Chatterjea type fixed point in Partial b -metric spaces
aa r X i v : . [ m a t h . GN ] F e b CHATTERJEA TYPE FIXED POINT IN PARTIAL b -METRIC SPACES YA´E ULRICH GABA , , † , COLLINS AMBURO AGYINGI , , AND DOMINI JOCEMA LEKO Abstract.
In this paper, we give and prove two Chatterjea type fixed point theoremson partial b -metric space. We propose an extension to the Banach contaction principle onpartial b -metric space which was already presented by Shukla and also study some relatedresults on the completion of a partial metric type space. In particular, we prove a jointChatterjea-Kannan fixed point theorem. We verify the T -stability of Picard’s iteration andconjecture the P property for such maps. We also give examples to illustrate our results. Introduction and Preliminaries
In literature, one finds numerous generalizations of metric spaces and Banach contractionprinciple (BCP). In this line, Czerwik [2] proposed b -metric spaces as a generalization ofmetric spaces and proved the famous BCP in such spaces. In this sequel, Gaba [3] introducedthe so-called “metric type space” and proved a common fixed point theorem with the helpwhat he called λ -sequence in that setting.After Matthews [10] introduced partial metric spaces as a generalization of the metric space,many authors have studied fixed point theorems on theses spaces (e.g. [1, 11]), in particular,Shukla[12] gave some analog of the Banach contraction principle as well as the Kannan typefixed point theorem in partial b -metric spaces.In this paper, analogs of the Chatterjea fixed point theorem are proved.First, we recall some definitions from partial b -metric spaces. Definition 1.1. (Compare [10]) A partial metric type on a set X is a function p : X × X → [0 , ∞ ) such that:(pm1) x = y iff ( p ( x, x ) = p ( x, y ) = p ( y, y ) whenever x, y ∈ X ,(pm2) 0 ≤ p ( x, x ) ≤ p ( x, y ) whenever x, y ∈ X ,(pm3) p ( x, y ) = p ( y, x ); whenever x, y ∈ X ,(pm4) There exists a real number s ≥ p ( x, y ) + p ( z, z ) ≤ s [ p ( x, z ) + p ( z, y )]for any points x, y, z ∈ X .The pair ( X, p ) is called a partial metric type space or a partial b -metric space.It is clear that, if p ( x, y ) = 0 , then, from (pm1) and (pm2), x = y .The family B ′ of sets B ′ p ( x, ε ) := { y ∈ X : p ( x, y ) < ε + p ( x, x ) } , x ∈ X, ε > , (1.1)is a basis for a topology τ ( p ) on X . The topology τ ( p ) is T . Mathematics Subject Classification.
Primary 47H05; Secondary 47H09, 47H10.
Key words and phrases. partial b -metric; fixed point. efinition 1.2. Let (
X, p ) be a partial b -metric space. Let ( x n ) n ≥ be any sequence in X and x ∈ X . Then:(1) The sequence ( x n ) n ≥ is said to be convergent with respect to τ ( p ) (or τ ( p )-convergent)and converges to x , if lim n →∞ p ( x, x n ) = p ( x, x ). We write x n p −→ x. (2) The sequence ( x n ) n ≥ is said to be a p -Cauchy sequence iflim n →∞ ,m →∞ p ( x n , x m )exists and is finite.( X, p ) is said to be complete if for every p -Cauchy sequence ( x n ) n ≥ ⊆ X , there exists x ∈ X such that: lim n →∞ ,m →∞ p ( x n , x m ) = lim n →∞ p ( x, x n ) = p ( x, x ) . We give these additional definitions, useful to characterize some specific complete partialmetric type spaces.
Definition 1.3.
Let (
X, p ) be a partial b -metric space.The sequence ( x n ) n ≥ ⊂ X is called 0-Cauchy iflim n,m →∞ p ( x n , x m ) = 0 . ( X, p ) is called 0-complete if for every 0-Cauchy sequence ( x n ) n ≥ ⊆ X , there exists x ∈ X such that: lim n,m →∞ p ( x n , x m ) = lim n →∞ p ( x n , x ) = p ( x, x ) = 0 . BCP extension
In this section, we show that if T is a self-map on a partial metric space type space ( X, p )and has a power which is a contraction, i.e. there exists n ∈ N , n > ≤ λ < p ( T n x, T n y ) ≤ λp ( x, y ) , then there is a transformation p ′ = φ ( p ) of p such that T a contraction on ( X, p ′ )). Moreover,we prove that the partial metric type space ( X, p ′ ) is 0-complete if T is uniformly continuous.Ideas for this section are merely copies of the results presented in [4]. We adjust them in thepartial metric type setting. We begin with the following definitions. Definition 2.1.
Two partial metrics type p and p on a set X are said to be equivalent ifthere exist α, β ≥ αp ( x, y ) ≤ p ( x, y ) ≤ βp ( x, y ) , for all x, y, z ∈ X. Definition 2.2.
Given two partial metric type spaces (
X, p ) and ( Y, p ) , we say that T : ( X, p ) → ( Y, p ) is uniformly continuous if for every real number ε > λ > δ = δ ( λ ) > x, y ∈ X with p ( x, y ) < δ , we have that p ( T x, T y ) < ε . heorem 2.3. ( [12, Theorem 1.] )Let ( X, p ) be a complete partial b -metric space with coefficient s ≥ and let T : X → X bea mapping such that there exists λ ∈ [0 , satisfying p ( T x, T y ) ≤ λ p ( x, y ) , (2.1) whenever x, y ∈ X. Then T has a unique fixed point. We give the following natural corollary:
Corollary 2.4.
