Rate of convergence under weak contractiveness conditions

Fixed Point Theory, 13(2012), No.1, ...-... http://www.math.ubbcluj.ro/∼nodeacj/sfptcj.html RATE OF CONVERGENCE UNDER WEAK CONTRACTIVENESS CONDITIONS DAVID ARIZA-RUIZ∗EYVIND MARTOL BRISEID∗∗, ANTONIO JIM´ ENEZ-MELADO∗∗∗ AND GENARO L´ OPEZ-ACEDO∗ ∗Dept. An´alisis Matem´atico, Fac. Matem´aticas, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain. E-mail: [email protected], glop[email protected] ∗∗ Dept. of Mathematics, The University of Oslo, Postboks 1053, Blindern, 0316 Oslo, Norway. E-mail address: eyvindm[email protected] ∗∗∗ Dept. An´alisis Matem´atico, Fac. Ciencias, Universidad de M´alaga, Apdo. 29071 M´alaga, Spain E-mail address: [email protected] Abstract. We introduce a new class of selfmaps Tof metric spaces, which generalizes the weakly Zamfirescu maps (and therefore weakly contraction maps, weakly Kannan maps, weakly Chatterjea maps and quasi-contraction maps with constant h < 1 2). We give an explicit Cauchy rate for the Picard iteration sequences {Tnx0}n∈Nfor this type of maps, and show that if the space is complete, then all Picard iteration sequences converge to the unique fixed point of T. Our Cauchy rate depends on the space (X, d), the map T, and the starting point x0∈Xonly through an upper bound b≥d(x0, T x0) and certain moduli θ,µ for the map, but is otherwise fully uniform. As a step on the way to proving our fixed point result we also calculate a modulus of uniqueness for this type of maps. Key Words and Phrases: Cauchy rate, weakly Zamfirescu maps, weakly contractive maps, quasi-contraction maps, modulus of uniqueness, rate of convergence, fixed points. 2000 Mathematics Subject Classification: Primary: 47H09, Secondary: 47H10, 54H25 1. Introduction Let (X, d) be a metric space and T:X→Xa map. The well-known Banach contraction principle states that if Tis contractive (that is, there 1 2 D. ARIZA-RUIZ, E.M. BRISEID, A. JIM´ ENEZ-MELADO, G. L´ OPEZ-ACEDO exists α∈[0,1) such that d(Tx, Ty)≤α d(x, y) for all x, y ∈X) and (X, d) is complete then Thas a unique fixed point pin X, and for any x0∈Xthe Picard iteration sequence {Tnx0}n∈Nconverges to p. Moreover, we have the following estimate of the rate of convergence d(Tnx0, p)≤αn 1−αd(x0, Tx0) for all n∈N. The rate of convergence for the Picard iteration sequences {Tnx0}n∈Nwhich we obtain from this is highly uniform – it depends on the space (X, d), the map T, and the starting point x0∈Xonly through a bound b≥d(x0, Tx0) and the constant α. It is of interest to obtain similar rates of convergence or Cauchy rates for other classes of mappings – where instead of the constant α the rates then depend on other constants or moduli governing the behavior of the mappings in question, e.g. a modulus of uniform continuity. In [3, 4, 5] such explicit rates of convergence were calculated for very general classes of