共查询到20条相似文献,搜索用时 687 毫秒
1.
Zuzana Pátíková 《Mathematica Slovaca》2010,60(2):223-236
We establish asymptotic formulas for nonoscillatory solutions of a special conditionally oscillatory half-linear second order
differential equation, which is seen as a perturbation of a general nonoscillatory half-linear differential equation
$
(r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = |x|^{p - 1} \operatorname{sgn} x,p > 1,
$
(r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = |x|^{p - 1} \operatorname{sgn} x,p > 1,
相似文献
2.
We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping under quite general conditions. These results are extensions of the recent results developed by Sun [Y.G. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations. 相似文献
3.
J. Sugie 《Acta Mathematica Hungarica》2008,118(4):369-394
This paper deals with the second-order half-linear differential equation
4.
Summary Oscillation criteria are obtained for vector partial differential equations of the type Δv+b(x, v)v=0, x∈G, v∈Em, where G is an exterior domain in En, and b is a continuous nonnegative valued function in G × Em. A solution v: G→Em is called h-oscillatory in G whenever the scalar product [v(x), h] (|h|=1) has zeros x in G with |x| arbitrarily large. It
is shown that the spherical mean of [v(x), h] over a hypersphere of radius r in En satisfies a nonlinear ordinary differential inequality. As a consequence, the main theorems give sufficient conditions on
b(x, t), depending upon the dimension n, for all solutions v to be h-oscillatory in G.
Entrata in Redazione il 26 giugno 1975. 相似文献
5.
A. Avantaggiati 《Annali di Matematica Pura ed Applicata》1972,93(1):271-297
Summary We pose the problems, that are in the title, for functions u(x) from Λ=R
+
2
\{(0, 0)} into a Banach space B: such functions are required to satisfy asintotical conditions for |x| →0 and |x| → + ∞. Existence and uniqueness theorems are proved (no.5) and the regularity of Cn,λ type up to vertex of Λ is discussed (no.6). The main treatment is the study about an integral function g joined with operators which define the differential conditions
(no.3). This function arises from the transformation of those problems by using a rapresentation's formula stated in the previous
paper [1].
Entrata in Redazione il 22 dicembre 1971. Lavoro eseguito con finanziamento del C.N.R. nell'ambito della attività del gruppo di Ricerca di Analisi Funzionale. 相似文献 6.
利用积分平均技巧,得到了半线性二阶阻尼微分方程[a(t)|x′(t)|α-1x′(t)]′+p(t)k(t,x(t),x′(t))x′(t)+q(t)|x(t)|α-1x(t)=0的一些新的振动定理.这些结果改进和推广了Manojlovic J V[5]的结果. 相似文献
7.
We consider a nonoscillatory half-linear second order differential equation
8.
Qingkai Kong 《Journal of Mathematical Analysis and Applications》2007,332(1):512-522
We study the oscillation problems for the second order half-linear differential equation ′[p(t)Φ(x′)]+q(t)Φ(x)=0, where Φ(u)=|u|r−1u with r>0, 1/p and q are locally integrable on R+; p>0, q?0 a.e. on R+, and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p≡1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712-723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180-188] on linear equations to the half-linear case for all r>0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39-47] on nonoscillation when 0<r<1 and on oscillation when r>1. The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363-373]. 相似文献
9.
Laurent Veron 《Journal d'Analyse Mathématique》1992,59(1):231-250
We prove the existence and the uniqueness of a solutionu of−Lu+h|u|
α-1u=f in some open domain ℝd, whereL is a strongly elliptic operator,f a nonnegative function, and α>1, under the assumption that ∂G is aC
2 compact hypersurface, lim
x→∂G
(dist(x, ∂G))2α/(α-1)
f(x)=0, and lim
x→∂G
u(x)=∞. 相似文献
10.
Manabu Naito 《Annali di Matematica Pura ed Applicata》2007,186(1):59-84
In this paper the even-order quasilinear ordinary differential equation
is considered under the hypotheses that n is even, D(α
i
)x = (|x|αi−1 x)′, α
i
> 0(i = 1,2,…, n), β > 0, and p(t) is a continuous, nonnegative, and eventually nontrivial function on an infinite interval [a, ∞), a > 0. The existence of positive solutions of (1.1) is discussed, and basic results to the classical equation
are extended to the more general equation (1.1). In particular, necessary and sufficient integral conditions for the existence
of positive solutions of (1.1) are established in the case α 1α2⋅s α
n
≠ β.
This research was partially supported by Grant-in-Aid for Scientific Research (No. 15340048), Japan Society for the Promotion
of Science.
Mathematics Subject Classification (2000) 34C10, 34C11 相似文献
11.
