The eigenvalue for a class of singular p-laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition

In this article, we consider the following boundary value problem of nonlinear fractional differential equation with p-Laplacian operator: $$\begin{aligned}& D^{\alpha}\bigl(\phi_{p}\bigl(D^{\alpha}u(t) \bigr)\bigr)= f\bigl(t, u(t)\bigr), \quad0< t< 1, \\& u(0)= u(1)= D^{\alpha}u(0)= D^{\alpha}u(1)=0, \end{aligned}$$ where $1<\alpha\leq2$
 is a real number, $D^{\alpha}$
 is the conformable fractional derivative, $\phi_{p}(s)=\vert s\vert ^{p-2}s$
 , $p>1$
 , $\phi_{p}^{-1}=\phi_{q}$
 , $1/p+1/q=1$
 , and $f:[0, 1]\times[0,+\infty)\to[0,+\infty)$
 is continuous. One of the difficulties here is that the corresponding Green’s function $G(t, s)$
 is singular at $s= 0$
 . By the use of an approximation method and fixed point theorems on cone, some existence and multiplicity results of positive solutions are acquired. Some examples are presented to illustrate the main results.

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Positive solutions to boundary value problems of p-Laplacian with fractional derivative

In this article, the author investigates the existence and multiplicity of positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator where and are the standard Riemann-Liouville derivatives with 1 < α ≤ 2, 0 < β ≤ 1, 0 < γ ≤ 1, 0 ≤ α - γ - 1, the constant σ is a positive number and p-Laplacian operator is defined as φ
 p
 (s) = |s|p-2s, p > 1. By means of the fixed point theorem on cones, some existence and multiplicity results of positive solutions are obtained. 2010 Mathematical Subject Classification: 34A08; 34B18.

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Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator

We are concerned with the existence of solutions for the singular fractional boundary value problem $^{c}\kern-1pt D^{\alpha}u = f(t,u)$
 , u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(ℝ∖{0})) and lim
 x→0
 f(t,x)=∞ for all t∈[0,1]. Here, $^{c}\kern-1pt D$
 is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of (0,1), that is, they “pass through” the singularity of f inside of (0,1). The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.

Fractional boundary value problems with singularities in space variables

This paper focuses on the maximum and minimum solutions for a fractional order differential system, involving a p-Laplacian operator and nonlocal boundary conditions, which arises from many complex processes such as ecological economy phenomena and diffusive interaction. By introducing new type growth conditions and using the monotone iterative technique, some new results about the existence of maximal and minimal solutions for a fractional order differential system is established, and the estimation of the lower and upper bounds of the maximum and minimum solutions is also derived. In addition, the iterative schemes starting from some explicit initial values and converging to the exact maximum and minimum solutions are also constructed.

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Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes

The upper and lower solutions method is used to study the p-Laplacian fractional boundary value problem D0

/pdf/upper-and-lower-solutions-method-for-a-class-of-singular-4fww1iesbg.pdf

Hongjun Xiang

Papers

Upper and Lower Solutions Method for a Class of Singular Fractional Boundary Value Problems with p-Laplacian Operator