Volume 10, Issue 1

The paper presents recent results concerning the problem of the existence of those selections, which preserve the properties of a given set-valued mapping of one real variable taking on compact values from a metric space. The properties considered are the boundedness of Jordan, essential or generalized variation, Lipschitz or absolute continuity. Selection theorems are obtained by virtue of a single compactness argument, which is the exact generalization of the Helly selection principle. For set-valued mappings with the above properties we obtain a Castaing-type representation and prove the existence of multivalued selections and selections which pass through the boundaries of the images of the set-valued mapping and which are nearest in variation to a given mapping. Multivalued Lipschitzian superposition operators acting on mappings of bounded generalized variation are characterized, and solutions of bounded generalized variation to functional inclusions and embeddings, including variable set-valued operators in the right hand side, are obtained.

Let ( X n ) be a sequence of independent not necessarily identically distributed random vectors belonging to the domain of attraction of a stable or semistable hemigroup, i.e. for an increasing sampling sequence ( k n ) such that k n+1 / k n → c ≥ 1 and linear operators A n , the normalized sums converge in distribution uniformly on compact subsets of {0 ≤ s < t } to some full probability μ s,t . Suppose that ( T n ) is a sequence of positive integer valued random variables such that T n /k n converges in probability to some positive random variable, where we do not assume ( X n ) and ( T n ) to be independent. Then weak limit theorems of random sums, where the sampling sequence ( k n ) is replaced by random sample sizes ( T n ), are presented.

For a constant k ∈ [0, ∞) a normalized function f , analytic in the unit disk, is said to be k -uniformly convex if Re (1 + zf ″( z ) /f ′( z )) > k | zf ″( z )/ f ′( z )| at any point in the unit disk. The class of k -uniformly convex functions is denoted k - ucv (cf. [Kanas, Wiśniowska, J. Comput. Appl. Math. 105: 327–336, 1999]). The function g is said to be k -starlike if g ( z ) = zf ′( z ) and f ∈ k - ucv . For analytic functions f , g , where f ( z ) = z + a 2 z 2 + ⋯ and g ( z ) = z + b 2 z 2 + ⋯, the integral convolution is defined as follows: In this note a problem of stability of the integral convolution of k -uniformly convex and k -starlike functions is investigated.

In this paper, we investigate a class of hyperbolic differential equations of neutral type and obtain some new sufficient conditions of the oscillation for such equations satisfying boundary condition

We consider a motion of a viscous compressible heat conducting fluid of a fixed mass bounded by a free surface. For a local solution of equations describing such a motion we derive some energytype inequalities which are necessary to prove the global existence of solutions.