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.

Existence of positive solutions for a class of fractional differential equation systems with multi-point boundary conditions

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Hussian Akrami ◽  
Gholam Hussian Erjaee
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Existence Of Solutions ◽  
Point Boundary ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
The Fixed Point Theorem

AbstractIn this article, we study the existence of positive solutions of a multi-point boundary value problem for some system of fractional differential equations. The fixed point theorem on cones will be applied to demonstrate the existence of solutions for this system. At the end, an example shows the application of the main results.

scholarly journals A Study of Nonlinear Boundary Value Problem

2021 ◽  
Author(s):  
Noureddine Bouteraa ◽  
Habib Djourdem
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Higher Order ◽  
Nonlinear Boundary Value Problem ◽  
Point Boundary ◽  
Error Estimations ◽  
Fractional Differential

In this chapter, firstly we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-order fractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions and the error estimations are also given. Secondly, we cover the multi-valued case of our problem. We investigate it for nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler. Two illustrative examples are presented at the end to illustrate the validity of our results.

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 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-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal control

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu Song ◽  
Lingling Zhang ◽  
Bibo Zhou ◽  
Nan Zhang
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Mixed Monotone Operator ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential

Abstract In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of φ-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main 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.

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.

The Existence of Positive Solutions for Boundary Value Problems of Fractional Differential Equation

2014 ◽  
Vol 711 ◽  
pp. 303-307 ◽  
Author(s):  
Jie Gao
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Boundary Value ◽  
Point Boundary ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

In this paper, by using Leggett-Williams fixed point theorem, we will study the existence of positive solutions for a class of multi-point boundary value problems of fractional differential equation on infinite interval.

scholarly journals Twin positive solutions for three-point boundary value problems of higher-order differential equations

2004 ◽  
Vol 2004 (39) ◽  
pp. 2049-2063
Author(s):  
Yuji Liu ◽  
Weigao Ge
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Higher Order ◽  
Point Boundary ◽  
Higher Order Differential Equations

A new fixed point theorem on cones is applied to obtain the existence of at least two positive solutions of a higher-order three-point boundary value problem for the differential equation subject to a class ofboundary value conditions. The associated Green's function is given. Some results obtained recently are generalized.

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