary condition involving the Caputo-Katugampola type. The existence and uniqueness results are established by applying the fixed point theorems of Krasnoselskii and Banach. Further, by using generalized Gronwall inequality the Ulam-Hyers stability results are proved. To demonstrate the effectiveness of the main results, appropriate examples are granted.

Impulsive boundary value problems for nonlinear implicit Caputo-exponential type fractional differential equations

This paper deals with existence and uniqueness of solutions to a class of impulsive boundary value problem for nonlinear implicit fractional differential equations involving the Caputo-exponential fractional derivative. The existence results are based on Schaefer’s fixed point theorem and the uniqueness result is established via Banach’s contraction principle. Two examples are given to illustrate the main results.

Impulsive initial value problems for a class of implicit fractional differential equations

In this article we consider, impulsive initial value problems for a class of implicit fractional differential equations involving the Caputo fractional derivative of order β in (1,2]. The solutions of this nonlinear equation are analyzed by establishing sufficient conditions for existence and uniqueness using Banach's contraction mapping principle and the Schaefer's fixed point theorem. In addition, using the Banach contraction principle, we establish uniqueness result. To demonstrate main results two examples are presented.

A survey on the initial value problems of fuzzy implicit fractional differential equations

The purpose of this paper is to present some basic theories of various classes of initial value problems of implicit fractional differential equation involving the Caputo-fuzzy-type, Riemann-Liouville-fuzzy-type and Hadamard-fuzzy-type concepts of fractional derivative in the case of the order α∈(0,1). In particular, the existence and uniqueness results of solution for the given initial value problems of implicit fractional differential equations is investigated. In addition, some results concerning continuous dependence of solutions on perturbations of the given parameters, namely, of the initial conditions, of the function and parameters on the right-hand side, of the order of fractional derivative are analyzed. Finally, some examples are given to illustrate our main results.

THEORY OF FRACTIONAL IMPLICIT DIFFERENTIAL EQUATIONS WITH COMPLEX ORDER

In this paper, we consider boundary value problems for the following nonlinear implicit differential equations with complex order D +x(t) = f t,x(t),D +x(t) , ? = m+i?, t ? J := [0,T], ax(0)+bx(T) = c,where D + is the Caputo fractional derivative of order ? ? C. Let ? ? R , 0 < ? < 1, m ? (0,1], and f : J ×R ? R is given continuous function. Here a,b,c are real constants with a+b = 0. We derive the existence and stability of solution for a class of boundary value problem(BVP) for nonlinear fractional implicit differential equations(FIDEs) with complex order. The results are based upon the Banach contraction principle and Schaefer’s fixed point theorem.

Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives

We present some results on the existence, uniqueness and Hyers–Ulam stability to the solution of an implicit coupled system of impulsive fractional differential equations having Hadamard type fractional derivative. Using a fixed point theorem of Kransnoselskii’s type, the existence and uniqueness results are obtained. Along these lines, different kinds of Hyers–Ulam stability are discussed. An example is given to illustrate the main theorems.

Investigation of Ulam Stability Results of a Coupled System of Nonlinear Implicit Fractional Differential Equations

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.