MULTIPLE POSITIVE PERIODIC SOLUTIONS FOR A NONLINEAR FIRST ORDER FUNCTIONAL DIFFERENCE EQUATION WITH APPLICATION TO HEMATOPOIESIS MODEL

SESHADEV PADHI; SMITA PATI

Title	MULTIPLE POSITIVE PERIODIC SOLUTIONS FOR A NONLINEAR FIRST ORDER FUNCTIONAL DIFFERENCE EQUATION WITH APPLICATION TO HEMATOPOIESIS MODEL
Publication Type	Journal Article
Year of Publication	2015
Authors	PADHI, SESHADEV, PATI, SMITA
Volume	19
Issue	3
Start Page	319
Pagination	10
Date Published	2015
ISSN	1083-2564
AMS	34K13, 34K15, 39A10, 39A12
Abstract	Sufficient conditions are obtained for the existence of at least three positive T - periodic solutions for the first order functional difference equation $$ { ∆x(n) = −a(n)x(n) + f(n, x(h(n))). }$$ The Leggett-Williams multiple fixed point theorem has been used to prove our results. We have applied our results to Hematopoiesis models in population dynamics and obtained an interesting result. The result is new in the literature.
URL	http://www.acadsol.eu/en/articles/19/3/1.pdf
Refereed Designation	Refereed
Full Text	REFERENCES [1] D. Bai and Y. Xu, Periodic solutions of first order functional differential equations with periodic deviations, Comp. Math. Appl. 53 (2007), 1361–1366. [2] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. [3] P. K. Eloe, Y. Raffoul, D. Reid and K. Yin, Positive solutions of nonlinear functional difference equation, Comp. Math. Appl. 42 (2001), 639–646. [4] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic Press, Boston, 1992. [5] K. Gopalsamy and P. Weng, Global attractivity and level crossing in model of hematopoiesis, Bull. Inst. Math. Acad. Sinica 22 (1994), 341–360. [6] W. S. C. Gurney, S. P. Blythe, R. M. Nisbet, Nicholson’s blowflies revisited, Nature, 287 (1980), 17–20. [7] D. Q. Jiang, J. J. Wei, B. Zhang, Positive periodic solution of functional differential equations and population models, Electron. J. Diff. Eqns. 2002 (2002), No.71, 1–13. [8] W. Joseph H. So and J. Yu, Global attractivity and uniformly persistence in Nicholsons blowflies, Diff. Eqns. Dynam. Syst. 2 (1994), No.1, 11–18. [9] W. Joseph H. So, J. Wu and X. Zhous, Structured population on two patches: modelling desparel and delay, J. Math. Biol. 43 (2001), 37–51. [10] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [11] R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673–688. [12] Y. Liu, Periodic solution of nonlinear functional difference equation at nonresonance case, J. Math. Anal. Appl. 327 (2007), 801–815. [13] M. Ma and J. S. Yu, Existence of multiple positive periodic solutions for nonlinear functional difference equations, J. Math. Anal. Appl. 305 (2005), 483–490. [14] M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1997), 287–289. [15] A. J. Nicholsons, The balance of animal population, J. Animal Ecol. 2(1993), 132–278. [16] S. Padhi and S. Srivastava, Multiple periodic solutions for a nonlinear first order functional differential equations with applications to population dynamics, Appl. Math. Comp. 203 (2008), 1–6. [17] S. Padhi, S. Srivastava and S. Pati, Three periodic solutions for a nonlinear first order functional differential equation, Appl. Math. Comp. 216 (2010), 2450–2456. [18] N. Raffoul, positive periodic solutions of nonlinear functional difference equation, Electron. J. Diff. Eqns. 2002 (2002), No.55, 1–8. [19] A. Wan, D. Jiang and X. Xu, A new existence theory for nonnegative periodic solutions of functional differential equations, Comp. Math. Appl. 47 (2004), 1257–1262. [20] P. Weng and M. Liang, The existence and behaviour of periodic solutions of hematopoiesis model, Math. Appl. 4 (1995), 434–439. [21] X. Wang and Z. Li, Dynamics for a class of general hematopoiesis model with periodic coeffi- cients, Appl. Math. Comp. 186 (2007), 460–468. [22] Z. Zeng, Existence of positive periodic solutions for a class of nonautonomous difference equations, Electron. J. Diff. Eqns. 2006 (2006), No.3, 1–18.