Investigating a Class of Nonlinear Fractional Differential Equations and Its Hyers-Ulam Stability by Means of Topological Degree Theory

References

• Samko, S. G., Kilbas, A. A., Marichev, O. I. (1993). Fractional Integrals and Derivatives (Theory and Applications). Switzerland: Gordon and Breach.
   Google Scholar
• Miller, K. S., Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley.
   Google Scholar
• Podlubny, I. (1999). Fractional Differential Equations, Mathematics in Science and Engineering. New York: Academic Press.
   Google Scholar
• Lakshmikantham, V., Leela, S., Vasundhara, J. (2009). Theory of Fractional Dynamic Systems. Cambridge, UK: Cambridge Academic Publishers.
   Google Scholar
• Hilfer, R. (2000). Applications of Fractional Calculus in Physics. Singapore: World Scientific.
   Google Scholar
• Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier.
   Google Scholar
• Wang, J., Xiang, H., Liu, Z. (2010). Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. 2010:12. Article ID 186928. DOI: 10.1155/2010/186928.
   Google Scholar
• Diethelm, K., Ford, N. J. (2002). Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2):229–248. DOI: 10.1006/jmaa.2000.7194.
   Web of Science ®Google Scholar
• Ahmad, B., Nieto, J. J. (2010). Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35:295–304.
   Web of Science ®Google Scholar
• Yang, W. (2012). Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Comput. Math. Appl. 63(1):288–297. DOI: 10.1016/j.camwa.2011.11.021.
   Web of Science ®Google Scholar
• Zhang, S. (2010). Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 59(3):1300–1309. DOI: 10.1016/j.camwa.2009.06.034.
   Web of Science ®Google Scholar
• Bai, Z., Wang, Y., Ge, W. (2004). Triple posive solutions for a class of two-point boundary value problem. Electron. J. Qual. Theory Differ. Equ. 6:1–8. DOI: 10.14232/ejqtde.2008.1.24.
   Google Scholar
• Shah, K. (2016). Multipoint boundary value problems for systems of fractional differential equations: existence theory and numerical simulations. Ph.D. dissertation. University of Malakand, Pakistan.
   Google Scholar
• Goodrich, S. C. (2010). Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23(9):1050–1055. DOI: 10.1016/j.aml.2010.04.035.
   Web of Science ®Google Scholar
• Khan, R. A., Rehman, M., Henderson, J. (2011). Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions. Fract. Differ. Cal. 1(1):29–43. DOI: 10.7153/fdc-01-02.
   Google Scholar
• Bai, Z. (2010). On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal: Theo. Meth. Appl. 72(2):916–924. DOI: 10.1016/j.na.2009.07.033.
   Web of Science ®Google Scholar
• Lv, L., Wang, J., Wei, W. (2011). Existence and uniqueness results for fractional differential equations with boundary value conditions. Opuscula Math. 31(4):629–643. DOI: 10.7494/OpMath.2011.31.4.629.
   Google Scholar
• Zhao, Y., Sun, S., Han, Z., Li, Q. (2011). The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16(4):2086–2097. DOI: 10.1016/j.cnsns.2010.08.017.
   Web of Science ®Google Scholar
• Agarwal, R. P., Benchohra, M., Hamani, S. (2010). A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109(3):973–1033. DOI: 10.1007/s10440-008-9356-6.
   Web of Science ®Google Scholar
• Khan, R. A., Shah, K. (2015). Existence and uniqueness of solutions to fractional order multi-point boundary value problems. Commun. Appl. Anal. 19:515–526.
   Google Scholar
• Mawhin, J. (1979). Topological degree methods in nonlinear boundary value problems. CMBS Regional Conference Series in Mathematics, Proceedings of the American Mathematical Society, Vol. 40.
   Google Scholar
• Isaia, F. (2006). On a nonlinear integral equation without compactness. Acta Math. Univ. Comen. 75:233–240.
   Google Scholar
• Wang, J., Zhou, Y., Wei, W. (2012). Study in fractional differential equations by means of topological degree methods. Numer. Funct. Anal. Optim. 33(2):216–238. DOI: 10.1080/01630563.2011.631069.
