A NEW SUZUKI TYPE COMMON COUPLED FIXED POINT RESULT FOR FOUR MAPS IN S b -METRIC SPACES

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1 JOURNAL OF INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) ISSN (o) wwwimviblorg /JOURNALS / JOURNAL Vol 8(208) 03-9 DOI: 0725/JIMVI8003R Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (o) ISSN X (p) A NEW SUZUKI TYPE COMMON COUPLED FIXED POINT RESULT FOR FOUR MAPS IN S b -METRIC SPACES K P R Rao E Taraka Ramudu and G N V Kishore Abstract In this paper we prove a Suzuki type unique common coupled fixed point theorem for two pairs of w-compatible mappings in S b -metric spaces We also furnish an example to support our main result Introduction In the year 2008 Suzuki [] generalized the Banach contraction principle [2] Theorem ([]) Let (X d) be a complete metric space and let T be a mapping on X Define a non-increasing function θ : [0 ) ( 2 ] by θ(r) = ( r)r 2 ( + r) Assume that there exists r [0 ) such that if 0 r ( 5 ) 2 if ( 5 ) 2 r if 2 2 r < 2 2 θ(r)d(x T x) d(x y) d(t x T y) rd(x y) for all x y X Then there exists a unique fixed point z of T Moreover for all x X lim n T n x = z 200 Mathematics Subject Classification 54H25 47H0 54E50 Key words and phrases Suzuki type S b -metric space w-compatible pairs S b -completeness coupled fixed points 03

2 04 KPRRAO ETARAKA RAMUDU AND GNVKISHORE Bhaskar and Lakshmikantham [4] introduced the notion of coupled fixed point and proved some coupled fixed point results Recently Sedghi et al [8] defined S b -metric spaces using the concept of S- metric spaces [9] The aim of this paper is to prove Suzuki type unique common coupled fixed point theorem for four mappings satisfying generalized contractive condition in S b -metric spaces Throughout this paper R R + and N denote the set of all real numbers non-negative real numbers and positive integers respectively First we recall some definitions lemmas and examples Definition ([9]) Let X be a non-empty set A S-metric on X is a function S : X 3 R + that satisfies the following conditions for each x y z a X (S) : 0 < S(x y z) for all x y z X with x y z (S2) : S(x y z) = 0 x = y = z (S3) : S(x y z) S(x x a) + S(y y a) + S(z z a) for all x y z a X Then the pair (X S) is called a S-metric space Definition 2 ([8]) Let X be a non-empty set and b be given real number Suppose that a mapping S b : X 3 R + be a function satisfying the following properties : (S b ) 0 < S b (x y z) for all x y z X with x y z (S b 2) S b (x y z) = 0 x = y = z (S b 3) S b (x y z) b(s b (x x a) + S b (y y a) + S b (z z a)) for all x y z a X Then the function S b is called a S b -metric on X and the pair (X S b ) is called a S b -metric space Remark ([8]) It should be noted that the class of S b -metric spaces is effectively larger than that of S-metric spaces Indeed each S-metric space is a S b -metric space with b = Following example shows that a S b -metric on X need not be a S-metric on X Example ([8]) Let (X S) be a S-metric space and S (x y z) = S(x y z) p where p > is a real number Note that S is a S b -metric with b = 2 2(p ) Also (X S ) is not necessarily a S-metric space Definition 3 ([8]) Let (X S b ) be a S b -metric space Then for x X and r > 0 we define the open ball B Sb (x r) and closed ball B Sb [x r] with center x and radius r as follows respectively: B Sb (x r) B Sb [x r] = y X : S b (y y x) < r = y X : S b (y y x) r Lemma ([8]) In a S b -metric space we have S b (x x y) b S b (y y x)

3 A NEW SUZUKI TYPE UNIQUE COMMON COUPLED FIXED POINT THEOREM 05 and S b (y y x) b S b (x x y) Lemma 2 ([8]) In a S b -metric space we have S b (x x z) 2b S b (x x y) + b 2 S b (y y z) Definition 4 ([8]) If (X S b ) be a S b -metric space A sequence x n in X is said to be: () S b -Cauchy sequence if for each ϵ > 0 there exists n 0 N such that S b (x n x n x m ) < ϵ for each m n n 0 (2) S b -convergent to a point x X if for each ϵ > 0 there exists a positive integer n 0 such that S b (x n x n x) < ϵ or S b (x x x n ) < ϵ for all n n 0 and we denote by lim n x n = x Definition 5 ([8]) A S b -metric space (X S b ) is called complete if every S b -Cauchy sequence is S b -convergent in X Lemma 3 ([8]) If (X S b ) be a S b -metric space with b and suppose that x n is a S b -convergent to x then we have (i) 2b S b(y x x) lim infs b(y y x n ) lim sups b(y y x n ) 2b S b (y y x) n n and (ii) b 2 S b (x x y) for all y X lim infs b(x n x n y) n lim sups b(x n x n y) b 2 S b (x x y) n In particular if x = y then we have lim n S b(x n x n y) = 0 Definition 6 ([4]) Let X be a non-empty set An element (x y) X X is called a coupled fixed point of a mapping F : X X X if x = F (x y) and y = F (y x) Definition 7 ([5]) Let X be a non-empty set An element (x y) X X is called (i) a coupled coincident point of mappings F : X X X and f : X X if fx = F (x y) and fy = F (y x) (ii) a common coupled fixed point of mappings F : X X X and f : X X if x = fx = F (x y) and y = fy = F (y x) Now we give our main result 2 Main Result Let Φ denote the class of all functions ϕ : R + R + such that ϕ is nondecreasing continuous ϕ(t) < t 4b for all t > 0 and ϕ(0) = 0 4 Theorem 2 Let (X S b ) be a S b -metric space Suppose that A B : X X X and P Q : X X be satisfying (2) A(X X) Q(X) B(X X) P (X)

