CC BY
49
УДК 517.988
GLUING OF FUNCTIONS AND LOCAL OPERATORS 1
© D. C. Djinja, Y. Nepomnyashchikh, A. Ponossov, A. Shindiapin
Keywords: local operator, gluing of funeions, gluing-invariant set.
Abstract: In 1976, I.V.Shragin introduced local operators as a natural generalization of the Nemytskii operators. The property of locality means that the values of the images (which are functions) restricted to some subset only depend on the values of the preimages restricted to the same subset. We study the property of locality of the nonlinear operators in spaces of measurable functions without assuming the continuity conditions, i. e. focussing on the property of gluing of functions.
In his work [6] I.V.Shragin introduced local operators as a natural generalization of the Nemytskii operators (also known as the superposition operators). The property of locality means that the values of the images (which are functions) restricted to some subset only depend on the values of the preimages restricted to the same subset. The importance of the local operators is stipulated by the fact that the property of locality is characteristic for ordinary differential equations [1]. Basic properties of the local operators in functional spaces (representations via Nemytskii operators, boundedness, continuity, etc.) have been studied by many authors (see [2] [8]).
Here we study the property of locality of the nonlinear operators in spaces of measurable functions without assuming the continuity conditions, i. e. focussing on the property of gluing. As far as we know the role of gluing in understanding the property of locality was first discussed in [3]. In this paper we systematize and extend some results from [5] and [8].
Let X, Y, Z be nonempty sets, (T, £, i) be a space with соmplete ст-finite measure; F(X) be the set of all classes of the i-equivalent fonctions from T to X. Let us fix some x0 G X and consider the projection operator Pe,x ■ F(X) ^ F(X), associating x G F(X) with Pe,xx G F(X) and being defined as follows: for each function x(-), representing the class of equivalence x and any function v(-), representing the class of equivalence Pe,xx, we put v(t) = x(t) almost everywhere on E and v(t) = x0 almost everywhere on T \ E. Similarly we define F (Y ) and the projections Pe , y- Let us fix an arbitrary nonempty set M С F(X).
D e f i n i t i о n 1. [4, 5]. An operator A ■ M ^ F (Y ) is called local, if for any E G £ and u,v G M the equality Pe,xu = Pe,xv implies Pe,yAu = Pe,y Av.
Proposition 1. Operator A ■ M ^ F (Y ) is local iff Pe , y FPe, x = Pe, y F (V E G £).
Proposition 2. If Ai ■ M ^ F (Y ) and A2 ■ A1(M ) ^ F (Z ) are local operators then their
product A = A2A1 ■ M ^ F(Z) is also a local operator.
Proposition3. If a local operator A ■ M ^ F (Y ) is injective, then the inverse operator A-1 ■ A(M) ^ M is local.
Proposition 4. If Y is a linear space over a field P, operato rs A1 ■ M ^ F (Y ) and
A2 ■ M ^ F (Y ) are local and c1,c2 G P, then the oper ator A = c1A1 + c2A2 is also local.
1 Работа поддержана Управлением науки университета Эдуардо Мондлане (Мозамбик), SIDA/SAREC (Швеция).
Proposition 5. If Y = R and the operators A1 : M — F(R), A2 : M — F(R) are local, then the operators max{A1, A2}, min{A1, A2}, |Ai| and A1 ■ A2 (defined by ((Ai ■ A2)x)(t) = (A1x)(t) ■ (A2x)(t) almost everywhere) are also local.
We will index the countable partitions of T by natural numbers and consider the finite partitions as special cases of the countable ones formally adding the empty sets as elements of the partition. By R(T) we denote the set of all nonempty countable measurable partitions of the set T and by S(M) we denote the set of all x E F(X) such that for some countable measurable partition {En}c^=1 E R(T) and some xn E M (n E N) we have PEn,xx = PEn,xxn (n E N).
Definition2[2j. The set M C F (X) is called gluing-invariant (g.i.) if i{En}c^'=1 E R(T) and ixn E M (n E N) the element x E F(X) defined (uniquely) by PEn,Xx = PEn,Xxn (n E N) belongs to M
Assume that S(M) = {M1 C F(X) | M1 D M, Mi g.i.}. It is evident that the set S(M) is closed
(M)
M
L e m m a 1. min S(M) = S(M), [M g.i.] & [M = S(M)].
Theoreml. Let A : M — F (Y) be a local operator.
1) There exists a unique local operator A* : S(M) — F(Y) such that A*Im = A;
2) A*(S(M)) = S(A(M));
3) There exists a local operator A** : F (X) — F (Y) such th at A**|^(M) = A*;
4) If Y consists of more than one element, then there exist local operators A** : F(X) — F(Y),
i = 1, 2 such th at A**IS(M) = A*, i = 1,2 m, d A1x = A2x (ix E F (X) \ S (M)).
