Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T06:49:17.028Z Has data issue: false hasContentIssue false
  • English
  • Français

SUCCESSIVE ITERATIONS FOR POSITIVE EXTREMAL SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS ON A HALF-LINE

Part of: Boundary value problems General theory in ordinary differential equations

Published online by Cambridge University Press:  27 August 2014

LIHONG ZHANG ,
BASHIR AHMAD  and
GUOTAO WANG
LIHONG ZHANG*
Affiliation:
School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, PR China email [email protected]
BASHIR AHMAD
Affiliation:
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia email [email protected]
GUOTAO WANG
Affiliation:
School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, PR China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, positive solutions of fractional differential equations with nonlinear terms depending on lower-order derivatives on a half-line are investigated. The positive extremal solutions and iterative schemes for approximating them are obtained by applying a monotone iterative method. An example is presented to illustrate the main results.

Keywords

monotone iterative methodfractional differential equationpositive extremal solutionshalf-line

MSC classification

Primary: 34A08: Fractional differential equations
Secondary: 34B10: Nonlocal and multipoint boundary value problems 34B18: Positive solutions of nonlinear boundary value problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 116 - 128
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Agarwal, R. P., Benchohra, M., Hamani, S. and Pinelas, S., ‘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), 235–244.Google Scholar
Arara, A., Benchohra, M., Hamidia, N. and Nieto, J. J., ‘Fractional order differential equations on an unbounded domain’, Nonlinear Anal. 72 (2010), 580–586.CrossRefGoogle Scholar
Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J. J., Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos (World Scientific, Boston, MA, 2012).CrossRefGoogle Scholar
Chen, F. and Zhou, Y., ‘Attractivity of fractional functional differential equations’, Comput. Math. Appl. 62 (2011), 1359–1369.Google Scholar
Hatano, Y., Nakagawa, J., Wang, S. and Yamamoto, M., ‘Determination of order in fractional diffusion equation’, J. Math-for-Ind. 5A (2013), 51–57.Google Scholar
Hu, C., Liu, B. and Xie, S., ‘Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments’, Appl. Math. Comput. 222 (2013), 72–81.Google Scholar
Jankowski, T., ‘Fractional equations of Volterra type involving a Riemann–Liouville derivative’, Appl. Math. Lett. 26 (2013), 344–350.CrossRefGoogle Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (Elsevier Science, Amsterdam, 2006).Google Scholar
Lakshmikantham, V., Leela, S. and Vasundhara Devi, J., Theory of Fractional Dynamic Systems (Cambridge Scientific Publishers, Cambridge, 2009).Google Scholar
Liang, S. and Zhang, J., ‘Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval’, Math. Comput. Modelling 54 (2011), 1334–1346.Google Scholar
Liang, S. and Zhang, J., ‘Existence of three positive solutions for m-point boundary value problems for some nonlinear fractional differential equations on an infinite interval’, Comput. Math. Appl. 61 (2011), 3343–3354.Google Scholar
Lin, L., Liu, X. and Fang, H., ‘Method of upper and lower solutions for fractional differential equations’, J. Differential Equations 2012(100) (2012), 13.Google Scholar
Liu, Z., Sun, J. and Szanto, I., ‘Monotone iterative technique for Riemann–Liouville fractional integro-differential equations with advanced arguments’, Results Math. 63 (2013), 1277–1287.CrossRefGoogle Scholar
Liu, S., Wang, G. and Zhang, L., ‘Existence results for a coupled system of nonlinear neutral fractional differential equations’, Appl. Math. Lett. 26 (2013), 1120–1124.CrossRefGoogle Scholar
McRae, F. A., ‘Monotone iterative technique and existence results for fractional differential equations’, Nonlinear Anal. 71 (2009), 6093–6096.Google Scholar
Nakagawa, J., Sakamoto, K. and Yamamoto, M., ‘Overview to mathematical analysis for fractional diffusion equations—new mathematical aspects motivated by industrial collaboration’, J. Math-for-Ind. 2A (2010), 99–108.Google Scholar
Podlubny, I., Fractional Differential Equations (Academic Press, San Diego, CA, 1999).Google Scholar
Ramirez, J. D. and Vatsala, A. S., ‘Monotone iterative technique for fractional differential equations with periodic boundary conditions’, Opuscula Math. 29 (2009), 289–304.Google Scholar
Sabatier, J., Agrawal, O. P. and Machado, J. A. T., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (Springer, Dordrecht, 2007).Google Scholar
Su, X., ‘Solutions to boundary value problem of fractional order on unbounded domains in a Banach space’, Nonlinear Anal. 74 (2011), 2844–2852.CrossRefGoogle Scholar
Su, X. and Zhang, S., ‘Unbounded solutions to a boundary value problem of fractional order on the half-line’, Comput. Math. Appl. 61 (2011), 1079–1087.Google Scholar
Wang, G., ‘Monotone iterative technique for boundary value problems of a nonlinear fractional differential equations with deviating arguments’, J. Comput. Appl. Math. 236 (2012), 2425–2430.Google Scholar
Wang, G., Agarwal, R. P. and Cabada, A., ‘Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations’, Appl. Math. Lett. 25 (2012), 1019–1024.Google Scholar
Wang, G., Ahmad, B. and Zhang, L., ‘A coupled system of nonlinear fractional differential equations with multi-point fractional boundary conditions on an unbounded domain’, Abstr. Appl. Anal. (2012), Art. ID 248709.Google Scholar
Wang, G., Baleanu, D. and Zhang, L., ‘Monotone iterative method for a class of nonlinear fractional differential equations’, Fract. Calc. Appl. Anal. 15 (2012), 244–252.Google Scholar
Wang, G., Cabada, A. and Zhang, L., ‘An integral boundary value problem for nonlinear differential equations of fractional order on an unbounded domain’, J. Integral Equations Appl. 26(1) (2014), 129.Google Scholar
Wang, G., Liu, S. and Zhang, L., ‘Neutral fractional integro-differential equation with nonlinear term depending on lower order derivative’, J. Comput. Appl. Math. 260 (2014), 167–172.CrossRefGoogle Scholar
Wei, Z., Li, G. and Che, J., ‘Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative’, J. Math. Anal. Appl. 367 (2010), 260–272.CrossRefGoogle Scholar
Zhang, S., ‘Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives’, Nonlinear Anal. 71 (2009), 2087–2093.Google Scholar
Zhang, L., Ahmad, B. and Wang, G. et al. , ‘Nonlocal integrodifferential boundary value problem for nonlinear fractional differential equations on an unbounded domain’, Abstr. Appl. Anal. (2013), Art. ID 813903.Google Scholar
Zhang, X., Liu, L., Wu, Y. and Lu, Y., ‘The iterative solutions of nonlinear fractional differential equations’, Appl. Math. Comput. 219 (2013), 4680–4691.Google Scholar
Zhang, L., Wang, G., Ahmad, B. and Agarwal, R. P., ‘Nonlinear fractional integro-differential equations on unbounded domains in a Banach space’, J. Comput. Appl. Math. 249 (2013), 51–56.CrossRefGoogle Scholar
Zhao, X. K. and Ge, W. G., ‘Unbounded solutions for a fractional boundary value problem on the infinite interval’, Acta Appl. Math. 109 (2010), 495–505.CrossRefGoogle Scholar