COMMON COUPLED FIXED POINTS FOR FOUR MAPS USING α - ADMISSIBLE FUNCTIONS IN PARTIAL METRIC SPACES

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1 JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) , ISSN (o) / JOURNALS /JOURNAL Vol. 6(06), 03-8 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (o), ISSN 986-5X (p) COMMON COUPLED FIXED POINTS FOR FOUR MAPS USING α - ADMISSIBLE FUNCTIONS IN PARTIAL METRIC SPACES K.P.R.Rao, P.Ranga Swamy, and K.R.K.Rao Abstract. In this paper, we introduce the generalized α-admissible functions associated with two pairs of partial ( ) compatible maps and obtain a unique common coupled fixed point theorem in partial metric spaces. We also give an example to illustrate our main theorem.. Introduction and Preliminaries There are many generalizations of the concept of metric spaces in the literature. One of them is a partial metric space introduced by Matthews [5] as a part of study of denotational semantics of data flow networks. After that fixed and common fixed point results in partial metric spaces were studied by many other authors, for example, [, 3, 4, 5, 7, 8, 9, 0,, 4, 6, 7]. Throughout this paper, R + and N denote the set of all non-negative real numbers and set of all positive integers respectively. First we recall some basic definitions and lemmas which play crucial role in the theory of partial metric spaces. Definition.. (See [5]) A partial metric on a nonempty set X is a function p : X X R + such that for all x, y, z X: (p ) x = y p(x, x) = p(x, y) = p(y, y), (p ) p(x, x) p(x, y), p(y, y) p(x, y), (p 3 ) p(x, y) = p(y, x), (p 4 ) p(x, y) p(x, z) + p(z, y) p(z, z). The pair (X, p) is called a partial metric space (PMS). 00 Mathematics Subject Classification. 54H5, 47H0. Key words and phrases. Complete metric spaces, α-admissible functions, Partial ( ) compatible mappings. 03

2 04 K.P.R.RAO, P.RANGA SWAMY, AND K.R.K.RAO If p is a partial metric on X, then the function d p : X X R + given by d p (x, y) = p(x, y) p(x, x) p(y, y) is a metric on X. It is clear that (i) p(x, y) = 0 x = y, (ii) x y p(x, y) > 0 and (iii) p(x, x) may not be 0. Example.. (See e.g. [4, 5, 7]) Consider X = R + with p(x, y) = maxx, y. Then (X, p) is a partial metric space. It is clear that p is not a (usual) metric. Note that in this case d p (x, y) = x y. We now state some basic topological notions (such as convergence, completeness, continuity) on partial metric spaces (see e.g. [4, 5, 8, 5, 7]). Definition.. (i) A sequence x n in the PMS (X, p) converges to the limit x if and only if p(x, x) = lim p(x, x n). n (ii) A sequence x n in the PMS (X, p) is called a Cauchy sequence if lim p(x n, x m ) exists and is finite. n,m (iii) A PMS (X, p) is called complete if every Cauchy sequence x n in X converges with respect to τ p, to a point x X such that p(x, x) = lim p(x n, x m ). n,m (iv) A mapping F : X X is said to be continuous at x X if for every ϵ > 0 there exists δ > 0 such that F (B p (x, δ)) B p (F x, ϵ). It is clear that if F is continuous at x X then F x n converges to F x whenever the sequence x n X converges to x. The following lemma is one of the basic results in PMS ([4, 5, 8, 5, 7]). Lemma.. (i) A sequence x n is a Cauchy sequence in the PMS (X, p) if and only if it is a Cauchy sequence in the metric space (X, d p ). (ii) A PMS (X, p) is complete if and only if the metric space (X, d p ) is complete. Moreover lim n d p(x, x n ) = 0 p(x, x) = lim n p(x, x n) = lim p(x n, x m ). n,m Next, we give a simple lemma which will be used in the proof of our main result. For the proof we refer to [7]. Lemma.. Assume x n z as n in a PMS (X, p) such that p(z, z) = 0. Then lim n p(x n, y) = p(z, y) for every y X. Bhaskar and Lakshmikantham [8] introduced the concept of coupled fixed points and Lakshmikantham and Ciric [9] defined the common coupled fixed points. Later several authors obtained coupled fixed point theorems in various spaces, for example, [,, 3, 4, 5, 6, 8] and the references their in. Definition.3. ([]) Let X be a nonempty set. Let F, G : X X X and f, g : X X be mappings. A point (x, y) X X is called a common

