scholarly journals Results of Positive Solutions for the Fractional Differential System on an Infinite Interval

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ying Wang ◽  
Jun Yang ◽  
Yumei Zi
Keyword(s):  
Positive Solutions ◽  
Differential System ◽  
Contraction Mapping Principle ◽  
Infinite Interval ◽  
Compactness Criterion ◽  
Fractional Differential System ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

The chief topic of this paper is to investigate the fractional differential system on an infinite interval. By introducing an appropriate compactness criterion in a special function space and applying the Schauder fixed-point theorem and the Banach contraction mapping principle, we established the results for the existence and uniqueness of positive solutions. An example is then given to show the utilization of the main results.

scholarly journals Monotone iterative schemes for positive solutions of a fractional differential system with integral boundary conditions on an infinite interval

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4399-4417
Author(s):  
Yaohong Li ◽  
Wei Cheng ◽  
Jiafa Xu
Keyword(s):  
Positive Solutions ◽  
Differential System ◽  
Integral Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Infinite Interval ◽  
Integral Boundary ◽  
Iterative Schemes ◽  
Monotone Iterative ◽  
Fractional Differential System ◽  
Fractional Differential

In this paper, using the monotone iterative technique and the Banach contraction mapping principle, we study a class of fractional differential system with integral boundary on an infinite interval. Some explicit monotone iterative schemes for approximating the extreme positive solutions and the unique positive solution are constructed.

scholarly journals On the Solvability of Caputo -Fractional Boundary Value Problem Involving -Laplacian Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hüseyin Aktuğlu ◽  
Mehmet Ali Özarslan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Laplacian Operator ◽  
Fractional Boundary Value Problem ◽  
Solvability Conditions ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

We consider the model of a Caputo -fractional boundary value problem involving -Laplacian operator. By using the Banach contraction mapping principle, we prove that, under some conditions, the suggested model of the Caputo -fractional boundary value problem involving -Laplacian operator has a unique solution for both cases of and . It is interesting that in both cases solvability conditions obtained here depend on , , and the order of the Caputo -fractional differential equation. Finally, we illustrate our results with some examples.

scholarly journals EXISTENCE AND STABILITY ANALYSIS OF SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATION WITH NONLOCAL INTEGRAL AND ANTI-PERIODIC-TYPE BOUNDARY CONDITIONS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040006 ◽  
Author(s):  
AMITA DEVI ◽  
ANOOP KUMAR ◽  
THABET ABDELJAWAD ◽  
AZIZ KHAN
Keyword(s):  
Boundary Conditions ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Banach Contraction ◽  
Existence And Stability ◽  
Fractional Differential ◽  
Type Boundary ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

In this paper, we deal with the existence and uniqueness (EU) of solutions for nonlinear Langevin fractional differential equations (FDE) having fractional derivative of different orders with nonlocal integral and anti-periodic-type boundary conditions. Also, we investigate the Hyres–Ulam (HU) stability of solutions. The existence result is derived by applying Krasnoselskii’s fixed point theorem and the uniqueness of result is established by applying Banach contraction mapping principle. An example is offered to ensure the validity of our obtained results.

scholarly journals Hilfer–Hadamard Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions

2021 ◽  
Vol 5 (4) ◽  
pp. 195
Author(s):  
Bashir Ahmad ◽  
Sotiris K. Ntouyas
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Existence Results ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

This paper is concerned with the existence and uniqueness of solutions for a Hilfer–Hadamard fractional differential equation, supplemented with mixed nonlocal (multi-point, fractional integral multi-order and fractional derivative multi-order) boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying the fixed point theorems due to Krasnoselskiĭ and Schaefer and Leray–Schauder nonlinear alternatives. We demonstrate the application of the main results by presenting numerical examples. We also derive the existence results for the cases of convex and non-convex multifunctions involved in the multi-valued analogue of the problem at hand.

