scholarly journals Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Imed Bachar ◽  
Habib Mâagli ◽  
Hassan Eltayeb
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Continuous Solution ◽  
Boundary Value ◽  
Banach Fixed Point Theorem ◽  
Uniqueness Of Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Boundary Value Problems

This paper deals with the following boundary value problem D α u t = f t , u t , t ∈ 0 , 1 , u 0 = u 1 = D α − 3 u 0 = u ′ 1 = 0 , where 3 < α ≤ 4 , D α is the Riemann-Liouville fractional derivative, and the nonlinearity f , which could be singular at both t = 0 and t = 1 , is required to be continuous on 0 , 1 × ℝ satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution u satisfying u t ≤ c ω t , for   t ∈ 0 , 1   and   c > 0 , where ω t ≔ t α − 2 1 − t 2 .

scholarly journals Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjie Liu ◽  
Xiao Fu ◽  
Liangping Qi
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Boundary Value Problems

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

scholarly journals RETRACTED ARTICLE: Existence and uniqueness of solutions for the Schrödinger integrable boundary value problem

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Jianjie Wang ◽  
Ali Mai ◽  
Hong Wang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Uniqueness Of Solutions ◽  
Linear Maps ◽  
Retracted Article

Abstract This paper is mainly devoted to the study of one kind of nonlinear Schrödinger differential equations. Under the integrable boundary value condition, the existence and uniqueness of the solutions of this equation are discussed by using new Riesz representations of linear maps and the Schrödinger fixed point theorem.

scholarly journals Existence and Uniqueness of Positive Solution for a Boundary Value Problem of Fractional Order

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
J. Caballero ◽  
J. Harjani ◽  
K. Sadarangani
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Partially Ordered Sets ◽  
Fractional Boundary Value Problem ◽  
Ordered Sets ◽  
Liouville Fractional Derivative

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional boundary value problem:D0+αu(t)+f(t,u(t))=0,0≤t≤1,3<α≤4,u(0)=u′(0)=u″(0)=u″(1)=0, whereD0+αdenotes the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also given to illustrate the results.

scholarly journals The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu ◽  
Ravi P. Agarwal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Infinite Delay ◽  
Banach Fixed Point Theorem ◽  
Uniqueness Of Solutions ◽  
Fractional Differential

We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem inΩ={y:(−∞,b]→ℝ:y|(−∞,0]∈ℬ}such thaty|[0,b]is continuous andℬis a phase space.

scholarly journals Blow-Up Solutions for a Singular Nonlinear Hadamard Fractional Boundary Value Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Imed Bachar ◽  
Said Mesloub
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Blow Up ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Fractional Boundary Value Problems

We consider singular nonlinear Hadamard fractional boundary value problems. Using properties of Green’s function and a fixed point theorem, we show that the problem has positive solutions which blow up. Finally, some examples are provided to explain the applications of the results.

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 Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Nichaphat Patanarapeelert ◽  
Thanin Sitthiwirattham
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Integral Boundary ◽  
Uniqueness Results ◽  
Fractional Hahn Integral

The existence and uniqueness results of two fractional Hahn difference boundary value problems are studied. The first problem is a Riemann-Liouville fractional Hahn difference boundary value problem for fractional Hahn integrodifference equations. The second is a fractional Hahn integral boundary value problem for Caputo fractional Hahn difference equations. The Banach fixed-point theorem and the Schauder fixed-point theorem are used as tools to prove the existence and uniqueness of solution of the problems.

scholarly journals Existence and Uniqueness of Positive Solutions for a Singular Fractional Three-Point Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
I. J. Cabrera ◽  
J. Harjani ◽  
K. B. Sadarangani
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Boundary Value ◽  
Point Boundary ◽  
Ordered Metric Spaces ◽  
Liouville Derivative

We investigate the existence and uniqueness of positive solutions for the following singular fractional three-point boundary value problemD0+αu(t)+f(t,u(t))=0, 0<t<1, u(0)=u′(0)=u′′(0)=0,u′′(1)=βu′′(η), where3<α≤4,D0+αis the standard Riemann-Liouville derivative andf:(0,1]×[0,∞)→[0,∞)withlim t→0+f(t,·)=∞(i.e.,fis singular att=0). Our analysis relies on a fixed point theorem in partially ordered metric spaces.

scholarly journals Existence of a unique solution for a third-order boundary value problem with nonlocal conditions of integral type

2021 ◽  
Vol 26 (5) ◽  
pp. 914-927
Author(s):  
Sergey Smirnov
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Unique Solution ◽  
Point Theorem ◽  
Boundary Value ◽  
Nonlocal Conditions ◽  
Integral Condition ◽  
Banach Fixed Point Theorem ◽  
Third Order

The existence of a unique solution for a third-order boundary value problem with integral condition is proved in several ways. The main tools in the proofs are the Banach fixed point theorem and the Rus’s fixed point theorem. To compare the applicability of the obtained results, some examples are considered.

scholarly journals Impulsive Antiperiodic Boundary Value Problems for Nonlinearqk-Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Lihong Zhang ◽  
Bashir Ahmad ◽  
Guotao Wang
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Uniqueness Of Solutions

We show the existence and uniqueness of solutions for an antiperiodic boundary value problem of nonlinear impulsiveqk-difference equations by applying some well-known fixed point theorems. An example is presented to illustrate the main results.

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