Banach - Steinhaus type theorem in locally convex spaces for \sigma-locally lipschitzian convex processes

Let (X, A) and (Y, ^) be two locally convex spaces. Assume that the locally convex topology ^ is generated by the family ( q e ) e e l of semi norms on Y . Let в ( Х \ ) denote the family of bounded sets in (X, A) and let ст С в (X a ) . For a linear mapping T : X ^ Y , a semi norm p on Y , and G ст set L(p, C )( T ) = sup p ( Th ) . According to [1], T is said to be ст-locally h ∈ C

Lipschitzian if

V C G ст, V в G I : L ( e,C ) = L ( q e ,C )( T ) < + ^ .

By Lip(X A , Y p , ст) we denote the vector space of ст-locally Lipschitzian operators. Note that Lip(X A , Y p , ст) is a locally convex space under the locally convex topology т (A, ^, ст) generated by the family of semi norms L ( в, C), в G I , C G ст.

The operator T : (X, A) ^ (Y, ^) is said to be sequentially continuous if for every sequence ( x n ) of X and every x G X such that x _n ^ x one has Tx n ^ 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 σ-locally Lipschitzian; but in general, converse implications fail. Let X ⁰ , X ^s , X ^b and X _σ ^L denote respectively the family of continuous linear functionals sequentially continuous linear functionals, linear bounded functionals and ст-locally Lipschitzian functionals on (X, A). In general, the inclusions X ⁰ ⊂ X ^s ⊂ ^b ⊂ _σ ^L are strict.

Let 9 ( X, X L ) denote the topology of uniform convergence on ст ( X ^^L ,X )-Cauchy sequences of X L . Note that if ст = в (X a ) , then X b = X L and consequently, 9 ( X, X L ) = 9 ( X, X ^b ).

It has been show in [1] that, if T n : (X, A) ^ (Y, ^), n G N is a sequence of ст-locally Lipschitzian operators admitting for each x G X a weak limit lim T n x = Tx, then the limit operator T maps 0(X, X L )-bounded sets into bounded sets.

Our objective in this paper is to generalize the above result to σ-locally Lipschitzian convex processes.

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) Ф : X ^ Y is a map from X to the set of subsets of Y . The domain of Ф is the set

D(Ф) = { x g X : ф(x) = 0 } .

We say Ф has nonempty images if its domain is X. For any subset C of X we write Ф(С) for the image IJx e C Ф(x) and the range of Ф is the set R(Ф) = Ф(X). We say Ф is surjective if its range is Y. The graph of Ф is the set

G(Ф) = {(x, У) G X x Y : y G Ф(x)}, and we define the inverse multifunction Ф-1 : Y ^ X by the relationship x G Ф-1(у) О y G Ф(х) 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 А-bounded sets have ^ bounded images, then we say that Ф is (А, ^) bounded.

A convex process Ф : (X, А) ^ (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 σ-locally Lipschitzian operators can be interpreted as σ-locally Lipschitzian convex processes. Our next step is to generalize all the results established by S. Lahrech in [1] to σ-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 σ -locally Lipschitzian convex processes in locally convex spaces

Let (X, А) and Y, ^ be two locally convex spaces. If Ф : (X, А) ^ R is a a-locally Lipschitzian convex process, then we say that Ф is a real a-locally Lipschitzian convex process with respect to the topology λ.

Denote by Lip _c ((X, a, А), R ) the class of a-locally Lipschitzian convex processes acting from X, λ into R .

Let (Ф _п ) be a sequence of multifunctions acting from X into Y such that Qn D(Ф _n ) = 0 .

We say that (Ф _п ) is a Cauchy sequence along the topology ^, if V x G X satisfying x G P| D(Ф _n ) 3 (Ф _{П к} ) a subsequence of (Ф _п ), 3 x k G Ф _{П к} (x), k = 1,2,... such that (x n k ) is a Cauchy sequence for the weak topology a(Y, Y 0 ) = a(Y, (Y, ^) 0 ). Denote by Lip _c ((X, a, А), (Y, ^)) the class of a-locally Lipschitzian convex processes acting from (X, А) into (Y, ^).

Let B С X, B = 0 . We say that B is bounded along the class Lip _c ((X, a, А), R ), if for every Cauchy sequence Ф _п along the topology ^ satisfying: Ф _п G Lip _c ((X, a, А), R ), B С QD(Ф _n ), and for every sequence (x k ) k of elements of B and for every double sequence у П G Ф _n (x k ), k = 1, 2,..., n = 1, 2,..., the double sequence (y k ) _n,k is bounded in R.

Let (Ф _п ) be a sequence of multifunctions acting from X into Y such that Qn D(Ф _n ) = 0 .

We define the upper limit (limsupФ _n ) of Ф _п with respect to the topology ^ by:

V x G X limsupФ n (x) = {y G Y : 3 (Ф п _к ) a subsequence of (Ф п ), 3 y _n k G Ф п _к (x) ( V k) such that y n k ^ y for the weak topology a(Y, (Y, ^) 0 )}.

Let Ф : X ^ Y be a multifunction. We say that Ф _п converges to Ф along the topology ц, if the following conditions holds:

• l)    limsup Ф _п = Ф,

• 2)    V x G X, Un Ф _n (x) is conditionally sequentially compact in (Y, ct(Y, (Y, ц) ⁰ )). In this case, we set limФ _n = lim sup Ф _п = Ф.

Proposition 1. Assume that (Ф _п ) is a sequence of convex processes converging to a some multifunction Ф along the topology ц. Then:

• 1)    T n D(Ф n ) C D(Ф) ,

• 2)    Ф is a convex process.

