Fixed point theory of cyclical generalized contractive conditions in partial metric spaces

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1 Chen Fixed Point Theory and Applications 013, 013:17 R E S E A R C H Open Access Fixed point theory of cyclical generalized contractive conditions in partial metric spaces Chi-Ming Chen * * Correspondence: ming@mail.nhcue.edu.tw Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu, Taiwan Abstract The purpose of this paper is to study fixed point theorems for a mapping satisfying the cyclical generalized contractive conditions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature. MSC: 47H10; 54C60; 54H5; 55M0 Keywords: fixed point; cyclic CW-contraction; cyclic MK-contraction; partial metric space 1 Introduction and preinaries Throughout this paper, by R +, we denote the set of all nonnegative real numbers, while N is the set of all natural numbers. Let (X, d) beametricspace,d be a subset of X and f : D X be a map. We say f is contractive if there exists α [0, 1) such that for all x, y D, d(fx, fy) α d(x, y). The well-known Banach fixed point theorem asserts that if D = X, f is contractive and (X, d) is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, in 1969, Boyd and Wong [] introduced the notion of -contraction. A mapping f : X X on a metric space is called -contraction if there exists an upper semi-continuous function :[0, ) [0, )such that d(fx, fy) ( d(x, y) ) for all x, y X. In 1994, Mattews [3] introduced the following notion of partial metric spaces. Definition 1 [3] A partial metric on a nonempty set X is a function p : X X R + such that for all x, y, z X, (p 1 ) x = y if and only if p(x, x)=p(x, y)=p(y, y); (p ) p(x, x) p(x, y); 013 Chen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 ChenFixed Point Theory and Applications 013, 013:17 Page of 15 (p 3 ) p(x, y)=p(y, x); (p 4 ) p(x, y) p(x, z)+p(z, y) p(z, z). A partial metric space is a pair (X, p)suchthatx is a nonempty set and p is a partial metric on X. Remark 1 It is clear that if p(x, y)=0,thenfrom(p 1 )and(p ), x = y. Butifx = y, p(x, y) may not be 0. Each partial metric p on X generates a T 0 topology τ p on X which has as a base the family of open p-balls {B p (x, γ ):x X, γ >0},whereB p (x, γ )={y X : p(x, y)<p(x, x)+γ } for all x X and γ >0.Ifp 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. We recall some definitions of a partial metric space as follows. Definition [3] Let(X, p) be a partial metric space. Then (1) a sequence {x n } in a partial metric space (X, p) converges to x X if and only if p(x, x)= p(x, x n ); () a sequence {x n } in a partial metric space (X, p) is called a Cauchy sequence if and only if m, p(x m, x n ) exists (and is finite); (3) a partial metric space (X, p) is said to be complete if every Cauchy sequence {x n } in X converges, with respect to τ p,toapointx X such that p(x, x)= m, p(x m, x n ); (4) a subset A of a partial metric space (X, p) is closed if whenever {x n } is a sequence in A such that {x n } converges to some x X,thenx A. Remark The it in a partial metric space is not unique. Lemma 1 [3, 4] (a) {x n } is a Cauchy sequence in a partial metric space (X, p) if and only if it is a Cauchy sequence in the metric space (x, d p ); (b) a partial metric space (X, p) is complete if and only if the metric space (X, d p ) is complete. Furthermore, d p (x n, x)=0if and only if p(x, x)= p(x n, x)= p(x n, x m ). In 003, Kirk, Srinivasan and Veeramani [5] introduced the following notion of the cyclic representation. Definition 3 [5] LetX be a nonempty set, m N and f : X X be an operator. Then X = m i=1 A i is called a cyclic representation of X with respect to f if (1) A i, i =1,,...,m are nonempty subsets of X; () f (A 1 ) A, f (A ) A 3,...,f(A m 1 ) A m, f (A m ) A 1. Kirk, Srinivasan and Veeramani [5] also proved the following theorem.