scholarly journals Iterative Positive Solutions to a Coupled Hadamard-Type Fractional Differential System on Infinite Domain with the Multistrip and Multipoint Mixed Boundary Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Xinran Du ◽  
Yuan Meng ◽  
Huihui Pang
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Differential System ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Infinite Interval ◽  
Infinite Domain ◽  
Complete Continuity ◽  
Fractional Differential System ◽  
Fractional Differential

This paper is devoted to the existence of positive solutions for a nonlinear coupled Hadamard fractional differential system, with multistrip and multipoint mixed boundary conditions on an infinite interval. Based on the Arzelá-Ascoli theorem, we establish an important lemma to prove the complete continuity of operators on the infinite interval. Using the monotone iterative technique, the existence criteria for positive extremal solutions can be acquired, and an example is given to illustrate the feasibility of the above study as well.

scholarly journals Existence and uniqueness of solutions for system of Hilfer–Hadamard sequential fractional differential equations with two point boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Warissara Saengthong ◽  
Ekkarath Thailert ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study existence and uniqueness of solutions for a system of Hilfer–Hadamard sequential fractional differential equations via standard fixed point theorems. The existence is proved by using the Leray–Schauder alternative, while the existence and uniqueness by the Banach contraction mapping principle. Illustrative examples are also discussed.

ON HADAMARD FRACTIONAL CALCULUS

Fractals ◽  
2017 ◽  
Vol 25 (03) ◽  
pp. 1750033 ◽  
Author(s):  
LI MA ◽  
CHANGPIN LI
Keyword(s):  
Fractional Calculus ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Fractional Operators ◽  
Initial Value ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping ◽  
Derivative Order

This paper is devoted to the investigation of the Hadamard fractional calculus in three aspects. First, we study the semigroup and reciprocal properties of the Hadamard-type fractional operators. Then, the definite conditions of certain class of Hadamard-type fractional differential equations (HTFDEs) are proposed through the Banach contraction mapping principle. Finally, we prove a novel Gronwall inequality with weak singularity and analyze the dependence of solutions of HTFDEs on the derivative order and the perturbation terms along with the proposed initial value conditions. The illustrative examples are presented as well.

scholarly journals Some Properties of Solutions to Multiterm Fractional Boundary Value Problems with p -Laplacian Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
KumSong Jong ◽  
HuiChol Choi ◽  
KyongJun Jang ◽  
SongGuk Jong ◽  
KyongSon Jon ◽  
...  
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Laplacian Operator ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Multipoint Boundary Value Problems ◽  
The Given ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

In this paper, we study some properties of positive solutions to a class of multipoint boundary value problems for nonlinear multiterm fractional differential equations with p -Laplacian operator. Using the Banach contraction mapping principle, the existence, the uniqueness, the positivity, and the continuous dependency on m -point boundary conditions of the solutions to the given problem are investigated. Also, two examples are presented to demonstrate our main results.

scholarly journals Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Asma ◽  
Sana Shabbir ◽  
Kamal Shah ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Stability Results ◽  
Rassias Stability ◽  
Banach Contraction Mapping ◽  
Implicit Fractional Differential Equations

AbstractSome fundamental conditions and hypotheses are established to ensure the existence, uniqueness, and stability to a class of implicit boundary value problems (BVPs) with Atangana–Baleanu–Caputo type derivative and integral. The required results are established by utilizing the Banach contraction mapping principle and fixed point theorem of Krasnoselskii. In addition, various types of stability results including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability are formulated for the problem under consideration. Pertinent examples are given to justify the results we obtain.

scholarly journals Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Systems with Nonlocal Coupled Fractional Integral Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 174
Author(s):  
Chanakarn Kiataramkul ◽  
Weera Yukunthorn ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Boundary Conditions ◽  
Fractional Integral ◽  
Integral Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Integral Boundary ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle

In this paper, we initiate the study of existence of solutions for a fractional differential system which contains mixed Riemann–Liouville and Hadamard–Caputo fractional derivatives, complemented with nonlocal coupled fractional integral boundary conditions. We derive necessary conditions for the existence and uniqueness of solutions of the considered system, by using standard fixed point theorems, such as Banach contraction mapping principle and Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.

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