共查询到20条相似文献,搜索用时 328 毫秒
1.
The following system considered in this paper:where \({p > 1, p^* > 1 (1/p + 1/p^* = 1)}\) and \({\phi_q(z) = |z|^{q-2}z}\) for q = p or q = p *. This system is referred to as a half-linear system. The coefficient f(t) is assumed to be bounded, but the coefficients e(t), g(t) and h(t) are not necessarily bounded. Sufficient conditions are obtained for global asymptotic stability of the zero solution. Our results can be applied to not only the case that the signs of f(t) and g(t) change like the periodic function but also the case that f(t) and g(t) irregularly have zeros. Some suitable examples are included to illustrate our results.
相似文献
$$x' = -\,e(t)x + f(t)\phi_{p^*}(y), \qquad y'= -\,(p-1)g(t)\phi_p(x) - (p-1)h(t)y,$$
2.
In this paper we study the existence of infinitely many periodic solutions for second-order Hamiltonian systems , where F(t, u) is even in u, and ?F(t, u) is of sublinear growth at infinity and satisfies the Ahmad-Lazer-Paul condition.
相似文献
$$\left\{ {\begin{array}{*{20}c} {\ddot u(t) + A(t)u(t) + \nabla F(t,u(t)) = 0,} \\ {u(0) - u(T) = \dot u(0) - \dot u(T) = 0,} \\ \end{array} } \right.$$
3.
V. Ya. Yakubov 《Differential Equations》2008,44(1):71-76
We present a number of unstable second-order equations of the form where the coefficient g(x) satisfies the conditions g(x) ∈ C(0, ∞) and limx→+∞ g(x)=0 but the maximum absolute values of solutions grow unboundedly (as power-law functions or even as exponentials).
相似文献
$$y'' + (1 + g(x))y = 0,$$
4.
S. A. Modina 《Russian Mathematics (Iz VUZ)》2009,53(4):31-33
In this paper we study the three-element functional equation , subject to We assume that the coefficients G(z) and g(z) are holomorphic in R and their boundary values G +(t) and g +(t) belong to H(Γ), G(t)G(t ?1) = 1. We seek for solutions Φ(z) in the class of functions holomorphic outside of \(\bar R\) such that they vanish at infinity and their boundary values Φ?(t) also belong to H(Γ). Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.
相似文献
$(V\Phi )(z) \equiv \Phi (iz) + \Phi ( - iz) + G(z)\Phi \left( {\frac{1}{z}} \right) = g(z), z \in R,$
$R: = \{ z:\left| z \right| < 1, \left| {\arg z} \right| < \frac{\pi }{4}\} .$
5.
For the first-order ordinary delay differential equation where p ∈ L loc(?+; ?+), τ ∈ C(?+; ?+), τ(t) ≤ t for t ∈ ?+, limt→+∞ τ(t) = +∞, and ?+:= [0, ∞), we obtain new criteria for the existence of sign-definite and oscillating solutions, thus generalizing some earlier-known results.
相似文献
$$u'(t) + p(t)u(r(t)) = 0,$$
6.
F. E. Lomovtsev 《Differential Equations》2008,44(6):866-871
We prove the well-posed solvability in the strong sense of the boundary value Problems where the unbounded operators A s (t), s > 0, in a Hilbert space H have domains D(A s (t)) depending on t, are subordinate to the powers A 1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A 0(t) and the symmetric operators A s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that , where the smoothing operators are defined by .
相似文献
$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$
$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$
$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$
7.
Adel Mahmoud Gomaa 《Czechoslovak Mathematical Journal》2017,67(2):339-365
We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation where {?(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([?d, 0],E) be the Banach space of continuous functions from [?d, 0] into E and f d : [a, b] × C([?d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ t x(s) = x(t + s) for each s ∈ [?d, 0]. We prove that, under certain conditions, the differential equation with delay has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.
相似文献
$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$
(P)
$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$
(Q)
8.
