Opuscula Math. 37, no. 3 (2017), 421-434

Opuscula Mathematica

Positive solutions of a singular fractional boundary value problem with a fractional boundary condition

Jeffrey W. Lyons
Jeffrey T. Neugebauer

Abstract. For \(\alpha\in(1,2]\), the singular fractional boundary value problem \[D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,\] satisfying the boundary conditions \(x(0)=D^{\beta}_{0^+}x(1)=0\), where \(\beta\in(0,\alpha-1]\), \(\mu\in(0,\alpha-1]\), and \(D^{\alpha}_{0^+}\), \(D^{\beta}_{0^+}\) and \(D^{\mu}_{0^+}\) are Riemann-Liouville derivatives of order \(\alpha\), \(\beta\) and \(\mu\) respectively, is considered. Here \(f\) satisfies a local Carathéodory condition, and \(f(t,x,y)\) may be singular at the value 0 in its space variable \(x\). Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.

Keywords: fractional differential equation, singular problem, fixed point.

Mathematics Subject Classification: 26A33, 34A08, 34B16.

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  1. R.P. Agarwal, D. O'Regan, S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl. 371 (2010), 57-68.
  2. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010.
  3. P.W. Eloe, J.W. Lyons, J.T. Neugebauer, An ordering on Green's functions for a family of two-point boundary value problems for fractional differential equations, Commun. Appl. Anal. 19 (2015), 453-462.
  4. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
  5. M.A. Krasonsel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Translated by A.H. Armstrong, translation edited by J. Burlak, A Pergamon Press Book, The Macmillan Co., New York, 1964.
  6. H. Mâagli, N. Mhadhebi, N. Zeddini, Existence and estimates of positive solutions for some singular fractional boundary value problems, Abstr. Appl. Anal. (2014), Art. ID 120781.
  7. K.S. Miller, B. Ross, A Introduction to the Fractional Calculus and Fractional Differetial Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1993.
  8. I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solutions and Some of Their Applications, Mathematics in Science and Enginnering, vol. 198, Academic Press, San Diego, 1999.
  9. S. Staněk, The existence of positive solutions of singular fractional boundary value problems, Comput. Math. Appl. 62 (2011), 1379-1388.
  10. X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal. 71 (2009), 4676-4688.
  11. C. Yuan, D. Jiang, X. Xu, Singular positone and semipositone boundary value problems of nonlinear fractional differential equations, Math. Probl. Eng. (2009), Art. ID 535209.
  12. X. Zhang, C. Mao, Y. Wu, H. Su, Positive solutions of a singular nonlocal fractional order differetial system via Schauder's fixed point theorem, Abstr. Appl. Anal. (2014), Art. ID 457965.
  • Jeffrey W. Lyons
  • Nova Southeastern University, Department of Mathematics, Fort Lauderdale, FL 33314 USA
  • Jeffrey T. Neugebauer
  • Eastern Kentucky University, Department of Mathematics and Statistics, Richmond, KY 40475 USA
  • Communicated by Theodore A. Burton.
  • Received: 2016-08-04.
  • Revised: 2016-10-29.
  • Accepted: 2016-10-29.
  • Published online: 2017-01-30.
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Cite this article as:
Jeffrey W. Lyons, Jeffrey T. Neugebauer, Positive solutions of a singular fractional boundary value problem with a fractional boundary condition, Opuscula Math. 37, no. 3 (2017), 421-434, http://dx.doi.org/10.7494/OpMath.2017.37.3.421

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