Über Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1)u′(t)=F(t,u) for )≤t≤∞ together with a generalized initial condition (2)u(t)=?(t) forr≤t≤0 or a generalized Nicoletti condition (3)N u=η. Here,N is a linear operator; in the case of a system ofn equations the classical Nicoletti operator is given byN u=(u ₁(t ₁),...,u _n(t _n)), with givent _i. The functionsu, F ? are Banach space valued, the functionF(t, z) is defined fort≥0 andz∈C ⁰[r,∞). The main point is that the value ofF(t, z) may depend on the values ofz(s) forr≤s≤t+σ(t), where σ(t)>0. Simple examples show that without a restriction on the magnitude of the advancement σ(t) there is neither existence nor uniqueness. Our results show that when σ(t) is properly bounded and when the solution is to satisfy a certain growth condition which depends on σ(t), then there exists exactly one solution, and it depends continuously on the given data. In the case of the Nicoletti problem (1), (3) there is convergence to the solution satisfyingu(0)=η if 0≤t _i≤T andT→0 (this holds in infinite-dimensional spaces, too). These results are true ifF satisfies a Lipschitz condition of the form $$\left| {F(t,z) - F(t,y)} \right| \leqslant h(t)\max \left\{ {\left| {z(s) - y(s)} \right|:r \leqslant s \leqslant t + \delta (t)} \right\}.$$ . In the case where (1) is a finite system andF is only continuous, an existence theory is developped based onSchauder's fixed point theorem. Again, growth conditions play an essential role here.     

Lower bounds for polynomials of many variables

A polynomial P(ξ) = P(ξ₁,..., ξ _n ) is said to be almost hypoelliptic if all its derivatives D ^ν P(ξ) can be estimated from above by P(ξ) (see [16]). By a theorem of Seidenberg-Tarski it follows that for each polynomial P(ξ) satisfying the condition P(ξ) > 0 for all ξ ∈ R ⁿ , there exist numbers σ > 0 and T ∈ R¹ such that P(ξ) ≥ σ(1 + |ξ|) ^T for all ξ ∈ R ⁿ . The greatest of numbers T satisfying this condition, denoted by ST(P), is called Seidenberg-Tarski number of polynomial P. It is known that if, in addition, P ∈ I _n , that is, |P(ξ)| → ∞ as |ξ| → ∞, then T = T(P) > 0. In this paper, for a class of almost hypoelliptic polynomials of n (≥ 2) variables we find a sufficient condition for ST(P) ≥ 1. Moreover, in the case n = 2, we prove that ST(P) ≥ 1 for any almost hypoelliptic polynomial P ∈ I₂.     

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(? ⁿ ) is a Banach space as well as an algebra, while B(? ⁿ , ? ^m ) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σ _r be the set of all operators of finite rank r in B(E, F) (or B(? ⁿ , ? ^m )). In fact, we have that 1) suppose Σ _r ∈ B(? ⁿ , ? ^m ), and then Σ _r is a smooth and path-connected submanifold of B(? ⁿ , ? ^m ) and dimΣ _r = (n + m)r ? r ², for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σ _r and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σ _r ∈ B(E, F), and dimF = ∞, and then Σ _r is a smooth and path-connected submanifold of B(E, F) with the tangent space T _A Σ _r = {B ∈ B(E, F): BN(A) ? R(A)} at each A ∈ Σ _r for 0 ? r ? ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(? ⁿ ), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.     

Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

Stationary distributions of the simplest dynamical systems in Rn perturbed by generalized Poisson process

Let z(t) ∈ Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) ∈ Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

Protecting convex sets

A point-setS is protecting a collection F =T ₁,T ₂,..., _n ofn mutually disjoint compact sets if each one of the setsT _i is visible from at least one point inS; thus, for every setT _i ∈F there are points xS andy T _i such that the line segment joining x to y does not intersect any element inF other thanT _i . In this paper we prove that [2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F ofn mutually disjoint compact convex sets. For an isothetic family F, consisting ofn mutually disjoint rectangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary to protect it. IfF is a family of triangles, [4n/7] points are always sufficient. To protect families ofn homothetic triangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.     

A limit theorem for order preserving nonexpansive operators inL 1

We prove the following theorem:Let T be an order preserving nonexpansive operator on L ₁ (μ) (or L ₁ ⁺ ) of a σ-finite measure, which also decreases theL _∞-norm, and let S=tI+(1?t)T for 0<t<1. Then for everyf ∈ L_p (1<p<∞),the sequence S ⁿf converges weakly in L_p. (The assumptions do not imply thatT is nonexpansive inL _p for anyp>1, even ifμ is finite.) For the proof we show that ∥S ⁿ⁺¹ f?S ⁿf∥ _p → 0 for everyf ∈L _p, 1<p<∞, and apply toS the following theorem:Let T be order preserving and nonexpansive in L ₁ ⁺ , and assume that T decreases theL _∞-norm. Then forg ∈L _p (1<p<∞) Tⁿg is weakly almost convergent. If forf ∈ L_p we have T ⁿ⁺¹ f?T ⁿ f → 0weakly, then T ⁿf converges weakly in L_p (1<p<∞).     

Pointwise convergence of the iterates of a harris-recurrent Markov operator

LetP be a Markov operator recurrent in the sense of Harris, withσ-finite invariant measureμ. (1) Ifμ is finite andP aperiodic, then forf ∈L ₁(μ),P ⁿf →f fdμ a.e. (2) Ifμ is infinite,P ⁿf → 0 a.e. for everyf ∈L _p (μ), 1≦p <∞.     

