Volume 15, Issue 1

We will study the properties of solutions 𝑓, {𝑔 𝑖 }, {ℎ 𝑖 } ∈ 𝐶 𝑏 (𝐺) of the functional equation where 𝐺 is a Hausdorff locally compact topological group, 𝐾 a compact subgroup of morphisms of 𝐺, χ a character on 𝐾, and μ a 𝐾-invariant measure on 𝐺. This equation provides a common generalization of many functional equations (D'Alembert's, Badora's, Cauchy's, Gajda's, Stetkaer's, Wilson's equations) on groups. First we obtain the solutions of Badora's equation [Aequationes Math. 43: 72–89, 1992] under the condition that (𝐺,𝐾) is a Gelfand pair. This result completes the one obtained in [Badora, Aequationes Math. 43: 72–89, 1992] and [Elqorachi, Akkouchi, Bakali and Bouikhalene, Georgian Math. J. 11: 449–466, 2004]. Then we point out some of the relations of the general equation to the matrix Badora functional equation and obtain explicit solution formulas of the equation in question for some particular cases. The results presented in this paper may be viewed as a continuation and a generalization of Stetkær's, Badora's, and the authors' works.

Using the notion of weighted and truncated sharings of sets we deal with the problem of uniqueness of meromorphic functions sharing two sets and obtain five theorems which not only improve a recent result of Yi and Lü [Acta Math. Sci. Ser. B Engl. Ed. 24: 83–90, 2004], but also improve and supplement the results of Lin and Yi [Kyungpook Math. J. 43: 73–85, 2003], Yi [Acta Math. Sinica (Chin. Ser.) 45: 75–82, 2002].

Let (𝑋, 𝑑) be a convex metric space, 𝐶 be a closed and convex subset of 𝑋 and let 𝐵(𝐶) be the family of all nonempty bounded subsets of 𝐶. In this paper some results are obtained on the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings 𝑆,𝑇 : 𝐶 → 𝐵(𝐶) which satisfy condition (2.1) below.

We provide the existence results for a nonlinear operator equation Λ*Φ′ (Λ𝑥) = 𝐹′(𝑥), in case 𝐹 – Φ is not necessarily convex. We introduce the dual variational method which is based on finding global minima of primal and dual action functionals on certain nonlinear subsets of their domains and on investigating relations between the minima obtained. The solution is a limit of a minimizng sequence whose existence and convergence are proved. The application for the non-convex Dirichlet problem with P.D.E. is given.

In this paper a theorem analogous to the Aleksandrov theorem is presented in terms of measure theory. Furthermore, we introduce the condensation rank of Hausdorff spaces and prove that any ordinal number is associated with the condensation rank of an appropriate locally compact totally imperfect space. This space is equipped with a probability Borel measure which is outer regular, vanishes at singletons, and is also inner regular in the sense of closed sets.

Some fixed and coincidence point theorems are obtained in the setting of single and multivalued mappings, whose images are not necessarily bounded. Our results generalize some recent results by Daffer and Kaneko [Sci. Math. 1: 125–131, 1998] and others.

Concerning the 𝐿 1 -convergence of modified cosine sums introduced by Kumari and Ram [Indian J. Pure Appl. Math. 19: 1101–1104, 1988], we generalise the results of Kumari, Ram [Indian J. Pure Appl. Math. 19: 1101–1104, 1988] and Bor [Proc. Indian Acad. Sci. Math. Sci. 102: 235–238, 1992]. Also a necessary and sufficient condition for the 𝐿 1 -convergence of cosine series has been deduced as a corollary for the said class.

In the present paper we obtain both the pointwise convergence and the rate of pointwise convergence theorems of a class of operators defined by as (𝑥, λ ) → (𝑥 0 , λ 0 ) in 𝐿 1 〈𝑎,𝑏〉, where 〈𝑎,𝑏〉 is an arbitrary interval in 𝑅. Here λ ∈ Λ and Λ is a nonempty set of indices.

