ON SOME FIXED POINT THEOREMS IN BANACH SPACES

In this paper, some fixed point theorems are proved for multi-mappings as well as a pair of mappings. These extend certain known results due to Kirk, Browder, Kanna, Ciric and Rhoades.

The interest in this result has been further enhanced due to simultaneous and in- dependent appearance of results of Browder [2] and GBhde [5] which are essentially special cases of the result of Kirk.Recently Kannan [6] and iri6 [4] have obtain- ed results in basically the same spirit by suitably modifying the non-expansive condition on the mapping and the condition of normal structure on the underlying set.
In this paper we give a fixed point result for multi-mappings (Theorem 2.1) and extend the results of Kannam [6] and Ciric [3] to a pair of mappings (Theorems 3.1 and 3.2).This enables us to establish convergence of Ishikawa iterates (cf.[9]) for a pair of mappings.
Let K be a closed, bounded and convex subset of a Banach space X.For x .X, let (x;K) denote sup lx-kll k K} and let (K) genote the diameter of K.
Recall that a point x K is called a non-diametral point of K if (x'K) < (K)   and that K is said to have normal structure whenever given any closed bounded con- vex subset C of K with more than one point, there exists a non-diametral x e C.
It is well-knocn (cf.[A]) that a compact convex subset of an arbitrary Banach space and a closed, bounded and convex subset of a uniformly convex Banach space have normal structure.With K as before, let r(K) denote the radius of K inf f(x K) x e K} and let K denote the Chebyshev centre of C K x e K r(K) (x,K) }.It is well known (cf.Opial [8]) that if K is n,n-empry weakly ,ompa,-t ,,,nvex ubset ,f 8ana'h =pa.X, then K is n,nempty I,.ed onvex stb.qet t K and, turtht-rmre tI K has normat structure, then

B}
Fheor,'m ..I. Let K be a nonempty weakly ompat cenvc subset of the Banach K space X.Assume K has normal structure.Let T:K 2 be a mapping satisfying: for each closed convex subset F of K invariant under T, there exists some a(F), 0 < a(F) < i, such that 5(Tx,Ty) < max (x,F), (F) 5 (F)   for each x, y F.
Then T has a fixed point x satisfying Tx {x }.

