scholarly journals Lower and Upper Solutions Method for Positive Solutions of Fractional Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
R. Darzi ◽  
B. Mohammadzadeh ◽  
A. Neamaty ◽  
D. Bǎleanu
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Positive Real Number ◽  
Boundary Value ◽  
Lower And Upper Solutions ◽  
Fractional Boundary Value Problem ◽  
Positive Real ◽  
Liouville Fractional Derivative ◽  
Fractional Boundary Value Problems

We apply the lower and upper solutions method and fixed-point theorems to prove the existence of positive solution to fractional boundary value problemD0+αut+ft,ut=0,0<t<1,2<α≤3,u0=u′0=0,D0+α−1u1=βuξ,0<ξ<1, whereD0+αdenotes Riemann-Liouville fractional derivative,βis positive real number,βξα−1≥2Γα, andfis continuous on0,1×0,∞. As an application, one example is given to illustrate the main result.

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 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 Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-8 ◽  
Author(s):  
Moustafa El-Shahed
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Fractional Differential Equation ◽  
Existence And Nonexistence ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We are concerned with the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem:D0+αu(t)+λa(t) f(u(t))=0, 0<t<1, u(0)=u′(0)=u′(1)=0,where2<α<3is a real number andD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. An example is also given to illustrate the main results.

scholarly journals Existence Results for the Solution of the Hybrid Caputo–Hadamard Fractional Differential Problems Using Dhage’s Approach

2021 ◽  
Vol 6 (1) ◽  
pp. 17
Author(s):  
Muhammad Yaseen ◽  
Sadia Mumtaz ◽  
Reny George ◽  
Azhar Hussain
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Numerical Examples ◽  
Fractional Differential

In this work, we explore the existence results for the hybrid Caputo–Hadamard fractional boundary value problem (CH-FBVP). The inclusion version of the proposed BVP with a three-point hybrid Caputo–Hadamard terminal conditions is also considered and the related existence results are provided. To achieve these goals, we utilize the well-known fixed point theorems attributed to Dhage for both BVPs. Moreover, we present two numerical examples to validate our analytical findings.

scholarly journals Existence of Solutions for Riemann-Liouville Fractional Boundary Value Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Wenzhe Xie ◽  
Jing Xiao ◽  
Zhiguo Luo
Keyword(s):  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Fixed Point Theorems ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Extreme Points ◽  
Fractional Boundary Value Problem ◽  
Liouville Fractional Derivative

By using the method of upper and lower solutions and fixed point theorems, the existence of solutions for a Riemann-Liouville fractional boundary value problem with the nonlinear term depending on fractional derivative of lower order is obtained under the classical Nagumo conditions. Also, some results concerning Riemann-Liouville fractional derivative at extreme points are established with weaker hypotheses, which improve some works in Al-Refai (2012). As applications, an example is presented to illustrate our main results.

Discrete fractional boundary value problems and inequalities

2021 ◽  
Vol 24 (6) ◽  
pp. 1777-1796
Author(s):  
Martin Bohner ◽  
Nick Fewster-Young
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Existence Theorem ◽  
Boundary Value ◽  
A Priori ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Fractional Boundary Value Problems

Abstract In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral representation, and, second, the establishment of solutions satisfying certain a priori bounds provided the key inequality holds. In order to establish the latter result, several novel discrete fractional inequalities are developed, each of them interesting in itself and of potential future use in different contexts. We illustrate the usefulness of our existence results by presenting two examples.

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 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 Fixed point theorems via Generalized WF-Contractions with applications

SeMA Journal ◽  
2021 ◽  
Author(s):  
Rosana Rodríguez-López ◽  
Rakesh Tiwari
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
New Class ◽  
Second Order Differential Equations ◽  
Fixed Point Result

AbstractThe aim of this paper is to introduce a new class of mixed contractions which allow to revise and generalize some results obtained in [6] by R. Gubran, W. M. Alfaqih and M. Imdad. We also provide an example corresponding to this class of mappings and show how the new fixed point result relates to the above-mentioned result in [6]. Further, we present an application to the solvability of a two-point boundary value problem for second order differential equations.

scholarly journals Analytical solutions to contact problem with fractional derivatives in the sense of Caputo

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 313-323
Author(s):  
Muhammad Noor ◽  
Muhammad Rafiq ◽  
Salah-Ud-Din Khan ◽  
Muhammad Qureshi ◽  
Muhammad Kamran ◽  
...  
Keyword(s):  
Boundary Value Problem ◽  
Iteration Method ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Contact Problems ◽  
Function Technique ◽  
Fractional Boundary Value Problem ◽  
Variational Iteration ◽  
Residual Errors ◽  
Fractional Boundary Value Problems

The current study extends the applications of the variational iteration method for the analytical solution of fractional contact problems. The problem involves Caputo sense while calculating the derivative of fractional order, we apply the Penalty function technique to transform it into a system of fractional boundary value problems coupled with a known obstacle. The variational iteration method is employed to find the series solution of fractional boundary value problem. For different values of fractional parameters, residual errors of solutions are plotted to make sure the convergence and accuracy of the solution. The reasonably accurate results show that one of the highly effective and stable methods for the solution of fractional boundary value problem is the method of variational iteration.

Sign in / Sign up

Export Citation Format

Share Document