3 ChenFixed Point Theory and Applications 013, 013:17 Page 3 of 15 Theorem 1 [5] Let (X, d) be a complete metric space, m N, A 1, A,...,A m, be closed nonempty subsets of X and X = m i=1 A i. Suppose that f satisfies the following condition: d(fx, fy) ψ ( d(x, y) ), for all x A i, y A i+1, i {1,,...,m}, where ψ :[0, ) [0, ) is upper semi-continuous from the right and 0 ψ(t) <tfor t >0.Then f has a fixed point z n i=1 A i. Recently, the fixed theorems for an operator f : X X defined on a metric space X with acyclicrepresentationofx with respect to f have appeared in the literature (see, e.g., [6 8]). In 010, Pǎcurar and Rus [7] introduced the following notion of a cyclic weaker ϕ-contraction. Definition 4 [7] Let(X, d) beametricspace,m N, A 1, A,...,A m be closed nonempty subsets of X and X = m i=1 A i.anoperatorf : X X is called a cyclic weaker ϕ-contraction if (1) X = m i=1 A i is a cyclic representation of X with respect to f ; () there exists a continuous, non-decreasing function ϕ :[0, ) [0, ) with ϕ(t)>0 for t (0, ) and ϕ(0) = 0 such that d(fx, fy) d(x, y) ϕ ( d(x, y) ) for any x A i, y A i+1, i =1,,...,m,whereA m+1 = A 1. And Pǎcurarand Rus [7] proved the following main theorem. Theorem [7] Let (X, d) be a complete metric space, m N, A 1, A,...,A m be closed nonempty subsets of X and X = m i=1 A i. Suppose that f is a cyclic weaker ϕ-contraction. Then f has a fixed point z n i=1 A i. In the recent years, fixed point theory has developed rapidly on cyclic contraction mappings, see [9 15]. The purpose of this paper is to study fixed point theorems for a mapping satisfying the cyclical generalized contractive conditions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature. Fixed point theorems (I) Inthesection,wedenoteby the class of functions ψ : R +3 R + satisfying the following conditions: (ψ 1 ) ψ is an increasing and continuous function in each coordinate; (ψ ) fort R +, ψ(t, t, t) t, ψ(t,0,0) t and ψ(0, 0, t) t. Next, we denote by the class of functions ϕ : R + R + satisfying the following conditions: (ϕ 1 ) ϕ is continuous and non-decreasing; (ϕ ) fort >0, ϕ(t)>0and ϕ(0) = 0.

4 ChenFixed Point Theory and Applications 013, 013:17 Page 4 of 15 And we denote by the class of functions φ : R + R + satisfying the following conditions: (φ 1 ) φ is continuous; (φ ) fort >0, φ(t)>0and φ(0) = 0. We now state a new notion of cyclic CW-contractions in partial metric spaces as follows. Definition 5 Let (X, p) be a partial metric space, m N, A 1, A,...,A m be nonempty subsets of X and Y = m i=1 A i.anoperatorf : Y Y is called a cyclic CW-contraction if (1) m i=1 A i is a cyclic representation of Y with respect to f ; () for any x A i, y A i+1, i =1,,...,m, ϕ ( p(fx, fy) ) ψ ( ϕ ( p(x, y) ), ϕ ( p(x, fx) ), ϕ ( p(y, fy) )) φ ( M(x, y) ), (.1) where ψ, ϕ, φ,andm(x, y)=max{p(x, y), p(x, fx), p(y, fy)}. Theorem 3 Let (X, p) be a complete partial metric space, m N, A 1, A,...,A m be nonempty closed subsets of X and Y = m i=1 A i. Let f : Y YbeacyclicCW-contraction. Then f has a unique fixed point z m i=1 A i. Proof Given x 0 and let x n+1 = fx n = f n x 0 for n =0,1,,...Ifthereexists n 0 N such that x n0 +1 = x n0,thenwefinishedtheproof.supposethatx n+1 x n for any n =0,1,,...Notice that for any n 0, there exists i n {1,,...,m} such that x n A in and x n+1 A in +1. Step 1. We will prove that p(x n, x n+1 )=0, thatis, d p(x n, x n+1 )=0. Using (.1), we have ϕ ( p(x n, x n+1 ) ) = ϕ ( p(fx n 1, fx n ) ) ψ ( ϕ ( p(x n 1, x n ) ), ϕ ( p(x n 1, fx n 1 ) ), ϕ ( p(x n, fx n ) )) φ ( M(x n 1, x n ) ) = ψ ( ϕ ( p(x n 1, x n ) ), ϕ ( p(x n 1, x n ) ), ϕ ( p(x n, x n+1 ) )) φ ( M(x n 1, x n ) ), where M(x n 1, x n )=max { p(x n 1, x n ), p(x n 1, fx n 1 ), p(x n, fx n ) } = max { p(x n 1, x n ), p(x n 1, x n ), p(x n, x n+1 ) }. If M(x n 1, x n )=p(x n, x n+1 ), then ϕ ( p(x n, x n+1 ) ) ψ ( ϕ ( p(x n, x n+1 ) ), ϕ ( p(x n, x n+1 ) ), ϕ ( p(x n, x n+1 ) )) φ ( p(x n, x n+1 ) ) ϕ ( p(x n, x n+1 ) ) φ ( p(x n, x n+1 ) ), which implies that φ(p(x n, x n+1 )) = 0, and hence p(x n, x n+1 ) = 0. This contradicts our initial assumption.

5 ChenFixed Point Theory and Applications 013, 013:17 Page 5 of 15 From the above argument, we have that for each n N, ϕ ( p(x n, x n+1 ) ) ϕ ( p(x n 1, x n ) ) φ ( p(x n 1, x n ) ), (.) and p(x n, x n+1 )<p(x n 1, x n ). And since the sequence {p(x n, x n+1 )} is decreasing, it must converge to some η 0. Taking it as n in (.) and by the continuity of ϕ and φ,weget ϕ(η) ϕ(η) φ(η), and so we conclude that φ(η)=0andη =0.Thus,wehave p(x n, x n+1 )=0. (.3) By (p ), we also have p(x n, x n )=0. (.4) Since d p (x, y) p(x, y) p(x, x) p(y, y) for all x, y X, using(.3) and(.4), we obtain that d p(x n, x n+1 )=0. (.5) Step. We show that {x n } is a Cauchy sequence in the metric space (Y, d p ). We claim that the following result holds. Claim For every ε >0,thereexistsn N such that if r, q n with r q =1mod m, then d p (x r, x q )<ε. Suppose the above statement is false. Then there exists ɛ > 0 such that for any n N, there are r n, q n N with r n > q n n with r n q n =1mod m satisfying d p (x qn, x rn ) ɛ. Now, we let n >m. Then corresponding to q n n use, we can choose r n in such a way it is the smallest integer with r n > q n n satisfying r n q n =1mod m and d p (x qn, x rn ) ɛ. Therefore, d p (x qn, x rn m) ɛ and ɛ d p (x qn, x rn ) m d p (x qn, x rn m)+ d p (x rn i, x rn i+1 ) i=1 m < ɛ + d p (x rn i, x rn i+1 ). i=1