O O O
Proof.We imitate in parts the proof of Kirk's theorem.Let denote the collection of non-empty closed convex subsets C of K that are left invariant by T(i.e., TC c C, where TC u[Tc c e C "). Order by set-inclusion.By weak compactness of K, we can apply Zorn's lemma to get a minimal element M. It suf- fices to show that M is a singleton.Suppose that M contains more than one ele- ment.
By the definition of normal structure there exists x M such that 1 If 6(Tx, Ty) S 6(x,M) for all x, y M, let M 6   {x E M: (x,:.[)-< i 6(M)}.
Hence T(M) is contained in a closed ball of arbitrary centre in Tx and radius 8(M).By the minimality of M, if m e Tx, then M c U( m 8(M)) (the closed ball of centre m and radius 8(M)), whence m M and T(M) c M. But (M6) < 86(M) < (M) which contradicts the minimality of M. Thus M is a singleton and this completes the proof.
Corollary 2.2.Let K be a nonempty weakly compact convex subset of the Banach space X.Assume K has normal structure.Let T be a mapping of K into itself which satisfies: for each closed convex subset F of K invariant under T there exists some (F), 0 < e(F) < i, such that max 6(x,F), e(F) for each x, y F. Then T has a fixed point.
Corollary 2.3.Let K be a nonempty weakly compact convex subset of the Banach space X.Assume K has normal structure.Let T be a mapping of K into itself which satisfies: for each closed convex subset F of K invariant under T there exists some e(F), 0 < e(F) < i, such that max lx-yll, r(F), a (F)   for each x, y F. Then T has a fixed point.
COMMON FIXED POINTS OF MAPPINGS.
Theorem 3.1.Let K be a weakly compact convex subset of the Banach space X.
Let T I, T 2 be two mappings of K into itself satisfying: (i) lrlx r2Yll <_ max (I Ix-rlxl I+I IY-T2Yl I)/2, (I Ix-TroY [+I lY-rlxl I)/3, (I !x-Yl I+l Ix-rlxl +:Iy-Tmyl I)/3} for each x, y e K, TIC C if and only if T2C C for each closed subset C of K; Then there exists a unique common fixed point of T I and T 2.
Proof.Let denote the family of all non-empty closed convex subsets of K, each of which is mapped into itself by T I and T 2. Ordering by set-inclusion, by weak compactness of K and Zorn's lemma, we obtain a minimal element F of K. With- out loss of generality, assume that zSPF lz-rmzll (F)/2.
(y e F).This gives that T2(F) c U(TIX r(F)) U, whence T2(F n U) F n U and by hypotheses (2) TI(F U) c F n U.By the minimality of F, we obtain F c U.
Consequently, if F contains more than one element, then F is a proper subset of F. c But this in view of above contradicts the minimality of F. Hence F contains exactly one element, say, x O, whence TlX0 x 0 T2xO.Assume there exists another element Y0 e K such that TlY0 YO T2Yo" Then using (I), we obtain 2 ITlX0 T2YoII <_ If TlX0 T2Y011 whence x 0 TlX0 T2Y0 Y0" THEOREM 3.2.Let K be a weakly compact convex subset of the Banach space X.
Assume K has normal structure.Let T I, T 2 be mappings of K into itself satisfying: (i) lTlX TmYll <_ max {(I x TlXll + IY-TmYl I)/2, (I Ix-rmYll + I!Y-TlXl I)/2, (I Ix-yl I+I IX-TlXl + Iy-Tmyl I)/3} for each x,y e K, TI C c C if and only if T2C c C for each closed convex subset C of K, Then there exists a unique common fixed point of T I and T 2.
PROOF.Let be as in Theorem 3.1.Exactly as in Theorem 3.1.,has a minimal element F. Without loss of generality, assume that sup lz-T2zll <_ r(F).This also yields the existence of a common fixed point of T and T 2. However, it need not be unique.
THEOREM 3.3.Let K be a weakly compact convex subset of the Banach space X.
Assume K has normal structure.Let T I, T 2 be mappings of K into itself satisfying (2) and (3) of the preceding theorem and, (i) lTlX T2Yll < max {I Ix-Yll, lX-TlXll, Ix-TIYlI, lx-T2xll, Ix-T2YlI}" Then there exists a common fixed point of T I and T 2.
The proof of the above theorem is similar to that of Theorem 3.2 and hence it is omitted.

ISHIKAWA ITERATION FOR COMMON FIXED POINTS.
A uniformly convex Banach space is reflexive.A bounded, closed and convex subset of a uniformly convex Banach space is therefore weakly compact; als% it has normal structure.Hence Theorems 2.1, 3.2 and 3.3 can be particularized to such a setting.Rhoades [9] has extended a result of iri (cf. [3],Theorem 2)   to a wider class of transformations by using Ishikawa iterative scheme.With a suitable modification of arguments, this extends to a pair of mappings of the type as in Theorem 3.2.
THEOREM 4.1.Let K be a non-empty closed bounded and convex subset of a uni- formly convex Banach space X.Let TI, T 2 be mappings of K into itself satisfying (i), ( 2) and (3) of Theorem 3.2.Let the sequence {x of iterates be defined by n (4) x 0 e K Yn (i 8n)Xn + 8n T 1 x n n _> 0 lim IIxnk T I xnkll 0 It would be sufficient, with (7), to show that limk lTlXnk-T2Ynkll O.
either zSPc lZ-TlZll _< (C)/2, or sp Iz-T2zll < (C)/2 z C holds for each closed convex subset C of K invariant under T I and T 2.
closed convex subset D of K invariant under T I and T 2.
r(F). (y e F) This gives exactly as in Theorem 3.1 that T l(F)c c F and T2(Fc)CF Since K has c c normal structure, one has (F < (F) if K contains more than one element, which c contradicts the minimality of F. Thus F contains Drecisely one element, which is the unique common fixed point of T I and T 2 as in Theorem 3.1.REMARK.One can replace condition (i) of Theorem 3.2 by (i) lrlx T2Yll <_max {I Ix-Yll, (I IX-TlXll + IY-TmYlI)/2 (I Ix-T2Yl + IY-TlXl I)/3, ([ Ix-Yl I+I IX-TlXl I+I IY-T2Yl I)/3} existence of the unique common fixed point of T 1 and T 2 results from Theorem 3.2.Let the unique common fixed point be v.From (1) lTlXn -vl[ < ]Ix n vll and Following exactly the same lines as in the proof of Theorem i of[9] we obtain subsequences Ynk, x of Yn'