4 06 KPRRAO ETARAKA RAMUDU AND GNVKISHORE (22) A P and B Q are w-compatible pairs (23) One of P (X) or Q(X) is S b -complete subspace of X S b (A(x y) A(x y) P x) S b (B(u v) B(u v) Qu) (24) 3 implies that S b (A(y x) A(y x) P y) S b (B(v u) B(v u) Qv) 2b 5 S b (A(x y) A(x y) B(u v)) Sb (P x P x Qu) S b (P y P y Qv) S b (P x P x Qu) S b (P y P y Qv) S b (A(x y) A(x y) P x) S b (A(y x) A(y x) P y) S b (B(u v) B(u v) Qu) S b (B(v u) B(v u) Qv) S b (A(xy)A(xy)Qu) S b (B(uv)B(uv)P x) +S b (P xp xqu) S b (A(yx)A(yx)Qv) S b (B(vu)B(vu)P y) +S b (P yp yqv) for all x y u v X ϕ Φ Then A B P and Q have a unique common coupled fixed point in X X Proof Let x 0 y 0 X From (2) we can construct the sequences x n y n z n and w n such that A(x 2n y 2n ) = Qx 2n+ = z 2n A(y 2n x 2n ) = Qy 2n+ = w 2n B(x 2n+ y 2n+ ) = P x 2n+2 = z 2n+ B(y 2n+ x 2n+ ) = P y 2n+2 = w 2n+ n = 0 2 Case (i) Suppose z 2m = z 2m+ and w 2m = w 2m+ for some m Assume that z 2m+ z 2m+2 or w 2m+ w 2m+2 Since 3 S b (A(x 2m+2 y 2m+2 ) A(x 2m+2 y 2m+2 ) P x 2m+2 ) ]S b (B(x 2m+ y 2m+ ) B(x 2m+ y 2m+ ) Qx 2m+ ) S b (A(y 2m+2 x 2m+2 ) A(y 2m+2 x 2m+2 ) P y 2m+2 ) S b (B(y 2m+ x 2m+ ) B(y 2m+ x 2m+ ) Qy 2m+ ) S b (P x 2m+2 P x 2m+2 Qx 2m+ ) S b (P y 2m+2 P y 2m+2 Qy 2m+ ) From (24) we have

5 A NEW SUZUKI TYPE UNIQUE COMMON COUPLED FIXED POINT THEOREM 07 S b (z 2m+2 z 2m+2 z 2m+ ) 2b 5 S b (A(x 2m+2 y 2m+2 ) A(x 2m+2 y 2m+2 ) B(x 2m+ y 2m+ )) S b (z 2m+ z 2m+ z 2m ) S b (w 2m+ w 2m+ w 2m ) S b (z 2m+2 z 2m+2 z 2m+ ) S b (w 2m+2 w 2m+2 w 2m+ ) S b (z 2m+ z 2m+ z 2m ) S b (w 2m+ w 2m+ w 2m ) S b (z 2m+2 z 2m+2 z 2m+ ) S b (z 2m+ z 2m+ z 2m ) +S b (z 2m+ z 2m+ z 2m ) S b (w 2m+2 w 2m+2 w 2m+ ) S b (w 2m+ w 2m+ w 2m ) +S b (w 2m+ w 2m+ w 2m ) = ϕ ( S b (z 2m+2 z 2m+2 z 2m+ ) S b (w 2m+2 w 2m+2 w 2m+ ) ) Similarly we can prove S b (w 2m+2 w 2m+2 w 2m+ ) ( Sb (z 2m+2 z 2m+2 z 2m+ ) S b (w 2m+2 w 2m+2 w 2m+ ) ) Thus Sb (z 2m+2 z 2m+2 z 2m+ ) S b (w 2m+2 w 2m+2 w 2m+ ) ( Sb (z 2m+2 z 2m+2 z 2m+ ) S b (w 2m+2 w 2m+2 w 2m+ ) ) It follows that z 2m+2 = z 2m+ and w 2m+2 = w 2m+ Continuing in this process we can conclude that z 2m+k = z 2m and w 2m+k = w 2m for all k 0 It follows that z n and w n are Cauchy sequences Case (ii) Assume that z 2n z 2n+ and w 2n w 2n+ for all n Put S n = S b (z n+ z n+ z n ) S b (w n+ w n+ w n ) Since 3 S b (A(x 2n+2 y 2n+2 ) A(x 2n+2 y 2n+2 ) P x 2n+2 ) S b (B(x 2n+ y 2n+ ) B(x 2n+ y 2n+ ) Qx 2n+ ) S b (A(y 2n+2 x 2n+2 ) A(y 2n+2 x 2n+2 ) P y 2n+2 ) S b (B(y 2n+ x 2n+ ) B(y 2n+ x 2n+ ) Qy 2n+ ) S b (P x 2n+2 P x 2n+2 Qx 2n+ ) S b (P y 2n+2 P y 2n+2 Qy 2n+ ) From (24) we have S b (z 2n+2 z 2n+2 z 2n+ )

