共查询到20条相似文献,搜索用时 62 毫秒
1.
Xiaojing Yang 《Journal of Mathematical Analysis and Applications》2004,296(2):708-718
In this paper, the unbounded solutions for the following nonlinear planar system:
x′=a+y+−a−y−+f(t),y′=−b+x++b−x−+g(t), 相似文献
2.
Kamil Jurczak 《Journal of Theoretical Probability》2018,31(3):1273-1302
In this article, we investigate high-dimensional band sample covariance matrices under the regime that the sample size n, the dimension p, and the bandwidth d tend simultaneously to infinity such that It is shown that the empirical spectral distribution of those matrices converges weakly to a deterministic probability measure with probability 1. The limiting measure is characterized by its moments. Certain restricted compositions of natural numbers play a crucial role in the evaluation of the expected moments of the empirical spectral distribution.
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$$\begin{aligned} n/p\rightarrow 0 \ \ \text {and} \ \ d/n\rightarrow y>0. \end{aligned}$$
3.
In this paper, we investigate the existence results for fractional differential equations of the form and where \(D_{c}^{q}\) is the Caputo fractional derivative. We prove the above equations have solutions in C[0, T). Particularly, we present the existence and uniqueness results for the above equations on \([0,+\infty )\).
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$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T)\left( 0<T\le \infty \right) , \quad q \in (1,2),\\ x(0)=a_{0},\quad x^{'}(0)=a_{1}, \end{array}\right. } \end{aligned}$$
(0.1)
$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T), \quad q \in (0,1),\\ x(0)=a_{0}, \end{array}\right. } \end{aligned}$$
(0.2)
4.
We prove the existence of positive ω-periodic solutions for the delayed differential equationwhere λ is a positive parameter, \({a,b,\tau \in C(\mathbb{R},\mathbb{R})}\) are ω-periodic functions with \({a,b\geq 0,a,b \not \equiv 0,f,g\in C([0,\infty ),[0,\infty ))}\), g does not need to be bounded above or bounded away from 0, and g(0) = 0 is allowed.
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$$x^{\prime}(t) = a(t)g(x(t))x(t) - \lambda b(t)f(x(t - \tau (t))),$$
5.
In this article, we establish some new criteria for the oscillation of fourth-order nonlinear delay differential equations of the formprovided that the second-order equationis nonoscillatory or oscillatory.
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$$(r_2(t)(r_1(t)(y''(t))^\alpha)')' + p(t)(y''(t))^\alpha + q(t)f(y(g(t))) = 0$$
$$(r_2(t)z'(t))') + \frac{p(t)}{r_1(t)}z(t) = 0$$
6.
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.
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$$\left\{ {\begin{array}{*{20}c} {\ddot u(t) + A(t)u(t) + \nabla F(t,u(t)) = 0,} \\ {u(0) - u(T) = \dot u(0) - \dot u(T) = 0,} \\ \end{array} } \right.$$
7.
8.
This paper describes the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation We call soliton a solution of (0.1) of the form u(t,x)=Q c (x?ct?y 0), where c>0, y 0∈? and \(Q_{c}''+Q_{c}^{4}=cQ_{c}\). Since (0.1) is not an integrable model, the general question of the collision of two given solitons \(Q_{c_{1}}(x-c_{1}t)\), \(Q_{c_{2}}(x-c_{2}t)\) with c 1≠c 2 is an open problem.
$\partial_tu+\partial_x(\partial_x^2u+u^4)=0,\quad t,x\in \mathbb{R}.$
(0.1)
We focus on the special case where the two solitons have nearly equal speeds: let U(t) be the solution of (0.1) satisfying for \(\mu_{0}=(c_{2}^{-}-c_{1}^{-})/(c_{1}^{-}+c_{2}^{-})>0\) small. By constructing an approximate solution of (0.1), we prove that, for all time t∈?, with y 1(t)?y 2(t)>2|ln?μ 0|+C, for some C∈?. These estimates mean that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable KdV equation in this regime.
