Existence and Asymptotic Behavior of Solutions for Weighted -Laplacian Integrodifferential System Multipoint and Integra
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Research Article Existence and Asymptotic Behavior of Solutions for Weighted pt-Laplacian Integrodifferential System Multipoint and Integral Boundary Value Problems in Half Line Yan Wang,1, 2 Yunrui Guo,3 and Qihu Zhang1, 2 1
Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China 2 School of Mathematical Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China 3 Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China Correspondence should be addressed to Qihu Zhang, [email protected] Received 5 February 2010; Revised 13 June 2010; Accepted 28 June 2010 Academic Editor: P. J. Y. Wong Copyright q 2010 Yan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the existence and asymptotic behavior of solutions for weighted ptLaplacian integro-differential system with multipoint and integral boundary value condition in half line. When the nonlinearity term f satisfies sub-p− − 1 growth condition or general growth condition, we give the existence of solutions via Leray-Schauder degree. Moreover, the existence of nonnegative solutions has been discussed.
1. Introduction In this paper, we consider the existence and asymptotic behavior of solutions for the following weighted pt-Laplacian integrodifferential system: −Δpt u δf t, u, wt1/pt−1 u , Su, T u 0,
t ∈ 0, ∞,
1.1
with the following multipoint and integral boundary value condition: u0
m−2
αi uξi e0 ,
i1
lim ut
t → ∞
∞ etutdt, 0
1.2
2
Journal of Inequalities and Applications
where u : 0, ∞ → RN ; S and T are linear operators defined by Sut
t
T ut
ψs, tusds,
∞
0
χs, tusds,
1.3
0
∞ where ψ ∈ CD, R, χ ∈ CD, R, D {s, t ∈ 0, ∞ × 0, ∞}; 0 |ψs, t|ds and ∞ |χs, t|ds are uniformly bounded with t; p ∈ C0, ∞, R, pt > 1, limt → ∞ pt exists 0 and limt → ∞ pt > 1; −Δpt u : −wt|u |pt−2 u is called the weighted pt-Laplacian; w ∈ C0, ∞, R satisfies 0 < wt, for all t ∈ 0, ∞, and wt−1/pt−1 ∈ L1 0, ∞; 0 < ξ1 < 1 · · · 0 see 6 , but the fact that λp > 0 is very important in the study of p-Laplacian problems. b If wt ≡ 1, pt ≡ p a constant and −Δp u > 0, then u is concave; this property is used extensively in the study of one-dimensional p-Laplacian problems, but it is invalid for −Δpt . It is another difference between −Δp and −Δpt . There are many results on the existence of solutions for p-Laplacian equation with integral boundary value conditions see 19–24 . On the existence of solutions for px-Laplacian systems boundary value problems, we refer to 4–7, 12–17 . On the pLaplacian equation multipoint problems, we refer to 25–27 and the references therein. In 25 , u
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