maps.1These rates of convergence are given in terms of appropriate moduli for the classes of mappings considered, and so one possible route to obtaining new rates of convergence for other classes of maps is to show that these are e.g. asymptotic contractions in the sense of Kirk, for which a rate of convergence is given in [3], and then to calculate the moduli for an asymptotic contraction from the moduli of the mapping at hand. However, since it often in this case is far from obvious how to turn one set of moduli into another, it is worthwhile to calculate explicit rates of convergence for more specific classes of maps directly in terms of the moduli through which these are given. This approach can also lead to better rates of convergence which can be expressed more simply than the rates of convergence for the more general classes of maps. An example of a class of maps for which it would be interesting to give a simple rate of convergence are the weakly contractive maps. This concept was introduced by Dugundji and Granas [8] by replacing the contraction constant αby a function α=α(x, y) compactly less than 1. In this paper we introduce a new type of maps that includes both the class of weakly contractive maps and several other well-known types of maps, such as the class of quasi-contraction 1In particular, Proposition 3.7 in [3] and Theorem 30 below together imply that in the setting of complete, bounded metric spaces the mappings for which we in this paper calculate a rate of convergence are included in the asymptotic contractions considered in [3]. RATE OF CONVERGENCE 3 maps with constant h < 1 2. The main result of the paper is an explicit and highly uniform Cauchy rate (which makes sense without assuming the completeness of the space) for this class of mappings, expressed in terms of certain moduli θand µ; and as a followup the result that if the space is complete, then all Picard iteration sequences converge to the unique fixed point. The structure of the paper is as follows: In Section 2 we introduce the condition (wEα, µ), and construct our Cauchy rate for mappings which satisfy this in addition to some extra condition. In Section 3 we show that the condition introduced in this paper includes several well-known types of maps such as weakly contractive maps, quasicontraction maps with constant h < 1 2, and others. In Section 4 we prove that the maps satisfying this new condition have the property of existence of a modulus of uniqueness. This is used to establish our fixed point result, which we give in Section 5, along with a local version of the theorem. To a certain extent our methodology of focussing on the moduli through which the mappings Tare given, rather than on the mappings themselves, as well as in particular some of the arguments involving a modulus of uniqueness, are inspired by so-called proof mining techniques (see [13]), which were the explicit background for the work on rates of convergence in [3, 4, 5]. Finally a notational point: We will throughout this paper let Ndenote the set of nonnegative integer numbers. 