3 COUPLED FIXED POINTS FOR FOUR MAPS USING α-admissible FUNCTIONS coupled fixed point of F, G, f and g if F (x, y) = G(x, y) = fx = gx = x and F (y, x) = G(y, x) = fy = gy = y. Samet et al. [] introduced partial compatible maps as follows Definition.4. ([]) Let (X, p) be a partial metric space and F, g : X X. Then the pair (F, g) is said to be partial compatible if the following conditions hold: (i) p(x, x) = 0 p(gx, gx) = 0 whenever x X, (ii) lim n p(f gx n, gf x n ) = 0 whenever there exists a sequence x n in X such that F x n t and gx n t for some t X. We observe that the Definition.4 seems to be insufficient. Hence we modify it as Definition.5. Let (X, p) be a partial metric space and F, g : X X.Then the pair (F, g) is said to be partial ( ) compatible if the following conditions hold: (i) p(x, x) = 0 p(gx, gx) = 0 whenever x X, (ii) lim n p(f gx n, gf x n ) = 0 whenever there exists a sequence x n in X such that F x n t and gx n t for some t X with p(t, t) = 0. Now we give an example in which the pair (F, g) is partial ( ) compatible, but not partial compatible. Example.. Let X = [0, ] and p(x, y) = maxx, y. Let F, g : X X as F x = x and gx = x. Clearly p(x, x) = 0 x = 0. Hence p(gx, gx) = 0. Let x n be any sequence in X such that F x n t and gx n t as n for some t X with p(t, t) = 0.Then clearly t = 0 and x n 0 as n. Hence lim p(f gx n, gf x n ) = 0. Thus the pair (F, g) is partial ( ) compatible. If n we take x n = and t =, then F x n t and gx n t as n. But lim p(f gx n, gf x n ) = n 0. Hence the pair (F, g) is not partial compatible. Now we extend this definition to the maps F : X X X and S : X X. Definition.6. Let (X, p) be a partial metric space and F : X X X and S : X X. Then the pair(f, S) is said to be partial ( ) compatible if (i) p(z, z) = 0 p(sz, Sz) = 0 whenever z X, (ii) lim n p(s(f (x n, y n )), F (Sx n, Sy n )) = 0 and lim n p(s(f (y n, x n )), F (Sy n, Sx n )) = 0 whenever there exist sequences x n and y n in X such that F (x n, y n ) t, Sx n t and F (y n, x n ) t, Sy n t for some t, t X with p(t, t) = 0 and p(t, t ) = 0. Samet et al. [] proved some fixed point theorems by introducing α- admissible mappings as follows Definition.7. ([]) Let X be a non empty set, T : X X and α : X X R + be mappings. Then T is called α- admissible if for all x, y X, we have α(x, y) implies α(t x, T y).

4 06 K.P.R.RAO, P.RANGA SWAMY, AND K.R.K.RAO Kaushik et al. [3] introduced the following concept which is a generalization of the concept introduced by Mursaleen et al. [7]. Definition.8. ([3]) Let X be a nonempty set and α : X X R + be a function. Let F : X X X, S : X X be mappings. Then F and S are said to be α-admissible if α((sx, Sy), (Su, Sv)) α((f (x, y), F (y, x)), (F (u, v), F (v, u))) for all x, y, u, v X. If S = I (Identity map), then the above definition is the concept of Mursaleen et al. [7]. In this paper, we give a generalization of the above definition. Definition.9. Let X be a nonempty set and α : X X R +. Let F, G : X X X and S, T : X X be mappings. Then we say that the pair (S, T ) is α-admissible with respect to the pair (F, G) if (i) α((sx, Sy), (T u, T v)) α((f (x, y), F (y, x)), (G(u, v), G(v, u))), (ii) α((t x, T y), (Su, Sv)) α((g(x, y), G(y, x)), (F (u, v), F (v, u))) for all x, y, u, v X. We say that the pair (S, T ) is triangular α-admissible with respect to the pair (F, G) if the pair (S, T ) is α-admissible with respect to the pair (F, G) and α((x, y ), (x, y ))), α((x, y ), (x 3, y 3 ))) α((x, y ), (x 3, y 3 ))) for all x, x, x 3, y, y, y 3 X. Note : maxa + b, c + d maxa, c + maxb, d for all a, b, c, d R +. Definition.0. A function f : R + R + is called an altering distance function if it satisfies the following properties (i) f is monotonically increasing and continuous and (ii) f(t) = 0 if and only if t = 0. Throughout this paper,let ψ, ϕ, φ : R + R + be such that ψ is an altering distance function, ϕ is continuous and φ is lower semi continuous, ϕ(0) = 0 = φ(0) and ψ(t) ϕ(t) + φ(t) > 0 for t > 0... (A) Now we prove our main result.. Main Result Theorem.. Let (X, p) be a complete partial metric space and α : X X R + be a function. Let F, G : X X X and S, T : X X be mappings on X satisfying the following : (..) F (X X) T (X), G(X X) S(X), (..) α((sx, Sy), (T u, T v)) ψ(p(f (x, y), G(u, v))) ϕ(m x, y u, v ) φ(m x, y u, v ) for all x, y X, where ψ, ϕ, φ are as in above and