6 08 KPRRAO ETARAKA RAMUDU AND GNVKISHORE 2b 5 S b (A(x 2n+2 y 2n+2 ) A(x 2n+2 y 2n+2 ) B(x 2n+ y 2n+ )) S b (z 2n+ z 2n+ z 2n ) S b (w 2n+ w 2n+ w 2n ) S b (z 2n+2 z 2n+2 z 2n+ ) S b (w 2n+2 w 2n+2 w 2n+ ) S b (z 2n+ z 2n+ z 2n ) S b (w 2n+ w 2n+ w 2n ) S b (z 2n+2 z 2n+2 z 2n ) S b (z 2n+ z 2n+ z 2n+ ) +S b (z 2n+ z 2n+ z 2n ) S b (w 2n+2 w 2n+2 w 2n ) S b (w 2n+ w 2n+ w 2n+ ) +S b (w 2n+ w 2n+ w 2n ) ( ) Sb (z 2n+ z 2n+ z 2n ) S b (z 2n+2 z 2n+2 z 2n+ ) = ϕ S b (w 2n+ w 2n+ w 2n ) S b (w 2n+2 w 2n+2 w 2n+ ) = ϕ ( ) S 2n+ S 2n Similarly we can prove that S b (w 2n+2 w 2n+2 w 2n+ ) ( S 2n+ S 2n ) Thus S 2n+ (S 2n S 2n+ ) If S 2n+ is imum then we get contradiction so that S 2n is imum Thus (2) S 2n+ (S 2n ) < S 2n Similarly we can conclude that S 2n < S 2n It is clear that S n is a non-increasing sequence of non-negative real numbers and must converge to a real number say r 0 Suppose r > 0 Letting n in (2) we have r (r) < r It is contradiction Hence r = 0 Thus (22) lim n S(z n+ z n+ z n ) = 0 and (23) lim n S(w n+ w n+ w n ) = 0 Now we prove that z 2n and w 2n are Cauchy sequences in (X S b ) On contrary we suppose that z 2n or w 2n is not Cauchy Then there exist ϵ > 0 and monotonically increasing sequence of natural numbers 2m k and 2n k such that n k > m k (24) S b (z 2mk z 2mk z 2nk ) S b (w 2mk w 2mk w 2nk ) ϵ and (25) S b (z 2mk z 2mk z 2nk 2 ) S b (w 2mk w 2mk w 2nk 2 ) < ϵ

7 A NEW SUZUKI TYPE UNIQUE COMMON COUPLED FIXED POINT THEOREM 09 From (24) and (25) we have (26) ϵ S b (z 2mk z 2mk z 2nk ) S b (w 2mk w 2mk w 2nk ) 2b S b (z 2mk z 2mk z 2mk +2) S b (w 2mk w 2mk w 2mk +2) +b S b (z 2nk z 2nk z 2mk +2) S b (w 2nk w 2nk w 2mk +2) 2b (2b S b (z 2mk z 2mk z 2mk +) S b (w 2mk w 2mk w 2mk +)) +2b (b S b (z 2mk +2 z 2mk +2 z 2mk +) S b (w 2mk +2 w 2mk +2 w 2mk +)) +b (2b S b (z 2nk z 2nk z 2nk +) S b (w 2nk w 2nk w 2nk +)) +b (b S b (z 2mk +2 z 2mk +2 z 2nk +) S b (w 2mk +2 w 2mk +2 w 2nk +)) 4b 3 S b (z 2mk + z 2mk + z 2mk ) S b (w 2mk + w 2mk + w 2mk ) +2b 2 S b (z 2mk +2 z 2mk +2 z 2mk +) S b (w 2mk +2 w 2mk +2 w 2mk +) +2b 3 S b (z 2nk + z 2nk z 2nk ) S b (w 2nk + w 2nk w 2nk ) +b 2 S b (z 2mk +2 z 2mk +2 z 2nk +) S b (w 2mk +2 w 2mk +2 w 2nk +) Now first we claim that (27) 3 S b (A(x 2mk +2 y 2mk +2) A(x 2mk +2 y 2mk +2) P x 2mk +2) S b (B(x 2nk + y 2nk +) B(x 2nk + y 2nk +) Qx 2nk +) S b (A(y 2mk +2 x 2mk +2) A(y 2mk +2 x 2mk +2) P y 2mk +2) S b (B(y 2nk + x 2nk +) B(y 2nk + x 2nk +) Qy 2nk +) Sb (P x 2mk +2 P x 2mk +2 Qx 2nk +) S b (P y 2mk +2 P y 2mk +2 Qy 2nk +) On contrary suppose that 3 S b (A(x 2mk +2 y 2mk +2) A(x 2mk +2 y 2mk +2) P x 2mk +2) S b (B(x 2nk + y 2nk +) B(x 2nk + y 2nk +) Qx 2nk +) S b (A(y 2mk +2 x 2mk +2) A(y 2mk +2 x 2mk +2) P y 2mk +2) S b (B(y 2nk + x 2nk +) B(y 2nk + x 2nk +) Qy 2nk +) > Sb (P x 2mk +2 P x 2mk +2 Qx 2nk +) S b (P y 2mk +2 P y 2mk +2 Qy 2nk +)