2. Cauchy rate We recall the two basic definitions that are the matter of interest of this paper. Definition 1. Let {xn}n∈Nbe a sequence in a metric space (X, d). We say that γ: (0,∞)→Nis a Cauchy rate for {xn}n∈Nif for all ε > 0 we have that d(xγ(ε), xγ(ε)+k)≤εfor each k∈N. Definition 2. Let {xn}n∈Nbe a sequence in a metric space (X, d). We say that γ: (0,∞)→Nis a rate of convergence for {xn}n∈Nto z∈Xif for all ε > 0 we have that n≥γ(ε) gives d(z, xn)≤ε. 4 D. ARIZA-RUIZ, E.M. BRISEID, A. JIM´ ENEZ-MELADO, G. L´ OPEZ-ACEDO In [10], J. Garcia-Falset et al. introduced a condition which is weaker than nonexpansiveness. This condition reads as follows: let (X, d) be a metric space, Da nonempty subset of Xand µ≥1 a real number. A map T:D→Xis said to satisfy the condition (Eµ) on Dif d(x, Ty)≤d(x, y) + µ d(x, Tx),(Eµ) for all x, y ∈D. In this paper, we shall consider a version of (Eµ) for weakly contractive mappings obtained by replacing in (Eµ) the nonexpansiveness condition by the weakly contractive condition. In order to do this, we need the following concept. Definition 3. Let Dbe a nonempty subset of a metric space (X, d). A map α:D×D→[0,1] is called compactly less than 1 if θ(a, b)<1 for all 0 < a ≤b, where θis given by θ(a, b) :=        sup a≤d(x,y)≤b α(x, y) if {(x, y)∈D×D:a≤d(x, y)≤b} 6=∅, 1 2otherwise. Remark 4.It is easy to check that α:D×D→[0,1] is compactly less than 1 if, and only if, there exists a function θ: (0,∞)×(0,∞)→[0,1) such that α(x, y)≤θ(a, b)<1 for all 0 < a ≤band all x, y ∈Dwith a≤d(x, y)≤b. This equivalent formulation is more convenient for our needs and will be used throughout this paper. So in a context where α:D×D→[0,1] is a map which is compactly less than 1 we will let θbe some function as given in this remark rather than necessarily the one appearing in Definition 3. Definition 5. Let Dbe a nonempty subset of a metric space (X, d). We say that T:D→Xsatisfies condition (wEα, µ)if d(x, Ty)≤α(x, y)d(x, y) + µ d(x, T x) (wEα, µ) for all x, y ∈D, where µ≥1 and α:D×D→[0,1] is compactly less than 1. Remark 6.Suppose that T:D→Xsatisfies condition (wEα, µ). If for any fixed σ∈(0,1) we define ασ:D×D→[0,1] by ασ(x, y) := max{α(x, y), σ}, RATE OF CONVERGENCE 5 then Talso satisfies the above condition with µ, ασ, and, moreover, we have that θσ(a, b) := sup{ασ(x, y) : a≤d(x, y)≤b}>0, for all 0 < a ≤b. Thus, from now on, when considering an αas in Definition 5 we will assume that for the corresponding θas in Definition 3 we have θ(a, b)> 0 for all 0 < a ≤b. Theorem 7. Let (X, d) be a metric space and T:X→Xbe a map which satisfies condition (wEα, µ) and the following condition d(Tx, T2x)≤α(x, Tx)d(x, Tx),(1) for all x∈X. Let