5 COUPLED FIXED POINTS FOR FOUR MAPS USING α-admissible FUNCTIONS Mu, x, v y = max p(sx, T u), p(sy, T v), p(sx, F (x, y)), p(sy, F (y, x)), p(t u, G(u, v)), p(t v, G(v, u)), [p(sx, G(u, v)) + p(t u, F (x, y))], [p(sy, G(v, u)) + p(t v, F (y, x))] (..3) (a) α((sx, Sy ), (F (x, y ), F (y, x ))), (b) α((sy, Sx ), (F (y, x ), F (x, y ))), (c) α((f (x, y ), F (y, x )), (Sx, Sy )) and (d) α((f (y, x ), F (x, y )), (Sy, Sx )) for some x, y X, (..4) The pair (S, T ) is triangular α-admissible with respect to the pair (F, G), (..5) The pairs (F, S) and (G, T ) are partial ( ) compatible and S and T are continuous on X, (..6) Whenever there exist sequences z n, w n in X such that α((z n, w n ), (z n+, w n+ )), α((z n+, w n+ ), (z n, w n )), α((w n, z n ), (w n+, z n+ )), α((w n+, z n+ ), (w n, z n )) with z n z and w n w, assume that (i) α((sz n, Sw n ), (z n+, w n+ )), α((sw n, Sz n ), (w n+, z n+ )), (ii) α((z n, w n ), (T z n+, T w n+ )), α((w n, z n ), (T w n+, T z n+ )), (iii) α((z, w), (z n+, w n+ )), α((w, z), (w n+, z n+ )), (iv) α((z n, w n ), (z, w)), α((w n, z n ), (w, z)) for all n N. Then F, G, S and T have a common coupled fixed point. (..7) Further if we assume that α((z, w), (z, w )) and α((w, z), (w, z )) whenever (z, w) and (z, w ) are common coupled fixed points of F, G, S and T then F, G, S and T have a unique common coupled fixed point in X X. Proof. Let x and y be in X satisfying (..3). Now define the sequences z n and w n from (..) as follows: z n+ = F (x n+, y n+ ) = T x n+, z n+ = G(x n+, y n+ ) = Sx n+3, w n+ = F (y n+, x n+ ) = T y n+, w n+ = G(y n+, x n+ ) = Sy n+3, for n = 0,,, 3, From (..3)(a), we have α((sx, Sy ), (F (x, y ), F (y, x ))) α((sx, Sy ), (T x, T y )), from definition of z n and w n α((f (x, y ), F (y, x )), (G(x, y ), G(y, x ))), from (..4) α((z, w ), (z, w )) from definition of z n and w n α((t x, T y ), (Sx 3, Sy 3 )), from definition of z n and w n α((g(x, y ), G(y, x )), (F (x 3, y 3 ), F (y 3, x 3 ))), from (..4) α((z, w ), (z 3, w 3 )).

6 08 K.P.R.RAO, P.RANGA SWAMY, AND K.R.K.RAO Continuing in this way, we have (.) α((z n, w n ), (z n+, w n+ )) n. Similarly from(..3)(c), (..3)(b) and (..3)(d) we can obtain (.) α((z n+, w n+ ), (z n, w n )). (.3) α((w n, z n ), (w n+, z n+ )). (.4) α((w n+, z n+ ), (w n, z n )). Let R n = max p(z n, z n+ ), p(w n, w n+ ). Case (i): Suppose R m = 0 for some m. Then z m = z m+ and w m = w m+. Now α((sx m+, Sy m+ ), (T x m+, T y m+ )) = α((z m, w m ), (z m+, w m+ )) from (.). Consider where ψ(p(z m+, z m+ )) = ψ(p(f (x m+, y m+ ), G(x m+, y m+ ))) α((sx m+, Sy m+ ), (T x m+, T y m+ )) ψ(p(f (x m+, y m+ ), G(x m+, y m+ ))) ϕ ( M x m+, y m+ ) ( x x m+, y m+ φ M m+, y m+ ) x m+, y m+ xm+, ym+ Mx m+, y m+ But = max p(z m, z m+ ), p(w m, w m+ ), p(z m, z m+ ), p(w m, w m+ ), p(z m+, z m+ ), p(w m+, w m+ ), [p(z m, z m+ ) + p(z m+, z m+ )], [p(w m, w m+ ) + p(w m+, w m+ )] [p(z m, z m+ ) + p(z m+, z m+ )] [p(z m, z m+ ) + p(z m+, z m+ )] max p(z m, z m+ ), p(z m+, z m+ ). Hence M x m+, y m+ x m+, y m+ = max R m, R m+ = R m+. Thus ψ(p(z m+, z m+ )) ϕ(r m+ ) φ(r m+ ). Similarly using (.3), we can show that Thus ψ(p(w m+, w m+ )) ϕ(r m+ ) φ(r m+ ). ψ(r m+ ) = ψ (max p(z m+, z m+ ), p(w m+, w m+ )) = max ψ(p(z m+, z m+ )), ψ(p(w m+, w m+ )) ϕ(r m+ ) φ(r m+ ) which in turn yields from (A)that R m+ = 0 so that z m+ = z m+ and w m+ = w m+. Continuing in this way, z m = z m+ = z m+ = and w m = w m+ = w m+ =. Hence z n and w n are Cauchy sequences in (X, p)..