Let ( X, p ) be a complete partial metric type space and let T : X → X be amapping such that there exists λ ∈ [0 , satisfying p ( T n x, T n y ) ≤ λ p ( x, y ) , for some n > , whenever x, y ∈ X. Then T has a unique fixed point.Proof. By Theorem 2.3, T n has a unique fixed point, say x ∈ X with T n x = x . Since T n +1 x = T ( T n x ) = T x = T n ( T x ) , it follows that T x is a fixed point of T n , and thus, by the uniqueness of x , we have T x = x ,that is, T has a fixed point. Since, the fixed point of T is necessarily a fixed point of T n , soit is unique. (cid:3) The main theorem of this section is as follows:
Theorem 2.5.
Let d be a partial metric type on a space X and T : ( X, p ) → ( X, p ) a selfmapping such that: p ( T n x, T n y ) ≤ Kp ( x, y ) , for some n > and < K < , whenever x, y, z ∈ X. If λ is a nonnegative real such that K n < λ < , then the application p ′ : X → [0 , ∞ ) defined by : p ′ ( x, y ) = n − X i =0 λ i p ( T i x, T i y ) , whenever x, y ∈ X, satisfies: i) p ′ is a partial metric type on the space X ; ii) T : ( X, p ′ ) → ( X, p ′ ) a self mapping such that: p ′ ( T x, T y ) ≤ λ p ′ ( x, y ) . Proof.
We first prove that p ′ is a partial metric type: i.e. T is a contraction with constant λ with respect to p ′ . pm1) Indeed for x, y ∈ X, if x = y , then p ′ ( x, y ) = p ′ ( x, x ) = p ′ ( y, y ) . Conversely, assume x, y ∈ X, are such that p ′ ( x, y ) = p ′ ( x, x ) = p ′ ( y, y ), which means n − X i =0 λ i p ( T i x, T i y ) = n − X i =0 λ i p ( T i x, T i x ) = n − X i =0 λ i p ( T i y, T i y ) . It is therefore obvious that p ( T i x, T i x ) = p ( T i y, T i x ) = p ( T i y, T i y ) for i = 0 , · · · , n − , in particular p ( x, x ) = p ( y, y ) = p ( x, y ), i.e. x = y. (pm2) For all x, y ∈ X and for all i = 0 , · · · , n − , we have0 ≤ p ( T i x, T i x ) ≤ p ( T i x, T i y ) , and hence n − X i =0 λ i p ( T i x, T i x ) ≤ n − X i =0 λ i p ( T i x, T i y )i.e. p ′ ( x, x ) ≤ p ′ ( x, y ) . (pm3) For all x, y ∈ X , p ′ ( x, y ) = n − X i =0 λ i p ( T i x, T i y ) = n − X i =0 λ i p ( T i y, T i x ) = p ′ ( y, x ) , that is p ′ ( x, y ) = p ′ ( y, x )for all x, y ∈ X. (pm4) For all x, y, a ∈ X , since λ i [ p ( T i x, T i y ) + p ( T i a, T i a )] ≤ λ i s [ p ( T i x, T i a ) + p ( T i a, T i y )] , we get p ′ ( x, y ) = n − X i =0 λ i p ( T i x, T i y ) ≤ n − X i =0 λ i s [ p ( T i x, T i a ) + p ( T i a, T i y )] − n − X i =0 λ i p ( T i a, T i a )= s n − X i =0 λ i [ p ( T i x, T i a ) + p ( T i a, T i y )] − n − X i =0 λ i p ( T i a, T i a )= s [ p ′ ( x, a ) + p ′ ( a, y )] − p ′ ( a, a ) . So p ′ ( x, y ) + p ′ ( a, a ) ≤ s [ p ′ ( x, a ) + p ′ ( a, y )] or any x, y, a ∈ X. Hence, p ′ is a partial metric type space on X .We now prove that T : ( X, p ′ ) → ( X, p ′ ) is a contraction with constant λ .It is readily seen, by a simple computation, that p ′ ( T x, T y ) = 1 λ [ p ′ ( x, y ) − p ( x, y )] + λ n − p ( T n x, T n y ) . Since T n : ( X, p ) → ( X, p ) is a contraction with constant K , it follows that p ′ ( T x, T y ) ≤ λ [ p ′ ( x, y ) − p ( x, y )] + Kλ n − p ( x, y )= 1 λ p ′ ( x, y ) + (cid:18) K − λ n (cid:19) λ n − p ( x, y ) ≤ λ p ′ ( x, y ) , because of the choice K n < λ . This completes the proof. (cid:3)
As observed in [4, Remark 2.2], under the assumptions of Theorem 2.5, it is readily seenthat p ′ ( x, y ) ≤ ∞ X i =0 λ i p ( T i x, T i y ) ≤ p ′ ( x, y ) + λ n Kp ′ ( x, y ) + λ n K p ′ ( x, y ) + · · · = 11 − λ n K p ′ ( x, y ) . The term h ( x, y ) := P ∞ i =0 λ i p ( T i x, T i y ) therefore defines a partial metric type, equivalent to p ′ , as long as the series happen to converge for some λ > T : ( X, p ) → ( X, p ) is uniformly continuousand the partial metric type p is 0-complete, then the partial metric type p ′ is also 0-complete. Theorem 2.6.
