凸度量空间中非扩张映象的不动点定理
Fixed Point Theorem of Nonexpansive Mappings in Convex Metric Spaces
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摘要: 设X是一凸度量空间,并且它的每一直径趋于零的非空闭子集的逆减序列具有非空交的性质.本文证明了,如果X的非空闭子集K的自映象T满足不等式:d(Tx,Ty)≤ad(x,y)+b{d(x,Tx)+d(y,Ty)}+c{d(x,Ty)+d(y,Tx)},∀x,y∈K其中0≤a<1,b≥0,c≥0,使得a+c≠0且a+2b+3c≤1.则T在K中存在唯一不动点.Abstract: Let X be a convex metric space with the property that every decreasing sequence of nonenply dosed subsets of X with diameters tending to has menemptyintersection. This paper proved that if T is a mapping of a elosed conver nonempty subset K of X into itself satisfying the inequality:d(Tx,Ty)≤ad(x,y)+b{d(x,Tx)+d(y,Ty)}+c{d(x,Ty)+d(y,Tx)},∀x,y∈Kfor all x,y in K,where 0≤a<1,b≥0,c≥0,a+c≠0 and a+2b+3c≤1, then T has a unique fixed point in K.
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[1] 张石生,《不动点理论及应用》,重庆出版社(1984). [2] Takahashi,W.,A convexity in metric space and nonexpansive mappings,I.Kodai Math.Sem.Rep.,22(1970),142-149. [3] Guay,M.D.,K.L.Singh and J.H.M.Whitfield,Fixed point theorems for nonexpansive mappings in convex metric space,Proceedings,Conference on Nonlinear Analysis(Ed.S.P.singh and J.H.Burry)Marcel Dekker,Inc.,New York,80(1982),179-189. [4] Beg,Ismat and Akbar Azam,(Quaid-i-Azam University,Pakistan)Fixed point theorems for Kannan mappings,Abstracts of International Congress of Mathematicians,Berkeley,California,U.S.A.(1986),3-11. [5] Kirk,W.A.,A fixed point theorem for mappings which do not increase distances,Amer.Math.Monthly,72(1965),1004-1006. -
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