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1、微分方程英文论文和翻译Differential Calculus Newton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successful accomplishments of these me
2、n were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus. In this article, we establish a result about controllability to the following class of partial neutral functional dierential equations with innit
3、e delay: Dxt=ADxt+Cu(t)+F(t,xt),t0 (1) tx0=fbwhere the state variablex(.)takes values in a Banach space(E,.)and the control u(.) is given in L2(0,T,U),T0,the Banach space of admissible control functions with U a Banach space. C is a bounded linear operator from U into E, A : D(A) E E is a linear ope
4、rator on E, B is the phase space of functions mapping (, 0 into E, which will be specied later, D is a bounded linear operator from B into E dened by Dj=j(0)-D0j,jB D0is a bounded linear operator from B into E and for each x : (, T E, T 0, and t 0, T , xt represents, as usual, the mapping from (, 0
5、into E dened by xt(q)=x(t+q),q(-,0 F is an E-valued nonlinear continuous mapping on+B. The problem of controllability of linear and nonlinear systems represented by ODE in nit dimensional space was extensively studied. Many authors extended the controllability concept to innite dimensional systems i
6、n Banach space with unbounded operators. Up to now, there are a lot of works on this topic, see, for example, 4, 7, 10, 21. There are many systems that can be written as abstract neutral evolution equations with innite delay to study 23. In recent years, the theory of neutral functional dierential e
7、quations with innite delay in innite. dimension was developed and it is still a eld of research (see, for instance, 2, 9, 14, 15 and the references therein). Meanwhile, the controllability problem of such systems was also discussed by many mathematicians, see, for example, 5, 8. The objective of thi
8、s article is to discuss the controllability for Eq. (1), where the linear part is supposed to be non-densely dened but satises the resolvent estimates of the Hille-Yosida theorem. We shall assume conditions that assure global existence and give the sucient conditions for controllability of some part
9、ial neutral functional dierential equations with innite delay. The results are obtained using the integrated semigroups theory and Banach xed point theorem. Besides, we make use of the notion of integral solution and we do not use the analytic semigroups theory. Treating equations with innite delay
10、such as Eq. (1), we need to introduce the phase space B. To avoid repetitions and understand the interesting properties of the phase space, suppose that (B,.B) is a (semi)normed abstract linear space of functions mapping (, 0 into E, and satises the following fundamental axioms that were rst introdu
11、ced in 13 and widely discussed in 16. +(A) There exist a positive constant H and functions K(.), M(.):,with K continuous and M locally bounded, such that, for any sand a0,if x : (, + a E, xsB and x(.)is continuous on , +a, then, for every t in , +a, the following conditions hold: (i) xtB. (ii) x(t)H
12、xtB,which is equivalent to j(0)HjBor everyjB. (iii) xtBK(t-s)supx(s)+M(t-s)xssstB(A) For the function x(.)in (A), t xt is a B-valued continuous function for t in , + a. (B) 1.The space B is complete. Throughout this article, we also assume that the operator A satises the Hille-Yosida condition : (H1
13、) There exist and w，such that (w,+)r(A) and n-n:nN,lwM sup(l-w)(lI-A) Let A0 be the part of operator A in D(A) dened by D(A0)=xD(A):AxD(A) A0x=Ax,for,xD(A0)It is well known that D(A0)=D(A)and the operator A0 generates a strongly continuous semigroup (T0(t)t0)on D(A). Recall that 19 for allxD(A) and
14、t0,one has f0tT0(s)xdsD(A0) and tA0T0(s)sds+x=T0(t)x. We also recall that (T0(t)t0coincides on D(A) with the derivative of the locally Lipschitz integrated semigroup (S(t)t0 generated by A on E, which is, according to 3, 17, 18, a family of bounded linear operators on E, that satises (i) S(0) = 0, (
15、ii) for any y E, t S(t)y is strongly continuous with values in E, (iii) S(s)S(t)=(S(t+r)-s(r)drfor all t, s 0, and for any 0 there exists a 0sconstant l() 0, such that S(t)-S(s)l(t)t-s or all t, s 0, . The C0-semigroup (S(t)t0 is exponentially bounded, that is, there exist two constants Mand w,such that S(t)Mewt for all t 0. Notice that the controllability of a class of non-densely dened functional dierential equatio
