Journal of Applied Mathematics

Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales

Yongkun Li and Chao Wang

Full-text: Open access

Abstract

Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale ( d / d t ) ( x ( t ) + c ( t ) x ( t - α ) ) = a ( t ) g ( x ( t ) ) x ( t ) - j = 1 n λ j f j ( t , x ( t - v j ( t ) ) ) , ( t , x ) T 0 ( x ) , Δ t | ( t , x ) S 2 i = Π i 1 ( t , x ) - t , Δ x | ( t , x ) S 2 i = Π i 2 ( t , x ) - x , where Π i 1 ( t , x ) = t 2 i + 1 + τ 2 i + 1 ( Π i 2 ( t , x ) ) and Π i 2 ( t , x ) = B i x + J i ( x ) + x ,    i = 1,2 , .    λ j    ( j = 1,2 , , n ) are parameters, T 0 ( x ) is a variable time scale with ( ω , p ) -property, c ( t ) ,    a ( t ) , v j ( t ), and f j ( t , x )    ( j = 1,2 , , n ) are ω -periodic functions of t , B i + p = B i ,    J i + p ( x ) = J i ( x ) uniformly with respect to i Z .

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 516476, 28 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1355495069

Digital Object Identifier
doi:10.1155/2012/516476

Mathematical Reviews number (MathSciNet)
MR2898062

Zentralblatt MATH identifier
1235.34243

Citation

Li, Yongkun; Wang, Chao. Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales. J. Appl. Math. 2012 (2012), Article ID 516476, 28 pages. doi:10.1155/2012/516476. https://projecteuclid.org/euclid.jam/1355495069


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