6 ChenFixed Point Theory and Applications 013, 013:17 Page 6 of 15 Letting n,weobtainthat d p(x qn, x rn )=ɛ. (.6) On the other hand, we can conclude that ɛ d p (x qn, x rn ) d p (x qn, x qn+1 )+d p (x qn+1, x rn+1 )+d p (x rn+1, x rn ) d p (x qn, x qn+1 )+d p (x qn+1, x qn )+d p (x qn, x rn )+d p (x rn, x rn+1 )+d p (x rn+1, x rn ). Letting n,weobtainthat d p(x qn+1, x rn+1 )=ɛ. (.7) Since d p (x, y)=p(x, y) p(x, x) p(y, y)andusing(.4), (.6)and(.7), we have that and p(x q n, x rn )= ɛ, (.8) p(x q n+1, x rn+1 )= ɛ. (.9) Since x qn and x rn lie in different adjacently labeled sets A i and A i+1 for certain 1 i m, by using the fact that f is a cyclic CW-contraction, we have ϕ ( p(fx qn +1, fx rn +1) ) = ϕ ( p(fx qn, fx rn ) ) ψ ( ϕ ( p(x qn, x rn ) ), ϕ ( p(x qn, fx qn ) ), ϕ ( p(x rn, fx rn ) )) φ ( M(x qn, x rn ) ) = ψ ( ϕ ( p(x qn, x rn ) ), ϕ ( p(x qn, x qn +1) ), ϕ ( p(x rn, x rn +1) )) φ ( M(x qn, x rn ) ), where M(x qn, x rn )=max { p(x qn, x rn ), p(x qn, x qn +1), p(x rn, x rn +1) }. Thus, letting n,wecanconcludethat ( ) ( ɛ ϕ ψ ϕ ( ɛ ) ), ϕ(0), ϕ(0) φ ( ɛ ) ϕ ( ɛ ) φ which implies φ( ɛ )=0,thatis,ɛ = 0. So, we get a contradiction. Therefore, our claim is proved. ( ɛ ),

7 ChenFixed Point Theory and Applications 013, 013:17 Page 7 of 15 In the sequel, we will show that {x n } is a Cauchy sequence in the metric space (Y, d p ). Let ε >0begiven.Byourclaim,thereexistsn 1 N such that if r, q n 1 with r q =1mod m, then d p (x r, x q ) ε. Since d p (x n, x n+1 )=0,thereexistsn N such that d p (x n, x n+1 ) ε m for any n n. Let r, q max{n 1, n } and r > q. Then there exists k {1,,...,m} such that r q = k mod m. Therefore, r q + j =1mod m for j = m k +1,andsowehave d p (x q, x r ) d p (x q, x r+j )+d p (x r+j, x r+j 1 )+ + d p (x r 1, x r ) ε + j ε m ε + m ε m = ε. Thus, {x n } is a Cauchy sequence in the metric space (Y, d p ). Step 3. We show that f has a fixed point ν in m i=1 A i. Since Y is closed, the subspace (Y, p) is complete. Then from Lemma 1, wehavethat (Y, d p )iscomplete.thus,thereexistsν X such that d p(x n, ν)=0. And it follows from Lemma 1 that we have p(ν, ν)= p(x n, ν)= p(x n, x m ). (.10) n,m On the other hand, since the sequence {x n } is a Cauchy sequence in the metric space (Y, d p ), we also have d p(x n, x m )=0. Since d p (x, y)=p(x, y) p(x, x) p(y, y), we can deduce that p(x n, x m )=0. (.11) Since Y = m i=1 A i is a cyclic representation of X with respect to f,thesequence{x n } has infinite terms in each A i for i {1,,...,m}. Now, for all i =1,,...,m,wemaytakeasubsequence {x nk } of {x n } with x nk A i 1 and also all converge to ν. Using(.10) and(.11), we have p(ν, ν)= p(x n, ν)= p(x n k, ν)=0.

