COMMON FIXED POINT THEOREMS AND APPLICATIONS

The purpose of this paper is to discuss the existence of common fixed points for mappings in general quasi-metric spaces. As applications, some common fixed point theorems for mappings in probabilistic quasi-metric spaces are given. The results presented in this paper generalize some recent results.


INTRODUCTION
In this paper, we show the existence of common fixed points for commuting mappings in general quasi-metric spaces.As applications, we give some fixed point theorems for commuting mappings in probabilistic quasi-metric spaces.Our main results generalize and improve some recent results in 1], [4], [] and [6].Let (G, _<, < be a partial order set satisfying the following conditions: (G-l) 0 is the minimal element in G, i.e., 0 < u for all z G, (G-2) for any , v G, sup{u, v} exists and belongs to G, (G-3) for any u /G, , (G-4) for any , v, w G, u < w and v < w = sup{z, v} < w, and < v, v _ w = u < w.DEFINITION 1.1 Let X be a nonempty set.(X, r) is called a general quasi-metric space if r-X x X --, G (G, _, < satisfies the following conditions: (QM-1) r(x, !/) 0 if and only ifx !/, (QM-2) r(x,l) It follows from the definition that every general quasi-metric space includes a metric space as its special case.DEFINITION 1.2 Let X be a nonempty set and let T be a self-mapping of X.A point x E X is called a periodic point of T if there exists a positive integer k such that Tkx x.The least positive integer satisfying this condition is called the periodic index of x.DEFINITION 1.3 A mapping F (-oo, oo) [0, oo) is called a distribution function if it is nondecreasing and left-continuous with infF(t) 0 and sup F(t) 1.In what follows we always denote by F(T) the set of all fixed points of T, P(T) the set of all periodic points of T and /) the set of all distribution functions, respectively, and let l) + {F 1): F(t) 0 for all t < 0}.
DEFINITION 1.4 (X, .') is called a probabilistic quasi-metric space if X is a nonempty set, " is a mapping from X x X into 1)+ (we shall denote the distribution function .'(x,y) by F=,u(t which represent the value of F=,u at t oo, oo)) satisfying the following conditions: (PQM-I) F=,u(0) 0, (PQM-2) F=,u(t) I for all t > 0 if and only if x y, (PQM-3) F,(t) Fux(t for all oo, oo).DEFINrrION 1.5 (X, ') is called a probabilistic metric space if (X, ') is a probabilistic quasi- metric space and the following condition is satisfied: (PQM-4) if F=,(tl) 1 and F.z( 1, then F,z(tl + z) 1.
For more details on probabilistic metric spaces, refer to [3] and [7].

2.
COMMON FIXED POINT TIOREMS Now, we give our main theorems.TitEOREM 2.1 Let (X, r) b a general quasi-metric space and let ' and T be two commuting self-mappings ofX.Iffor any x X and any two positive integers n, q _> 2 with Tix T:x, O < < j <_ n -1, (2.1) SexSJ'x, 0_<i'<._<q-1,,(T"S"=, ST=) < max sup r(TJz, S =), sup r(T:=, x), sup r(S =, x) 1 <_j<_n, 1 <_] <_q 1 <_j<_n 1 <_] <_q (2.2) for 1, 2, n 1 and j 1, 2, q 1, then S and T have a common fixed point in X if and only if there exist integers m, n, p, q, m > n > 0, p > q > 0, and a point x X such that Tx SP= Tnx sqx (2.3) and either F(S) or F(T) 7 O.If this condition is satisfied, then either T"x or Sqx is a common fixed point of S and T. PROOF.Let x* E X be a common fixed point ofT, i.e., x* Sx* Tx*.Then (2.3) is true in case rn p l and n q O. Conversely, suppose that there exist a point x X and four integers rn, n, p, q, rn > n >_ 0, p > q > 0, such that (2.3) is satisfied.Without loss of generality, we can assume that x _ F(S) and m is the minimal integer satisfying Tx Tx, k > n.Putting y T"x and p rn-n, we have T y y and Pl is the minimal integer satisfying Tky y, k > 1.
