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| Some Common Fixed Point Theorems of Commuting Maps |
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Citation： |
Ding Xieping.Some Common Fixed Point Theorems of Commuting Maps[J].Chinese Annals of Mathematics B,1982,3(5):631~638 |
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Net amount： 802 |
Authors： |
Ding Xieping; |
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| Abstract： |
In this paper we obtain several new results on Common fixed point of commuting maps in L-space and metric space by introducing new contractive type conditions.
Our main results are the following:
Theorem 2. Let f, g be continuous self-mappings of a separated L-space (X, \rightarrow) which is d-complete for some semi-metric d on X. Then f and g have a common fixed point in X if and only if there exist continuous mappings T_1: X\rightarrow g^t_2(X) and $T_2:X\rightarrow f^t_1(X)$ such that $T_1f=fT_1,T_2g=gT_2$ and for all x,y \in X
$d(T_1^px,T_2^py)\leq \Phi(d(f^t_1x,g^t_2y),d(f^t_1,T_1^px),d(g^t_2y,T_2^qy)),$ where t_1, t_2, p, q\in N and $\Phi:R__^3 \rightarrow R_+$ which is nondecreasing in each coordinate variable and satisfy $\phi(t)=\Phi(t,t,t),\sum\limits_{n=1}^\infty(\phi^n(t)<\infty,\forall t>0$, . Indeed,each of pairs (T_1,f(and (T_2, g)have a u-nique common fixed point and these two points coincide.
Theorem 3. Let f, g be continuous self-mappings of a L-spaces (X, \rightarrow), T_1, T_2 be any self-mappings of X such that T_1(X)\subset g^t_2(X), T_2(X)\subset f^t_2(X),T_1f=fT_1,T_2g=gT_2,where t_1,t_2\in N.suppose(X,\rightarrow) is d-complete for some continuous demi-metric d.If there exist p,q\in N and the function \Phi:R_+^3\rightarrow R_+ Satisfying the supposition in Theorem 2 such that for all x,y \in X
$d(Ux,Vy)\leq \Phi(d(Sx,Ry),d(Sx,Ux),d(Ry,Vy))$
where U-T_1^p,V-T_2^\alpha,S=f^t_1 and R=g^t_2.Then each of pairs (T_1,f) and (T_2,g) have a unique common fixed point and these two points coincide.
Theorem 5. Let f, g be self-mappings of a complete metric space (X, d). For
some fixed m, k\in N, f^m and g^k are continuous, suppose {T_n}_{n\in N} a sequence of selfmappings of X such that T_n( f^m-l (X) \cap g^k-1(X)) \subset f(f^m-1(X) \cap g^k-1(X))\capg(f^m-1(X)\\cap g^k-1(X)),T_nf=fT_n,T_ng-gT_n,\forall n \in N.
If there exist an upper semi-continuous function \Phi:R_+4\rightarrowR_+ which is nondecreasing in oo each coordinate variable such that \phi(t)=\phi(t,t,t,t) satisfies \sum\limits_{n=0}^\infty \phi^n(t)<\infty,\forall t>0 and such that for all x, y\in X ,
i, j\in N,i \ne j,
d(T_ix, T_jy)\leq\Phi(d(fx, gy), d(fx, T_ix), d(gy, T_jy),
1/2[ d(fx, T_jy) +d(gy, T_ix)] .
Then each of pairs ({T_n}n\in N,f) and ({T_n}_n\in N,g) have a unique common fixed point
and these two points ooinoide.
some important rCT’^a of [3—8, 10, 11] are the speoial cases of our results. |
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