x0∈Xbe the starting point of the sequence {xn}n∈Ndefined by xn+1 := Txn. Let b > 0 satisfy d(x0, x1)≤b, and define ρ: (0,∞)→(0,∞) by ρ(ε) := min ε 2µ1−θ(ε 2, ε), b. Define γ: (0,∞)→Nby γ(ε) := log ρ(ε)−log b log θ(ρ(ε), b)+ 1,(2) where d·e denotes the ceiling function given by dxe:= min{n∈Z:x≤n}. Then γis a Cauchy rate for {xn}n∈N. Remark 8.Note that γdepends only on θ,µ,band ε. Thus, we should consider γ: (0,1)(0,∞)×(0,∞)×[1,∞)×(0,∞)×(0,∞)→N and γ(θ, µ, b, ε) instead of γ(ε). We shall use both notations interchangeably. Observe also that if (X, d) is complete, then γ(θ, µ, b, ·) is a rate of convergence for {xn}n∈Nto z:= limn→∞ xn. Proof. Let x0∈Xand define the Picard iteration sequence {xn}n∈Nby xn+1 := Txnfor n∈N. Note that (1) gives d(xn+1, xn+2)≤α(xn, xn+1)d(xn, xn+1) (3) for all n∈N, so since 0 ≤α(xn, xn+1)≤1 it follows that the sequence {d(xn, xn+1)}n∈Nis nonincreasing. Let now ε > 0. We will first prove that d(xγ(ε), xγ(ε)+1)< ρ(ε). In order to do this, we assume d(xγ(ε), xγ(ε)+1)≥ρ(ε) 6 D. ARIZA-RUIZ, E.M. BRISEID, A. JIM´ ENEZ-MELADO, G. L´ OPEZ-ACEDO and we shall get a contradiction. Note that ρ(ε)≤d(xi, xi+1)≤bfor all 0≤i≤γ(ε). Thus, α(xi, xi+1)≤θ(ρ(ε), b) for 0 ≤i≤γ(ε). Then, ρ(ε)≤d(xγ(ε), xγ(ε)+1) ≤ γ(ε) Y i=1 α(xγ(ε)−i, xγ(ε)+1−i)·d(x0, x1) ≤θ(ρ(ε), b)γ(ε)·b < ρ(ε), which is a contradiction. Here the last step θ(ρ(ε), b)γ(ε)·b<ρ(ε) follows from θ(ρ(ε), b)γ(ε)·b<ρ(ε)⇐⇒ γ(ε)·log θ(ρ(ε), b) + log b < log ρ(ε)⇐⇒ γ(ε)>log ρ(ε)−log b log θ(ρ(ε), b) and the definition of γ. Therefore, we have d(xγ(ε), xγ(ε)+1)< ρ(ε).(4) Next, we will prove by induction that d(xγ(ε), xγ(ε)+k)≤εfor k∈N. This inequality is true for k= 1 by (4), since ρ(ε)< ε. Assuming d(xγ(ε), xγ(ε)+k)< ε, let us see d(xγ(ε), xγ(ε)+k+1)< ε. Taking x=xγ(ε)and y=xγ(ε)+k, from condition (wEα,µ) we obtain d(xγ(ε), xγ(ε)+k+1)≤α(xγ(ε), xγ(ε)+k)d(xγ(ε), xγ(ε)+k) + µ d(xγ(ε), xγ(ε)+1). (5) If d(xγ(ε), xγ(ε)+k)< ε/2, using that 0 ≤α(xγ(ε), xγ(ε)+k)≤1 and (4), we have d(xγ(ε), xγ(ε)+k+1)<ε 2+µ ρ(ε)≤ε 2+µε 2µ=ε. And if d(xγ(ε), xγ(ε)+k)≥ε/2, applying the induction hypothesis, we have ε 2≤d(xγ(ε), xγ(ε)+k)< ε. RATE OF CONVERGENCE 7 Then, α(xγ(ε), xγ(ε)+k)≤θ(ε 2, ε). Thus, from (4) and (5), we conclude that d(xγ(ε), xγ(ε)+k+1)≤θ(ε 2, ε)ε+µ ρ(ε) ≤θ(ε 2, ε)ε+µε 2µ(1 −θ(ε 2, ε)) =ε 2(1 + θ(ε 2, ε)) < ε.  3. Some examples The aim of this section is to show that some well-known types of maps satisfy condition (wEα, µ). 3.1. Weakly Zamfirescu maps. In 1972 Zamfirescu [16], combining the contractive condition, Kannan’s condition [11] and Chatterjea’s condition [6], defined the following type of maps. Definition 9. Let Dbe a nonempty subset of a metric space (X, d). A map T:D→Xis a Zamfirescu map if there exists a constant α∈[0,1) such that d(Tx, Ty)≤α MT(x, y) (Z) for all x, y ∈D, where MT(x, y) := max d(x, y),1 2[d(x, Tx) + d(y, Ty)],1 2[d(x, Ty) + d(y, Tx)]. In [2], the authors