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1.
By using the Z
p
geometrical index theory, some sufficient conditions on the multiplicity results of periodic solutions to the second-order difference equations
are obtained. By two examples, we show that our results are the best possible in the sense that the lower bound of the number of periodic solutions cannot be improved. 相似文献
2.
Alexander L. Skubachevskii Hans-Otto Walther 《Journal of Dynamics and Differential Equations》2006,18(2):257-355
For periodic solutions to the autonomous delay differential equation
with rational periods we derive a characteristic equation for the Floquet multipliers. This generalizes a result from an earlier paper where only periods larger than 2 were considered. As an application we obtain a criterion for hyperbolicity of certain periodic solutions, which are rapidly oscillating in the sense that the delay 1 is larger than the distance between consecutive zeros. The criterion is used to find periodic orbits which are unstable and hyperbolic. An example of a non-autonomous periodic linear delay differential equation with a monodromy operator which is not hyperbolic shows how subtle the conditions of the hyperbolicity criteria in the present paper and in its predecessor are. We also derive first results on Floquet multipliers in case of irrational periods. These are based on approximations by periodic solutions with rational periods. 相似文献
3.
In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles.
We find a fractional Lagrangian L(x(t), where
a
c
D
t
α
x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange
At last, exact solutions for some Euler–Lagrange equations are presented. In particular, we consider the following equations
where g(t) and f(t) are suitable functions.
D. Baleanu is on leave of absence from Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele-Bucharest, Romania. e-mail:
baleanu@venus.nipne.ro. 相似文献
| (1) |
| (2) |
| (3) |
4.
彭奇林 《应用数学和力学(英文版)》2001,22(7):842-845
Ｕｐｔｏｎｏｗ ,ｔｈｅｑｕａｌｉｔａｔｉｖｅｂｅｈａｖｉｏｒｏｆｔｈｅｓｅｃｏｎｄ＿ｏｒｄｅｒｎｏｎｌｉｎｅａｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓａｎｄｔｈｅｉｒａｐｐｌｉｃａｔｉｏｎｓｈａｖｅｂｅｅｎｄｉｓｃｕｓｓｅｄｅｘｔｅｎｓｉｖｅｌｙｉｎｌｉｔｅｒａｔｕｒｅ.ＷｅｒｅｆｅｒｔｏｔｈｅＲｅｆｓ .[1 -8] .Ｃｏｎｖｅｒｓｅｌｙ ,ｔｈｅｄｉｓｃｕｓｓｉｏｎｆｏｒｔｈｅｑｕａｌｉｔａｔｉｖｅｂｅｈａｖｉｏｒｏｆｔｈｅｄｅｌａｙｓｙｓｔｅｍｉｓｌｅｓｓ .Ｉｎｔｈｉｓｐａｐｅｒ,ｗｅｓｔｕｄｙｔ… 相似文献
5.
David N. Cheban 《Journal of Dynamics and Differential Equations》2008,20(3):669-697
In the present article we consider a special class of equations
when the function (E is a strictly convex Banach space) is V-monotone with respect to (w.r.t.) , i.e. there exists a continuous non-negative function , which equals to zero only on the diagonal, so that the numerical function α(t):= V(x
1(t), x
2(t)) is non-increasing w.r.t. , where x
1(t) and x
2(t) are two arbitrary solutions of (1) defined on . The main result of this article states that every V-monotone Levitan almost periodic (almost automorphic, Bohr almost periodic) Eq. (1) with bounded solutions admits at least
one Levitan almost periodic (almost automorphic, Bohr almost periodic) solution. In particulary, we obtain some new criterions
of existence of almost recurrent (Levitan almost periodic, almost automophic, recurrent in the sense of Birkgoff) solutions
of forced vectorial Liénard equations.
相似文献
6.
We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension
where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate
solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial)
viscous parabolic approximation
as . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u
t
+ F(u)
x
= 0, satisfying the Liu admissibility conditions.
These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic
fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23].
Dedicated to Prof. Tai Ping Liu on the occasion of his 60
th
birthday 相似文献
7.
Under certain assumptions on f and g we prove that positive, global and bounded solutions u of the non-autonomous heat equation
in
(N ≥ 3) converge to a steady state.
Dedicated to Prof. Pavol Brunovsky on the occasion of his 70th birthday. 相似文献
8.
We study the stationary Dirac equation
where M(x) is a matrix potential describing the external field, and R(x, u) stands for an asymptotically quadratic nonlinearity modeling various types of interaction without any periodicity assumption.
For ħ fixed our discussion includes the Coulomb potential as a special case, and for the semiclassical situation (ħ → 0), we handle the scalar fields. We obtain existence and multiplicity results of stationary solutions via critical point
theory. 相似文献
9.
10.
Let E be a Banach space. We prove the instability of the trivial solution of the differential equation
where f: [0, +∞) × E → ℝ is a continuous mapping for which
__________
Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 404–414, July–September, 2005. 相似文献
11.
