Dynamic Systems and Applications 17 (2008) 325-330 ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS YEOL JE CHO, DONAL O REGAN, AND SVATOSLAV STANEK Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University, Chinju 660-701, Korea Department of Mathematics, National University of Ireland, Galway, Ireland Department of Mathematical Analysis, Faculty of Science, Palacky University, Tomkova 40, 779 00 Olomouc, Czech Republic ABSTRACT. New antipodal fixed point theorems for compact Kakutani maps between Fréchet spaces are presented. Keywords. Fixed point theory, projective limits. AMS (MOS) Subject Classification. 47H10. 1. INTRODUCTION This paper presents new antipodal fixed point theorem for compact Kakutani maps between Fréchet spaces. The proofs rely on an antipodal fixed point theorem for compact Kakutani maps due to O Regan and Peran [4] and viewing a Fréchet space as the projective limit of a sequence of Banach spaces. In the literature [1, 2] one usually assumes the map F is defined on a subset X of a Fréchet space E and its restriction (again called F ) is well defined on X n (see Section 2). In general of course for Volterra operators the restriction is always defined on X n and in most applications it is in fact defined on X n and usually even on E n (see Section 2). In this paper we make use of the fact that the restriction is well defined on X n and we only assume it admits an extension (satisfying certain properties) on X n. We also show in Section 2 how easily one can extend antipodal fixed point theory in Banach spaces to fixed point theory in Fréchet spaces. Existence in Section 2 will be based on an antipodal fixed point theorem for compact Kakutani maps due to O Regan and Peran [4]. We state a particular case of it here for the convenience of the reader. Let X and Y be topological vector spaces. We say F : X CK(Y ) is a Kakutani map if F is upper semicontinuous; here CK(Y ) denotes the family of nonempty compact convex subsets of Y. We write F K(X, Y ) if F is a compact Kakutani map. Received September 14, 2007 1056-2176 $15.00 c Dynamic Publishers, Inc.
326 Y. J. CHO, D. O REGAN, AND S. STANEK Theorem 1.1. Let E be a Banach space and M a closed, bounded, symmetric subset of E with 0 M and let F K(M, E) with F (x) ( F ( x)) for all x M. Then F has a fixed point in M. Now let I be a directed set with order and let E α } α I be a family of locally convex spaces. For each α I, β I for which α β let π α,β : E β E α continuous map. Then the set } x = (x α ) α I E α : x α = π α,β (x β ) α, β I, α β be a is a closed subset of α I E α and is called the projective limit of E α } α I and is denoted by lim E α (or lim E α, π α,β } or the generalized intersection [3, pp. 439] α I E α.) 2. FIXED POINT THEORY IN FRÉCHET SPACES Let E = (E, n } n N ) be a Fréchet space with the topology generated by a family of seminorms n : n N}; here N = 1, 2,... }. We assume that the family of seminorms satisfies (2.1) x 1 x 2 x 3 for every x E. A subset X of E is bounded if for every n N there exists r n > 0 such that x n r n for all x X. For r > 0 and x E we denote B(x, r) = y E : x y n r n N}. To E we associate a sequence of Banach spaces (E n, n )} described as follows. For every n N we consider the equivalence relation n defined by (2.2) x n y iff x y n = 0. We denote by E n = (E / n, n ) the quotient space, and by (E n, n ) the completion of E n with respect to n (the norm on E n induced by n and its extension to E n are still denoted by n ). This construction defines a continuous map µ n : E E n. Now since (2.1) is satisfied the seminorm n induces a seminorm on E m for every m n (again this seminorm is denoted by n ). Also (2.2) defines an equivalence relation on E m from which we obtain a continuous map µ n,m : E m E n since E m / n can be regarded as a subset of E n. Now µ n,m µ m,k = µ n,k if n m k and µ n = µ n,m µ m if n m. We now assume the following condition holds: for each n N, there exists a Banach space (E n, n ) (2.3) and an isomorphism (between normed spaces) j n : E n E n. Remark 2.1. (i). For convenience the norm on E n is denoted by n. (ii). In our applications E n = E n for each n N.
ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS 327 (iii). Note if x E n (or E n ) then x E. However if x E n then x is not necessaily in E and in fact E n is easier to use in applications (even though E n is isomorphic to E n ). For example if E = C[0, ), then E n E which coincide on the interval [0, n] and E n = C[0, n]. Finally we assume E 1 E 2 and for each n N, (2.4) j n µ n,n+1 jn+1 1 x n x n+1 x E n+1 consists of the class of functions in (here we use the notation from [3] i.e. decreasing in the generalized sense). lim E n (or 1 E n where 1 is the generalized intersection [3]) denote the projective limit of E n } n N (note π n,m = j n µ n,m jm 1 : E m E n for m n) and note lim E n = E, so for convenience we write E = lim E n. For each X E and each n N we set X n = j n µ n (X), and we let X n, int X n and X n denote respectively the closure, the interior and the boundary of X n with respect to n in E n. Also the pseudo-interior of X is defined by Let pseudo int (X) = x X : j n µ n (x) X n \ X n for every n N}. The set X is pseudo-open if X = pseudo int (X). For r > 0 and x E n we denote B n (x, r) = y E n : x y n r}. Let M E and consider the map F : M 2 E. Assume for each n N and x M that j n µ n F (x) is closed. Let n N and M n = j n µ n (M). Since we only consider Volterra type operators we assume (2.5) if x, y E with x y n = 0 then H n (F (x), F (y)) = 0; here H n denotes the appropriate generalized Hausdorff distance (alternatively we could assume n N, x, y M if j n µ n (x) = j n µ n (y) then j n µ n F (x) = j n µ n F (y) and of course here we do not need to assume that j n µ n F (x) is closed for each n N and x M). Now (2.5) guarantees that we can define (a well defined) F n on M n as follows: For y M n there exists a x M with y = j n µ n (x) and we let F n (y) = j n µ n F (x) (we could of course call it F y since it is clear in the situation we use it); note F n : M n C(E n ) and note if there exists a z M with y = j n µ n (z) then j n µ n F (x) = j n µ n F (z) from (2.5) (here C(E n ) denotes the family of nonempty closed subsets of E n ). In this paper we assume F n will be defined on M n i.e. we assume the F n described above admits an extension (again we call it F n ) F n : M n 2 En (we will assume certain properties on the extension). We now show how easily one can extend fixed point theory in Banach spaces to applicable fixed point theory in Fréchet spaces.
328 Y. J. CHO, D. O REGAN, AND S. STANEK Theorem 2.1. Let E and E n be as described in the beginning of Section 2, M a bounded symmetric subset of E, F : Y 2 E with Y E and with j n µ n (0) M n and M n Y n for each n N. Also assume for each n N and x Y that j n µ n F (x) is closed and in addition for each n N that F n : M n 2 En is as described above. Suppose the following conditions are satisfied: (2.6) for each n N, F n K(M n, E n ) (2.7) for each n N, we have F n (x) ( F n ( x)) for x M n and (2.8) for each n 2, 3,... } if y M n solves y F n y in E n then j k µ k,n jn 1 (y) M k for k 1,..., n 1}. Then F has a fixed point in E. Remark 2.2. Note in Theorem 2.1 if x M n then x Y n so there exists a y Y with x = j n µ n (y) and so F n (x) = j n µ n F (y). PROOF: Fix n N. We would like to apply Theorem 1.1. To do so we need to show (2.9) M n is a bounded symmetric subset of E n. To see that M n is symmetric let ˆx µ n (M). Then for every x µ 1 n (ˆx) we have x M since M is symmetric and of course ˆx = µ n (x). Now its easy to see µ n (x) = µ n ( x) so ˆx = µ n (x) = µ n ( x) µ n (M). As a result µ n (M) is symmetric and since j n is linear (in