Nonlocal initial and boundary value problems via fractional calculus with exponential singular kernel

Ntouyas, Sotiris K. ¹ ; Tariboon, Jessada ² ; Sawaddee, Chalong ³

¹ Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece;Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
² Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand;Centre of Excellence in Mathematics, CHE, Ayutthaya Rd., Bangkok 10400, Thailand
³ Department of Applied Mathematics and Statistics, Faculty of Sciences and Liberal Arts, Rajamangala University of Technology Isan, Nakhon Ratchasima 30000, Thailand

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 9, p. 1015-1030.

Résumé

In this paper, we investigate the existence and uniqueness of solutions for nonlocal initial and boundary value problems of exponential fractional differential equations, by applying standard fixed point theorems. Enlightening examples are also presented.

Détail

DOI : 10.22436/jnsa.011.09.01

Classification : 34A08, 34A12, 34B15
Keywords: Exponential fractional integral, exponential fractional derivative, nonlocal initial value problems, nonlocal boundary value problems, fixed point theorems

Bibliographie

[1] Agarwal, R. P.; Zhou, Y.; He, Y. Y. Existence of fractional neutral functional differential equations, Comput. Math. Appl., Volume 59 (2010), pp. 1095-1100 | DOI

[2] Agarwal, R. P.; Zhou, Y.; Wang, J. R.; Luo, X. N. Fractional functional differential equations with causal operators in Banach spaces, Math. Comput. Modelling, Volume 54 (2011), pp. 1440-1452 | DOI | Zbl

[3] Ahmad, B.; Ntouyas, S. K. Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions, Electron. J. Differential Equations, Volume 2012 (2012), pp. 1-22 | Zbl

[4] Ahmad, B.; S. K. Ntouyas Integro-differential equations of fractional order with nonlocal fractional boundary conditions associated with financial asset model , Electron. J. Differential Equations, Volume 2013 (2013), pp. 1-10 | Zbl

[5] Ahmad, B.; Ntouyas, S. K.; A. Alsaedi New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Difference Equ., Volume 2011 (2011), pp. 1-11 | DOI

[6] Ahmad, B.; Sivasundaram, S. On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order, Appl. Math. Comput., Volume 217 (2010), pp. 480-487 | DOI | Zbl

[7] Anderson, D. R.; Boucherif, A. Nonlocal initial value problem for first-order dynamic equations on time scales, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., Volume 16 (2009), pp. 222-226 | Zbl

[8] Boucherif, A.; Precup, R. On the nonlocal initial value problem for first order differential equations , Fixed Point Theory, Volume 4 (2003), pp. 205-212

[9] Boucherif, A.; Ntouyas, S. K. Nonlocal initial value problems for first order fractional differential equations, Dynam. Systems Appl., Volume 20 (2011), pp. 247-259

[10] Diethelm, K. The Analysis of Fractional Differential Equations, Springer-verlag, Berlin, 2010 | DOI

[11] A. M. A. El-Sayed Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal., Volume 33 (1998), pp. 181-186 | DOI

[12] Granas, A.; Dugundji, J. Fixed Point Theory , Springer-Verlag, New York, 2003 | DOI

[13] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006

[14] Krasnoselskii, M. A. Two remarks on the method of successive approximations, Uspekhi Mat. Nauk, Volume 10 (1955), pp. 123-127

[15] Lakshmikantham, V.; Leela, S.; J. V. Devi Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009

[16] Miller, K. S.; Ross, B. An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York, 1993

[17] Niyom, S.; Ntouyas, S. K.; Laoprasittichok, S.; Tariboon, J. Boundary value problems with four orders of Riemann-Liouville fractional derivatives , Adv. Difference Equ., Volume 2016 (2016), pp. 1-14 | DOI

[18] Ntouyas, S. K.; Tariboon, J.; Thaiprayoon, C. Nonlocal boundary value problems for Riemann-Liouville fractional differential inclusions with Hadamard fractional integral boundary conditions, Taiwanese J. Math., Volume 20 (2016), pp. 91-107 | Zbl | DOI

[19] Podlubny, I. Fractional Differential Equations, Academic Press, San Diego, 1999

[20] Samko, S. G.; Kilbas, A. A.; Marichev, O. I. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993

[21] Tariboon, J.; Ntouyas, S. K.; Thiramanus, P. Riemann-Liouville fractional differential equations with Hadamard fractional integral conditions, Inter. J. Appl. Math. Stat., Volume 54 (2016), pp. 119-134

[22] Tariboon, J.; Ntouyas, S. K.; Sudsutad, W. Fractional integral problems for fractional differential equations via Caputo derivative, Adv. Difference Equ., Volume 2014 (2014), pp. 1-17 | DOI

[23] Tariboon, J.; Ntouyas, S. K. Initial and boundary value problems via exponential fractional calculus, International Electronic Journal of Pure and Applied Mathematics , Volume 12 (2018), pp. 1-19 | DOI

[24] Yu, C.; Gao, G. Z. Some results on a class of fractional functional differential equations, Comm. Appl. Nonlinear Anal., Volume 11 (2004), pp. 67-75

[25] Yu, C.; Gao, G. Z. Existence of fractional differential equations, J. Math. Anal. Appl., Volume 310 (2005), pp. 26-29 | DOI