8 0 KPRRAO ETARAKA RAMUDU AND GNVKISHORE Now from (24) we have ϵ S b (z 2mk z 2mk z 2nk ) S b (w 2mk w 2mk w 2nk ) 2b S b (z 2mk z 2mk z 2mk +) S b (w 2mk w 2mk w 2mk +) +b S b (z 2nk z 2nk z 2mk +) S b (w 2nk w 2nk w 2mk +) 2b 2 S b (z 2mk + z 2mk + z 2mk ) S b (w 2mk + w 2mk + w 2mk ) +b 2 S b (z 2mk + z 2mk + z 2nk ) S b (w 2mk + w 2mk + w 2nk ) < 2b 2 S b (z 2mk + z 2mk + z 2mk ) S b (w 2mk + w 2mk + w 2mk ) S b (z 2mk +2 z 2mk +2 z 2mk +) +b 2 3 S b (w 2mk +2 w 2mk +2 w 2mk +) S b (z 2nk + z 2nk + z 2nk ) S b (w 2nk + w 2nk + w 2nk ) Letting k we have ϵ 0 It is a contradiction Hence (27) holds Now from (24) we have 2b 5 S b (z 2mk +2 z 2mk +2 z 2mk +) S b (z 2mk + z 2mk + z 2nk ) S b (w 2mk + w 2mk + w 2nk ) S b (z 2mk +2 z 2mk +2 z 2mk +) S b (w 2mk +2 w 2mk +2 w 2mk +) S b (z 2nk + z 2nk + z 2nk ) S b (w 2nk + w 2nk + w 2nk ) Similarly S b (z 2mk +2z 2mk +2z 2nk ) S b (z 2nk +z 2nk +z 2mk +) +S b (z 2mk +z 2mk +z 2nk ) S b (w 2mk +2w 2mk +2w 2nk ) S b (w 2nk +w 2nk +w 2mk +) +S b (w 2mk +w 2mk +w 2nk ) 2b 5 S b (w 2mk +2 w 2mk +2 w 2mk +) S b (z 2mk + z 2mk + z 2nk ) S b (w 2mk + w 2mk + w 2nk ) S b (z 2mk +2 z 2mk +2 z 2mk +) S b (w 2mk +2 w 2mk +2 w 2mk +) S b (z 2nk + z 2nk + z 2nk ) S b (w 2nk + w 2nk + w 2nk ) Thus S b (z 2mk +2z 2mk +2z 2nk ) S b (z 2nk +z 2nk +z 2mk +) +S b (z 2mk +z 2mk +z 2nk ) S b (w 2mk +2w 2mk +2w 2nk ) S b (w 2nk +w 2nk +w 2mk +) +S b (w 2mk +w 2mk +w 2nk ) (28) 2b 5 S b (z 2mk +2 z 2mk +2 z 2mk +) S b (w 2mk +2 w 2mk +2 w 2mk +) S b (z 2mk + z 2mk + z 2nk ) S b (w 2mk + w 2mk + w 2nk ) S b (z 2mk +2 z 2mk +2 z 2mk +) S b (w 2mk +2 w 2mk +2 w 2mk +) S b (z 2nk + z 2nk + z 2nk ) S b (w 2nk + w 2nk + w 2nk ) S b (z 2mk +2z 2mk +2z 2nk ) S b (z 2nk +z 2nk +z 2mk +) +S b (z 2mk +z 2mk +z 2nk ) S b (w 2mk +2w 2mk +2w 2nk ) S b (w 2nk +w 2nk +w 2mk +) +S b (w 2mk +w 2mk +w 2nk )