8 ChenFixed Point Theory and Applications 013, 013:17 Page 8 of 15 By (.1), ϕ ( p(x nk+1, f ν) ) = ϕ ( p(fx nk, f ν) ) ψ ( ϕ ( p(x nk, ν) ), ϕ ( p(x nk, fx nk ) ), ϕ ( p(ν, f ν) )) φ ( M(x nk, ν) ) = ψ ( ϕ ( p(x nk, ν) ), ϕ ( p(x nk, x nk +1) ), ϕ ( p(ν, f ν) )) φ ( M(x nk, ν) ), where M(x nk, ν)=max { p(x nk, ν), p(x nk, x nk +1), p(ν, f ν) }. Letting k,wehave ϕ ( p(ν, f ν) ) ψ ( ϕ(0), ϕ(0), ϕ ( p(ν, f ν) )) φ ( p(ν, f ν) ) ϕ ( p(ν, f ν) ) φ ( p(ν, f ν) ), which implies φ(p(ν, f ν)) = 0, that is, p(ν, f ν)=0.so,ν = f ν. Step 4. Finally, to prove the uniqueness of the fixed point, suppose that μ, ν are fixed points of f. Then using the inequality (.1), we obtain that ϕ ( p(μ, ν) ) = ϕ ( p(f μ, f ν) ) ψ ( ϕ ( p(μ, ν) ), ϕ ( p(μ, f μ) ), ϕ ( p(ν, f ν) )) φ ( M(μ, ν) ), where M(μ, ν)=max { p(μ, ν), p(μ, f μ), p(ν, f ν) } = p(μ, ν). So, we also deduce that ϕ ( p(μ, ν) ) ψ ( ϕ ( p(μ, ν), 0, 0 )) ϕ ( p(μ, ν) ) φ ( p(μ, ν) ), which implies that φ(p(μ, ν)) = 0, and hence p(μ, ν)=0,thatis,μ = ν. So, we complete the proof. The following provides an example for Theorem 3. Example 1 Let X = [0, 1] and A = [0, 1], B =[0, 1 ], C =[0, 1 ]. We define the partial metric 4 p on X by p(x, y)=max{x, y} for all x, y X, and define the function f : X X by x f (x)= 1+x for all x X.

9 ChenFixed Point Theory and Applications 013, 013:17 Page 9 of 15 Now, we let ϕ, φ : R + R + and ψ : R +3 R + be ϕ(t)=t, φ(t)= t 5(1 + t) and ψ(t)= 4 5 max{t 1, t, t 3 }. Then f is a cyclic CW-contraction and 0 is the unique fixed point. Proof We claim that f is a cyclic CW-contraction. (1) Note that f (A)=[0, 1 ] B, f (B)=[0, 1 6 ] C and f (C)=[0, 1 0 ] A.Thus,A B C is a cyclic representation of X with respect to f ; () For x A and y B (or, x B and y C), without loss of generality, we may assume that x y,thenwehave and Since we have ϕ ( p(fx, fy) ) ( ( x = ϕ p 1+x, y )) ( x ) = ϕ = x 1+y 1+x 1+x, ψ ( ϕ ( p(x, y) ), ϕ ( p(x, fx) ), ϕ ( p(y, fy) )) = ψ ( ϕ ( p(x, y) ) ( (, ϕ p = ψ ( ϕ(x), ϕ(x), ϕ(y) ) = ψ(x,x,y)= 8x 5, x, φ ( max { p(x, y), p(x, fx), p(y, fy) }) ( { ( = φ max p(x, y), p x, = φ ( max{x, x, y} ) = x 1+x 8x 5 x 5(1 + x), x )) ( ( y ))), ϕ p y, 1+x 1+y x ) (, p y, 1+x x 5(1 + x). y )}) 1+y ϕ ( p(fx, fy) ) ψ ( ϕ ( p(x, y) ), ϕ ( p(x, fx) ), ϕ ( p(y, fy) )) φ ( max { p(x, y), p(x, fx), p(y, fy) }). On the other hand, for x C and y A, without loss of generality, we may assume that x y,thenitiseasytogettheaboveinequality. Note that Example 1 satisfies all of the hypotheses of Theorem 3,andwegetthat0isthe unique fixed point. 3 Fixed point theorems (II) In this article, we also recall the notion of a Meir-Keeler function (see [16]). A function φ :[0, ) [0, ) is said to be a Meir-Keeler function if for each η >0,thereexistsδ >0

