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.     

Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation

Sufficient conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the second-order nonlinear differential equation $$x^{\prime\prime}(t)+q(t)|x(t)|^{\gamma}\,{\rm sgn}\, x(t)=0, \quad \quad (A)$$ where γ is a positive constant different from 1 and q : [a, ∞) → (0, ∞) is a continuous integrable function. We show how an application of the theory of regular variation gives the possibility of determining the precise asymptotic behavior of solutions of both superlinear and sublinear equation (A).     

Algebras with skew-symmetric identity of degree 3

Algebras with one of the following identities are considered:

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

Some Properties of Essential Spectra of a Positive Operator

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ_ew(T) of the operator T is a set , where is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum

Some properties of essential spectra of a positive operator, II

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ_ew(T), in particular, the equality , where is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ_ef(T), then for every a ≠ 0, where T₋₁ is a residue of the resolvent R(., T) at r(T). The new conditions for which implies , are derived. The question when the relation holds, where is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands such that if , where , then an operator on D_i is band irreducible.      

The metric projection on the classH(Ω) satisfies Lipschitz condition

Устанавливается лиш шщивость и односторо нняя дифференцируемость метрической проекции $$P_H^t :x \to P_H^t \left( x \right) = \left\{ {y \in H: \parallel x - y\parallel \leqq t + \varrho \left( {x, H} \right)} \right\}, t \geqq 0$$ на класс из пространстваB(Q) огр аниченных на множест веQ функцийx c нормой ∥x∥=sup {∣x(q)∣:q∈Q}, гдеΩ—метрика наQ. В частн ости, дляy _i ∈B(Q),t _i ≧0, метрикΩ _i и \(P_i = P_{H(\Omega _i )}^{t_i } \left( {y_i } \right), i = 1, 2\) доказано неравенство $$h\left( {P_1 , P_2 } \right) \leqq \frac{3}{2}\mathop {\sup }\limits_{q_1 , q_2 \in Q} \left| {\Omega _1 \left( {q_1 , q_2 } \right) - \Omega _2 \left( {q_1 , q_2 } \right)} \right| + 2\parallel y_1 - y_2 \parallel + 1\left| {t_1 - t_2 } \right|,$$ гдеh — расстояние Хау сдорфа. Константы 3/2, 2, 1 в э том неравенстве неулучш аемы.     

Existence of Nondecreasing and Continuous Solutions of an Integral Equation with Linear Modification of the Argument

We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0, 1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation x（t） = a（t） ＋ （Tx）（t） ∫0^σ（t） u（t, s, x（s）, x（λs））ds 0 〈 λ 〈 1 which can be considered in connection with the following Cauchy problem x＇（t） = u（t, s, x（t）, x（λt））, t ∈ [0, 1], 0 〈 λ 〈 1 x（0） = u0.     

On the structure of positive solutions of a class of fourth order nonlinear differential equations

A detailed analysis is made of the structure of positive solutions of fourth-order differential equations of the form

On second order sublinear oscillation

The basic purpose of this paper is to present a new oscillation criterion for second order sublinear ordinary differential equations of the formx(t) +a(t)f[x(t)] = 0,t t ₀>0, wherea is a continuous function on [t ₀, ) without any restriction on its sign andf is a continuous function on the real line, which is continuously differentiable, except possibly at 0, and satisfiesyf(y)>0 andf(y)>0 fory 0, and . The results obtained include the average behavior of the integral of the coefficienta.     

Decreasing Regularly Varying Solutions of Sublinearly Perturbed Superlinear Thomas–Fermi Equation

We consider the perturbed Thomas–Fermi equation $$\begin{array}{ll} x^{\prime \prime}\, =\, p(t)|x|^{\gamma-1}x\, +\, q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (A) \end{array}$$ where δ and γ are positive constants with \({\delta < 1 < \gamma}\) and p(t) and q(t) are positive continuous functions on \({[a,\infty), a > 0}\) . Our aim here is to establish criteria for the existence of positive solutions of (A) decreasing to zero as \({t \to \infty}\) in the case where p(t) and q(t) are regularly varying functions (in the sense of Karamata). Generalization of the obtained results to equations of the form $$\begin{array}{ll} \left(r(t)x^{\prime}\right)^{\prime}\, =\, p(t)|x|^{\gamma-1}x \,+ \,q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (B) \end{array}$$ is also given.     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

Existence of positive solutions for a class of second-order two-point boundary value problem

By using different convex functionals to compute fixed point index, the existence of positive solutions for a class of second-order two-point boundary value problem

On the Fučik point spectrum for Schrödinger operators on {\mathbb{R}}^{N}

We investigate the Fučik point spectrum of the Schr?dinger operator when the potential V_λ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that S_λ has essential spectrum with finitely many eigenvalues below the infimum of . We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schr?dinger equations with jumping nonlinearity.      

Some inequalities for gradients of harmonic functions

For a function u(x, y) harmonic in the upper half-plane y>0 and represented by the Poisson integral of a function v(t) ∈ L ₂ (−∞,∞), we prove that the inequality is true. A similar inequality is obtained for a function harmonic in a disk. Odessa Marine University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1135–1136, August, 1997.     

Multivalued evolution equations with nonlocal initial conditions in Banach spaces

We prove the existence of integral solutions to the nonlocal Cauchy problem in a Banach space X, where is m-accretive and such that –A generates a compact semigroup, has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and . The case when A depends on time is also considered.      

Rate of Type II blowup for a semilinear heat equation

A solution u of a Cauchy problem for a semilinear heat equation