Existence of solutions for nonlinear impulsive \(q_k\)-difference equations with first-order \(q_k\)-derivatives

Volume 9, Issue 5, pp 2615--2630 http://dx.doi.org/10.22436/jnsa.009.05.58

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Authors

Changlong Yu - College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P. R. China. Jufang Wang - College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P. R. China. Yanping Guo - College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P. R. China.

Abstract

In this paper, we study the nonlinear second-order impulsive \(q_k\)-difference equations with Sturm-Liouville type, in which nonlinear team and impulsive teams are dependent on first-order \(q_k\)-derivatives. We obtain the existence and uniqueness results of solutions for the problem by Banach's contraction mapping principle and Schaefer's fixed point theorems. Finally, we give two examples to demonstrate the use of the main results.

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Keywords

• boundary value problem
• \(q_k\)-derivative
• \(q_k\)-integral
• impulsive \(q_k\)-difference equation
• fixed point theorem.

MSC

•  39A13
•  34B15
•  34B37
•  81Q99

References

• [1] C. R. Adams, On the linear ordinary q-difference equation, Ann. Math., 30 (1928), 195-205.


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


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


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


• [5] B. Ahmad, J. J. Nieto, Basic theory of nonlinear third-order q-difference equations and inclusions, Math. Model. Anal., 18 (2013), 122-135.


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


• [7] 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.


• [8] G. Bangerezako, Variational q-calculus , J. Math. Anal. Appl., 289 (2004), 650-665.


• [9] M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive differential equations and inclusions, vol.2, Hindawi Publishing Corporation, New York (2006)


• [10] R. D. Carmichael , The general theory of linear q-difference equations , Amer. J. Math., 34 (1912), 147-168.


• [11] A. Dobrogowska, A. Odzijewicz, Second order q-difference equations solvable by factorization method, J. Comput. Appl. Math., 193 (2006), 319-346.


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


• [13] F. H. Jackson , q-difference equations , Amer. J. Math., 32 (1910), 305-314.


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


• [15] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, World Scientific, Singapore-London (1989)


• [16] A. Lavagno, P. N. Swamy , q-deformed structures and nonextensive statistics: a comparative study, Non extensive thermodynamics and physical applications Phys. A, 305 (2002), 310-315.


• [17] T. E. Mason , On properties of the solutions of linear q-difference equations with entire function coefficients, Amer. J. Math., 37 (1915), 439-444.


• [18] D. N. Page, Information in black hole radiation, Phys. Rev. Lett., 71 (1993), 3743-3746.


• [19] A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Scientific, Singapore-London (1985)


• [20] D. R. Smart , Fixed Point Theorems , Cambridge Univ. Press, Cambridge (1974)


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


• [22] C. Thaiprayoon, J. Tariboon, S. K. Ntouyas, Separated boundary value problems for second-order impulsive q- integro-difference equations, Adv. Differ. Equ., 2014 (2014), 23 pages.


• [23] D. Youm, q-deformed conformal quantum mechanics, ArXiv, 2000 (2000), 9 pages.


• [24] C. Yu, J. Wang, Existence of solutions for nonlinear second-order q-difference equations with first-order q- derivatives, Adv. Differ. Equ., 2013 (2013), 11 pages.


• [25] C. Yu, J. Wang, Positive solutions of nonlocal boundary value problem for high-order nonlinear fractional q- difference equations, Abstr. Appl. Anal., 2013 (2013), 9 pages.


• [26] C. Yu, J. Wang, Eigenvalue of boundary value problem for nonlinear singular third-order q-difference equations, Adv. Differ. Equ., 2014 (2014), 8 pages.


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