7 COUPLED FIXED POINTS FOR FOUR MAPS USING α-admissible FUNCTIONS Case (ii): Assume that R n 0 for all n. As in Case (i), we have ψ(r n+ ) ϕ(maxr n, R n+ ) φ(maxr n, R n+ ). If maxr n, R n+ = R n+ then ψ(r n+ ) ϕ(r n+ ) φ(r n+ ) which in turn yields from (A) that R n+ = 0. It is a contradiction. Hence (.5) ψ(r n+ ) ϕ(r n ) φ(r n ) < ψ(r n ), from (A). Since ψ is increasing, we have R n+ R n. Using (.) and (.4), we can show that R n R n. Continuing in this way, we can conclude that R n is a non-increasing sequence of non-negative real numbers and hence must converge to a real number, say, r 0. Suppose r > 0. Letting n in (.5), Hence r = 0. Thus ψ(r) ϕ(r) φ(r) < ψ(r), from (A). (.6) lim n p(z n, z n+ ) = 0 = lim n p(w n, w n+ ) Hence from (p ), we have (.7) lim n p(z n, z n ) = 0 = lim n p(w n, w n ) From (.6), (.7) and from the definition of d p, we have (.8) lim n d p(z n, z n+ ) = 0 = lim n d p(w n, w n+ ) Now we will show that z n and w n are Cauchy sequences in the metric space (X, d p ). On the contrary, suppose that z n or w n is not Cauchy. This implies that maxd p (z m, z n ), d p (w m, w n ) 0 as m, n. Then there exist an ϵ > 0 and monotone increasing sequences of natural numbers m k and n k such that n k > m k > k, (.9) maxd p (z mk, z nk ), d p (w mk, w nk ) ϵ and (.0) maxd p (z mk, z nk ), d p (w mk, w nk ) < ϵ From (.9), ϵ maxd p (z mk, z nk ), d p (w mk, w nk ) dp (z max mk, z nk ) + d p (z nk, z nk ) + d p (z nk, z nk ), d p (w mk, w nk ) + d p (w nk, w nk ) + d p (w nk, w nk ) max d p (z mk, z nk ), d p (w mk, w nk ) dp (z + max nk, z nk ) + d p (z nk, z nk ),, from Note. d p (w nk, w nk ) + d p (w nk, w nk )

8 0 K.P.R.RAO, P.RANGA SWAMY, AND K.R.K.RAO Letting k and using (.0) and (.8), (.) lim k maxd p(z mk, z nk ), d p (w mk, w nk ) = ϵ Using definition of d p and (.7), (.) lim maxp(z m k, z nk ), p(w mk, w nk ) = ϵ k Letting k and then using (.), (.8) in so that max dp (z mk +, z nk ), d p (w mk +, w nk ) max dp (z mk, z nk ), d p (w mk, w nk ) dp (z max mk +, z mk ), d p (w mk +, w mk ) lim maxd p(z mk +, z nk ), d p (w mk +, w nk ) = ϵ k (.3) lim maxp(z m k +, z nk ), p(w mk +, w nk ) = ϵ k Letting k and then using (.), (.8) in so that max dp (z mk +, z nk ), d p (w mk +, w nk ) max dp (z mk, z nk ), d p (w mk, w nk ) dp (z max mk +, z mk ) + d p (z nk, z nk ), d p (w mk +, w mk ) + d p (w nk, w nk ) lim maxd p(z mk +, z nk ), d p (w mk +, w nk ) = ϵ k (.4) lim maxp(z m k +, z nk ), p(w mk +, w nk ) = ϵ k Letting k and then using (.), (.8) in so that max dp (z mk, z nk ), d p (w mk, w nk ) max dp (z mk, z nk ), d p (w mk, w nk ) maxd p (z nk, z nk ), d p (w nk, w nk ) lim maxd p(z mk, z nk ), d p (w mk, w nk ) = ϵ k (.5) lim k maxp(z m k, z nk ), p(w mk, w nk ) = ϵ Now by using (.) and triangular property of α, we have α((sx mk +, Sy mk +), (T x nk, T y nk )) = α((z mk, w mk ), (z nk, w nk )). Consider