$\lim_{t\to-\infty}\|{U}(t)-Q_{c_1^-}(.-c_1^-t)-Q_{c_2^-}(.-c_2^-t)\|_{H^1}=0,$
$\begin{array}{l}\displaystyle{U}(t)={Q}_{c_1(t)}(x-y_1(t))+{Q}_{c_2(t)}(x-y_2(t))+{w}(t)\\[6pt]\displaystyle\quad\mbox{where }\|w(t)\|_{H^1}\leq|\ln\mu_0|\mu_0^2,\end{array}$
However, unlike in the integrable case, we prove that the collision is not perfectly elastic, in the following sense, for some C>0, 相似文献
$\lim_{t\to+\infty}c_1(t)>c_2^-\biggl(1+\frac{\mu_0^5}{C}\biggr),\quad \lim_{t\to+\infty}c_2(t)
and \({w}(t)\not\to0\) in H 1 as t→+∞.9.
We further develop the method, devised earlier by the authors, which permits finding closed-form expressions for the optimal controls by elastic boundary forces applied at two ends, x = 0 and x = l, of a string. In a sufficiently large time T, the controls should take the string vibration process, described by a generalized solution u(x, t) of the wave equation from an arbitrary initial state to an arbitrary terminal state
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$$u_{tt} (x,t) - u_{tt} (x,t) = 0,$$
$$\{ u(x,0) = \varphi (x), u_t (x,0) = \psi (x)$$
$$\{ u(x,T) = \hat \varphi (x), u_t (x,T) = \hat \psi (x).$$
10.
M. M. Kabardov 《Vestnik St. Petersburg University: Mathematics》2009,42(2):90-94
In this paper, we consider a method for inverting the Laplace transform F(s) = \(\int\limits_0^\infty {e^{ - st} f(t)dt} \), which consists in representing the original function by the Laguerre series
$f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt).} $
(1)
First, we perform a conformal mapping of the plane (s), which depends on parameter ξ. The value of the parameter is determined by the location of the singular points of the given representation. Under this mapping, series (1) takes the form
$f(t) = \frac{{b - \xi }}{b}\exp (\xi t)\sum\limits_{k = 0}^\infty {c_k L_k ((b - \xi )t).} $
It is demonstrated that such inverting scheme is equivalent to applying the Picone-Tricomi method with further acceleration of the rate of convergence of series (1) using the Euler-Knopp nonlinear procedure
$\sum\limits_{k = 0}^\infty {a_k z^k } = \sum\limits_{k = 0}^\infty {A_k (p)\frac{{z^k }}{{(1 - pz)^{k + 1} }},} A_k (p) = \sum\limits_{j = 0}^k {\left( \begin{gathered} k \hfill \\ j \hfill \\ \end{gathered} \right)( - p)^{k - j} a_j } .$
Under this approach, the original function is represented by the series where parameters ξ and p are related by the formula p = x/(ξ ? b). Unlike many other methods for summation of series, in the scheme suggested, there is no need to investigate the regularity conditions.
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$f(t) = \exp \left( {\frac{{bpt}}{{p - 1}}} \right)\sum\limits_{k = 0}^\infty {\frac{{A_k (p)}}{{(1 - p)^{k + 1} }}L_k } \left( {\frac{{bpt}}{{1 - p}}} \right),$
11.
We prove the existence of positive \(\omega \)-periodic solutions for the double-delayed differential equation where \(\lambda \) is a positive parameter, \(a,b,c,\tau ,\nu \in C(\mathbb {R}, \mathbb {R})\) are \(\omega \)-periodic functions with \(a,b\ge 0,a,b\not \equiv 0,f,g,h\in C([0,\infty ),\mathbb {R})\) with \(g>0\) on \((0,\infty ),\) \(\ h\) is bounded, f is either superlinear or sublinear at \(\infty \) and could change sign.