It follows from (G-4) that sup {r(y,T:y)} < sup {r(T:y,y)}, l<j<p-1 l<j<_p-1 which is a contradiction.Therefore, y Tx is a fixed point of T. Further, since S and T are commuting, we have V T"x T"Sx ST"x Sy, i.e., y Tx is a common fixed point of S and T. In this case, when x E F(T), we have, by terchanging the role of S and T, that y Sqx is a common fixed point of S and T. This completes the proof On the other hand, by using Theorem 3 of [6], we have the following: THEOREM 2.2 Let (X, r) be a general quasi-metric space and let S and T be two commuting self-mappings of X. Assume that for any x, y E X, xy, there exists a positive integer p(x, y) such that r((ST)'u)x, (ST)'U)y) < sup{r(x, y), r(x, (ST)'')x), r(y, (ST)')y), r(x, (ST)")y), r(y, (ST) ''u) x)} Then S and T have a common fixed point in X if and only if there exists a periodic poim x X of ST with periodic index k such that for any u, v A {x, STx,...,(ST)k-x}, u v, there exist x', y' A, x y', satisfying the following conditions: (ST)e'z' u, (ST)' v and either F(S) Q P(ST) # or F(T) Q P(St) # @.If these conditions are satisfied, then the point x is the unique common fixed point of S and T.
PROOF.The necessity condition is obvious.
The sufficiency condition follows from Theorem 3 of [6] as follows: In Theorem 3 of [6], if we replace T by ST, we can conclude that ST has a unique fixed point x in X.Since S and T are commuting, Sx S(STx) ST(Sx) and so Sx is also a fixed point of ST.Uniqueness gives Sx x.Similarly, Tx x.This completes the proof The following is a special case of Theorem 2.2 by setting p(x, y) p(x): COROLLARY 2.3 Let (X, r) be a general quasi-metric space and let S and T be two commuting self-mappings of X. Assume that for any x X, there exists a positive integer p(x) such that every yEX, xy, r((ST)=)x, (ST)(=)y) < sup(r(x, y), r(x, (ST)=)x), r(y, (ST)K=)x), r(y, (ST)=)y), r(x, (ST)')y), r(y, (ST)=)x)} Then S and T has a common fixed point in X if and only if there exists a periodic point x E X of ST with periodic index k such that for any u, v E A {x, STx,...,(ST)k-lx}, u :fi v, there exist x', y' E A, x' =f= y', satisfying the following conditions: (ST) ()z' u, (ST)e)y v and either F(S) N P(ST) 0 or F(T) N P(ST) O.If these conditions are satisfied, then the point x is the unique common fixed point of S and T.
The following is obtained from Corollary 2.3 by setting p(x) p: COROLLARY 2.4 Let (X, r) be a general quasi-metric space and let S and T be two commuting self-mappings of X. Assume that there exists a positive integer p such that for any x, y X, xy, r((ST)'x, (ST)y) < sup{r(x, y), r(x, (ST)"x), r(y, (ST)y), r(x, (ST)y), r(y, (ST)x)} Then S and T have a common fixed point in X if and only if there exists a periodic point x X of ST and either F(S) f'l P(ST) t or F(T) P(ST) : .If this condition is satisfied, then the point x is the unique common fixed point of S and T. By setting p 1 in Corollary 2.4, we have the following: COROLLARY 2.5 Let (X, r) be a general quasi-metric space and let S and T be two commuting self-mappings of X. Assume that for any x, y X, x :fy, r(STx, STy) < sup[r(x, y), r(x, STx), r(y, STy), r(x, STy), r(y, STx)}.
Then S and T have a common fixed point in X if and only if there exists a periodic point x X of ST and either F(S) q P(ST) or F(T) P(ST) :/: .If this condition is satisfied, then the point x is the unique common fixed point of S and T. By using Theorem 5 of [6], we have the following: THEOREM 2.6 Let (X, r) be a general quasi-metric space and let S and T be two commuting self-mappings ofX.Assume that there exist positive integers p, q such that for any x, y, E X, x t y, r((ST)x, (ST)qy) < sup(v(x, y), r(x, (ST)"x), r(y, (ST)qy), r(x, (ST)qy), r(y, Then S and T have a fixed point in X if and only if there exists a periodic point x 6 X of ST with periodic index k which satisfies the following condition: where p =/hk + P2, q ql k + q2, 0 _< P2, q2 < k and Pl, ql are non-negative integers and either F(S) gl P(ST) or F(T) ; P(ST) -f}.If this condition is satisfied, then the point x is the unique common fixed point of S and T. PROOF.The necessity condition is obvious.
To prove converse, if we use Theorem 5 of [6] by replacing T with ST, then ST has a unique fixed point x in X.Therefore, employing the same argument as in the proof of Theorem 2.2, it follows that the point x is the unique fixed point of S and T. This completes the proof.
