Existence of local solutions for differential equations with arbitrary fractional order. (English) Zbl 1359.34011
Summary: In this paper, we establish sufficient conditions for the existence of local solutions for a class of Cauchy type problems with arbitrary fractional order. The results are established by the application of the contraction mapping principle and Schaefer’s fixed point theorem. An example is provided to illustrate the applicability of the results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
References:
| [1] | Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003 |
| [2] | Alsaedi, A., Ahmad, B., Assolami, A.: On antiperiodic boundary value problems for higher-order fractional differential equations. Abstract and Applied Analysis, (2012), Article ID 325984, p. 15 · Zbl 1251.34009 |
| [3] | Granas A., Dugundji J.: Fixed Point Theory. Springer. New York (2005) · Zbl 1025.47002 |
| [4] | Ahmad B.: Existence of solutions for fractional differential equations of order \[{q \in (2,3]}\] q∈(2,3] with anti periodic conditions. J. Appl. Math. Comput. 24, 822-825 (2011) |
| [5] | Ahmad B., Nieto J.: Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan. J. Math. 15, 981-993 (2011) · Zbl 1270.34034 |
| [6] | Lee C., Chang F.: Fractional-order PID controller optimization via improved electromagnetism-like algorithm. Exp. Syst. Appl. 37, 8871-8878 (2010) · doi:10.1016/j.eswa.2010.06.009 |
| [7] | Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional calculus models and numerical methods. Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2012) · Zbl 1248.26011 |
| [8] | Delbosco D., Rodino L.: Existence and uniqueness for a fractional differential equation. J. Math. Anal. Appl. 204, 609-625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456 |
| [9] | Ahmed E., El-Sayed A., El-Saka H.: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 325, 542-553 (2007) · Zbl 1105.65122 · doi:10.1016/j.jmaa.2006.01.087 |
| [10] | Liu F., Burrage K.: Novel techniques in parameter estimation for fractional dynamical models arising from biological systems. Comput. Math. Appl. 62, 822-833 (2011) · Zbl 1228.93114 · doi:10.1016/j.camwa.2011.03.002 |
| [11] | Meral F., Royston T., Magin R.: Fractional calculus in viscoelasticity: an experimental study. Communications in Nonlinear Science and Numerical Simulation. 15, 939-945 (2010) · Zbl 1221.74012 · doi:10.1016/j.cnsns.2009.05.004 |
| [12] | Mophou G.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61, 68-78 (2011) · Zbl 1207.49006 · doi:10.1016/j.camwa.2010.10.030 |
| [13] | N’guerekata G.M.: A Cauchy problem for some fractional abstract differential equation with nonlocal condition. Nonlinear Anal. 70, 1873-1876 (2009) · Zbl 1166.34320 · doi:10.1016/j.na.2008.02.087 |
| [14] | Sokolov I., Chechkin A., Klafter J.: Fractional diffusion equation for a power-lawtruncated Levy process. Physica A. 336, 245-251 (2004) · doi:10.1016/j.physa.2003.12.044 |
| [15] | Sabatier, J., Agarwal, O.P., Machado, J.A.T.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht Netherlands (2007) · Zbl 0881.34005 |
| [16] | Wang J., Zhou Y., Wei W.: Optimal feedback control for semilinear fractional evolution equations in Banach spaces. Syst. Control Lett. 61, 472-476 (2012) · Zbl 1250.49035 · doi:10.1016/j.sysconle.2011.12.009 |
| [17] | Oldham K.: Ractional differential equations in electrochemistry. Adv. Eng. Softw. 41, 9-12 (2010) · Zbl 1177.78041 · doi:10.1016/j.advengsoft.2008.12.012 |
| [18] | Benchohra M., Hamani S., Ntouyas S.K.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2391-2396 (2009) · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073 |
| [19] | Matar M.: On existence of solution to nonlinear fractional differential equations for \[{0< \alpha \leq 3}0\]<α≤3. Journal of Fractional Calculus and Applications. 3, 1-7 (2011) |
| [20] | Matar, M.: Existence of integral and anti-periodic boundary valued problem of fractional order \[{0< \alpha \leq 3}0\]<α≤3. Bull. Malaysian Math. Sci. Soc. Accepted · Zbl 1372.34017 |
| [21] | Matar M.: On existence of solution to some nonlinear differential equations of fractional order \[{2< \alpha \leq 3}2\]<α≤3. Int. J. Math. Anal. 6(34), 1649-1657 (2012) · Zbl 1267.34010 |
| [22] | Matar, M.:, El-Bohisie, F.A.: On existence of solution for higher-order fractional differential inclusions with anti-periodic type boundary conditions. Br. J. Math. Comput. Sci. doi:10.9734/BJMCS/2015/15367 |
| [23] | Agrawal O.P.: Generalized variational problems and Euler-Lagrange equations. Comput. Math. Appl. 59, 1852-1864 (2010) · Zbl 1189.49029 · doi:10.1016/j.camwa.2009.08.029 |
| [24] | Gorenflo R., Mainardi F.: Some recent advances in theory and simulation of fractional diffusion processes. J. Comput. Appl. Math. 229, 400-415 (2009) · Zbl 1166.45004 · doi:10.1016/j.cam.2008.04.005 |
| [25] | Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publisher. Connecticut, Conn USA (2006) |
| [26] | Nigmatullin R., Omay T., Baleanu D.: On fractional filtering versus conventional filtering in economics. Commun. Nonlinear Sci. Numer. Simul. 15, 979-986 (2010) · Zbl 1221.91040 · doi:10.1016/j.cnsns.2009.05.027 |
| [27] | Agrawal R.P., Benchohra M., Hamani S.: Boundary value problems for fractional differential equations. Georgian Math. J. 16(3), 401-411 (2009) · Zbl 1179.26011 |
| [28] | Agarwal R.P., Benchohra M., Hamani S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109(3), 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6 |
| [29] | Jiang X., Xu M., Qi H.: The fractional diffusion model with an absorption term andmodified Fick’s law for non-local transport processes. Nonlinear Anal. Real World Appl. 11, 262-269 (2010) · Zbl 1196.37120 · doi:10.1016/j.nonrwa.2008.10.057 |
| [30] | Odibat Z.: A note on phase synchronization in coupled chaotic fractional order systems. Nonlinear Anal. Real World Appl. 13, 779-789 (2012) · Zbl 1238.34121 · doi:10.1016/j.nonrwa.2011.08.016 |
| [31] | Xu Y., He Z.: Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition. J. Appl. Math. Comput. 44, 417-435 (2014) · Zbl 1296.34042 · doi:10.1007/s12190-013-0700-2 |
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