Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian

Abstract

In this article, we investigate the Sturm-Liouville boundary value problems of fractional differential equations with p-LaplacianBy means of the properties of the Green's function, Leggett-Williams fixed-point theorems, and fixed-point index theory, several new sufficient conditions for the existence of at least two or at least three positive solutions are obtained. As an application, an example is given to demonstrate the main result.

On Coupledp-Laplacian Fractional Differential Equations with Nonlinear Boundary Conditions

“…Since -Laplacian operators have been greatly applied in the mathematical modeling of large numbers of real world phenomenons devoted to physics, mechanics, dynamical systems, electrodynamics, and so forth, therefore researchers paid much attention to study such type of differential equation dealing with -Laplacian operators from different aspects including existence theory, multiplicity results, and stability analysis. For instance, Lu et al [7] discussed Sturm-Liouville boundary value problems (BVP) of FDEs with -Laplacian operator for existence of two or three positive solutions by using fixed point theory. By applying Leggett-William fixed point theorem, the mentioned author studied the following problem:…”

Multiplicity of positive solutions for coupled system of fractional differential equation with p-Laplacian two-point BVPs

“…Later, these results are further extended to fractional order boundary value problems Ahmad and Nieto [2], Anderson and Avery [3], Bai and Sun [7], Goodrich [8], Rao [13], and Su [18] by applying various fixed point theorems on cones. Recently,researchers are concentrating on the theory of fractional order boundary value problems associated with p-Laplacian operator Lu [11] and Prasad and Krushna [16]. The above papers that are motivated to this work.…”

Existence and iteration of positive solution for fractional integral boundary value problems with p-Laplacian operator

“…This paper studies the existence of extremal solutions for the boundary value problem of a fractional p-Laplacian equation with the following form: The p-Laplacian operator is defined as φ p (t) = |t| p-2 t, p > 1, and (φ p ) -1 = φ q , 1 p + 1 q = 1. Recently, much attention has been paid to the study of the existence of extremal solutions, for fractional differential equations with corresponding initial or boundary conditions; see [1][2][3][4][5][6][7][8][9]. The monotone iterative technique, combined with the method of upper and lower solutions, provides an effective mechanism to prove constructive existence results for nonlinear differential equations, the advantage and importance of the technique needs no special emphasis [10,11].…”