THEOREM 3.1 (Embedding Theorem) Let (X, ') be a probabilistic quasi-metric space.Then (X, .') is a general quasi-metric space, where G (D +, _<, < is the partial order set induced by the way as above. PROOF.Let r(x, y) Fx,y for all x, y e X.It is easy to see that r satisfies the conditions (QM- 2) and (QM-2) ofDefinition 1.1.
The following results are obtained from Theorems 2.1 2.6 and Theorem 3.1 immediately: THEOREM 3.2 Let (X,.T') be a probabilistic quasi-metric space and let S and T be two commuting self-mappings of X.If for any x X and positive integer n, q _> 2 with TixTTx, O<_i<j<_n-1, S'x Sx, O < < j <_ q- FT, s,x.s,z (T) > rffm ( rain Fr,: s,, (t) rain FT, (t) nfin Fs,,.x (t) ) 1 <_j<_n, 1 <_j' <_q '1 <_j<_n 1 <_3 <_q for 1, 2, n 1 and j 1, 2, q 1, then S and T have a common fixed point in X if and only if there exist integers m, p, q, m > n _> 0, p, q >_ 0, and a point x E X such that Tx Sx T"x and either F(S) : O or F(T) :/: .If this condition is satisfied, then either T"x or Sqx is a common fixed point of S and T. THEOREM 3.3 Let (X,Y') be a probabilistic quasi-metric space and let S and T be two commuting self-mappings of X.If there exist positive integers p, q such,that for any.x, y e X, x : y, and for all t > 0, F(sr),.(sr),()> min{F.(),F.(ST),(t), F.(ST),(), F.ST),(), F.(ST),() }, then S and T have a common fixed point in X if and only if there exists a periodic point x X of ST with periodic index k which satisfies the condition (2.5) in Theorem 2.6 and either F(S) P(ST) 7 1 or F(T) P(ST) 7k $.If this condition is satisfied, then the point x is the unique common fixed point of S and T.
The followin is a special case of Theorem 3.3 obtained by settin p q 1 COROLLARY 3.4 Let (X,.') be a probabilistic quasi-metric space and let S and T be two commuting self-mappings ofX.Iffor any x, y X, x y, and t > 0, FST,STu(t) > min{F,u(t), F,ST(t), Fu,ST(t), F,STu(t), Fu,sr(t)} then S and T have a common fixed point in X if and only if there exists a periodic point x G X of ST and either F(S) Iq P(ST) :fi 0 or F(T) f3 P(ST) $.If this condition is satisfied, then the point z is the unique common fixed point ofS and T. THEOREM 3.5 Let (X,.T) be a probabilistic quasi-metric space and let S and T be two commuting self-mappings of X.If for any x X, z # y, there exists a positive integer p(z, V) such that for all t > 0, F(sT).),(sr)z.%t > min{F.v(t),F,(ST)., t Fv.(ST).%t F.(sr),.% (t), F.(ST),,, (t) }, then S and T have a common fixed point in X if and only if there exists a periodic point x X of ST with periodic index k such that for any u, v A {x, (ST)x,..., (ST)k-I}, u v, there exist x , y A, x' ://: y, satisfying the following conditions: (St) (e'/)x' u, (ST)e'V')y v and either F(S) n P(ST) or F(T) fq P(ST) .If these conditions are satisfied, then the point x is the unique common fixed point of S and T. By setting p(x, y) p(x) in Theorem 3.5, we have the following: COROLLARY :}.6 Let (X,.T') be a probabilistic quasi-metric space and let S and T be two commuting self-mappings of X.If there exists a positive integer p(x) such that for every y X, x #: y, and for all t > 0, F(ST),)z,(ST)Z)v(t > min{F,v(t), Fz,ST),)z(t), then S and T have a common fixed point in X if and only if there exists a periodic point x X of ST with periodic index k such that for any u, v A {x, (ST)x (ST)-Ix}, u # v, there exist :d, y' e A, Z' # y, satisfying the following conditions: (:T)ez u, (ST)eV' v and either F(S) fq P(ST) -#-or F(T) f P(ST) #-t3.If these conditions are satisfied, then the point x is the unique common fixed point of T. REMARK.Theorems 3.3 3.6 include Theorems 5 in [4] and Theorems 3.3 3.6 in [5] as special cases.