CC BY
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Владикавказский математический журнал 2008, Том 10, Выпуск 2, С. 32-35
УДК 517.95
BANACH-STEINHAUS TYPE THEOREM IN LOCALLY CONVEX SPACES FOR a-LOCALLY LIPSCHITZIAN CONVEX PROCESSES
S. Lahrech, A. Jaddar, J. Hlal, A. Ouahab, A. Mbarki
The main purpose of this paper is to generalize the Banach-Steinhaus theorem in locally convex spaces for c-locally Lipschitzian operators established by S. Lahrech in [1] to c-locally Lipschitzian convex processes.
Mathematics Subject Classification: 49J52, 49J50, 47B37, 46A45.
Key words: c-Locally Lipschitzian convex processes, locally convex spaces, Banach-Steinhaus theorem.
1. Introduction
Let (X, A) and (Y, be two locally convex spaces. Assume that the locally convex topology ^ is generated by the family (q^)ве/ of semi norms on Y. Let в(Хл) denote the family of bounded sets in (X, A) and let a С в(Хл). For a linear mapping T : X — Y, a semi norm
p on Y, and £ a set L(p, C)(T) = sup p(Th). According to [1], T is said to be a-locally
hec
Lipschitzian if
VC £ a, Vв £ I : L(e,C) = L(qe,C)(T) <
By Lip(Xл, Y^, a) we denote the vector space of a-locally Lipschitzian operators. Note that Lip(Xл, Y^, a) is a locally convex space under the locally convex topology т(A, a) generated by the family of semi norms L(e, C), в £ I, C £ a.
The operator T : (X, A) — (Y, is said to be sequentially continuous if for every sequence
(xn) of X and every x £ X such that xn — x one has Txn Tx. T is said to be bounded if T sends bounded sets in (X, A) into bounded sets in (Y, Clearly, continuous operators are sequentially continuous, sequentially continuous operators are bounded, and linear bounded operators are a-locally Lipschitzian; but in general, converse implications fail. Let XXs, Xb and X^ denote respectively the family of continuous linear functionals sequentially continuous linear functionals, linear bounded functionals and a-locally Lipschitzian functionals on (X, A). In general, the inclusions X' С Xs С b С L are strict.
Let 0(X, X^) denote the topology of uniform convergence on a(X^, X)-Cauchy sequences of XL. Note that if a = в(X^, then Xb = X^ and consequently, 0(X, XL) = 0(X, X6).
It has been show in [1] that, if Tn : (X, A) — (Y, n £ n is a sequence of a-locally Lipschitzian operators admitting for each x £ X a weak limit lim Tnx = Tx, then the limit operator T maps 0(X, XL)-bounded sets into bounded sets.
Our objective in this paper is to generalize the above result to a-locally Lipschitzian convex processes.
© 2008 Lahrech S., Jaddar A., Hlal J., Ouahab A., Mbarki A.
These are multifunctions which maps every set C of a into bounded set in (Y, and whose graphs are convex cones.
Recall that a multifunction (or set-valued map) $ : X ^ Y is a map from X to the set of subsets of Y. The domain of $ is the set
D($) = {x G X : $(x) = 0}.
We say $ has nonempty images if its domain is X. For any subset C of X we write $(C) for the image (JxeC $(x) and the range of $ is the set R($) = $(X). We say $ is surjective if its range is Y. The graph of $ is the set
G($) = {(x,y) G X x Y : y G $(x)},
and we define the inverse multifunction $-1 : Y ^ X by the relationship
x G $-1(y) ^ y G $(x) for x G X and y G Y.
A multifunction is convex, or closed if its graph is likewise. A process is a multifunction whose graph is a cone. For example, we can interpret linear closed operators as closed convex processes in the obvious way. If A-bounded sets have ^ bounded images, then we say that $ is (A, bounded.
A convex process $ : (X, A) ^ (Y, is said to be a-locally Lipschitzian if it maps every set C of a into bounded set in (Y, Clearly, bounded convex processes are a-locally Lipschitzian. Note also that a-locally Lipschitzian operators can be interpreted as a-locally Lipschitzian convex processes. Our next step is to generalize all the results established by S. Lahrech in [1] to a-locally Lipschitzian convex processes. But before proving the main results, we pause to recall some terminologies and definitions which will be used later.
