共查询到20条相似文献,搜索用时 754 毫秒
1.
Mircea Voda 《Journal of Mathematical Analysis and Applications》2011,375(1):58-186
We find a solution to the Loewner chain equation in the case when the infinitesimal generator satisfies h(0,t)=0, Dh(0,t)=A for any A∈L(Cn,Cn) with m(A)>0. We also study the related classes of spirallike mappings, mappings with parametric representation and asymptotically spirallike mappings. 相似文献
2.
We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions of the Loewner differential equation that is satisfied by univalent subordination chains of the form ${f(z, t)=e^{tA}z+\cdots,}We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions
of the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z, t)=etAz+?,{f(z, t)=e^{tA}z+\cdots,} where
A ? L(\mathbbCn, \mathbbCn){A\in L(\mathbb{C}^n, \mathbb{C}^n)} has the property k
+(A) < 2m(A). Here k+(A)=max{?l:l ? s(A)}{k_+(A)=\max\{\Re\lambda:\lambda\in \sigma(A)\}} and m(A)=min{?áA(z), z ?: ||z||=1}{m(A)=\min\{\Re\langle A(z), z \rangle: \|z\|=1\}} . (The notion of parametric representation has a useful generalization under these conditions, so that one has a canonical
solution of the Loewner differential equation.) In particular, we determine the form of the univalent solutions. The results
are applied to subordination chains generated by spirallike mappings on the unit ball in
\mathbbCn{\mathbb{C}^n} . Finally, we determine the form of the solutions in the presence of certain coefficient bounds. 相似文献
3.
Let B be the unit ball of with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f(z,t) and the transition mapping v(z,s,t) associated to f(z,t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈B. Moreover, we prove that a mapping f∈H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e−tf(z,t)}t?0 is a normal family on B and f(z)=f(z,0) for z∈B. Also we show that univalent solutions f(z,t) of the generalized Loewner differential equation in higher dimensions are unique when {e−tf(z,t)}t?0 is a normal family on B. Finally we show that the set S0(B) of mappings which have parametric representation on B is compact. 相似文献
4.
Anne Greenbaum 《Linear algebra and its applications》2009,430(1):52-2352
Given an n by n matrix A, we look for a set S in the complex plane and positive scalars m and M such that for all functions p bounded and analytic on S and throughout a neighborhood of each eigenvalue of A, the inequalities
m·inf{‖f‖L∞(S):f(A)=p(A)}?‖p(A)‖?M·inf{‖f‖L∞(S):f(A)=p(A)} 相似文献
5.
In this paper we are concerned with solutions, in particular with univalent solutions, of the Loewner differential equation
associated with non-normalized subordination chains on the Euclidean unit ball B
n
in
\mathbbCn{\mathbb{C}^n}. The main result is a generalization to higher dimensions of a well known result due to Becker. Various particular cases
of this result have been recently obtained for subordination chains with normalization Df(0,t)=etIn{Df(0,t)=e^tI_n} or Df(0, t) = e
tA
, t ≥ 0, where
A ? L(\mathbbCn,\mathbbCn){A\in L(\mathbb{C}^n,\mathbb{C}^n)}. We also determine the form of the standard solutions to the Loewner differential equation associated with generalized spirallike
mappings. In the last section we obtain the form of the solution in the presence of coefficient bounds. 相似文献
6.
In this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x′(t)=Ax([t])+f(t), t∈R, where A is a bounded linear operator in X and f is an X-valued almost automorphic function. 相似文献
7.
In this paper we study the maximal regularity property for non-autonomous evolution equations t∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m. 相似文献
8.
Jia-Feng Tang 《Journal of Mathematical Analysis and Applications》2007,334(1):517-527
In this paper, we study the differential equations of the following form w2+R(z)2(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w2+P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w2+(z−z0)P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w2+A(z−z0)2(w′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab2=1, a2=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B2=C and q′(z)=±P(z). 相似文献
9.
William Dimbour 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(6):2351-2357
Using spectral theory we obtain sufficient conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x′(t)=A(t)x([t])+f(t),t∈R, where A(t) is an almost automorphy operator, f(t) is an X-valued almost automorphic function and X is a finite dimensional Banach space. 相似文献
10.
Khalid Koufany 《Journal of Functional Analysis》2006,236(2):546-580
Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We consider the Poisson transform Psf(z) for a hyperfunction f on S defined by the Poisson kernel Ps(z,u)=s(h(z,z)n/r/2|h(z,u)n/r|), (z,u)×Ω×S, s∈C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in Mn,m(C) (n?m), we prove that an eigenvalue equation for the second order Mn,n-valued Hua operator characterizes the image. 相似文献
11.