Let now X, Y be linear spaces over the field R, and M be a linear subset of X. Let x0 and y0 be
the zero elements of the spaces X and Y, respectively: x0 = 0x, y0 = 0^. By R(T) we denote the
T
Proposition 6. The operator A : M — F (Y) is local iff FPe,x = Pe,y F + Ps\e,y F 0x (iE E Z).
Corollary 1. Let an operator A : M — F (Y) satisfy the condition F 0x = 0y (for example,
A is linear). Then A is local iff FPe,x = Pe,yF (iE E £).
L e m m a 2. S(M) = {^1 Pe„,xxn : {En}^=1 E R(T), xn E M (n E N)}.
Theorem 2. Let A : M — F (Y) be a local operator. Then the one-to-one local extension A* : S (M) — F (Y) of the opera tor A is given by
A*x = ^ PEn,YAxn, x E S(M),
n=1
where {EnE R(T) and xn E M (n E N) satisfy x = ^2^^ PEn,xxn-
Corollary 1. If the set M g.i. and A : M — F (Y) is a local operator, then for any
{En}c^'=1 E R(T) and xn E M (n E N) we have
\ ^
y, PEn, x xnj =^2 PEn, Y Axn n=1 / n=1
References
1. Azbelev N.V., Maksimov V.P. and Rakhmatullina L.F. Introduction to the theory of functional differential equations // World Federation Publishers, Atlanta. 1996. 172 p.
2. Appell J., Zabrejko P.P. Nonlinear superposition operators // Cambridge: Cambridge University Press. 1990.
3. Krasnosel’skii M.A., Pokrovskii A.V. The discontinuous superposition operators j j Uspehi Matem. Nauk. 1977, V. 32, No 1, P.169-170.
4. Ponosov A.V. On the Nemytskii Conjecture // Dokl. Acad. Nauk SSSR. 1986, V. 289, No 6, P. 1308-1311.
5. Nepomnyashchikh Yu.V., Ponosov A.V. Local operators in some subspaces of the space Lo (Russian) j j Izv. Vysh. Uchebn. Zaved. Mathem., 1999, No 6, P.50-64.
6. Sragi/n I. V. Abstract Nemytskii Operators (Locally Defined Operators) (Russian) // Dokl. Acad. Nauk SSSR. 1976, V. 227, No 1, P. 47-49.
7. Shragin I. V. Superpositional measurability and the superposition operator. Selected themes. // Feniks, Odessa. 2006, 103 p.
8. Shragin I. V., Nepomnyashchikh Yu. V. On extension of measurable functions and local operators. // Proc. of the Sixth Conf. "Function Spaces", Wroclaw, Poland. 3-8 Sept., 2001. World Scientiïc. New-Jersey-London-Singapore-Hong Kong. 2003. P. 251-262.
Аннотация: В 1976 году И.В.Шрагин ввёл понятие локального оператора как естественное обобщение оператора Немыцкого. Свойство локальности означает, что сужение функции-образа на некотором множестве зависит только от значений функции-прообраза на том же множестве. Мы исследуем свойство локальности нелинейных операторов в пространствах измеримых функций без предположения непрерывности, с точки зрения склеек функций.
Ключевые слова: локальный оператор, склейки функций, инвариантное относительно склейки множество.
Djinja Domingos Celso
doctor of phys.-math. sciences, full professor Eduardo Mondlane University Mozambique Maputo celsodjinja@yahoo.com.br
Джиниа Домингос Цельсо д. ф.-м. н., профессор Университет Эдуардо Мондлане Мозамбик, Мапуту celsodjinja@yahoo.com.br
Yury Nepomnyashchikh
doctor of phys.-math. sciences, full professor
Eduardo Mondlane University
Mozambique, Maputo
e-mail: yuvn2@yandex.ru
Непомнящих Юрий д. ф.-м. н., профессор Университет Эдуардо Мондлане Мозамбик, Мапуту e-mail: yuvn2@yandex.ru
Arkadi Ponossov
doctor of phys.-math. sciences, professor Norwegian University of Life Sciences Norway, Aas e-mail: arkadi@umb.no
Поносов Аркадий Владимирович д. ф.-м. п., профессор Норвежский университет естественных наук Норвегия, Ос e-mail: arkadi@umb.no
Andrei Shindiapin
doctor of phys.-math. sciences, full professor Eduardo Mondlane University Mozambique Maputo e-mail: andrei.olga@tvcabo.co.mz
Шиндяпин Андрей Игоревич д. ф.-м. п., профессор Университет Эдуардо Мондлане Мозамбик, Мапуту e-mail: andrei.olga@tvcabo.co.mz