2. Banach—Steinhaus theorem for a-locally Lipschitzian convex processes in locally convex spaces
Let (X, A) and Y, ^ be two locally convex spaces. If $ : (X, A) ^ R is a a-locally Lipschitzian convex process, then we say that $ is a real a-locally Lipschitzian convex process with respect to the topology A.
Denote by Lipc((X, a, A), r) the class of a-locally Lipschitzian convex processes acting from X, A into r.
Let ($n) be a sequence of multifunctions acting from X into Y such that P|n D($n) = 0.
We say that ($n) is a Cauchy sequence along the topology if V x G X satisfying x G P| D($n) 3 ($nk) a subsequence of ($n), 3 G $nk (x), k = 1,2,... such that
n
(xnk) is a Cauchy sequence for the weak topology a(Y, Y') = a(Y, (Y, ). Denote by Lipc((X, a, A), (Y, the class of a-locally Lipschitzian convex processes acting from (X, A) into (Y,
Let B C X, B = 0. We say that B is bounded along the class Lipc((X, a, A), r), if for every Cauchy sequence $n along the topology ^ satisfying: $n G Lipc((X, a, A),r), B c f|D($n), and for every sequence (x&)k of elements of B and for every double sequence y^n G $n(x&), k = 1, 2,..., n = 1, 2,..., the double sequence (yn)n,k is bounded in R.
Let ($n) be a sequence of multifunctions acting from X into Y such that P|n D($n) = 0.
We define the upper limit (limsup$n) of $n with respect to the topology ^ by:
Vx G X limsup$n(x) = {y G Y : 3 ($nk) a subsequence of ($n), 3 yrik G $nk (x) (V k) such that ynk ^ y for the weak topology a(Y, (Y, ^)')}.
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Lahrech S., Jaddar A., Hlal J., Ouahab A., Mbarki A.
Let Ф : X — Y be a multifunction. We say that ФП converges to Ф along the topology if the following conditions holds:
1) limsup ФП = Ф,
2) Vx £ X, (Jn ФП^) is conditionally sequentially compact in (Y, a(Y, (Y, ^)')). In this case, we set 1™Фп = limsupФn = Ф.
Proposition 1. Assume that (Фп) is a sequence of convex processes converging to a some multifunction Ф along the topology ^. Then:
1)Пn ДФп) с Я(Ф),
2) Ф is a convex process.
< Let x £ P|n ^(Фп). Then, there is a sequence xn £ Фn(x) (n £ n). On the other hand, (Jn Фп^) is conditionally sequentially compact in (Y, a(Y, Y')). Hence, there exists a subsequence (xnk) £ ФПк (x) of (xn) converging to some y for the weak topology a(Y, Y'). Consequently, y £ limФn(x) = Ф(x). This implies that Ф^) = 0. Thus, x £ ^(Ф), and the desired inclusion follows.
Let now (x, y) £ С(Ф) and A ^ 0. Then, there is a subsequence (ФПк ) of (Фп) and
ynk £ ФПк(x) such that ynk y. Therefore, Ay £ Ф(Ax). Hence, A(x, y) £ С(Ф). So, using
the same argument, we prove that С(Ф) is convex. Thus, we achieve the proof. >
It follows from the above proposition that if 1шФп exists, then (Ф) is a Cauchy sequence.
Let us remark also that if the topology is separated and if (An) is a sequence of linear operators converging to some operator A acting from X into Y at each x £ X for the weak topology a(Y, Y'), then (Фп) converge to Ф in our sense, where (Фп) and Ф are the convex processes defined by Фп^) = {Anx}, and Ф^) = {Ax}.
For a multifunction Ф : X — Y and y' £ Y' = (Y, we define y' о Ф to be the multifunction Ф1 : X — r defined by Ф^) = y'^(x)).
Now we are ready to prove the main results of our paper.
Theorem 2 (Banach-Steinhaus theorem for a-locally Lipschitzian convex processes). Let Фп : (X, A) — (Y, n = 1, 2,... be a sequence of a-locally Lipschitzian convex processes converging along the topology ^ to some multifunction Ф : X — Y. Then the limit convex process Ф is (Lipc((X, a, A), r), ^-bounded. That is for any В С X such that В is bounded along the class Lipc((X, a, A), r), Ф(В) is bounded in (Y,
< Let y' £ Y' = (Y, Then, y' о Фп converge to y' along the topology Therefore, (y' о Фп) is a Cauchy sequence along the topology Assume that В is a bounded set along the class Lipc((X, a, A), r). Let (x&) be a sequence of elements of В and let yk £ Ф^к), k = 1, 2,...
