8/2/2019 Fixed Point Therorems in Metric Spaces

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29 2 ARCH. MATH,

F i x P o i n t T h e o r e m s i n M e t r i c S p a c e sB y

TU DO R ZA_~IFIRESCU

I n t h i s p a p e r r e s u l t s s im i l a r t o t h e w e l l -k n o w n c o n t r a c t i o n t h e o r e m s o f B A ~A C Ha r e o b t a i n e d * ) . T h e s e c la s s ic a l t h e o r e m s , o u r m e n t i o n e d s i m i la r r e s u l ts a n d t h e o r e m so f K ~ N . ~ , E D E L S T E ~ a n d S r ~ G H a r e al l, i n f a c t , c o r o ll a ri e s o f a fe w m o r e g e n e r a lt h e o r e m s , a s w e s h a l l s e e .

T h e o r e m 1 . Le t M be a co~ p le te me tr ic space , ~., 8 , 7 rea l nu m ber s wi th ~ . < 1 , ~ < ~-,7 < ~,~ an d / : M --+ M a func t ion such tha t fo r each coup le o f d i f f e ren t po in ts x , y e M ,a t l e a s t o n e o f t h e f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d :

1) d ( f ( x ) , f ( y ) ) ~ ~ . d ( x, y ) * * ) ,2 ) d ( f ( x ) , / ( y ) ) g f l ( d ( x , / ( x ) ) + d ( y , / ( y ) ) ) ,3 ) d ( / ( x ) , ] ( y ) ) ~ 7 ( d ( x , / ( y ) ) d ( y , / ( x ) ) ) .

T h e n f h a s a u n i q u e f i x e d p o i n t .P r o o f . C o n s i de r t h e n u m b e r

6 = m a x a , - 1 - f l , 1 ?, "O b v i o u s l y , d < 1 .

N o w , c h o o s e x 0 e M a r b i t r a r i l y a n d f i x a n i n t e g e r n u m b e r n > = O . T a k e x = f n (x 0)a n d y = ]n+l (xo). S u p p o s e x . y ; o t h e r w i s e x is a f i xe d p o i n t o f / . I f f o r t h e s e t w op o i n t s c o n d i t i o n 1 ) i s s a t is f ie d , t h e n

d (1~+1 (xo), f~+ 2 (xo)) =< ~d (f~ (xo), 1~+1 (xo )).I f f o r x , y , c o n d i t i o n 2 ) i s v e r i fi e d , t h e n

d (fn- ,1 (xo), fn+ 2 (xo)) ~ f l (d ( fn (xo) , fn+ l (xo)) q- d ( fn+l (xo), fn+2 ( x o ) ) ) ,w h i c h i m p l i e s

( / n + l ( x o ) , / n + 2 (xo)) ~ ~ d ( fn (xo) ,/n+~ (xo)) ~ 5 d ( /n ( x o ) , /n + 1 (xo)) 9I n c a s e c o n d i t i o n 3 ) i s sa t is f ie d ,

d (/'+~ (xo),/,~+2 (xo)) =< ~ d (/~ (xo),/~+ " (xo)) =