9 COUPLED FIXED POINTS FOR FOUR MAPS USING α-admissible FUNCTIONS... ψ(p(z mk +, z nk )) = ψ(p(f (x mk +, y mk +), G(x nk, y nk ))) α((sx mk +, Sy mk +), (T x nk, T y nk ))ψ(p(f (x mk +, y mk +), G(x nk, y nk ))) ϕ ( M x m k +, y mk + x nk, y nk ) φ M x m k +, y mk + x nk, y nk ( M x m k +, y mk + = max ϵ x nk, y nk ), where p(z mk, z nk ), p(w mk, w nk ), p(z mk, z mk +), p(w mk, w mk +), p(z nk, z nk ), p(w nk, w nk ), [p(z m k, z nk ) + p(z nk, z mk +)], [p(w m k, w nk ) + p(w nk, w mk +)] from (.5), (.6), (.), (.4). Thus lim k ψ(p(z m k +, z nk )) ϕ( ϵ ) φ( ϵ ). Similarly using (.3) and (..) we can show that lim ψ(p(w m k +, w nk )) ϕ( ϵ k ) φ( ϵ ). Since ψ is continuous, from (.3), we have ψ( ϵ ) = lim ψ(max p(z m k +, z nk ), p(w mk +, w nk )) k = max lim ψ(p(z m k +, z nk )), lim ψ(p(w m k +, w nk )) k k ϕ( ϵ ) φ( ϵ ) < ψ( ϵ ), from (A). It is a contradiction. Hence z n and w n are Cauchy sequences in the metric space(x, d p ). Letting n, m and using (.), (.8) in maxd p (z n+, z m+ ), d p (w n+, w m+ ) maxd p (z n, z m ), d p (w n, w m ) dp (z max n+, z n ) + d p (z m, z m+ ), d p (w n+, w n ) + d p (w m, w m+ ) lim d p(z n+, z m+ ) = 0 = lim d p(w n+, w m+ ). n,m n,m Thus z m+ and w m+ are Cauchy in (X, d p ). Cauchy in (X, d p ). Hence lim d p(z n, z m ) = 0 = lim d p(w n, w m ) n,m n,m From the definition of d p and (.7), it folows that (.6) lim p(z n, z m ) = 0 = lim p(w n, w m ) n,m n,m Hence z n and w n are Thus z n and w n are Cauchy in (X, p). Since (X, p) is a complete partial metric space, there exist z, w X such that p(z, z) = lim p(z n, z) = lim p(z n, z m ) n n,m

10 K.P.R.RAO, P.RANGA SWAMY, AND K.R.K.RAO and From (.6), we have p(w, w) = lim p(w n, w) = lim p(w n, w m ) n n,m (.7) p(z, z) = 0 = p(w, w) Hence (.8) and (.9) p(z, z) p(w, w) = lim p(f (x n+, y n+ ), z) = lim p(g(x n+, y n+ ), z) n n = lim p(t x n, z) = lim p(sx n+, z) = 0 n n = lim p(f (y n+, x n+ ), w) = lim p(g(y n+, x n+ ), w) n n = lim p(t y n, w) = lim p(sy n+, w) = 0 n n Since the pair (F, S) is partial ( ) compatible, from (.7), we have and p(sz, Sz) = 0 = p(sw, Sw) (.0) lim n p(s(f (x n+, y n+ )), F (Sx n+, Sy n+ )) = 0 and (.) lim n p(s(f (y n+, x n+ )), F (Sy n+, Sx n+ )) = 0 Since S is continuous at z and w we have (.) lim n p(ssx n+, Sz) = p(sz, Sz) = 0 = lim n p(s(f (x n+, y n+ )), Sz) (.3) lim n p(ssy n+, Sw) = p(sw, Sw) = 0 = lim n p(s(f (y n+, x n+ )), Sw) Now from (..6)(i) we have (.4) α((ssx n+, SSy n+ ), (T x n+, T y n+ )) = α((sz n, Sw n ), (z n+, w n+ )) Letting n and using (.) and (.8) in p(ssx n+, T x n+ ) p(sz, z) p(ssx n+, Sz) + p(z, T x n+ ) (.5) lim n p(ssx n+, T x n+ ) = p(sz, z) Letting n and using (.3) and (.9) in p(ssy n+, T y n+ ) p(sw, w) p(ssy n+, Sw) + p(w, T y n+ ) (.6) lim n p(ssy n+, T y n+ ) = p(sw, w)