Since yk £ Ф^к), then without loss of generality we can assume that there is a sequence yn £ Фп^к), k,n £ n such that, for any k, уп converges to yk with respect to the weak topology a(Y, Y'). On the other hand, В is bounded along the class Lipc((X, a, A), r). Consequently, the sequence ((y',yn))n,k is bounded in
r. Therefore, lim 1 (y', yj^ = 0
uniformly in n £ n.
Now fix e > 0. Then, there is an integer ko such that |(y', уп)1 < §ko for all n £ n and all k ^ ko. Fix а k ^ ko. Since уп — Ук as n — then there is an no £ n such that
1(у',уп0 )-(у',Ук )| < § • Therefore, |(y', yk )| < |(у',Ук )-(у',уп° )l + 1(у',уп° )l < § + 2 ko. This shows that the set {(y', y) : x £ B, y £ Ф^)} is bounded. Since y' £ (Y, is arbitrary, Ф(В) is ^-bounded by the classical Mackey theorem. Thus, we achieve the proof. >
The next result give us a sufficient conditions to guarantee that the limit convex process Ф in the above theorem is a-locally Lipschitzian.
We say that Lipc((X, a, A), r) is sequentially complete, if every Cauchy sequence in Lipc((X, a, A), r) converges in Lipc((X, a, A), r).
Theorem 3. Let $n : (X, A) ^ (Y, y), n = 1, 2,... be a sequence of a-locally Lipschitzian convex processes converging along the topology y to some multifunction $ : X ^ Y. Assume that Lipc((X, a, A), r) is sequentially complete. Then the limit convex process $ is a-locally Lipschitzian from (X, A) into (Y, y).
< Let C G a. We must prove that the set $(C) is bounded in (Y, y). Let y' G (Y, y)'. Then y' o $n ^ y' o $ along the topology y. Consequently, (y' o $n) is a Cauchy sequence in Lipc((X, a, A), r).
On the other hand, Lipc((X, a, A), r) is sequentially complete. Therefore, y' o $ G Lipc((X, a, A), r). Thus, y'($(C)) is bounded. Hence, $(C) is y-bounded by the classical Mackey theorem. >
Remark 4. Let us remark that if An : X ^ Y is a sequence of linear a-locally Lipschitzian operators converging to A at each x G X with respect to the weak topology a(Y, Y'), and if moreover, the topology y is separated, then the multifunction $n : X ^ Y defined by $n(x) = {Anx} is a-locally Lipschitzian convex process converging to the convex process $ defined by $(x) = {Ax}. Therefore, we recapture all the results given by S. Lahrech in [1] using our theorems.
References
1. Lahrech S. Banach-Steinhaus type theorem in locally convex spaces for linear a-locally Lipschitzian operators // Miskolc Mathematical Notes.—2005.—V. 6.—P. 43-45.
2. Rockafellar R. T. Monotone processes of Convex and Concave type // Memoirs of the Amer. Math. Soc.—1967.—№ 77.
3. Rockafellar R. T. Convex Analysis.—Princeton: Princeton Univ. press, 1970.
4. Borwein J. M., Lewis A. S. Convex Analysis and Nonlinear Optimization // CMS Books in Mathematics.—Gargnano, 1999.
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6. Kothe G. Toplogical vector spaces, I.—Berlin etc.: Springer, 1983.
7. Brezis H. Analyse fonctionnelle, Theorie et applications.—Masson, 1983.
Received October 26, 2006. Lahrech S.
Mohamed first university Oujda, Morocco (GAFO Laboratory) E-mail: [email protected] Jaddar A.
National School of Managment Oujda, Morocco (GAFO Laboratory) E-mail: [email protected] Hlal J.
Mohamed first university
Oujda, Morocco (GAFO Laboratory)
E-mail: [email protected]
Ouahab A.
Mohamed first university
Oujda, Morocco (GAFO Laboratory)
E-mail: [email protected]
Mbarki A.
National School of Applied Sciences Oujda, Morocco (GAFO Laboratory) E-mail: [email protected]
