Volume 17, Issue 3

The existence of a unique mild solution on a semiinfinite interval for a first order semilinear neutral functional differential equations involving evolution operators in Fréchet spaces is investigating using a nonlinear alternative of Leray–Schauder type for contractive maps, combined with semigroup theory.

In this paper, we consider a problem for a semilinear hyperbolic equation. The existence, uniqueness and continuous dependence of a solution upon the data are proved by Rothe's method. Besides, some convergence results are established for approximations.

In this paper, based on -convergence, we introduce the -core of sequences of complex numbers and determine a class of matrices A for which the -core( Ax ) is a subset of Knopp's core for all bounded sequences x .

Some modifed direct localized boundary-domain integral equations (LBDIEs) systems associated with the mixed boundary value problem (BVP) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed. The main results established in the paper are the LBDIEs equivalence to the original variable-coefficient BVPs and the invertibility of the corresponding localized boundary-domain integral operators in appropriately chosen function spaces.

We consider by a dual variational method the existence of solutions to certain fourth order Dirichlet problems with nonlinearities corresponding to the derivatives of a sum of a convex and a concave function. The growth conditions are imposed only on the convex part.

In the present paper, special functional spaces are constructed, which are naturally connected with sufficiently wide classes of irregular Carleman–Vekua equations. Their properties used essentially for the investigation of irregular generalized analytic functions, are studied. In constructing these functional spaces, we have to investigate the existence of -primitive functions for special classes.

We introduce and describe the spaces BMO p(·) (𝔻), 1 ≤ p ( z ) < ∞, of functions of bounded mean oscillation over the unit disc in the complex plane in the hyperbolic Bergman metric with respect to the Lebesgue measure and the variable exponent p = p ( z ).

We consider the initial value problem for a two-dimensional semi-linear wave equation with exponential type nonlinearity. We obtain global well-posedness in the energy space. We also establish the linearization of bounded energy solutions in the spirit of Gérard [J. Funct. Anal. 141: 60–98, 1996]. The proof uses Moser–Trudinger type inequalities and the energy estimate.

We investigate immunity properties of the s-degrees. In particular we show that neither the immune nor the hyperimmune s-degrees are upwards closed since there exist s-degrees a ≤ s b such that a is hyperimmune, but b is immune free. We also show that there is no hyperhyperimmune set A such that , where is the complement of the halting set and ≤ ŝ denotes the finite-branch version of s-reducibility.

We study properties of the form (linear functional) u = λ ( x – a ) –1 ν + δ b , where ν is a regular form. We give a necessary and sufficient condition for the regularity of the form u . The coefficients of a three-term recurrence relation, satisfied by the corresponding sequence of orthogonal polynomials, are given explicitly. The semi-classical character of the founded families is studied. We conclude by giving some examples.

We establish the parametrization of extremals for the problem which is one of the known generalizations of Chebotarev's problem. The parametrization of this problem essentially differs from that of Chebotarev's problem and Grötzsch's problem, which is a generalization of Chebotarev's problem.

We study the boundedness and compactness of products of Volterra operators and composition operators acting between weighted Bergman spaces of infinite order and weighted Bloch type spaces B w .