11 COUPLED FIXED POINTS FOR FOUR MAPS USING α-admissible FUNCTIONS... 3 (.7) p(ssx n+, F (Sx n+, Sy n+ )) p(ssx n+, Sz) + p(sz, S(F (x n+, y n+ ))) +p(s(f (x n+, y n+ )), F (Sx n+, Sy n+ )) 0 as n from (.) and (.0) (.8) p(ssy n+, F (Sy n+, Sx n+ )) p(ssy n+, Sw) + p(sw, S(F (y n+, x n+ ))) +p(s(f (y n+, x n+ )), F (Sy n+, Sx n+ )) 0 as n from (.3) and (.) Letting n and using (.) and (.8) in p(ssx n+, G(x n+, y n+ )) p(sz, z) p(ssx n+, Sz) + p(z, G(x n+, y n+ )) (.9) lim n p(ssx n+, G(x n+, y n+ )) = p(sz, z) Letting n and using (.8), (.), (.0) in p(t x n+, F (Sx n+, Sy n+ )) p(sz, z) p(t x n+, z) + p(sz, S(F (x n+, y n+ )) +p(s(f (x n+, y n+ )), F (Sx n+, Sy n+ )) (.30) lim n p(t x n+, F (Sx n+, Sy n+ )) = p(sz, z) Letting n and using (.3) and (.9) in p(ssy n+, G(y n+, x n+ )) p(sw, w) p(ssy n+, Sw) + p(w, G(y n+, x n+ )) (.3) lim n p(ssy n+, G(y n+, x n+ )) = p(sw, w) Letting n and using (.9), (.3) and (.) in p(t y n+, F (Sy n+, Sx n+ )) p(sw, w) p(t y n+, w) + p(sw, S(F (y n+, x n+ )) +p(s(f (y n+, x n+ )), F (Sy n+, Sx n+ )) (.3) lim n p(t y n+, F (Sy n+, Sx n+ )) = p(sw, w) Since ψ is monotonically increasing we have ψ(p(sz, z)) ψ p(sz, S(F (x n+, y n+ )))+ p(s(f (x n+, y n+ )), F (Sx n+, Sy n+ ))+ p(f (Sx n+, Sy n+ ), G(x n+, y n+ ))+ p(g(x n+, y n+ ), z)

12 4 K.P.R.RAO, P.RANGA SWAMY, AND K.R.K.RAO Letting n and using continuity of ψ and (..), where ψ(p(sz, z)) lim n ψ(p(f (Sx n+, Sy n+ ), G(x n+, y n+ ))) lim n α((ssx n+, SSy n+ ), (T x n+, T y n+ )) M Sx n+, Sy n+ x n+, y n+ Thus lim n [ ϕ = max ( ψ(p(f (Sx n+, Sy ) n+ ( ), G(x n+, y n+ )] ))), from(4) Mx Sxn+,Syn+ Sxn+, Syn+ n+,y n+ φ Mx n+, y n+ p(ssx n+, T x n+ ), p(ssy n+, T y n+ ), p(ssx n+, F (Sx n+, Sy n+ )), p(ssy n+, F (Sy n+, Sx n+ )), p(t x n+, G(x n+, y n+ )), p(t y n+, G(y n+, x n+ )), [ ] p(ssx n+, G(x n+, y n+ )), [ +p(t x n+, F (Sx n+, Sy n+ )) ] p(ssy n+, G(y n+, x n+ )) +p(t y n+, F (Sy n+, Sx n+ )) maxp(sz, z), p(sw, w) from (.5) (.3) and (.6). ψ(p(sz, z)) ϕ(maxp(sz, z), p(sw, w)) φ(maxp(sz, z), p(sw, w)). Similarly using (..6)(i) and proceeding as above, we can show that ψ(p(sw, w)) ϕ(maxp(sz, z), p(sw, w)) φ(maxp(sz, z), p(sw, w)). Thus ψ(maxp(sz, z), p(sw, w)) ϕ(maxp(sz, z), p(sw, w)) φ(maxp(sz, z), p(sw, w)) which from (A) gives that p(sz, z) = 0 = p(sw, w) so that Sz = z and Sw = w. Since the pair (G, T ) is partial ( ) compatible and T is continuous we can show that T z = z and T w = w by using (..6)(ii). Now from (..6)(iii) we have α((sz, Sw), (T x n+, T y n+ )) = α((z, w), (z n+, w n+ )). Since ψ is continuous and by Lemma., we have ψ(p(f (z, w), z)) = lim ψ(p(f (z, w), G(x n+, y n+ ))) n lim α((sz, Sw), (T x n+, T y n+ )) ψ(p(f (z, w), G(x n+, y n+ ))) n [ ( ) ( )] lim ϕ Mx z, w n n+, y n+ φ Mx z, w n+, y n+

