Volume 19, Issue 1

Abstract. A Hölder type stability estimate in the second order Sobolev space of functions is established for the solution of the elliptic obstacle problem with respect to variations of the coefficients of the corresponding differential operator. In modern finance, this estimate provides results on the sensitivity of a perpetual American option price with respect to variations of the volatilities in the underlying assets prices. To meet the objective of this paper, the stability result with respect to external force functions, which was proved by J.-F. Rodrigues under the nondegeneracy condition, is generalized to the case of arbitrary functions belonging to the space , .

Abstract. Some two-point boundary value problems are considered for linear impulsive systems with singularities. The singularity is understood in the sense that the matrix- and vector-functions of a continuous (as well as of a discrete) argument which correspond to this system maybe nonsummable. The effective conditions are established for each considerable problem to be uniquely solvable. Analogous problems are also considered for similar impulsive systems with small parameters.

Abstract. The purpose of this paper is to consider a two-dimensional version of the Aifantis equation of statics of the theory of consolidation with double porosity and effectively solve the basic BVPs for the half-plane. We extend the potential method and the theory of integral equations to BVPs of the theory of consolidation with double porosity. For all problems, we construct Fredholm type integral equations. For the Aifantis equation of statics we construct a particular solution and reduce the solution of the basic BVPs of the theory of consolidation with double porosity to the solution of the basic BVPs for the equation for an isotropic body. A Poisson type formula is constructed for the solution of the first and of the second boundary value problem for the half-plane.

Abstract. In this paper we introduce a new kind of modification of Durrmeyer type -Baskakov operators based on the concept of -integral. These operators are different from those proposed by Aral and Gupta in [Nonlinear Anal. 72 (2010), 1171–1180]. We establish direct and local approximation theorems of operators and obtain the estimates on the rate of convergence and weighted approximation of operators .

Abstract. Backward stochastic differential equations (BSDEs) with convex generators of quadratic growth are considered. The existence of a solution is proved for such equations driven by a continuous martingale with unbounded characteristic. A suitable optimization problem is formulated and it is shown that the corresponding value process satisfies BSDEs.

Abstract. We prove that the ideal defined in the Cantor space is not generated. This answers a question of Z. Grande and E. Strońska in the case of the Cantor space.

Abstract. It is proved that the maximal operator of the triangular-Fejér-means of a two-dimensional Walsh–Fourier series is bounded from the dyadic Hardy space to for all and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular-Fejér-means of a function converge a.e. to . The maximal operator is bounded from the Hardy space to the space weak- and is not bounded from the Hardy space to the space .

Abstract. In this paper some recent results by Feng and Liu and by Klim and Wardowski are generalized to the setting of single-valued and multi-valued mappings. We also introduce the notion of lower -semicontinuity.

Abstract. A sufficient condition for the nonexistence of blowing-up mild solutions of a nonlinear evolution fractional functional-differential equation associated with a strongly continuous semigroup and with a nonlinearity containing the Riemann–Liouville fractional integral is established. We prove a result on a new type of nonlinear integral inequalities with weakly singular kernels and delay and apply it in the proof of the result on the nonexistence of blowing-up solutions. This result is applied to a fractionally damped pendulum equation with a time delay forcing term (a feedback control).

Abstract. We give sufficient conditions such that multilinear maximal operator and multilinear fractional maximal operators satisfy strong type and weak type two-weight norm inequalities. These results extend Sawyer's theorem and other well-known results.

Abstract. Cancellous bone can be regarded as a lattice of asymptotically small rods and plates. In this paper, a thin rod of an orthotropic elastic material with a cross-section where is considered. This thin cross-section leads to asymptotic expansions of the displacement vector, stress and strain tensors, and of the temporal change in the volumetric fraction of solid bone in the bone matrix. A rate equation describing this change in the volumetric fraction governs the remodeling process of bone deposition and reabsorption. It has linear and quadratic terms of strain tensors and thus, its own asymptotic expansion. This expansion's leading term is used in some simple numerical simulations.

Abstract. Problems of approximation nature are considered for some spaces of real and complex valued functions and also measures defined on a locally compact abelian group. The approximating subspaces are analogs of the spatial entire functions of exponential type. The problem of convergence of a sequence of Fejér type positive operators is studied.