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1.
It is well known that the recurrence relations
are periodic, in the sense that they define periodic sequences for all choices of the initial data, and lead to sequences
with periods 2, 5 and 8, respectively. In this paper we determine all periodic recursions of the form
where are complex numbers, are non-zero and . We find that, apart from the three recursions listed above, only
lead to periodic sequences (with periods 6 and 8). The non-periodicity of (R) when (or and ) depends on the connection between (R) and the recurrence relations
and
We investigate these recursions together with the related
Each of (A), (B), and (C) leads to periodic sequences if k = 1 (with periods 6, 5, and 9, respectively). Also, for k = 2, (B) leads to periodicity with period 8. However, no other cases give rise to periodicity. We also prove that every real
sequence satisfying any of (A), (B), and (C) must be bounded. As a consequence, we find that for an arbitrary k, every rational sequence satisfying any of (A), (B), and (C) must be periodic.
(Received 27 June 2000; in revised form 5 January 2001) 相似文献
3.
Let us consider the linear boundary value problem
where
and
is defined by
Classical Lyapunov inequality states that
for any function
where
The constant 4/ L is optimal. Let us note that Lyapunov inequality is given in terms of
the usual norm in the space L1(0, L). In this paper we review some recent results on Lp Lyapunovtype inequalities,
, for ordinary and partial differential equations on a bounded and regular domain in
In the last case, it is showed that the relation between the quantities p and N/2 plays a crucial role, pointing out a deep difference with respect to the ordinary case. In the proof, the best constants
are obtained by using a related variational problem and Lagrange multiplier theorem. Finally, the linear results are combined
with Schauder fixed point theorem in the study of resonant nonlinear problems.
The authors have been supported by the Ministry of Science and Technology of Spain MTM2005- 01331 and by Junta de Andalucia
(FQM116). 相似文献
4.
We consider the nonlinear Schr?dinger equation where W( x) = V( x) − E.
We establish the existence of the least energy solutions. We also formulate conditions guaranteeing the existence of multiple
solutions in terms of the Lusternik–Schnirelemann category.
Received March 30, 2001; in revised form May 29, 2002 相似文献
5.
We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation
posed in a bounded domain , with pure Neumann boundary conditions Under the assumption that with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution if ∫ Ω
f
dx = 0, and also that the kernel is generated by a function , unique up to a multiplicative constant, which satisfies a.e. on Ω. We also prove that the equation has a unique solution for all ν > 0 and the map is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation
The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure
data and to parabolic problems. 相似文献
6.
In this paper we study the Dirichlet problem in Q
T
= Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:
The principal part of the operator degenerates in u = 0 according to a nonnegative increasing real function α( u), and the term grows quadratically with respect to the gradient. We prove an existence result for solutions to this problem in the framework
of the distributional solutions under the hypotheses that both f and the initial datum u
0 are bounded nonnegative functions. Moreover as further results we get an existence result for the model problem
in the case that the principal part of the operator is of fast-diffusion type, i.e. α( u) = u
m
, with −1 < m < 0.
相似文献
7.
The authors consider quasilinear parabolic systems in two space dimensions. The function a has p-growth behaviour, 1< p < ∞, and the ellipticity “constant” behaves like (1+|∇ u|)
p
− 2. The author prove full regularity of the weak solution on interior subdomains, but globally in time. The key idea in the
proof is a technique to obtain boundedness of the gradient based on logarithmic estimates.
Received: 8 July 1998 / Revised version: 19 February 1999 相似文献
8.
We consider the following Liouville equation in
For each fixed and a
j
> 0 for 1 ≤ j ≤ k, we construct a solution to the above equation with the following asymptotic behavior:
相似文献
9.
In this paper we study the jumping nonlinear problem
together with its energy functional
Convexity and concavity of J
(b,a)( u) in the case where Ky Fan’s minimax theorem does not directly work is studied, existence of type (II) regions is verified,
and unique solvability of the problem
is investigated.
