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2018 On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay
Nai-Sher Yeh
Abstr. Appl. Anal. 2018: 1-6 (2018). DOI: 10.1155/2018/5321314

Abstract

For each x 0 [ 0,2 π ) and k N , we obtain some existence theorems of periodic solutions to the two-point boundary value problem u ( x ) + k 2 u ( x - x 0 ) + g ( x , u ( x - x 0 ) ) = h ( x ) in ( 0 , 2 π ) with u ( 0 ) - u ( 2 π ) = u ( 0 ) - u ( 2 π ) = 0 when g : ( 0,2 π ) × R R is a Caratheodory function which grows linearly in u as u , and h L 1 ( 0,2 π ) may satisfy a generalized Landesman-Lazer condition ( 1 + s i g n ( β ) ) 0 2 π h ( x ) v ( x ) d x < v ( x ) > 0 g β + ( x ) v x 1 - β d x + v ( x ) < 0 g β - ( x ) v x 1 - β d x for all v N ( L ) \ { 0 } . Here N ( L ) denotes the subspace of L 1 ( 0,2 π ) spanned by sin k x and cos k x , - 1 < β 0 , g β + ( x ) = l i m i n f u ( g x , u u / u 1 - β ) , and g β - ( x ) = l i m i n f u - ( g x , u u / u 1 - β ) .

Citation

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Nai-Sher Yeh. "On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay." Abstr. Appl. Anal. 2018 1 - 6, 2018. https://doi.org/10.1155/2018/5321314

Information

Received: 24 October 2017; Accepted: 28 January 2018; Published: 2018
First available in Project Euclid: 8 May 2018

zbMATH: 06929588
MathSciNet: MR3786304
Digital Object Identifier: 10.1155/2018/5321314

Rights: Copyright © 2018 Hindawi

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