Opuscula Math. 37, no. 2 (2017), 265-280
Fractional boundary value problems on the half line
Abstract. In this paper, we focus on the solvability of a fractional boundary value problem at resonance on an unbounded interval. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin. The obtained results are illustrated by an example.
Keywords: boundary value problem at resonance, existence of solution, unbounded interval, coincidence degree of Mawhin, fractional differential equation.
Mathematics Subject Classification: 34B40, 34B15.
- R.P. Agarwal, M. Benchohra, S. Hamani, S. Pinelas, Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half line, Dyn. Contin. Discrete Impuls. Syst. Ser. A. Math. Anal. 18 (2011) 2, 235-244.
- R.P. Agarwal, D. O'Regan, Infinity Interval Problems for Difference and Integral Equations, Kluwer Academic Publisher Dordrecht, 2001.
- R.P. Agarwal, D. O'Regan, Infinite interval problems modeling phenomena which arise in the theory of plasma and electrical potential theory, Stud. Appl. Math. 111 (2003) 3, 339-358.
- A. Arara, M. Benchohra, N. Hamidi, J.J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Anal. 72 (2010), 580-586.
- Y. Chen, X. Tang, Positive solutions of fractional differential equations at resonance on the half-line, Boundary Value Problems (2012) 2012:64, 13 pp.
- C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973.
- Y. Cui, Solvability of second-order boundary-value problems at resonance involving integral conditions, Electron. J. Differ. Equ. (2012) 2012:45, 9 pp.
- W. Feng, J.R.L. Webb, Solvability of three-point boundary value problems at resonance, Nonlinear Anal. Theory, Methods and Appl. 30 (1997), 3227-3238.
- D. Franco, G. Infante, M. Zima, Second order nonlocal boundary value problems at resonance, Math. Nachr. 284 (2011) 7, 875-884.
- A. Guezane-Lakoud, A. Kilickman, Unbounded solution for a fractional boundary value problem, Advances in Difference Equations (2014) 2014:154.
- C.P. Gupta, S.K. Ntouyas, P.Ch. Tsamatos, On an \(m\)-point boundary-value problem for second-order ordinary differential equations, Nonlinear Anal. 23 (1994), 1427-1436.
- A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, vol. 204, 2006.
- N. Kosmatov, Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear Anal. 68 (2008) 8, 2158-2171.
- R. Ma, Existence of positive solutions for second-order boundary value problems on infinity intervals, Appl. Math. Lett. 16 (2003) 1, 33-39.
- J. Mawhin, Topological degree methods in nonlinear boundary value problems, NSFCBMS Regional Conference Series in Mathematics, Am. Math. Soc, Providence, 1979.
- D. O'Regan, B. Yan, R.P. Agarwal, Solutions in weighted spaces of singular boundary value problems on the half-line, J. Comput. Appl. Math. 205 (2007), 751-763.
- X. Su, S. Zhang, Unbounded solutions to a boundary value problem of fractional order on the half-line, Comput. Math. Appl. 61 (2011), 1079-1087.
- Communicated by Jean Mawhin.
- Received: 2016-03-11.
- Revised: 2016-05-13.
- Accepted: 2016-06-02.
- Published online: 2017-01-03.