We repeat the assumptions of Theorem 2.5. If T is uniformly continuousand the partial metric type p is -complete, then so is the partial metric type p ′ .Proof. Since p ( x, y ) ≤ p ′ ( x, y ) for any x, y ∈ X , any 0-Cauchy sequence in ( X, p ′ ) is also a0-Cauchy sequence in ( X, p ). It is therefore enough to prove that, under uniform continuityof T , any convergent sequence ( x n ) n ≥ ⊂ ( X, p ) such that there exists x ∗ ∈ X withlim n →∞ p ( x n , x ∗ ) = lim n,m →∞ p ( x n , x m ) = p ( x ∗ , x ∗ ) = 0 , is such that there exists y ∗ ∈ X withlim n →∞ p ′ ( x n , y ∗ ) = lim n,m →∞ p ′ ( x n , x m ) = p ′ ( y ∗ , y ∗ ) = 0 . o let ( x n ) n ≥ be a sequence in the G -metric space ( X, p ) such that ( x n ) n ≥ converges tosome ξ ∈ ( X, p ) . and p ( ξ, ξ ) = 0 . Set M = max { λ i , i = 1 , · · · , n − } and observe that M ≥ λ > . Since all the powers of T are also uniformly continuous in ( X, d ), we can write that, for any ε >
0, there exists η > x, y ∈ X, and i = 1 , · · · , n − p ( x, y ) < η = ⇒ p ( T i x, T i y ) < εM n . Since { x n } converges to some ξ ∈ ( X, p ) , and p ( ξ, ξ ) = 0 there exists n ∈ N such that k ≥ n = ⇒ p ( ξ, x k ) < η. Then k > n = ⇒ p ( T i ξ, T i x k ) < εM n for i = 1 , · · · , n − , i.e. p ′ ( ξ, x k , ) < εn (cid:20) M + λM + · · · + λ n − M (cid:21) < ε. Thus ( x n ) n ≥ converges to ξ with respect to the partial metric space p ′ and p ′ ( ξ, ξ ) = 0 . This completes the proof. (cid:3)
In concluding this section, we introduce what we call partial ultra-metrics and conjecturethat the construction of Frink[5] could be used to obtain a modular metric from an ultra-modular metric. Taking inspiration from the theory of ultra-metric space and that of metrictype spaces (see [3]), we can define:
Definition 2.7.
A partial ultra-metric on the set X is is a function p : X × X → [0 , ∞ )such that:(pm1) x = y iff ( p ( x, x ) = p ( x, y ) = p ( y, y ) whenever x, y ∈ X ,(pm2) 0 ≤ p ( x, x ) ≤ p ( x, y ) whenever x, y ∈ X ,(pm3) p ( x, y ) = p ( y, x ); whenever x, y ∈ X ,(pm4) here exists a real number s ≥ p ( x, y ) + p ( z, z ) ≤ max { p ( x, z ) + p ( z, y ) } for any points x, y, z ∈ X .The pair ( X, p ) is called a partial ultra-metric space .We are interested in the following question:
Problem 2.8.
Given a partial ultra metric ω on a non empty set X , can we construct apartial metric type ω ′ on X such that ω and ω ′ are equivalent? If not, are there conditionswhich guarantee the existence of such a partial metric type ω ′ on X ?The authors plan to take up this investigation [8] by using “ the chain construction ” as atool. . Main results
In this section, we present some fixed point results for Chatterjea type mapping in the settingof a partial b -metric space. Following theorem is an analog to Chatterjea fixed point theoremin partial b -metric space. Theorem 3.1.
Let ( X, p ) be a complete partial b -metric space with coefficient s ≥ and T : X → X be a mapping satisfying the following condition: p ( T x, T y ) ≤ λ [ p ( x, T y ) + p ( y, T x )] . (Ch) for all x, y ∈ X , where λ ∈ (cid:2) , s (cid:1) . Then T has a unique fixed point u ∈ X and p ( u, u ) = 0 .Proof. Let us first show that if T has a fixed point u , then it is unique and p ( u, u ) = 0.From (Ch), we have p ( u, u ) = p ( T u, T u ) ≤ λ [ p ( u, T u ) + p ( u, T u )] = 2 λp ( u, T u ) < p ( u, u ) , a contradiction, unless p ( u, u ) = 0 . Suppose u, v ∈ X are two distinct fixed points of T , that is, T u = u , T v = v and u = v. .Then it follows from (Ch) that p ( u, v ) = p ( T u, T v ) ≤ λ [ p ( u, T v ) + p ( v, T u )] ≤ λp ( u, v ) < p ( u, v )a contradiction, unless p ( u, v ) = 0, i.e. u = v. Thus if a fixed point of T exists, thenit is unique. For existence of fixed point, let x ∈ X be arbitrary; set x n = T n x and b n = p ( x n , x n +1 ). Without loss of generality, we may assume that b n > n ≥ x n is a fixed point of T for at least one n ≥ n ∈ N , it follows from (Ch) that b n = p ( x n , x n +1 ) = p ( T x n − , T x n ) ≤ λ [ p ( x n − , x n +1 ) + p ( x n , x n )] ≤ λ [ p ( x n − , x n ) + p ( x n , x n +1 ) − p ( x n , x n ) + p ( x n , x n )]= λ [ p ( x n − , x n ) + p ( x n , x n +1 )]= λ [ b n − + b n ] , therefore b n ≤ µb n − where µ = λ − λ < λ ∈ (cid:2) , s (cid:1) ⊂ (cid:2) , (cid:1) ). On repeating this, oneobtains b n ≤ µ n b (3.1)hence lim n →∞ b n = 0 . For m, n ∈ N with m > n , we obtain ( x n , x m ) ≤ s [ p ( x n , x n +1 ) + p ( x n +1 , x m )] − p ( x n +1 , x n +1 ) ≤ sp ( x n , x n +1 ) + s [ p ( x n +1 , x n +2 ) + p ( x n +2 , x m )] − sp ( x n +2 , x n +2 ) ≤ sp ( x n , x n +1 ) + s p ( x n +1 , x n +2 ) + s p ( x n +2 , x n +2 )+ · · · + s m − n p ( x m − , x m ) . Using (3.1) in the above inequality, p ( x n , x m ) ≤ sµ n [1 + sµ + ( sµ ) + · · · ] p ( x , x ) ≤ sµ n − sµ p ( x , x ) . As λ ∈ (cid:2) , s (cid:1) ⊂ (cid:2) , s (cid:1) and s >
1, it follows from the above inequality thatlim n,m →∞ p ( x n , x m ) = 0 . Therefore, ( x n ) is a Cauchy sequence in X . By completeness of X there exists x ∗ ∈ X suchthat lim n →∞ p ( x ∗ , x n ) = lim n,m →∞ p ( x n , x m ) = p ( x ∗ , x ∗ ) = 0 . (3.2)We shall show that x ∗ is a fixed point of T .For any n ∈ N it follows from (Ch) that p ( x ∗ , T x ∗ ) ≤ s [ p ( x ∗ , x n +1 ) + p ( x n +1 , T x ∗ )] − p ( x n +1 , x n +1 ) ≤ s [ p ( x ∗ , x n +1 ) + p ( T x n , T x ∗ )] ≤ s [ p ( x ∗ , x n +1 ) + λ ( p ( x n , T x ∗ ) + p ( x ∗ , x n +1 ))] ≤ sp ( x ∗ , x n +1 ) + sλp ( x ∗ , x n +1 )+ s λ [ p ( x n , x ∗ ) + p ( x ∗ , T x ∗ )] − sλp ( x ∗ , x ∗ ) . Taking limit as n → ∞ , as p ( x ∗ , x ∗ ) = 0 , we have p ( x ∗ , T x ∗ ) ≤ s λp ( x ∗ , T x ∗ ) < p ( x ∗ , T x ∗ ) , –a contradiction, unless p ( x ∗ , T x ∗ ) = 0 , that is, T x ∗ = x ∗ . Thus, x ∗ is the unique fixed pointof T . (cid:3) Theorem 3.2.
Let ( X, p ) be a complete partial b -metric space with coefficient s > and T : X → X be a mapping satisfying the following condition: p ( T x, T y ) ≤ λ max { p ( x, y ) , p ( x, T y ) , p ( y, T x ) } . (Ch2) for all x, y ∈ X , where λ ∈ (cid:2) , s (cid:1) . Then T has a unique fixed point u ∈ X and p ( u, u ) = 0 .Proof. Let us first show that if T has a fixed point u , then it is unique and p ( u, u ) = 0.Suppose u, v ∈ X are two distinct fixed points of T , that is, T u = u , T v = v and u = v. .Then it follows from (Ch2) that ( x n +1 , x n ) = p ( T x n , T x n − ) ≤ λ max { p ( x n , x n − ) , p ( x n , x n ) , p ( x n − , x n +1 ) }≤ max { p ( x n , x n − ) , p ( x n − , x n +1 ) } since p ( x, x ) ≤ p ( x, y ) whenever x, y ∈ X. At this point, we distinguish between two cases.Case 1. max { p ( x n , x n − ) , p ( x n − , x n +1 ) } = p ( x n , x n − ) p ( x n +1 , x n ) ≤ λp ( x n − , x n ) . Iterating this process, we get p ( x n +1 , x n ) ≤ λ n p ( x , x ) , for all n ∈ N .From the proof of the previous theorem, we can easily establish that for m, n ∈ N with m > n , p ( x n , x m ) ≤ sλ n − sλ p ( x , x ) . As λ ∈ (cid:2) , s (cid:1) and s >
1, it follows from the above inequality thatlim n,m →∞ p ( x n , x m ) = 0 . Therefore, ( x n ) is a Cauchy sequence in X . By completeness of X there exists x ∗ ∈ X such thatlim n →∞ p ( x ∗ , x n ) = lim n,m →∞ p ( x n , x m ) = p ( x ∗ , x ∗ ) = 0 . (3.3)We shall show that x ∗ is a fixed point of T . For any n ∈ N , we have p ( x ∗ , T x ∗ ) ≤ s [ p ( x ∗ , x n +1 ) + p ( x n +1 , T x ∗ )] − p ( x n +1 , x n +1 ) ≤ s [ p ( x ∗ , x n +1 ) + p ( x n +1 , T x ∗ )] ≤ sp ( x ∗ , x n +1 ) + sλp ( x ∗ , x n ) . Using (3.3) in the above inequality we obtain p ( x ∗ , T x ∗ ) = 0, that is, T x ∗ = x ∗ .Thus, x ∗ is the unique fixed point of T .Case 2. If max { p ( x n , x n − ) , p ( x n − , x n +1 ) } = p ( x n +1 , x n − ), a similar argument as in the Case1 leads to the existence of a unique fixed point of T . (cid:3) Problem 3.3.
Theorem 3.1 advocates for the existence of a fixed point for a Chatterjeacontraction in a complete partial b -metric space for which the constant s is such that s ≥ < s < λ. Of course heorem 3.1 remains true for the sharp inequality s ≥ √ ≤ s < √ . We conclude this section by presenting a joint Chatterjea-Kannan fixed point leading to theexistence of a unique fixed point.
