Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup _i→∞{sup _y∈K ‖t ⁱx−Tⁱy∼−‖x−y‖}≦0. IfT ^N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tⁱx−Tⁱy‖≦k _i‖x−y∼,x,y∈K, wherek _i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ₀ (X), be less than one. Research supported by National Science Foundation Grant GP 18045.     

Optimal properties of contraction semigroups in banach spaces

Suppose that(T _t )_t>0 is aC ₀ semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ ⁻¹(Jx)={x}, whereJ is the duality mapping fromX toP(X ^*). If |<T _t x,f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible. If the hypothesisJ ⁻¹(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ ⁻¹(Jx) is weakly compact, then if |<T _t x, f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1, there existsy∈J ⁻¹(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ₁ orL ₁.     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

On the ratio of the expected maximum of a martingale and theL p-norm of its last term

For eachp>1, the supremum,S, of the absolute value of a martingale terminating at a random variableX inL _p, satisfiesES≦(Γ(q))^1/q‖X‖_p (q=p(p-1)^-1).The maximum,M, of a mean-zero martingale which starts at zero and terminates atX, satisfiesES≦(Γ(q))^1/q‖X‖_p (q=p(p-1)^-1), whereσ _q is the unique solution of the equationt = ‖Z −t ‖ _q for an exponentially distributed random variableZ with mean 1.σ _p has other characterizations and satisfies lim _p‖ q ^{− 1} σ _q =c withc determined byce ^c+1 = 1. Equalities in (1) and (2) are attainable by appropriate martingales which can be realized as stopped segments of Brownian motion. A presumably new property of the exponential distribution is obtained en route to inequality (2).     

The bicompletion of an asymmetric normed linear space

A biBanach space is an asymmetric normed linear space (X,‖·‖) such that the normed linear space (X,‖·‖^s) is a Banach space, where ‖x‖^s= max {‖x‖,‖-x‖} for all x∈X. We prove that each asymmetric normed linear space (X,‖·‖) is isometrically isomorphic to a dense subspace of a biBanach space (Y,‖·‖_Y). Furthermore the space (Y,‖·‖_Y) is unique (up to isometric isomorphism). This revised version was published online in August 2006 with corrections to the Cover Date.     

Extremal properties of contraction semigroups onc 0

For any complex Banach spaceX, letJ denote the duality mapping ofX. For any unit vectorx inX and any (C ₀) contraction semigroup (T _t ) _t>0 onX, Baillon and Guerre-Delabriere proved that ifX is a smooth reflexive Banach space and if there isx ^*∈J(x) such that ÷〈(T(t)x, J(x)〈÷→1 ast→∞, then there is a unit vectory∈X which is an eigenvector of the generatorA of (T _t ) _t>0 associated with a purely imaginary eigenvalue. They asked whether this result is still true ifX is replaced byc ₀. In this article, we show the answer is negative Partial results of this paper were obtained when the author attended the International Conference of Convexity at the University of Marne-La-Vallée. He would like to express his gratitude for the kind hospitality offered to him. He would also like to thank Profs. Goldstein and Jamison for their valuable suggestions.     

Large deviations of theL p-norm of a wiener process with drift

In this paper we calculate the exact asymptotics of the probability P{‖w(t)+uct‖ _p >u},u→∞, wherew(t) is the standard Wiener process and ‖x‖ _p is the ordinary norm in the spaceL ^p[0,1],p≥2. The result is obtained on the basis of a general theorem due to the author on the asymptotics of the Gaussian measureP(uD),u→∞, for a Borel setD belonging to a separable Banach space. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 429–436, March, 1999.     

The classification of injective Banach lattices

LetE be a 1-injective Banach lattice,X any Banach space andT: E ← X a norm bounded linear operator. Then eitherT is an isomorphism on some copy ofl _∞ inE or for all σ > 0 there is φ ∈E ₊ ^′ such that ‖Tu‖≦φ (|u|)+σ ‖u‖ for allu ∈E. We deduce the theorem that: A norm order continuous injective Banach lattice is order isomorphic to an (AL)-space.     