13 COUPLED FIXED POINTS FOR FOUR MAPS USING α-admissible FUNCTIONS... 5 where M z, w x n+, y n+ = max p(z, z n+ ), p(w, w n+ ), p(z, F (z, w)), p(w, F (w, z)), p(z n+, z n+ ), p(w n+, w n+ ), [p(z, z n+) + p(z n+, F (z, w))], [p(w, w n+) + p(w n+, F (w, z))] Thus ψ(p(f (z, w), z)) ϕ maxp(z, F (z, w)), p(w, F (w, z)) from (.8), (.9) and (.6) and from Lemma.. ( p(z, F (z, w)), max p(w, F (w, z)) Similarly by using (..6)(iii), we can show that ( p(z, F (z, w)), ψ(p(f (w, z), w)) ϕ max p(w, F (w, z)) Hence ( ) p(z, F (z, w)), ψ max p(w, F (w, z)) ( p(z, F (z, w)), ϕ max p(w, F (w, z)) ) ( p(z, F (z, w)), φ max p(w, F (w, z)) ) φ ( p(z, F (z, w)), max p(w, F (w, z)) ) ( p(z, F (z, w)), φ max p(w, F (w, z)) ) ). ). which from (A) gives that F (z, w) = z and F (w, z) = w. Similarly by using (..6)(iv), we will prove that G(z, w) = z and G(w, z) = w. Thus Sz = T z = z = F (z, w) = G(z, w) and Sw = T w = w = F (w, z) = G(w, z). Hence (z, w) is a common fixed point of F, G, S and T. Suppose (z, w ) is another common fixed point of F, G, S and T. ψ(p(z, z )) = ψ(p(f (z, w), G(z, w ))) α((sz, ( Sw), ) (T z (, T w ))ψ(p(f ) (z, w), G(z, w ))), from(..7) ϕ M z,w z,w φ M z,w z,w ( ) ( ) p(z, z = ϕ max ), p(z, z p(w, w φ max ), ) p(w, w. ) Similarly from (..7), we have ( ψ(p(w, w p(z, z ) ϕ max ), p(w, w ) Thus ( p(z, z ψ max ), p(w, w ) ) ( p(z, z ϕ max ), p(w, w ) ) ( p(z, z φ max ), p(w, w ) ). ) ( p(z, z φ max ), p(w, w ) which from (A) gives that w = z and w = z. Thus (z, w) is the unique common coupled fixed point of F, G, S and T. Now we give an example to illustrate Theorem.. )

14 6 K.P.R.RAO, P.RANGA SWAMY, AND K.R.K.RAO Example.. Let X = [0, ] and p(x, y) = maxx, y, for all x, y X. Define F, G : X X X by and S, T : X X by F (x, y) = x +y 4 and G(x, y) = x +y 7 Sx = x x and T x = 3. Define α : X X R + by, if x, y, u, v [0, 3], α((x, y), (u, v)) = 0, otherwise Let ψ, ϕ, φ : R + R + be defined by ψ(t) = 4t, ϕ(t) = 7t and φ(t) = 7 t. Clearly F (X X) T (X) and G(X X) S(X). To verify (..) we consider the following two cases. Case (a): Suppose x, y, u, v [0, 3]. Then α((sx, Sy), (T u, T v)) = α(( x u ), ( )) =., y 3, v 3 α((sx, Sy), (T u, T v))ψ(p(f (x, y), G(x, y))) = 4 max = max max max x +y 4, u +v 7 x +y 6, u +v 8 x +y 6, u +v 9 x 6, u 9 + max y 6, v 9 = 3 p(sx, T u) + 3p(Sy, T v) maxp(sx, T u), p(sy, T v) 3 Case (b): Atleast one of x, y, u, v / [0, 3]. Then α((sx, Sy), (T u, T v)) = α(( x 3 M x,y u,v 7 M x,y u,v = ϕ ( M x,y u,v, y ) ( ) φ M x,y u,v ), ( u 3, v 3 )). Sub case: If x, y, u 3, v 3 (, 3] then α((sx, Sy), (T u, T v)) =. The inequality (..) holds as in case (a). Sub case: If at least one of x, y, u 3, v 3 ( 3, ] then α((sx, Sy), (T u, T v)) = 0. Hence (..) holds, since 0 7 M u,v x,y. By definition of α, the condition (..3) with x = 0 = y and (..4) are satisfied clearly. To verify the partial ( ) compatibility of the pair(f, S), let us consider the sequences x n, y n in X such that F (x n, y n ) t, Sx n t, F (y n, x n ) t and Sy n t for some t, t X with p(t, t) = 0 and p(t, t ) = 0. Then t = 0 and t = 0. Clearly F (x n, y n ) t and Sx n t implies lim p(f (x n, y n ), t) = p(t, t) = 0 and n lim p( x n n x, t) = 0. Thus lim n +y n n 4 = 0 = lim n x n which in turn yields that

