A sufﬁcient and necessary condition for the convergence of the sequence of successive approximations to a unique ﬁxed point II


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1 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 DOI /s RESEARCH Open Access A sufﬁcient and necessary condition for the convergence of the sequence of successive approximations to a unique ﬁxed point II Tomonari Suzuki1,2* and Badriah Alamri2 * Correspondence: 1 Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu, , Japan 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Abstract Using the concept of a BoydWong contraction, we obtain a simple, suﬃcient, and necessary condition for the convergence of the sequence of successive approximations to a unique ﬁxed point. MSC: 54H25 Keywords: Banach contraction principle; BoydWong contraction; contraction; ﬁxed point; successive approximation 1 Introduction The Banach contraction principle is a very forceful tool in nonlinear analysis. Theorem (Banach [ ] and Caccioppoli [ ]) Let (X, d) be a complete metric space and let T be a contraction on X, that is, there exists r [, ) such that d(tx, Ty) rd(x, y) for all x, y X. Then the following holds: (A) T has a unique ﬁxed point z and {T n x} converges to z for any x X. In [, ] we studied (A) and obtained the following. See also [, ]. Theorem ([, ]) Let T be a mapping on a complete metric space (X, d). Then (A), (B), and (C) are equivalent: (B) T is a strong Leader mapping, that is, the following hold:  For x, y X and ε >, there exist δ > and ν N such that d T i x, T j y < ε + δ d T i+ν x, T j+ν y < ε for all i, j N { }, where T is the identity mapping on X.  For x, y X, there exist ν N and a sequence {αn } in (, ) such that d T i x, T j y < αn d T i+ν x, T j+ν y < /n for all i, j N { } and n N Suzuki and Alamri; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
2 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 2 of 10 (C) There exist a τ distance p and r (0, 1) such that p(tx, T 2 x) rp(x, Tx) and p(tx, Ty)<p(x, y) for all x, y X with x y. WecannottellthatTheorem2 issimple.motivatedbythisfact,inthispaper,weobtain a simpler condition equivalent to (A). In 1969, Boyd and Wong proved a very interesting fixed point theorem. See [7]. The concept of a BoydWong contraction plays an important role in this paper. Indeed, using this concept, we give a condition equivalent to (A); see Theorem 9 below. We will find that Theorem 3 is an essential generalization of Theorem 1 in some sense; see Theorem 11 and Example 12 below. Theorem 3 (Boyd and Wong [8]) Let (X, d) be a complete metric space and let T be a Boyd Wong contraction on X, that is, there exists a function ϕ from [0, ) into itself satisfying the following: (i) ϕ is upper semicontinuous. (ii) ϕ(t)<t for every t (0, ). (iii) d(tx, Ty) ψ(d(x, y)) for all x, y X. Then (A) holds. Later, in 1975, Matkowski proved the following generalization of Theorem 1. Interestingly, while Theorem 3 and Theorem 4 look similar, we will find that Theorem 4 is similar to Theorem 1,notTheorem3, in some sense; see Theorem 13 below. Theorem 4 (Matkowski [9]) Let (X, d) be a complete metric space and let T be a Matkowski contraction on X, that is, there exists a function ψ from [0, ) into itself satisfying the following: (i) ψ is nondecreasing. (ii) lim n ψ n (t)=0for every t (0, ). (iii) d(tx, Ty) ψ(d(x, y)) for all x, y X. Then (A) holds. We introduce two more interesting theorems. Theorem 5 is a generalization of Theorem 3 and Theorem 6 is a generalization of Theorems 4 and 5. Theorem 5 (Meir and Keeler [10]) Let (X, d) be a complete metric space and let T be a MeirKeeler contraction on X, that is, for every ε >0,there exists δ >0such that d(x, y)<ε + δ d(tx, Ty)<ε for all x, y X. Then (A) holds. Theorem 6 (Ćirić [11], Jachymski [12] and Matkowski [13, 14]) Let (X, d) be a complete metric space and let T be a CJM contraction on X, that is, the following hold: (i) For every ε > 0, there exists δ > 0such that d(x, y) < ε + δ implies d(tx, Ty) ε. (ii) x y implies d(tx, Ty) <d(x, y). Then (A) holds.
