Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. (English) Zbl 1254.34128

Summary: Let \(\mathbb{T}\) be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay \[ x^\Delta(t)= -a(t) x^3(\sigma(t))+ c(t) x^{\widetilde\Delta}(t- r(t))+ G(t, x^3(t), x^3(t- r(t))),\quad t\in\mathbb{T}, \] where \(f^\Delta\) is the \(\Delta\)-derivative on \(\mathbb{T}\) and \(f^{\widetilde\Delta}\) is the \(\Delta\)-derivative on \((\mathrm{id}-r)(\mathbb{T})\). We invert this equation to construct a sum of a compact map and a large contraction which is suitable for applying the Burton-Krasnoselskii’s theorem.


34N05 Dynamic equations on time scales or measure chains
34K13 Periodic solutions to functional-differential equations
34K40 Neutral functional-differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI


[1] Adıvar, M.; Raffoul, Y. N., Existence of periodic solutions in totally nonlinear delay dynamic equations, Electron J Qual Theory Differ Equat, 1, 1-20 (2009) · Zbl 1195.34138
[2] Ardjouni, A.; Djoudi, A., Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale, Rend Sem Mat Univ Politec Torino, 68, 4, 349-359 (2010) · Zbl 1226.34062
[4] Bohner, M.; Peterson, A., Dynamic equations on time scales. Dynamic equations on time scales, An introduction with applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[5] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[6] Burton, T. A., Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud, 9, 2, 181-190 (2002) · Zbl 1084.47522
[7] Burton, T. A., Stability by fixed point theory for functional differential equations (2006), Dover: Dover New York · Zbl 1160.34001
[8] Deham, H.; Djoudi, A., Periodic solutions for nonlinear differential equation with functional delay, Georgian Math J, 15, 4, 635-642 (2008) · Zbl 1171.47061
[9] Deham, H.; Djoudi, A., Existence of periodic solutions for neutral nonlinear differential equations with variable delay, Electron J Differ Equat, 2010, 127, 1-8 (2010) · Zbl 1203.34110
[10] Kaufmann, E. R.; Raffoul, Y. N., Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J Math Anal Appl, 319, 1, 315-325 (2006) · Zbl 1096.34057
[11] Kaufmann, E. R.; Raffoul, Y. N., Periodicity and stability in neutral nonlinear dynamic equation with functional delay on a time scale, Electron J Differ Equat, 27, 1-12 (2007) · Zbl 1118.34058
[12] Smart, D. R., Fixed point theorems, Camb Tracts in Math, vol. 66 (1974), Cambridge University Press: Cambridge University Press London-New York · Zbl 0297.47042
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