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1.
In this paper, we establish maximal Lp−Lq estimates for non-autonomous parabolic equations of the type u′(t)+A(t)u(t)=f(t), u(0)=0 under suitable conditions on the kernels of the semigroups generated by the operators −A(t), t∈[0,T]. We apply this result on semilinear problems of the form u′(t)+A(t)u(t)=f(t, u(t)), u(0)=0. 相似文献
2.
This paper gives lower estimates for the frequency modules of almost periodic solutions to equations of the form
, where A generates a strongly continuous semigroup in a Banach space
, F(t,x) is 2π-periodic in t and continuous in (t,x), and f is almost periodic. We show that the frequency module ℳ(u) of any almost periodic mild solution u of (*) and the frequency module ℳ(f) of f satisfy the estimate e
2π
iℳ(f)⊂e
2π
iℳ(u). If F is independent of t, then the estimate can be improved: ℳ(f)⊂ℳ(u). Applications to the nonexistence of quasi-periodic solutions are also given. 相似文献
3.
P. S. Kelevedjiev D. O’Regan N. Popivanov R. P. Agarwal 《Aequationes Mathematicae》2010,79(3):281-291
Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem ${x''=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,}Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem x"=f(t,x), t ? (0,1), x(0)=A, x¢(1)=0,{x'=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,} where the scalar function f(t, x) may be singular at x = A. 相似文献
4.
Dr. A. Mukherjea 《Probability Theory and Related Fields》1969,11(2):142-146
Summary In this paper, idempotent probability measures have been considered on semigroups which are locally compact or metric and satisfy: (*) A
–1
B and Ax
–1 are compact whenever A and B are so, for every x in the semigroup. Such semigroups are more general than compact semigroups which do admit of such measures. On such semigroups we can construct such measures by the usual process if there is a compact sub-semigroup. It is shown in this paper that if such a measure exists in such semigroups, then it must be such an extension measure. Some related results concerning the conditions (*) are also discussed here. 相似文献
5.
S. L. Wismath 《Algebra Universalis》1996,36(1):1-7
Iterative hyperidentities are hyperidentities of the special formF
a
(x
1,...,x
k
=F
a+b
(x
1,...,x
k
). This type of hyperidentity has been considered by Denecke and Pöschel, and by Schweigert. Here we consider iterative hyperidentities for the variety An,m of commutative semigroups satisfyingx
n
=x
n+m
,n,m 1. We introduce two parameters(m, n) and(m) associated withn andm, and show thatA
nn,m
satisfies the iterative hyperidentitiesF
(x
1,...,x
k
=F
+b
(x
1,...,x
k
) for every arityk. Moreover, the numbers and are minimal, making these hyperidentities irreducible in the sense of Schweigert. We also show how these hyperidentities for An,m may be used to prove that no non-trivial proper variety of commutative semigroups can have a finite hyperidentity basis.Presented by W. Taylor.Research supported by NSERC of Canada 相似文献
6.
LetA be a closed linear operator such that the abstract Cauchy problemu″(t)=Au(t), t∈R; u(0)=x, u′(0)=y, is well-posed. We present some multiplicative perturbation theorems which give conditions on an operatorC so that the abstract Cauchy problems for differential equationsu″(t)=ACu(t) andu″(t)=CAu(t) also are well-posed. Some new or known additive perturbation theorems and mixed-type perturbation theorems are deduced as
corollaries. Applications to characterization of the infinitesimal comparison of two cosine operator functions are also discussed.
Research supported in part by the National Science Council of Taiwan. 相似文献
7.
E. I. Lin'kov 《Mathematical Notes》1968,3(4):267-271
In a real Hilbert space H we consider the nonlinear operator equation P(x)=0 and the continuous gradient methodx (t)= –P (x)*
P
(x), x (0) = x0. Two theorems on the convergence of the process (*) to the solution of the equation P(x)=0 are proved.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 421–426, April, 1968. 相似文献
8.
Pitambar Das 《Proceedings Mathematical Sciences》1993,103(3):341-347
In this paper, we obtain the sufficient and necessary conditions for all solutions of the odd-order nonlinear delay differential equation.x (n)+Q(t)f(x(g(t)))=0 to be oscillatory. In particular, ifn=1, Q(t)>0, f(x)=x α, where α∈(0,1) and is a ratio of odd integers andg(t)=t?? for some ?>0, then every solution of (*) oscillates if and only if ∫∞Q(s)ds=∞. 相似文献
9.
In the space of functions B
a3+={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T3/2)=g(x, −t)}, we establish that if the condition aT
3
(2s−1)=4πk, (4πk, a (2s−1))=1, k ∈ ℤ, s ∈ ℕ, is satisfied, then the linear problem u
u
−a
2
u
xx
=g(x, t), u(0, t)=u(π, t)=0, u(x, t+T
3
)=u(x, t), ℝ2, is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator.
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
Feburary, 1997
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
Feburary, 1997 相似文献
10.
We study a periodic problem for the equation u
tt−uxx=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ2 and establish conditions of the existence and uniqueness of the classical solution.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 558–565, April, 1997. 相似文献
11.
