Volume 22, Issue 1

In this paper, we establish some new Ostrowski type inequalities for s -logarithmically convex functions. Some applications of our results to PDFs and in numerical integration are given.

Let R be a *-ring. In this paper we introduce the notion of *- n -derivation in R . An additive mapping x ↦ x * of R into itself is called an involution on R if it satisfies the conditions: (i) ( x * ) * = x , (ii) ( x y ) * = y * x * for all x , y ∈ R . A ring R equipped with an involution `*' is called a *-ring. It is shown that if a prime *-ring R admits a nonzero *- n -derivation (resp. a reverse *- n -derivation) D , then R is commutative. Further, some related properties of *- n -derivation in a semiprime *-ring have also been investigated. Finally, a structure theorem for *- n -derivation has been obtained.

Using variational methods, we prove the existence of positive solutions to an elliptic equation involving critical nonlinearities and multiple inverse square potentials.

We define the classes of almost weak p ω·2+ n -projective and almost ω 1 -weak p ω·2+ n -projective p -primary abelian groups and establish some of their crucial properties. Important generalizations and extensions of classical results are also obtained in this new framework.

Let G be a finite group. In this paper we introduce the generalized conjugate graph Γ ( G , n ) c which is a graph whose vertices are all the non-central subsets of G with n elements and two distinct vertices X and Y joined by an edge if X = Y g for some g ∈ G . General properties of the graph such as the number of edges, clique, chromatic, dominating, independence numbers, automorphism and energy of the graph are discussed. We also present a condition under which two generalized conjugate graphs are isomorphic. Moreover, the generalized conjugate graph is a key to define the probability that two subsets of the group G with the same cardinality are conjugate. We obtain some upper and lower bounds for this probability.

This paper studies some new existence and uniqueness results for boundary value problems for nonlinear fractional differential equations by using a variety of fixed point theorems. Some illustrative examples are also presented.

We consider the stationary oscillation case of the theory of linear thermoelasticity with microtemperatures of microstretch materials. A representation formula of a general solution of a homogeneous system of differential equations is written in terms of eight metaharmonic functions. Such formulas are very convenient and useful in many specific problems of concrete geometry. We demonstrate an application of these formulas to Dirichlet and Neumann type boundary value problems in a ball. Explicit solutions in the form of absolutely and uniformly convergent series are constructed.

Duality is obtained for the space L w p ( I , X ), which is the weighted Lebesgue space that consists of functions having values in a Banach space X .

The Cauchy and Cauchy–Darboux problems for semilinear wave equations in the class of continuous functions are investigated. The questions of existence, uniqueness and nonexistence of global solutions of the problems are considered. The local solvability of the problems is also discussed.

A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D , and the set V ( G )∖ D is independent. The double outer-independent domination number of a graph G , denoted by γ d oi ( G ), is the minimum cardinality of a double outer-independent dominating set of G . We prove that for every nontrivial tree T of order n , with l leaves and s support vertices we have γ d oi ( T ) ≤ (2 n + l + s )/3, and we characterize the trees attaining this upper bound.

We study the analytical properties of a dynamic value function and of an optimal solution to the utility maximization problem in incomplete markets for utility functions defined on the whole real line. It was shown by Kramkov and Sirbu [Ann. Appl. Probab. 16 (2006), no. 3, 1352–1384] that if the relative risk-aversion coefficient of the utility function defined on the half real line is uniformly bounded away from zero and infinity, then the value function at time t = 0 of utility maximization problem is two-times differentiable and the optimal wealth is differentiable in probability with respect to the initial capital. Similar results are true for utility functions defined on the whole real line if instead of relative risk-aversion the same condition on the risk-aversion coefficient is imposed. Besides, assuming the continuity of the filtration F we prove that the derivative of the optimal wealth is strictly positive and that the derivative exists also in the sense of L 1 -convergence. This enables us to show the existence of a strictly increasing (with respect to the initial capital) and absolutely continuous modification of the optimal wealth. We also study the differentiability properties of the value function V ( t , x ) and of the optimal wealth process X t ( x ) for any t ∈ [0, T ] and give the conditions under which the second derivative of the value function and the derivative of the optimal wealth process are continuous. We need these properties to show that the value function satisfies a certain backward stochastic partial differential equation used to characterize the optimal wealth process.

The main aim of this paper is to discuss the properties of the maximal operator of two-dimensional Walsh–Kaczmarz–Fejér means, where the set of indices is inside a cone-like set. We prove that the maximal operator σ L κ,* is bounded from H p to L p for 1/2 < p and is of weak type (1,1). As a corollary, we get Simon's theorem [Monatsh. Math. 131 (2000), no. 4, 321–334] on the convergence almost everywhere of cone-restricted two-dimensional Fejér means of integrable functions. In the endpoint case p = 1/2, we show that the maximal operator σ L κ,* is not bounded from the Hardy space H 1/2 to the space L 1/2 .

In this paper we study compactness of subsets of grand Lebesgue spaces (also known as Iwaniec–Sbordone spaces ) and grand Sobolev spaces. Namely, we prove a Kolmogorov–Riesz compactness theorem for grand Lebesgue spaces and the corresponding version of a result of Sudakov. In addition, the validity of a Rellich–Kondrachov type compact embedding is shown to hold in the context of grand Sobolev spaces.