Chong Li was supported by NSFC(10601058), NSFC(10471098), NSFC(10571096), and TYF(10526027)
Shujie Li was supported by NSFC(10471098) and NSFB(KZ200610028015)
Zhaoli Liu was supported by NSFC(10571123), NSFB(KZ200610028015), and PHR(IHLB). 相似文献
10.
We consider the micropolar fluid system in a bounded domain of
and prove the existence and the uniqueness of a global strong solution with initial data being a perturbation of the stationary
solution, whose existence is also obtained. We prove that these solutions converge uniformly to the stationary solutions with
exponential decay rate. The technique of our analysis is the semigroups approach in L
p
-spaces.
M. A. Rodríguez-Bellido is partially supported by D.G.E.S. and M.C. y T. (Spain), Projet BFM2003-06446-C02-01 and by M.E.C.
(Spain), Projet MTM2006-07932. 相似文献
11.
Abstract. We study the existence of a holomorphic generalized solution u of the PDE
where f is a given holomorphic generalized function and .
(Received 4 December 2000; in revised form 9 May 2001) 相似文献
12.
We study the limit cases p and p 1 of the functionals
where u 1 on a given compact set
and a > 0 is also given. Minimizers up of these functionals have uniformly bounded support p := { up > 0} and satisfy
Received: 19 October 2004 相似文献
13.
Abstract. Let be a semidirect product of semitopological semigroups S and T . If S and T act on topological spaces X and Y , respectively, then under suitable conditions there is a natural action of on X × Y . In this paper we characterize the almost periodic and strongly almost periodic compactification of the flow , in terms of related compactifications of (S,X) and (T,Y) . 相似文献
14.
For L a second order linear elliptic differential operator on
, one is usually interested in finding positive solutions of the heat equation
where V is a nonnegative potential. But for L the Laplacian, it was discovered by [BG] in 1984 that positive solutions may not exist if V is too singular. We use Gaussian estimates to extend this result to the case when L is not symmetric.
Helmut H. Schaefer in memoriam 相似文献
15.
Let L( x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H( x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution
of the Hamilton-Jacobi equation
is an action function in the large, i.e.,
for all
Received: 13 June 2003 相似文献
16.
§１ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｉｎｉｔｉａｌａｎｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｓｆｏｌｏｗｓ：ｕｔ＝Δ（ｇｒａｄφ（ｕ））＋αΔｂ（ｕ）＋ｆ（ｘ,ｔ,ｕ）,（ｘ,ｔ）∈ＱＴ＝Ω×（０,Ｔ］（... 相似文献
18.
Denoting Δ ? the Laplacian operator on the (2 N+1)-dimensional Heisenberg group ?
N
, we prove some nonexistence results for solutions of inequalities of the three types
in ?
N
and ?
N
×ℝ} +, with a∈ L
∞, when 1< p≤ p
0, where p
0 depends on N and the type of equation.
Received: 17 June 1999 相似文献
19.
We investigate spectral properties of operators on H2 of the form
We describe the fine spectrum when g is a rational function. We also provide useful relations for these operators in the Calkin algebra.Submitted: August 21, 2003 相似文献
20.
A result of Barbashin ([1], [15]) states that an exponentially bounded evolution family defined on a Banach space and satisfying some measurability conditions is uniformly exponentially stable if and only if for
some 1 ≤ p < ∞, we have that:
Actually the Barbashin result was formulated for non-autonomous differential equations in the framework of finite dimensional
spaces. Here we replace the above ”uniform” condition be a ”strong” one.
Among others we shall prove that the evolution family is uniformly exponentially stable if there exists a non-decreasing function with for all r > 0 such that for each , one has:
In particular, the family U is uniformly exponentially stable if and only if for some 0 < p < ∞ and each , the inequality
is fulfilled. The latter result extends a similar one from the recent paper [4]. Related results for periodic evolution families
are also obtained.
相似文献
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