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1.
We consider vector differential equtions x″ + g(t, x, x′) = f(t), x?Rn, and show the existence of periodic solutions under conditions which show that growth conditions on g required in earlier results are not needed.  相似文献   

2.
Sufficient conditions are given so that all solutions of the nonlinear differential equation u″ + φ(t, u, u′)u′ + p(t) gf(u) g(u′) = h(t, u, u′) are continuable to the right of an initial t-value t0 ? 0. These conditions are then extended so that all solutions u of the equation in question together with their derivative u′ are bounded for t ? t0 .  相似文献   

3.
The existence of solutions in a weak sense of x′ + (A + B(t, x))x = f(t, x), x(0) = x(T) is established under the conditions that A generates a semigroup of compact type on a Hilbert space H; B(t,x) is a bounded linear operator and f(t, x) a function with values in H; for each square integrable ?(t) the problem with B(t, ?(t)) and f(t, ?(t)) in place of B(t, x) and f(t, x) has a unique solution; and B and f satisfy certain boundedness and continuity conditions.  相似文献   

4.
A general uniqueness theorem for the abstract Cauchy problem x′(t) = f(t, x(t)); x(t) = P(t) + o(μ(t)) as t ↓ 0 is proved extending a result due to Goldstein. Applications of this theorem to a partial differential equation of a generalized Euler-Poisson-Darboux type are also discussed.  相似文献   

5.
When m = qt, g(xt+1, x2t+1,…, x(q?1)t+1) is a linear combination of only odd (or only even) elementary symmetric functions, then every cycle of the nonlinear shift register with feedback function f(x1, x2,…, xm) = x1 + g(xt+1, x2t+1,…, x(q?1)t+1) has a minimal period dividing m(q+1). It is also shown that when g is derived from a cyclic code with minimum distance ?3, every cycle of this shift register has a minimal period dividing m(q + 1).  相似文献   

6.
It is shown that the first order multivalued equation for V = V(t, x, y, z) involving the sum of two subdifferentials composed with the partials of V (Vt +f(t, x, y, z) · ▽xV + β(Vy) + γ(Vz) + h(t, x, y, z) ? 0 a.e.) has a Lipschitz solution. This solution is shown to be the value of a differential game in which the players are restricted to choosing monotone nondecreasing functions of time. Accordingly, the multivalued equation is interpreted as the corresponding Hamilton-Jacobi equation of the game.  相似文献   

7.
The initial value problem on [?R, R] is considered: ut(t, x) = uxx(t, x) + u(t, x)γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥ → ∞ as tT. For the specific initial value ? = , where ψ ? 0, ψxx + ψγ = 0, ψR) = 0, k is sufficiently large, it is shown that if x ≠ 0, then limtTu(t, x) and limtTux(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.  相似文献   

8.
This paper deals with the construction of analytic-numerical solutions with a priori error bounds for systems of the type ut = Auxx, u(0,t) + ux(0,t) = 0, Bu(1,t) + Cux(1,t) = 0, 0 < x < 1, t > 0, u(x,0) = f(x). Here A, B, C are matrices for which no diagonalizable hypothesis is assumed. First an exact series solution is obtained after solving appropriate vector Sturm-Liouville-type problems. Given an admissible error ε and a bounded subdomain D, after appropriate truncation an approximate solution constructed in terms of data and approximate eigenvalues is given so that the error is less than the prefixed accuracy ε, uniformly in D.  相似文献   

9.
Some results are given concerning positive solutions of equations of the form x(n) + P(t) G(x) = Q(t, x).Let class I (II) consist of all n-times differentiable functions x(t), such that x(t)>0 and x(n ? 1)(t) ? 0 (x(n ? 1)(t) ? 0) for all large t. Two theorems are given guaranteeing the nonexistence of solutions in class I and II, respectively, and three theorems ensure the convergence to zero of positive solutions. A recent result of Hammett concerning the second-order case is extended to the general case.  相似文献   

10.
Oscillation criteria for the class of forced functional differential inequalities x(t){Lnx(t) + f(t, x(t), x[g1(t)],…, x[gm(t)]) ? h(t)} ? 0, for n even, and x(t){Lnx(t) ? f(t, x(t), x[g1(t)],…, x[gm(t)]) ? h(t)} ? 0, for n odd, are established.  相似文献   

11.
Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝0ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝0tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝0ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f0tr(t, s) f(s) ds maps a weighted L1 space continuously into another weighted L1 space, and a weighted L space into another weighted L space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.  相似文献   

12.
The purpose of this paper is to obtain sufficient conditions for oscillation of all solutions of the equation x(t) = f(t) + ∝at K(t, s, x(s), x(g(s))) ds to study the behaviour of its oscillatory solutions in a dependence on the distance between their consecutive zeros and to establish a theorem for localization of the zeros of its solutions.  相似文献   

13.
This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?p(x′))′ + b(t)?p(x′) + c(t)?p(x) = 0, where ?p(x) = |x|p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.  相似文献   

14.
Sufficient conditions are developed for the null-controllability of the nonlinear delay process (1) x?(t) = L(t, xt) + B(t) u(t) + f(t, xt, u(t)) when the values of the control functions u lie in an m-dimensional unit cube Cm of Em. Conditions are placed on f which guarantee that if the uncontrolled system x?(t) = L(t, xt) is uniformly asymptotically stable and if the linear control system x(t) = L(t, xt) + B(t) u(t) is proper, then (1) is null-controllable.  相似文献   

15.
This paper extends a result of Fujita [On the blowing up of solutions to the Cauchy problem for ut = Δu + u1 + a, J. Faculty Science, U. of Tokyo 13 (1966), 109–124] to show that solutions u = u(t, x) for t > 0 and x?R2 to the equation ut = Δu + u2 with u(0, x) = a(x) must grow at a rate faster than exp(∥x2) at some finite time t, as long as a(x) is nonnegative and not almost everywhere zero.  相似文献   

16.
Several oscillation criteria are given for the second-order damped nonlinear differential equation (a(t)[y′(t)]σi +p(t)[y′(t)]σ +q(t)f(y(t)) = 0, where σ > 0 is any quotient of odd integers, a ϵ C(R, (0, ∞)), p(t) and q(t) are allowed to change sign on [to, ∞), and f ϵ Cl (R, R) such that xf (x) > 0 for x≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.  相似文献   

17.
We examine the asymptotic stability of the zero solution of the first-order linear equation x′(t) = Ax(t) + ∝0tB(t ? s) x(s) ds, where B(t) is integrable and does not change sign on [0, ∞). The results are applied to an examination of the stability of equilibrium of some nonlinear population models.  相似文献   

18.
Oscillation and nonoscillation criteria for the nth order nonlinear functional differential equation Lnx(t) + f(t, x(t), x[g(t)]) = h(t) are established. Some illustrative examples are also included.  相似文献   

19.
We consider the pure initial value problem for the system of equations νt = νxx + ?(ν) ? w, wt= ε(ν ? γw), ε, γ ? 0, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here ?(v) = ?v + H(v ? a), where H is the Heaviside step function and a ? (0, 12). This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if limt→∞ s(t) = ∞. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as ¦x¦ → ∞.  相似文献   

20.
The problem of determining the source term F(x, t) in the linear parabolic equation u t = (k(x)u x (x, t)) x + F(x, t) from the measured data at the final time u(x, T) = µ(x) is formulated. It is proved that the Fréchet derivative of the cost functional J(F) = ‖µ T (x) ? u(x, T)‖ 0 2 can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.  相似文献   

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