• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Measure Functional Differential Equations with Infinite Delay: Differentiability of Solutions with Respect to Initial Conditions

    2017-05-15 11:09:30LIBaolinWANGBaodi
    關(guān)鍵詞:西北師范大學(xué)初值國(guó)家自然科學(xué)基金

    LI Baolin, WANG Baodi

    (College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu)

    Measure Functional Differential Equations with Infinite Delay: Differentiability of Solutions with Respect to Initial Conditions

    LI Baolin, WANG Baodi

    (CollegeofMathematicsandStatistics,NorthwestNormalUniversity,Lanzhou730070,Gansu)

    In this paper, we consider a measure functional differential equation with infinite delay,which can be changed into a generalized ordinary differential equation. By differentiability of solutions with respect to initial condition for the generalized ODE, we obtain the differentiability for the measure functional differential equation.

    measure functional differential equation; differentiability of solutions; Kurzweil integral; generalized ordinary differential equation

    1 Introduction

    There are many sources that describe the differentiability of solutions with respect to initial conditions for ordinary differential equations, such as [1-2]. From [3], we can see the description of a similar type for ordinary differential equations, and for dynamic equations on time scales. Similar work as [3] was also carried out in [4]. In this paper, we consider the measure differential equations.

    When a system described by ordinary differential equation

    (1)

    is acted upon by perturbation, the resultant perturbed system is generally given by ordinary differential equation of the form dx/dt=f(t,x)+G(t,x), where we assume the perturbation termG(t,x) is well-behaved, i.e.,G(t,x) is continuous or integrable and as such the state of the system changes continuously with respect to time. However, in some system, the perturbations are impulsive, so we cannot expect the perturbation is always well-behaved. Therefore, the following equation

    (2)

    was defined in [5], whereDudenotes the distributional derivative of functionu. Ifuis a function of bounded variation,Ducan be identified with a Stieltjes measure, it will suddenly change the state of the system at a discontinuity ofu. In [5], equations of the form (2) are called measure differential equations, also a special case of the equation (2). Inspired by [5], the authors of [6] have generalized a very useful functional differential equation as following

    (3)

    wherextrepresents the restriction of the functionx(·) (x(·) denotes a solution of equation (2)) means a function of bounded variation whose distributional derivativeDxsatisfies the equation (2) on the interval [m(t),n(t)],mandnbeing functions with the propertym(t)≤n(t)≤t.

    Moreover, in [7], an important theorem was proved. The main contents are as following:

    x(·) is a solution of (2) through (t0,x0) on an intervalI, with left end pointt0, if and only ifx(·) satisfies the following equations

    Authors of [8] especially proved the following measure functional differential equation with infinite delay

    (4)

    is equivalent to the generalized ordinary differential equation under some conditions. Also, equations (4) is the integral form of the following measure equation

    Dx=G(s,xs)dg(s),

    whereg(s) is a nondecreasing function, and the integral on the right-hand side of (4) is the Kurzweil-Stieltjes integral.

    In this paper, we shall consider differentiability of initial value problem for measure differential equations

    (5)

    wherexis an unknown function with values inRnandthesymbolxsdenotesthefunctionxs(τ)=x(s+τ)definedon(-∞,0],whichcorrespondingtothelengthofthedelay, f:P×[t0,t0+σ]→Rnis a function satisfies the following conditions (A)-(C):

    (B)ThereexistsafunctionM:[t0,t0+σ] →R+,whichisKurzweil-Stieltjiesintegrablewithrespecttog,suchthat

    wheneverx∈Oand[a,b]?[t0,to+σ].

    (C)ThereexistsafunctionL :[t0,to+σ] →R+,which is Kurzweil-Stieltjies integrable with respect tog, such that

    wheneverx,y∈Oand [a,b]?[t0,to+σ].(we assume that the integral on the right-hand side exists).

    Andg:[t0,t0+σ]→Risanondecreasingfunction, P={xt:x∈O,t∈[t0, t0+σ]}? H0,H0? G((-∞,0],Rn) is a Banach space equipped with a norm denoted by ‖·‖. We assumeH0satisfies the following conditions (H1)-(H6):

    (H1)H0is complete.

    (H2) Ifx∈H0andt<0, thenxt∈H0.

    (H3) There exists a locally bounded functionk1:(-∞,0]→R+suchthatifx∈H0andt≤0,then‖x(t)‖≤k1(t)‖x‖.