10 Chen Fixed Point Theory and Applications 013, 013:17 Page 10 of 15 such that for t [0, )withη t < η +δ,wehaveφ(t)<η. We now introduce a new notion of a weaker Meir-Keeler function φ :[0, ) [0, ) in a partial metric space (X, p) as follows. Definition 6 Let (X, p) be a partial metric space. We call φ :[0, ) [0, ) aweaker Meir-Keeler function in X if for each η >0,thereexistsδ > 0 such that for x, y X with η p(x, y)<η + δ,thereexistsn 0 N such that φ n 0(p(x, y)) < η. Inthesection,wedenoteby the class of weaker Meir-Keeler functions φ : R + R + in a partialmetric space in (X, p) satisfying the following conditions: (φ 1 ) φ(t)>0for t >0, φ(0) = 0; (φ ) {φ n (t)} n N is decreasing; (φ 3 ) fort n [0, ), (a) if t n = γ >0,then φ(t n )<γ and (b) if t n =0,then φ(t n )=0. And we denote by the class of functions ψ : R + R + a continuous function satisfying ψ(t)>0fort >0,ψ(0) = 0. First, we state a new notion of cyclic MK-contractions in partial metric spaces as follows. Definition 7 Let (X, p) be a partial metric space, m N, A 1, A,...,A m be nonempty subsets of X and Y = m i=1 A i.anoperatorf : Y Y is called a cyclic MK-contraction if (1) m i=1 A i is a cyclic representation of Y with respect to f ; () for any x A i, y A i+1, i =1,,...,m, p(fx, fy) φ ( p(x, y) ) ψ ( p(x, y) ), (3.1) where A m+1 = A 1, φ and ψ. Theorem 4 Let (X, p) be a complete partial metric space, m N, A 1, A,...,A m be nonempty closed subsets of X and Y = m i=1 A i. Let f : Y YbeacyclicMK-contraction. Then f has a unique fixed point z m i=1 A i. Proof Given x 0 and let x n+1 = fx n = f n x 0,forn =0,1,,...Ifthereexistsn 0 N such that x n0 +1 = x n0,thenwefinishedtheproof.supposethatx n+1 x n for any n =0,1,,...Notice that for any n 0, there exists i n {1,,...,m} such that x n A in and x n+1 A in +1. Then by (3.1), we have p(x n, x n+1 )=p(fx n 1, fx n ) φ ( p(x n 1, x n ) ) ψ ( p(x n 1, x n ) ). Step 1. We will prove that p(x n, x n+1 )=0, thatis, d p(x n, x n+1 )=0.

11 Chen Fixed Point Theory and Applications 013, 013:17 Page 11 of 15 Since f is a cyclic MK-contraction, we can conclude that p(x n, x n+1 ) φ ( p(x n 1, x n ) ) φ(φ ( p(x n, x n 1 ) ) = φ ( p(x n, x n 1 ) ) φ n( p(x 0, x 1 ) ). Since {φ n (p(x 0, x 1 ))} n N is decreasing, it must converge to some η 0. We claim that η =0. On thecontrary,assumethat 0<η. Then by the definition of a weaker Meir-Keeler function φ, thereexistsδ > 0 such that for x 0, x 1 X with η p(x 0, x 1 )<δ + η, thereexists n 0 N such that φ n 0(p(x 0, x 1 )) < η.since φ n (p(x 0, x 1 )) = η,thereexistsk 0 N such that η φ k (p(x 0, x 1 )) < δ + η, for all k k 0.Thus,weconcludethatφ k 0+n 0 (p(x 0, x 1 )) < η. So, we get a contradiction. Therefore, φ n (p(x 0, x 1 )) = 0, and so we have p(x n, x n+1 )=0. (3.) By (p ), we also have p(x n, x n )=0. (3.3) Since d p (x, y) p(x, y) p(x, x) p(y, y) for all x, y X, using(3.) and(3.3), we obtain that d p(x n, x n+1 )=0. (3.4) Step. We show that {x n } is a Cauchy sequence in the metric space (Y, d p ). We claim that the following result holds. Claim For every ε >0,thereexistsn N such that if r, q n with r q =1mod m, then d p (x r, x q )<ε. Suppose the above statement is false. Then there exists ɛ > 0 such that for any n N, there are r n, q n N with r n > q n n with r n q n =1mod m satisfying d p (x qn, x rn ) ɛ. Now, we let n >m. Then corresponding to q n n use, we can choose r n in such a way it is the smallest integer with r n > q n n satisfying r n q n =1mod m and d p (x qn, x rn ) ɛ. Therefore, d p (x qn, x rn m) ɛ and ɛ d p (x qn, x rn ) m d p (x qn, x rn m)+ d p (x rn i, x rn i+1 ) i=1 m < ɛ + d p (x rn i, x rn i+1 ). i=1