Theorem 3.4.
Let ( X, p ) be a -complete partial b -metric space with coefficient s ≥ and T : X → X be a self mapping satisfying the following condition: p ( T x, T y ) ≤ λ p ( x, y ) + λ p ( x, T x ) p ( y, T y )1 + p ( x, y ) + λ p ( x, T y ) p ( y, T x )1 + p ( x, y ) (Ch-Ka)+ λ p ( x, T x ) p ( x, T y )1 + p ( x, y ) + λ p ( y, T y ) p ( y, T x )1 + p ( x, y ) , for all x, y ∈ X , where λ , λ , λ , λ , λ are nonnegative real numbers satisfying: λ + λ + 2 sλ + sλ + sλ < . Then T has a unique fixed point u in X and p ( u, u ) = 0 . In proving this theorem, we shall need the following lemma.
Lemma 3.5.
Let ( X, p ) be a partial b -metric space with coefficient s ≥ and T : X → X bea self mapping. Suppose that ( x n ) is a sequence in X constructed as x n +1 = T x n and suchthat p ( x n , x n +1 ) ≤ λp ( x n − , x n ) , for all n ∈ N , where λ ∈ [0 , is a constant. Then ( x n ) is a -Cauchy sequence.Proof. Let x ∈ X and construct a Picard iterative sequence ( x n ) by x n +1 = T x n , ( n ∈ N ).We distinguish the following three cases.Case 1. λ ∈ [0 , s ) ( s > p ( x n , x n +1 ) ≤ λp ( x n − , x n ), we have p ( x , x n +1 ) ≤ λ n p ( x , x ) . Thus, for any n > m and n, m ∈ N , we have, by following the proof of Theorem 3.1 p ( x m , x n ) ≤ s [ p ( x m , x m +1 ) + p ( x m +1 , x n )] − p ( x m +1 , x m +1 ) ≤ s [ p ( x m , x m +1 ) + p ( x m +1 , x n ) ≤ p ( x m , x m +1 ) + s p ( x m +1 , x m +2 ) + s [ p ( x m +2 , x m +3 ) + p ( x m +3 , x n )]... ≤ sλ m (1 + sλ + s λ ) + · · · + s n − m − λ n − m − ) p ( x , x ) ≤ sλ m " ∞ X i =0 ( sλ ) i p ( x , x )= sλ m − sλ p ( x , x ) → m → ∞ ) , which implies that ( x n ) is a 0-Cauchy sequence. ase 2. Let λ ∈ [ s ,
1) ( s > λ n → n → ∞ . So there is n o ∈ N such that λ n o < s . Thus, by Case 1, we claim that { ( T n o ) x } n ≥ := { x n o , x n o +1 , · · · , x n o + n , · · · } is a 0-Cauchy sequence. Then ( x n ) is a 0-Cauchy sequence.Case 3. Let s = 1. Similar to the process of Case 1, the claim holds. (cid:3) Now, we prove the Theorem 3.4.
Proof.
Choose x ∈ X and construct a Picard iterative sequence ( x n ) by x n +1 = T x n , ( n ∈ N ). If there exists n o ∈ N such that x n o = x n o +1 , then x n o = x n o +1 = T x n o i.e. x n o is afixed point of T . Next, without loss of generality, let x n = x n +1 for all n ∈ N . By (Ch-Ka),we have p ( x n , x n +1 ) = ( T x n − , T x n ) ≤ λ p ( x n − , x n ) + λ p ( x n − , T x n − ) p ( x n , T x n )1 + p ( x n − , x n )+ λ p ( x n − , T x n ) p ( x n , T x n − )1 + p ( x n − , x n ) + λ p ( x n − , T x n − ) p ( x n − , T x n )1 + p ( x n − , x n )+ λ p ( x n , T x n ) p ( x n , T x n − )1 + p ( x n − , x n )= λ p ( x n − , x n ) + λ p ( x n − , x n ) p ( x n , x n +1 )1 + p ( x n − , x n ) + λ p ( x n − , x n +1 ) p ( x n , x n )1 + p ( x n − , x n )+ λ p ( x n − , x n ) p ( x n − , x n +1 )1 + p ( x n − , x n ) + λ p ( x n , x n +1 ) p ( x n , x n )1 + p ( x n − , x n ) . In view of axioms (pm2) and (pm4), we have λ p ( x n − , x n +1 ) p ( x n , x n )1 + p ( x n − , x n ) ≤ λ p ( x n − , x n +1 ) p ( x n , x n ) p ( x n − , x n ) ≤ λ p ( x n − , x n +1 ) ≤ sλ [ p ( x n − , x n ) + p ( x n , x n +1 )] , i.e. p ( x n − , x n +1 ) p ( x n , x n )1 + p ( x n − , x n ) ≤ sλ [ p ( x n − , x n ) + p ( x n , x n +1 )] . We also have λ p ( x n − , x n ) p ( x n − , x n +1 )1 + p ( x n − , x n ) ≤ λ p ( x n − , x n ) p ( x n − , x n +1 ) p ( x n − , x n ) ≤ λ p ( x n − , x n +1 ) ≤ sλ [ p ( x n − , x n ) + p ( x n , x n +1 )] , .e. λ p ( x n − , x n ) p ( x n − , x n +1 )1 + p ( x n − , x n ) ≤ sλ [ p ( x n − , x n ) + p ( x n , x n +1 )] . Hence p ( x n , x n +1 ) = ( T x n − , T x n ) ≤ λ p ( x n − , x n ) + λ p ( x n , x n +1 ) + sλ [ p ( x n − , x n ) + p ( x n , x n +1 )]+ sλ [ p ( x n − , x n ) + p ( x n , x n +1 )] + λ p ( x n , x n +1 ) . It follows that (1 − λ − sλ − sλ ) p ( x n , x n +1 ) ≤ ( λ + sλ + sλ ) p ( x n − , x n ) . (3.4)Again, by (Ch-Ka), and exploiting the symmetry of p , i.e. p ( x n , x n +1 ) = p ( T x n , T x n − ), weare led to (1 − λ − sλ − sλ ) p ( x n , x n +1 ) ≤ ( λ + sλ + sλ ) p ( x n − , x n ) (3.5)Adding up (3.4) and (3.5) yields p ( x n , x n +1 ) ≤ λ + 2 sλ + sλ + sλ − λ − sλ − sλ − sλ p ( x n − , x n )Put λ = λ +2 sλ + sλ + sλ − λ − sλ − sλ − sλ . In view of λ + λ + 2 sλ + sλ + sλ < , then 0 ≤ λ < . Thus, by Lemma 3.5, ( x n ) is a 0-Cauchy sequence in X . Since ( X, p ) is 0-complete, thenthere exists some point x ∗ ∈ X such that:lim n,m →∞ p ( x n , x m ) = lim n →∞ p ( x n , x ∗ ) = p ( x ∗ , x ∗ ) = 0 . By (Ch-Ka), it is easy to see that p ( x n +1 , T x ∗ ) = p ( T x n , T x ∗ ) ≤ λ p ( x n , x ∗ ) + λ p ( x n , x n +1 ) p ( x ∗ , T x ∗ )1 + p ( x n , x ∗ )+ λ p ( x n , T x ∗ ) p ( x ∗ , x n +1 )1 + p ( x n , x ∗ ) + λ p ( x n , x n +1 ) p ( x n , T x ∗ )1 + p ( x n , x ∗ )+ λ p ( x ∗ , T x ∗ ) p ( x ∗ , x n +1 )1 + p ( x n , x ∗ )Taking the limit as n → ∞ , we get lim n →∞ p ( x n +1 , T x ∗ ) = 0On another side, p ( x ∗ , T x ∗ ) ≤ s [ p ( x ∗ , x n +1 ) + p ( x n +1 , T x ∗ )] − p ( x n +1 , x n +1 )Taking the limit on both sides as n → ∞ , we get p ( x ∗ , T x ∗ ) = 0 . It gives that
T x ∗ = x ∗ . In other words, x ∗ is a fixed point of T . or uniqueness of the fixed point, assume y ∗ is another fixed point of T , then by (Ch-Ka),it is easy to check that p ( x ∗ , y ∗ ) = p ( T x ∗ , T y ∗ ) ≤ λ p ( x ∗ , y ∗ ) + λ p ( x ∗ , y ∗ )= ( λ + λ ) p ( x ∗ , y ∗ ) . Because λ + λ + 2 sλ + sλ + sλ < λ + λ < , we conclude that x ∗ = y ∗ since p ( x ∗ , y ∗ ) = 0 . (cid:3) Corollary 3.6.
Let ( X, p ) be a complete partial metric space with coefficient s ≥ and T : X → X be a self mapping satisfying the following condition: p ( T x, T y ) ≤ λ p ( x, y ) + λ p ( x, T x ) p ( y, T y )1 + p ( x, y ) + λ p ( x, T y ) p ( y, T x )1 + p ( x, y ) (Ch-Ka)+ λ p ( x, T x ) p ( x, T y )1 + p ( x, y ) + λ p ( y, T y ) p ( y, T x )1 + p ( x, y ) . for all x, y ∈ X , where λ , λ , λ , λ , λ are nonnegative real numbers satisfying: λ + λ + λ + λ + λ < . Then T has a unique fixed point in X .Proof. Take s = 1 in Theorem 3.4, thus the claim holds. (cid:3) Remark . Take λ = λ = λ = λ = 0 in Theorem 3.4 or in Corollary 3.6, then Theorem3.4 and Corollary 3.6 are reduced to [12, Theorem 2.4] and Banach contraction principle,respectively. From this point of view, our results are genuine generalizations of the previousresults.Recently, Qing and Rhoades [13] established the notion of T -stability of Picard’s iterationin metric space. In the following, we modify their definition and introduce the concept of T -stability of Picard’s iteration in partial b -metric space. Definition 3.8.
Let (
X, p ) be a partial b -metric space, x ∈ X and T : X → X be amapping with F ( T ) = ∅ , where F ( T ) denotes the set of all fixed points of T . Then Picard’siteration x n +1 = T x n is said to be T -stable with respect to T if x n p −→ q, q ∈ F ( T ) andwhenever ( y n ) is a sequence in X with lim n →∞ p ( y n +1 , T y n ) = 0, we have y n p −→ q .What follows is a useful lemma for the proof of our main result in this section. Lemma 3.9. [9]
Let ( a n ) , ( c n ) be nonnnegative sequences satisfying a n +1 ≤ ha n + c n for all n ∈ N , ≤ h < , lim n →∞ c n = 0 . Then lim n →∞ a n = 0 . Now we state our main result on T -stability. Theorem 3.10.
