Fixed Points of Geraghty - Type Mappings in Various Generalized Metric Spaces

and Applied Analysis 3 A partial metric space is a pair X, p such that X is a nonempty set and p is a partial metric on X. It is clear that, if p x, y 0, then from p1 and p2 x y. But if x y, p x, y may not be 0. Each partial metric p on X generates a T0 topology τp on X which has as a base the family of open p-balls {Bp x, ε : x ∈ X,ε > 0}, where Bp x, ε {y ∈ X : p x, y < p x, x ε} for all x ∈ X and ε > 0. A sequence {xn} in X, p converges to a point x ∈ X, with respect to τp, if limn→∞p x, xn p x, x . This will be denoted as xn → x, n → ∞ or limn→∞xn x. If p is a partial metric on X, then the function p : X ×X → R given by p ( x, y ) 2p ( x, y ) − p x, x − py, y 2.1 is a metric on X. Furthermore, limn→∞p xn, x 0 if and only if p x, x lim n→∞ p xn, x lim n,m→∞ p xn, xm . 2.2 Example 2.2. 1 A basic example of a partial metric space is the pair R , p , where p x, y max{x, y} for all x, y ∈ R . The corresponding metric is p ( x, y ) 2max { x, y } − x − y ∣x − y∣. 2.3 2 If X, d is a metric space and c ≥ 0 is arbitrary, then p ( x, y ) d ( x, y ) c 2.4 defines a partial metric on X and the corresponding metric is p x, y 2d x, y . Other examples of partial metric spaces which are interesting from a computational point of view may be found in 5, 15 . Remark 2.3. Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function p ·, · need not be continuous in the sense that xn → x and yn → y implies p xn, yn → p x, y . For example, ifX 0, ∞ and p x, y max{x, y} for x, y ∈ X, then for {xn} {1}, p xn, x x p x, x for each x ≥ 1 and so, for example, xn → 2 and xn → 3 when n → ∞. Definition 2.4 see 8 . Let X, p be a partial metric space. Then one has the following 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} in X converges, with respect to τp, to a point x ∈ X such that p x, x limn,m→∞p xn, xm . 4 Abstract and Applied Analysis Lemma 2.5 see 5, 6 . 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 2.6. Let X be a nonempty set. Then X, p, is called an ordered partial metric space if: i X, p is a partial metric space and ii X, is a partially ordered set. The space X, p, is called regular 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. 2.2. Some Auxiliary Results Assertions similar to the following lemma see, e.g., 16 were used and proved in the course of proofs of several fixed point results in various papers. Lemma 2.7. Let X, d be a metric space, and let {xn} be a sequence in X such that lim n→∞ d xn 1, xn 0. 2.5 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 . 2.6 As a corollary we obtain the following. Lemma 2.8. Let X, p be a partial metric space, and let {xn} be a sequence in X such that lim n→∞ p xn 1, xn 0. 2.7 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 . 2.8 Proof. Suppose that {xn} is a sequence in X, p satisfying 2.7 such that {x2n} is not Cauchy. According to Lemma 2.5, it is not a Cauchy sequence in the metric space X, p , either. Applying Lemma 2.7 we get the sequences p x2mk , x2nk , p s x2mk , x2nk 1 , p s x2mk−1, x2nk , p s x2mk−1, x2nk 1 2.9 tending to some 2ε > 0 when k → ∞. Using definition 2.1 of the associated metric and 2.7 which by p2 implies that also limn→∞p xn, xn 0 , we get that the sequences 2.8 tend to ε when k → ∞. Abstract and Applied Analysis 5 2.3. Property (P) Let X be a nonempty set and f : X → X a self-map. As usual, we denote by F f the set of fixed points of f . Following Jeong and Rhoades 17 , we say that the map f has property P if it satisfies F f F f for each n ∈ N. The proof of the following lemma is the same as in the metric case 17, Theorem 1.1 . Lemma 2.9. Let X, p be a partial metric space, and let f : X → X be a selfmap such that F f / ∅. Then f has property (P ) ifand Applied Analysis 5 2.3. Property (P) Let X be a nonempty set and f : X → X a self-map. As usual, we denote by F f the set of fixed points of f . Following Jeong and Rhoades 17 , we say