We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f. 相似文献
12.
Svatoslav Staněk 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):e153
The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u″))′=λf(t,u,u′,u″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y. 相似文献
13.
Vesselin Petkov 《Journal of Functional Analysis》2006,235(2):357-376
We obtain global Strichartz estimates for the solutions u of the wave equation for time-periodic potentials V(t,x) with compact support with respect to x. Our analysis is based on the analytic properties of the cut-off resolvent Rχ(z)=χ(U−1(T)−zI)ψ1, where U(T)=U(T,0) is the monodromy operator and T>0 the period of V(t,x). We show that if Rχ(z) has no poles z∈C, |z|?1, then for n?3, odd, we have a exponential decal of local energy. For n?2, even, we obtain also an uniform decay of local energy assuming that Rχ(z) has no poles z∈C, |z|?1, and Rχ(z) remains bounded for z in a small neighborhood of 0. 相似文献
14.
Existence of the mild solution for some fractional differential equations with nonlocal conditions 总被引:1,自引:0,他引:1
We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential
equation with nonlocal conditions: D
q
x(t)=Ax(t)+t
n
f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ+, x(0)+g(x)=x
0, where 0<q<1, A is the infinitesimal generator of a C
0-semigroup of bounded linear operators on a Banach space X. 相似文献
15.
We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L2(R,X), is Fredholm if and only if the not well-posed equation u′(t)=D(t)u(t), t∈R, has exponential dichotomies on R+ and R− and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems. 相似文献
16.
Sergiu Aizicovici 《Journal of Mathematical Analysis and Applications》2007,331(1):308-328
The solvability of the abstract implicit nonlinear nonautonomous differential equation (A(t)u′(t))+B(t)u(t)+C(t)u(t)∋f(t) will be investigated in the case of a measure as an initial value. It will be shown that this problem has a solution if the inner product of A(t)x and B(t)x+C(t)x is bounded below. 相似文献
17.
Raúl Naulin 《Journal of Mathematical Analysis and Applications》2002,274(1):305-318
The unstable properties of the null solution of the nonautonomous delay system x′(t)=A(t)x(t)+B(t)x(t−r1(t))+f(t,x(t),x(t−r2(t))) are examined; the nonconstant delays r1, r2 are assumed to be continuous bounded functions. The case A=constant is reviewed, where a theorem, recalling the Perron instability theorem for ordinary differential equations, is obtained. 相似文献
18.
Ciprian G. Gal 《Journal of Mathematical Analysis and Applications》2007,333(2):971-983
In this paper we consider the nonlinear differential equation with deviated argument u′(t)=Au(t)+f(t,u(t),u[φ(u(t),t)]), t∈R+, in a Banach space (X,‖⋅‖), where A is the infinitesimal generator of an analytic semigroup. Under suitable conditions on the functions f and φ, we prove a global existence and uniqueness result for the above equation. 相似文献
19.
John H. Clifford Michael G. Dabkowski 《Journal of Mathematical Analysis and Applications》2005,305(1):183-196
We find the singular values and corresponding Schmidt pairs of a compact composition operator Cφ induced by φ(z)=az+b, where |a|+|b|<1, on the classical Hardy space. We do so by solving a functional equation that is a generalization of Schröder's equation: find a function f, holomorphic on the open unit disc, and a complex number λ such that G(z)f(ψ(z))=λf(ψ(z)), where ψ is a holomorphic self-map of the open unit disc with an interior fixed point and G is a bounded holomorphic function on the open unit disc. In addition, we find the spectrum of the weighted composition operator MGCψ. 相似文献
20.
In this paper we introduce the concept of a weak solution for second order differential inclusions of the form u″(t) ∈ Au(t) + f(t), where A is a maximal monotone operator in a Hilbert space H. We prove existence and uniqueness of weak solutions to two point boundary value problems associated with such kind of equations. Furthermore, existence of (strong and weak) solutions to the equation above which are bounded on the positive half axis is proved under the optimal condition tf(t) ∈ L 1(0, ∞; H), thus solving a long-standing open problem (for details, see our comments in Section 3 of the paper). Our treatment regarding weak solutions is similar to the corresponding theory related to the first order differential inclusions of the form f(t) ∈ u′(t) + Au(t) which has already been well developed. 相似文献