scholarly journals The existence of a ground state solution for a class of fractional differential equation with p-Laplacian operator

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Zhigang Hu ◽  
Wenbin Liu ◽  
Taiyong Chen
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Laplacian Operator ◽  
Fractional Differential

scholarly journals Ground State Solutions for a Class of Fractional Differential Equations with Dirichlet Boundary Value Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhigang Hu ◽  
Wenbin Liu ◽  
Jiaying Liu
Keyword(s):  
Differential Equation ◽  
Ground State ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Fractional Integrals ◽  
Fractional Differential ◽  
Left And Right ◽  
The Nehari Manifold

In this paper, we apply the method of the Nehari manifold to study the fractional differential equation(d/dt)((1/2) 0Dt-β(u′(t))+(1/2) tDT-β(u′(t)))=  f(t,u(t)), a.e.t∈[0,T],andu0=uT=0,where 0Dt-β, tDT-βare the left and right Riemann-Liouville fractional integrals of order0≤β<1, respectively. We prove the existence of a ground state solution of the boundary value problem.

scholarly journals Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation withp-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Ya-ling Li ◽  
Shi-you Lin
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Continuous Function ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fixed Point Theorems ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential

We study the following nonlinear fractional differential equation involving thep-Laplacian operatorDβφpDαut=ft,ut,1<t<e,u1=u′1=u′e=0,Dαu1=Dαue=0, where the continuous functionf:1,e×0,+∞→[0,+∞),2<α≤3,1<β≤2.Dαdenotes the standard Hadamard fractional derivative of the orderα, the constantp>1, and thep-Laplacian operatorφps=sp-2s. We show some results about the existence and the uniqueness of the positive solution by using fixed point theorems and the properties of Green's function and thep-Laplacian operator.

scholarly journals Eigenvalue of Fractional Differential Equations withp-Laplacian Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Wenquan Wu ◽  
Xiangbing Zhou
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Upper And Lower Solutions ◽  
Laplacian Operator ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

We investigate the existence of positive solutions for the fractional order eigenvalue problem withp-Laplacian operator-𝒟tβ(φp(𝒟tαx))(t)=λf(t,x(t)),  t∈(0,1),  x(0)=0,  𝒟tαx(0)=0,  𝒟tγx(1)=∑j=1m-2‍aj𝒟tγx(ξj), where𝒟tβ,  𝒟tα,  𝒟tγare the standard Riemann-Liouville derivatives andp-Laplacian operator is defined asφp(s)=|s|p-2s,  p>1.f:(0,1)×(0,+∞)→[0,+∞)is continuous andfcan be singular att=0,1andx=0.By constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of fractional differential equation is established.

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