沈文國

(蘭州工業(yè)學(xué)院基礎(chǔ)學(xué)科部,甘肅蘭州730050)

奇異高階積分邊值問題正解的全局結(jié)構(gòu)

沈文國

(蘭州工業(yè)學(xué)院基礎(chǔ)學(xué)科部,甘肅蘭州730050)

本文研究了帶Riemann-Stieltjes積分邊值條件的奇異高階積分邊值問題正解的全局分歧結(jié)構(gòu).利用相關(guān)文獻(xiàn),獲得了此類問題的格林函數(shù)并推證其滿足的性質(zhì),同時可獲得此類問題等價于一個全連續(xù)算子方程;其次,在滿足所給的條件時,利用Krein-Rutmann定理建立了此類問題對應(yīng)的線性問題存在簡單的主特征值;最后,當(dāng)非線性項在零和無窮遠(yuǎn)處滿足非漸進(jìn)線性增長條件、參數(shù)滿足不同范圍的值時,利用Dancer全局分歧定理、Zeidler全局分歧定理和序列集取極限的方法,建立了此類問題正解的全局結(jié)構(gòu),進(jìn)而獲得了正解的存在性,推廣了文獻(xiàn)[8]中的主要結(jié)果.

奇異高階積分邊值問題;全局分岐;正解

1 引言

利用錐上不動點理論,文獻(xiàn)[1-6]研究了邊值問題正解的存在性;文獻(xiàn)[7-8]研究了帶Riemann-Stieltjes積分邊值條件的高階問題,其中2012年,當(dāng)ra(t)f(x)=λf(t,x)時,文獻(xiàn)[8]研究了下列奇異高階問題

其中f(t,x)在t=0,t=1處奇異,α,β：[0,1]→?分別是有界變差函數(shù).

應(yīng)用分歧方法,文獻(xiàn)[9-11]研究了二階邊值問題;文獻(xiàn)[12-14]研究了四階邊值問題;文獻(xiàn)[15]研究了高維問題;文獻(xiàn)[16-17]研究了帶Riemann-Stieltjes積分邊值條件問題.

受上述文獻(xiàn)的啟發(fā),本文研究奇異高階含Riemann-Stieltjes積分邊值條件的問題(1.1)正解的存在性問題.本文做如下假設(shè)

k(τ(s),s),ki(τi(s),s)分別由引理2.2與引理2.3給出;

(H4)f(·)∈C([0,∞),[0,∞)),對任何s＞0,都有f(s)＞0成立; (H5)f0,f∞∈(0,+∞);

(H6)f0∈(0,+∞)且f∞=∞; (H7)f0=0且f∞=∞; (H8)f0=∞且f∞=∞,

其中

本章安排如下：在第二部分給出格林函數(shù)及其性質(zhì);第三部分給出預(yù)備知識;第四部分給出問題(1.1)至少存在一個正解的主要定理及證明.

2 格林函數(shù)及其性質(zhì)

引理2.1(見文獻(xiàn)[8,引理1])假設(shè)條件(H1)和(H2)成立.對于任何y∈C[0,1],則問題(2.1)存在唯一解

其中

引理2.2(見文獻(xiàn)[8,引理2])由(2.4)式定義的k(t,s)滿足下列性質(zhì)

其中

引理2.3 k(t,s)由(2.4)式定義,i=2,···,n,下式成立

并且ki(t,s)滿足

證相似于文獻(xiàn)[7]第1937-1938頁中定理3.1的證明方法,易得引理2.3,故證明略.

引理2.4(見文獻(xiàn)[8,引理3])假設(shè)條件(H1)和(H2)成立.由(2.3)式定義的K(t,s)滿足下列性質(zhì)

(i)K(t,s)在[0,1]×[0,1]上連續(xù)且K(t,s)≥0;

(ii)對于任意t,s∈[0,1]都有K(t,s)≤K(s)成立,對于任意t,s∈[0,1],下式成立

引理2.5(見文獻(xiàn)[8,引理4])假設(shè)條件(H1)和(H2)成立.則對于y∈C[0,1]且y≥0, (2.1)式的唯一解滿足

其中q(t)由引理2.3(ii)給出.

3 預(yù)備知識

容易驗證L為閉算子且L-1：Y→D(L)是全連續(xù)算子.

令Σ為(1.1)在[0,∞)×E上正解集合的閉包.

