Coupled Coincidence Points of Mappings in Ordered Partial Metric Spaces

and Applied Analysis 3 Definition 2.3. Let X, p be a partial metric space. Then, 1 a sequence {xn} in X, p is called a Cauchy sequence if limn,m→∞p xn, xm exists and is finite ; 2 the space X, p is said to be complete if every Cauchy sequence {xn} inX converges, with respect to τp, to a point x ∈ X such that p x, x limn,m→∞ p xn, xm . Lemma 2.4. Let X, p be a partial metric space. a {xn} is a Cauchy sequence in X, p if and only if it is a Cauchy sequence in the metric space X, p . b The space X, p is complete if and only if the metric space X, p is complete. Definition 2.5. Let X be a nonempty set. Then X, , p is called an ordered partial metric space if i X, is a partially ordered set, and ii X, p is a partial metric space. We will say that the space X, , p satisfies the ordered-regular condition abr. ORC if the following holds: if {xn} is a nondecreasing sequence in X with respect to such that xn → x ∈ X as n → ∞, then xn x for all n ∈ N. Definition 2.6 see 13, 14 . Let X, be a partially ordered set, F : X ×X → X, and g : X → X. 1 F is said to have g-mixed monotone property if the following two conditions are satisfied: ∀x1, x2, y ∈ X ) gx1 gx2 ⇒ F ( x1, y ) Fx2, y ) , ∀x, y1, y2 ∈ X ) gy1 gy2 ⇒ F ( x, y1 ) Fx, y2 ) . 2.4 If g iX the identity map , we say that F has the mixed monotone property. 2 A point x, y ∈ X ×X is said to be a coupled coincidence point of F and g if F x, y gx and F y, x gy and their common coupled fixed point if F x, y gx x and F y, x gy y. Definition 2.7 see 20 . Let X, d be a metric space, and let F : X ×X → X and g : X → X. The pair F, g is said to be compatible if lim n→∞ d ( gF ( xn, yn ) , F ( gxn, gyn )) 0, lim n→∞ d ( gF ( yn, xn ) , F ( gyn, gxn )) 0 2.5 whenever {xn} and {yn} are sequences in X such that limn→∞ F xn, yn limn→∞g xn x and limn→∞F yn, xn limn→∞g yn y for some x, y ∈ X. 2.2. Some Auxiliary Results Lemma 2.8. (i) Let X, , p be an ordered partial metric space. If relation is defined on X2 by Y V ⇐⇒ x u ∧ y v, Y x, y, V u, v ∈ X2, 2.6 4 Abstract and Applied Analysis and P : X2 ×X2 → R is given by P Y, V p x, u p ( y, v ) , Y ( x, y ) , V u, v ∈ X2, 2.7 then X2, , P is an ordered partial metric space. The space X2, , P is complete iff X, , p is complete. (ii) If F : X × X → X and g : X → X, and F has the g-mixed monotone property, then the mapping TF : X2 → X2 given by TFY ( F ( x, y ) , F ( y, x )) , Y ( x, y ) ∈ X2 2.8 is Tg-nondecreasing with respect to , that is, TgY TgV ⇒ TFY TFV, 2.9 where TgY Tg x, y gx, gy . (iii) If g is continuous in X, p (i.e., with respect to τp), then Tg is continuous in X2, P (i.e., with respect to τP ). If F is continuous from X2, P to X, p (i.e., xn → x and yn → y imply F xn, yn → F x, y ), then TF is continuous in X2, P . Proof. i Relation is obviously a partial order on X2. To prove that P is a partial metric on X2, only conditions P1 and P4 are nontrivial. P1 If Y V ∈ X2, then obviously p Y, Y p Y, V p V, V holds. Conversely, let p Y, Y p Y, V p V, V , that is, p x, x p ( y, y ) p x, u p ( y, v ) p u, u p v, v . 