Existence results for mixed Hadamard and Riemann-Liouville fractional integro-differential inclusions

Volume 9, Issue 12, pp 6333--6347 http://dx.doi.org/10.22436/jnsa.009.12.34

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)

• 1554 Downloads
• 2740 Views

Authors

Bashir Ahmad - 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. Sotiris K. Ntouyas - 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. - Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece. Jessada Tariboon - 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.

Abstract

In this paper, we investigate a new class of mixed initial value problems of Hadamard and Riemann- Liouville fractional integro-differential inclusions. The existence of solutions for convex valued (the upper semicontinuous) case is established by means of Krasnoselskii's fixed point theorem for multivalued maps and nonlinear alternative criterion, while the existence result for non-convex valued maps (the Lipschitz case) relies on a fixed point theorem due to Covitz and Nadler. Illustrative examples are also included.

Share and Cite

Keywords

• Fractional differential inclusions
• Hadamard derivative
• Riemann-Liouville derivative
• fixed point theorem.

MSC

•  34A08
•  34A12
•  34A60

References

• [1] M. P. Aghababa, Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique, Nonlinear Dynam., 69 (2012), 247--261


• [2] M. P. Aghababa, A novel terminal sliding mode controller for a class of non-autonomous fractional-order systems, Nonlinear Dynam., 73 (2013), 679--688


• [3] B. Ahmad, A. Alsaedi, S. Z. Nazemi, S. Rezapour, Some existence theorems for fractional integro-differential equations and inclusions with initial and non-separated boundary conditions, Bound. Value Probl., 2014 (2014), 40 pages


• [4] B. Ahmad, S. K. Ntouyas, Existence of solutions for fractional differential inclusions with four-point nonlocal Riemann-Liouville type integral boundary conditions, Filomat, 27 (2013), 1027--1036


• [5] B. Ahmad, S. K. Ntouyas, A. Alsaedi, New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions, Bound. Value Probl., 2013 (2013), 14 pages


• [6] B. Ahmad, S. K. Ntouyas, J. Tariboon, A study of mixed Hadamard and Riemann-Liouville fractional integro- differential inclusions via endpoint theory, Appl. Math. Lett, 52 (2016), 9--14


• [7] S. Balochian, M. Nazari, Stability of particular class of fractional differential inclusion systems with input delay, Control Intell. Syst., 42 (2014), 279--283


• [8] C. Bardaro, P. L. Butzer, I. Mantellini, The foundations of fractional calculus in the Mellin transform setting with applications, J. Fourier Anal. Appl., 21 (2015), 961--1017


• [9] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl., 269 (2002), 387--400


• [10] J. Cao, C. Ma, Z. Jiang, S. Liu, Nonlinear dynamic analysis of fractional order rub-impact rotor system, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1443--1463


• [11] C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York (1977)


• [12] L.-C. Ceng, C.-M. Chen, C.-T. Pang, On a finite family of variational inclusions with the constraints of generalized mixed equilibrium and fixed point problems, J. Inequal. Appl., 2014 (2014), 38 pages


• [13] H. Covitz, S. B. Nadler, Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 8 (1970), 5--11


• [14] M. F. Danca, Approach of a class of discontinuous dynamical systems of fractional order: existence of solutions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3273--3276


• [15] K. Deimling, Multivalued differential equations, de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin (1992)


• [16] M. O. Efe, Fractional order systems in industrial automation-a survey, IEEE Trans. Ind. Inform., 7 (2011), 582--591


• [17] M. A. Ezzat, Theory of fractional order in generalized thermoelectric MHD, Appl. Math. Model., 35 (2011), 4965--4978


• [18] V. Gafiychuk, B. Datsko, V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math., 220 (2008), 215--225


• [19] Z. Gao, Active disturbance rejection control for nonlinear fractional-order systems, Internat. J. Robust Nonlinear Control, 26 (2016), 876--892


• [20] J. R. Graef, J. Henderson, A. Ouahab, Fractional differential inclusions in the Almgren sense, Fract. Calc. Appl. Anal., 18 (2015), 673--686


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


• [22] J. Hadamard, Essai sur l'etude des fonctions, donnees par leur developpement de Taylor, J. Mat. Pure Appl. (Ser. 4), 8 (1892), 10--186


• [23] S. H. Hosseinnia, R. Ghaderi, A. Ranjbar, M. Mahmoudian, S. Momani, Sliding mode synchronization of an uncertain fractional order chaotic system, Comput. Math. Appl., 59 (2010), 1637--1643


• [24] S.-C. Hu, N. S. Papageorgiou, Handbook of multivalued analysis, Vol. I, Theory. Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (1997)


• [25] A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191--1204


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


• [27] M. Kisielewicz, Differential inclusions and optimal control, Mathematics and its Applications (East European Series), Kluwer Academic Publishers Group/PWN|Polish Scientific Publishers, Dordrecht/Warsaw (1991)


• [28] M. Kisielewicz, Stochastic differential inclusions and applications, Springer Optimization and Its Applications, Springer, New York (2013)


• [29] N. Laskin, Fractional market dynamics, Economic dynamics from the physics point of view, Bad Honnef, (2000), Phys. A, 287 (2000), 482--492


• [30] A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys., 13 (1965), 781--786


• [31] Y. Luo, Y.-Q. Chen, Y.-G. Pi, Experimental study of fractional order proportional derivative controller synthesis for fractional order systems, Mechatronics, 21 (2011), 204--214


• [32] S. S. Majidabad, H. T. Shandiz, A. Hajizadeh, Nonlinear fractional-order power system stabilizer for multimachine power systems based on sliding mode technique, Internat. J. Robust Nonlinear Control, 25 (2015), 1548--1568


• [33] S. K. Ntouyas, S. Etemad, J. Tariboon, Existence results for multi-term fractional differential inclusions, Adv. Difference Equ., 2015 (2015), 15 pages


• [34] N. Özdemir, D. Avcı, Optimal control of a linear time-invariant space-time fractional diffusion process, J. Vib. Control, 20 (2014), 370--380


• [35] I. Petráš, R. L. Magin, Simulation of drug uptake in a two compartmental fractional model for a biological system, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4588--4595


• [36] A. Petrusel, Fixed points and selections for multi-valued operators, Semin. Fixed Point Theory Cluj-Napoca, 2 (2001), 3--22


• [37] A. Pisano, M. R. Rapaić, Z. D. Jeličić, E. Usai, Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics, Internat. J. Robust Nonlinear Control, 20 (2010), 2045--2056


• [38] J.-W. Sun, Q. Yin, Robust fault-tolerant full-order and reduced-order observer synchronization for differential inclusion chaotic systems with unknown disturbances and parameters, J. Vib. Control, 21 (2015), 2134--2148


• [39] M. S. Tavazoei, M. Haeri, Synchronization of chaotic fractional-order systems via active sliding mode controller, Phys. A, 387 (2008), 57--70


• [40] G. Toufik, Existence and controllability results for fractional stochastic semilinear differential inclusions, Differ. Equ. Dyn. Syst., 23 (2015), 225--240


• [41] X. Wang, P. Schiavone, Harmonic three-phase circular inclusions in finite elasticity, Contin. Mech. Thermodyn., 27 (2015), 739--747


• [42] C. Yin, S. Dadras, S.-M. Zhong, Y.-Q. Chen, Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach, Appl. Math. Model., 37 (2013), 2469--2483


• [43] X. Zhao, H.-T. Yang, Y.-Q. He, Identification of constitutive parameters for fractional viscoelasticity, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 311--322