12 Chen Fixed Point Theory and Applications 013, 013:17 Page 1 of 15 Letting n,weobtainthat d p(x qn, x rn )=ɛ. (3.5) On the other hand, we can conclude that ɛ d p (x qn, x rn ) d p (x qn, x qn+1 )+d p (x qn+1, x rn+1 )+d p (x rn+1, x rn ) d p (x qn, x qn+1 )+d p (x qn+1, x qn )+d p (x qn, x rn )+d p (x rn, x rn+1 )+d p (x rn+1, x rn ). Letting n,weobtainthat d p(x qn+1, x rn+1 )=ɛ. (3.6) Since d p (x, y)=p(x, y) p(x, x) p(y, y)andusing(3.5)and(3.6), we have that and p(x q n, x rn )= ɛ, (3.7) p(x q n+1, x rn+1 )= ɛ. (3.8) Since x qn and x rn lie in different adjacently labeled sets A i and A i+1 for certain 1 i m, by using the fact that f is a cyclic MK-contraction, we have p(x qn+1, x rn+1 )=p(fx qn, fx rn ) φ ( p(x qn, x rn ) ) ψ ( p(x qn, x rn ) ). Letting n, by using the condition φ 3 of the function φ,weobtainthat ɛ ɛ ( ) ɛ ψ, and consequently, ψ( ɛ ) = 0. By the definition of a function ψ, wegetɛ =0whichisa contraction. Therefore, our claim is proved. In the sequel, we will show that {x n } is a Cauchy sequence in the metric space (Y, d p ). Let ε >0begiven.Byourclaim,thereexistsn 1 N such that if r, q n 1 with r q =1mod m, then d p (x r, x q ) ε. Since d p (x n, x n+1 )=0,thereexistsn N such that d p (x n, x n+1 ) for any n n. ε m

13 Chen Fixed Point Theory and Applications 013, 013:17 Page 13 of 15 Let r, q max{n 1, n } and r > q. Then there exists k {1,,...,m} such that r q = k mod m. Therefore, r q + j =1mod m for j = m k +1,andsowehave d p (x q, x r ) d p (x q, x r+j )+d p (x r+j, x r+j 1 )+ + d p (x r 1, x r ) ε + j ε m ε + m ε m = ε. Thus, {x n } is a Cauchy sequence in the metric space (Y, d p ). Step 3. We show that f has a fixed point ν in m i=1 A i. Since Y is closed, the subspace (Y, p) is complete. Then from Lemma 1, wehavethat (Y, d p )iscomplete.thus,thereexistsν X such that d p(x n, ν)=0. And it follows from Lemma 1 that we have p(ν, ν)= p(x n, ν)= p(x n, x m ). (3.9) n,m On the other hand, since the sequence {x n } is a Cauchy sequence in the metric space (Y, d p ), we also have d p(x n, x m )=0. Since d p (x, y)=p(x, y) p(x, x) p(y, y), we can deduce that p(x n, x m ) = 0. (3.10) Since Y = m i=1 A i is a cyclic representation of X with respect to f,thesequence{x n } has infinite terms in each A i for i {1,,...,m}. Now, for all i =1,,...,m,wemaytakeasubsequence {x nk } of {x n } with x nk A i 1 and also all converge to ν.using(3.9)and(3.10), we have By (3.1), p(ν, ν)= p(x n, ν)= p(x n k, ν)=0. p(x nk+1, f ν)=p(fx nk, f ν) φ ( p(x nk, ν) ) ψ ( p(x nk, ν) ) φ ( p(x nk, ν) ). Letting k,wehave p(ν, f ν) 0, and so ν = f ν.

