Basic results on nonlinear eigenvalue problems of fractional order

Mohammed Al-Refai

Research output: Contribution to journalArticlepeer-review

29 Citations (Scopus)

Abstract

In this article, we discuss the basic theory of boundary-value problems of fractional order 1 < δ < 2 involving the Caputo derivative. By applying the maximum principle, we obtain necessary conditions for the existence of eigenfunctions, and show analytical lower and upper bounds estimates of the eigenvalues. Also we obtain a sufficient condition for the non existence of ordered solutions, by transforming the problem into equivalent integrodifferential equation. By the method of lower and upper solution, we obtain a general existence and uniqueness result: We generate two well defined monotone sequences of lower and upper solutions which converge uniformly to the actual solution of the problem. While some fundamental results are obtained, we leave others as open problems stated in a conjecture.

Original languageEnglish
JournalElectronic Journal of Differential Equations
Volume2012
Publication statusPublished - Oct 31 2012

Keywords

  • Boundary-value problems
  • Caputo fractional derivative
  • Fractional differential equations
  • Lower and upper solutions
  • Maximum principle

ASJC Scopus subject areas

  • Analysis

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