We conjecture that the Gauss–Bonnet formula gives a necessary and sufficient condition for the existence of a metric deformation to obtain a positive/negative Gaussian curvature on a disk. The conjecture is reduced to a problem of a partial differential inequality with Dirichlet–Neumann boundary condition. We give a partial answer and an example.

We give a characterization of all cardinal numbers representable as the cardinality of the group of all automorphisms of an ω -tree. As a consequence, we obtain some statements., in terms of ω -trees and their automorphisms, which are related to the continuum hypothesis.

We study the growth of functions which are harmonic in any number of variables. The results are expressed in terms of spherical harmonic coefficients as well as by the approximation error of the harmonic function with (𝑝, 𝑞)-growth.

The boundary value problem at resonance 𝑥″ + 𝑥 = 𝑞 sin 𝑡 + 𝑓(𝑡,𝑥,𝑥′), 𝑥(0) = 0, 𝑥(π) = 0, is considered, where 𝑓 : [0,π] × 𝑹 2 → 𝑹 is a bounded Carathéodory function, 𝑞 is a parameter. We state the multiplicity results without assuming that 𝑓 has limits.

We present the theory of Gröbner bases for the submodules of the free module 𝐴 𝑚 , 𝑚 ≥ 1, where 𝐴 = 𝑅[𝑥 1 ,…,𝑥 𝑛 ] and 𝑅 is a Noehterian commutative ring. This generalizes the theory of Gröbner bases for the ideals of 𝐴 and the submodules of (𝐾[𝑥 1 ,…,𝑥 𝑛 ]) 𝑚 , where 𝐾 is a field, see [Möller, J. Symbolic Comput. 6: 345–359, 1988], [Möller, J. Algebra 100: 138–178, 1986] and [Zacharias, Generalized Gröbner bases in commutative polynomial rings, MIT, 1978]. This generalization to the submodules of 𝐴 𝑚 was also partially considered in [Rutman, J. Symbolic Comput. 14: 483–503, 1992], however our presentation is more complete and detailed because we have included all algorithms, proofs and examples omitted in [Rutman, J. Symbolic Comput. 14: 483–503, 1992].

We present a new definition of a discrete 2D behavior, which generalizes the standard notion by including trajectories from all quarter- and half-planes in ℤ 2 . Behaviors then admit certain natural homology spaces, which can be used to define controllability and autonomy, and describe finite trajectories. We discuss the regularity, controllability and autonomy of these models, and demonstrate the equivalence of suitable generalizations of controllability in the senses of Willems and of Rosenbrock, and similarly for autonomy.

Let 𝔻 be the open unit disk in the complex plane. For ε > 0 we consider the sector ∑ ε = {𝑧 : |arg 𝑧| < ε }. We prove that for certain classes of functions 𝑓 in the Besov space (𝔻) such that 𝑓(0) = 0, the norm is obtained by integration over 𝑓 –1 (∑ ε ).

Using the rational splitting of a cosine operator-function, the fourth order accuracy decomposition scheme is constructed for hyperbolic equation when the principal operator is self-adjoint positively defined and is represented as a sum of two summands. Stability of the constructed scheme is shown and the error of an approximate solution is estimated.

This paper is concerned with the definition and study of semipartial isometries on Banach spaces. This class of operators, which is a natural generalization of partial isometries from Hilbert to general Banach spaces, contains in particular the class of partial isometries recently introduced by M. Mbekhta [Acta Sci. Math. (Szeged) 70: 767–781, 2004]. First of all, we establish some basic properties of semi-partial isometries. Next, we introduce the notion of pseudo Moore–Penrose inverse as a natural generalization of the Moore–Penrose inverse from Hilbert spaces to arbitrary Banach spaces. This concept is used to carry out a classification for semi-partial isometries in Banach spaces and to provide a characterization for Hilbert spaces.

We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸 𝑑 , 𝑓 𝑐 (𝑤) = 𝑤(𝑤 + 𝑐) 𝑑 . Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤 2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸 𝑑 , 𝑓 𝑐 (𝑤) = 𝑤(𝑤 + 𝑐) 𝑑 , is connected.