defined a new class of map which generalizes the class of Zamfirescu maps and the class of weakly contraction maps [8, 9] and weakly Kannan maps [1]. Definition 10. Let (X, d) be a metric space, D⊆Xand T:D→X. We say that Tis a weakly Zamfirescu map if there exists α:D×D→[0,1] compactly less than 1 such that d(Tx, Ty)≤α(x, y)MT(x, y) (wZ) for all x, y ∈D, where MT(x, y) := max d(x, y),1 2d(x, Tx) + d(y, Ty),1 2d(x, Ty) + d(y, Tx). 8 D. ARIZA-RUIZ, E.M. BRISEID, A. JIM´ ENEZ-MELADO, G. L´ OPEZ-ACEDO It is clear that every Zamfirescu map is a weakly Zamfirescu map. Next we shall show with Example 21 of [2] that the converse is not true. Example 11. Consider the metric space (X, d), where X= [0,1] and dis the usual metric. The map T: [0,1] →[0,1] given as Tx =     2 3xif 0 ≤x < 1, 0 if x= 1, is a weakly Zamfirescu map, since Tis a weakly Chatterjea map (for more details, see [2]). Moreover, Tis not a Zamfirescu map, since lim x→1− d(Tx, T1) MT(x, 1) = lim x→1− 2 3x max{1−x, 1 2(x 3+ 1),1 2(x+ 1 −2 3x)}= 1. The following result assures that every weakly Zamfirescu map satisfies condition (wEα, µ) and inequality (1). Proposition 12. Let (X, d) be a metric space. If T:X→Xis a weakly Zamfirescu map, then Tsatisfies condition (wEα, µ) and inequality (1) with µ= 3. Proof. First we will show that Tsatisfies condition (wEα, µ). Let x, y ∈X. Case 1: If MT(x, y) = d(x, y), then d(x, Ty)≤d(x, Tx) + d(Tx, Ty)≤d(x, Tx) + α(x, y)d(x, y). Case 2: If MT(x, y) = 1 2d(x, Tx) + d(y, T y), then d(x, Ty)≤d(x, Tx) + d(Tx, Ty) ≤d(x, Tx) + α(x, y) 2d(x, Tx) + d(y, T y) ≤α(x, y) 2+ 1d(x, Tx) + α(x, y) 2[d(x, y) + d(x, T y)] . Hence, d(x, Ty)≤α(x, y) 2−α(x, y)d(x, y) + 2 + α(x, y) 2−α(x, y)d(x, T x). RATE OF CONVERGENCE 9 Case 3: If MT(x, y) = 1 2d(x, Ty) + d(y, Tx), then d(x, Ty)≤d(x, Tx) + d(Tx, Ty) ≤d(x, Tx) + α(x, y) 2d(x, Ty) + d(y, Tx) ≤α(x, y) 2+ 1d(x, Tx) + α(x, y) 2[d(x, y) + d(x, T y)] . Hence, d(x, Ty)≤α(x, y) 2−α(x, y)d(x, y) + 2 + α(x, y) 2−α(x, y)d(x, T x). Since t 2−t≤tand 1 ≤2+t 2−t≤3 for all t∈[0,1], we get d(x, Ty)≤α(x, y)d(x, y)+3d(x, Tx). Next, we will prove that Tsatisfies inequality (1). Let x∈X. We note that MT(x, Tx) = max d(x, Tx),1 2d(x, Tx) + d(Tx, T2x),1 2d(x, T2x) ≤max d(x, Tx),1 2d(x, Tx) + d(Tx, T2x), 1 2d(x, Tx) + d(Tx, T2x) = max d(x, T x),1 2d(x, Tx) + d(Tx, T2x). So, since Tis a weakly Zamfirescu map, d(Tx, T2x)≤α(x, Tx) max d(x, T x),1 2d(x, Tx) + d(Tx, T2x) If the maximum is equal to d(x, Tx), we have finished. Otherwise, we have d(Tx, T2x)≤α(x, Tx) 2d(x, Tx) + d(Tx, T2x). Hence, d(Tx, T2x)≤α(x, Tx) 2−α(x, Tx)d(x, Tx)≤α(x, Tx)d(x, Tx).  