2.10 We know by P2 that p x, x ≤ p x, u and p y, y ≤ p y, v . Adding up, we obtain that p x, x p y, y ≤ p x, u p y, v , and since in fact equality holds, we conclude that p x, x p x, u and p y, y p y, v . Similarly, we get that p u, u p x, u and p v, v p y, v . Hence, p x, x p x, u p u, u , p ( y, y ) p ( y, v ) p v, v , 2.11 and applying property P1 of partial metric p, we get that x u and y v, that is, Y V . P4 Let Y x, y , V u, v , Z w, z ∈ X2. Then P Y, V p x, u p ( y, v ) ≤ p x,w p w,u − p w,w ) py, z p z, v − p z, z ) p x,w p ( y, z ) p w,u p z, v − p w,w p z, z ] P Y,Z P Z,V − P Z,Z . 2.12 ii, iii The proofs of these assertions are straightforward. Abstract and Applied Analysis 5 Remark 2.9. Let p be the metric associated with the partial metric p as in 2.1 . It is easy to see that, with notation as in the previous lemma, P Y, V p x, u p ( y, v ) , Y ( x, y ) , V u, v ∈ X2 2.13and Applied Analysis 5 Remark 2.9. Let p be the metric associated with the partial metric p as in 2.1 . It is easy to see that, with notation as in the previous lemma, P Y, V p x, u p ( y, v ) , Y ( x, y ) , V u, v ∈ X2 2.13 is the associated metric to the partial metric P on X2. We note, however, that when we speak about continuity of mappings, we always assume continuity in the sense of the partial metric p, that is, in the sense of the respective topology τp. This should not be confused with the approach given by O’Neill in 3 where both pand p-continuity were assumed. It is easy to see that using notation as in the previous lemma , the mappings F and g are p-compatible in the sense of Definition 2.7 if and only if the mappings TF and Tg are P-compatible in the usual sense i.e., limn→∞ d TFTgYn, TgTFYn 0, whenever {Yn} is a sequence in X2 such that limn→∞ TFYn limn→∞TgYn . Assertions similar to the following lemma see, e.g., 21 were used and proved in the course of proofs of several fixed point results in various papers. Lemma 2.10. Let X, d be a metric space, and let {xn} be a sequence in X such that {d xn 1, xn } is decreasing and lim n→∞ d xn 1, xn 0. 2.14 If {x2n} is not a Cauchy sequence, then there exist ε > 0 and two sequences {mk} and {nk} of positive integers such that the following four sequences tend to ε when k → ∞: d x2mk , x2nk , d x2mk , x2nk 1 , d x2mk−1, x2nk , d x2mk−1, x2nk 1 . 2.15 As a corollary applying Lemma 2.10 to the associated metric p of a partial metric p, and using Lemma 2.4 we obtain the following. Lemma 2.11. Let X, p be a partial metric space, and let {xn} be a sequence in X such that {p xn 1, xn } is decreasing and lim n→∞ p xn 1, xn 0. 2.16 If {x2n} is not a Cauchy sequence in X, p , then there exist ε > 0 and two sequences {mk} and {nk} of positive integers such that the following four sequences tend to ε when k → ∞: p x2mk , x2nk , p x2mk , x2nk 1 , p x2mk−1, x2nk , p x2mk−1, x2nk 1 . 2.17 3. Coupled Coincidence and Fixed Points under Geraghty-Type Conditions Let G denote the class of real functions γ : 0, ∞ → 0, 1 satisfying the condition γ tn −→ 1 implies tn −→ 0. 