Skip to main content Accessibility help
×
Home

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

  • LIHONG ZHANG (a1), BASHIR AHMAD (a2) and GUOTAO WANG (a3)

Abstract

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.

Copyright

Corresponding author

References

Hide All
[1]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), 235244.
[2]Arara, A., Benchohra, M., Hamidia, N. and Nieto, J. J., ‘Fractional order differential equations on an unbounded domain’, Nonlinear Anal. 72 (2010), 580586.
[3]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).
[4]Chen, F. and Zhou, Y., ‘Attractivity of fractional functional differential equations’, Comput. Math. Appl. 62 (2011), 13591369.
[5]Hatano, Y., Nakagawa, J., Wang, S. and Yamamoto, M., ‘Determination of order in fractional diffusion equation’, J. Math-for-Ind. 5A (2013), 5157.
[6]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), 7281.
[7]Jankowski, T., ‘Fractional equations of Volterra type involving a Riemann–Liouville derivative’, Appl. Math. Lett. 26 (2013), 344350.
[8]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).
[9]Lakshmikantham, V., Leela, S. and Vasundhara Devi, J., Theory of Fractional Dynamic Systems (Cambridge Scientific Publishers, Cambridge, 2009).
[10]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), 13341346.
[11]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), 33433354.
[12]Lin, L., Liu, X. and Fang, H., ‘Method of upper and lower solutions for fractional differential equations’, J. Differential Equations 2012(100) (2012), 13.
[13]Liu, Z., Sun, J. and Szanto, I., ‘Monotone iterative technique for Riemann–Liouville fractional integro-differential equations with advanced arguments’, Results Math. 63 (2013), 12771287.
[14]Liu, S., Wang, G. and Zhang, L., ‘Existence results for a coupled system of nonlinear neutral fractional differential equations’, Appl. Math. Lett. 26 (2013), 11201124.
[15]McRae, F. A., ‘Monotone iterative technique and existence results for fractional differential equations’, Nonlinear Anal. 71 (2009), 60936096.
[16]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), 99108.
[17]Podlubny, I., Fractional Differential Equations (Academic Press, San Diego, CA, 1999).
[18]Ramirez, J. D. and Vatsala, A. S., ‘Monotone iterative technique for fractional differential equations with periodic boundary conditions’, Opuscula Math. 29 (2009), 289304.
[19]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).
[20]Su, X., ‘Solutions to boundary value problem of fractional order on unbounded domains in a Banach space’, Nonlinear Anal. 74 (2011), 28442852.
[21]Su, X. and Zhang, S., ‘Unbounded solutions to a boundary value problem of fractional order on the half-line’, Comput. Math. Appl. 61 (2011), 10791087.
[22]Wang, G., ‘Monotone iterative technique for boundary value problems of a nonlinear fractional differential equations with deviating arguments’, J. Comput. Appl. Math. 236 (2012), 24252430.
[23]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), 10191024.
[24]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.
[25]Wang, G., Baleanu, D. and Zhang, L., ‘Monotone iterative method for a class of nonlinear fractional differential equations’, Fract. Calc. Appl. Anal. 15 (2012), 244252.
[26]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.
[27]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), 167172.
[28]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), 260272.
[29]Zhang, S., ‘Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives’, Nonlinear Anal. 71 (2009), 20872093.
[30]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.
[31]Zhang, X., Liu, L., Wu, Y. and Lu, Y., ‘The iterative solutions of nonlinear fractional differential equations’, Appl. Math. Comput. 219 (2013), 46804691.
[32]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), 5156.
[33]Zhao, X. K. and Ge, W. G., ‘Unbounded solutions for a fractional boundary value problem on the infinite interval’, Acta Appl. Math. 109 (2010), 495505.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed