Volume 13, Issue 2

A new result is obtained on the vanishing of the local time of a non-negative continuous semimartingale at zero. Based on this result, an early exercise premium representation of a value function of the American put option is obtained in a one-dimensional general diffusion model.

In this paper, the existence of solutions and extremal solutions for a second order perturbed nonlinear boundary value problem with integral boundary conditions is proved under the mixed generalized Lipschitz and Carathéodory conditions.

This paper presents a generalization of nonlinear integral inequalities of the Gronwall–Bellman–Bihari type for discontinuous functions and its application to the investigation of the practical stability of solutions of systems of integro-differential equations with impulse perturbations at fixed moments of time.

We are interested in existence results for nonconvex functional differential inclusions. First, we prove an existence result, in separable Hilbert spaces, for first order nonconvex sweeping processes with perturbation and with delay. Then, by using this result and a fixed point theorem we prove an existence result for second order nonconvex sweeping processes with perturbation and with delay of the form 𝑢˙ (𝑡) ∈ 𝐶(𝑢(𝑡)), 𝑢¨(𝑡) ∈ –𝑁 𝑃 (𝐶(𝑢(𝑡)); 𝑢˙(𝑡)) + 𝐹(𝑡, 𝑢˙ 𝑡 ) when 𝐶 is a nonconvex bounded Lipschitz set-valued mapping and 𝐹 is a set-valued mapping with convex compact values taking their values in finite dimensional spaces.

This paper is concerned with a screen type boundary value problem arising from the wave diffraction problem with a reactance condition. We consider the problem in a weak formulation within Bessel potential spaces, and where both cases of a complex and a pure real wave number are analyzed. Using the potential method, the boundary value problem is converted into a system of integral equations. The invertibility of the corresponding matrix pseudodifferential operator is shown in appropriate function spaces which allows the conclusion about the existence and uniqueness of a weak solution to the original problem. Higher regularity properties of solutions are also proved to exist in some scale of Bessel potential spaces, upon the corresponding smoothness improvement of given data. In particular, the 𝐶 α -smoothness of solutions in a neighbourhood of the screen edge is established with arbitrary α < 1 in the two-dimensional case and α < 1/2 in the three-dimensional case.

We provide a generalized sufficient condition for the continuity of the Nemytskij operator between 𝐿 𝑝 1 (𝑋) and 𝐿 𝑝 2 (𝑋) . An example is given.

We define and study monadic 𝐵𝐿-algebras as a pair of 𝐵𝐿-algebras one of which is a special case of relatively complete subalgebra.

In the present note, we study a certain Durrmeyer type integral modification of Bernstein polynomials. We investigate simultaneous approximation and estimate the rate of convergence in simultaneous approximation.

Some combinatorial results concerning finite decompositions (dissections) of a 𝑘-dimensional cube into cubes (respectively, simplexes) of the same dimension are presented in the paper. In connection with such decompositions, the notion of a decomposability number is introduced and the problem of description of all these numbers is discussed.

We give positive answer to Velichko's question in which the quotient and 𝑠-map is replaced by a sequence-covering and 𝑐𝑠-map. In addition, let 𝑋 have a star-countable 𝑘-network, then 𝑋 is a sequence-covering and 𝑐𝑠-image of a locally separable metric space if and only if 𝑋 is a sequencecovering and 𝑐𝑠-image of a metric space.

In this paper we study the modified Beta operators. We extend the result of [Gupta and Ahmad, İstanbul niv. Fen Fak. Mat. Derg. 54: 11–22 1995] and obtain an inverse result for the linear combination of these modified Beta operators in simultaneous approximation.

Let 𝐴 be a bounded linear operator acting on a Hilbert space 𝐻. The 𝐵-Weyl spectrum of 𝐴 is the set σ 𝐵𝑤 (𝐴) of all ⋋ ∈ ℂ such that 𝐴 – ⋋𝐼 is not a 𝐵-Fredholm operator of index 0. Let 𝐸(𝐴) be the set of all isolated eigenvalues of 𝐴. Recently, in [Berkani and Arroud, J. Aust. Math. Soc. 76: 291–302, 2004] the author showed that if 𝐴 is hyponormal, then 𝐴 satisfies the generalized Weyl's theorem σ 𝐵𝑤 (𝐴) = σ (𝐴) \ 𝐸(𝐴), and the 𝐵-Weyl spectrum σ 𝐵𝑤 (𝐴) of 𝐴 satisfies the spectral mapping theorem. Lee [Han, Proc. Amer. Math. Soc. 128: 2291–2296, 2000] showed that Weyl's theorem holds for algebraically hyponormal operators. In this paper the above results are generalized to an algebraically (𝑝, 𝑘)-quasihyponormal operator which includes an algebraically hyponormal operator.

We rephrase Fridli's result [Fridli, Acta Math. Hungar. 45: 393–396, 1985] on the modulus of continuity with respect to a Vilenkin group in the Lebesgue space. We show that this result is valid in the logarithm space and for Vilenkin-like systems. In addition, we prove that there is a strong connection between the best approximation of Fourier series and the modulus of continuity, not only in the Lebesgue space [Gát, Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 17: 161–169, 2001] but in the logarithm space too. We formulate two variable generalizations of the obtained results, which have not been known till now even in the Walsh case.

Exact distributions of |𝑋𝑌| and |𝑋/𝑌| are derived when 𝑋 and 𝑌 are Pearson type VII and Bessel function random variables distributed independently of each other.

An automatically stable and order three split rational approximation scheme is proposed for solving the Cauchy abstract problem. The third order precision is obtained by introducing the complex parameter and performing a rational approximation of the semigroup. For the considered scheme, an explicit a priori estimate is derived.

The paper gives an extension of the fundamental principles of selfadjoint operators in Fréchet–Hilbert spaces, countable-Hilbert and nuclear Fréchet spaces. Generalizations of the well known theorems of von Neumann, Hellinger–Toeplitz, Friedrichs and Ritz are obtained. Definitions of generalized central and generalized spline algorithms are given. The restriction 𝐴 ∞ of a selfadjoint operator 𝐴 defined on a dense set 𝐷(𝐴) of the Hilbert space 𝐻 to the Frechet space 𝐷(𝐴 ∞ ) is substantiated. The extended Ritz method is used for obtaining an approximate solution of the equation 𝐴 ∞ 𝑢 = 𝑓 in the Frechet space 𝐷(𝐴 ∞ ). It is proved that approximate solutions of this equation constructed by the extended Ritz method do not depend on the number of norms that generate the topology of the space 𝐷(𝐴 ∞ ). Hence this approximate method is both a generalized central and generalized spline algorithm. Examples of selfadjoint and positive definite elliptic differential operators satisfying the above conditions are given. The validity of theoretical results in the case of a harmonic oscillator operator is confirmed by numerical calculations.

By using the general means, we establish some oscillation theorems for the even order delay differential equation (𝑟(𝑡)|𝑥 (𝑛–1) (𝑡)| α –1 𝑥 (𝑛–1) (𝑡))′ + 𝐹(𝑡, 𝑥[𝑔(𝑡)]) = 0, where α > 0 is a constant, , and 𝑔 ∈ 𝐶([𝑡 0 , ∞), ℝ). The results obtained extend and improve some results known in the literature.

The corrections to our paper [Wu and Sun, Georgian Math J. 12: 551–559, 2005] are made.