Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian

In this paper we consider the multiplicity of positive solutions for the one-dimensional p-Laplacian differential equation (?p^′(u^′))+q(t)f(t,u,u^′)=0, t∈(0,1), subject to some boundary conditions. By means of a fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to some multipoint boundary value problems.     

T- and Hayman T-points of meromorphic functions for small functions in the unit disk

In this article, we consider the singular points of meromorphic functions in the unit disk. We prove the second fundamental theorem for the Ahlfors-Shimizu's characteristic in the unit disk in terms of Nevanlinna theory in the angular domains, and obtain the existence of T-points and Hayman T-points dealing with small functions as target.     

On the tensor products of operators in lebesgue spaces

The following question concerning the computation of the norms of the tensor products of operators in the Lebesgue spaces is studied: Is it true that the norm of the tensor product A?B: L^p(μ?μ)→L^q(ν?ν) of operators A: L^p(μ)→L^q(ν) and B: L^p(μ)→L^q(ν) coincides with the product ‖A‖ ‖B‖ of their norms? An answer is positive if and only if 1≤p≤q≤+∞. Bibliography: 26 titles.     

An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

Existence of three positive pseudo-symmetric solutions for a one dimensional p-Laplacian

We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the three point boundary value problem, (g(u′))′+a(t)f(u)=0, u(0)=0, and u(ν)=u(1), where g(v)=|v|^p−2v, with p>1 and ν∈(0,1).     

Gradients of inner functions in strictly pseudoconvex domains

We investigate the unbalanced ordinary partition relations of the form λ → (λ, α)² for various values of the cardinal λ and the ordinal α. For example, we show that for every infinite cardinal κ, the existence of a κ+-Suslin tree implies κ⁺ ? (κ⁺, log _κ (κ⁺) + 2)². The consistency of the positive partition relation b → (b, α)2 for all α < ω₁ for the bounding number b is also established from large cardinals.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

On an Iteration Leading to a q-Analogue of the Digamma Function

We show that the q-Digamma function ψ _q for 0<q<1 appears in an iteration studied by Berg and Durán. This is connected with the determination of the probability measure ν _q on the unit interval with moments $1/\sum_{k=1}^{n+1} (1-q)/(1-q^{k})$ , which are q-analogues of the reciprocals of the harmonic numbers. The Mellin transform of the measure ν _q can be expressed in terms of the q-Digamma function. It is shown that ν _q has a continuous density on ]0,1], which is piecewise C ^∞ with kinks at the powers of q. Furthermore, (1?q)e ^?x ν _q (e ^?x ) is a standard p-function from the theory of regenerative phenomena.     

Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian

We consider the boundary value problems: (?p(x^′′(t)))+q(t)f(t,x(t),x(t−1),x^′(t))=0, ?p(s)=|s|^p−2s, p>1, t∈(0,1), subject to some boundary conditions. By using a generalization of the Leggett-Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems.     

Exact number of positive solutions for a class of semipositone problems

This paper is concerned with the boundary value problems y″+λ(y^p−y^q)=0 and y(−1)=y(1)=0, where p>q>−1 and λ>0 is a positive parameter. We discuss the existence of positive solutions and give a complete study.     

Full automorphism group of the generalized symplectic graph

Let Fq be a finite field of odd characteristic, m, ν the integers with 1≤m≤ν and Ka 2ν× 2ν nonsingular alternate matrix over Fq. In this paper, the generalized symplectic graph GSp2ν (q, m) relative to K over Fq is introduced. It is the graph with m-dimensional totally isotropic subspaces of the 2ν-dimensional symplectic space F(2ν)q as its vertices and two vertices P and Q are adjacent if and only if the rank of PKQT is 1 and the dimension of P ∩ Q is m-1. It is proved that the full automorphism group of the graph GSp2ν(q, m) is the projective semilinear symplectic group PΣp(2ν, q).     

Periodic solutions of non-autonomous second order systems with (q(t), p(t))-Laplacian

Using the least action principle in critical point theory we obtain some existence results of periodic solutions for (q(t), p(t))-Laplacian systems which generalize some existence results.     

Eigenfunction expansions for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schrödinger equation -f″ + q_κf = Ef on the positive half-axis with the potential q_κ(r) = (κ² - 1/4)r^-2. For each complex number ν, we construct a solution u_ν^κ(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the “singular” point κ = 0. For -1 < κ < 1 and real ν, the solutions u_ν^κ(E) determine a unitary eigenfunction expansion operator U_κ,ν: L₂(0,∞) → L₂(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -?_r² + q_κ(r) for the Hamiltonian is diagonalized by the operator U_κ,ν for some ν ∈ R. Using suitable singular Titchmarsh–Weyl m-functions, we explicitly find the measures V_κ,ν and prove their continuity in κ and ν.     

Some results concerning Lp(μ)-spaces

It is proved that L^p(μ) does not have an unconditional basis if the cardinality of L^p(μ) is sufficiently large and μ is a finite measure. It is also shown that L^p(μ) has a weaker kind of basis for arbitrary μ and 1 < p < ∞. A new truncation lemma concerning sequences in L^p equivalent to the usual l^p-basis is given. This lemma is used in solving the problem of when l^p(Γ) imbeds in L^r(μ) for uncountable sets Γ and finite measures μ. It may also be used to give a nonprobabilistic proof of the fact (due to Schwartz-Kwapien) that there exist non-q-absolutely summing operators from L^∞to L^qfor 2 < q < ∞. It is again used in proving that basic sequences in L^p equivalent to the usual l^p-basis admit subsequences with a complemented linear span. Other applications of the techniques introduced are also given.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

Fractional Covers for Convolution Products

A non-zero vector-valued sequence u ∈ ?^q(X′) is a cover for a subset M of ?_P(X) if, for some 0 < α ≤ 1, ∥u * h∥ _∞ ≥ α ∥u∥_q ∥h∥_p for all h ∈ M. Covers of ?¹ = ?¹(R) are important in worst case system identification in ?¹ and in the reconstruction of elements in a normed space from corrupted functional values. We investigate the existence of covers for certain naturally occurring subspaces of ?^p(X). We show that there exist finitely supported covers for some subspaces, and obtain lower bounds for their ’lengths’. We also obtain similar results for covers associated with convolution products for spaces of measurable vector-valued functions defined on the positive real axis.     

Global attractors for p-Laplacian equation

The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|^p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c₁q|u|−k?f(u)u?c₂q|u|+k and f^′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.     

Geometry of Cubic and Quartic Hypersurfaces over Finite Fields

Let YPⁿ be a cubic hypersurface defined over GF(q). Here, we study the Finite Field Nullstellensatz of order [q/3] for the set Y(q) of its GF(q)-points, the existence of linear subspaces of PG(n,q) contained in Y(q) and the possibility to join any two points of Y(q) by the union of two lines of PG(n,q) entirely contained in Y(q). We also study the existence of linear subspaces defined over GF(q) for the intersection of Y with s quadrics and for quartic hypersurfaces.     

The pointwise multipliers of Bloch type space β and Dirichlet type space Dq on the unit ball of C

In the paper, we will discuss the pointwise multipliers from Dirichlet type space D_p to Bloch type space β^q on the unit ball of Cⁿ. The multiplier spaces M(D_p,β^q) are fully characterized for all p, q.