×

Asymptotic convergence of solutions of a scalar \(q\)-difference equation with double delays. (English) Zbl 1413.39006

Summary: We obtain sufficient conditions for the asymptotic convergence of all solutions of a scalar \(q\)-difference equation with double delays. Moreover, we prove that the limits of the solutions could be formulated in terms of the initial functions and the solution of a corresponding sum equation.

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams C. R.: On the linear ordinary q-difference equation. Ann. Math. 30, 195-205 (1928) · JFM 55.0263.01 · doi:10.2307/1968274
[2] Arino O., Pituk M.: Convergence in asymptotically autonomous functional differential equations. J. Math. Anal. Appl. 237, 376-392 (1999) · Zbl 0936.34064 · doi:10.1006/jmaa.1999.6489
[3] Atkinson F. V., Haddock J. R.: Criteria for asymptotic constancy of solutions of functional differential equations. J. Math. Anal. Appl. 91, 410-423 (1983) · Zbl 0529.34065 · doi:10.1016/0022-247X(83)90161-0
[4] G. Bangerezako, An Introduction to q-Difference Equations, University of Burundi Preprints (Bujumbura, 2008).
[5] Bastinec J., Diblik J., Smarda Z.: Convergence tests for one scalar differential equation with vanishing delay. Arch. Math. (Brno) 36, 405-414 (2000) · Zbl 1090.34596
[6] Bereketoglu H., Huseynov A.: Convergence of solutions of nonhomogeneous linear difference systems with delays. Acta Appl. Math. 110, 259-269 (2010) · Zbl 1204.39003 · doi:10.1007/s10440-008-9404-2
[7] Bereketoglu H., Karakoc F.: Asymptotic constancy for impulsive delay differential equations. Dynam. Systems Appl. 17, 71-83 (2008) · Zbl 1159.34052
[8] Bereketoglu H., Karakoc F.: Asymptotic constancy for a system of impulsive pantograph equations. Acta Math. Hungar. 145, 1-12 (2015) · Zbl 1363.34207 · doi:10.1007/s10474-014-0470-9
[9] Bereketoglu H., Oztepe G. S.: Convergence of the solution of an impulsive differential equation with piecewise constant arguments. Miskolc Math. Notes 14, 801-815 (2013) · Zbl 1299.34247
[10] Bereketoglu H., Oztepe G. S.: Asymptotic constancy for impulsive differential equations with piecewise constant argument. Bull. Math. Soc. Sci. Math. 57, 181-192 (2014) · Zbl 1389.34253
[11] H. Bereketoglu and M. Pituk, Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, in: Dynamical Systems and Differential Equations (Wilmington, NC, 2002), Discrete Contin. Dyn. Syst., (2003), pp. 100-107. · Zbl 1071.34080
[12] Berezansky L., Diblik J., Ruzickova M., Suta Z.: Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case. Abstr. Appl. Anal. 2011, 15 pp. (2011) · Zbl 1220.39004
[13] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Springer Science & Business Media (Boston, 2001). · Zbl 0978.39001
[14] M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, Springer Science & Business Media (Boston, 2002). · Zbl 1025.34001
[15] Carmichael R. D.: The general theory of linear q-difference equations. Amer. J. Math. 34, 147-168 (1912) · JFM 43.0411.02 · doi:10.2307/2369887
[16] Diblik J.: A criterion of asymptotic convergence for a class of nonlinear differential equations with delay. Nonlinear Anal. 47, 4095-4106 (2001) · Zbl 1042.34580 · doi:10.1016/S0362-546X(01)00527-2
[17] Diblik J., Ruzickova M.: Convergence of the solutions of the equation \[{\dot{y} (t) =\beta (t) [y( t-\delta ) -y( t-\tau )] }\] y˙(t)=β(t)[y(t-δ)-y(t-τ)] in the critical case. J. Math. Anal. Appl. 331, 1361-1370 (2007) · Zbl 1125.34059 · doi:10.1016/j.jmaa.2006.10.001
[18] Diblik J., Ruzickova M., Smarda Z., Suta Z.: Asymptotic convergence of the solutions of a dynamic equation on discrete time scales. Abstr. Appl. Anal. 2012, 20 pp. (2012) · Zbl 1232.39006
[19] Györi I., Karakoc F., Bereketoglu H.: Convergence of solutions of a linear impulsive differential equations system with many delays. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18, 191-202 (2011) · Zbl 1227.34077
[20] Jackson F. H.: On q-difference equations. Amer. J. Math. 32, 305-314 (1910) · JFM 41.0502.01 · doi:10.2307/2370183
[21] V. Kac and P. Cheung, Quantum Calculus, Springer-Verlag (New York, 2002). · Zbl 0986.05001
[22] Lavagno A., Swamy P. N.: q-Deformed structures and nonextensive statistics: a comparative study. Phys. A 305, 310-315 (2002) · Zbl 0984.82006 · doi:10.1016/S0378-4371(01)00680-X
[23] Mason T. E.: On properties of the solutions of linear q-difference equations with entire function coefficients. Amer. J. Math. 37, 439-444 (1915) · JFM 45.0509.01 · doi:10.2307/2370216
[24] Oztepe G. S., Bereketoglu H.: Convergence in an impulsive advanced differential equations with piecewise constant argument. Bull. Math. Anal. Appl. 4, 57-70 (2012) · Zbl 1314.34150
[25] Strominger A.: Information in black hole radiation. Phys. Rev. Lett. 71, 3743-3746 (1993) · Zbl 0972.83567 · doi:10.1103/PhysRevLett.71.3397
[26] Youm D.: q-Deformed Conformal Quantum Mechanics. Phys. Rev. D 62, 095009 (2000) · doi:10.1103/PhysRevD.62.095009
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.