    (H4)Thereexistsafunctionk2: (0,∞) →[1,∞)suchthatifσ > 0andx∈H0isafunctionwhosesupportiscontainedin[-σ,0],then

    (H5) There exists a locally bounded functionk3:(-∞,0]→R+suchthatifx∈H0andt≤0,then

    (H6)Ifx∈H0,thenthefunctiont |→‖xt‖isregulatedon(-∞,0].

    t0∈R,σ>0,O?Ht0+σis a space satisfying conditions 1)-6) of Lemma 2.7.G((-∞,0],Rn)denotesthesetofallregulatedfunctionsf:(-∞,0]→Rn.

    Our main result is to derive the differentiability of solutions with respect to initial conditions for measure function differential equations with infinite delay.

    2 Preliminaries

    We start this section with a short summary of Kurzweil integral.

    Letδ:[a,b]→R+beafunction,andτbeapartitionofinterval[a,b]withdivisionpointsa=α0≤α1≤…≤αk=b.Thetagsτi∈[αi-1,αi]iscalledδ-fineif[αi-1,αi]?[τi-δ(τi),τi+δ(τi)],i=1,2,…,k.

    Definition2.1[2]Amatrix-valuedfunctionF:[a,b]×[a,b]→Rn×mis called Kurzweil integrable on [a,b], if there is a matrixI∈Rn×msuchthatforeveryε>0,thereisagaugeδon[a,b]suchthat

    AnimportantspecialcaseistheKurzweil-Stieltjesintegralofafunctionf:[a,b]→Rnwith respect to a functiong:[a,b]→R, which corresponds to the choice

    Definition 2.2[1]G?Rn× R,(x,t)∈G, a functionx:[a,b]→Bis called a solution of the generalized ordinary differential equation

    (7)

    whenever

    Definition 2.3[8]LetXbe a Banach space. Consider a setO?X. A functionF:O×[t0,t0+σ] →Xbelongs to the classF(O × [t0,t0+σ] ,h,k),ifthefollowingconditionsaresatisfied:

    (F1)Thereexistsanondecreasingfunctionh:[t0,t0+σ]→R such thatF:O×[t0,t0+σ] →Xsatisfies

    for everyx∈Oands1,s2∈[t0,t0+σ],

    (F2) There exists a nondecreasing functionk:[t0,t0+σ]→RsuchthatF:O×[t0,t0+σ] →Xsatisfies

    (9)

    foreveryx,y∈Oands1,s2∈[t0,t0+σ],

    Lemma 2.2[2]LetU:[a,b]×[a,b]→Rn×nbeaKurzweilintegrablefunction,assumethereexistsapairoffunctionsf:[a,b]→Rnandg:[a,b]→Rsuchthatfisregulated, gisnondecreasing,and

    (10)

    Then

    Lemma2.3[9]AssumethatU:[a,b]×[a,b]→Rn×mis Kurzweil integrable andu:[a,b]→Rn×misitsprimitive,i.e.,

    IfUisregulatedinthesecondvariable,thenuisregulatedandsatisfies

    Moreover,ifthereexistsanondecreasingfunctionh:[a,b]→R such that

    then

    Lemma 2.4[9]Leth:[a,b]→[0,+∞) be a nondecreasing left-continuous function,k>0,l≥0. If thatψ:[a,b]→[0,+∞) is bounded and satisfies

    thenψ(ξ)≤kel(h(ξ)-h(a))for everyξ∈[a,b].

    Lemma 2.5[2]Assume thatF:[a,b]×[a,b]→Rn×nsatisfies(8).Lety,z :[a,b]→Rnbe a pair of functions such that

    Then,zis regulated on [a,b].

    Lemma 2.6[2]Assume thatF:[a,b]×[a,b]→Rn×nisKurzweilintegrableandsatisfies(8)withaleft-continuousfunctionh.Thenforeveryz0∈Rn, the initial value problem

    (12)

    has a unique solutionz:[a,b]→Rn.

    Toestablishthecorrespondencebetweenmeasurefunctionaldifferentialequationsandgeneralizedordinarydifferentialequations,wealsoneedasuitablespaceHaofregulatedfunctionsdefinedon(-∞,a],wherea∈R, the next lemma shows that the spacesHainherit all important properties ofH0.

    Lemma 2.7[8]IfH0?G((-∞,0],Rn)isaspacesatisfyingconditions1)-6),thenthefollowingstatementsaretrueforeverya∈R:

    1)Hais complete; 2) Ifx∈Haandt≤a, thenxt∈H0; 3) Ift≤aandx∈Ha, then ‖x(t)‖≤k1(t-a)‖x‖; 4) Ifσ> 0 andx∈Ha+σis a function whose support is contained in [a,a+σ], then

    5) Ifx∈Ha+σandt≤a+σ, then ‖xt‖≤k3(t-a-σ)‖x‖; 6) Ifx∈Ha+σ, then the functiont|→‖xt‖is regulated on (-∞,a+σ].