Copositive pointwise approximation

We prove that if a functionf ∈C ⁽¹⁾ (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP _n =P _n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω _k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP _n =P _n (x) of degree ≤n such thatP _n (x) ≥ 0,x ∈I, and |f(x) ?P _n (x)| ≤c(k)ω _k (f;n ^?2 +n ^?1 √1 ?x ²),x ∈I.     

F-limit points in dynamical systems defined on the interval

Given a free ultrafilter p on ? we say that x ∈ [0, 1] is the p-limit point of a sequence (x _n ) _n∈? ? [0, 1] (in symbols, x = p -lim _n∈? x _n ) if for every neighbourhood V of x, {n ∈ ?: x _n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f ^p : [0, 1] → [0, 1] is defined by f ^p (x) = p -lim _n∈? f ⁿ (x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f ^p . For a filter F we also define the ω ^F -limit set of f at x. We consider a question about continuity of the multivalued map x → ω _f ^F (x). We point out some connections between the Baire class of f ^p and tame dynamical systems, and give some open problems.     

A weak-type inequality for uniformly bounded trigonometric polynomials

This paper is devoted to refining the Bernstein inequality. Let D be the differentiation operator. The action of the operator Λ = D/n on the set of trigonometric polynomials T _n is studied: the best constant is sought in the inequality between the measures of the sets {x ∈ T: |Λt(x)| > 1} and {x ∈ T: |t(x)| > 1}. We obtain an upper estimate that is order sharp on the set of uniformly bounded trigonometric polynomials T _n ^C = {t ∈ T _n : ‖t‖ ≤ C}.     

On oscillation of solutions of a nonautonomous quasilinear second-order equation

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for t=ω) solutions of the nonautonomous quasilinear equation $$y'' \pm \lambda (t)y = F(t,y,y'),$$ wheret ∈ Δ=[a, ω[,-∞ <a < ω ≤+ ∞, λ(t) > 0, λ(t) ∈ C _Δ ⁽¹⁾ , |F((t,x,y))|≤L(t)(|x|+|y|)^1+α, L(t) ≥-0, α ∈ [0,+∞[, F: Δ × R² →R,F ∈C _Δ×R ²,R is the set of real numbers, and R² is the two-dimensional real Euclidean space.     

Iterative Methods for Fixed Points of Asymptotically Weakly Contractive Maps

Suppose K is a closed convex nonexpansive retract of a real uniformly smooth Banach space E with P as the nonexpansive retraction. Suppose T : K → E is an asymptotically d-weakly contractive map with sequence {k_n }, k_n ≥ 1, lim k_n = 1 and with F(T) _n int (K) ≠ ø F(T):= {x ∈ K: Tx = x}. Suppose {x _n } is iteratively defined by x _n+1 = P((l ? k_nα_n )x _n +k _n α _n T(PT) ^n?l x_n ), n = 1,2,...,x ₁ ∈ K, where α_n∈ (0,l) satisfies lim α_n = 0 and Σα_n = ∞. It is proved that {x _n } converges strongly to some x ^* ∈ F(T)∩ int K. Furthermore, if K is a closed convex subset of an arbitrary real Banach space and T is, in addition uniformly continuous, with F(T) ≠ ø, it is proved that {x_n } converges strongly to some x ^* ∈ F(T).     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

On two-sided and asymptotic estimates for the norms of embedding operators of into L q (dμ)

Explicit upper and lower estimates are given for the norms of the operators of embedding of , n ∈ ?, in L _q (dµ), 0 < q < ∞. Conditions on the measure µ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ_{1, n, β} , β > 0, of the boundary value problems (?d ²/dx ²) ⁿ u(x) = λ|x| ^β?1, x ∈ (?1, 1), u ^(k)(±1) = 0, k ∈ {0, 1, ..., n ? 1}.     

Balanced extensions of graphs and hypergraphs

For a hypergraphG withv vertices ande _i edges of sizei, the average vertex degree isd(G)= ∑ie ₁/v. Callbalanced ifd(H)≦d(G) for all subhypergraphsH ofG. Let $$m(G) = \mathop {\max }\limits_{H \subseteqq G} d(H).$$ A hypergraphF is said to be abalanced extension ofG ifG?F, F is balanced andd(F)=m(G), i.e.F is balanced and does not increase the maximum average degree. It is shown that for every hypergraphG there exists a balanced extensionF ofG. Moreover everyr-uniform hypergraph has anr-uniform balanced extension. For a graphG let ext (G) denote the minimum number of vertices in any graph that is a balanced extension ofG. IfG hasn vertices, then an upper bound of the form ext(G) ₁ n ² is proved. This is best possible in the sense that ext(G)>c ₂ n ² for an infinite family of graphs. However for sufficiently dense graphs an improved upper bound ext(G) ₃ n can be obtained, confirming a conjecture of P. Erdõs.     

Minimal polynomials for compact sets of the complex plane

LetW _N(z)=a_Nz^N+... be a complex polynomial and letT _n be the classical Chebyshev polynomial. In this article it is shown that the polynomials (2a_N)^?n+1T_n(W_N), n ∈N, are minimal polynomials on all equipotential lines for {z∈C:|W _N(z)|≤1 Λ ImW _N(z)=0}