偏序度量空間上壓縮映像不動點定理在分數(shù)階微分方程邊值問題上的應(yīng)用

湯 宇,許曉婕,趙 虹

(1. 吉林工商學(xué)院 基礎(chǔ)部,長春 130062;2. 中國石油大學(xué)(華東) 理學(xué)院,山東 青島 266555；3. 長春師范學(xué)院 數(shù)學(xué)學(xué)院， 長春 130032)

分數(shù)階微分方程在流體力學(xué)、 流變學(xué)、 黏彈性力學(xué)、 分數(shù)控制系統(tǒng)和分數(shù)控制器、 各種電子回路、 電分析化學(xué)、 生物系統(tǒng)的電傳導(dǎo)、 神經(jīng)的分數(shù)模型及回歸模型,特別是與分形維數(shù)有關(guān)的物理與工程問題上應(yīng)用廣泛[1-8]. 文獻[9-15]研究偏序度量空間上壓縮映像不動點的存在性,分別給出了該不動點定理在度量空間、 常微分方程和積分方程中的一些應(yīng)用. 文獻[16]給出了分數(shù)階微分方程邊值問題:

(1)

顯然,在給定距離

d(x,y)=sup{|x(t)-y(t)|:t∈[0,1],x,y∈C[0,1]}

條件下,空間C[0,1]是一個完備度量空間.C[0,1]中的偏序定義為:x,y∈C[0,1],x?y?x(t)≤y(t),t∈[0,1].

令A(yù)表示滿足下列條件的函數(shù)集φ: [0,∞) → [0,∞):

1)φ為單調(diào)增函數(shù);

2) 對任意的x>0,φ(x)

3)β(x)=φ(x)/x∈B,其中B表示滿足條件“當β(tn) → 1時,則有tn→ 0”的函數(shù)β: [0,∞) →[0,1)構(gòu)成的集合.

證明: ?t0∈[0,1],需要證明H(t)在t0上連續(xù).

情形1) 假設(shè)t0=0. 因為tσF(t)是[0,1]上的連續(xù)函數(shù),故存在常數(shù)M>0,使得|tσF(t)|≤M,t∈[0,1]. 又因為(s-t)+(α-2)(1-t)s≤(α-1)s,所以有

情形2) 假設(shè)0t0,則有

其中:

I2=(tn-s)α-1-(t0-s)α-1.

顯然

易證當tn→t0時,I1→0,I2→0.

進一步,有

令tn→t0,由上述表達式,有|H(tn)-H(t0)| → 0,n→ ∞.

0≤tσ[f(t,y)-f(t,x)]≤λφ(y-x),

其中φ∈A. 則方程(1)存在唯一解.

下面證明偏序集上壓縮映像原理的條件均成立. 先證明映射T是單調(diào)增的. 由假設(shè),對u≥v,有tσf(t,u)≥tσf(t,v),t∈[0,1]. 應(yīng)用G(t,s)>0,t,s∈(0,1),有

此外,對u≥v及u≠v,有

顯然,當u=v時,最后一個不等式成立.

由偏序集上的弱壓縮映像不動點定理可知T存在唯一不動點,即方程(1)存在唯一非負解u(t)∈C[0,1]. 事實上,方程(1)存在唯一正解. 反證法. 若存在0

則由G(t,s)≥0和f(t,y)≥0,有G(t*,s)f(s,u(s))=0 a.e.(s),且當G(t,s)>0,t∈(0,1)時,有f(s,u(s))=0 a.e.(s).

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