Volume 13, Issue 1

In this paper, we prove a common fixed point theorem for two pairs of weakly compatible self-mappings of a complete metric space without requiring continuity. Our result generalizes a theorem obtained by V. Popa and H. K. Pathak in 1998 and a result obtained by B. Fisher and S. Sessa in 1986.

Let 𝑋 be a holomorphically convex complex manifold and Exc(𝑋) ⊆ 𝑋 the union of all positive dimensional compact analytic subsets of 𝑋. We assume that Exc(𝑋) ≠ 𝑋 and 𝑋 is not a Stein manifold. Here we prove the existence of a holomorphic vector bundle 𝐸 on 𝑋 such that is not holomorphically trivial for every open neighborhood 𝑈 of Exc(𝑋) and every integer 𝑚 ≥ 0. Furthermore, we study the existence of holomorphic vector bundles on such a neighborhood 𝑈, which are not extendable across a 2-concave point of ∂ (𝑈).

In this paper, sufficient conditions are established for the controllability of Sobolev type semilinear functional differential and integrodifferential inclusions with an unbounded delay in Banach spaces. The main results are obtained by using the fixed point theorem for condensing maps due to Martelli.

We use the contraction mapping theorem to obtain stability results of the scalar nonlinear neutral differential equation with functional delay 𝑥′(𝑡) = –𝑎𝑥(𝑡) + 𝑏(𝑡)𝑥 2 (𝑡 – 𝑟(𝑡)) + 𝑐(𝑡)𝑥(𝑡 – 𝑟(𝑡))𝑥′(𝑡 – 𝑟(𝑡)).

A general representation of solutions by six metaharmonic functions is obtained for a system of homogeneous equations of oscillation of two-component mixtures. The boundary value problem of oscillation of two-component mixtures is investigated when the normal components of partial displacement vectors and the tangent components of partial rotation vectors are given on the boundary. Uniqueness theorems of the considered problem are proved. Solutions are obtained in terms of absolutely and uniformly convergent series.

Bases of the spaces of Eisenstein series 𝐸 𝑘 (Γ 0 (4𝑁), χ) (𝑘 ∈ ℕ, 𝑘 ≥ 3, 𝑁 is an odd natural and square-free) and , ℕ is an odd natural and square-free) are constructed for any Dirichlet character mod 4𝑁 and Fourier expansions of these series are obtained.

A cotorsion hull of the separable 𝑝-group 𝑇 is considered when 𝑇 is a direct sum of torsion-complete groups. It is proved that in the considered case its cotorsion hull is fully transitive if and only if 𝑇 is a direct sum of cyclic groups or is a torsion-complete group.

In this paper, generating relations involving generalized Gegenbauer polynomials are obtained by constructing a three-dimensional Lie algebra isomorphic to special linear algebra sl(2). Further, a number of new interesting relations involving various generalized polynomials are obtained as applications of these generating relations.

Several combinatorial questions and facts connected with certain types of mutual positions of finitely many hyperplanes in a finite-dimensional affine space are considered. An application of one of such facts to a multi-dimensional version of the well-known Sylvester theorem is presented.

We study the boundedness of the maximal operator in the spaces 𝐿 𝑝(·) (Ω, ρ ) over a bounded open set Ω in 𝑅 𝑛 with the weight , where 𝑤 𝑘 has the property that belongs to a certain Zygmund-type class. Weight functions 𝑤 𝑘 may oscillate between two power functions with different exponents. It is assumed that the exponent 𝑝(𝑥) satisfies the Dini–Lipschitz condition. The final statement on the boundedness is given in terms of index numbers of functions 𝑤 𝑘 (similar in a certain sense to the Boyd indices for the Young functions defining Orlicz spaces).

Let 𝑋 be an 𝑙-connected space and 𝑈 a connected CW complex with dim 𝑈 ≤ 𝑙. Let 𝐹(𝑈,𝑋) be the space of continuous maps from 𝑈 to 𝑋. In this paper, an algebraic model for the evaluation map 𝐹(𝑈,𝑋) × 𝑈 → 𝑋 is considered in terms of the model for the function space due to Brown and Szczarba [Trans. Amer. Math. Soc. 349: 4931-4951, 1997]. It turns out that the Brown and Szczarba model for the function space coincides with Haefliger's model.

A pseudo-spectral collocation method for a class of equations describing resonant long-short wave interaction is studied. Semi-discrete and fully discrete Fourier pseudo-spectral collocation schemes are given. In fully discrete case we establish a three-level explicit scheme which is convenient and saves time in real computation. We use energy estimation methods to obtain error estimates for the approximate solutions.

We consider the vector-valued extension of the Fefferman–Stein–Strömberg sharp maximal inequality under growth condition. As an application we obtain a vector-valued extension of the boundedness of the commutator. Furthermore, we prove the boundedness of the commutator.

We obtain a version of Noether's invariance theorem for optimal control problems with a finite number of cost functionals. The result is obtained by formulating E. Noether's result for optimal control problems subject to isoperimetric constraints, and then using the unimprovable (Pareto) notion of optimality. It was A. Gugushvili who drew the author's attention to a result of this kind that was posed as an open mathematical question of a great interest in applications of control engineering.

Let 𝑚 ∈ 𝑛 and 𝑎 1 , . . . , 𝑎 𝑚 be real numbers such that for each 𝑖, 𝑎 𝑖 ≠ 0 and 𝑎 𝑖 ≠ 𝑎 𝑗 if 𝑖 ≠ 𝑗. In this paper we study integral operators of the form 𝑇𝑓 (𝑥) = ∫ 𝑘 1 (𝑥 – 𝑎 1 𝑦) . . . 𝑘 𝑚 (𝑥 – 𝑎 𝑚 𝑦) 𝑓 (𝑦) 𝑑𝑦, with . If φ 𝑖,𝑗 satisfy certain uniform regularity conditions out of the origin, we obtain the boundedness of 𝑇 : 𝐿 𝑝 (𝑤) → 𝐿 𝑝 (𝑤) for all power weights 𝑤 in adequate Muckenhoupt classes.

Let 𝑓 and 𝑔 be 2 π periodic functions. If 𝑓 ∈ 𝐿 1 [0, 2 π ] and 𝑔 is from ∧ 𝐵𝑉 ( 𝑝 ) [0, 2 π ] or, Lip( α , 𝑝)[0, 2 π ] or, 𝑟 – 𝐵𝑉 [0, 2 π ], then 𝑓 convolution 𝑔 inherit the same property.