Vol 26, No 135 (2021)
Articles
On Chaplygin’s theorem for an implicit differential equation of order n
Abstract
We consider the Cauchy problem for the implicit differential equation of order n g t, x, x, …, x (n) =0, t ∈ 0; T , x 0 = A. It is assumed that A = A 0 ,…, A n -1 ∈ Rn , the function g :[0, T ] × Rn +1 → R is measurable with respect to the first argument t ∈[0, T ] , and for a fixed t , the function g t , ∙× Rn +1 → R is right continuous and monotone in each of the first n arguments, and is continuous in the last n +1 -th argument. It is also assumed that for some sufficiently smooth functions η , ν there hold the inequalities ν i 0 ≥ A i ≥ η i 0 , i= 0, n-1, ν n t ≥ η n t , t∈[0; T]; g t; νt , ν t , ..., ν n t ≥ 0, g t, ηt , η t ,…, η (n) (t) ≤ 0, t∈0; T . Sufficient conditions for the solvability of the Cauchy problem are derived as well as estimates of its solutions. Moreover, it is shown that under the listed conditions, the set of solutions satisfying the inequalities η n t ≤ x n t ≤ ν n t , is not empty and contains solutions with the largest and the smallest n -th derivative. This statement is similar to the classical Chaplygin theorem on differential inequality. The proof method uses results on the solvability of equations in partially ordered spaces. Examples of applying the results obtained to the study of second-order implicit differential equations are given.
Russian Universities Reports. Mathematics. 2021;26(135):225-233
225-233
Eckland and Bishop-Phelps variational principles in partially ordered spaces
Abstract
In this paper, an assertion about the minimum of the graph of a mapping acting in partially ordered spaces is obtained. The proof of this statement uses the theorem on the minimum of a mapping in a partially ordered space from [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Caristi-like condition and the existence of minima of mappings in partially ordered spaces // Journal of Optimization Theory and Applications. 2018. V. 180. Iss. 1, 48-61]. It is also shown that this statement is an analogue of the Eckland and Bishop-Phelps variational principles which are effective tools for studying extremal problems for functionals defined on metric spaces. Namely, the statement obtained in this paper and applied to a partially ordered space created from a metric space by introducing analogs of the Bishop-Phelps order relation, is equivalent to the classical Eckland and Bishop-Phelps variational principles.
Russian Universities Reports. Mathematics. 2021;26(135):234-240
234-240
Perturbation of the fixed point problem for continuous mappings
Abstract
We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.
Russian Universities Reports. Mathematics. 2021;26(135):241-249
241-249
On the existence of a solution for a periodic boundary value problem for semilinear fractional-order differential inclusions in Banach spaces
Abstract
In this paper, we study a periodic boundary value problem for a class of semilinear differential inclusions of fractional order in a Banach space for which the multivalued nonlinearity satisfies the regularity condition expressed in terms of measures of noncompactness. To prove the existence of solutions to the problem, we first construct the corresponding Green function. Then we introduce into consideration a multivalued resolving operator in the space of continuous functions and reduce the posed problem to the existence of fixed points of the resolving multioperator. To prove the existence of a fixed point, a generalized theorem of B.N. Sadovskii type for a condensing multivalued map is used.
Russian Universities Reports. Mathematics. 2021;26(135):250-270
250-270
Existence and stability of periodic solutions in a neural field equation
Abstract
We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the existence of a special class of solutions that we call 1-bump periodic solutions. We then analyze the stability of these solutions by studying the spectrum of the Frechet derivative of the corresponding Hammerstein operator. We prove that the spectrum of this operator agrees up to zero with the spectrum of a block Laurent operator. We show that the non-zero spectrum consists of only eigenvalues and obtain an analytical expression for the eigenvalues and the eigenfunctions. The results are illustrated by multiple examples.
Russian Universities Reports. Mathematics. 2021;26(135):271-295
271-295
Symbols in Berezin quantization for representation operators
Abstract
The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F ♮ defined on M . These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization - we call it the polynomial quantization - is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G . In this case symbols turn out to be polynomials on the Lie algebra g . In this paper we offer a new theme in the Berezin quantization on G/H : as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G= SL (2;R) , H - the subgroup of diagonal matrices, G/H - a hyperboloid of one sheet in R3 ; b) G - the pseudoorthogonal group SO0 (p ; q ) , the subgroup H covers with finite multiplicity the group SO0 (p -1, q -1)× SO0 (1;1) ; the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G .
Russian Universities Reports. Mathematics. 2021;26(135):296-304
296-304
Superpositional measurability of a multivalued function under generalized Caratheodory conditions
Abstract
For a multivalued mapping F :[a ; b ] × Rm → comp ( Rm ) , the problem of superpositional measurability and superpositional selectivity is considered. As it is known, for superpositional measurability it is sufficient that the mapping F satisfies the Caratheodory conditions, for superpositional selectivity it is sufficient that F(·, x) has a measurable section and F(t; ·) is upper semicontinuous. In this paper, we propose generalizations of these conditions based on the replacement, in the definitions of continuity and semicontinuity, of the limit of the sequence of coordinates of points in the images of multivalued mappings to a one-sided limit. It is shown that under such weakened conditions the multivalued mapping F possesses the required properties of superpositional measurability / superpositional selectivity. Illustrative examples are given, as well as examples of the significance of the proposed conditions. For single-valued mappings, the proposed conditions coincide with the generalized Caratheodory conditions proposed by I.V. Shragin (see [Bulletin of the Tambov University. Series: natural and technical sciences, 2014, 19:2, 476-478]).
Russian Universities Reports. Mathematics. 2021;26(135):305-314
305-314
Homogeneous spaces yielding solutions of the k[S] -hierarchy and its strict version
Abstract
The k[S] -hierarchy and its strict version are two deformations of the commutative algebra k[S] , k=R or C ; in the N×N -matrices, where S is the matrix of the shift operator. In this paper we show first of all that both deformations correspond to conjugating k[S] with elements from an appropriate group. The dressing matrix of the deformation is unique in the case of the k[S] -hierarchy and it is determined up to a multiple of the identity in the strict case. This uniqueness enables one to prove directly the equivalence of the Lax form of the k[S] -hierarchy with a set of Sato-Wilson equations. The analogue of the Sato-Wilson equations for the strict k[S] -hierarchy always implies the Lax equations of this hierarchy. Both systems are equivalent if the setting one works in, is Cauchy solvable in dimension one. Finally we present a Banach Lie group G( S 2 ) , two subgroups P + (H) and U + (H) of G( S 2 ) , with U + (H)⊂ P + (H) , such that one can construct from the homogeneous spaces G S 2 / P + (H) resp. G( S 2 )/ U + (H) solutions of respectively the k[S] -hierarchy and its strict version.
Russian Universities Reports. Mathematics. 2021;26(135):315-336
315-336