9 A NEW SUZUKI TYPE UNIQUE COMMON COUPLED FIXED POINT THEOREM But S b (z 2mk + z 2mk + z 2nk ) S b (w 2mk + w 2mk + w 2nk ) 2b S b (z 2mk + z 2mk + z 2mk ) S b (w 2mk + w 2mk + w 2mk ) +b S b (z 2nk z 2nk z 2mk ) S b (w 2nk w 2nk w 2mk ) 2b S b (z 2mk + z 2mk + z 2mk ) S b (w 2mk + w 2mk + w 2mk ) +b 2 S b (z 2mk z 2mk z 2nk ) S b (w 2mk w 2mk w 2nk ) 2b S b (z 2mk + z 2mk + z 2mk ) S b (w 2mk + w 2mk + w 2mk ) +b 2 ( 2b S b (z 2mk z 2mk z 2nk 2 ) S b (w 2mk w 2mk w 2nk 2 ) ) +b 2 ( b S b (z 2nk z 2nk z 2nk 2 ) S b (w 2nk w 2nk w 2nk 2 ) ) < 2b S b (z 2mk + z 2mk + z 2mk ) S b (w 2mk + w 2mk + w 2mk ) +2b 3 ϵ + b 3 (2b S b (z 2nk z 2nk z 2nk ) S b (w 2nk w 2nk w 2nk )) +b 3 ( b S b (z 2nk 2 z 2nk 2 z 2nk ) S b (w 2nk 2 w 2nk 2 w 2nk ) ) 2b S b (z 2mk + z 2mk + z 2mk ) S b (w 2mk + w 2mk + w 2mk ) +2b 3 ϵ + 2b 4 S b (z 2nk z 2nk z 2nk ) S b (w 2nk w 2nk w 2nk ) +b 5 S b (z 2nk z 2nk z 2nk 2) S b (w 2nk w 2nk w 2nk 2) Letting k we have (29) lim k S b(z 2mk + z 2mk + z 2nk ) S b (w 2mk + w 2mk + w 2nk ) 2b 3 ϵ Also lim k S b (z 2mk +2 z 2mk +2 z 2nk ) S b (z 2nk + z 2nk + z 2mk +) + S b (z 2mk + z 2mk + z 2nk ) [ [2bSb (z 2mk +2 z 2mk +2 z 2mk +) + bs b (z 2nk z 2nk z 2mk +)] ] lim k [2bS b (z 2nk + z 2nk + z 2nk ) + bs b (z 2mk + z 2mk + z 2nk )] + S b (z 2mk + z 2mk + z 2nk ) b 3 S b (z 2mk + z 2mk + z 2nk ) S b (z 2mk + z 2mk + z 2nk ) lim k + S b (z 2mk + z 2mk + z 2nk ) lim k b3 S b (z 2mk + z 2mk + z 2nk ) 2b 6 ϵ from(29) Similarly S b (z 2mk +2 z 2mk +2 z 2nk ) S b (w 2nk + w 2nk + w 2mk +) lim 2b 6 ϵ k + S b (w 2mk + w 2mk + w 2nk )

10 2 KPRRAO ETARAKA RAMUDU AND GNVKISHORE Letting k in (28) we have (20) lim k S b(z 2mk +2 z 2mk +2 z 2nk +) S b (w 2mk +2 w 2mk +2 w 2nk +) 2b 5 ϕ ( 2b 3 ϵ b 6 ϵ 2b 6 ϵ ) = 2b 5 ϕ(2b6 ϵ) Now letting n in (26) from (22)(23) and (20) we have ϵ b 2 2b 5 ϕ(2b6 ϵ) < ϵ It is a contradiction Hence z 2n and w 2n are S b -Cauchy sequences in (X S b ) In addition S b (z 2n+ z 2n+ z 2m+ ) S b (w 2n+ w 2n+ w 2m+ ) 2b S b (z 2n+ z 2n+ z 2n ) S b (w 2n+ w 2n+ w 2n ) +b S b (z 2m+ z 2m+ z 2n ) S b (w 2m+ w 2m+ w 2n ) 2b S b (z 2n+ z 2n+ z 2n ) S b (w 2n+ w 2n+ w 2n ) +2b 2 S b (z 2m+ z 2m+ z 2m ) S b (w 2m+ w 2m+ w 2m ) +b 2 S b (z 2n z 2n z 2m ) S b (w 2n w 2n w 2m ) Since z 2n and w 2n are Cauchy and using (22) (23) we have z 2n+ and w 2n+ are also S b -Cauchy sequences in (X S b ) Thus z n and w n are S b - Cauchy sequences in (X S) Suppose assume that P (X) is a S b - complete subspace of (X S b ) Then the sequences z n and w n converge to α and β in P (X) Thus there exist a and b in P (X) such that (2) lim n z n = α = P a and lim w n = β = P b n Before going to prove common coupled fixed point for the mappings A B P and Q first we claim that for each n at least one of the following assertions holds Sb (z 2n+ z 2n+ z 2n ) S 3 S b (w 2n+ w 2n+ w 2n ) b (α α z 2n ) S b (β β w 2n ) Sb (z 2n z 2n z 2n ) 3 S b (w 2n w 2n w 2n ) On contrary suppose that Sb (z 2n+ z 2n+ z 2n ) 3 S b (w 2n+ w 2n+ w 2n ) Sb (z 2n z 2n z 2n ) 3 S b (w 2n w 2n w 2n ) or S b (α α z 2n 2 ) S b (β β w 2n 2 ) > S b (α α z 2n ) S b (β β w 2n ) and > S b (α α z 2n ) S b (β β w 2n )

