Existence of Positive Periodic Solutions for Two Types of Third-order Nonlinear Neutral Differential Equations with Variable Coefficients
Автор(ы):
Bouzid Mansouri
Faculty of Sciences,
Department of Mathematics,
University Annaba,
P.O. Box 12, Annaba
23000, Algeria
mansouri.math@yahoo.fr
Abdelouaheb Ardjouni
Faculty of Sciences and Technology,
Department of Mathematics and Informatics,
University Souk Ahras,
P.O. Box 1553, Souk Ahras,
41000, Algeria
abd_ardjouni@yahoo.fr
Ahcene Djoudi
Faculty of Sciences,
Department of Mathematics,
University Annaba,
P.O. Box 12, Annaba
23000, Algeria
Аннотация:
In this work, we study the existence of positive periodic solutions for two
types of third-order nonlinear neutral differential equations with variable
coefficients. The results are established by using the Krasnoselskii's fixed
point theorem. The results obtained here extend the work of Ren, Siegmund and
Chen. Two examples are given to illustrate this work.
Ключевые слова
- differential equations
- fixed point
- positive periodic solutions
- third-order neutral
Ссылки:
- A. Ardjouni and A. Djoudi, Existence of periodic solutions for a second-order nonlinear neutral differential equation with variable delay, Palestine Journal of Mathematics, Vol. 3(2) (2014), 191-197
- A. Ardjouni, A. Djoudi and A. Rezaiguia, Existence of positive periodic solutions for two types of third-order nonlinear neutral differential equations with variable delay, Applied Mathematics E-Notes, 14 (2014), 86-96
- A. Ardjouni and A. Djoudi, Existence of positive periodic solutions for a nonlinear neutral differential equations with variable delay, Applied Mathematics E-Notes, 12 (2012), 94-101
- A. Ardjouni and A. Djoudi, Existence of periodic solutions for a second order nonlinear neutral differential equation with functional delay, Electronic Journal of Qualitative Theory of Differential Equations, 2012, No. 31, 1-9
- A. Ardjouni and A. Djoudi, Periodic solutions for a second-order nonlinear neutral differential equation with variable delay, Electron. J. Differential Equations, Vol. 2011 (2011), No. 128, pp. 1-7
- A. Ardjouni and A. Djoudi, Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale, Rend. Sem. Mat. Univ. Politec. Torino Vol. 68, 4(2010), 349-359
- C. Avramescu, On a fixed point theorem, Studii si Cercetari Matematice, 9, Tome 22, 2 (1970), pp. 215-220
- C. Avramescu and C. Vladimirescu, Some remarks on Krasnoselskii's fixed point theorem, Fixed Point Theory, Volume 4, No. 1, 2003, 3-13
- T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii's theorem, Nonlinear Stud. 9 (2002), No. 2, 181-190
- T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006
- T. Candan, Existence of positive periodic solutions of first-order neutral differential equations, Math. Methods Appl. Sci. 40 (2017), 205-209
- T. Candan, Existence of positive periodic solutions of first-order neutral differential equations with variable coefficients, Applied Mathematics Letters 52 (2016), 142-148
- F. D. Chen, Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput. 162 (2005), No. 3, 1279-1302
- F. D. Chen and J. L. Shi, Periodicity in a nonlinear predator-prey system with state dependent delays, Acta Math. Appl. Sin. Engl. Ser. 21 (2005), no. 1, 49-60
- Z. Cheng and J. Ren, Existence of positive periodic solution for variable coefficient third-order differential equation with singularity, Math. Meth. Appl. Sci. 2014, 37, 2281-2289
- Z. Cheng and Y. Xin, Multiplicity Results for variable-coefficient singular third-order differential equation with a parameter, Abstract and Applied Analysis, Vol. 2014, Article ID 527162, 1-10
- S. Cheng and G. Zhang, Existence of positive periodic solutions for non-autonomous functional differential equations, Electron. J. Differential Equations, Vol. 2001 (2001), No. 59, 1-8
- H. Deham and A. Djoudi, Periodic solutions for nonlinear differential equation with functional delay, Georgian Mathematical Journal 15 (2008), No. 4, 635-642
- H. Deham and A. Djoudi, Existence of periodic solutions for neutral nonlinear differential equations with variable delay, Electronic Journal of Differential Equations, Vol. 2010 (2010), No. 127, pp. 1-8
- Y. M. Dib, M. R. Maroun and Y. N. Rafoul, Periodicity and stability in neutral nonlinear differential equations with functional delay, Electronic Journal of Differential Equations, Vol. 2005 (2005), No. 142, pp. 1-11
- M. Fan and K. Wang, P. J. Y. Wong and R. P. Agarwal, Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sin. Engl. Ser. 19 (2003), no. 4, 801-822
- H. I. Freedman, J. Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal. 23 (1992) 689-701
- M. Gregus, Third Order Linear Differential Equations, Reidel, Dordrecht, 1987
- Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993
- M. A. Krasnoselskii, Some problems of nonlinear analysis, American Mathematical Society Translations, Ser. 2, 10 (1958), pp. 345-409
- W. G. Li and Z. H. Shen, An constructive proof of the existence theorem for periodic solutions of Duffing equations, Chinese Sci. Bull. 42 (1997), 1591-1595
- Y. Liu, W. Ge, Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients, Tamsui Oxf. J. Math. Sci. 20 (2004) 235-255
- F. Nouioua, A. Ardjouni, A. Djoudi, Periodic solutions for a third-order delay differential equation, Applied Mathematics E-Notes, 16 (2016), 210-221
- D. O'Regan, Fixed-point theory for the sum of two operators, Appl. Math. Lett. 9 (1) (1996), 1-8
- J. Ren, S. Siegmund and Y. Chen, Positive periodic solutions for third order nonlinear differential equations, Electron. J. Differential Equations, Vol. 2011 (2011), No. 66, 1-19
- D. R. Smart, Fixed Points Theorems, Cambridge University Press, Cambridge, 1980
- Q. Wang, Positive periodic solutions of neutral delay equations (in Chinese), Acta Math. Sinica (N. S. ) 6(1996), 789-795
- Y. Wang, H. Lian and W. Ge, Periodic solutions for a second order nonlinear functional differential equation, Applied Mathematics Letters 20 (2007) 110-115
- E. Zeidler, Nonlinear analysis and its applications I: Fixed point theorems, Springer-Verlag, 1985
- W. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sinica (N. S. ) 13(1997), 373-380
- G. Zhang, S. Cheng, Positive periodic solutions of non-autonomous functional differential equations depending on a parameter, Abstr. Appl. Anal. 7 (2002) 279-286