Volume 5, Issue 3

The impulsive differential equation with several retarded arguments is considered, where p i ( t ) ≥ 0, 1 + b k > 0 for i = 1, . . . , m , t ≥ 0, k ∈ ℕ. Sufficient conditions for the oscillation of all solutions of this equation are found.

Among the functions defined on the two-dimensional unit sphere we distinguish functions generalizing the conjugate integral, the conjugate function, and the conjugate series which depend on one variable. We establish the properties of these functions whose structures essentially differ from those of integrals, functions, and series based on the theory of analytic functions of two complex variables.

Connections between the spectral radius and the radius of boundedness are studied. Different characterizations of algebras ( Q -property, strong saquentiality) are given in terms of these radii. Examples and applications are also provided.

The contact problem of the plane theory of elasticity is studied for an elastic orthotropic half-plane supported by periodically located (infinitely many) stringers of equal resistance. Using the methods of the theory of a complex variable, the problem is reduced to the Keldysh–Sedov type problem for a circle. The solution of the problem is constructed.

Sufficient conditions are established for the solvability of the boundary value problem where p : C ( I ; R n ) × C ( I ; R n ) → L ( I ; R n ), q : C ( I ; R n ) → L ( I ; R n ), l : C ( I ; R n ) × C ( I ; R n ) → R n , and c n : C ( I ; R n ) → R n are continuous operators, and p ( x , ·) and l ( x , ·) are linear operators for any fixed x ∈ C ( I ; R n ).

We equip the category of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In particular, a Leibniz algebra becomes a Lie object in and the universal enveloping algebra functor UL from Leibniz algebras to associative algebras factors through the category of cocommutative Hopf algebras in . This enables us to prove a Milnor–Moore type theorem for Leibniz algebras.

For functions of several variables a property is established similar to the well-known result of S. B. Stechkin about the metric property of almost increasing functions. In the case of one variable the proof is easier than the known one.

The paper deals with the contact problems of the theory of elasticity. The problems are reduced to Prandtl-type integral differential equations with a coefficient at the singular operator which has higher-order zeros at the ends of the integration interval. In some concrete cases the solution is constructed efficiently. Asymptotic representations are obtained.