蔣玲芳

(西北師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,甘肅 蘭州 730070)

非線性一階常微分方程解的存在惟一性

蔣玲芳

(西北師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,甘肅 蘭州 730070)

討論了一類非線性一階常微分方程邊值問(wèn)題解的存在惟一性.得到了當(dāng)參數(shù)在一定的范圍取值時(shí)解存在惟一的充分條件,并包含了一些已知結(jié)果.主要結(jié)果基于Leray-Schauder非線性抉擇理論和Banach不動(dòng)點(diǎn)定理.

一階非線性微分方程;存在性;惟一性;Leray-Schauder非線性抉擇; Banach不動(dòng)點(diǎn)定理

1 引言及主要結(jié)果

許多物理現(xiàn)象都與微分方程的周期解密切相關(guān),從而周期邊值問(wèn)題引起了許多作者的研究,見(jiàn)文獻(xiàn)[1-6]及所附文獻(xiàn).特別地,2004年,文獻(xiàn)[5]在引理1中討論了一階周期邊值問(wèn)題

解的存在性,當(dāng)M＞0,則對(duì)任意的σ∈C[0,1],問(wèn)題(*1)存在一個(gè)周期解.

反周期問(wèn)題解的存在性等價(jià)于所研究問(wèn)題倍周期變號(hào)解的存在性,因此,反周期問(wèn)題解的存在性在近十幾年來(lái)也得到了許多學(xué)者的廣泛關(guān)注[710].特別地,2003年,文獻(xiàn)[8]研究了一階非線性微分方程反周期邊值問(wèn)題

解的存在性.并得到了下面的結(jié)論.

則(*2)式至少存在一個(gè)解.

可見(jiàn),周期解與反周期解存在密切的聯(lián)系,一個(gè)自然的問(wèn)題,能否對(duì)周期邊值問(wèn)題與反周期邊值問(wèn)題間建立更一般的聯(lián)系?本文將給出一個(gè)統(tǒng)一的論述.利用Leray-Schauder非線性抉擇理論和Banach不動(dòng)點(diǎn)定理考慮一階非線性微分方程

解的存在惟一性.其中,α∈?,ρ∈?,λ∈?為參數(shù),且α/=eρT.

注1當(dāng)α=1時(shí),(1)式為周期邊值問(wèn)題;當(dāng)α=-1時(shí),(1)式為反周期邊值問(wèn)題.本文的結(jié)論對(duì)周期[5]和反周期問(wèn)題[8]同樣成立.

注2對(duì)于一階泛函微分方程:

其中λ∈?是參數(shù),a(·)∈C([0,T],[0,+∞)),b(·),τ(·)∈C([0,T],?),f,g∈C([0,T],?).其中Green函數(shù)G1(t,s)為:

當(dāng)a,g,b滿足一定的條件時(shí),本文的結(jié)論對(duì)(*)式仍然成立.

本文總假定:

(H)f:[0,T]×?→?為L(zhǎng)1-Carathˊeodory函數(shù).即滿足:

(1)?x∈?,f(·,x)在[0,T]上可測(cè);

(2)對(duì)幾乎所有的t∈[0,T],f(t,·)在?上連續(xù);

(3)?r＞0,?φr(t)∈L1[0,T].使得對(duì)幾乎所有的t∈[0,T]及x∈?,當(dāng)‖u‖＜r時(shí),有|f(t,u(t)|≤φr(t).

Leray-Schauder非線性抉擇理論:

定理A設(shè)B是Banach空間,E?B是有界閉凸集,若U是E中的相對(duì)開(kāi)球,且θ∈U, S:fiU→E全連續(xù).則下面結(jié)論之一成立:

(a)S在fiU中至少存在一個(gè)不動(dòng)點(diǎn);

2 預(yù)備知識(shí)

3 主要結(jié)果的證明

由Banach不動(dòng)點(diǎn)定理,算子A在X中存在唯一的不動(dòng)點(diǎn),即(1)式在X中存在惟一解.

[1] Cheng S, Zhang G. Existence of positive periodic solutions for non-autonomous functional differential equations[J]. Electron. J. Differential Equations, 2001,59:1-8.

[2] Wan A, Jiang D. Existence of positive periodic solutions for functional differential equations[J]. Kyushu J. Math., 2002,56:193-202.

[3] Zhang G, Cheng S. Positive periodic solutions of non-autonomuos functional differential equations depending on aparameter[J]. Abstr. Appl. Anal., 2002,7:279-286.

[4] Padhi S, Shilpee Srivastava, Smita Pati. Three periodic solutions for a nonlinear first order functional differential equation[J]. Comput. Math. Appl., 2010,216:2450-2456.

[5] Peng S. Positive solutions for first order periodic boundary value problem[J]. Applied Mathmatics and Computation, 2004,158:345-351.

[6] 廖新元.非自治時(shí)滯微分方程正周期解的存在性(英文)[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2003,19(3):268-273.

[7] Okochi H. On the existence of anti-periodic solutions to nonlinear evolution equations with associated with odd subdifferential operators[J]. J. Funct. Nonlinear Anal., 1990,91:246-258.

[8] 孫晉易,蔣玲芳,楊變霞.一類泛函微分方程反周期解的存在性[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2011,27(2):280-284.

[9] 馬如云.非線性常微分方程非局部問(wèn)題[M].北京:科學(xué)出版社,2004.

[10] Franco D, Nieto J J, O'Regan D. Anti-periodic boundary value problem for nonlinear first-order differential equations[J]. Math. Inequal. Appl., 2003,6:477-485.

[11]O′Regan D.Existence Theory for Nonlinear Ordinary Diferential Equations[M].K luwer:Dordrecht,1997.

Existence and uniqueness of solutions for first-order nonlinear ordinary differential equations

Jiang Lingfang

(College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China)

In this paper, we show the existence and uniqueness of solutions of first-order nonlinear ordinary differential equation boundary value problem. The sufficient conditions for the existence and uniqueness of solutions are obtained when the parameter belongs to appropriate intervals, and we include some known results. The main results are based upon Leray-Schauder nonlinear alternative theorem and Banach's fixed point theorem.

first-order nonlinear differential equation, existence, uniqueness,Leray-Schauder nonlinear alternative,Banach′s fixed point theorem

O178

A

1008-5513(2012)02-0256-06

2011-06-29.

國(guó)家自然科學(xué)基金(10671158);甘肅省自然科學(xué)基金(3ZS051-A 25-016).

蔣玲芳(1989-),碩士生,研究方向：常微分方程邊值問(wèn)題.

2010 MSC:15A42