   Web of Science ®Google Scholar
• Shah, K., Ali, A., Khan, R. A. (2016). Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems. Bound. Value Probl. 2016:1–12.
   Web of Science ®Google Scholar
• Trigeassou, J. C., Maamri, N., Sabatier, J., Oustaloup, A. (2011). A lyapunov approach to the stability of fractional differential equations. Signal Process. 91(3):437–445. DOI: 10.1016/j.sigpro.2010.04.024.
   Web of Science ®Google Scholar
• Stamova, I. (2015). Mittag-Leffler stability of impulsive differential equations of fractional order. Quart. Appl. Math. 73(3):525–535. DOI: 10.1090/qam/1394.
   Web of Science ®Google Scholar
• Gao, L., Wang, D., Wang, G. (2015). Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects. Appl. Math. Comput. 268:186–200. DOI: 10.1016/j.amc.2015.06.023.
   Web of Science ®Google Scholar
• Ulam, S. M. (1960). A Collection of the Mathematical Problems. New York, USA: Interscience Publishers.
   Google Scholar
• Hyers, D. H. (1941). On the stability of the linear functional equations. Proc. Natl. Acad. Sci. USA. 27(4):222–224. DOI: 10.1073/pnas.27.4.222.
   PubMedGoogle Scholar
• Rassias, T. M. (1978). On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72:297–300, DOI: 10.2307/2042795.
   Google Scholar
• Ger, R., Semrl, P. (1996). The stability of the exponential equation. Proc. Am. Math. Soc. 124:779–787. DOI: 10.1090/S0002-9939-96-03031-6.
   Google Scholar
• Jung, S. M. (2004). Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17(10):1135–1140. DOI: 10.1016/j.aml.2003.11.004.
   Web of Science ®Google Scholar
• Rassias, T. M. (2000). On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 62:23–130. DOI: 10.1023/A:1006499223572.
   Web of Science ®Google Scholar
• Castro, L. P., Ramos, A. (2009). Hyers-Ulam-Rassias stability for a class of nonlinear volterra integral equations. Banach J. Math. Anal. 3(1):36–43. DOI: 10.15352/bjma/1240336421.
   Web of Science ®Google Scholar
• Wang, J., Fec˘kan, M., Zhou, Y. (2012). Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395(1):258–264. DOI: 10.1016/j.jmaa.2012.05.040.
   Web of Science ®Google Scholar
• Wang, J., Zhou, Y., Fec˘kan, M. (2012). Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64(10):3389–3405. DOI: 10.1016/j.camwa.2012.02.021.
   Web of Science ®Google Scholar
• Wang, J., Li, X. (2016). A uniformed method to Ulam-Hyers stability for some linear fractional equations. Mediterr. J. Math. 13(2):625–635. DOI: 10.1007/s00009-015-0523-5.
   Web of Science ®Google Scholar
• Wang, J., Fečkan, M., Tian, Y. (2017). Stability analysis for a general class of non-instantaneous impulsive differential equations. Mediterr. J. Math. 14:1–21.
   Web of Science ®Google Scholar
• Castro, L. P., Simões, A. M. (2018). Hyers-Ulam-Rassias stability of nonlinear integral equations through the bielecki metric. Math. Method Appl. Sci. 41(17):7367–7383. DOI: 10.1002/mma.4857.
   Web of Science ®Google Scholar
• Timoshenko, S. P. (1961). Theory of Elastic Stability. New York: McGraw-Hill.
   Google Scholar
• Soedel, W. (1993). Vibrations of Shells and Plates. New York: Dekker.
   Google Scholar
• Usmani, R. A. (1978). Discrete variable methods for a boundary value problem with engineering applications. Math. Comput. 32(144):1087–1096. DOI: 10.1090/S0025-5718-1978-0483496-5.
   Web of Science ®Google Scholar
• Ma, T. F. (2001). Boundary stabilization for a non-linear beam on elastic bearings. Math. Methods Appl. Sci. 24:583–594.
   Web of Science ®Google Scholar