In this paper, necessary and sufficient conditions are obtained for every bounded solution of to oscillate or tend to zero as t → ∞ for different ranges of p(t). It is shown, under some stronger conditions, that every solution of (*) oscillates or tends to zero as t → ∞. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.
相似文献
$$\left( * \right)\quad \quad \quad \quad \quad \quad \left[ {y\left( t \right) - p\left( t \right)y\left( {t - \tau } \right)^{\left( n \right)} } \right] + Q\left( t \right)G\left( {y\left( {t - \sigma } \right)} \right) = f\left( t \right),\quad t \geqslant 0,$$
9.
We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case λ ? 1 and if the corresponding second order differential equation h″ + q(t)h = 0 is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.
相似文献
$$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$
10.
For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, T_u) = P{inf (t∈[0,S]_(s∈[t,t+T_u])) sup R_u(s) 0}, S, T_u 0.For X being a general Gaussian process we derive approximations of P_S(u, T_u) as u →∞. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval. 相似文献
11.
This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation on a time scale \(\mathbb{T}\) where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on \(\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}\) and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty \). We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.
相似文献
$$(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$$
12.
Salih Jawad 《Monatshefte für Mathematik》2016,180(4):765-783
In dieser Arbeit befassen wir uns mit der inhomogenen Differentialgleichung: im abstrakten Hilbertraum und weisen eindeutige starke Lösungen aus der Klasse der lipschitzstetigen Funktionen nach, falls f(t) von beschränkter Variation ist, sowie klassische Lösungen, falls f(t) zusätzlich stetig ist. Das bedeutet den Verzicht auf die bisher übliche Forderung der Lipschitzstetigkeit von f(t) für den direkten Nachweis der klassischen Lösbarkeit.
相似文献
$$\begin{aligned} u'(t)+A(t)u(t)+f(t)= & {} 0,\quad t\in (t_1,t_2)\\ u(t_1)= & {} \varphi \end{aligned}$$
13.
Óscar Ciaurri T. Alastair Gillespie Luz Roncal José L. Torrea Juan Luis Varona 《Journal d'Analyse Mathématique》2017,132(1):109-131
It is well known that the fundamental solution of with u(n, 0) = δ nm for every fixed m ∈ Z is given by u(n, t) = e ?2t I n?m (2t), where I k (t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series W t f(n) = Σ m∈Z e ?2t I n?m (2t)f(m). This formula allows us to analyze some operators associated with the discrete Laplacian using semigroup theory. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ? p (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. We also show that the Riesz transforms essentially coincide with the so-called discrete Hilbert transform defined by D. Hilbert at the beginning of the twentieth century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.
相似文献
$${u_t}\left( {n,t} \right) = u\left( {n + 1,t} \right) - 2u\left( {n,t} \right) + u\left( {n - 1,t} \right),n \in \mathbb{Z},$$
14.
N. A. Izobov 《Differential Equations》2008,44(5):618-631
We prove the conditional exponential stability of the zero solution of the nonlinear differential system with L p -dichotomous linear Coppel-Conti approximation .x = A(t)x whose principal solution matrix X A (t), X A (0) = E, satisfies the condition where P 1 and P 2 are complementary projections of rank k ∈ {1, …, n ? 1} and rank n ? k, respectively, and with a higher-order infinitesimal perturbation f:[0, ∞) × U → R n that is piecewise continuous in t ≥ 0 and continuous in y in some neighborhood U of the origin.
相似文献
$$\dot y = A(t)y + f(t,y),{\mathbf{ }}y \in R^n ,{\mathbf{ }}t \geqslant 0,$$
$$\mathop \smallint \limits_0^t \left\| {X_A (t)P_1 X_A^{ - 1} (\tau )} \right\|^p d\tau + \mathop \smallint \limits_t^{ + \infty } \left\| {X_A (t)P_2 X_A^{ - 1} (\tau )} \right\|^p d\tau \leqslant C_p (A) < + \infty ,{\mathbf{ }}p \geqslant 1,{\mathbf{ }}t \geqslant 0,$$
15.