    Theorem 2.8[8]Assume thatOis a subset ofHt0+σhaving the prolongation property fort≥t0,P={xt:x∈O,t∈[t0,t0+σ]},?∈P,g:[t0,t0+σ]→Risanondecreasingfunction, f:P×[t0,t0+σ]→Rnsatisfies conditions (A), (B), (C), andF:O×[t0,t0+σ]→G((-∞,t0+σ],Rn)givenby(13)hasvaluesinHa+σ.Ify∈Oisasolutionofthemeasurefunctionaldifferentialequation

    then the functionx:[t0,t0+σ]→Ogiven by

    is a solution of the generalized ordinary differential equation

    Wherextakes values inO, andF:O×[t0,t0+σ]→G((-∞,t0+σ],Rn)isgivenby

    (13)

    for everyx∈Oandt∈[t0,t0+σ].

    Proof The statement follows easily from Theorem 3.6 in [8]

    3 Main result

    Now, we discuss the differentiability theorem of solutions with respect to initial conditions for equation (5).

    Theorem 3.1 Letf:P×[t0,t0+σ]→RnbeacontinuousfunctionwhosederivativefxexistsandiscontinuousonP×[t0,t0+σ],andsatisfiestheaforementionedconditions(A)-(C),whereP={xt:x∈O, t∈[t0, t0+σ]}? H0,andH0? G((-∞,0],Rn) be a Banach space satisfying the aforementioned conditions (H1)-(H6),t0∈{R},σ>0, O? Ht0+σ.Ifg : [t0,t0+σ]→R is a nondecreasing function andλ0∈Rl,σ>0,Λ={λ∈Rl; ‖λ-λ0‖<σ},x0:Λ→O× [t0,t0+σ] for everyλ∈Λ, the initial value problem of the measure functional differential equations with infinite delay (5) is equivalent to the initial value problem

    (14)

    then (14) has a solution defined on [t0,t0+σ]. Letx(t,λ) be the value of that solution att∈[t0,t0+σ].

    Moreover, let the following conditions be satisfied:

    1) For every fixedt∈[t0,t0+σ], the functionx|→F(x,t) is continuously differentiable onO× [t0,t0+σ].

    2) The functionx0is differentiable atλ0.

    Then the functionλ|→x(t,λ) is differentiable atλ0, uniformly for allt∈[t0,t0+σ]. Moreover, its derivativeZ(t)=xλ(t,λ0),t∈[t0,t0+σ] is the unique solution of the generalized differential equation

    (15)

    Proof Our proof is based on the idea from [2].

    According to the assumptions, there exist positive constantsA1,A2such that

    for everyx,y∈O,t∈[t0,t0+σ], andt0≤t1

    for everyx∈O, the fourth statement of Lemma 2.7 implies

    where

    by the fifth statement of Lemma 2.7. The last expression is smaller than or equal to

    where

    i.e.,Fx∈F(O × [t0,t0+σ],h,k).

    BecauseofO × [t0,t0+σ]isclosed,accordingtothemean-valuetheoremforvectorvaluedfunctionandFx∈F(O× [t0,t0+σ],h,k)

    (16)

    By the assumptions, we have

    According to the Lemma 2.3, every solutionxis a regulated and left-continuous function on [t0,t0+σ]. If Δλ∈Rlissuchthat‖Δλ‖<σ,then

    where

    By(16),weobtain

    andbyusingLemma2.2,foreverys∈[t0,t0+σ],weobtain

    Consequently,byusingLemma2.4,wehave

    SowecanseethatwhenΔλ→0, x(s,λ0+Δλ)→x(s,λ0)uniformlyforalls∈[t0,t0+σ].

    LetW(τ,t)=Fx(x(τ,λ0),t).BecauseFx∈F(O× [t0,t0+σ] ,h,k),W(τ,t) satisfies (16), by Lemma 2.6, (15) has a unique solutionZ:[t0,t0+σ]→Rn× n.ByusingLemma2.5,thesolutionisregulated.SothereexistsaconstantN>0suchthat‖Z(t)‖≤N,t∈[t0,t0+σ].ForeveryΔλ∈Rlsuch that ‖Δλ‖<σ, let

    Next, we will prove that if Δλ→0, thenφ(r,Δλ)→0 uniformly forr∈[t0,t0+σ].

    Letε>0 be given, there exists aδ>0 such that if Δλ∈Rland‖Δλ‖<σ,then

    and

    It is obvious that

    where

    Thus,

    Because of the functionx|→F(x,t) is continuously differentiable onO×[t0,t0+σ] and the definition of theφ(r,Δλ), so for any givenε>0,t,s∈[t0,t0+σ], we have

    and thus (usingFx∈F(O × [t0,t0+σ] ,h,k) )

    Consequently

    Finally,Gronwall’sinequalityleadstotheestimate

    Sinceε→0+,wehavethatifΔλ→0,thenφ(r,Δλ)→0uniformlyforanyr∈[t0,t0+σ].