As a consequence of the above result and Theorem 7 we get a new result concerning a Cauchy rate for weakly Zamfirescu maps. 16 D. ARIZA-RUIZ, E.M. BRISEID, A. JIM´ ENEZ-MELADO, G. L´ OPEZ-ACEDO Proposition 23. There does not exist a function φ: [0,1)(0,∞)×(0,∞)×[1,∞)× (0,∞)→(0,∞) such that φ(θ, µ, ·) : (0,∞)→(0,∞) works as a modulus of uniqueness for all metric spaces (X, d) and all maps T:X→Xsatisfying (wEα, µ) with µand θsatisfying the requirements in Definition 5 and Definition 3. Proof. Assume for a contradiction that φ(θ, µ, ·) : (0,∞)→(0,∞) satisfies the condition ∀ε > 0∀x, y ∈Xd(x, T x)≤φ(θ, µ, ε)∧d(y, T y)≤φ(θ, µ, ε)⇒d(x, y)≤ε for all metric spaces (X, d) and maps T:X→Xwhich satisfy (wEα, µ) with µand θsatisfying the requirements in Definition 5 and Definition 3. Let {x1 n}n∈Nand {x2 n}n∈Nbe two sequences of real numbers, with xi n6=xj m if n6=mor i6=j. Let δ∈(0,1) and k0∈Nwith k0≥1 4δ. We equip X={x1 n}n≥k0∪ {x2 n}n≥k0with the metric d(xi n, xj m) =      0 if n=mand i=j, δif n=mand i6=j, n+mif n6=m. Let T:X→Xbe a map given by Tx1 n=x2 nand Tx2 n=x1 nfor each n≥k0. Next we will check that Tsatisfies condition (wEα, µ) with µ:= 2, α:X×X→[0,1] given by α(x, y) :=      max n1 2,1−1 d(x,y)2oif x6=y, 0 if x=y, and with θ(as in Remark 4) given by θ(a, b) := max 1 2,1−1 b2. We consider the non-trivial case x=xi nand y=xj mwith n, m ≥k0,n6=m and i, j ∈ {1,2}. Since 1 n+m≤1 2k0 ≤2δ RATE OF CONVERGENCE 17 we get d(x, Ty) = n+m ≤n+m−1 n+m+ 2 δ =d(x, y)−1 d(x, y)+ 2 d(x, T x) ≤α(x, y)d(x, y)+2d(x, T x). Note that µand also θare the same for all choices of δ∈(0,1). Let now δ:= min{1/2, φ(θ, µ, 1)}. By taking n=k0+ 1 and m=k0+ 2 we get d(x1 n, Tx1 n) = d(x1 n, x2 n) = δ≤φ(θ, µ, 1) and d(x1 m, Tx1 m) = d(x1 m, x2 m) = δ≤φ(θ, µ, 1). So, d(x1 n, x1 m)≤1, which is a contradiction, since d(x1 n, x1 m) = n+m≥2.  We can use Theorem 21 to improve Theorem 7 in the following way: Theorem 24. Let (X, d) be a metric space and T:X→Xbe a map which satisfies conditions (wEα, µ) and (1), with µand θas in Definition 5 and Definition 3. Let x0∈Xbe the starting point of the sequence {xn}n∈N defined by xn+1 := Txn. If in addition to the starting point x0∈Xwe have another starting point y0∈X, then lim n→∞ d(xn, yn)=0,(6) where {yn}n∈Nis defined by yn=Tny0, for every n∈N. Proof. In order to show (6), suppose this equality does not hold. Then there exist ε > 0 and k0∈Nsuch that for every n≥k0we have d(xk0, xn)<ε 4, d(yk0, yn)<ε 4 and d(xk0, yk0) = ε. 18 D. ARIZA-RUIZ, E.M. BRISEID, A. JIM´ ENEZ-MELADO, G. L´ OPEZ-ACEDO So, ε 2< d(xn, yn)<3ε 2 for all n≥k0. Now, we can take b=3ε 2in Theorem 21 and n≥k0so big that d(xn, xn+1)≤ψ(θ, µ, b, ε 2) and d(yn, yn+1)≤ψ(θ, µ, b, ε 2) to get d(xn, yn)≤ε 2, which is a contradiction.  Using the above theorem, Theorem 7, and Theorem 21 we can show that for maps satisfying conditions (wEα, µ) and (1) there exists a modulus of uniqueness which does not depend on a bound b > 0 on the diameter of the space. Proposition 23 shows that this is in sharp contrast with the situation for maps satisfying only condition (wEα, µ). Before proving this result, we need two lemmas. Lemma 25. Let (X, d) be a metric space, ε > 0, x0∈X, and