11 A NEW SUZUKI TYPE UNIQUE COMMON COUPLED FIXED POINT THEOREM 3 Now consider Sb (z 2n z 2n z 2n ) S b (w 2n w 2n w 2n ) 2bSb (z 2n z 2n α) + b 2 S b (α α z 2n ) 2bS b (w 2n w 2n β) + b 2 S b (β β z 2n ) Sb 2b 2 (α α z 2n ) Sb + b 2 (α α z 2n ) S b (β β w 2n ) S b (β β z 2n ) Sb < 4b (z 2n+ z 2n+ z 2n ) Sb + S b (w 2n+ w 2n+ w 2n ) (z 2n z 2n z 2n ) S b (w 2n w 2n w 2n ) Sb 4b (z 2n z 2n z 2n ) Sb + S b (w 2n w 2n w 2n ) (z 2n z 2n z 2n ) S b (w 2n w 2n w 2n ) Sb = 3 (z 2n z 2n z 2n ) S b (w 2n w 2n w 2n ) Sb (z 2n z 2n z 2n ) < S b (w 2n w 2n w 2n ) It is a contradiction Hence our assertion holds Sub case(a) 3 Sb (z 2n+ z 2n+ z 2n ) S b (w 2n+ w 2n+ w 2n ) Sb (α α z 2n ) S b (β β w 2n ) holds Since 3 S b (A(a b) A(a b) α) S b (z 2n+ z 2n+ z 2n ) S b (A(b a) A(b a) β) S b (w 2n+ w 2n+ w 2n ) Sb (P a P a z 2n ) S b (P b P b w 2n ) That is 3 Sb (A(a b) A(a b) α) S b (z 2n+ z 2n+ z 2n ) S b (A(b a) A(b a) β) S b (w 2n+ w 2n+ w 2n ) Sb (α α z 2n ) S b (β β w 2n ) From (24) and Lemma () we have

12 4 KPRRAO ETARAKA RAMUDU AND GNVKISHORE S 2b b (A(a b) A(a b) α) lim inf S b(a(a b) A(a b) B(x 2n+ y 2n+ )) n lim inf 2 n b5 S b (A(a b) A(a b) B(x 2n+ y 2n+ )) S b (α α z 2n ) S b (β β w 2n ) S b (A(a b) A(a b) α) S b (A(b a) A(b a) β) S lim inf ϕ b (z 2n+ z 2n+ z 2n) S b (w 2n+ w 2n+ w 2n) n [ Sb (A(a b) A(a b) z 2n)S b (z 2n+ z 2n+ α) ] +S b (ααz 2n ) [ Sb (A(b a) A(b a) w 2n )S b (w 2n+ w 2n+ β) ] +S b (ββw 2n ) = ϕ ( 0 0 S b (A(a b) A(a b) α) S b (A(b a) A(b a) β) ) = ϕ ( S b (A(a b) A(a b) α) S b (A(b a) A(b a) β) ) Similarly ( 2b S Sb (A(a b) A(a b) α) b(a(b a) A(b a) β) S b (A(b a) A(b a) β) ) Thus 2b Sb (A(a b) A(a b) α) S b (A(b a) A(b a) β) ( Sb (A(a b) A(a b) α) S b (A(b a) A(b a) β) ) By the definition of ϕ it follows that A(a b) = α = P a and A(b a) = β = P b Since (A P ) is w-compatible pair we have A(α β) = P α and A(β α) = P β From the definition of S b -metric it is clear that S b (A(α β) A(α β) P α) S b (A(β α) A(β α) P β) S 3 b (B(x 2n+ y 2n+ ) B(x 2n+ y 2n+ ) Qx 2n+ ) S b (B(y 2n+ x 2n+ ) B(y 2n+ x 2n+ ) Qy 2n+ ) = 0 S b (P α P α Qx 2n+ ) S b (P β P β Qy 2n+ )