15 COUPLED FIXED POINTS FOR FOUR MAPS USING α-admissible FUNCTIONS... 7 x n 0 and y n 0. Consider p(s(f (x n, y n )), F (Sx n, Sy n )) = max ( x n +y n 4 ), 4 ( x n as n. Similarly we can show that p(s(f (y n, x n )), F (Sy n, Sx n )) y 4 n 4 ) 0 Thus the pair (F, S) is partial ( ) compatible. Similarly we can show that the pair (G, T ) is partial ( ) compatible. Clearly S and T are are continuous on X. Thus (..5) is satisfied. By definition of α, one can easily verify the conditions (..6) and (..7). Thus all the conditions of Theorem. are satisfied and (0,0) is the unique common fixed point of F, G, S and T. References [] B.Samet, M. Rajović, R. Lazović and R. Stojiljković. Common fixed point results for nonlinear contractions in ordered partial metric spaces, Fixed Point Theory and Applications, 0, (0):7,4 pages,doi:0.86/ [] B.Samet, C.Vetro and P.Vetro. Fixed point theorems for α ψ-contractive type mappings, Nonlinear Analysis, 75(0), [3] D.Ilić, V. Pavlović and V. Rakočević. Some new extensions of Banach s contraction principle to partial metric spaces, Appl. Math. Letters, 4(8)(0), , doi:0.06/j.aml [4] E.Karapınar and I.M. Erhan. Fixed point theorems for operators on partial metric spaces, Applied Mathematics Letters, 4()(0), [5] E.Karapınar. Generalizations of Caristi Kirk s Theorem on Partial metric Spaces, Fixed Point Theory and. Appl., 0, 0:4,7 pages,doi:0.86/ [6] G.V.R.Babu, K.K.M.Sarma and V.A.Kumari. Common fixed points of (α η)-geraghty contraction maps, Journal of Advanced Research in Pure Mathematics, 7()(05), 43-53, doi:0.5373/jarpm [7] H.Aydi. Fixed point results for weakly contractive mappings in ordered partial metric spaces, Journal of Advanced Mathematical Studies, 4()(0), -. [8] I. Altun, F. Sola and H. Simsek. Generalized contractions on partial metric spaces, Topology and its Applications, 57(8)(00), [9] I. Altun and A. Erduran. Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory and Applications 0,(0), Article ID , 0 pages, doi:0.55/0/ [0] K.P.R.Rao and G.N.V. Kishore. A unique common fixed point theorem for four maps under ψ ϕ contractive condition in partial metric spaces, Bulletin of Mathematical Analysis and Applications, 3(3)(0), [] K.P.R.Rao, I. Altun and S.Hima Bindu. Common coupled fixed point theorems in generalized fuzzy metric spaces, Advances in Fuzzy Systems, 0, Article ID , (0), 6 pages, doi : 0.55/0/ [] K.P.R.Rao, S.Hima Bindu and Md.Mustaq Ali. Coupled fixed point theorems in d-complete topological spaces, J.Nonlinear Sci.Appl., 5(0), [3] K.P.R.Rao, G.N.V.Kishore and V.C.C.Raju. A coupled fixed point theorem for two pairs of W-compatible maps using altering distance function in partial metric spaces, Journal of Advanced Research in Pure Mathematics, 4(4)(0), (doi:0.4373/jarpm.86.) [4] K.P.R.Rao, G.N.V.Kishore and Nguyen Van Luong. A Unique Common Coupled fixed point theorem for four maps under ψ ϕ contractive condition in partial metric spaces, CUBO A Mathematical Journal, 4(03)(0), 5-7. [5] K.P.R.Rao, K.R.K.Rao and S.Sedghi. Common coupled fixed point theorems in fuzzy metric spaces, Gazi University Journal of Science, 7()(04),

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