14 Chen Fixed Point Theory and Applications 013, 013:17 Page 14 of 15 Step 4. Finally, to prove the uniqueness of the fixed point, let μ be another fixed point of f in m i=1 A i. By the cyclic character of f,wehaveμ, ν n i=1 A i.sincef is a cyclic weaker MK-contraction, we have p(ν, μ)=p(ν, f μ) = p(x n k+1, f μ) = p(fx n k, f μ) [ ( φ p(xnk, μ) ) ψ ( p(x nk, μ) )] p(ν, μ) ψ ( p(ν, μ) ), and we can conclude that ψ ( p(ν, μ) ) =0, which implies p(ν, μ) = 0. So, we have μ = ν. We complete the proof. The following provides an example for Theorem 4. Example Let X = [0, 1] and A = [0, 1], B =[0, 1 ], C =[0, 1 ]. We define the partial metric 4 p on X by p(x, y)=max{x, y} for all x, y X, and define the function f : X X by x f (x)= 1+x for all x X. Now, we let ψ, φ : R + R + be φ(t)= 4t 5 and ψ(t)= t 5(1 + t). Then f is a cyclic MK-contraction and 0 is the unique fixed point. By Theorem 4, it is easy to get the following corollary. Corollary 1 Let (X, p) be a complete partial metric space, m N, A 1, A,...,A m nonempty closed subsets of X, Y = m i=1 A i and let f : Y Y. Assume that (1) m i=1 A i is a cyclic representation of Y with respect to f ; () for any x A i, y A i+1, i =1,,...,m, be p(fx, fy) φ ( p(x, y) ), where A m+1 = A 1 and φ. Then f has a unique fixed point z m i=1 A i.

15 Chen Fixed Point Theory and Applications 013, 013:17 Page 15 of 15 Competing interests The author declares that they have no competing interests. Acknowledgements The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper. Received: 6 November 01 Accepted: 13 January 013 Published: 8 January 013 References 1. Banach, S: Sur les operations dans les ensembles abstraits et leur application aux equations integerales. Fundam. Math. 3, (19). Boyd, DW, Wong, SW: On nonlinear contractions. Proc. Am. Math. Soc. 0, (1969) 3. Mattews, SG: Partial metric topology. In: Proc. 8th Summer of Conference on General Topology and Applications. Ann. New York Acad. Sci., vol. 78, pp (1994) 4. Oltra, S, Valero, O: Banach s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste 36, 17-6 (004) 5. Kirk, WA, Srinivasan, PS, Veeramani, P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory Appl. 4(1), (003) 6. Karapinar, E: Fixed point theory for cyclic weaker φ-contraction. Appl. Math. Lett. 4(6),8-85 (011) 7. Pǎcurar, M, Rus, IA: Fixed point theory for cyclic ϕ-contractions. Nonlinear Anal. 7(3-4), (010) 8. Rus, IA: Cyclic representations and fixed points. Ann. Tiberiu Popoviciu Sem. Funct. Equ. Approx. Convexity 3, (005) 9. Agarwal, RP, Alghamdi, MA, Shahzad, N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 01,ArticleID40 (01) 10. Di Bari, C, Suzuki, T, Vetro, C: Best proximity for cyclic Meir-Keeler contractions. Nonlinear Anal. 69, (008) 11. Karpagam, S, Agrawal, S: Best proximity point theorems for p-cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 009, Article ID (009) 1. Karpagam, S, Agrawal, S: Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps. Nonlinear Anal. 74, (010) 13. Karapinar, E, Erhan, IM: Best proximity point on different type contraction. Inf. Sci. Appl. Math. 5, (011) 14. Karapinar, E, Erhan, IM: Fixed point theorem for cyclic maps on partial metric spaces. Inf. Sci. Appl. Math. 6, (01) 15. Karapinar, E, Erhan, IM: Cyclic contractions and fixed point theorems. Filomat 6, (01) 16. Meir,A,Keeler,E:Atheoremoncontractionmappings.J.Math.Anal.Appl.8, (1969) doi: / Cite this article as: Chen: Fixed point theory of cyclical generalized contractive conditions in partial metric spaces. Fixed Point Theory and Applications :17.