We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n
u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2
n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31). 相似文献
12.
13.
D. Fortunato 《Annali di Matematica Pura ed Applicata》1979,119(1):317-331
Summary Let A=
be an elliptic differential operator inR
u, If, for |α|=l, the coefficients aα are ? nearly constant ? and, for |α|<l, they tend to zero at infinity with a certain swiftness, it is proved that A is a
Fredholm operator with indexx(A)=0 between a suitable weighted Sobolev space M contained in Wl,p (R n) and Lp(R
n, (1+|x|)lp)==
. It is shown, by counterexamples, that the above result, holds only if n>l, p>n/(n−l) and that isomorphism results can be
obtained, in general only if the coefficients aα(|α|<l) are assumed to be ? sufficiently small ? also on compact sets. Then a Sturm-Liouville type problem is studied and
a class of negative and falling off at infinity potentials V(x) is constructed in such a way that the Schr?dinger operator
H=−Δ+V(x), in L2(R
n), has a zero eigenvalue.
Sunto Sia un operatore differenziale ellittico inR n. Se, per |α|=l, i coefficienti aα sono ? quasi costanti ? e, per |α|<l, tendono a zero all'infinito con una certa rapidità, si dimostra che A è un operatore di Fredholm con indiceX(A)=0 tra un opportuno spazio di Sobolev con peso M contenuto in Wl,p(R n) ed Lp(R n, (1+|x|)lp)== . Si prova, mediante controesempi, che tale risultato è valido solo se n>l, p>n/(n−l) e che teoremi di isomorfismo si possono ottenere, in generale, solo se si assume che i coefficienti aα (|α|<l) sono ? sufficientemente piccoli ? anche su insiemi compatti. Si studia quindi un problema del tipo Sturm-Liouville e si costruisce una classe di potenziali V(x) negativi e convergenti a zero all'infinito, tali che l'operatore di Schr?dinger H=−δ+V(x) in L2(R n) abbia un autovalore nullo. Entrata in Redazione il 10 agosto 1977. Work supported by C.N.R. (G.N.A.F.A.). 相似文献 14.
Approximation to the function |x| plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals |x| if α = 1. We construct a Newman Type Operator rn(x) and prove max |x|≤1|xαsgn x-rn(x)|~Cn1/4e-π1/2(1/2)αn. 相似文献
15.
Ante Mimica 《Potential Analysis》2010,32(3):275-303
In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ
d
. We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process
in ℝ
d
with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr
− d − α
, ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion
with subordinator with Laplace exponent ϕ(λ) = λ
α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2). 相似文献
16.
On weighted approximation by Bernstein-Durrmeyer operators 总被引:6,自引:0,他引:6
Zhang Zhenqiu 《分析论及其应用》1991,7(2):51-64
In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in Lp[0, 1] (1≤p≤∞), where the weight function w(x)=xα(1−x)β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals.
Supported by Zhejiang Provincial Science Foundation. 相似文献
17.
S.M. Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant
nodes in [−1,1], In 2000, M. Rever generalized S.M. Lozinskii’s result to |x|α(0≤α≤1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|α(1<α<2). 相似文献
18.
B. Wróbel 《Acta Mathematica Hungarica》2009,124(4):333-351
Imaginary powers associated to the Laguerre differential operator $
L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1}
{{x_i^2 }}(\alpha _i^2 - 1/4)
$
L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1}
{{x_i^2 }}(\alpha _i^2 - 1/4)
are investigated. It is proved that for every multi-index α = (α1,...α
d
) such that α
i
≧ −1/2, α
i
∉ (−1/2, 1/2), the imaginary powers $
\mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R}
$
\mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R}
, of a self-adjoint extension of L
α, are Calderón-Zygmund operators. Consequently, mapping properties of $
\mathcal{L}_\alpha ^{ - i\gamma }
$
\mathcal{L}_\alpha ^{ - i\gamma }
follow by the general theory. 相似文献
19.
刘斌 《高校应用数学学报(英文版)》2002,17(2):135-144
§ 1 IntroductionWe are interested in the existence ofthree-solutions ofthe following second-order dif-ferential equations with nonlinear boundary value conditionsx″=f( t,x,x′) , t∈ [a,b] ,( 1 .1 )g1 ( x( a) ,x′( a) ) =0 , g2 ( x( b) ,x′( b) ) =0 ,( 1 .2 )where f:[a,b]×R1 ×R1 →R1 ,gi:R1 ×R1 →R1 ( i=1 ,2 ) are continuous functions.The study ofthe existence of three-solutions ofboundary value prolems forsecond or-der differential equations was initiated by Amann[1 ] .In[1 … 相似文献
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