奇異高階積分邊值問(wèn)題正解的全局結(jié)構(gòu)

沈文國(guó)

(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(wèn)題正解的全局結(jié)構(gòu)

沈文國(guó)

(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(wèn)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(lèi)問(wèn)題的格林函數(shù)并推證其滿足的性質(zhì),同時(shí)可獲得此類(lèi)問(wèn)題等價(jià)于一個(gè)全連續(xù)算子方程;其次,在滿足所給的條件時(shí),利用Krein-Rutmann定理建立了此類(lèi)問(wèn)題對(duì)應(yīng)的線性問(wèn)題存在簡(jiǎn)單的主特征值;最后,當(dāng)非線性項(xiàng)在零和無(wú)窮遠(yuǎn)處滿足非漸進(jìn)線性增長(zhǎng)條件、參數(shù)滿足不同范圍的值時(shí),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(lèi)問(wèn)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(wèn)題;全局分岐;正解

1 引言

利用錐上不動(dòng)點(diǎn)理論,文獻(xiàn)[1-6]研究了邊值問(wèn)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(wèn)題,其中2012年,當(dāng)ra(t)f(x)=λf(t,x)時(shí),文獻(xiàn)[8]研究了下列奇異高階問(wèn)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(wèn)題;文獻(xiàn)[12-14]研究了四階邊值問(wèn)題;文獻(xiàn)[15]研究了高維問(wèn)題;文獻(xiàn)[16-17]研究了帶Riemann-Stieltjes積分邊值條件問(wèn)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(wèn)題(1.1)正解的存在性問(wèn)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(duì)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(shí);第四部分給出問(wèn)題(1.1)至少存在一個(gè)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(jiàn)文獻(xiàn)[8,引理1])假設(shè)條件(H1)和(H2)成立.對(duì)于任何y∈C[0,1],則問(wèn)題(2.1)存在唯一解

其中

引理2.2(見(jiàn)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(yè)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(jiàn)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(duì)于任意t,s∈[0,1]都有K(t,s)≤K(s)成立,對(duì)于任意t,s∈[0,1],下式成立

引理2.5(見(jiàn)文獻(xiàn)[8,引理4])假設(shè)條件(H1)和(H2)成立.則對(duì)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(shí)

容易驗(yàn)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(duì)于r＞0,令Ωr={x∈P|kxkE＜r}.首先考慮線性問(wèn)題

由Krein-Rutmann定理(見(jiàn)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(gè)對(duì)應(yīng)于第一特征值的正的特征函數(shù)φ1∈intP,它是簡(jiǎn)單的并且再?zèng)]有別的特征值對(duì)應(yīng)正的特征函數(shù).

引理3.2設(shè)(H1)-(H4)成立,則問(wèn)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(wèn)題(1.1)的一個(gè)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(jiàn)文獻(xiàn)[17])設(shè)X是一個(gè)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(jiàn)文獻(xiàn)[21]).

4 主要結(jié)果

首先考慮下列特征值問(wèn)題

事實(shí)上,對(duì)所有(t,s)∈[0,1]×[0,1],由引理2.1-2.3可得

其中

則對(duì)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(duì)于問(wèn)題(4.2),可得如下結(jié)論.

引理4.1令(H1)-(H5)成立,(rλ

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

注4.1問(wèn)題(4.1)的形如(1,x)的任何解將產(chǎn)生問(wèn)題(1.1)的一個(gè)解x.為了獲得結(jié)論,僅僅證明C在[0,∞)×P中穿過(guò)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(wèn)題(1.1)至少有一個(gè)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(wèn)題(1.1)至少有一個(gè)正解.

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

沈文國(guó)(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