Under the conditions of Theorem 3.4, if sλ + 2 λ + ( s + s )( λ + λ ) < ,then Picard’s iteration is T -stable. roof. From Theorem 3.4, we know that T has a unique fixed point x ∗ ∈ X and p ( x ∗ , x ∗ ) = 0.Assume that ( y n ) is a sequence in X with lim n →∞ p ( y n +1 , T y n ) = 0. Taking advantage of(Ch-Ka), on the one hand, we have p ( T y n , x ∗ ) = p ( T y n , T x ∗ ) ≤ λ p ( y n , x ∗ ) + λ p ( y n , T y n ) p ( x ∗ , T x ∗ )1 + p ( y n , x ∗ ) + λ p ( y n , T x ∗ ) p ( x ∗ , T y n )1 + p ( y n , x ∗ )+ λ p ( y n , T y n ) p ( y n , T x ∗ )1 + p ( y n , x ∗ ) + λ p ( x ∗ , T x ∗ ) p ( x ∗ , T y n )1 + p ( y n , x ∗ ) ≤ λ p ( y n , x ∗ ) + λ p ( x ∗ , T y n ) + λ p ( y − n, T y n ) ≤ ( λ + sλ ) p ( y n , x ∗ ) + ( λ + sλ ) p ( x ∗ , T y n ) , which means (1 − λ − sλ ) p ( T y n , x ∗ ) ≤ ( λ + sλ ) p ( y n , x ∗ ) . (3.6)On the other hand, owing to the symmetry of p , we have p ( T y n , x ∗ ) = p ( T x ∗ , T x n ) , which yields 1 − λ − sλ ) p ( T y n , x ∗ ) ≤ ( λ + sλ ) p ( y n , x ∗ ) . (3.7)Combining (3.6) and (3.7), we get(2 − λ − sλ − sλ ) p ( x ∗ , T y n ) ≤ (2 λ + sλ + sλ ) p ( x ∗ , y n ) , leading to p ( x ∗ , T y n ) ≤ λ + sλ + sλ − λ − sλ − sλ p ( x ∗ , y n ) . (3.8)If we set l = s (2 λ + sλ + sλ )2 − λ − sλ − sλ , it follows from 2 sλ + 2 λ + ( s + s )( λ + λ ) < ≤ l < a n = p ( y n , x ∗ ) , c n = sp ( y n +1 , T y n ), and owing to (3.8), we have a n +1 = p ( y n +1 , x ∗ ) ≤ s [ p ( y n +1 , T y n ) + p ( T y n , x ∗ )] ≤ ha n + c n . Thus, lim n →∞ p ( y n , x ∗ ) = 0 = p ( x ∗ , x ∗ ), i.e. have y n p −→ x ∗ . As a consequence, Picard’siteration is T -stable. (cid:3) Corollary 3.11.
Under the conditions of Corollary 3.6, Picard’s iteration is T -stable.Proof. Just notice that Corollary 3.6 is a special case of Theorem 3.4 where we take s = 1. (cid:3) Corollary 3.12. ( [12, Theorem 1.] ) Let ( X, p ) be a complete partial b -metric space withcoefficient s ≥ and T : X → X be a mapping satisfying the following condition: p ( T x, T y ) ≤ λp ( x, y ) , (Ch3) or all x, y ∈ X , where λ ∈ [0 , . The Picard’s iteration is T -stable.Proof. Just notice that Corollary 3.12 is a special case of Corollary 3.6 where we take λ = λ = λ = λ = 0. (cid:3) Problem 3.13.
The authors plan, in [8], to study the T -stability of both the Kannan andthe Chatterjea contractions for the Picard iteration for a self mapping defined on partial b -metric space.The Corollary (2.4) illustrates the idea of the so-called P property . If a map T satisfies F ( T ) = F ( T n ) for each n ∈ N , then it is said to have the P property (see [7]). The followingresults are generalizations of the corresponding results in partial b -metric spaces. Theorem 3.14.
Let ( X, p ) be a partial b -metric space with coefficient s ≥ . Let T : X → X be a mapping such that F ( T ) = ∅ and that p ( T x, T ) ≤ λp ( x, T x ) (3.9) for all x ∈ X , where ≤ λ < is a constant. Then T has the P property.Proof. We always assume that n >
1, since the statement for n = 1 is trivial. Let z ∈ F ( T n ).It is clear that p ( z, T z ) ≤ p ( T T n − z, T T n − z ) ≤ λp ( T n − z, T n z ) = λp ( T T n − z, T T n − z ) ≤ λ p ( T n − z, T n − z ) ≤ · · · ≤ λ n p ( z, T z ) → n → ∞ ) . Hence, p ( z, T z ) = 0, that is., T z = z . (cid:3) In concluding this section, we make a conjecture with respect to P property with regards toTheorem 3.4 and Corollary 3.6. They are yet to be proved. Conjecture 3.15.
Under the conditions of Theorem 3.4, T has the P property. For theproof, it is enough to check if the mapping T satisfies (3.9).Also Conjecture 3.16.
Under the conditions of Corollary 3.6, T has the P property.We conclude this paper by giving examples to illustrate Theorem 3.4. Example 3.17.
Let X = { , , , } and p : X × X → R be defined by p ( x, y ) = | x − y | + max { x, y } , if x = y ; x, if x = y = 1;0 , if x = y = 1 . Then (
X, p ) is a complete partial b -metric space with coefficient s = 4 > T X → X by T T T T . A simple computation gives: p ( T , T
2) = p (1 ,
1) = 0 ≤ p (1 , p ( T , T
3) = p (1 ,
2) = 3 ≤ p (1 , p ( T , T
4) = p (1 ,
2) = 3 ≤
13 = p (1 , p ( T , T
3) = p (1 ,
2) = 3 ≤ p (2 , p ( T , T
4) = p (1 ,
2) = 3 ≤ p (2 , p ( T , T
4) = p (2 ,
2) = 2 ≤ p (3 , T satisfies all the conditions of Theorem 3.4, with λ ∈ (cid:2) , (cid:1) , λ = λ = λ = λ = 0and obviously λ + λ + 2 sλ + sλ + sλ < . Now, by Theorem 3.4, T has a unique fixedpoint , which in this case is 1. Example 3.18.
