scholarly journals Positive Solutions for Multipoint Boundary Value Problem of Fractional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Wenyong Zhong
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions ◽  
Multipoint Boundary Value Problem

We study the existence and multiplicity of positive solutions for the fractionalm-point boundary value problemD0+αu(t)+f(t,u(t))=0,0<t<1,u(0)=u'(0)=0,u'(1)=∑i=1m-2aiu'(ξi), where2<α<3,D0+αis the standard Riemann-Liouville fractional derivative, andf:[0,1]×[0,∞)↦[0,∞)is continuous. Here,ai⩾0fori=1,…,m-2,0<ξ1<ξ2<⋯<ξm-2<1, andρ=∑i=1m-2aiξiα-2withρ<1. In light of some fixed point theorems, some existence and multiplicity results of positive solutions are obtained.

scholarly journals Positive Solutions to a Fractional-Order Two-Point Boundary Value Problem with p-Laplacian Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Xiangshan Kong ◽  
Haitao Li
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Operator Equation ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Laplacian Operator ◽  
Point Boundary ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

This paper systematically investigates positive solutions to a kind of two-point boundary value problem (BVP) for nonlinear fractional differential equations with p-Laplacian operator and presents a number of new results. First, the considered BVP is converted to an operator equation by using the property of the Caputo derivative. Second, based on the operator equation and some fixed point theorems, several sufficient conditions are presented for the nonexistence, the uniqueness, and the multiplicity of positive solutions. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are effective.

scholarly journals Monotone and Concave Positive Solutions to Three-Point Boundary Value Problems of Higher-Order Fractional Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Wenyong Zhong ◽  
Lanfang Wang
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Higher Order ◽  
Point Boundary ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Fractional Differential

We study the three-point boundary value problem of higher-order fractional differential equations of the formDc0+ρut+ft, ut=0,0<t<1,2⩽n-1<ρ<n,u′(0)=u′′(0)=⋯=un-1(0)=0,u(1)+pu′(1)=qu′(ξ), where cD0+ρis the Caputo fractional derivative of orderρ, and the functionf:[0,1]×[0,∞)↦[0,+∞)is continuously differentiable. Here,0⩽q⩽p,0<ξ<1,2⩽n-1<ρ<n. By virtue of some fixed point theorems, some sufficient criteria for the existence and multiplicity results of positive solutions are established and the obtained results also guarantee that the positive solutions discussed are monotone and concave.

scholarly journals Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 107
Author(s):  
Daliang Zhao ◽  
Juan Mao
Keyword(s):  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Integral Boundary ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

scholarly journals Positive Solutions for Nonlinear Fractional Differential Equations with Boundary Conditions Involving Riemann-Stieltjes Integrals

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Jiqiang Jiang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Integral Boundary ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
Integral Boundary Value Problems

We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existence and multiplicity results of positive solutions are obtained. The results obtained in this paper improve and generalize some well-known results.

scholarly journals Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth-Order Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Main Tool ◽  
Bifurcation Theorem ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

we show the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problemΔ4ut-2=λhtfut,t∈T2,u1=uT+1=Δ2u0=Δ2uT=0, whereλ>0,h:T2→(0,∞)is continuous, andf:R→[0,∞)is continuous,T>4,T2=2,3,…,T. The main tool is the Dancer's global bifurcation theorem.

scholarly journals Existence of multiple positive solutions for semipositone fractional boundary value problems

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 749-759 ◽  
Author(s):  
Şerife Ege ◽  
Fatma Topal
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions ◽  
Fractional Boundary Value Problems

In this paper, we study the existence and multiplicity of positive solutions to the four-point boundary value problems of nonlinear semipositone fractional differential equations. Our results extend some recent works in the literature.

scholarly journals Positive Solutions for a Mixed-Order Three-Point Boundary Value Problem forp-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Francisco J. Torres
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Main Tool ◽  
Index Theory ◽  
Laplacian Operator ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions

The author investigates the existence and multiplicity of positive solutions for boundary value problem of fractional differential equation withp-Laplacian operator. The main tool is fixed point index theory and Leggett-Williams fixed point theorem.

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