COINCIDENCE POINT THEOREMS FOR MULTIVALUED MAPPINGS

Some new coincidence point and fixed point theorems for multivalued mappings in complete metric space are proved. The results presented in this paper enrich and extend the corresponding results in [5-16, 20-25, 29].


INTRODUUHON AND PRELIMINARIES.
In recent years, the existence and uniqueness of coincidence points and fixed points for commuting mappings, weakly commuting mappings and compatible mappings have been considered by several authors (see [2,3,6,8,[17][18][19][20][21][22][23][24][25][26][27][28]). The purpose of this paper is to study the existence of coincidence points and fixed point for multivalued mappings in complete metric space from different aspects. The results presented in this paper enrich and extend the corresponding results in [5-16, 20-25, 29]. Throughout this paper, letR [0, +oo)  it is easy to verify that p(.,.) is a metric onR z. Therefore (R2,p) is also a metric space, and it is bounded. Now we consider the following subsets of (R z,p): for all x, y in X, where 0 h < 1. Then there exists a point x. (E X such that fx. E Txo. As an improvement and generalization of Theorem 1, we have the following THEOREM 2. Let F: X CC(X), $, T: X CB(X) be three multivalued mappings such that S(X) U T(X) C F(X), F(X) is closed and H(Sx, Ty @(max l d(Fx,Fy), d(Fx,Sx), d(Fy, Ty ), 1/2(d(Fx, Ty + d(Fy,Sx ))[) for all x, y in X, where O: R R is an increasing function satisfying conditions (1.1) and (1.2). Then there exists a point z EX such that Fz NSz CI Tz # .
(2.2) Since T(X) C F(X), for y Tx C F(X), Sere exism a int X such Sat y F. is implies Sat we n find an y F Tx such at (2.2) holds.
If d(y, Y2) > 0, in view of condition (1.4), we know that :E-td(yl, Y2)) is convergent. It follows from (2.6) that Y_,d(y,, y, t) is convergent too. This implies that {y, } is a Cauchy sequence in X. Let it converge to some point y. in X. Since y, _ Fx, C F(X) and F(X) is closed, this shows that y. F(X). Hence there exists z X such that y. _Fz. By (2.1) and (2.3) we have aty.,S) afy., y./)+d(y./, Sz) (ii) Even if the mapping F in Theorem 2 is assumed to satisfy the condition "F(X) is closed", Theorem 2 still weakens the continuity and compatibility conditions on T in Theorem I. This can be seen from the following Example: EXAMPLE 2. Let X R and f and g be two functions from R" into R defined by g(x) x(x + 1)-1.
It is easy to see that f(X) is closed, f and g are continuous, but they are not compatible (see [8, Example
PROOF. For the sake of convenience we prove the conclusions of Corollary only for the case of i= and j=2. By Theorem 2, there exists an z X such that z Tz T z. How we prove that the fixed point sets of T1 and T are equal to each other. In fact, if u is a fixed point of T, i.e. u Tzu, then we have d(u,T_u)'H (Tlu,T_u) m. d (.,I, d(.,rI, (.,r.I, g(d(.,rl/(d(.,r,.llt -(max{0, 0, d(u,T2u ), 1/2d(u,T2u)}) q)(d(u, T u )). REMARK 3. If all the mapping T, 1, 2 in Corollary I are single-valued, then T, 1, 2 have a unique common fixed point in X.
In fact, let u, v X be two common fixed points of T,  (t,t,t, at, bt) l(t), Vt O, a + b 3, a, b 1,2. THEOREM 3. Let F: X CC(X), $, T: X CB(X) be three multivalued mappings such that S(X) U T(X)(7. F(X), F(X) is closed and satisfies the following conditions: H (Sx, Ty)