Volume 19, Issue 2

Abstract. We give a characterization of pseudo-Einstein real hypersurfaces of a complex space form. We prove that if the Ricci tensor S of a real hypersurface M of a complex space form with constant principal curvatures satisfies for any , where a is a function and T 0 is the holomorphic distribution on M , then M is a pseudo-Einstein real hypersurface.

Abstract. Motivated by the two-step directional Newton method considered by Argyros and Hilout (2010) for approximating a zero of a differentiable function F defined on a convex set of a Hilbert space H , we consider a two-step Newton–Lavrentiev method (TSNLM) for obtaining an approximate solution to the nonlinear ill-posed operator equation , where is a nonlinear monotone operator defined on a real Hilbert space X . It is assumed that and that the only available data are with . We prove that the TSNLM converges cubically to a solution of the equation (such solution is an approximation of ) where x 0 is the initial guess. Under a general source condition on , we derive order optimal error bounds by choosing the regularization parameter α according to the balancing principle considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

Abstract. For a one-dimensional discrete time system S on the closed unit interval I we introduce an operator Q S , which transforms a distribution function G into a distribution function under the backward operation of S . We study the operator Q S for piecewise linear transformations S and give conditions under which the iterations of the distribution function G 0 converge to a Q S -invariant distribution function G , which conjugates S and the corresponding piecewise linear transformation T of constant slope.

Abstract. In this paper, we introduce a new explicit iterative scheme for approximation of fixed points of demicontinuous pseudocontractive mappings in uniformly smooth Banach spaces and prove strong convergence of our proposed iterative scheme. Furthermore, we modify our explicit iterative scheme for approximation of zeroes of bounded demicontinuous accretive mappings in uniformly smooth Banach spaces. Our result improves, extends and unifies most of the results that have been proved for this class of mappings.

Abstract. We consider the operator T defined by , , where is an unbounded domain, S is a positive definite selfadjoint operator defined on a domain and is a bounded complex measurable function with the property for a . We derive an estimate for the norm of the resolvent of T . In addition, we prove that T is invertible, and the inverse operator is a sum of a normal operator and a quasinilpotent one, having the same invariant subspaces. By the derived estimate, spectrum perturbations are investigated. Moreover, a representation for the resolvent of T by the multiplicative integral is established. As examples, we consider the Schrödinger operators on the positive half-line and orthant.

Abstract. In this paper, new classes of nondifferentiable generalized invex functions are introduced. Further, nonsmooth vector optimization problems with functions belonging to the introduced classes of (generalized) (Φ,ρ)-type I functions are considered. Sufficient optimality conditions and duality results for such classes of nonsmooth vector optimization problems are established. It turns out that the presented results are proved also for such nonconvex vector optimization problems in which not all functions constituting them possess the fundamental property of invexity.

Abstract. In this paper, we answer the question: for any with , what are the greatest value and the least value such that the double inequality holds for all with ? Here and are the logarithmic and p th generalized logarithmic means of a and b , respectively.

Abstract. We present a further generalization of the 𝒯 𝒜 d -density topology introduced in [Real Anal. Exchange 32 (2006/07), 349–358] as a generalization of the density topology. We construct an ascending sequence of density topologies which leads to the -density topology including all previous topologies. We examine several basic properties of the topologies.