Fractional Order Functional Differential Equations Research Articles

Some sufficient conditions of existence and trajectory controllability for impulsive and initial value fractional order functional differential equations

For a class of impulsive fractional differential equation with fractional order r∈(1,2) by means of solution operator Sr(t), we develop the mild solution and deal with the existence results of mild solution for the considered equation with state-dependent delay through three fixed point theorems, namely Banach, Nonlinear Leray–Schauder Alternative and Krasnoselskii’s fixed point theorem. Next, we define the mild solution via an operator function Wr,β(τ) for abstract version of the considered equation. Further, we study the trajectory controllability (which is a stronger notion than other controllability) of the equation. In the last, we present some examples to illustrate the established results through partial fractional differential equations.

Some critical remarks on “Some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations”

In this manuscript, we generalize, improve, and enrich recent results established by Budhia et al. [L. Budhia, H. Aydi, A.H. Ansari, D. Gopal, Some new fixed point results in rectangular metric spaces with application to fractional-order functional differential equations, Nonlinear Anal. Model. Control, 25(4):580–597, 2020]. This paper aims to provide much simpler and shorter proofs of some results in rectangular metric spaces. According to one of our recent lemmas, we show that the given contractive condition yields Cauchyness of the corresponding Picard sequence. The obtained results improve well-known comparable results in the literature. Using our new approach, we prove that a Picard sequence is Cauchy in the framework of rectangular metric spaces. Our obtained results complement and enrich several methods in the existing state-ofart. Endorsing the materiality of the presented results, we also propound an application to dynamic programming associated with the multistage process.

Non-instantaneous impulsive fractional-order delay differential systems with Mittag-Leffler kernel

<abstract><p>The existence of fractional-order functional differential equations with non-instantaneous impulses within the Mittag-Leffler kernel is examined in this manuscript. Non-instantaneous impulses are involved in such equations and the solution semigroup is not compact in Banach spaces. We suppose that the nonlinear term fulfills a non-compactness measure criterion and a local growth constraint. We further assume that non-instantaneous impulsive functions satisfy specific Lipschitz criteria. Finally, an example is given to justify the theoretical results.</p></abstract>

Some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations

In this paper, we establish some new fixed point theorems for generalized ϕ–ψ-contractive mappings satisfying an admissibility-type condition in a Hausdorff rectangular metric space with the help of C-functions. In this process, we rectify the proof of Theorem 3.2 due to Budhia et al. [New fixed point results in rectangular metric space and application to fractional calculus, Tbil. Math. J., 10(1):91–104, 2017]. Some examples are given to illustrate the theorems. Finally, we apply our result (Corollary 3.6) to establish the existence of a solution for an initial value problem of a fractional-order functional differential equation with infinite delay.

Solutions of linear uncertain fractional-order delay differential equations

Uncertain fractional-order delay differential equation is a class of fractional-order functional differential equations driven by Liu process. This paper devotes to studying linear uncertain fractional-order delay differential equation. The explicit representation and iterative formula of the solution to linear uncertain fractional-order delay differential equations are obtained. Meanwhile, the inverse uncertainty distribution of the solution to linear uncertain fractional-order delay differential equation by the $$\alpha $$ -path is presented.

Existence and uniqueness for fractional order functional differential equations with Hilfer derivative

Existence and uniqueness for fractional order functional differential equations with Hilfer derivative

Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials

There is an increasing interest in the field of functional and fractional differential equations. The lack of closed-form analytical solutions motivates the development of numerical methods for solving mixed-type fractional-order functional differential equations (MFFDEs) with retarded and neutral terms. This paper studies the solution of MFFDEs by a collocation technique with modified Lucas polynomials. The proposed method uses operational matrices to obtain an approximate solution by means of a system of linear algebraic equations. The accuracy of the proposed algorithm is verified with three illustrative examples.

Oscillation of fractional order functional differential equations with nonlinear damping

Abstract In this paper, we are concerned with the oscillatory behavior of a class of fractional differential equations with functional terms. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. Based on a certain variable transformation, by using a generalized Riccati transformation, generalized Philos type kernels, and averaging techniques we establish new interval oscillation criteria. Illustrative examples are also given.

Integrable solutions for implicit fractional order functional differential equations with infinite delay

In this paper we study the existence of integrable solutions for initial value problem for implicit fractional order functional differential equations with infinite delay. Our results are based on Schauder type fixed point theorem and the Banach contraction principle fixed point theorem.

Solutions to fractional functional differential equations with nonlocal conditions

Abstract In this paper, we discuss the solutions to nonlocal initial value problems of fractional order functional differential equations in a Banach space. In particular, we prove the existence and uniqueness of mild and classical solutions assuming that −A generates a resolvent operator family and nonlinear part is a Lipschitz continuous function. We also investigate the global existence of the solution. At the end, a fractional order partial differential equation is given to illustrate the obtained abstract results.

POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF SINGULAR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION

In this paper, we consider a class of singular fractional order functional differential equations with delay. By means of a fixed point theorem on cones, some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established. We also give examples to illustrate the applicability of our main results.

WEIGHTED PSEUDO ALMOST AUTOMORPHIC SOLUTIONS OF FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper we discuss the existence of weighted pseudo almost automorphic solution of fractional order functional differential equations. Using the fixed point theorem we establish existence and uniqueness of solution to the problem under consideration. The results obtained extend the theory of almost automorphic solutions to a more general class of weighted pseudo almost automorphic solutions. These extensions allow to treat infinite dimensional dynamics such as fractional wave and heat equation which are presented in the paper. At the end we give several example to illustrate the analytical findings.

Existence results for fractional order functional differential equations with impulse

In this paper, we use the analytic semigroup theory of linear operators and the fixed point method to prove the existence of mild solutions for the semilinear fractional order functional differential equations with impulse in a Banach space.

Existence of Positive Solutions for a Functional Fractional Boundary Value Problem

We study the existence of positive solutions for a boundary value problem of fractional‐order functional differential equations. Several new existence results are obtained.

Existence results for fractional order functional differential equations with infinite delay

The Banach fixed point theorem and the nonlinear alternative of Leray–Schauder type are used to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay.

Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces

In this article, a recent nonlinear alternative for contraction maps in Fréchet spaces due to Frigon and Granas [1998, Résultats de type Leray-Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22, 161–168] is used to investigate the existence and uniqueness of solutions for fractional order functional differential equations with infinite delay.