C Let x G Un D(Ф _n ). Then, there is a sequence x _n G Ф _n (x) (n G N ). On the other hand, Un Ф _п (х) is conditionally sequentially compact in (Y, ct(Y, Y 0 )). Hence, there exists a subsequence ( x n k ) G Ф _{П к} (x) of (x n ) converging to some y for the weak topology ct(Y, Y 0 ). Consequently, y G limФ _n (x) = Ф(x). This implies that Ф(x) = 0 . Thus, x G D(Ф), and the desired inclusion follows.

Let now (x,y) G G(Ф) and A >  0. Then, there is a subsequence (Ф _{П к} ) of (Ф _п ) and ct ( Y,Y 0 )

ynk G ФПк(x) such that ynk  —»  y. Therefore, Ay G Ф(Ax). Hence, A(x, y) G G(Ф). So, using the same argument, we prove that G(Ф) is convex. Thus, we achieve the proof. B

It follows from the above proposition that if lim Ф _п exists, then (Ф) is a Cauchy sequence.

Let us remark also that if the topology ц, is separated and if (A n ) is a sequence of linear operators converging to some operator A acting from X into Y at each x ∈ X for the weak topology ct(Y, Y 0 ), then (Ф _п ) converge to Ф in our sense, where (Ф _п ) and Ф are the convex processes defined by Ф _n (x) = { A n x } , and Ф(x) = { Ax } .

For a multifunction Ф : X ^ Y and y ⁰ G Y ⁰ = (Y, ц) ⁰ , we define y ⁰ ◦ Ф to be the multifunction Ф 1 : X ^ R defined by Ф ₁ (x) = y ⁰ (Ф(x)).

Now we are ready to prove the main results of our paper.

Theorem 2 (Banach–Steinhaus theorem for σ -locally Lipschitzian convex processes). Let Ф _п : (X, A) ^ (Y, ц) , n = 1, 2,... be a sequence of ст-locally Lipschitzian convex processes converging along the topology ц to some multifunction Ф : X ^ Y . Then the limit convex process Ф is (Lip _c ((X, ст, A), R ), ^-bounded. That is for any B C X such that B is bounded along the class Lip _c ((X, ст, A), R ) , Ф(B) is bounded in (Y, ц) .

C Let y ⁰ G Y ⁰ = (Y, ц ) . Then, y ⁰ ◦ Ф _п converge to y ⁰ along the topology ц. Therefore, (y ⁰ ◦ Ф _п ) is a Cauchy sequence along the topology ц. Assume that B is a bounded set along the class Lip _c ((X, ст, A), R ). Let (x k ) be a sequence of elements of B and let y k G Ф(x k ), k = 1, 2,...

Since y k G Ф(x k ), then without loss of generality we can assume that there is a sequence у П G Ф _n (x k ), k,n G N such that, for any k, y nn converges to y k with respect to the weak topology ct(Y, Y 0 ). On the other hand, B is bounded along the class Lip _c ((X, ст, A), R ). Consequently, the sequence ( h y ⁰ ,y n i ) n,k is bounded in R . Therefore, lim ¹ h y ⁰ , y nn i = 0 ,                                 k,s ^ + ^

uniformly in n ∈ N .

Now fix e >  0. Then, there is an integer k g such that | h y ⁰ , y nn i| <  | k g for all n G N and all k >  k g . Fix а k >  k g . Since у П ^ y k as n ^ + ro , then there is an n g G N such that Ih y ⁰ ,y n 0 i-h y ⁰ ,y k il <  2 . Therefore, |h y ⁰ ,y k i| 6 |h y ⁰ ,y k i - hy ,y n i| + y y. i| 6 | + | k g . This shows that the set {h y ⁰ , y i : x G B, y G Ф(x) } is bounded. Since y⁰ G (Y,ц) ⁰ is arbitrary, Ф(B) is ц-bounded by the classical Mackey theorem. Thus, we achieve the proof. B

The next result give us a sufficient conditions to guarantee that the limit convex process Ф in the above theorem is σ-locally Lipschitzian.

We say that Lip c ((X, a, A), R ) is sequentially complete, if every Cauchy sequence in Lip _c ((X, a, A), R ) converges in Lip _c ((X, a, A), R ).

Theorem 3. Let Ф _п : (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. Assume that Lip _c ((X, a, A), R ) is sequentially complete. Then the limit convex process Ф is a-locally Lipschitzian from (X, A) into (Y, ^) .

C Let C E a. We must prove that the set Ф(С) is bounded in (Y, ^). Let y 0 E (Y, ^) 0 . Then y ⁰ о Ф _п ^ y ⁰ о Ф along the topology ^. Consequently, (y ⁰ о Ф _п ) is a Cauchy sequence in Lip c ((X, a, A), R ).

On the other hand, Lip _c ((X, a, A), R ) is sequentially complete. Therefore, y 0 о Ф e Lip _c ((X, a, A), R ). Thus, у⁰(Ф(С )) is bounded. Hence, Ф(С) is ^-bounded by the classical Mackey theorem. B

Remark 4. Let us remark that if A n : X ^ Y is a sequence of linear a-locally Lipschitzian operators converging to A at each x E X with respect to the weak topology a(Y, Y 0 ), and if moreover, the topology ^ is separated, then the multifunction Ф _п : X ^ Y defined by Ф _п (х) = { A n x } is a-locally Lipschitzian convex process converging to the convex process Ф defined by Ф(х) = { Ax } . Therefore, we recapture all the results given by S. Lahrech in [1] using our theorems.