We introduce a notion of disjoint envelope functions to study asymptotic structures of Banach spaces. The main result gives a new characterization of asymptotic-${\ell }_p$ spaces in terms of the ${\ell }_p$-behavior of “disjoint-permissible” vectors of constant coefficients. Applying this result to Tirilman spaces we obtain a negative solution to a conjecture of Casazza and Shura. Further investigation of the disjoint envelopes leads to a finite-representability result in the spirit of the Maurey-Pisier...

We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f(x,y)=px+(1-p)y-{\sum }_{s=m}^{\infty }s(x,y)$ uniformly in a neighborhood of the origin, where m > 1, ${}_s(x,y)={\sum }_{i=0}^s{a}_{i,s}{x}^{s-i}{y}^i$; (c) $ₘ(1,1)={\sum }_{i=0}^m{a}_{i,m}>0$. Let x₀,x₁ ∈ (0,α) and $x_{n+1}=f(xₙ,x_{n-1})$, n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $xₙ\sim {((2-p)/\left((m-1){\sum }_{i=0}^m{a}_{i,m}\right))}^{1/(m-1)}1/\sqrt[m-1]{n}$.

In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p-$Laplacian, where $1<p<2$. We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t\to \infty$.

A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${\left(x_j^i\right)}_{j=1}^{\infty }\subseteq X$, 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $li{m}_{n₁,₁}...li{m}_{nₘ,ₘ}\left|\right|{\sum }_{i=1}^m{x}_{n_i}^i\left|\right|\le Cli{m}_{n_{\sigma \left(1\right)}{,}_{\sigma \left(1\right)}}...li{m}_{n_{\sigma \left(m\right)}{,}_{\sigma \left(m\right)}}\left|\right|{\sum }_{i=1}^m{x}_{n_i}^i\left|\right|$. We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${\left(x_j^i\right)}_{j=1}^{\infty }$. Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then...

Our main purpose is to establish the existence of a positive solution of the system ⎧$-{∆}_{p\left(x\right)}u=F(x,u,v)$, x ∈ Ω, ⎨$-{∆}_{q\left(x\right)}v=H(x,u,v)$, x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded domain with C² boundary, $F(x,u,v)={\lambda }^{p\left(x\right)}[g\left(x\right)a\left(u\right)+f\left(v\right)]$, $H(x,u,v)={\lambda }^{q(x})[g\left(x\right)b\left(v\right)+h\left(u\right)]$, λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $-{∆}_{p\left(x\right)}u=-{div\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\overline{linf\left(\right[0,1\left]\right)}$ contains an order isomorphic copy of D(0,1), (2) $\overline{linf\left(Q\right)}$ contains an isomorphic copy of C([0,1]), (3) $\overline{linf\left(\right[0,1\left]\right)}/\overline{linf\left(Q\right)}$ contains an isomorphic copy of c₀(Γ) for some uncountable...

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f:{[0,1]}^m\to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f:{[0,1]}^m\to X$ with respect to any norming subset there exists a separately increasing function $g:{[0,1]}^m\to ℝ$ such that the sets of...

In this note we survey the partial results needed to show the following general theorem: $l_p\left(l_q\right):1\le p,q\le +\infty$ is a family of mutually non isomorphic Banach spaces. We also comment some related facts and open problems.

The results of the first part concern the existence of higher order ℓ₁ spreading models in asymptotic ℓ₁ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space T[(ₙ,θₙ)ₙ], ${\theta }_{n+m}\ge \theta ₙ\theta ₘ$ and $li{m}_n\theta {ₙ}^{1/n}=1$, admits an $\ell {₁}^{\omega }$ spreading model in every block subspace. We also prove that if X is a Banach space with a basis, with the property that there exists a sequence (θₙ)ₙ ⊂ (0,1) with $li{m}_n\theta {ₙ}^{1/n}=1$, such that, for every n ∈ ℕ, $\left|\right|{\sum }_{k=1}^m{x}_k\left|\right|\ge \theta ₙ{\sum }_{k=1}^m\left|\right|x_k\left|\right|$ for every ₙ-admissible block sequence ${\left(x_k\right)}_{k=1}^m$ of vectors in X, then there exists c...

A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_k}$ for which $su{p}_{f\in S_{X*}}limsu{p}_{k\to \infty }f\left(x_{n_k}\right)$ is attained at some f in the dual unit sphere $S_{X*}$. A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f\in S_{X*}$, there exists $g\in S_{X*}$ such that $limsu{p}_{n\to \infty }f\left(xₙ\right)<limin{f}_{n\to \infty }g\left(xₙ\right)$. Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega$ is a smooth bounded domain of ${ℝ}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_p)$, as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $\epsilon cₜ=k_c\Delta c-f\left(c\right)c+g(a,c)$, x ∈ Ω, t > 0, for $\Omega \subset {ℝ}^N$, completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^p$-estimates.