    [1] KEllEY W G, PETERSON A C. The Theory of Differential Equations[M]. 2nd ed. New York:Springer-Verlag,2010.

    [3] LAKSHMIKANTHAM V, BAINOV D D, SIMEONOV P S. Theory of Impulsive Differential Equations[M]. Singapore:World Scientific,1989.

    [4] HILSHCER R, ZEIDAN V, KRATZ W. Differentiation of solutions of dynamic equations on time scales with respect to parameters[J]. Adv Dyn Syst Appl,2009,4(1):35-54.

    [5] SCHMAEDEKE W W. Optimal control theory for nonlinear vector differential equations containing measures[J]. SIAM Control,1965,3(2):231-280.

    [6] DAS P C, SHARMA R R. On optimal comtrols for measure delay-differential equations[J]. SIAM Control,1971,9(1):43-61.

    [7] PURNA C D, RISHI R S. Existence and stability of measure differential equations[J]. Czechoslovak Math J,1972,22(97):145-158.

    [12] VERHUST F. Nonlinear Differential Equations and Dynamical Systems[M]. 2nd ed. New York:Springer-Verlag,2000.

    [13] KURZWEIL J. Generalized ordinary differential equations and continuous dependence on a parameter[J]. Czechoslovak Math,1957,82(7):418-449.

    [14] 朱雯雯,徐有基. 帶非線性邊界條件的一階微分方程多個(gè)正解的存在性[J]. 四川師范大學(xué)學(xué)報(bào)(自然科學(xué)版),2016,39(2):226-230.

    [15] KURZWEIL J. Generalized ordinary differential equations[J]. Czechoslovak Math J,1958,83(8):360-389.

    無(wú)限滯后測(cè)度泛函微分方程的解關(guān)于初值條件的可微性

    李寶麟, 王保弟

    (西北師范大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 甘肅 蘭州 730070)

    利用廣義常微分方程的解關(guān)于初值條件的可微性,考慮可以轉(zhuǎn)化為廣義常微分方程的無(wú)限時(shí)滯測(cè)度泛函微分方程,得到這類方程的解關(guān)于初值條件的可微性.

    測(cè)度泛函微分方程; 解的可微性; Kurzweil 積分; 廣義常微分方程

    O175.12

    A

    1001-8395(2017)01-0061-07

    2016-07-01

    國(guó)家自然科學(xué)基金(11061031)

    李寶麟(1963—)男,教授,主要從事常微分方程和拓?fù)鋭?dòng)力系統(tǒng)的研究,E-mail:libl@nwnu.edu.cn

    Foundation Items:This work is supported by National Natural Science Foundation of China (No.11061031)

    10.3969/j.issn.1001-8395.2017.01.010

    (編輯 陶志寧)

    Received date:2016-07-01

    2010 MSC:26A39; 30G30; 34A20; 34G10

    猜你喜歡
    西北師范大學(xué)初值國(guó)家自然科學(xué)基金
    西北師范大學(xué)作品
    大眾文藝(2023年9期)2023-05-17 23:55:52
    西北師范大學(xué)美術(shù)學(xué)院作品選登
    具非定常數(shù)初值的全變差方程解的漸近性
    常見(jiàn)基金項(xiàng)目的英文名稱(一)
    西北師范大學(xué)美術(shù)學(xué)院作品選登
    西北師范大學(xué)美術(shù)學(xué)院作品選登
    一種適用于平動(dòng)點(diǎn)周期軌道初值計(jì)算的簡(jiǎn)化路徑搜索修正法
    三維擬線性波方程的小初值光滑解
    我校喜獲五項(xiàng)2018年度國(guó)家自然科學(xué)基金項(xiàng)目立項(xiàng)
    2017 年新項(xiàng)目
    崇礼县| 周宁县| 陕西省| 清水县| 盐池县| 沁阳市| 晋江市| 镇原县| 通河县| 湘阴县| 临夏市| 浦东新区| 吉安县| 田阳县| 八宿县| 巴林右旗| 洪湖市| 大荔县| 太仆寺旗| 晋江市| 特克斯县| 望都县| 抚松县| 平潭县| 资阳市| 建昌县| 明溪县| 若羌县| 泽库县| 神农架林区| 龙门县| 弥渡县| 盐亭县| 确山县| 宝清县| 长岭县| 尖扎县| 孟村| 富川| 仙桃市| 平顺县|