E. Di Benedetto 《Annali di Matematica Pura ed Applicata》1979,121(1):249-262
Sunto In questo lavoro si considera il problema di Cauchy per l'equazione di filtrazione ∂u/∂t=∂
2
ϕ(u)/∂x
2
nella regioneR×(0,T],0<T<∞. Sotto opportune ipotesi sulla funzione ϕ(u) si determina una stima dell'incremento temporale della soluzione u(x, t)
(intesa nel senso debole). Nel caso politropico (ϕ(u)=um), quando m>2 si trova in particolare un comportamento h?lderiano di u(x, t) rispetto a t con l'esponente1/(m−1); viene anche dimostrato che questo esponete è effettivamente assunto da una particolare soluzione, per cui la stima ottenuta
è la migliore possibile.
Entrata in Redazione il 18 luglio 1978. 相似文献
Entrata in Redazione il 18 luglio 1978. 相似文献
12.
In this work, we use integrated semigroups to state results on the existence and uniqueness of integral solutions and solutions
for the abstract Cauchy problem x′(t)=Bx(t)+Lxt, t⩾0, where B is a nondensely defined linear operator on a Banach space X.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
13.
Summary In this paper we discuss the construction of a spline function for a class of singular two-point boundary value problemx
–(x
u)=f (x, u),u(0)=A,u(1)=B, 0<<1 or =1,2. The boundary conditions may also be of the formu(0)=0,u(1)=B. Three point finite difference methods, using the above splines, are obtained for the solution of the boundary value problem. These methods are of second order and are illustrated by four numerical examples. 相似文献
14.
We investigate the linear periodic problem u
tt
−u
xx
=F(x, t), u(x+2π, t)=u(x, t+T)=u(x, t), ∈ ℝ2, and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry
spaces.
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
February, 1997. 相似文献
15.
J. R. Cannon 《Annali di Matematica Pura ed Applicata》1964,66(1):155-165
Summary Let u(x, t) satisfy the heat equation in 0<x<1, 0<t≤T. Let u(x, 0)=0 for 0<x<1 and let |u(0, t)|<ε, | ux(0, t) |<ε, and | u(1, t) |<M for 0≤t≤T. Then,
, where M1 and β(x) are given explicitly by simple formulas. The application of the a priori bound to obtain error estimates for a numerical
solution of the Cauchy problem for the heat equation with u(x, 0)=h(x), u(0, t)=f(t), and ux(0, t)=g(t) is discussed.
Work performed under the auspices of the U. S. Atomic Energy Commission. 相似文献
16.
S. Kumagai 《Journal of Optimization Theory and Applications》1980,31(2):285-288
In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:R
n ×R
m R
n, withF(x
0,y
0)=0, that requires neither differentiability ofF nor nonsingularity of
x
F(x
0,y
0). In the proof, the local one-to-one condition forF(·,y):A R
n R
n for ally B is consciously or unconsciously treated as implying thatF(·,y) mapsA one-to-one ontoF(A, y) for ally B, and the proof is not perfect. A proof can be given directly, and the theorem is shown to be the strongest, in the sense that the condition is truly if and only if. 相似文献
17.
An approach to inverse problems based on the boundary control theory is developed. The dynamic problem to recover a density
of an inhomogeneous string via its free endopoint oscillations generated by an instantaneous force source is proposed. The
problem is to determine the coefficient ρ(x)>0 in the equation ρ(x)utt(x, t)−uxx(x, t)=0(x, t>0) with the conditions u|<0=0, ux(0, t)=δ(t) by using a known function (response) u(0, t)=r(t) (t>0). The authors propose an algorithm based upon the approach
and demonstrate its numerical efficiency in the test problems including those for nonmonotone ρ(x)'s. Bibliography: 12 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 37–49, 1990.
Translated by T. N. Surkova. 相似文献
18.
In ℝ
m
×ℝ
n−m
, endowed with coordinates x=(x′,x″), we consider bounded solutions of the PDE
We prove a geometric inequality, from which a symmetry result follows.
相似文献
19.
We study the asymptotic behaviour of the trajectories of the second order equation ${\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)}We study the asymptotic behaviour of the trajectories of the second order equation [(x)\ddot](t)+g[(x)\dot](t)+?f(x(t))+e(t)x(t)=g(t){\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)} , where γ > 0, g ? L1([0,+¥[;H){g \in L^1([0,+\infty[;H)}, Φ is a C
2 convex function and e{\varepsilon} is a positive nonincreasing function. 相似文献
20.
We consider an Abel equation (*)y’=p(x)y
2 +q(x)y
3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane)
is thaty
0=y(0)≡y(1) for any solutiony(x) of (*).
We introduce a parametric version of this condition: an equation (**)y’=p(x)y
2 +εq(x)y
3
p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1).
We show that the parametric center condition implies vanishing of all the momentsm
k
(1), wherem
k
(x)=∫
0
x
pk
(t)q(t)(dt),P(x)=∫
0
x
p(t)dt. We investigate the structure of zeroes ofm
k
(x) and on this base prove in some special cases a composition conjecture, stated in [10], for a parametric center problem.
The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the
Minerva Foundation. 相似文献