13 A NEW SUZUKI TYPE UNIQUE COMMON COUPLED FIXED POINT THEOREM 5 From (24) and Lemma () we have S 2b b (A(α β) A(α β) α) lim sup S b(a(α β) A(α β) B(x 2n+ y 2n+ )) n lim sup 2 n b5 S b (A(α β) A(α β) B(x 2n+ y 2n+)) S b (A(α β) A(α β) z 2n) S b (A(β α) A(β α) w 2n) lim sup ϕ 0 0 S b (z 2n+ z 2n+ z 2n ) S b (w 2n+ w 2n+ w 2n ) n S b (A(αβ)A(αβ)z 2n ) S b (z 2n+ z 2n+ A(αβ)) +S b (A(αβ)A(αβ)z 2n ) S b (A(βα)A(βα)w 2n ) S b (w 2n+ w 2n+ A(βα)) +S b (A(βα)A(βα)w 2n ) S lim sup ϕ b (A(α β) A(α β) z 2n ) S b (A(β α) A(β α) w 2n ) n S b (z 2n+ z 2n+ z 2n ) S b (w 2n+ w 2n+ w 2n ) S b (z 2n+ z 2n+ A(α β)) S b (w 2n+ w 2n+ A(β α)) ( ) 2bSb (A(α β) A(α β) α) 2bS b (A(β α) A(β α) β) 0 0 b 2 S b (α α A(α β)) b 2 S b (β β A(β α)) ( 2b 2 S b (A(α β) A(α β) α) S b (A(β α) A(β α) β) ) Similarly ( 2b S b(a(β α) A(β α) β) 2b 2 S( A(α β) A(α β) α) S b (A(β α) A(β α) β) ) Thus 2b Sb (A(α β) A(α β) α) ( S b (A(β α) A(β α) β) ) 2b 2 Sb (A(α β) A(α β) α) S b (A(β α) A(β α) β) By the definition of ϕ it follows that A(α β) = α = P α and A(β α) = β = P β Therefore (α β) is a common coupled fixed point of A and P Since A(X X) Q(X) there exist x and y in X such that A(α β) = α = Qx and A(β α) = β = Qy Since we have that 3 From (24) we have Sb (A(α β) A(α β) P α) S b (A(β α) A(β α) P β) S b (B(x y) B(x y) Qx) S b (B(y x) B(y x) Qy) Sb (P α P α Qx) = 0 S b (P β P β Qy) 2 b 5 S b (A(α β) A(α β) B(x y))

14 6 KPRRAO ETARAKA RAMUDU AND GNVKISHORE S b (P α P α Qx) S b (P β P β Qy) S b (A(α β) A(α β) P α) S b (A(β α) A(β α) P β) S b (B(x y) B(x y) Qx) S(B(y x) B(y x) Qy) S b (A(αβ)A(αβ)Qx) S b (B(xy)B(xy)P α) +S b (P αp αqx) S b (A(βα)A(βα)Qy)S b (B(yx)B(yx)P β) +S b (P βp βqy) = ϕ ( S b (B(x y) B(x y) α) S b (B(y x) B(y x) β) 0 0 ) ( b S b (α α B(x y)) S b (β β B(y x)) ) Similarly Thus 2 b 5 S b (β β B(y x)) ( b S b (α α B(x y)) S b (β β B(y x)) ) 2 b 5 Sb (α α B(x y)) S b (β β B(y x)) ( Sb (α α B(x y)) b S b (β β B(y x)) ) It follows that B(x y) = α = Qx and B(y x) = β = Qy Since (B Q) is w-compatible pair we have B(α β) = Qα and B(β α ) = Qβ Since we have that Sb (A(α β) A(α β) P α) S b (A(β α) A(β α) P β) 3 From (24) we have S b (B(α β) B(α β) Qα) S b (B(β α) B(β α) Qβ) Sb (P α P α Qα) = 0 S b (P β P β Qβ) 2 b 5 S b (A(α β) A(α β) B(α β)) S b (P α P α Qα) S b (P β P β Qβ) S b (A(α β) A(α β) P α) S b (A(β α) A(β α) P β) S b (B(α β) B(α β) Qα) S b (B(β α) B(β α) Qβ) Similarly Thus S b (A(αβ)A(αβ)Qα)S b (B(αβ)B(αβ)P α) +S b (P αp αqα) S b (A(βα)A(βα)Qβ)S b (B(βα)B(βα)P β) +S b (P βp βqβ) ( ) S = ϕ b (α α B(α β)) S b (β β B(β α)) S b (B(α β) B(α β) α) S b (B(β α) B(β α) β) ( b S b (α α B(α β)) S b (β β B(β α)) ) 2 b 5 S b (β β B(β α)) ( b S b (α α B(α β)) S b (β β B(β α)) ) Sb (α α B(α β)) S b (β β B(β α)) ( Sb (α α B(α β)) b S b (β β B(β α)) ) It follows that B(α β) = α = Qα and B(β α) = β = Qβ Thus (α β) is a common coupled fixed point of A B P and Q

