A new general forward difference operator and some applications

Year 2018, Volume: 9, 117 - 124, 28.12.2018

Yaşar Bolat Ömer Akın

Abstract

In this study, the forward difference operator is defined in the most general form. As an application we give some criteria on the behavior of solutions of some first-order difference equations involving this operator. To do this, we use a lemma firstly constructed here that gives the relationship between ordinary difference operator and our new operator. Our main theorem improves the known results in the literature, since the potential function in this equation is of a wider function class, including potential functions in equivalent equations existing in the literature. Also some examples are provided to illustrate our main results.

Keywords

Difference equations, Generalized difference operator, Variable difference operator, Oscillation, Oscillatory

References

• Agarwal, R.P., Difference Equations and Inequalities, Theory, Methods and Applications, Marcel Dekker, New York, 2000.
• Agarwal, R.P., Bohner, M., Grace, S.R., Regan, D.O., Discrete Oscillation Theory, Hindawi, New York, 2005.
• Agarwal, R.P., Grace, S.R., Regan, D.O., Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 2000.
• Agarwal, R.P., Wong, P.J.Y., Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht, 1997.
• Bolat, Y., Trichotomy of nonoscillatory solutions to second-order neutral difference equation with generalized difference operators, International Conference on Mathematics and Mathematics Education (ICMME-2016), Elazığ-Turkey.
• Bolat, Y, Akın, Ö., Oscillation criteria for higher order half linear delay difference equations involving generalized dierence, Math. Slovaca., 66(3)(2016), 1–10.
• Chatzarakis, G,E., Koplatadze, R., Stavroulakis, I.P., Oscillation criteria of first order linear difference equations with delay argument, Nonlinear Analysis, 68(2008), 994–1005.
• Chatzarakis, G.E., Stavroulakis, I.P., Oscillations of first order linear delay difference equations, Aust. J. Math. Anal. Appl., 3(1)(2006), Art.14, 11pp.
• Chen, M.P., Yu, Y.S., Oscillations of delay difference equations with variable coecients, Proc. First Intl. Conference on Dierence Equations (Edited by S.N. Elaydi et al), Gordon and Breach, 1995, 105–114.
• Domshlak, Y., Discrete version of Sturmian Comparison Theorem for non-symmetric equations Doklady Azerb. Acad. Sci., 37(1981), 12–15.
• Erbe, L.H., Zhang, B.G., Oscillation of discrete analogues of delay equations, Differential Integral Equations, 2(1989), 300–309.
• Györi, I., Ladas, G., Oscillation Theory of Delay Differential Equations With Applications, Clarendon Press, Oxford, p. 167, 1991.
• Ladas, G., Philos, C.G., Sficas, Y.G., Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation., 2(1989), 101–112.
• Parhi, N., Oscillations of first order difference equations, Proc. Indian Acad. Sci. (Math. Sci.), 110(2)(2000), 147–155.
• Parhi, N., Oscillation and non-oscillation of solutions of second order difference equations involving generalized dierence, Applied Mathematics and Computation, 218(2011), 458–468.
• Parhi, N., Panda, A., Nonoscillation and oscillation of solutions of a class of third order difference equations, J. Math. Anal. Appl., 336(2007), 213–223.
• Stavroulakis, I.P., Oscillations of delay difference equations, Comput. Math. Applic., 29(1995), 83–88.
• Stavroulakis, I.P., Oscillation criteria for first order delay difference equations, Mediterr. J. Math., 1(2004), 231–240.
• Tan, M.C., Yang, E.H., Oscillation and nonoscillation theorems for second order difference equations, J. Math. Anal. Appl., 276(2002), 239–247.