Fractional Calculus and Applied Analysis

, Volume 17, Issue 4, pp 1175–1187 | Cite as

On the existence of blow up solutions for a class of fractional differential equations

  • Zhanbing Bai
  • YangQuan Chen
  • Hairong Lian
  • Sujing Sun
Research Paper

Abstract

In this paper, by using fixed-point theorems, and lower and upper solution method, the existence for a class of fractional initial value problem (FIVP) \(\begin{gathered} D_{0 + }^\alpha u(t) = f(t,u(t)),t \in (0,h), \hfill \\ t^{2 - \alpha } u(t)|_{t = 0} = b_1 D_{0 + }^{\alpha - 1} u(t) = |_{t = 0} = b_2 , \hfill \\ \end{gathered} \) is discussed, where fC([0, hR,R), D0+αu(t) is the standard Riemann-Liouville fractional derivative, 1 < α < 2. Some hidden confusion and fallacy in the literature are commented. A new condition on the nonlinear term is given to guarantee the equivalence between the solution of the FIVP and the fixed-point of the operator. Based on the new condition, some new existence results are obtained and presented as example.

Key Words and Phrases

fractional initial value problem fixed point theorem inequalities existence 

MSC 2010

Primary 34B15 Secondary 34B15, 34A08 

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Copyright information

© Versita Warsaw and Springer-Verlag Wien 2014

Authors and Affiliations

  • Zhanbing Bai
    • 1
  • YangQuan Chen
    • 2
  • Hairong Lian
    • 3
  • Sujing Sun
    • 4
  1. 1.College of Mathematics and System ScienceShandong University of Science and TechnologyQingdaoPR China
  2. 2.School of EngineeringUniversity of CaliforniaMercedUSA
  3. 3.School of ScienceChina University of GeosciencesBeijingPR China
  4. 4.College of Information Science and EngineeringShandong University of Science and TechnologyQingdaoPR China

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