Attractivity for differential equations of fractional order and ψ-Hilfer type

2020 ◽  
Vol 23 (4) ◽  
pp. 1188-1207
Author(s):  
J. Vanterler da C. Sousa ◽  
Mouffak Benchohra ◽  
Gaston M. N’Guérékata
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Broad Class ◽  
Hilfer Fractional Derivative ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Fractional Differential

AbstractThis paper investigates the overall solution attractivity of the fractional differential equation involving the ψ-Hilfer fractional derivative and using the Krasnoselskii’s fixed point theorem. We highlight some particular cases of the results presented here, especially involving the Riemann-Liouville, thus illustrating the broad class of fractional derivatives to which these results can be applied.

scholarly journals Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jun-Rui Yue ◽  
Jian-Ping Sun ◽  
Shuqin Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Nonlinear Fractional Differential Equation ◽  
Krasnoselskii Fixed Point Theorem ◽  
Fractional Differential

We consider the following boundary value problem of nonlinear fractional differential equation:(CD0+αu)(t)=f(t,u(t)),  t∈[0,1],  u(0)=0,   u′(0)+u′′(0)=0,  u′(1)+u′′(1)=0, whereα∈(2,3]is a real number, CD0+αdenotes the standard Caputo fractional derivative, andf:[0,1]×[0,+∞)→[0,+∞)is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.

scholarly journals The Ulam Stability of Fractional Differential Equation with the Caputo-Fabrizio Derivative

2022 ◽  
Vol 2022 ◽  
pp. 1-9
Author(s):  
Shuyi Wang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Differential Equation ◽  
Point Theorem ◽  
Ulam Stability ◽  
Integral Boundary ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Uniqueness Results ◽  
Fractional Differential

The aim of this paper is to establish the Ulam stability of the Caputo-Fabrizio fractional differential equation with integral boundary condition. We also present the existence and uniqueness results of the solution for the Caputo-Fabrizio fractional differential equation by Krasnoselskii’s fixed point theorem and Banach fixed point theorem. Some examples are provided to illustrate our theorems.

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

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Kemei Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Differential Equation ◽  
Point Theorem ◽  
Fractional Derivatives ◽  
Space Variable ◽  
Higher Order ◽  
Existence Results ◽  
Fractional Differential

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.

scholarly journals A Study of Fractional Differential Equation with a Positive Constant Coefficient via Hilfer Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Hoa Ngo Van ◽  
Vu Ho
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Positive Constant ◽  
Constant Coefficient ◽  
Initial Conditions ◽  
Hilfer Fractional Derivative ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Rassias Stability

The aim of the paper is to consider the existence and uniqueness of solution of the fractional differential equation with a positive constant coefficient under Hilfer fractional derivative by using the fixed-point theorem. We also prove the bounded and continuous dependence on the initial conditions of solution. Besides, Hyers–Ulam stability and Hyers–Ulam–Rassias stability are discussed. Finally, we provide an example to demonstrate our main results.

scholarly journals Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 765
Author(s):  
Zhifu Jia ◽  
Xinsheng Liu ◽  
Cunlin Li
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Differential Equation ◽  
Growth Condition ◽  
Lipschitz Condition ◽  
Point Theorem ◽  
Linear Growth ◽  
Linear Growth Condition ◽  
Fractional Differential

No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, for the one-dimensional case, we give two types of uncertain FDEs with jump that are symmetric in terms of form. The next, for the multidimensional case, when the coefficients of the equations satisfy Lipschitz condition and linear growth condition, we establish an existence and uniqueness theorems of uncertain FDEs with jump of Riemann-Liouville type by Banach fixed point theorem. A symmetric proof in terms of form is suitable to the Caputo type. When the coefficients do not satisfy the Lipschitz condition and linear growth condition, we just prove an existence theorem of the Caputo type equation by Schauder fixed point theorem. In the end, we present an application about uncertain interest rate model.

scholarly journals Uniqueness of Positive Solutions for a Perturbed Fractional Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Chen Yang ◽  
Jieming Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Point Boundary ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem:D0+αu(t)+f(t,u,u',…,u(n-2))+g(t)=0, 0<t<1, n-1<α≤n, n≥2,u(0)=u'(0)=⋯=u(n-2)(0)=u(n-2)(1)=0, whereD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem of generalized concave operators. An example is given to illustrate the main result.

scholarly journals Implicit Hybrid Fractional Boundary Value Problem via Generalized Hilfer Derivative

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1937
Author(s):  
Abdellatif ‬Boutiara ◽  
Mohammed S. ‬Abdo ◽  
Mohammed A. ‬Almalahi ◽  
Hijaz Ahmad ◽  
Amira Ishan
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivatives ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Banach Algebras ◽  
Fractional Operators ◽  
Fractional Boundary Value Problem ◽  
Fractional Differential

This research paper is dedicated to the study of a class of boundary value problems for a nonlinear, implicit, hybrid, fractional, differential equation, supplemented with boundary conditions involving general fractional derivatives, known as the ϑ-Hilfer and ϑ-Riemann–Liouville fractional operators. The existence of solutions to the mentioned problem is obtained by some auxiliary conditions and applied Dhage’s fixed point theorem on Banach algebras. The considered problem covers some symmetry cases, with respect to a ϑ function. Moreover, we present a pertinent example to corroborate the reported results.

scholarly journals Existence and Nonexistence of Positive Solutions for Fractional Three-Point Boundary Value Problems with a Parameter

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yunhong Li ◽  
Weihua Jiang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Boundary Value ◽  
Point Boundary ◽  
Existence And Nonexistence ◽  
Fractional Differential

In this work, we investigate the existence and nonexistence of positive solutions for p-Laplacian fractional differential equation with a parameter. On the basis of the properties of Green’s function and Guo-Krasnosel’skii fixed point theorem on cones, the existence and nonexistence of positive solutions are obtained for the boundary value problems. We also give some examples to illustrate the effectiveness of our main results.

Existence of positive solutions for systems of nonlinear fractional differential equation with p-Laplacian

2019 ◽  
Vol 13 (05) ◽  
pp. 2050089 ◽  
Author(s):  
S. Nageswara Rao ◽  
Meshari Alesemi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Nonlinear Fractional Differential Equation ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

In this paper, we establish sufficient conditions for the existence of positive solutions for a system of nonlinear fractional [Formula: see text]-Laplacian boundary value problems under different combinations of superlinearity and sublinearity of the nonlinearities via the Guo–Krasnosel’skii fixed point theorem. Moreover, an example is given to illustrate our results.

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