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

• Published in: Boundary value problems Vol. 2012; no. 1; pp. 1 - 20
• Main Author: Chai, Guoqing
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 15.02.2012; Hindawi Limited; BioMed Central Ltd
• Subjects: Analysis; Approximations and Expansions; Boundary value problems; Classification; Derivatives; Difference and Functional Equations; Differential equations; Existence theorems; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators; Ordinary Differential Equations; Partial Differential Equations; positive solution; fixed point index; fractional differential equations; multiplicity of solutions; Laplacian operator
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/1687-2770-2012-18

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 D 0 + β φ p D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) + σ D 0 + γ u ( 1 ) = 0 , D 0 + α u ( 0 ) = 0 , where D 0 + β , D 0 + α and D 0 + γ 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 -2 s , 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.