Existence and approximation of solutions of fractional order iterative differential equations

• Authors: JianHua Deng , JinRong Wang
• Languages of publication: EN

Abstracts

EN

In this paper, we investigate existence and approximation of solutions of fractional order iterative differential equations by virtue of nonexpansive mappings, fractional calculus and fixed point methods. Three existence theorems as well as convergence theorems for a fixed point iterative method designed to approximate these solutions are obtained in two different work spaces via Chebyshev’s norm, Bielecki’s norm and β norm. Finally, an example is given to illustrate the obtained results.

Keywords

EN

fractional order; iterative differential equations; existence; approximation; nonexpansive mappings

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 10
• Pages: 1377-1386

Dates

published

1 - 10 - 2013

online

19 - 12 - 2013

Contributors

author

JianHua Deng

• Department of Mathematics, Guizhou University Guiyang, Guizhou, 550025, P.R. China

author

JinRong Wang

References

• [1] C. T. Kelley, Iterative methods for linear and nonlinear equations (Society for Industrial and Applied Mathematics, Philadelphia, 1995) http://dx.doi.org/10.1137/1.9781611970944[Crossref]
• [2] K. Wang, Funkcialaj Ekvacioj 33, 405 (1990)
• [3] M. Medveď, Ann. Polonici Math. LIV. 3, 263 (1991)
• [4] M. Fečkan, Math. Slovaca 43, 39 (1993)
• [5] J. Si, Xi. Wang, J. Math. Anal. Appl. 226, 377 (1998) http://dx.doi.org/10.1006/jmaa.1998.6086[Crossref]
• [6] S. Stanek, Funct. Diff. Eqs. 5, 463 (1998)
• [7] J. Liu, Results Math. 55, 129 (2009) http://dx.doi.org/10.1007/s00025-009-0388-7[Crossref]
• [8] E. Egri, I. A. Rus, Mathematica 52, 67 (2007)
• [9] V. Muresan, Novi Sad J. Math. 33, 1 (2003)
• [10] V. Berinde, Miskolc Math. Notes 11, 13 (2010)
• [11] M. Lauran, Filomat, 25, 21 (2011) http://dx.doi.org/10.2298/FIL1102021L[Crossref]
• [12] D. Baleanu, J. A. T. Machado, A.C.-J. Luo, Fractional dynamics and control (Springer, Berlin, 2012) http://dx.doi.org/10.1007/978-1-4614-0457-6[Crossref]
• [13] K. Diethelm, Lect. Notes Math. 2004 (2010)
• [14] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations (Elsevier Science B.V., Amsterdam, 2006)
• [15] V. Lakshmikantham, S. Leela, J. V. Devi, Theory of fractional dynamic systems (Cambridge Scientific Publishers, Cambridge, 2009)
• [16] K. S. Miller, B. Ross, An introduction to the fractional calculus and differential equations (John Wiley, New York, 1993)
• [17] M. W. Michalski, Derivatives of noninteger order and their applications, Dissertationes Mathematicae, CCCXXVIII (Inst. Math., Polish Acad. Sci., Warsaw, 1993)
• [18] I. Podlubny, Fractional differential equations (Academic Press, New York, 1999)
• [19] V. E. Tarasov, Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media (Springer, HEP, New York, 2011)
• [20] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus models and numerical methods, Series on complexity, nonlinearity and chaos (World Scientific, Singapore, 2012)
• [21] S. Abbas, M. Benchohra, G. M. N’Guérékata, Topics in fractional differential equations (Springer, 2012) http://dx.doi.org/10.1007/978-1-4614-4036-9[Crossref]
• [22] J. Klafter, S. C. Lim, R. Metzler, Fractional dynamics in physics: Recent advances (World Scientific, Singapore, 2012)
• [23] A. Debbouche, D. Baleanu, R. P. Agarwal, Bound. Value Probl. 2012, 78 (2012) http://dx.doi.org/10.1186/1687-2770-2012-78[Crossref]
• [24] D. Baleanu, O. G. Mustafa, R. P. Agarwal, Abstr. Appl. Anal. 2010, 865139 (2010)
• [25] D. Baleanu, O. G. Mustafa, R. P. Agarwal, Comput. Math. Appl. 62, 1492 (2011) http://dx.doi.org/10.1016/j.camwa.2011.03.021[Crossref]
• [26] D. Baleanu, O. G. Mustafa, R. P. Agarwal, Appl. Math. Lett. 23, 1129 (2010) http://dx.doi.org/10.1016/j.aml.2010.04.049[Crossref]
• [27] B. Ahmad, J. J. Nieto, Topol. Methods Nonlinear Anal. 35, 295 (2010)
• [28] Z. Bai, Nonlinear Anal.:TMA 72, 916 (2010) http://dx.doi.org/10.1016/j.na.2009.07.033[Crossref]
• [29] Y. K. Chang, J. J. Nieto, Math. Comput. Model. 49, 605 (2009) http://dx.doi.org/10.1016/j.mcm.2008.03.014[Crossref]
• [30] Y. Zhou, F. Jiao, Nonlinear Analysis:RWA 11, 4465 (2010) http://dx.doi.org/10.1016/j.nonrwa.2010.05.029[Crossref]
• [31] E. Hernández, D. O’Regan, K. Balachandran, Nonlinear Anal.:TMA 73, 3462 (2010) http://dx.doi.org/10.1016/j.na.2010.07.035[Crossref]
• [32] M. Fečkan, Y. Zhou, J. Wang, Commun. Nonlinear Sci. Numer. Simulat. 17, 3050 (2012) http://dx.doi.org/10.1016/j.cnsns.2011.11.017[Crossref]
• [33] J. Wang, M. Fečkan, Y. Zhou, Appl. Math. Model. 37, 6055 (2013) http://dx.doi.org/10.1016/j.apm.2012.12.011[Crossref]
• [34] J. Wang, M. Fečkan, Y. Zhou, J. Optim. Theory Appl. 156, 13 (2013) http://dx.doi.org/10.1007/s10957-012-0170-y[Crossref]
• [35] T. F. Nonnenmacher, W. G. Glockle, Philosophical Magazine Letters 64, 89 (1991) http://dx.doi.org/10.1080/09500839108214672[Crossref]
• [36] H. Schiessel, A. Blumen, J. Physics A: Mathematical and General 26, 5057 (1993) http://dx.doi.org/10.1088/0305-4470/26/19/034[Crossref]
• [37] T. F. Nonnenmacher, R. Metzler, Applications of fractional calculus techniques to problems in biophysics, Applications of Fractional Calculus in Physics (World Scientific Publishing, Singapore, 2000) 377 http://dx.doi.org/10.1142/9789812817747_0008[Crossref]
• [38] R. Metzler, T. F. Nonnenmacher, Int. J. Plasticity 19, 941 (2003) http://dx.doi.org/10.1016/S0749-6419(02)00087-6[Crossref]
• [39] S. M. Jung, T. S. Kim, K. S. Lee, Bull. Korean Math. Soc. 43, 531 (2006) http://dx.doi.org/10.4134/BKMS.2006.43.3.531[Crossref]

Identifiers

DOI

10.2478/s11534-013-0270-9