×

zbMATH — the first resource for mathematics

Multiple positive periodic solutions for a nonlinear first order functional difference equation. (English) Zbl 1204.39013
The authors investigate the existence of positive periodic solutions for the following first order functional difference equation \[ x(n+1)-x(n)=-a(n)x(n)+f(n,x(h_1(n)),\dots,x(h_m(n))),\quad n=0,1,2,\dots, \]
where \(a(n),h_i(n)\) are periodic sequences with a common period \(T\geq 1\), and \(f(n,x_1,\dots,x_m)\) is a real-valued function which is \(T\)-periodic on \(n\) and continuous on \(x_i\geq 0\).
By using the Leggett-Williams fixed point theorem, the authors establish three theorems, which ensure that the above functional difference equation has at least three non-negative periodic solutions. The main results of this paper provide a partial answer to a problem proposed by Y. N. Raffoul [Electron. J. Differ. Equ. 2002, Paper No. 55, 8p. (2002; Zbl 1007.39005)]. In addition, the authors apply their results to a hematopoiesis model in population dynamics.

MSC:
39A23 Periodic solutions of difference equations
39A10 Additive difference equations
39A12 Discrete version of topics in analysis
34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1016/j.camwa.2006.12.012 · Zbl 1123.34328 · doi:10.1016/j.camwa.2006.12.012
[2] DOI: 10.1016/S0898-1221(01)00183-3 · Zbl 0998.39003 · doi:10.1016/S0898-1221(01)00183-3
[3] Gopalsamy K., Bull. Inst. Math. Acad. Sin. 22 pp 341– (1994)
[4] DOI: 10.1038/287017a0 · doi:10.1038/287017a0
[5] DOI: 10.1016/S0898-1221(03)00103-2 · Zbl 1052.39008 · doi:10.1016/S0898-1221(03)00103-2
[6] Jiang D.Q., Electron. J. Differ. Equ. 2002 pp 1– (2002)
[7] Joseph W., Differential Equations Dynam. Systems 2 pp 11– (1994)
[8] Joseph W., J. Math. Biol. 43 pp 7– (2001)
[9] Kuang Y., Delay Differential Equations with Applications in Population Dynamics (1993) · Zbl 0777.34002
[10] DOI: 10.1512/iumj.1979.28.28046 · Zbl 0421.47033 · doi:10.1512/iumj.1979.28.28046
[11] DOI: 10.1016/j.jmaa.2006.04.027 · Zbl 1113.39006 · doi:10.1016/j.jmaa.2006.04.027
[12] DOI: 10.1016/j.jmaa.2004.11.010 · Zbl 1070.39019 · doi:10.1016/j.jmaa.2004.11.010
[13] DOI: 10.1126/science.267326 · Zbl 1383.92036 · doi:10.1126/science.267326
[14] Nicholsons A.J., J. Anim. Ecol. 2 pp 132– (1993)
[15] Padhi S., Appl. Math. Comput. (2008)
[16] S. Padhi, S. Srivastava, and J.G. Dix, Existence of three nonnegative periodic solutions for functional differential equations and applications to hemaopoiesis, submitted for publication · Zbl 1187.34119
[17] S. Padhi, S. Srivastava, and S. Pati, Three periodic solutions for a nonlinear first order functional differential equation, communicated · Zbl 1198.34140
[18] Raffoul N., Electron. J. Differ. Equ. 2002 pp 1– (2002) · Zbl 1029.34021 · doi:10.14232/ejqtde.2002.1.15
[19] DOI: 10.1016/S0898-1221(04)90120-4 · Zbl 1073.34082 · doi:10.1016/S0898-1221(04)90120-4
[20] Wang X., Appl. Math. Comput. (2006)
[21] Weng P., Math. Appl. 4 pp 434– (1995)
[22] DOI: 10.1155/ADE/2006/90479 · Zbl 1134.39008 · doi:10.1155/ADE/2006/90479
[23] Zhang R.Y., Funct. Differ. Equ. 7 pp 223– (2000)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.