particular j n ( y) = j n (y) for y E n ) we have that M n = j n µ n (M) is symmetric and so M n is symmetric. Also M n is bounded (so M n is bounded) since M is bounded (note if y M n then there exists x M with y = j n µ n (x)). Thus (2.9) holds. For each n N (see Theorem 1.1) there exists y n M n with y n F n (y n ). Lets look at y n } n N. Notice y 1 M 1 and j 1 µ 1,k j 1 k (y k ) M 1 for k N\1} from (2.8). Note j 1 µ 1,n jn n) F 1 (j 1 µ 1,n jn n)) in E 1 ; to see note for n N fixed there exists a x E with y n = j n µ n (x) so j n µ n (x) F n (y n ) = j n µ n F (x) on E n so on E 1 we have j 1 µ 1,n jn 1 (y n) = j 1 µ 1,n jn 1 j n µ n (x) j 1 µ 1,n jn 1 j n µ n F (x) = j 1 µ 1,n µ n F (x) = j 1 µ 1 F (x) = F 1 (j 1 µ 1 (x)) = F 1 (j 1 µ 1,n j 1 n j n µ n (x)) = F 1 (j 1 µ 1,n jn 1 (y n)). As a result j 1 µ 1,n jn n) F 1 (j 1 µ 1,n jn n)) in E 1, j 1 µ 1,n jn 1 (y n) M 1 for n N, together with (2.6) implies there is a subsequence N1 of N and a z 1 M 1 with j 1 µ 1,n jn 1 (y n) z 1 in E 1 as n in N1 and z 1 F 1 (z 1 ) since F 1 is upper
ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS 329 semicontinuous. Let N 1 = N 1 \ 1}. Now j 2 µ 2,n j 1 n (y n) M 2 for n N 1 together with (2.6) guarantees that there exists a subsequence N 2 of N 1 and a z 2 M 2 with j 2 µ 2,n jn 1 (y n) z 2 in E 2 as n in N2 and z 2 F 2 (z 2 ). Note from (2.4) and the uniqueness of limits that j 1 µ 1,2 j2 1 z 2 = z 1 in E 1 since N2 N 1 (note j 1 µ 1,n jn 1 (y n ) = j 1 µ 1,2 j2 1 j 2 µ 2,n jn 1 (y n ) for n N2 ). Let N 2 = N2 \ 2}. Proceed inductively to obtain subsequences of integers N 1 N 2, N k k, k + 1,... } and z k M k with j k µ k,n jn 1 (y n ) z k in E k as n in Nk and z k F k (z k ). Note j k µ k,k+1 j 1 k+1 z k+1 = z k in E k for k 1, 2,... }. Also let N k = Nk \ k}. Fix k N. Now z k F k (z k ) in E k. Note as well that z k = j k µ k,k+1 j 1 k+1 z k+1 = j k µ k,k+1 j 1 k+1 j k+1 µ k+1,k+2 j 1 k+2 z k+2 = j k µ k,k+2 j 1 k+2 z k+2 = = j k µ k,m jm 1 z m = π k,m z m for every m k. We can do this for each k N. As a result y = (z k ) lim E n = E and also note z k M k Y k for each k N. Thus for each k N we have so y F (y) in E. j k µ k (y) = z k F k (z k ) = j k µ k F (y) in E k In Theorem 2.1 it is possible to replace M n Y n with M n a subset of the closure of Y n in E n provided Y is a closed subset of E so in this case we could have Y = M if M was closed. To see this note from y = (z k ) lim E n = E and π k,m (y m ) z k in E k as m we can conclude that y Y = Y (note q Y iff for every k N there exists (x k,m ) Y, x k,m = π k,n (x n,m ) for n k with x k,m j k µ k (q) in E k as m ). Thus z k = j k µ k (y) Y k and so j k µ k (y) j k µ k F (y) in E k as before. For completeness we state these results. Theorem 2.2. Let E and E n be as described in the beginning of Section 2, M a bounded symmetric subset of E, F : Y 2 E with Y a closed subset of E and with j n µ n (0) M n and M n a subset of the closure of Y n in E n for each n N. Also assume for each n N and x Y each n N that F n : M n 2 En (2.8) hold. Then F has a fixed point in E. that j n µ n F (x) is closed and in addition for is as described above. Suppose (2.6), (2.7) and Corollary 2.1. Let E and E n be as described in the beginning of Section 2, M a closed bounded symmetric subset of E, F : M 2 E with j n µ n (0) M n for each n N. Also assume for each n N and x M that j n µ n F (x) is closed and in addition for each n N that F n : M n 2 En is as described above. Suppose (2.6), (2.7) and (2.8) hold. Then F has a fixed point in E.
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