Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations

oleh: Mujeeb Ur Rehman, Rahmat Ali Khan

Format: Article
Diterbitkan: Wiley 2010-01-01

Deskripsi

We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type π‘π’Ÿπ›Ώ0+𝑒(𝑑)+𝑓(𝑑,𝑒(𝑑))=0, π‘‘βˆˆ(0,1), 0<𝑑<1. 𝑒(1)=𝛽𝑒(πœ‚)+πœ†2, π‘’ξ…ž(0)=π›Όπ‘’ξ…ž(πœ‚)βˆ’πœ†1, π‘’ξ…Ÿ(0)=0, 𝑒(0)=0⋯𝑒(π‘›βˆ’1)(0)=0, where, π‘›βˆ’1<𝛿<𝑛, 𝑛(β‰₯3)βˆˆβ„•, 0<πœ‚,𝛼,𝛽<1, the boundary parameters πœ†1,πœ†2βˆˆβ„+ and 𝑐𝐷𝛿0+ is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.