that the map f has property P if it satisfies F f F f for each n ∈ N. The proof of the following lemma is the same as in the metric case 17, Theorem 1.1 . Lemma 2.9. Let X, p be a partial metric space, and let f : X → X be a selfmap such that F f / ∅. Then f has property (P ) if p ( fx, f2x ) ≤ λpx, fx 2.10 holds for some λ ∈ 0, 1 and either i for all x ∈ X or ii for all x / fx. 2.4. Metric Type Spaces Definition 2.10 see 11 . Let X be a nonempty set, K ≥ 1 a real number, and let a function D : X ×X → R satisfy the following properties: a D x, y 0 if and only if x y; b D x, y D y, x for all x, y ∈ X; c D x, z ≤ K D x, y D y, z for all x, y, z ∈ X. Then X,D,K is called a metric type space. Obviously, for K 1, metric type space is simply a metric space. The notions such as convergent sequence, Cauchy sequence, and complete space are defined in an obvious way. A metric type space may satisfy some of the following additional properties: d D x, z ≤ K D x, y1 D y1, y2 · · · D yn, z for arbitrary points x, y1, y2, . . . , yn, z ∈ X; e function D is continuous in two variables, that is, xn −→ x and yn −→ y in X,D,K implies D ( xn, yn ) −→ Dx, y. 2.11 The last condition is in the theory of symmetric spaces usually called “property HE ”. Condition d was used instead of c in the original definition of a metric type space by Khamsi 11 . Note that weaker version of property e : e′ xn → x and yn → x in X,D,K implies that D xn, yn → 0 is satisfied in an arbitrary metric type space. It can also be proved easily that the limit of a sequence in a metric type space is unique. Indeed, if xn → x and xn → y in X,D,K and D x, y ε > 0, then 0 ≤ Dx, y ≤ KD x, xn D ( xn, y )) < K ( ε 2K ε 2K ) ε 2.12 for sufficiently large n, which is impossible. 6 Abstract and Applied Analysis 3. Results 3.1. Results in Partial Metric Spaces Theorem 3.1. Let X, p be a complete partial metric space, and let f : X → X be a self-map. Suppose that there exists β ∈ S such that p ( fx, fy ) ≤ βpx, ypx, y 3.1 holds for all x, y ∈ X. Then f has a unique fixed point z ∈ X and for each x ∈ X the Picard sequence {fnx} converges to z when n → ∞. Proof. Let x1 ∈ X be arbitrary, and let xn 1 fxn for n ∈ N. Consider the following two cases: 1 p xn0 1, xn0 0 for some n0 ∈ N; 2 p xn 1, xn > 0 for each n ∈ N. Case 1. Under this assumption we get that p xn0 2, xn0 1 p ( fxn0 1, fxn0 ) ≤ βp xn0 1, xn0 ) p xn0 1, xn0 β 0 · 0 0, 3.2 and it follows that p xn0 2, xn0 1 0. By induction, we obtain that p xn 1, xn 0 for all n ≥ n0 and so xn xn0 for all n ≥ n0. Hence, {xn} is a Cauchy sequence, converging to xn0 which is a fixed point of f . Case 2. Wewill prove first that in this case the sequence p xn 1, xn is decreasing and tends to 0 as n → ∞. For each n ∈ N we have that 0 < p xn 2, xn 1 p ( fxn 1, fxn ) ≤ βp xn 1, xn ) p xn 1, xn < p xn 1, xn . 3.3 Hence, p xn 1, xn is decreasing and bounded from below, thus converging to some q ≥ 0. Suppose that q > 0. Then, it follows from 3.3 that p xn 2, xn 1 p xn 1, xn ≤ βp xn 1, xn ) < 1, 3.4 where from, passing to the limit when n → ∞, we get that limn→∞β p xn 1, xn 1. Using property 1.1 of the function β, we conclude that limn→∞p xn 1, xn 0, that is, q 0, a contradiction. Hence, limn→∞p xn 1, xn 0 is proved. In order to prove that {xn} is a Cauchy sequence in X, p , suppose the contrary. As was already proved, p xn 1, xn → 0 as n → ∞, and so, using p2 , p xn, xn → 0 as n → ∞. Hence, using 2.1 , we get that p xn 1, xn → 0 as n → ∞. Using Lemma 2.8, 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 x2mk , x2nk , p x2mk , x2nk 1 , p x2mk−1, x2nk , p x2mk−1, x2nk 1 . 3.5 Abstract and Applied Analysis 7 Putting in the contractive condition x x2mk−1 and y x2nk , it follows that p x2mk , x2nk 1 ≤ β ( p x2mk−1, x2nk ) p x2mk−1, x2nk < p x2mk−1, x2nk . 3.6and Applied Analysis 7 Putting in the contractive condition x x2mk−1 and y x2nk , it follows that p x2mk , x2nk 1 ≤ β ( p x2mk−1, x2nk ) p x2mk−1, x2nk < p x2mk−1, x2nk . 3.6