Let X = [0 , , k > p : X × X → R + by p ( x, y ) = | x − y | k for all x, y ∈ X . Then ( X, p ) is a complete partial b -metric space with coefficient s = 2 k >
1. Define a mapping T : X → X by T x = e x − λ , where λ > x, y ∈ X and x = y , there exists somereal number ξ belonging to between x and y such that | e x − λ − e y − λ | k = ( e ξ − λ ) k | x − y | k ≤ ( e − λ ) k | x − y | k . Hence p ( T x, T y ) = | e x − λ − e y − λ | k ≤ ( e − λ ) k | x − y | k ≤ ( e − λ ) k p ( x, y ) . Then, T satisfies all the conditions of Theorem 3.4, with λ = ( e − λ ) k , λ = λ = λ = λ = 0and obviously λ + λ + 2 sλ + sλ + sλ < . Now, by Theorem 3.4, T has a unique fixedpoint in u ∈ X .In view of λ > λ = ( e − λ ) k < − p = s , so 2 sλ +2 λ +( s + s )( λ + λ ) < T -stable.To see exactly what this T -stability means, consider the sequence y n = nn +1 u ∈ X . It followsthat p ( y n +1 , T y n ) = (cid:12)(cid:12)(cid:12)(cid:12) n + 1 n + 2 u − e nn +1 u − λ (cid:12)(cid:12)(cid:12)(cid:12) → | u − e u − λ | = 0 ( n → ∞ ) . Note that y n = nn +1 u → u ( n → ∞ ) . Going further
Recently, Zheng et al.[14] introduced the so-called θ - φ contraction in complete metric spacesand this technique was successfully applied to Kannan type mapping in partial metric spaces(see [6]). The results of the present paper will be applied in future investigations by the au-thors regarding θ - φ contraction in complete partial b -metric spaces. Hence the continuationof this research is considering θ - φ -Chatterjea type contraction in partial b -metric spaces andinvestigate the existence of fixed points. We have a definition for θ - φ -Chatterjea type contrac-tion and we must verify that it follows the idea of Chatterjea contractions and generalizesthem in a way that keeps their properties and their relationship with other contractions. oreover, a natural question is to check whether this new type of contraction is T -stableand has the P property. References
1. T. Abdeljawad, E. Karapnar and K. Ta¸s;
A generalized contraction principle with control functions onpartial metric spaces , Computers and Mathematics with Applications, 63 (2012), 716–719.2. S. Czerwik;
Contraction mappings in b -metric spaces . Atti Sem. Mat. Univ. Modena 46, 263–276 (1998).3. Y. U. Gaba; Metric type spaces and λ -sequences , Quaestiones Mathematicae, Vol. 40, Iss. 1, pp. 49–55,2017.4. Y. U. Gaba; Related G -metrics and Fixed Points , Analele Universit˘at¸ii de Vest, Timi¸soara Seria Matem-atic˘a– Informatic˘a, LVI, 1, (2018), 64–72.5. A. H. Frink, Distance functions and the metrization problem , Bull. AMS 43 (1937), 133- 142.6. T. Hu, D. Zheng and J. Zhou;
Some New Fixed Point Theorems on Partial Metric Spaces , InternationalJournal of Mathematical Analysis Vol. 12, 2018, no. 7, 343–352.7. G. S. Jeong, B. E. Rhoades;
Maps for which F ( T ) = F ( T n ), Fixed Point Theory Appl. 6, 71–105 (2005).8. D. J. Leko, Y. U. Gaba; Fixed point for θ - φ -Chatterjea contraction in complete partial b -metric spaces ,in preparation.9. Q. Liu; A convergence theorem of the sequence of Ishikawa iterates for quasi- contractive mappings , J.Math. Anal. Appl. 146(2), 301305 (1990).10. S. G. Matthews;
Partial metric topology , in: Proceedings of the 8th Summer Conference on Topologyand its Applications, Ann. New York Acad. Sci. 728 (1994) 183–197.11. S. Oltra, O. Valero;
Banach’s fixed point theorem for partial metric spaces , Rend. Istit. Mat. Univ. Trieste, Spanish Ministry of Science and Technol- ogy, (2004), 17–26.12. S. Shukla;
Partial b -Metric Spaces and Fixed Point Theorems , Mediterr. J. Math. (2014) 11: 703–711.13. Y. Qing, B. E. Rhoades; T -Stability of Picard iteration in metric spaces , Fixed Point Theory Appl. 2008,418971 (2008).14. D. Zheng, Z. Cai and P. Wang; New fixed point theorems for θ - φ contraction in complete metric spaces ,Journal of Nonlinear Sciences and Applications, 10 (2017), 2662–2670. Institut de Math´ematiques et de Sciences Physiques (IMSP)/UAC, 01 BP 613 Porto-Novo,B´enin. Department of Mathematics and Applied Mathematics, Nelson Mandela University, P.O.Box 77000, Port Elizabeth 6031, South Africa. African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon. † Corresponding author.
E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected]@gmail.com