We study the leading order behaviour of positive solutions of the equation $-\Delta u+ϵu-{\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u=0,x\in {ℝ}^N$, where $N\ge 3$, $q>p>2$ and when $ϵ>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*}=\frac{2N}{N-2}$. For $p<2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>2^{*}$ the solution asymptotically...

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower-${\ell }_q$-tree estimate and let T be a bounded linear operator from X which satisfies an upper-${\ell }_p$-tree estimate. Then T factors through a subspace of ${\left(\sum Fₙ\right)}_{{\ell }_r}$, where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $({\ell }_p,{\ell }_q)$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic...

A new derivation of the classic asymptotic expansion of the $n$-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with ${li}^{-1}\left(n\right)$, after having retained the first $m$ terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n\&gt;s_3\left(n\right)$ where $s_3\left(n\right)$ is the sum of the first four terms of the asymptotic...

We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure $m_k\left(P\right)$ of a polynomial $P,$ where $m_k\left(P\right)$ is the integral of $lo{g}^k\left|P\right|$ over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding $m_k\left(P\right)$, in particular $|m_k\left(P\right)|/k!\to 1/\pi$ as k → ∞.

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

Let $c_q\left(n\right)$ be the Ramanujan sum, i.e. $c_q\left(n\right)={\sum }_{d\left|\right(q,n)}d\mu (q/d)$, where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for ${\sum }_{n\le y}{\left({\sum }_{q\le x}{c}_q\left(n\right)\right)}^k$ (k = 1,2) are obtained. As an analogous problem, we evaluate ${\sum }_{n\le y}{\left({\sum }_{n\le x}c{̂}_q\left(n\right)\right)}^k$ (k = 1,2), where $c{̂}_q\left(n\right):={\sum }_{d\left|\right(q,n)}d\left|\mu (q/d)\right|$.

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

We generalize some results concerning the classical notion of a spreading model to spreading models of order ξ. Among other results, we prove that the set $S{M}_{\xi }^w\left(X\right)$ of ξ-order spreading models of a Banach space X generated by subordinated weakly null ℱ-sequences endowed with the pre-partial order of domination is a semilattice. Moreover, if $S{M}_{\xi }^w\left(X\right)$ contains an increasing sequence of length ω then it contains an increasing sequence of length ω₁. Finally, if $S{M}_{\xi }^w\left(X\right)$ is uncountable, then it contains an antichain...

Geometric structure of Cesàro function spaces $Ce{s}_p\left(I\right)$, where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $Ce{s}_p[0,1]$ contains isomorphic and complemented copies of $l_q$-spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $Ce{s}_p[0,1]$.

We give a full solution of the following problems concerning the spaces $M_{\phi }\left(X⃗\right)$: (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{\phi }\left(X⃗\right)\ne M_{\psi }\left(X⃗\right)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{\phi }\left(X⃗\right)⊊M_{\psi }\left(X⃗\right)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{\phi }\left(X⃗\right)$-space for some (unknown beforehand) Banach couple X⃗.

Let ${\beta \left(n\right)}_{n=0}^{\infty }$ be a sequence of positive numbers and 1 ≤ p < ∞. We consider the space ${\ell }^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }f̂\left(n\right)zⁿ$ such that ${\sum }_{n=0}^{\infty }{\left|f̂\left(n\right)\right|}^p{\left|\beta \left(n\right)\right|}^p<\infty$. We give some sufficient conditions for the multiplication operator, $M_z$, to be unicellular on the Banach space ${\ell }^p\left(\beta \right)$. This generalizes the main results obtained by Lu Fang [1].

Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and ${\ell }^{\infty }$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then ${\ell }^{\infty }$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X{\otimes }_{\gamma }Y*$. Applications to embeddings of c₀ in various spaces of operators are given.

We introduce higher order spreading models associated to a Banach space X. Their definition is based on ℱ-sequences ${\left(x_s\right)}_{s\in ℱ}$ with ℱ a regular thin family and on plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy ${\left({ℳ}_{\xi }\left(X\right)\right)}_{\xi <\omega ₁}$. Each ${ℳ}_{\xi }\left(X\right)$ contains all spreading models generated by ℱ-sequences ${\left(x_s\right)}_{s\in ℱ}$ with order of ℱ equal to ξ. We also study the fundamental properties of this hierarchy.

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an $L_p$-space, then it is either an $L_p$-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non-commutative $L_p$-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which...

About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let $p$ be a prime, and let $N(k;p)$ denote the number of all $1\le a_i\le p-1$ such that $a_1{a}_2\cdots a_k\equiv 1modp$ and $2\mid a_i+{\overline{a}}_i+1,$ $i=1,2,\cdots ,k$. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function $N(k;p),$ and give an interesting asymptotic formula...