# Impulsive first-order functional $q_k$-integro-difference inclusions with boundary conditions

Volume 9, Issue 1, pp 46--60
• 1555 Views

### Authors

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. Sotiris K. Ntouyas - 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. Weerawat Sudsutad - Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok, 10800, Thailand.

### Abstract

In this paper, we discuss the existence of solutions for a first order boundary value problem for impulsive functional $q_k$-integro-difference inclusions. Some new existence results are obtained for convex as well as non-convex multivalued maps with the aid of some classical fixed point theorems. Illustrative examples are also presented.

### Share and Cite

##### ISRP Style

Jessada Tariboon, Sotiris K. Ntouyas, Weerawat Sudsutad, Impulsive first-order functional $q_k$-integro-difference inclusions with boundary conditions, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 1, 46--60

##### AMA Style

Tariboon Jessada, Ntouyas Sotiris K., Sudsutad Weerawat, Impulsive first-order functional $q_k$-integro-difference inclusions with boundary conditions. J. Nonlinear Sci. Appl. (2016); 9(1):46--60

##### Chicago/Turabian Style

Tariboon, Jessada, Ntouyas, Sotiris K., Sudsutad, Weerawat. "Impulsive first-order functional $q_k$-integro-difference inclusions with boundary conditions." Journal of Nonlinear Sciences and Applications, 9, no. 1 (2016): 46--60

### Keywords

• $q_k$-derivative
• $q_k$-integral
• impulsive $q_k$-difference inclusions
• existence
• fixed point theorem.

•  34A60
•  26A33
•  39A13
•  34A37

### References

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

• [2] B. Ahmad, Boundary-value problems for nonlinear third-order q-difference equations, Electron. J. Differential Equ., 2011 (2011), 7 pages.

• [3] B. Ahmad, A. Alsaedi, S. K. Ntouyas , A study of second-order q-difference equations with boundary conditions, Adv. Difference Equ., 2012 (2012), 10 pages.

• [4] B. Ahmad, J. J. Nieto , Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl., 2011 (2011), 9 pages.

• [5] B. Ahmad, J. J. Nieto, On nonlocal boundary value problems of nonlinear q-difference equations , Adv. Difference Equ., 2012 (2012), 10 pages.

• [6] B. Ahmad, J. J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces Appl., 2013 (2013), 8 pages.

• [7] B. Ahmad, S. K. Ntouyas, Boundary value problems for q-difference inclusions, Abstr. Appl. Anal., 2011 (2011), 15 pages.

• [8] B. Ahmad, S. K. Ntouyas, A. Alsaedi , New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Difference Equ., 2011 (2011), 11 pages.

• [9] B. Ahmad, S. K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng., 2013 (2013), 9 pages.

• [10] B. Ahmad, S. K. Ntouyas, I. K. Purnaras, Existence results for nonlinear q-difference equations with nonlocal boundary conditions, Comm. Appl. Nonlinear Anal., 19 (2012), 59-72.

• [11] D. Băleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus models and numerical methods, Series on complexity, nonlinearity and chaos, World Scientific, Boston (2012)

• [12] D. Băleanu, O. G. Mustafa, R. P. Agarwal, On $L^p$-solutions for a class of sequential fractional differential equations, Appl. Math. Comput., 218 (2011), 2074-2081.

• [13] M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York (2006)

• [14] A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values, Studia Math., 90 (1988), 69-86.

• [15] C. Castaing, M. Valadier , Convex analysis and measurable multifunctions , Lecture Notes in Mathematics 580, Springer-Verlag, New York (1977)

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

• [17] K. Deimling, Multivalued differential equations, Walter De Gruyter, Berlin-New York (1992)

• [18] M. El-Shahed, H. A. Hassan , Positive solutions of q-difference equation, Proc. Amer. Math. Soc., 138 (2010), 1733-1738.

• [19] M. Frigon , Théorémes d'existence de solutions d'inclusions différentielles, Topological methods in differential equations and inclusions (edited by A. Granas and M. Frigon), NATO ASI Series C, Kluwer Acad. Publ. Dordrecht, 472 (1995), 51-87.

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

• [21] S. Hu, N. Papageorgiou , Handbook of multivalued analysis, Theory I, Kluwer, Dordrecht (1997)

• [22] V. Kac, P. Cheung, , Quantum calculus, Springer, New York (2002)

• [23] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo , Theory and applications of fractional differential equations, NorthHolland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam (2006)

• [24] M. Kisielewicz, Differential inclusions and optimal control, Kluwer, Netherlands (1991)

• [25] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov , Theory of impulsive differential equations, World Scientific, Teaneck, NJ (1989)

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

• [27] X. Liu, M. Jia, W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Difference Equ., 2013 (2013), 12 pages.

• [28] D. O'Regan, S. Staněk , Fractional boundary value problems with singularities in space variables, Nonlinear Dynam., 71 (2013), 641-652.

• [29] I. Podlubny, Fractional differential equations, Academic Press, San Diego (1998)

• [30] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, theory and applications, Gordon and Breach, Yverdon (1993)

• [31] A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Scientific, Singapore (1995)

• [32] J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Difference Equ., 2013 (2013), 19 pages.

• [33] J. Tariboon, S. K. Ntouyas, Boundary value problem for first-order impulsive functional q-integrodifference equations, Abstr. Appl. Anal., 2014 (2014), 11 pages.

• [34] W. Zhou, H. Liu, Existence solutions for boundary value problem of nonlinear fractional q-difference equations, Adv. Difference Equ., 2013 (2013), 12 pages.