15 A NEW SUZUKI TYPE UNIQUE COMMON COUPLED FIXED POINT THEOREM 7 To prove uniqueness let us take (α β ) as another common coupled fixed point of A B P and Q Since it is clear that S b (A(α β) A(α β) P α) S b (A(β α) A(β α) P β) 3 S b (B(α β ) B(α β ) Qα = 0 ) S b (B(β α ) B(β α ) Qβ ) From (24) we have Sb (P α P α Qα ) S b (P β P β Qβ ) 2 b 5 S b (α α α ) = 2 b 5 S b (A(α β) A(α β) B(α β )) S b (α α α ) S(β β β ) S b (α α α) S b (β β β) S b (α α α ) S b (β β β ) Similarly Thus S b (ααα )S b (α α α) +S b (ααα ) S b(βββ )S b (β β β) +S b (βββ ) (b S b (α α α ) S b (β β β )) 2 b 5 S(β β β ) (bs(α α α ) bs b (β β β )) 2 b 5 S b (α α α ) S b (β β β ) ( b S b (α α α ) S b (β β β ) ) It follows that α = α and β = β Hence (α β) is unique common coupled fixed point of A B P and Q Sub case(b) Sb (z 2n z 2n z 2n ) 3 S b (w 2n w 2n w 2n ) Sb (α α z 2n ) S b (β β w 2n ) holds By proceeding as in Sub case(a) we can prove that (α β) is unique common coupled fixed point of A B P and Q Similarly the theorem holds when Q(X) is a S b -complete subspace of (X S b ) Now we give an example to illustrate the Theorem 2 Example 2 Let X = [0 ] and S : X X X R + by S b (x y z) = ( y + z 2x + y z ) 2 then (X S b ) is a S b -metric space with b = 4 P Q : X X by A(x y) = x2 + y B = x2 + y 2 Define A B : X X X and 4 9 P (x) = x2 4

16 8 KPRRAO ETARAKA RAMUDU AND GNVKISHORE and Q(x) = x2 6 Let ϕ : R+ R + be defined by ϕ(t) = t 4 6 Consider 2b 5 S b (A(x y) A(x y) B(u v)) = 2 ( 4 6) ( A(x y) B(u v) ) 2 = 2(4 6 ) = 2(4 6 ) x2 +y 2 4 u2 +v x2 u y2 v (4 6 4x ) 2 u y 2 v ( 2(46 ) 2 (4 7 ) 2 = 2(4 6 ) x 2 4x 2 u u y 2 v 2 6 y2 4 v2 6 ) 2 2 = 2(4 7 ) S(P x P x Qu) S(P y P y Qv) S(P x P x Qu) S(P y P y Qv) S(A(x y) A(x y) P x)s(a(y x) A(y x) P y) S(B(u v) B(u v) Qu) S(B(v u) B(v u) Qv) S(A(xy)A(xy)Qu) S(B(uv)B(uv)P x) +S(P xp xqu) S(A(yx)A(yx)Qv) S(B(vu)B(vu)P y) +S(P yp yqv) Thus the condition (24) is satisfiedone can easily verify the remaining conditions of Theorem 2 In this example (0 0) is the unique common coupled fixed point of A B P and Q References [] M Abbas and M Ali khan and S Randenović Common coupled fixed point theorems in cone metric spaces for w-compatible mappings Appl Math Comput 27()(200) [2] S Banach Theorie des Operations lineaires Manograic Mathematic Zne Warsaw Poland 932 [3] S Czerwik Contraction mapping in b-metric spaces Communications in Mathematics Formerly Acta Mathematica et Informatica Universitatis Ostraviensis ()(993) 5 - [4] T Gnana Bhaskar and V Lakshmikantham Fixed point theorems in partially ordered metric spaces and applications Nonlinear Analysis Theory Methods and Applications 65(7)(2006)

17 A NEW SUZUKI TYPE UNIQUE COMMON COUPLED FIXED POINT THEOREM 9 [5] V Lakshmikantham and Lj Ćirić Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces Nonlinear Analysis Theory Methods and Applications 70(2)(2009) [6] Z Mustafa and B Sims A new approach to generalized metric spaces J Nonlinear Convex Anal 7(2)(2006) [7] S Sedghi I Altun N Shobe and M Salahshour Some properties of S-metric space and fixed point results Kyungpook Math J 54()(204) 3-22 [8] S Sedghi Gholidahneh T Došenović J Esfahani and S Radenović Common fixed point of four maps in S b -metric spaces J Linear Top Algebra 5(2)(206) [9] S Sedghi N Shobe and A Aliouche A generalization of fixed point theorem in S-metric spaces Mat Vesnik 64(3)(202) [0] S Sedghi N Shobe and T Došenović Fixed point results in S-metric spaces Nonlinear Functional Analysis and Applications 20()(205) [] T Suzuki A generalized Banach contraction principle which characterizes metric completeness Proc Amer Math Soc 36(2008) Receibed by editors ; Revised version 25207; Available online Department of Mathematics Acharya Nagarjuna University Nagarjuna Nagar Guntur Andhra Pradesh India address: kprrao2004@yahoocom Amrita Sai Institute of Science and Technology Paritala-5280 Andhra Pradesh India address: tarakaramudu32@gmailcom Department of Mathematics K L University Vaddeswaram Guntur Andhra Pradesh India address: gnvkishore@kluniversityin kishoreapr2@gmailcom