一類(lèi)帶有成對(duì)邊界條件的奇異半正分?jǐn)?shù)階差分系統(tǒng)的正解
艾尚明,盧源秀,高 鵬,葛 琦
(延邊大學(xué)理學(xué)院 數(shù)學(xué)系,吉林 延吉 133002)
針對(duì)一類(lèi)帶有成對(duì)分?jǐn)?shù)階邊界條件的奇異半正分?jǐn)?shù)階差分系統(tǒng)正解的存在性,首先分析該系統(tǒng)的格林函數(shù)的一些性質(zhì),然后利用Banach空間錐上的不動(dòng)點(diǎn)定理,證明當(dāng)參數(shù)λ屬于不同范圍時(shí),該系統(tǒng)正解的存在性,最后舉例表明其結(jié)果的正確性.
奇異半正分?jǐn)?shù)階差分系統(tǒng);Green函數(shù);不動(dòng)點(diǎn)定理;成對(duì)邊界條件
2095-4107(2014)
04-0103-16
近年來(lái),隨著分?jǐn)?shù)階微分學(xué)理論在眾多科學(xué)領(lǐng)域的廣泛應(yīng)用,分?jǐn)?shù)階差分方程理論作為新的研究領(lǐng)域越來(lái)越受到關(guān)注[1-2].目前,關(guān)于分?jǐn)?shù)階差分方程解的存在性研究取得很大進(jìn)展[3-15].其中Goodrich C S[8]研究階數(shù)在(1,2]內(nèi)的分?jǐn)?shù)階差分系統(tǒng)正解的存在性.基于此,筆者研究分?jǐn)?shù)階差分系統(tǒng)正解的存在性,即
關(guān)于分?jǐn)?shù)階差分理論的相關(guān)基本概念和性質(zhì)見(jiàn)文獻(xiàn)[9-12].
將邊值條件x(ν-3)=y(ν-3)=0分別代入式(9)和式(10)得出C3=3=0.由于
則由邊值條件[Δαx(t)]|t=ν-α-2=0,得出C2=0.同理2=0.再由邊值條件
證明 由定理2.1和注2.5知定理2.6成立.
為了方便證明,構(gòu)造Banach空間:
由定理2.1分?jǐn)?shù)階差分系統(tǒng)式(22)可表示為
因此,分?jǐn)?shù)階差分系統(tǒng)式(22)與系統(tǒng)式(24)同解,所以要求分?jǐn)?shù)階差分系統(tǒng)式(1)的解,只需求系統(tǒng)式(24)的解.為此,對(duì)于(x,y)∈P 定義算子平凡完全連續(xù)算子A:A(x,y)=(A1(x,y),A2(x,y)),且
其中i=1,2,i+j=3.由于
為了方便,固定整數(shù)δ,σ(0≤δ<σ≤T-1)?[0,T-1]N0,且記
因此,對(duì)于?(x,y)∈P∩?Ω4,有
例4.1 考慮奇異分?jǐn)?shù)階差分系統(tǒng):
利用Banach空間錐上的不動(dòng)點(diǎn)定理,研究一類(lèi)帶有成對(duì)分?jǐn)?shù)階邊界條件的奇異半正分?jǐn)?shù)階差分系統(tǒng)正解的存在性,獲得當(dāng)參數(shù)λ屬于不同范圍時(shí),該系統(tǒng)存在正解的充分條件.
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O175.6
A
DOI 10.3969/j.issn.2095-4107.2014.04.016
2014-03-26;
關(guān)開(kāi)澄
國(guó)家自然科學(xué)基金項(xiàng)目(11161049);延邊大學(xué)本科生科研立項(xiàng)基金項(xiàng)目(2013~2014)
艾尚明(1990-),男,碩士研究生,主要從事微分方程理論及應(yīng)用方面的研究.
葛 琦,E-mail:geqi9688@163.com