# Positive periodic solutions of functional discrete systems with a parameter

- Youssef N. Raffoul yraffoul1@udayton.edu
- Ernest Yankson ernestoyank@gmail.com

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## DOI:

https://doi.org/10.4067/S0719-06462019000100079## Abstract

The existence of multiple positive periodic solutions of the system of difference equations with a parameter

x(n + 1) = A(n, x(n))x(n) + Î»f(n, x_{n}),

is studied. In particular, we use the eigenvalue problems of completely continuous operators to obtain our results. We apply our results to a well-known model in population dynamics.

## Keywords

[1] A. Datta and J. Henderson, Differences and smoothness of solutions for functional difference equations, Proceedings Difference Equations, 1 (1995), 133-142.

[2] Y. Chen, B. Dai and N. Zhang, Positive periodic solutions of non-autonomous functional differential systems, J. Math. Anal. Appl. 333 (2007) 667-678.

[3] S. N. Elaydi, An Introduction to Difference Equations, 2nd ed., Undergraduate Texts in Math- ematics, Springer-Verlag, New York, 1999.

[4] D.J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics and Science and Engineering, vol. 5, Academic Press Inc., Boston, MA, 1988, pp. 2-99.

[5] J. Henderson and A. Peterson, Properties of delay variation in solutions of delay difference equations, Journal of Differential Equations, 1 (1995), 29-38.

[6] R.P. Agarwal and P.J.Y. Wong, On the existence of positive solutions of higher order difference equations, Topological Methods in Nonlinear Analysis, 10 (1997) 2, 339-351.

[7] P.W. Eloe, Y. Raffoul, D. Reid and K. Yin, Positive solutions of nonlinear Functional Difference Equations, Computers and Mathematics With applications, 42 (2001) , 639-646.

[8] J. Henderson and W. N. Hudson, Eigenvalue problems for nonlinear differential equations, Communications on Applied Nonlinear Analysis, 3 (1996), 51-58.

[9] M. A. Krasnosel‘skii, Positive solutions of operator Equations, Noordhoff, Groningen, (1964).

[10] Y. Li and L. Zhu, Positive periodic solutions of higher-dimensional functional difference equations with a parameter, J. Math. Anal. Appl. 290 (2004) 654-664.

[11] F. Merdivenci, Two positive solutions of a boundary value problem for difference equations, Journal of Difference Equations and Application, 1 (1995), 263-270.

[12] Y.N. Raffoul, Positive periodic solutions of nonlinear functional difference equations, Electron. J. Differential Equations, 55 (2002) 1-8.

[13] Y.N. Raffoul, Periodic solutions for scalar and vector nonlinear difference equations, Pan- American Journal of Mathematics, 9 (1999), 97-111.

[14] W. Yin, Eigenvalue problems for functional differential equations, Journal of Nonlinear Differential Equations, 3 (1997), 74-82.

[2] Y. Chen, B. Dai and N. Zhang, Positive periodic solutions of non-autonomous functional differential systems, J. Math. Anal. Appl. 333 (2007) 667-678.

[3] S. N. Elaydi, An Introduction to Difference Equations, 2nd ed., Undergraduate Texts in Math- ematics, Springer-Verlag, New York, 1999.

[4] D.J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics and Science and Engineering, vol. 5, Academic Press Inc., Boston, MA, 1988, pp. 2-99.

[5] J. Henderson and A. Peterson, Properties of delay variation in solutions of delay difference equations, Journal of Differential Equations, 1 (1995), 29-38.

[6] R.P. Agarwal and P.J.Y. Wong, On the existence of positive solutions of higher order difference equations, Topological Methods in Nonlinear Analysis, 10 (1997) 2, 339-351.

[7] P.W. Eloe, Y. Raffoul, D. Reid and K. Yin, Positive solutions of nonlinear Functional Difference Equations, Computers and Mathematics With applications, 42 (2001) , 639-646.

[8] J. Henderson and W. N. Hudson, Eigenvalue problems for nonlinear differential equations, Communications on Applied Nonlinear Analysis, 3 (1996), 51-58.

[9] M. A. Krasnosel‘skii, Positive solutions of operator Equations, Noordhoff, Groningen, (1964).

[10] Y. Li and L. Zhu, Positive periodic solutions of higher-dimensional functional difference equations with a parameter, J. Math. Anal. Appl. 290 (2004) 654-664.

[11] F. Merdivenci, Two positive solutions of a boundary value problem for difference equations, Journal of Difference Equations and Application, 1 (1995), 263-270.

[12] Y.N. Raffoul, Positive periodic solutions of nonlinear functional difference equations, Electron. J. Differential Equations, 55 (2002) 1-8.

[13] Y.N. Raffoul, Periodic solutions for scalar and vector nonlinear difference equations, Pan- American Journal of Mathematics, 9 (1999), 97-111.

[14] W. Yin, Eigenvalue problems for functional differential equations, Journal of Nonlinear Differential Equations, 3 (1997), 74-82.

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## Published

2019-04-01

## How to Cite

[1]

Y. N. . Raffoul and E. . Yankson, “Positive periodic solutions of functional discrete systems with a parameter”,

*CUBO*, vol. 21, no. 1, pp. 79–90, Apr. 2019.## Issue

## Section

Articles