Abstract
We consider the following boundary-value problem of nonlinear fractional differential equation withp-Laplacian operatorD0+β(ϕp(D0+αu(t)))+a(t)f(u)=0,0<t<1,u(0)=γu(h)+λ,u′(0)=μ,ϕp(D0+αu(0))=(ϕp(D0+αu(1)))′=(ϕp(D0+αu(0)))′′=(ϕp(D0+αu(0)))′′′=0, where1<α⩽2,3<β⩽4are real numbers,D0+α,D0+βare the standard Caputo fractional derivatives,ϕp(s)=|s|p-2s,p>1,ϕp-1=ϕq,1/p+1/q=1,0⩽γ<1,0⩽h⩽1,λ,μ>0are parameters,a:(0,1)→[0,+∞),andf:[0,+∞)→[0,+∞)are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parametersλandμare obtained. The uniqueness of positive solution on the parametersλandμis also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.
Highlights
In 1695, L’Hopital asked Leibniz: what if the order of the derivative is 1/2? To which Leibniz considered in a useful means, it follows that will be equal to x√dx : x, an obvious paradox
There have been some papers dealing with the existence and multiplicity of solutions of nonlinear initial fractional differential equations by the use of techniques of nonlinear analysis [10,11,12,13,14,15,16,17,18,19,20,21], upper and lower solutions method [22,23,24], fixed point index [25, 26], coincidence theory [27], Banach contraction mapping principle [28], and so forth
The fractional differential equation boundaryvalue problems have been studied by several authors, very little is known in the literature on the existence and nonexistence of positive solutions of fractional differential equation boundary-value problems with p-Laplacian operator when a parameter λ is involved in the boundary conditions
Summary
In 1695, L’Hopital asked Leibniz: what if the order of the derivative is 1/2? To which Leibniz considered in a useful means, it follows that will be equal to x√dx : x, an obvious paradox. Chai [11] investigated the existence and multiplicity of positive solutions for a class of boundary-value problem of fractional differential equation with p-Laplacian operator. The fractional differential equation boundaryvalue problems have been studied by several authors, very little is known in the literature on the existence and nonexistence of positive solutions of fractional differential equation boundary-value problems with p-Laplacian operator when a parameter λ is involved in the boundary conditions. There is very little known about the uniqueness of the solution of fractional differential equation boundaryvalue problems with p-Laplacian operator on the parameter λ. Han et al [29] studied the existence and uniqueness of positive solutions for the fractional differential equation with p-Laplacian operator. The uniqueness of positive solution on the parameters λ and μ, is studied
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