BOUNDEDNESS, PERIODIC SOLUTIONS AND STABILITY IN NEUTRAL FUNCTIONAL DELAY EQUATIONS WITH APPLICATION TO BERNOULLI TYPE DIFFERENTIAL EQUATIONS

TitleBOUNDEDNESS, PERIODIC SOLUTIONS AND STABILITY IN NEUTRAL FUNCTIONAL DELAY EQUATIONS WITH APPLICATION TO BERNOULLI TYPE DIFFERENTIAL EQUATIONS
Publication TypeJournal Article
Year of Publication2015
AuthorsRAFFOUL, YOUSSEFN, UNAL, MEHMET
Volume19
Issue1
Start Page149
Pagination13
Date Published2015
ISSN1083-2564
AMS34K20, 45D05, 45J05
Abstract

We use Two fixed point theorems to prove the existence of Bounded solution, periodic solution and stability of solutions of the functional neutral differential equation $${ \frac{d}{dt} [x(t) − cx(t − τ)] = −a(t)x(t − r_1) + b(t)f(x(t − r_2(t)). }$$ Then we apply our results to the neutral Bernoulli differential equation $${ \frac{d}{dt} [x(t) − cx(t − τ)] = −a(t)x(t − r_1) + b(t)x^{\frac{1}{3}} (t − r_2(t)). }$$

URLhttp://www.acadsol.eu/en/articles/19/1/11.pdf
Refereed DesignationRefereed
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REFERENCES
[1] E. Beretta, F. Solimano, and Y. Takeuchi, A mathematical model for drug administration by using the phagocytosis of red blood cells, J Math Biol. 1996 Nov; 35(1), 1–19.
[2] T. A. Burton and T. Furumochi, Asymptotic behavior of solutions of functional differential
equations by fixed point theorems, Dynamic Systems Appl., 11 (2002), 459–521.
[3] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover, New York, 2006.
[4] Y. Chen, New results on positive periodic solutions of a periodic integro-differential competition system, Appl. Math. Comput., 153 (2) (2004), 557–565.
[5] F. D. Chen, Positive periodic solutions of neutral Lotka-Volterra system with feedback control,
Appl. Math. Comput., 162 (3) (2005), 1279–1302.
[6] F. D. Chen, Periodicity in a nonlinear predator-prey system with state dependent delays, Acta
Math. Appl. Sinica English Series, 21 (1) (2005), 49–60.
[7] F. D. Chen and S. J. Lin, Periodicity in a Logistic type system with several delays, Comput. Math. Appl. 48 (1-) (2004), 35–44.
[8] F. D. Chen, F. X. Lin, and X. X. Chen, Sufficient conditions for the existence of positive periodic
solutions of a class of neutral delay models with feedback control, Appl. Math. Comput., 158 (1) (2004), 45–68.
[9] S. N. Chow, Existence of periodic solutions of autonomous functional differential equations, J. Differential Equations, 15(1974), 350–378.
[10] M. Fan and K. Wang, Global periodic solutions of a generalized n-species Gilpin-Ayala competition model, Comput. Math. Appl., 40 (2000), 1141–1151.
[11] M. Fan, P. Y. Wong, and R. P. Agarwal, Periodicity and stability in a periodic n-species
Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica, 19 (4) (2003), 801–822.
[12] M. E. Gilpin and F. J. Ayala, Global models of growth and competition, Proc. Natl. Acad. Sci., USA 70 (1973) 3590–3593.
[13] K. Gopalsamy, Stability and Oscillations in delay differential equations of population dynamics,
Kluwer Academic Press, Boston, 1992.
[14] K. Gopalsamy, X. He, and L. Wen, On a periodic neutral logistic equation, Glasgow Math. J.,
33 (1991), 281–286.
[15] H. F. Huo and W. T. Li, Periodic solutions of a periodic Lotka-Volterra system with delay,
Appl. Math. Comput., 156 (3) (2004), 787–803.
[16] D. Q. Jiang and J. J. Wei, Existence of positive periodic solutionsfor Volterra integro-differential
equations, Acta Mathematica Scientia, 21B(4)(2002), 553–560.
[17] L. Y. Kun, Periodic solution of a periodic neutral delay equation, J. Math. Anal. Appl., 214
(1997), 11–21.
[18] Y. K. Li and Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and
Systems, J. Math. Anal. Appl., 255 (1) (2001), 260–280.
[19] Y. Li, G. Wang, and H. Wang, Positive periodic solutions of neutral logistic equations with
distributed delays, Electron. J. Diff. Eqns., Vol. 2007(2007), No. 13, 1–10.
[20] E. R. Kaufmann and Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation
on a time scale, J. Math. Anal. Appl.,319 (2006), 315–325.
[21] Y. K. Li, Periodic solutions for delay Lotka-Volterra competition systems, J. Math. Anal. Appl.,
246 (1) (2000) 230–244.
[22] Z. Li and X. Wang, Existence of positive periodic solutions for neutral functional differential
equations, Electron. J. Diff. Eqns., Vol. 2006(2006), No. 34, 1–8.
[23] Y. N. Raffoul, Periodic solutions for neutral nonlinear differential equations with functional
delay, Electron. J. Diff. Eqns., Vol. 2003(2003), No. 102, 1–7.
[24] Y. N. Raffoul, Stability in neutral nonlinear differential equations with functional delays using
fixed point theory, Math. Comput. Modelling, 40 (2004), no. 7-8, 691–700.
[25] Y. N. Raffoul, Existence of positive periodic solutions in neutral nonlinear equations with functional
delay, Rocky Mountain Journal of Mathematics, 42 (2012), no. 6, 1–11.
[26] Y. N. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, Electronic
Journal of Qualitative Theory of Differential Equations, Vol. 2007(2007), No. 10, 1–10.
[27] W. J. H. So and J. Wu, X. Zou, Structured population on two patches: modeling desperal and
delay, J. Math. Biol., 43 (2001), 37–51.
[28] Y. Song, Positive periodic solutions of a periodic survival red blood cell model, Applicable
Analysis, Volume 84, Number 11 / November 2005.
[29] M. Wazewska-Czyzewska and A. Lasota, Mathematical models of the red cell system, Matematyta
Stosowana, 6(1976), 25–40.
[30] P. Weng and M. Liang, The existence and behavior of periodic solution of Hematopoiesis model,
Mathematica Applicate., 8(4) (1995), 434–439.
[31] W. Xu and J. Li, Global attractivity of the model for the survival of red blood cells with several
delays, Ann. Differential Equations, 14 (1998), 257–263.