Positive Solutions for a Higher-Order Nonlinear Neutral Delay Differential Equation

Zeqing Liu, Ming Jia, Jeong Sheok Ume and Shin Min Kang

Abstract and Applied Analysis, 2011, vol. 2011, 1-16

Abstract:

This paper deals with the higher-order nonlinear neutral delay differential equation ( ð ‘‘ ð ‘› / ð ‘‘ ð ‘¡ ð ‘› ) [ ð ‘¥ ( ð ‘¡ ) + ∑ ð ‘š ð ‘– = 1 ð ‘ ð ‘– ( ð ‘¡ ) ð ‘¥ ( 𠑇 ð ‘– ( ð ‘¡ ) ) ] + ( ð ‘‘ ð ‘› − 1 / ð ‘‘ ð ‘¡ ð ‘› − 1 ) ð ‘“ ( ð ‘¡ , ð ‘¥ ( ð ›¼ 1 ( ð ‘¡ ) ) , … , ð ‘¥ ( ð ›¼ 𠑘 ( ð ‘¡ ) ) ) + â„Ž ( ð ‘¡ , ð ‘¥ ( ð ›½ 1 ( ð ‘¡ ) ) , … , ð ‘¥ ( ð ›½ 𠑘 ( ð ‘¡ ) ) ) = ð ‘” ( ð ‘¡ ) , ð ‘¡ ≥ ð ‘¡ ð ‘œ , where ð ‘› , ð ‘š , 𠑘 ∈ â„• , ð ‘ ð ‘– , ð œ ð ‘– , ð ›½ ð ‘— , ð ‘” ∈ ð ¶ ( [ ð ‘¡ ð ‘œ , + ∞ ) , â„ ) , ð ›¼ ð ‘— ∈ ð ¶ ð ‘› − 1 ( [ ð ‘¡ ð ‘œ , + ∞ ) , â„ ) , ð ‘“ ∈ ð ¶ ð ‘› − 1 ( [ ð ‘¡ ð ‘œ , + ∞ ) × ℠𠑘 , â„ ) , â„Ž ∈ ð ¶ ( [ ð ‘¡ ð ‘œ , + ∞ ) × ℠𠑘 , â„ ) , and l i m ð ‘¡ → + ∞ ð œ ð ‘– ( ð ‘¡ ) = l i m ð ‘¡ → + ∞ ð ›¼ ð ‘— ( ð ‘¡ ) = l i m ð ‘¡ → + ∞ ð ›½ ð ‘— ( ð ‘¡ ) = + ∞ , ð ‘– ∈ { 1 , 2 , … , ð ‘š } , ð ‘— ∈ { 1 , 2 , … , 𠑘 } . By making use of the Leray-Schauder nonlinear alterative theorem, we establish the existence of uncountably many bounded positive solutions for the above equation. Our results improve and generalize some corresponding results in the field. Three examples are given which illustrate the advantages of the results presented in this paper.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:328956

DOI: 10.1155/2011/328956

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