一類分?jǐn)?shù)階積分多點(diǎn)邊值問(wèn)題正解的存在性

戴 琛

(蘇州高等幼兒師范學(xué)校，江蘇蘇州 215008)

一類分?jǐn)?shù)階積分多點(diǎn)邊值問(wèn)題正解的存在性

戴 琛

(蘇州高等幼兒師范學(xué)校，江蘇蘇州 215008)

本文運(yùn)用上下解和不動(dòng)點(diǎn)定理的方法對(duì)一類分?jǐn)?shù)階積分多點(diǎn)邊值問(wèn)題進(jìn)行研究，通過(guò)對(duì)格林公式性質(zhì)的研究和分析得到了該問(wèn)題正解的存在性。

分?jǐn)?shù)階微分方程；不動(dòng)點(diǎn)定理；上下解；正解

我們研究如下形式的一類帶有p-Laplacian算子的分?jǐn)?shù)階m多點(diǎn)邊值問(wèn)題：

近年來(lái)，分?jǐn)?shù)階積分理論因其在物理、化學(xué)、機(jī)械、工程等領(lǐng)域的廣泛應(yīng)用而受到國(guó)內(nèi)外許多學(xué)者的關(guān)注.分?jǐn)?shù)階微分方程解的存在性研究是微分方程定性理論的重要內(nèi)容之一，也是進(jìn)一步研究分?jǐn)?shù)階微分方程理論及其應(yīng)用的重要基礎(chǔ).隨著分?jǐn)?shù)階微分方程理論的不斷發(fā)展，關(guān)于分?jǐn)?shù)階微分方程初值問(wèn)題解的在存性研究有比較豐富的結(jié)果[1-11]，文獻(xiàn)[2]運(yùn)用錐拉伸與錐壓縮不動(dòng)點(diǎn)理論討論了含參數(shù)的分?jǐn)?shù)階兩點(diǎn)邊值問(wèn)題正解的存在與不存在性，文獻(xiàn)[9]運(yùn)用上下解方法和不動(dòng)點(diǎn)定理，研究了一類分?jǐn)?shù)階兩點(diǎn)邊值問(wèn)題正解的存在性，BashirAhmad在文獻(xiàn)[5]中用壓縮映射不動(dòng)點(diǎn)定理，研究了一類分?jǐn)?shù)階反周期邊值問(wèn)題解的存在性.張淑琴在文[7]中利用不動(dòng)點(diǎn)定理，研究了一類分?jǐn)?shù)階高階奇異微分方程邊值問(wèn)題正解的存在性.

引理1 設(shè)0<ηα<α(α-1). 如果h∈C[0,1]，那么

有唯一的解

(3)

其中G(t,s)=G1(t,s)+G2(t,s)，且

引理2 邊值問(wèn)題(1)～(2)等價(jià)于下面的積分方程

.

(4)

在這部分，通過(guò)上下解的方法建立問(wèn)題(1)～(2)中正解的存在性. 假設(shè)f:[0,1]×[0,+∞)→[0,+∞)是連續(xù)函數(shù).

注：顯然，由引理1.1和引理1.2可知，當(dāng)h(t)≥0時(shí)，u(t)≥0.

定義4 如果θ(t)∈C[0,1]且θ(t)滿足

稱θ(t)是問(wèn)題(1)～(2)中的下解.

定義5 如果γ(t)∈C[0,1]且γ(t)滿足

稱γ(t)是定義問(wèn)題(1)～(2)中的上解.

本文的主要結(jié)論如下：

定理6 如果u(t)滿足下列條件：(Hf)f(t,u)∈C([0,1]×[0,+∞),R+)對(duì)變量u是單調(diào)非增函數(shù)，f(t,ρ(t))≠0，?t∈(0,1)，存在一個(gè)常數(shù)μ<1，使得kμf(t,u)≤f(t,ku) ，?0≤k≤1，那么，問(wèn)題(1)～(2)存在正解u(t).

由引理2，可知g(t)是下列方程的解

從引理3的結(jié)論可知a1ρ(t)≤g(t)≤a2ρ(t)，?t∈[0,1].

因此，利用定理6的假設(shè)條件可知

這意味著

顯然，函數(shù)θ(t)=k1g(t)和γ(t)=k2g(t)滿足邊界條件(2).因此，α(t)=k1g(t)，β(t)=k2g(t)分別是問(wèn)題(1)～(2)的下解和上解.

下面證明問(wèn)題

(5)

(6)

有解，其中

算子T:C[0,1]→[0,1]是連續(xù)的，那么G(t,s)和g(t,u(t))也是連續(xù)的函數(shù). 由Arzela-Ascoli定理可知，T是一個(gè)緊算子.因此，由Leray-Schauder的不定點(diǎn)定理可以知道，算子T是有一個(gè)不動(dòng)點(diǎn)，即問(wèn)題(5)～(6)有解.

最后，證明問(wèn)題(1)～(2)存在正解.

假設(shè)u*(f)是問(wèn)題(5)～(6)的一個(gè)解. 由于f(t,u)是關(guān)于u的單調(diào)非增函數(shù)，則

那么

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Existence of Positive Solutions to the Multi-point Boundary Value Problems of Fractional Integral of the Same Category

DAI Chen

(Suzhou Higher Infant Normal School, Suzhou Jiangsu 215008, China)

This paper applies the methods of upper and lower solutions and the fixed point theorem to the study of multi- point boundary value problems of fractional integral of the same category. Through the research and the analysis of the Green formula properties, the existence of positive solutions to the problems is obtained.

Fractional differential equation; Fixed-point theorem; Lower and upper solution; Positive solutions

2014-01-07

戴 琛(1980- )，女，江蘇蘇州人，蘇州高等幼兒師范學(xué)校講師，從事應(yīng)用數(shù)學(xué)研究。

O175

A

2095-7602(2014)04-0003-04