相似文献
$$\begin{aligned} x^{\prime }(t)-a(t)g(x(t))x(t)=-\lambda (b(t)f(x(t-\tau (t))+c(t)h(x(t-\nu (t))), \end{aligned}$$
12.
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 .
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$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$
$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$
$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$
13.
In this work, we prove the Cauchy–Kowalewski theorem for the initial-value problem where in the space \(P_{D}\left( E\right) \) of Pseudo Q-holomorphic functions.
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$$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$
$$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$
14.
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.
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$$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$
15.
Başak Karpuz 《Journal of Fixed Point Theory and Applications》2016,18(4):889-903
We study nonoscillation/oscillation of the dynamic equationwhere \({t_0 \in \mathbb{T}}\), \({{\rm sup} \mathbb{T} = \infty}\), \({r \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, \mathbb{R}^+)}\), \({p \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, {\mathbb{R}^+_0})}\). By using the Riccati substitution technique, we construct a sequence of functions which yields a necessary and sufficient condition for the nonoscillation of the equation. In addition, our results are new in the theory of dynamic equations and not given in the discrete case either. We also illustrate applicability and sharpness of the main result with a general Euler equation on arbitrary time scales. We conclude the paper by extending our results to the equationwhich is extensively discussed on time scales.
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$${(rx^\Delta)}^{\Delta}(t) + p(t)x(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$
$${(rx^\Delta)}^{\Delta}(t) + p(t)x^\sigma(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$
16.
Abdulkadir Dogan 《Positivity》2018,22(5):1387-1402
This paper deals with the existence of positive solutions of nonlinear differential equation subject to the boundary conditions where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality
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$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$
$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$
$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$
17.
Ri-An Yan Shu-Rong Sun Dian-Wu Yang 《Journal of Applied Mathematics and Computing》2015,48(1-2):187-203
In this paper, we study the existence of solutions for the boundary value problems of fractional perturbation differential equations or subject to where \(1<\alpha <2,\,D^{\alpha }\) is the standard Caputo fractional derivatives. Using some fixed point theorems, we prove the existence of solutions to the two types. For each type we give an example to illustrate our results.
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$$\begin{aligned} D^{\alpha }\left( \frac{x(t)}{f(t,x(t))}\right) =g(t,x(t)),\;\;a.e.\;t\in J=[0,1], \end{aligned}$$
$$\begin{aligned} D^{\alpha }\left( x(t)-f(t,x(t))\right) =g(t,x(t)),\;\;a.e.\;t\in J, \end{aligned}$$
$$\begin{aligned} x(0)=y(x),\;\;x(1)=m, \end{aligned}$$
18.
We consider parabolic systemsin two space dimensions with initial and Dirichlet boundary conditions. The elliptic part including a 0 is derived from a potential with quadratic growth in ?u and is coercive and monotone. The term a 0 may grow quadratically in ?u and satisfies a sign condition a 0 · u ≥ ?K. We prove the existence of a regular long time solution verifying a regularity criterion of Arkhipova. No smallness is assumed on the data.
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$u_{t} - {\rm div} \left( a(t, x, u, \nabla u)\right) + a_{0}(t, x, u, \nabla u) = 0$
19.
We find conditions under which the system of root functions of the operator is a Riesz basis in L 2[0, 1].
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$$L_y = l[y] = ay'(x) + y'(1 - x) + p_1 (x)y(x) + p_2 (x)y(1 - x),x \in [0,1],U_1 (y) = \int\limits_0^1 {y(t)d\sigma (t) = 0,} $$
20.
G. A. Grigoryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2007,42(4):184-197
The paper studies some properties of solutions of the Riccati equation on a semiaxis [t 0, +∞) for different types of initial value sets. Two types of solutions are singled out: normal, that are in a sense stable, and extremal, that are non-stable in the Lyapunov sense. Relations expressing the extremal solutions by means of a given normal solution in quadratures and elementary functions are obtained and some relations between solutions the extendable to [t 0, +∞) are derived.
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$y'(t) + a(t)y^2 (t) + b(t)y(t) + c(t) = 0$