偏序度量空間上壓縮映像不動點定理在分數階微分方程邊值問題上的應用

湯 宇,許曉婕,趙 虹

(1. 吉林工商學院 基礎部,長春 130062;2. 中國石油大學(華東) 理學院,山東 青島 266555；3. 長春師范學院 數學學院， 長春 130032)

分數階微分方程在流體力學、 流變學、 黏彈性力學、 分數控制系統和分數控制器、 各種電子回路、 電分析化學、 生物系統的電傳導、 神經的分數模型及回歸模型,特別是與分形維數有關的物理與工程問題上應用廣泛[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用. 文獻[16]給出了分數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表示滿足下列條件的函數集φ: [0,∞) → [0,∞):

1)φ為單調增函數;

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β: [0,∞) →[0,1)構成的集合.

證明: ?t0∈[0,1],需要證明H(t)在t0上連續.

情形1) 假設t0=0. 因為tσF(t)是[0,1]上的連續函數,故存在常數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是單調增的. 由假設,對u≥v,有tσf(t,u)≥tσf(t,v),t∈[0,1]. 應用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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