Abstract and Applied Analysis

Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

Haidong Liu and Cuiqin Ma

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Abstract

We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: ( p ( t ) x Δ n - 1 ( t ) α - 1 x Δ n - 1 ( t ) ) Δ + f ( t , x ( δ ( t ) ) )  +  a σ ( b ) k ( t , s ) x ( g ( t , s ) ) θ ( s ) s g n ( x ( g ( t , s ) ) ) Δ ξ ( s ) = 0 , where t [ t 0 , ) 𝕋 : = [ t 0 , ) 𝕋 , 𝕋 a time scale which is unbounded above, n 2 is even, f ( t , u ) q ( t ) u α , α > 0 is a constant, and θ : [ a , b ] 𝕋 1 is a strictly increasing right-dense continuous function; p , q : [ t 0 , ) 𝕋 , k : [ t 0 , ) 𝕋 × [ a , b ] 𝕋 1 , δ : [ t 0 , ) 𝕋 [ t 0 , ) 𝕋 , and g : [ t 0 , ) 𝕋 × [ a , b ] 𝕋 1 [ t 0 , ) 𝕋 are right-dense continuous functions; ξ : [ a , b ] 𝕋 1 is strictly increasing. Our results extend and supplement some known results in the literature.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 395381, 8 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412273243

Digital Object Identifier
doi:10.1155/2014/395381

Mathematical Reviews number (MathSciNet)
MR3186963

Zentralblatt MATH identifier
07022304

Citation

Liu, Haidong; Ma, Cuiqin. Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals. Abstr. Appl. Anal. 2014 (2014), Article ID 395381, 8 pages. doi:10.1155/2014/395381. https://projecteuclid.org/euclid.aaa/1412273243


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