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

On the chromatic number of random geometric graphs

Given independent random points X ₁,...,X _n ∈ℝ ^d with common probability distribution ν, and a positive distance r=r(n)>0, we construct a random geometric graph G _n with vertex set {1,..., n} where distinct i and j are adjacent when ‖X _i −X _j ‖≤r. Here ‖·‖ may be any norm on ℝ ^d , and ν may be any probability distribution on ℝ ^d with a bounded density function. We consider the chromatic number χ(G _n ) of G _n and its relation to the clique number ω(G _n ) as n→∞. Both McDiarmid [11] and Penrose [15] considered the range of r when $r \ll \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}$r \ll \left( {\tfrac{{\ln n}} {n}} \right)^{1/d} and the range when $r \gg \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}$r \gg \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}, and their results showed a dramatic difference between these two cases. Here we sharpen and extend the earlier results, and in particular we consider the ‘phase change’ range when $r \sim \left( {\tfrac{{t\ln n}} {n}} \right)^{1/d}$r \sim \left( {\tfrac{{t\ln n}} {n}} \right)^{1/d} with t>0 a fixed constant. Both [11] and [15] asked for the behaviour of the chromatic number in this range. We determine constants c(t) such that $\tfrac{{\chi (G_n )}} {{nr^d }} \to c(t)$\tfrac{{\chi (G_n )}} {{nr^d }} \to c(t) almost surely. Further, we find a “sharp threshold” (except for less interesting choices of the norm when the unit ball tiles d-space): there is a constant t ₀>0 such that if t≤t ₀ then $\tfrac{{\chi (G_n )}} {{\omega (G_n )}}$\tfrac{{\chi (G_n )}} {{\omega (G_n )}} tends to 1 almost surely, but if t>t ₀ then $\tfrac{{\chi (G_n )}} {{\omega (G_n )}}$\tfrac{{\chi (G_n )}} {{\omega (G_n )}} tends to a limit >1 almost surely.     

Eigenvalues and power growth

LetX be a complex Banach space and letT be a bounded linear operator onX. Denote by σ _p (T) the point spectrum ofT and by the unit circle. We investigate how the growth of the sequence ‖T ⁿ ‖ is influenced by the size of the set (T) and by the geometry of the spaceX. We also prove analogous results forC ₀-semigroups(T _t )_t≥0. Research partially supported by grants from NSERC, FQRNT and the Canada research chairs program.     

A Functional LIL for <Emphasis Type="Italic">d</Emphasis>-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes

We establish a functional LIL for the maximal process M(t) :=sup _0≤s≤t ‖X(s)‖ of an ℝ ^d -valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X. Supported in part by NSF Grant DMS 02-05034.     

On a criterion for the uniform boundedness of a <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup of operators in a Hilbert space

Let T(t), t ≥ 0, be a C ₀-semigroup of linear operators acting in a Hilbert space H with norm ‖·‖. We prove that T(t) is uniformly bounded, i.e., ‖T(t)‖ ≤ M, t ≥ 0, if and only if the following condition is satisfied:

The uniform approximation property in Orlicz spaces

It is proved that for every reflexive Orlicz spaceX there is a functionn(k,ε) so that wheneverE is ak-dimensional subspace ofX there exists an operatorT: X→X such thatT _1E=identity, ‖T‖≦1+ε and dimTX≦n(k,ε). Some general facts concerning the uniform approximation property are also presented. Research of the first named author was partially supported by NSF Grant MPS 74-07509-A01.     

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.     

Any Banach space has an equivalent norm with trivial isometries

For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖ _G onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖ _G ) is isomorphic toG × {−1, + 1}.     

Closed sectorial forms and one-parameter contraction semigroups

Suppose thats[u, v] is a closed sesquilinear sectorial form with vertex at zero, half-angle α ∈ [0, π/2), and dense domainD(s) in a Hilbert spaceH, S is them-sectorial operator associated withs, S _R is the real part ofS, andT(t)=exp(−tS) is the contraction semigroup with generator −S, holomorphic in the sector |argt|<π/2−α. We characterizes in terms ofT(t). In particular, we prove that the following conditions a`2) the function ‖T(t)u‖ is differentiable at zero; 3) the function (T(t)u, u) is differentiable at zero. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 643–654, May, 1997. Translated by V. E. Nazaikinskii     

On multivariate Gaussian tails

Let {X _{n, n} ≥1} be a sequence of standard Gaussian random vectors in ℝ _d ,d ≥ 2. In this paper we derive lower and upper bounds for the tail probabilityP{X _n >t _n } witht _n ∈ ℝ _d some threshold. We improve for instance bounds on Mills ratio obtained by Savage (1962,J. Res. Nat. Bur. Standards Sect. B,66, 93–96). Furthermore, we prove exact asymptotics under fairly general conditions on bothX _n andt _n , as ‖t _n ‖→∞ where the correlation matrix Σ _n ofX _n may also depend onn.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.