T:X→X a map satisfying conditions (wEα, µ) and (1), with µand θas in Definition 5 and Definition 3. Then, we have the following implication: d(y0, xk0)> b ·γ(θ, µ, b, ε)+3ε⇒d(y0, Ty0)> b for all y0∈Xand b∈(0,∞), where k0=γ(θ, µ, d(x0, x1), ε) and γis the Cauchy rate from Theorem 7. Proof. Let y0∈Xand b∈(0,∞). Suppose that d(y0, T y0)≤b. Using the triangle inequality, we have d(y0, yk1)≤k1b, where k1=γ(θ, µ, b, ε). By Theorem 7, d(yk1, yn)≤εfor all n≥k1. So, we have that d(y0, yn)≤d(y0, yk1) + d(yk1, yn)≤k1b+ε for all n≥k1. Again, using Theorem 7, we get d(xk0, xn)≤εfor all n≥k0, where k0=γ(θ, µ, d(x0, x1), ε). On the other hand, by Theorem 24, there RATE OF CONVERGENCE 19 exists k2∈Nsuch that d(xn, yn)< ε for all n≥k2. Therefore we can take n≥max{k0, k1, k2}to get d(y0, xk0)≤d(y0, yn) + d(yn, xn) + d(xn, xk0) < k1b+ 3 ε.  Lemma 26. Let (X, d) be a metric space, T:X→Xa map satisfying conditions (wEα, µ) and (1), with µand θas in Definition 5 and Definition 3. Let x, y ∈Xand b > 0. If d(x, Tx)≤band d(y, Ty)≤b, then d(x, y)≤2(b γ(θ, µ, b, ε)+3ε) for all ε > 0. Proof. By Lemma 25.  Theorem 27. Let (X, d) be a metric space, x0∈Xand T:X→Xbe a map satisfying conditions (wEα, µ) and (1), with µand θas in Definition 5 and Definition 3. Define ψ(θ, µ, ·) : (0,∞)→(0,∞) by ψ(θ, µ, t) := min 1,t[1 −θ(t, 2 (γ(θ, µ, 1,1) + 3))] µ+ 1 . Then for all ε∈(0,∞) and for all x, y ∈Xwith d(x, Tx)≤ψ(θ, µ, ε) and d(y, T y)≤ψ(θ, µ, ε), we have that d(x, y)≤ε. Proof. Let ε > 0 and x, y ∈Xwith d(x, Tx)≤ψ(θ, µ, ε)≤1 and d(y, Ty)≤ ψ(θ, µ, ε)≤1. By Lemma 26, we have d(x, y)≤2γ(θ, µ, 1,1) + 3.(7) Now the result follows by Theorem 21.  If we exchange being compactly less than 1 for another condition (type Rakotch), we can delete the boundedness hypothesis in Theorem 21 without requiring Tto satisfy condition (1). Theorem 28. Let Da nonempty subset of a metric space (X, d) and T:D→ Xa map such that d(x, Ty)≤α(ε)d(x, y) + µ d(x, Tx) (wE0 α, µ) 20 D. ARIZA-RUIZ, E.M. BRISEID, A. JIM´ ENEZ-MELADO, G. L´ OPEZ-ACEDO for all ε > 0 and for all x, y ∈Dwith d(x, y)> ε, where µ≥1 and α: (0,∞)→ [0,∞) satisfies that 0 ≤α(t)<1 for all t > 0. If we define ψ: (0,∞)→(0,∞) by ψ(t) := t(1 −α(t)) µ+ 1 , then for all ε∈(0,∞) and for all x, y ∈X, if d(x, Tx)≤ψ(ε) and d(y, T y)≤ ψ(ε), then d(x, y)≤ε. Remark 29.Actually Rakotch [14] considered that αmust also be a monotonically decreasing function. Proof. Assume that x, y ∈Xsuch that d(x, Tx)≤ψ(ε) and d(y, Ty)≤ψ(ε). We will prove that d(x, y)≤ε. To do this, we will argue by contradiction. Suppose that d(x, y)> ε. We note d(x, y)≤d(x, T y) + d(y, T y) ≤α(ε)d(x, y) + µ d(x, T x) + d(y, Ty) ≤α(ε)d(x, y)+(µ+ 1)ψ(ε). Since α(ε)<1, we have that d(x, y)≤µ+ 1 1−α(ε)ψ(ε) = ε, which is a contradiction.  