Multivalued nonlinear equations on the half line: A fixed point approach

New fixed point theorems of the authors are used to establish the existence of one (or more) C[0, ∞) solutions to the nonlinear integral inclusion y(t)∈ ∫₀^∞ K(t, s) F(s, y(s))ds fort ∈ [0,∞).     

Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

Some Oscillation Theorems for Second Order Differential Equations

In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation

Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation

In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE)

二阶非线性矩阵微分方程的振动性

Some new oscillation criteria are established for the second-order matrix differential system(r(t)Z′(t))′ p(t)Z′(t) Q(t)F(Z′(t))G(Z(t)) = 0, t ≥ to ＞ 0,are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation.     

An abstract nonlinear Volterra equation

The existence, uniqueness, regularity and dependence upon data of solutions of the abstract Volterra equation u(t)+∫ ₀ ^t a(t-s)A(u(s))ds∈f(t), t≧0 are studied in a real Banach space. The nonlinear operatorA is assumed to bem-accretive and the assumptions on the kernela do not exclude the possibility that lim _t→0+ a(t)=+∞.     

Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values

We realize the Perron effect of change of values of characteristic exponents: for arbitrary parameters λ ₁ <- λ ₂ < 0, β ₂ ≥ β ₁ ≥ λ ₂, and m > 1, we prove the existence of a linear differential system $ \dot x $ \dot x = A(t)x, x ∈ R ₂, t ≥ t ₀, with bounded infinitely differentiable coefficients and with characteristic exponents λ ₁(A) = λ ₁ <- λ ₂(A) = λ ₂ and of an m-perturbation f: [t ₀,+∞) × R ₂ → R ₂ infinitely differentiable in time, continuously differentiable with respect to the phase variables y ₁ and y ₂, (y ₁, y ₂) = y ∈ R ₂ (infinitely differentiable with respect to the variables y ₁ ≠ 0 and y ₂ ≠ 0 and with respect to all of these variables in the case of a positive integer m > 1), satisfying the condition ‖f(t, y)‖ ≤ const × ‖y‖ ^m , y ∈ R ², t ≥ t ₀, and such that all nontrivial solutions y(t, c) of the perturbed system

An analysis of the solution set to a homotopy equation between polynomials with real coefficients

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h ₁(x, y, t)+ih ₂(x, y, t), whereh ₁ andh ₂ are polynomials from ℝ² × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh ₃ such thath ₂(x, y, t)=yh₃(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h ₁(x, 0, t)=0} and {(x+iy, t)∶h ₁(x, y, t)=0,h ₃(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.     

A fragmentation process connected to Brownian motion

Let (B _s , s≥ 0) be a standard Brownian motion and T ₁ its first passage time at level 1. For every t≥ 0, we consider ladder time set ℒ ^(t) of the Brownian motion with drift t, B ^(t) _s = B _s + ts, and the decreasing sequence F(t) = (F ₁(t), F ₂(t), …) of lengths of the intervals of the random partition of [0, T ₁] induced by ℒ ^(t) . The main result of this work is that (F(t), t≥ 0) is a fragmentation process, in the sense that for 0 ≤t < t′, F(t′) is obtained from F(t) by breaking randomly into pieces each component of F(t) according to a law that only depends on the length of this component, and independently of the others. We identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by Aldous and Pitman [3]. Received: 19 February 1999 / Revised version: 17 September 1999 /?Published online: 31 May 2000     

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

On the stability of the quadratic functional equation on bounded domains

A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]ⁿ →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]ⁿ withx + y, x - y ∈ [-t, t]ⁿ, then there exists a quadratic functionq: ℝⁿ →E such that ∥f(x) -q(x)∥ < (2912n² + 1872n + 334)δ for anyx ∈ [-t, t] ⁿ .     

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

Karhunen-Loève expansions of <Emphasis Type="Italic">α</Emphasis>-Wiener bridges

We study Karhunen-Loève expansions of the process(X _t ^(α)) _t∈[0,T) given by the stochastic differential equation $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) , with the initial condition X ₀^(α) = 0, where α > 0, T ∈ (0, ∞), and (B _t )t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X ^(α). As applications, we calculate the Laplace transform and the distribution function of the L ²[0, T]-norm square of X ^(α) studying also its asymptotic behavior (large and small deviation).     

Extensions of the measurable choice theorem by means of forcing

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S _t ^def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf _n: T→T such thatS _t={f_n(t)|nεω} for alltε[0,1],W _x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW _x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.     

A method for indefinite integration of oscillatory and singular functions

We propose a general method for computing indefinite integrals of the form