FIXED POINTS AND CONTINUITY FOR MULTIVALUED MAPPINGS

Many of the contractive definitions do not require continuity of the map. How- ever, in a previous paper the second author has shown in (1) that, in most cases, the function is continuous at a fixed point. In this paper we show that the same behavior is exhibited for many multivalued mappings.

In Proposition 2, we obtain the same conclusions by replacing CL(X) with B(X), the set of all nonempty bounded subsets of X, and replacing D by ,(A,B), where ,(A,B) := sup{d(a,b): a E A,b B}.We can also replace CL(X) with CB(X), the collection of nonempty closed, bounded subsets of X, and replace D with H, the Hausdorff metric.
A similar proof show that G is continuous at z. THEOREM 7. Let F, G :X B(X), X bounded, F continuous, F commutes with G, and satisfying ,(Ft'Gt'x,Gy) <_ cmax{$(F G x,a y),,5(F G x,F a'x),,5(y,Gy) 0 <_ r,r,s,s <_ p;i 0,1} (7) for all x, y in X, 0 < c < 1, p a fixed positive integer Then F and G have a unique common fixed point z, Fz Gz {z}, and G is continuous at z.
A similar argument shows that the assumption that F and J are continuous leads to the continuity of I and J at z.The special case of Theorem 9 in which I is the identity map on X yields the result in Fisher [12].
THEOREM 10.Let F, G" X --, B(X), I, d," X --, X satisfying 6(Fx,Gy) < cmax{d(Ix,Jy),6(Ix,Fx),*(Jy,Gy)} (10) for all x,V in X,0 _< c < 1.If F commutes with I and G commutes with J, G(X) C_ I(X),F(X) C__ J(X) and, if I or J is continuous, then F, G,I, and J have a unique common fixed point z.Further, Fz Gz {z}, and z is the unique common fixed point of F, G, I, and J. Further, the continuity of I implies that F is continuous at z, and the continuity of J implies that G is continuous at z.
The assumption that J is continuous leads to the continuity of G at z.
The special case of Theorem 10 with I J Ix yields the result of Fisher [10], and the continuity of both F and G at z. THEOREM 11.Let F, G X -B(X),I, J,: X -, X satisfying 6(F"x,Gy) <_ cmax{(Frx,Gy),6(Frx,y) 0 <_ r < p} (II) for all x, y in X, 0 _< c < 1, p a fixed positive integer.If F also maps B(X) into itself, then F and G have a unique common fixed point z.Further, z is the unique fixed point of F and G, Fz Gz {z}, and G is continuous at z.The existence and uniqueness of z come from Theorem 2 in Fisher [14].To prove the continuity of F, let {y.} C X, y,, z, and set x z, y y, in (11) to get 6(F p z ,Gy.<_ c max {6(F z,Gy.), 6(F z,y.)" 0 < r <_ p}; 6(z,Gy,) <_ cmax{6(z,Gy,,), 6(z,yn)}, which implies that 6(z,Gy,) <_ c 6(z,y,) -0 as n --, oo, and G is continuous at z.
A similar calculation verifies that G is continuous at z.We now establish continuity for multivalued mappings with metric defined by the Hausdorff metric.THEOREM 12. Let T X CB(X) satisfying H(Tx,Ty) < a(x,y)D(x,Tx) + a'(x,y)D(y,Ty) + b(x,y)D(x,Ty)+ b'(x,y)D(y,Tx) + c(x,y)d(x,y) (12) for all x, y in X, a, a', b, b', c X x X --R+ and (a + a' + b + b' + c)(x,y) < 1 for all x, y in X.
If limsup(a +a' +b+b' +c)(x,y) < 1, d(,)o then for each X in X there exists a sequence of iterates converging to a fixed point z of T, and T is continuous at z.
COROLLARY 1.Let f be the identity map on X.Then, under the hypotheses of Theorem 13, T has a fixed point z, and T is continuous at z.
The fixed point portion of Corollary 1 is essentially due to Ciric [17].The theorem of Ciric also contains a result of Reich [18] as a special ease.Kaneko [19] proves the Cirie result under the weaker conditions that X be a reflexive space and the range of T is the family of all nonempty weakly compact subsets of X.He is apparently unaware that the two standard definitions of the Hausdorff metric are equivalent.
Also S and T are continuous at z.
A similar argument verifies that F2 is continuous at z. THEOREM 20.Let X be a complete metric space, F, :X --C(X).Suppose that there exists a function satisfying the conditions of Theorem 19 and such that H(Fix,Fjy) _ {D(x,Fix),D(y,Fjy),D(x,Fy),D(y,Fix), d(x,y)} (20) for each x, y in X, for each i, j, :/: j.Then {F, } has a common fixed point, and each of the Fi is continuous at this fixed point.
The existence of a common fixed point is Theorem 2.5 of Guay et al [28].The continuity is proved in the same way as in Theorem 19.
Taking the limit as n oo we obtain lim, Hx(Fy,,FI z) O, and F is continuous at z. Similarly, Fz is continuous at z.
The result in Mishra and Singh [30] is a special case of Theorem 3.1 of Mishra [29].
THEOREM 22.Let X be a reflexive Banach space, K a nonempty closed bounded convex subset of X.Let T be a mapping of K into the family of nonempty weakly compact convex subsets of K satisfying H(Tx,Ty) < (max{D(x,Tx), D(y,Ty)} (22) for each x, y in X, where 4: [0,oo) -, [0,oo), nondecreasing, fight continuous, such that (t) < for each > 0. Then there exists a nonempty subset M of K such that Tx M for each x 6 M.Moreover, T is continuous at each point of M. The fact that a subset M exists with the stated properties is Theorem 1 of Kaneko [31].To prove the continuity of T, let z 6 M. Let {y.} C X,y. --, z, and set x y.,y z in (22) to get H(Ty.,Tz)<_ (max{D(y.,Ty.),D(z,Tz)}).