定義錐

其中q(t)由引理2.3(ii)給出,且對于r＞0,令Ωr={x∈P|kxkE＜r}.首先考慮線性問題

由Krein-Rutmann定理(見文獻(xiàn)[18,定理2.5],亦可參考文獻(xiàn)[19]或[20]),可得下列引理.

引理3.1設(shè)(H1)-(H3)成立,r(Lλ)是Lλ的譜半徑.則r(Lλ)6=0且Lλ有一個對應(yīng)于第一特征值的正的特征函數(shù)φ1∈intP,它是簡單的并且再沒有別的特征值對應(yīng)正的特征函數(shù).

引理3.2設(shè)(H1)-(H4)成立,則問題(1.1)的解x(t)滿足

結(jié)論獲證.

引理3.3設(shè)(H1)-(H4)成立.假設(shè){(μk,xk)}?(0,∞)×P是問題(1.1)的一個正解序列,存在常數(shù)c0＞0,使得kμkk≤c0,且

由(H3)可得

結(jié)合引理3.2,存在常數(shù)M2＞0滿足kxk(t)kE≤M2.與已知條件矛盾,結(jié)論獲證.

引理3.4(見文獻(xiàn)[17])設(shè)X是一個Banach空間且令{Cn|n=1,2,···}是X中的閉連通分支序列.假設(shè)

(i)存在zn∈Cn,n=1,2,···和z?∈X,使得zn→z?;

(ii)rn=sup{kxk|x∈Cn}=∞;

和xni∈Cni,使得xni→x}(見文獻(xiàn)[21]).

4 主要結(jié)果

首先考慮下列特征值問題

事實上,對所有(t,s)∈[0,1]×[0,1],由引理2.1-2.3可得

其中

則對于任意t,s∈[0,1],i=1,···,n,都有

其中

由(4.4)式,i=1,···,n,可得

由L-1的緊性結(jié)合(H3),i=1,···,n,可得進(jìn)而i=1,···,n,k(L-1[a(·)ζ(x(·))])(i-1)k∞=o(kxkE).即(4.3)式得證.

由引理3.1和全局分岐定理(可參考Dancer[22]和Zeidler[23]推論15.12),對于問題(4.2),可得如下結(jié)論.

引理4.1令(H1)-(H5)成立,(rλ

f10,0)是問題(4.2)的一個分岐點.進(jìn)而,存?在式正解的一個連通分支C,滿足C(?[0,∞)×E),并且C在[0,∞)×P中連接和

注4.1問題(4.1)的形如(1,x)的任何解將產(chǎn)生問題(1.1)的一個解x.為了獲得結(jié)論,僅僅證明C在[0,∞)×P中穿過超平面{1}×E即可.

下面是本文主要結(jié)果.

定理4.1令(H1)-(H5)成立.要么λ1/f∞＜r＜λ1/f0成立,要么λ1/f0＜r＜λ1/f∞成立.則問題(1.1)至少有一個正解.

證由引理4.1易得結(jié)論,故證明略.

定理4.2令(H1)-(H4)和(H6)成立.假設(shè)r∈(0,λf01).則問題(1.1)至少有一個正解.

證受文獻(xiàn)[24]的啟發(fā),可以定義截斷函數(shù)f如下

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GLOBAL BIFURCATION OF POSITIVE SOLUTIONS FOR SINGULAR HIGH-ORDER PROBLEMS INVOLVING STIELTJES INTEGRAL CONDITIONS

SHEN Wen-guo
(Department of Basic Courses,Lanzhou Institute of Technology,Lanzhou 730050,China)

In this paper,we establish global bifurcation structure of positive solutions for a class of singular higher-order boundary value problems.First,according to the relevant literature,we obtain that the Green fuction and its property for the above problem.Meanwhile, we can obtain that the above problem is equivalent to the completely continuous operator equation.Second,we have that the above linear problem exists simple principal eigenvalue by the Krein-Rutman theorem.Finally,we establish the global bifurcation structure of positive solutions with non-asymptotic nonlinearity at or by Dancer and Zeidler global bifurcation theorems and the approximation of connected components which extends and improves the corresponding results of Shen[8].

high-order singular boundary problems;global bifurcation;positive solutions

O175.8

A

0255-7797(2017)05-1054-11

2016-01-04接收日期：2016-04-22

國家自然科學(xué)基金(11561038);甘肅省自然科學(xué)基金(145RJZA087)

沈文國(1963-),男,甘肅景泰,教授,主要研究方向：分歧理論及非線性微分方程.

2010 MR Subject Classif i cation：34B15;34K18