Anti-periodic BVP for Volterra integro-differential equation of fractional order \(1<\alpha \leq 2\), involving Mittag-Leffler function in the kernel

Volume 9, Issue 2, pp 452--460 http://dx.doi.org/10.22436/jnsa.009.02.11

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1486 Downloads
• 2579 Views

Authors

Hüseyin Aktuğlu - Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey. Mehmet Ali Özarslan - Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey.

Abstract

In this paper, we consider an anti-periodic Boundary Value Problem for Volterra integro-differential equation of fractional order \(1<\alpha \leq 2\); with generalized Mittag-Leffler function in the kernel. Some existence and uniqueness results are obtained by using some well known fixed point theorems. We give some examples to exhibit our results.

Share and Cite

Keywords

• Fractional derivative
• fractional integral
• Caputo fractional derivative
• boundary value problem
• Caputo fractional boundary value problem
• integral operators
• Mittag-Leffler functions.

MSC

•  34A08
•  34B15

References

• [1] T. Abdeljawad, D. Baleanu, Caputa q-fractional initial value problems and a q-analogue Mittag-Leffler function, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 4682-4688.


• [2] T. Abdeljawad, D. Baleanu , Fractional differences and integration by parts, J. Comput. Anal. Appl., 13 (2011), 574-582.


• [3] R. P. Agarwal, B. Ahmad , Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, Comput. Math. Appl., 62 (2011), 1200-1214.


• [4] P. Agarwal, J. Choi, R. B. Paris, Extended Riemann-Liouville fractional derivative operator and its applications, J. Nonlinear Sci. Appl., 8 (2015), 451-466.


• [5] R. P. Agarwal, G. Wang, A. Hobiny, L. Zhang, B. Ahmad, Existence and nonexistence of solutions for nonlinear second order q-integro-difference equations with non-separated boundary conditions, J. Nonlinear Sci. Appl., 8 (2015), 976-985.


• [6] A. Aghajani, Y. Jalilian, J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc, Appl. Anal., 15 (2012), 44-69.


• [7] B. Ahmad, Existence of soultions for fractional differential equations of order \(q \in (2; 3]\) with anti-periodic boundary conditions, J. Appl. Math. Comput., 34 (2010), 385-391.


• [8] B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 58 (2009), 1838-1843.


• [9] B. Ahmad, J. J. Nieto, Anti-periodic fractional boundary value problems, Comput. Math. Appl., 62 (2011), 1150-1156.


• [10] A. Alsaedi, B. Ahmad, A. Assolami, On Antiperiodic Boundary Value Problems for Higher-Order Fractional Differential Equations, Abstr. Appl. Anal., 2012 (2012), 15 pages.


• [11] H. Aktuğlu, M. A. Özarslan , On the solvability of Caputo q-fractional boundary value problem involving p-Laplacian operator, Abstr. Appl. Anal., 2013 (2013), 8 pages.


• [12] H. Aktuğlu, M. A. Özarslan, Solvability of differential equations of order \(2 < \alpha <= 3\) involving the p-Laplacian operator with boundary conditions, Adv. Differ. Eqn., 2013 (2013), 13 pages.


• [13] T. Chen, W. Liu, An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator, Appl. Math. Lett., 25 (2012), 1671-1675.


• [14] X. L. Ding, Y. L. Jiang , Semilinear fractional differential equations base on a new integral operator approach, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 5143-5150.


• [15] A. Granas, J. Dugundji, Fixed point theory, Springer Monographs in Mathematics, Springer-Verlag, New York (2003)


• [16] A. A. Kilbas, M. Saigo, R. K. Saxena, Solutions of volterra integro-differential equations with generalized Mittag-Leffler function in the kernels, J. Int. Equ. Appl., 14 (2002), 377-396.


• [17] A. A. Kilbas, M. Saigo, R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct., 15 (2004), 31-49.


• [18] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Amsterdam (2006)


• [19] C. P. Li, D. L. Qian, Y. Q. Chen, On Riemann-Liouville and Caputo derivatives, Discrete Dyn. Nat. Soc., 2011 (2011), 15 pages.


• [20] X. Liu, M. Jia, X. Hiang, On the solvability of a fractional differential equation model involving the p-Laplacian operator, Comput. Math. Appl., 64 (2012), 3267-3275.


• [21] G. M. Mittag-Leffler, Sur la nouvelle fonction \(E_\alpha(x)\), C. R. Acad. Sci. Paris, 137 (1903), 554-558.


• [22] I. Podlubny, Fractional differential equations, Academy Press, San Diego (1999)


• [23] T. R. Prabhakar, A singular integral equation with a general Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.


• [24] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integral and derivatives, Gordon and Breach Science, Yverdon (1993)


• [25] H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.


• [26] Z. Tomovski, Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonlinear Anal., 75 (2012), 3364-3384.


• [27] Z. Tomovski, R. Garra , Analytic solutions of fractional integro-differential equations of volterra type with variable coeficients, Fract. Calc. Appl. Anal., 17 (2014), 38-60.


• [28] Z. Tomovski, R. Hilfer, H. M. Srivastava, Fractional operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Int. Trans. Spec. Func., 21 (2010), 797-814.


• [29] G.Wang, B. Ahmad, L. Zhang, Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal., 74 (2011), 792-804.