3.1 6 Abstract and Applied Analysis An example of a function in G may be given by γ t e−2t for t > 0 and γ 0 ∈ 0, 1 . In an attempt to generalize the Banach contraction principle, Geraghty proved in 1973 the following. Theorem 3.1 see 10 . Let X, d be a complete metric space, and let T : X → X be a self-map. Suppose that there exists γ ∈ G such that d ( Tx, Ty ) ≤ γdx, ydx, y 3.2 holds for all x, y ∈ X. Then T has a unique fixed point z ∈ X and for each x ∈ X the Picard sequence {Tnx} converges to z when n → ∞. Subsequently, several authors proved such results, including the very recent paper of D − ukić et al. 22 . We begin with the following auxiliary result. Lemma 3.2. Let X, , p be an ordered partial metric space which is complete. Let T, S : X → X be self-maps such that S is continuous, TX ⊂ SX, and one of these two subsets of X is closed. Suppose that T is S-nondecreasing (with respect to ) and there exists x0 ∈ X with Sx0 Tx0 or Tx0 Sx0. Assume also that there exists γ ∈ G such that p ( Tx, Ty ) ≤ γpSx, SypSx, Sy 3.3 holds for all x, y ∈ X such that Sx and Sy are comparable. Assume that either 1◦: T is continuous and the pair T, S is pcompatible or 2◦: X satisfies (ORC). Then, T and S have a coincidence point in X. Proof. The proof follows the lines of proof of 22, Theorems 3.1 and 3.5 . Take x0 ∈ X with, say, Sx0 Tx0, and using that T is S-nondecreasing and that TX ⊂ SX form the sequence {xn} satisfying Txn Sxn 1, n 0, 1, 2, . . ., and Sx0 Tx0 Sx1 Tx1 Sx2 · · · Txn Sxn 1 · · · . 3.4 Since Sxn−1 and Sxn are comparable, we can apply the contractive condition to obtain p Sxn 1, Sxn p Txn, Txn−1 ≤ γ ( p Sxn−1, Sxn ) p Sxn−1, Sxn ≤ p Sxn−1, Sxn . 3.5 Consider the following two cases: 1 p Sxn0 1, Sxn0 0 for some n0 ∈ N; 2 p Sxn 1, Sxn > 0 for each n ∈ N. Abstract and Applied Analysis 7 Case 1. Under this assumption, we get thatand Applied Analysis 7 Case 1. Under this assumption, we get that p Sxn0 2, Sxn0 1 p Txn0 1, Txn0 ≤ γ ( p Sxn0 1, Sxn0 ) p Sxn0 1, Sxn0 γ 0 · 0 0, 3.6 and it follows that p Sxn0 2, Sxn0 1 0. By induction, we obtain that p Sxn 1, Sxn 0 for all n ≥ n0 and so Sxn Sxn0 for all n ≥ n0. Hence, {Sxn} is a Cauchy sequence, converging to Sxn0 , and xn0 is a coincidence point of S and T . Case 2. We will prove first that in this case the sequence {p Sxn 1, Sxn } is strictly decreasing and tends to 0 as n → ∞. For each n ∈ N we have that 0 < p Sxn 2, Sxn 1 p Txn 1, Txn ≤ γ ( p Sxn 1, Sxn ) p Sxn 1, Sxn < p Sxn 1, Sxn . 3.7 Hence, p Sxn 1, Sxn is strictly decreasing and bounded from below, thus converging to some q ≥ 0. Suppose that q > 0. Then, it follows from 3.7 that p Sxn 2, Sxn 1 p Sxn 1, Sxn ≤ γp Sxn 1, Sxn ) < 1, 3.8 wherefrom, passing to the limit when n → ∞, we get that limn→∞γ p Sxn 1, Sxn 1. Using property 3.1 of the function γ , we conclude that limn→∞p Sxn 1, Sxn 0, that is, q 0, a contradiction. Hence, limn→∞p Sxn 1, Sxn 0 is proved. In order to prove that {Sxn} is a Cauchy sequence in X, p , suppose the contrary. As was already proved, p Sxn 1, Sxn → 0 as n → ∞, and so, using P2 , p Sxn, Sxn → 0 as n → ∞. Hence, using 2.1 , we get that p Sxn 1, Sxn → 0 as n → ∞. Using Lemma 2.11, we obtain that there exist ε > 0 and two sequences {mk} and {nk} of positive integers such that the following four sequences tend to ε when k → ∞: p Sx2mk , Sx2nk , p Sx2mk , Sx2nk 1 , p Sx2mk−1, Sx2nk , p Sx2mk−1, Sx2nk 1 . 3.9 Putting in the contractive condition x x2mk−1 and y x2nk , it follows that p Sx2mk , Sx2nk 1 ≤ γ ( p Sx2mk−1, Sx2nk ) p Sx2mk−1, Sx2nk < p Sx2mk−1, Sx2nk . 3.10