Nonlocal Fractional Differential Equations Research Articles

An analysis on approximate controllability results for impulsive fractional differential equations of order 1 < r < 2 with infinite delay using sequence method

In this work, we discuss the existence and approximate controllability results for impulsive fractional differential systems of order with infinite delay. Sectorial operator of type , the nonlinear alternative of the Leray–Schauder fixed point theorem, sequence method, and impulsive systems have been used to establish these results. First, we investigate the existence of mild solutions for impulsive fractional differential equations of order . We also establish the approximate controllability results for the nonlocal fractional delay differential equations. An example is also given to illustrate our main results.

A study of nonlocal fractional delay differential equations with hemivariational inequality

<abstract><p>In this paper, we study an abstract system of fractional delay differential equations of order $ 1 < q < 2 $ with a hemivariational inequality in Banach spaces. To establish the existence of a solution to the abstract inequality, we employ the Rothe technique in conjunction with the surjectivity of multivalued pseudomonotone operators and features of the Clarke generalized gradient. Further, to show the existence of the fractional differential equation, we use the fractional cosine family and fixed point theorem. Finally, we include an example to elaborate the effectiveness of the findings.</p></abstract>

Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

<abstract><p>This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.</p></abstract>

Fractional differential equations with nonlocal conditions on weight space

This paper studies the existence of solutions for nonlocal fractional differential equations on the half-line in a special Banach space. The main result is based on a fixed point theorem combining with a suitable measure of non-compactness for condensing operator, an example is given to illustrate our approach.

Nonlocal semilinear Ф-Caputo fractional evolution equation with a measure of noncompactness in Banach space

The aim of this work is to study the existence of solutions for nonlocal fractional differential equations inclusions involving ?-Caputo fractional dirivative in Banach space. The proofs are based on the noncompactness measure method. As application, we give an example is given to illustrate the theoretical results.

ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS TO NONLOCAL DIFFERENTIAL EQUATIONS

We consider the asymptotic behavior of solutions on the half-line to nonlocal fractional differential equations of the form k∗u′=Au(t)+f(t,u(t)), where k∗u′ stands for the nonlocal derivative of u in Caputo’s sense corresponding to a singular kernel k, A is a positively definite, selfadjoint operator; the nonlinearity f is either locally Lipschitz continuous with respect to the second variable or of sublinear growth for small time and Lipschitz continuous for large time. Our main results show that if f is an asymptotically almost periodic function on t then the problem possesses asymptotically almost periodic solutions.

Nonlocal Fractional Differential Equation On The Half Line in Banach Space

Our aim in this paper is to study the existence of solution sets and its topological structure for non-localfractional differential equations on the half-line in a Banach space using Riemann-Liouville definition. Themain result is based on Meir-Keeler fixed point theorem for condensing operators combined with measure ofnon-compactness. An example is given to illustrate the feasibility of our main result.

Regularity properties of nonlocal fractional differential equations and applications

Abstract The regularity properties of nonlocal fractional elliptic and parabolic equations in vector-valued Besov spaces are studied. The uniform B p , q s B_{p,q}^{s} -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. In particular, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, the maximal regularity properties of the nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional parabolic equations are studied.

Solvability of nonlinear fractional integro-differential equation with nonlocal condition

PurposeThis study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.Design/methodology/approachThe functional analysis method is the a priori estimate method or energy inequality method.FindingsThe results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.Research limitations/implicationsThe authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.Originality/valueThe authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

On a class of nonlinear nonlocal fractional differential equations

We investigate the existence of extremal solutions for a class of fractional differential equations in the area of fluid dynamics. By establishing a new comparison theorem and applying the classical monotone iterative approach, we establish sufficient conditions to ensure the existence of the extremal solutions and construct twin convergent monotone explicit iterative schemes. Generalized nonlinear nonlocal Bagley-Torvik equation and generalized Basset equation with nonlinear source functions are some special cases of our discussed problem.

Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

In this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competent definition of a mild solution. On this basis, we verify the well-posedness of the mild solution. Meanwhile, with a construction of Lagrange problem, we elaborate the existence of optimal pairs of the fractional evolution systems. The main tools are the fractional calculus, cosine family, multivalued analysis, measure of noncompactness method, and fixed point theorem. Finally, an example is propounded to illustrate the validity of our main results.

Solvability for a class of nonlocal singular fractional differential equations of Lane Emden type

In this paper, we study a class of nonlocal singular differential equations of fractional orders. The considered class involves Caputo derivatives and Riemann-Liouville integral in its linearities. It is a more general problem for which the question of existence and uniqueness of solutions is investigated. For this class, the question of existence of at least one solution is also discussed. At the end, an illustrative example is presented.

Nonlocal Fractional Differential Equations and Applications

The regularity properties of nonlocal fractional differential equations in Banach spaces are studied. Uniform $$L_{p}$$ -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, maximal regularity properties of nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional differential equations are studied.

Neutral fractional stochastic partial differential equations with Clarke subdifferential

ABSTRACT By using fractional calculus, stochastic analysis theory and fixed point theorems, sufficient conditions for approximate controllability of nonlocal Sobolev-type neutral fractional stochastic differential equations with fractional Brownian motion and Clarke subdifferential are established. Finally, an example is given to illustrate the obtained results.

Existence of Mild Solution of a Class of Nonlocal Fractional order Differential Equation with not Instantaneous Impulses

Abstract In this article, we establish a set of sufficient conditions for the existence of mild solution of a class of fractional differential equations with not instantaneous impulses. The results are obtained by using Banach fixed point theorem and Krasnoselskii’s fixed point theorem. An example is presented for validation of result.

Positive Solutions for a Higher-Order Semipositone Nonlocal Fractional Differential Equation with Singularities on Both Time and Space Variable

In this paper, we consider the following higher-order semipositone nonlocal Riemann-Liouville fractional differential equation D0+αx(t)+f(t,x(t),D0+βx(t))+e(t)=0, 0<t<1,D0+βx(0)=D0+β+1x(0)=⋯=D0+n+β-2x(0)=0, and D0+βx(1)=∑i=1m-2ηiD0+βx(ξi), where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives. The existence results of positive solution are given by Guo-krasnosel’skii fixed point theorem and Schauder’s fixed point theorem.

Positive Solutions for Higher Order Nonlocal Fractional Differential Equation with Integral Boundary Conditions

In this paper, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, some properties of the first eigenvalue of a fractional differential equation were obtained; combining fixed point index theorem, sufficient conditions for the existence of positive solutions are established. An example is given to demonstrate the application of our main results.

Uniqueness of Successive Positive Solution for Nonlocal Singular Higher-Order Fractional Differential Equations Involving Arbitrary Derivatives

In this article, by means of fixed point theorem on mixed monotone operator, we establish the uniqueness of positive solution for some nonlocal singular higher-order fractional differential equations involving arbitrary derivatives. We also give iterative schemes for approximating this unique positive solution.

The Exact Iterative Solution of Fractional Differential Equation with Nonlocal Boundary Value Conditions

We deal with a singular nonlocal fractional differential equation with Riemann-Stieltjes integral conditions. The exact iterative solution is established under the iterative technique. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have been derived. An example is also given to demonstrate the results.

Monotone Solution of Cauchy Type Weighted Nonlocal Fractional Differential Equation

In this paper we will consider a nonlinear fractional di fferential equation withweighted initial and nonlocal conditions and will obtain monotone solution by thesequence of successive approximations starting at a lower solution converges monotonicallyto the solution of the related cauchy type weighted nonlocal fractionaldi fferential equation under some suitable conditions.