Emanuele Nunzio Spadaro 《Archive for Rational Mechanics and Analysis》2009,193(3):659-678
In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557–611, 1982) about the uniqueness
of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals,
where Ω is homeomorphic to a ball.
We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global
minimizers, both analytic up to the boundary. All these examples are suggested by the theory of minimal surfaces. 相似文献
12.
It is well-known that a KAM torus can be considered as a graph of smooth viscosity solution. Salamon and Zehnder (Comment
Math Helv 64:84–132, 1989) have proved that there exist invariant tori having prescribed Diophantine frequencies for nearly
integrable and positively definite Lagrangian systems with associated Hamiltonian H, whose Diophantine index is τ. If the invariant torus is represented as in the cotangent bundle , then we can show that for any viscosity solution u (x, P), which satisfies the H-J Eq. (1.1),
when is small enough.
For the more exact form, please see Theorem 2 for details. 相似文献
13.
We investigate the dynamics of the semiflow φ induced on H01(Ω) by the Cauchy problem of the semilinear parabolic equation
on Ω. Here
is a bounded smooth domain, and
has subcritical growth in u and satisfies
. In particular we are interested in the case when f is definite superlinear in u. The set
of attraction of 0 contains a decreasing family of invariant sets
distinguished by the rate of convergence
. We prove that the Wk’s are global submanifolds of
, and we find equilibria in the boundaries
. We also obtain results on nodal and comparison properties of these equilibria. In addition the paper contains a detailed exposition of the semigroup approach for semilinear equations, improving earlier results on stable manifolds and asymptotic behavior near an equilibrium.Supported by DFG Grant BA 1009/15-1. 相似文献
14.
Robert Jensen Changyou Wang Yifeng Yu 《Archive for Rational Mechanics and Analysis》2008,190(2):347-370
For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:
For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and
that of the Hamilton–Jacobi equation .
Supported by NSF DMS 0601162.
Supported by NSF DMS 0601403. 相似文献
15.
We prove the following statement:
Theorem 1.
Let
E and
be an arbitrary infinite-dimensional Banach space and a continuous mapping, respectively. Then, for every
and > 0, there exists a continuous mapping
such that
and the Cauchy problem
does not have a solution for every > 0. 相似文献
16.
The divergence identity for punctured domain B1(0)\ {0}
suggest a viewpoint on describing the behavior of a function uC2(B1(0)\{0}) near the origin. This is useful especially on describing the singular behavior of solutions of polyharmonic equations. In this paper we mainly show that the solution u of the equation
satisfies the identity that, letting
vi=(–)iu
provided there exist s0>0 and t0 0 such that f(x,t) c|x|–tq for 0<|x|<s0 and t t0 with n–q(n–2p) and q>1.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday. 相似文献
17.
Bin Liu 《Journal of Dynamics and Differential Equations》2006,18(4):975-990
In this paper, we study the stability of the equilibrium of planar systems
where X and Y are real analytic in x, y, and t, and quasi-periodic in t with frequencies (ω1,...,ω
n
). Under some reasonable assumptions, we obtain a sufficient and necessary condition. 相似文献
18.
Iterative solution of nonlinear equations with strongly accretive operators in Banach spaces 总被引：1，自引：0，他引：1
周海云 《应用数学和力学(英文版)》1999,20(3):282-289
1IntroductionandPreliminariesLetXbearealBanachspacewithnormIJ'11andadualX'.ThenormalizeddualitymappingJ:X~ZxisdefinedbyJx={x'eX*I(x,x')=11x112=11x if'},where',')denotesthegeneralizeddualitypairing.Itiswell-knownthatifX isstrictlyconvex,Jissingle-valuedandJ(tx)=tjxforallt201xeX.IfX*isuniformlyconvex,thenJisuniformlycontinuousonanyboundedsubsetSofX(of.Browde,fljandBarbuL2]).AnoperatorTwithdomainD(T)andrangeR(T)inXissaidtobeaccretiveifforeveryx,y6D(T),thereexistsajeJ(x--y)suchthat(T… 相似文献
19.
On the existence of periodic solutions to higher dimensional periodic system with delay 总被引：1，自引：0，他引：1
Ｃｏｎｓｉｄｅｒｉｎｇｔｈｅｈｉｇｈｅｒｄｉｍｅｎｓｉｏｎａｌｐｅｒｉｏｄｉｃｓｙｓｔｅｍｓｗｉｔｈｄｅｌａｙｏｆｔｈｅｆｏｒｍｘ′（ｔ）＝Ａ（ｔ,ｘ（ｔ））ｘ（ｔ）＋ｆ（ｔ,ｘ（ｔ－τ））,（１）ｘ′（ｔ）＝ｇｒａｄＧ（ｘ（ｔ））＋ｆ（ｔ,ｘ（ｔ－... 相似文献
20.
Michael Winkler 《Journal of Dynamics and Differential Equations》2005,17(2):331-351
The article deals with positive solutions of the Dirichlet problem for
where f(s)>0 for s>0 and f(0)=0. The asymptotic behavior of solutions is discussed for a rather large class of g. For g regular near zero, stability properties of equilibria are investigated. 相似文献