In this paper, we construct a continuous positive periodic function p(t) such that the corresponding superlinear Duffing equation x′′+ a(x)~(x2n+1)+p(t)x~(2m+1)= 0, n + 2≤2 m+12n+1 possesses a solution which escapes to infinity in some finite time, and also has infinitely many subharmonic and quasi-periodic solutions, where the coefficient a(x) is an arbitrary positive smooth periodic function defined in the whole real axis. 相似文献
16.
Nikos Frantzikinakis 《Journal d'Analyse Mathématique》2010,112(1):79-135
We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for “typical” choices of Hardy field functions a(t) with polynomial growth, the averages converge in mean and we determine their limit. For example, this is the case if a(t) = t 3/2, t log t, or t 2 + (log t)2. Furthermore, if {a 1(t), …, a ? (t)} is a “typical” family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages converge in mean to the product of the integrals of the corresponding functions. For example, this is the case if the functions a i (t) are given by different positive fractional powers of t. We deduce several results in combinatorics. We show that if a(t) is a non-polynomial Hardy field function with polynomial growth, then every set of integers with positive upper density contains arithmetic progressions of the form {m,m + [a(n)], …, m + ?[a(n)]}. Under suitable assumptions, we get a related result concerning patterns of the form {m,m + [a 1(n)], …,m + [a ? (n)]}.
相似文献
${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[a(n)]}}x) \cdots {f_\ell }({T^{\ell [a(n)]}}x)} $
${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[{a_1}(n)]}}x) \cdots {f_\ell }({T^{[{a_\ell }(n)]}}x)} $
17.
A. K. Demenchuk 《Differential Equations》2008,44(2):186-191
We consider the nonlinear quasiperiodic Pfaff system Let K (j) be a frequency basis with respect to t j of the functions F (1),...,F (m), and let L (j) be a frequency basis with respect to t j of the functions G (1),...,G (m). Suppose that the set K (j) ∪ L (j) of numbers is rationally linearly independent. We obtain necessary and sufficient conditions for the existence of quasiperiodic solutions with frequency bases L (1),..., L (m).
相似文献
$$\frac{{\partial x}}{{\partial t_j }} = F^{(j)} (t,x) + G^{(j)} (t,x)(j = 1,...,m).$$
18.
Mohamed Nor Frioui Sofiane El-hadi Miri Tarik Mohamed Touaoula 《Journal of Applied Mathematics and Computing》2018,58(1-2):47-73
The aim of this paper is to provide a unified Lyapunov functional for an age-structured model describing a virus infection. Our main contribution is to consider a very general nonlinear infection function, gathering almost all usual ones, for the following problem: where T(t), i(t, a) and V(t) are the populations of uninfected cells, infected cells with infection age a and free virus at time t respectively. The functions \(\delta (a),\) p(a), are respectively, the age-dependent per capita death, and the viral production rate of infected cells with age a. The global asymptotic analysis is established, among other results, by the use of compact attractor and strongly uniform persistence. Finally some numerical simulations illustrating our results are presented.
相似文献
$$\begin{aligned} \left\{ \begin{array}{lll} T'(t)=A- dT(t)-f(T(t),V(t)) \;\;\ t \ge 0,\\ i_t(t,a)+i_a(t,a)=-\delta (a) i(t,a), \\ V'(t)=\int _0^{\infty } p(a)i(t,a)da-cV(t), \end{array} \right. \end{aligned}$$
(0.1)
19.
In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systemswhere \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.
相似文献
$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$
20.
A. V. Lasunskii 《Differential Equations》2009,45(3):460-463
For the nonautonomous Lotka-Volterra model where some part φ(x, y) of the prey population is out of reach of the predator, we obtain sufficient conditions for the existence of a positive asymptotically stable equilibrium in the domain of admissible values of the variables x and y. We consider the cases in which φ(x, y) = m, φ(x, y) = mx, and φ(x, y) = my.
相似文献
$\dot x = \alpha (t)(x - M^{ - 1} x^2 - K^{ - 1} (x - \phi (x,y))y),\dot y = \beta (t)y(L^{ - 1} (x - \phi (x,y)) - 1),$