Impulsive antiperiodic boundary value problems for nonlinear \(q_k\)-difference equations. (English) Zbl 1470.39022

Summary: We show the existence and uniqueness of solutions for an antiperiodic boundary value problem of nonlinear impulsive \(q_k\)-difference equations by applying some well-known fixed point theorems. An example is presented to illustrate the main results.


39A13 Difference equations, scaling (\(q\)-differences)
34B37 Boundary value problems with impulses for ordinary differential equations
Full Text: DOI


[1] Jackson, F. H., On \(q\)-difference equations, American Journal of Mathematics, 32, 305-314, (1910) · JFM 41.0502.01
[2] Kac, V.; Cheung, P., Quantum Calculus, (2002), Springer · Zbl 0986.05001
[3] Ernst, T., A Comprehensive Treatment of q-Calculus, (2012), Springer · Zbl 1256.33001
[4] Aral, A.; Gupta, V.; Agarwal, R. P., Applications of q-Calculus in Operator Theory, (2013), Springer · Zbl 1273.41001
[5] El-Shahed, M.; Hassan, H. A., Positive solutions of \(q\)-difference equation, Proceedings of the American Mathematical Society, 138, 5, 1733-1738, (2010) · Zbl 1201.39003
[6] Ahmad, B.; Ntouyas, S. K., Boundary value problems for \(q\)-difference inclusions, Abstract and Applied Analysis, 2011, (2011) · Zbl 1216.39012
[7] Ahmad, B.; Ntouyas, S. K.; Purnaras, I. K., Existence results for nonlinear \(q\)-difference equations with nonlocal boundary conditions, Communications on Applied Nonlinear Analysis, 19, 59-72, (2012) · Zbl 1278.39010
[8] Ahmad, B.; Alsaedi, A.; Ntouyas, S. K., A study of second-order \(q\)-difference equations with boundary conditions, Advances in Difference Equations, 2012, article 35, (2012) · Zbl 1302.39002
[9] Bohner, M.; Chieochan, R., Floquet theory for \(q\)-difference equations, Sarajevo Journal of Mathematics, 8, 21, 355-366, (2012) · Zbl 1321.39010
[10] Ahmad, B.; Nieto, J. J., Basic theory of nonlinear third-order \(q\)-difference equations and inclusions, Mathematical Modelling and Analysis, 18, 122-135, (2013) · Zbl 1264.34027
[11] Pongarm, N.; Asawasamrit, S.; Tariboon, J., Sequential derivatives of nonlinear \(q\)-difference equations with three-point \(q\)-integral boundary conditions, Journal of Applied Mathematics, 2013, (2013) · Zbl 1266.39009
[12] Area, I.; Godoy, E.; Nieto, J. J., Fixed point theory approach to boundary value problems for second-order difference equations on non-uniform lattices, Advances in Difference Equations, 2014, article 14, (2014) · Zbl 1343.34071
[13] Area, I.; Atakishiyev, N.; Godoy, E.; Rodal, J., Linear partial q-difference equations on \(q\)-linear lattices and their bivariate \(q\)-orthogonal polynomial solutions, Applied Mathematics and Computation, 223, 520-536, (2013) · Zbl 1329.39011
[14] Bangerezako, G., \(q\)-Difference linear control systems, Journal of Difference Equations and Applications, 17, 9, 1229-1249, (2011) · Zbl 1234.39001
[15] Bangerezako, G., Variational \(q\)-calculus, Journal of Mathematical Analysis and Applications, 289, 2, 650-665, (2004) · Zbl 1043.49001
[16] Logan, J. D., First integrals in the discrete variational calculus, Aequationes Mathematicae, 9, 2-3, 210-220, (1973) · Zbl 0268.49022
[17] Guermah, S.; Djennoune, S.; Bettayeb, M., Controllability and observability of linear discrete-time fractional-order systems, International Journal of Applied Mathematics and Computer Science, 18, 2, 213-222, (2008) · Zbl 1234.93014
[18] Bartosiewicz, Z.; Pawłuszewicz, E., Realizations of linear control systems on time scales, Control and Cybernetics, 35, 4, 769-786, (2006) · Zbl 1133.93033
[19] Mozyrska, D.; Bartosiewicz, Z., On observability concepts for nonlinear discrete-time fractional order control systems, New Trends in Nanotechnology and Fractional Calculus Applications, 4, 305-312, (2010) · Zbl 1222.93139
[20] Abdeljawad, T.; Jarad, F.; Baleanu, D., Vartiational optimal-control problems with delayed arguments on time scales, Advances in Difference Equations, 2009, (2009) · Zbl 1184.49023
[21] Mainardi, F.; Carpinteri, A.; Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics, (1997), New York, NY, USA: Springer, New York, NY, USA · Zbl 0917.73004
[22] Agrawal, O., Some generalized fractional calculus operators and their applications in integral equations, Fractional Calculus and Applied Analysis, 15, 700-711, (2012) · Zbl 1312.26010
[23] Ahmad, B.; Sivasundaram, S., Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems, 3, 3, 251-258, (2009) · Zbl 1193.34056
[24] Tian, Y.; Bai, Z., Existence results for the three-point impulsive boundary value problem involving fractional differential equations, Computers and Mathematics with Applications, 59, 8, 2601-2609, (2010) · Zbl 1193.34007
[25] Mophou, G. M., Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Analysis: Theory, Methods and Applications, 72, 3-4, 1604-1615, (2010) · Zbl 1187.34108
[26] Wang, G.; Ahmad, B.; Zhang, L., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Analysis: Theory, Methods and Applications, 74, 3, 792-804, (2011) · Zbl 1214.34009
[27] Ahmad, B.; Nieto, J. J., Anti-periodic fractional boundary value problems, Computers and Mathematics with Applications, 62, 3, 1150-1156, (2011) · Zbl 1228.34010
[28] Fečkan, M.; Zhou, Y.; Wang, J. R., On the concept and existence of solution for impulsive fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 17, 3050-3060, (2012) · Zbl 1252.35277
[29] Zhou, X. F.; Liu, S.; Jiang, W., Complete controllability of impulsive fractional linear time-invariant systems with delay, Abstract and Applied Analysis, 2013, (2013) · Zbl 1291.34129
[30] Wang, G.; Ahmad, B.; Zhang, L.; Nieto, J. J., Comments on the concept of existenc e of solution for impulsive fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 19, 401-403, (2014)
[31] Chauhan, A.; Dabas, J., Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition, Communications in Nonlinear Science and Numerical Simulation, 19, 821-829, (2014)
[32] Tariboon, J.; Ntouyas, S. K., Quantum calculus on finite intervals and applications to impulsive difference equations, Advances in Difference Equations, 2013, article 282, (2013) · Zbl 1391.39017
[33] Tariboon, J.; Ntouyas, S. K., Boundary value problems for first-order impulsive functional \(q\)-integro- difference equations, Abstract and Applied Analysis, 2014, (2014) · Zbl 1469.39006
[34] Zhang, L.; Baleanu, D.; Wang, G., Nonlocal boundary value problem for nonlinear \(q_k\) integro-difference equation, Abstract and Applied Analysis, 2014, (2014) · Zbl 1470.39023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.