Stability And Boundedness In Nonlinear And Neutral Difference Equations using New Variation of Parameters Formula And Fixed Point Theory

  • Youssef N Raffoul Department of Mathematics, University of Dayton, Dayton, OH 45469-2316 U.S.A.
Keywords: New variation of parameters, Difference, Neutral, Sability, Boundedness, Fixed point theorems, Contraction mapping, equi-boundedness

Abstract

In the case of nonlinear problems, whether in differential or difference equations, it is difficult and in some cases impossible to invert the problem and obtain a suitable mapping that can be effectively used in fixed point theory to qualitatively analyze its solutions. In this paper we consider the existence of a positive sequence and utilize it in the capacity of integrating factor to obtain a new variation of parameters formula. Then, we will use the obtained new variation of parameters formula and revert to the contraction principle to arrive at results concerning, boundedness, periodicity and stability. The author is working on parallel results for the continuous case.

References

[1] Adivar, M,. Islam, M. and Raffoul, Y., Separate contraction and existence of periodic solutions in totally nonlinear delay differential equations, Hacettepe Journal of Mathematics and Statistics, 41 (1) (2012), 1-13.
[2] T. Burton, Volterra Integral and Differential Equations, Academic Press, New York, 1983.
[3] T. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, New York, 1985.
[4] T. Burton and T. Furumochi, Fixed points and problems in stability theory, Dynam. Systems Appl. 10(2001), 89-116.
[5] Burton, T.A. Integral equations, implicit functions, and fixed points, Proc. Amer. Math. Soc. 124 (1996), 2383-2390.
[6] S. Elaydi, An Introduction to Difference Equations, Springer, New York, 1999.
[7] S. Elaydi, Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl., 181(1994), 483-492.
[8] S. Elaydi and S. Murakami, Uniform asymptotic stability in linear Volterra difference equations, J. Differ. Equations Appl., 3(1998), 203-218.
[9] P. Eloe, M. Islam and Y. Raffoul, Uniform asymptotic stability in nonlinear Volterra discrete systems, Special Issue on Advances in Difference Equations IV, Computers Math. Appl., 45(2003), 1033-1039.
[10] Y. Hino and S. Murakami, Total stability and uniform asymptotic stability for linear Volterra equations, J. London Math. Soc., 43(1991), 305-312.
[11] M. Islam and Y. Raffoul, Exponential stability in nonlinear difference equations , J. Differ. Equations Appl., 9(2003), 819-825.
[12] M. Islam and E. Yankson Boundedness and stability in nonlinear delay difference equations employing fixed point theory, Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 26, 1-18.
[13] W. Kelley and A. Peterson, Difference Equations: An Introduction with Applications, Harcourt Academic Press, San Diego, 2001.
[14] J. Liu, A First Course In The Qualitative Theory of Differential Equations,Pearson Education, Inc., Upper Saddle River, New Jersey 07458, 2003.
[15] M. Maroun and Y. Raffoul, Periodic solutions in nonlinear neutral difference equations with functional delay, J. Korean Math. Soc. 42 (2005), no. 2, 255-268.
[16] R. Medina, Asymptotic behavior of Volterra difference equations, Computers Math. Appl. 41(2001), 679-687.
[17] R. Medina, The asymptotic behavior of the solutions of a Volterra difference equation, Computers Math. Appl. 181(1994), 19-26.
[18] Y. Raffoul, Qualitative Theory of Volterra Difference Equations, Springer, New York, 2018.
[19] Y. Raffoul, Stability and periodicity in discrete delay equations, J. Math. Anal. Appl. 324 (2006) 1356-1362.
[20] Y. Raffoul, Stability in neutral nonlinear differential equations with functional delays using fixed point theory, Mathematical and Computer Modelling, 40(2004), 691-700.
[21] Y. Raffoul, General theorems for stability and boundedness for nonlinear functional discrete systems, J. Math. Anal. Appl. ,279(2003), 639-650.
Published
2020-01-20
How to Cite
Raffoul, Y. N. (2020). Stability And Boundedness In Nonlinear And Neutral Difference Equations using New Variation of Parameters Formula And Fixed Point Theory. CUBO, A Mathematical Journal, 21(3), 39–61. https://doi.org/10.4067/S0719-06462019000300039