杜睿娟

(甘肅政法學(xué)院計算機(jī)科學(xué)學(xué)院,甘肅蘭州 730070)

共振情形下四點泛函邊值問題解的存在性

杜睿娟

(甘肅政法學(xué)院計算機(jī)科學(xué)學(xué)院,甘肅蘭州 730070)

運(yùn)用Mawhin重合度理論,討論了一類二階四點泛函邊值問題解的存在性和多解性.分別在非線性項f有界和無界的條件下,獲得了此類泛函邊值問題解的存在性結(jié)果.

泛函邊值問題;存在性;Caratheodory條件;共振

1 引言

微分方程多點邊值問題在經(jīng)濟(jì)學(xué)、人口動力學(xué)、生態(tài)學(xué)等方面有著廣泛的應(yīng)用,例如動物血紅細(xì)胞存在模型、人口動力系統(tǒng)模型等.因此,對微分方程邊值問題解存在性的研究就更具有現(xiàn)實意義.近年來,諸多學(xué)者對微分方程邊值問題解的存在性進(jìn)行了深入的研究,并取得了豐富的研究成果[1-5].特別地,文獻(xiàn)[6]討論了如下非線性二階四點微分方程邊值問題:

解的存在性.顯然,上述非線性項僅與x,x′有關(guān),而不依賴于x,x′的函數(shù)F(x),G(x′).因此,自然地,當(dāng)非線性項不僅與x,x′有關(guān),而且還依賴于x,x′的函數(shù)F(x),G(x′)時,上述微分方程邊值問題的解是否仍然存在?基于此,本文討論了如下共振情形下二階四點泛函邊值問題:

解的存在性,其中a,b,c,d∈R,a＜c≤d＜b,f:[a,b]×R4→R滿足Caratheodory條件, F,G∈D,D為全體連續(xù)有界算子.得到了上述邊值問題有解以及多解的充分條件,豐富了多點泛函邊值問題解的存在性理論.本文用到的非線性工具為Mawhin[7]重合度理論.

2 預(yù)備知識

3 非線性項f有界情形下解的存在性

4 非線性項f無界情形下解的存在性

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[3]An Y.Existence of solution for a three-point boundary problem at resonance[J].Non.Anal.,2006,65(3):1633-1643.

[4]莫宜春,孫晉易,王珍燕.一類泛函微分方程半正問題的正周期解[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2012,28(1):137-143.

[5]姚慶六.線性增長限制下一類三階邊值問題的可解性[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2005,21(2):164-167.

[6]Rachunkove,Stanek.Topological degree method in functional boundary value problems at resonance[J]. Nonlinear Anal.,1996,27:271-285.

[7]Mawhin J.Topological degree methods in nonlinear boundary value problems[C]//NSF-CBMS Regional Conference series in Mathematics.Providence R I:American Mathematical Society,1997.

Solvability of functional differential equations with four-point boundary value problem at resonance

Du Ruijuan

(Department of Computer Science,Gansu Political Science and Law Institute,Lanzhou730070,China)

In this paper,by using Mawhin coincidence degree theorem,we study the solvability and multiplicity of the seconder-order functional differential equations with four-point boundary value problems at resonance. Under the conditions of boundary and unboundary of the nonlinearity f respectively,the existence of solution for the above functional boundary value problems are obtained.

functional value problem,existence,Caratheodory conditions,resonance

O175.8

A

1008-5513(2013)03-0255-09

10.3969/j.issn.1008-5513.2013.03.006

2012-11-16.

甘肅省自然科學(xué)基金(1107RJZA233);甘肅政法學(xué)院科研資助青年項目(GZF2013XQNW).

杜睿娟(1981-),碩士,講師,研究方向:非線性常微分方程邊值問題.

2010 MSC:34B15