5. A fixed point result Under the assumption of completeness, we use Theorem 7 and Theorem 24 to obtain the following fixed point theorem. Theorem 30. Let (X, d) be a nonempty, complete metric space, and let T:X→Xbe a map satisfying the conditions (wEα, µ) and (1). Then Thas a unique fixed point pin X, and all Picard iteration sequences converge to p, with rates of convergence given by Theorem 7. Proof. Let x0∈X. By Theorem 7 we know that {Tnx0}n∈Nis a Cauchy sequence, and since (X, d) is complete, there exists pin Xsuch that Tnx0→p as n→ ∞. Assume that pis not a fixed point of T. Then there exist ε > 0 and y0∈Xwith Tp =y0and d(p, y0) = ε. By Theorem 24 we know that lim n→∞ yn=p, RATE OF CONVERGENCE 21 where {yn}n∈Nis defined by yn=Tny0for each n∈N. By taking x=ynand y=pin (wEα, µ) we get that for each n∈N d(yn, Tp)≤α(yn, p)d(yn, p) + µ d(yn, Tyn) ≤d(yn, p) + µ d(yn, yn+1). Taking limits as n→ ∞, we obtain that ε≤0, which is a contradiction. Finally, we shall prove the uniqueness of the fixed point p. Suppose that qis another fixed point of T, with p6=q. By Theorem 24 we know that limn→∞ Tnq=p, which is a contradiction.  Remark 31.Note that if we check the above proof carefully, we can show that any map satisfying the conditions (wEα, µ) and (1) is continuous at its unique fixed point. As a consequence of the previous theorem, we obtain the following local result. Theorem 32. Let (X, d) be a complete metric space, x0∈Xand r > 0. Assume that T:B(x0, r)→Xsatisfies the conditions (wEα, µ) and (1). If θ is defined as usual, and d(x0, Tx0)≤r µmin 1 2,1−θr 2, r, then Thas a unique fixed point in B(x0, r). Proof. Using the above result, we only have to show that B(x0, r) is invariant under T. In order to do this, let x∈B(x0, r). We shall consider two cases. If d(x0, x)≤r 2, by (wEα, µ), we have d(x0, Tx)≤α(x0, x)d(x0, x) + µ d(x0, Tx0) ≤r 2+µr 2µ=r. Otherwise, r 2≤d(x0, x)≤r. So, α(x0, x)≤θ(r 2, r)<1. Then, d(x0, Tx)≤α(x0, x)d(x0, x) + µ d(x0, Tx0) ≤θ(r 2, r)r+µr µ1−θ(r 2, r)=r.  22 D. ARIZA-RUIZ, E.M. BRISEID, A. JIM´ ENEZ-MELADO, G. L´ OPEZ-ACEDO Acknowledgments The research of the first author is supported by Junta de Andaluc´ıa, Grant FQM-3543. The second author is supported by the Research Council of Norway, Project 204762/V30. The third author is partially supported by DGES, Grant MTM2007-60854 and Junta de Andaluc´ıa, Grant FQM-210 and FQM- 1504. The fourth author is partially supported by DGES, Grant MTM2009- 13997-C02-01 and Junta de Andaluc´ıa, Grant FQM-127. References [1] D. Ariza-Ruiz, A. Jim´ enez-Melado, A continuation method for weakly Kannan maps, Fixed Point Theory and Applications 2010, Art. ID 321594, 12 pp. [2] D. Ariza-Ruiz, A. Jim´ enez-Melado, G. L´ opez-Acedo, A fixed point theorem for weakly Zamfirescu mappings, Nonlinear Anal. 74 (2011), no. 5, 1628–1640. [3] E.M. Briseid, A rate of convergence for asymptotic contractions, Journal of Mathematical Analysis and Applications 330 (2007), 364–376. [4] E.M. 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