scholarly journals Positive Solution to a Fractional Boundary Value Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
A. Guezane-Lakoud ◽  
R. Khaldi
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
Positivity Of Solutions ◽  
The Green Function

A fractional boundary value problem is considered. By means of Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function, and Guo-Krasnosel'skii fixed point theorem on cone, some results on the existence, uniqueness, and positivity of solutions are obtained.

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 Existence and Positivity of Solutions for a Second-Order Boundary Value Problem with Integral Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Assia Guezane-Lakoud ◽  
Hamidane Nacira ◽  
Khaldi Rabah
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Integral Condition ◽  
Second Order ◽  
Nonlinear Alternative ◽  
Existence Of Positive Solutions ◽  
Banach Contraction ◽  
Positivity Of Solutions ◽  
Uniqueness And Existence

This work is devoted to the study of uniqueness and existence of positive solutions for a second-order boundary value problem with integral condition. The arguments are based on Banach contraction principle, Leray Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in cone. Two examples are also given to illustrate the 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 Existence of Positive Solution for the Eighth-Order Boundary Value Problem Using Classical Version of Leray–Schauder Alternative Fixed Point Theorem

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 129 ◽  
Author(s):  
Thenmozhi Shanmugam ◽  
Marudai Muthiah ◽  
Stojan Radenović
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Eighth Order ◽  
Classical Version ◽  
Nonlinear Alternative

In this work, we investigate the existence of solutions for the particular type of the eighth-order boundary value problem. We prove our results using classical version of Leray–Schauder nonlinear alternative fixed point theorem. Also we produce a few examples to illustrate our results.

scholarly journals Existence Results for Fractional Difference Equations with Three-Point Fractional Sum Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Thanin Sitthiwirattham ◽  
Jessada Tariboon ◽  
Sotiris K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Difference Equations ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Fractional Difference ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Nonlinear Alternative

We consider a discrete fractional boundary value problem of the formΔαu(t)=f(t+α-1,u(t+α-1)),  t∈[0,T]ℕ0:={0,1,…,T},  u(α-2)=0,  u(α+T)=Δ-βu(η+β),where1<α≤2,β>0,η∈[α-2,α+T-1]ℕα-2:={α-2,α-1,…,α+T-1}, andf:[α-1,α,…,α+T-1]ℕα-1×ℝ→ℝis a continuous function. The existence of at least one solution is proved by using Krasnoselskii's fixed point theorem and Leray-Schauder's nonlinear alternative. Some illustrative examples are also presented.

scholarly journals Positive Solutions for Boundary Value Problem of Nonlinear Fractional -Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Farah M. Al-Askar
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Fractional Boundary Value Problem ◽  
Fractional Difference ◽  
Fractional Difference Equation

We investigate the existence of multiple positive solutions to the nonlinear -fractional boundary value problem , , by using a fixed point theorem in a cone.

scholarly journals Multiple Positive Solutions for a Fractional Boundary Value Problem with Fractional Integral Deviating Argument

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
A. Guezane-Lakoud ◽  
R. Khaldi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Point Theorem ◽  
Fractional Integral ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Fractional Boundary Value Problem ◽  
Deviating Argument

This work is devoted to the existence of positive solutions for a fractional boundary value problem with fractional integral deviating argument. The proofs of the main results are based on Guo-Krasnoselskii fixed point theorem and Avery and Peterson fixed point theorem. Two examples are given to illustrate the obtained results, ending the paper.

scholarly journals Existence of Positive Solutions for m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Wafa M. Shammakh
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential ◽  
The Green Function

We investigate an m-point boundary value problem for nonlinear fractional differential equations. The associated Green function for the boundary value problem is given at first, and some useful properties of the Green function are obtained. By using the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed point theorem, the existence of multiple positive solutions is obtained.

L 1-solutions of the boundary value problem for implicit fractional order differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Benoumran Telli ◽  
Mohammed Said Souid
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Banach Contraction ◽  
The Banach Contraction Principle

Abstract The aim of this paper is to present new results on the existence of solutions for a class of the boundary value problem for fractional order implicit differential equations involving the Caputo fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

scholarly journals Multiple Positive Solutions for Nonlinear Fractional Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Daliang Zhao ◽  
Yansheng Liu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Multiple Positive Solutions ◽  
Fractional Boundary Value Problem ◽  
Krasnoselskii’S Fixed Point Theorem

This paper is devoted to the existence of multiple positive solutions for fractional boundary value problemDC0+αu(t)=f(t,u(t),u′(t)),0<t<1,u(1)=u′(1)=u′′(0)=0, where2<α≤3is a real number,CD0+αis the Caputo fractional derivative, andf:[0,1]×[0,+∞)×R→[0,+∞)is continuous. Firstly, by constructing a special cone, applying Guo-Krasnoselskii’s fixed point theorem and Leggett-Williams fixed point theorem, some new existence criteria for fractional boundary value problem are established; secondly, by applying a new extension of Krasnoselskii’s fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions to the considered boundary value problem from its auxiliary problem. Finally, as applications, some illustrative examples are presented to support the main results.

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