田元生

(湘南學院數(shù)學系,湖南郴州 423000)

帶p-Laplacian算子三點邊值問題擬對稱正解的多重性

田元生

(湘南學院數(shù)學系,湖南郴州 423000)

應用Avery-Peterson不動點定理,討論了一類帶p-Laplacian算子三點邊值問題在非線性項f依賴于未知函數(shù)的一階導數(shù)的情況下擬對稱正解的多重性,得到了這類邊值問題至少存在三個擬對稱正解的充分條件.

p-Laplacian算子;三點邊值問題;擬對稱正解;錐

1 引言

帶p-Laplacian算子微分方程的邊值問題,在非牛頓力學,宇宙物理,血漿問題和彈性理論等諸多領域都有廣泛的應用.人們對此類邊值問題正解的存在性進行了深入的研究,取得很多有意義的成果[14].最近幾年,對邊值問題的對稱正解和擬對稱正解的存在性及多重性的關注逐漸增多.文[5]通過建立一個線性算子,利用線性算子的譜理論,研究了一類邊值問題的對稱正解的存在性和多重性.文[6]給出了擬對稱函數(shù)的定義,應用五個泛函的不動點定理研究了下面邊值問題擬對稱正解的存在性.本文應用Avery-Peterson不動點定理,把文[7]中得到擬對稱正解的存在性和多重性結論,推廣到帶p-Lpalcian算子的邊值問題(Q)上.

在本文中,總假定以下條件成立

2 預備知識

為了證明本文的主要結果,需要如下的預備知識.

定義2.1設E是一個實Banach空間,如果P是E中某非空凸閉子集,并且滿足下面兩個條件:

定義2.2設η∈(0,1),稱函數(shù)u∈C1[0,1]關于η是擬對稱的,如果

定義2.3映射α稱為P上的非負連續(xù)凹泛函,只要α:P?→[0,∞)連續(xù)且

對所有的x,y∈P以及0≤t≤1成立.類似地,稱映射γ是P上的非負連續(xù)凸泛函,只要γ:P?→[0,∞)連續(xù)且

對所有的x,y∈P以及0≤t≤1成立.

定義2.4給定正數(shù)a,b,c,和d,設γ和θ是P上的非負連續(xù)凸泛函,α是P上的非負連續(xù)凹函數(shù),ψ是P上的非負連續(xù)泛函.定義凸集

本文主要結論的證明要用下面的Avery-Peterson不動點定理.

定理2.1[8]令P是實Banach空間E中的錐,設γ和θ是P上的非負連續(xù)凸泛函,α 是P上的非負連續(xù)凹泛函,ψ是P上的非負連續(xù)泛函且滿足ψ(λx)≤λψ(x),0≤λ≤1,存在正數(shù)M和d,使得

3 主要結論及其證明

于是,根據(jù)A rzela-Ascoli定理知T(D)為列緊集.其次由勒貝格控制收斂定理不難證明T在P上是連續(xù)的.因此T:P?→P是全連續(xù)的.

為了方便,引用下面記號由條件(A1)知:f(t,u,v)≤φp(d/H).另一方面,對u∈P,有Tu∈P,則Tu在[0,1]上凹,于是

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The multip licity of positive pseudo-symm etricsolutions to a three-point boundary value problem of p-Laplacian equations

TIAN Yuan-sheng

(Department of Mathem atics,X iangnan University,Chenzhou,Hunan,423000,China)

By using Avery-Peterson theorem on convex cone,we consider the three-point boundary value problem for p-Lap lace equations with the nonlinear term depending on the first order derivative,the resutlt of three positive pseudo-symm etric solutions are obtained.

p-Laplacian opertor,three-point boundary value porblem,positive pseudo-symm etric solution, cone

O175.8

A

1008-5513(2009)02-0294-08

2008-07-08.

湖南省教育廳科研項目(08C826),湖南省重點建設學科項目,湖南省高?？萍紕?chuàng)新團隊計劃項目.

田元生(1962-),副教授,研究方向：微分方程.

2000M SC:34B15