Existence and continuous dependence for weighted fractional differential equations with infinite delay

Dong, Qixiang

doi:10.1186/1687-1847-2014-190

Cited by 5 publications

(14 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Existence and continuous dependence results of solutions are obtained in finite dimensional spaces. In this paper, we continue the work of [6], to study the weighted fractional differential equations with infinite delay in Banach spaces. The difficulty is that the bounded subsets in Banach spaces are not compact in general.…”

Section: Introductionmentioning

confidence: 99%

“…Fractional differential equations in Banach spaces are also wildly studied [1,[11][12][13][14]. Among these works, some authors study functional fractional differential equations [2,3,6,8,11]. For example, in [2], Benchohra et al studied fractional order differential equations D˛y.t / D f .t; y t /; t 2 OE0; b; 0 <˛< 1; with infinite delay y.t / D .t /; t 2 .…”

Section: Introductionmentioning

confidence: 99%

“…For this reason, when y.0/ ¤ 0, the solutions to the functional fractional differential equations given in the mentioned papers may not be well-defined. An example can be found in [6]. In order to remedy this defect, in [6], the author modified the model and studied the weighted fractional differential equations with infinite delay.…”

Section: Introductionmentioning

confidence: 99%

“…An example can be found in [6]. In order to remedy this defect, in [6], the author modified the model and studied the weighted fractional differential equations with infinite delay. Existence and continuous dependence results of solutions are obtained in finite dimensional spaces.…”

Section: Introductionmentioning

confidence: 99%

“…There has been a significant development in fractional differential equations in the past decades. See, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein. Fractional differential equations in Banach spaces are also wildly studied [1,[11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Weighted fractional differential equations with infinite delay in Banach spaces

Dong

Liu²,

Fan³

2016

Open Mathematics

Self Cite

View full text Add to dashboard Cite

This paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schauder's and Banach's fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved, and the comparison property of fractional integral is discussed.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Weighted fractional differential equations with infinite delay in Banach spaces

Dong

Liu²,

Fan³

2016

Open Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

An analysis on the optimal control and approximate controllability for Hilfer fractional neutral integro‐differential systems with finite delay

Ma,

Kavitha,

Shukla

et al. 2023

Optim Control Appl Methods

View full text Add to dashboard Cite

The existence and uniqueness of solutions to Hilfer fractional neutral delay integro‐differential equations subject to nonlocal conditions are discussed in this study. The main results of this study are based in part on the fixed point techniques of Banach contraction and Krasnoselskii's fixed point theorem from calculus theory. First, we determine whether or not the fractional system has a mild solution. The uniqueness of the mild solution is further illustrated by expanding our results. The optimal control problems are governed by a new class of neutral delay integro‐differential equations in Banach spaces, and we also develop sufficient conditions for the approximate controllability of a nonlinear fractional system. An example is given at the end to strengthen the compatibility of the results.

show abstract

Weighted fractional stochastic integro-differential equation with infinite delay

Arioui

2023

Arab. J. Math.

View full text Add to dashboard Cite

In this paper, we consider a weighted fractional stochastic integro-differential equation with infinite delay and nonzero initial values involving a Riemann–Liouville fractional derivative of order $$1/2<\alpha <1$$ 1 / 2 < α < 1 . The existence of a mild solution is investigated using fractional calculus, stochastic analysis, and the fixed point theorem. An example is also provided to illustrate the obtained result.

show abstract

Existence and continuous dependence for weighted fractional differential equations with infinite delay

Abstract: This paper is concerned with weighted fractional differential equations with infinite delay, the Riemann-Liouville derivative, and nonzero initial values. Existence and continuous dependence results of solutions are obtained. An illustrative example is also presented.

Cited by 5 publications

References 8 publications

Weighted fractional differential equations with infinite delay in Banach spaces

Weighted fractional differential equations with infinite delay in Banach spaces

An analysis on the optimal control and approximate controllability for Hilfer fractional neutral integro‐differential systems with finite delay

Weighted fractional stochastic integro-differential equation with infinite delay

Contact Info

Product

Resources

About