3 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 3 of 10 2 Preliminaries Throughout this paper, we denote by N, Z, andr the sets of positive integers, integers, and real numbers, respectively. For t R, wedenoteby[t] themaximumintegernotexceeding t. For an arbitrary set A,wedenoteby A the cardinal number of A. Let (X, d) be a metric space. We denote by Cont(X, d), BWC(X, d), MC(X, d), MKC(X, d), and CJMC(X, d) the sets of all contractions, all BoydWong contractions, all Matkowski contractions, all MeirKeeler contractions, and all CJM contractions on (X, d), respectively. We know Cont(X, d) BWC(X, d) MKC(X, d) CJMC(X, d) and Cont(X, d) MC(X, d) CJMC(X, d). In the proof of our main result, we use the following. Lemma 7 Let X be a set, let z be an element of X and let f be a function from X \{z} into (0, ). Define a function ρ from X Xinto[0, ) by 0 if x = y, f (x) if x y, y = z, ρ(x, y)= f (y) if x y, x = z, max{f (x), f (y)} if x y, x z, y z. (1) Then (X, ρ) is a complete metric space. Proof It is obvious that ρ(x, y)=0 x = y,andρ(x, y)=ρ(y, x). We also note that ρ(x, y)=max { ρ(x, z), ρ(y, z) } for all x, y X with x y. Let x, y, w be three distinct elements of X \{z}.wehave ρ(x, z) max { ρ(x, z), ρ(y, z) } = ρ(x, y) ρ(x, y)+ρ(z, y), ρ(x, y)=max { ρ(x, z), ρ(y, z) } ρ(x, z)+ρ(y, z), ρ(x, y)=max { ρ(x, z), ρ(y, z) } max { max { ρ(x, z), ρ(w, z) }, max { ρ(y, z), ρ(w, z) }} = max { ρ(x, w), ρ(y, w) } ρ(x, w)+ρ(y, w). So ρ satisfies the triangle inequality. Therefore (X, ρ) is a metric space. Finally, in order to show the completeness of (X, ρ), let {x n } be a Cauchy sequence in X.Inthecasewhere {n : x n = y} = for some y X, {x n } obviously converges to y.inthecasewhere {n : x n = y} < for any y X, we can choose a subsequence {x g(n) } of {x n } such that x g(n) are all
4 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 4 of 10 different. Then we have 0= lim m n m,n ρ(x g(m), x g(n) )= lim m n m,n max { ρ(x g(m), z), ρ(x g(n), z) }. Thus, {x g(n) } converges to z.hence{x n } converges to z. Therefore (X, ρ)iscomplete. We denote by the set of all functions ψ satisfying (i) and (ii) of Theorem 4. Lemma 8 Let ϕ and ψ belong to. Then a function η from [0, ) defined by η(t) = max{ϕ(t), ψ(t)} also belongs to. Proof It is obvious that η is nondecreasing. Since ϕ(t)<t and ψ(t)<t for t (0, ), we have η(t)<t for t (0, ). Fix t (0, ). Define a mapping ν from N into N {0} by ν(n)= { k Z :0 k < n, ψ ( η k (t) ) = η k+1 (t) }, where η 0 (t)=t. Without loss of generality, we may assume lim n ν(n)=. Sinceη n (t) ψ ν(n) (t), we obtain lim n η n (t) = 0. Therefore η. 3 Main results We prove our main results. Theorem 9 Let (X, d) be a complete metric space and let T be a mapping on X. Then (A) and (D) are equivalent. (D) There exists a complete metric ρ on X such that ρ d and T BWC(X, ρ). Proof (D) (A): We assume (D). Then Theorem 3 shows that there exists a unique fixed point z of T and that for every x X, {T n x} converges to z in (X, ρ). Since the topology of (X, ρ) is stronger than that of (X, d), {T n x} converges to z in (X, d). Hence (A) holds. (A) (D): We assume (A). Define functions α from X \{z} into (0, ), l from X \{z} into Z, k from X \{z} into N {0} and f from X \{z} into (0, )by α(x)=max { d ( T n x, z ) : n N {0} }, l(x)= [ log 4 α(x) ], k(x)=max { n N {0} : T n x z, l ( T n x ) = l(x) }, f (x)=2 2l(x)+4 2 2l(x)+3 k(x). We note and d(x, z) α(x), α(tx) α(x), lim n α( T n x ) =0 l(tx) l(x), lim n l( T n x ) =
5 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 5 of 10 for x X \{z}.defineametricρ on X by (1). Then by Lemma 7,(X, ρ)isacompletemetric space. We shall show ρ d.inthecasewherex z,wehave ρ(x, z)=2 2l(x)+4 2 2l(x)+3 k(x) 2 2l(x)+4 2 2l(x)+3 =2 4 l(x)+1 >2 4 log 4 α(x) =2α(x) 2d(x, z)>d(x, z). In the case where x y, x z,andy z,wehave ρ(x, y)=max { ρ(x, z), ρ(y, z) } > max { 2d(x, z), 2d(y, z) } d(x, z)+d(y, z) d(x, y). In both cases, we obtain ρ d. Next, we shall show that T BWC(X, ρ). Define a function ϕ from [0, )intoitselfby 0 ift =0, 0 if2 2l+2 t <2 2l+3 for some l Z, 2 2l+2 if 2 2l+4 2 2l+3 t <2 2l+4 2 2l+2 ϕ(t)= for some l Z, 2 2l+4 2 2l+4 k if 2 2l+4 2 2l+3 k t <2 2l+4 2 2l+2 k for some l Z, k N. We note that ϕ is well defined because [0, )={0} (0, )={0} l Z [ 2 2l+2,2 2l+4) = {0} l Z ([ 2 2l+2,2 2l+3) [ 2 2l+3,2 2l+4)) ( [2 2l+2,2 2l+3) [ 2 2l+4 2 2l+3 k,2 2l+4 2 2l+2 k)), = {0} l Z k N {0} where represents disjoint union. It is obvious that ϕ(t)<t for t (0, ) and ϕ is right continuous. We note that ϕ is strictly increasing on the range of ρ becausetherangeofρ is a subset of {0} { 2 2l 4 2 2l 3 k : l Z, k N {0} } and 2 2l 4 2 2l 3 k <2 2l 4 2 2l 3 k for l, l Z and k, k N {0, } with l < l (l = l k < k ), where 2 =0, represents logical or and represents logicaland.letx and y be two distinct elements of X \{z}. We consider the following three cases: Tx = z;
6 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 6 of 10 Tx z and k(x)=0; Tx z and k(x) N. In the first case, we have ρ(tx, z)=0 ϕ ( ρ(x, z) ). In the second case, noting l(tx)<l(x),we have ρ(tx, z)=2 2l(Tx)+4 2 2l(Tx)+3 k(tx) <2 2l(Tx)+4 2 2l(x)+2 = ϕ ( 2 2l(x)+4 2 2l(x)+3) = ϕ ( ρ(x, z) ). In the third case, noting l(tx)=l(x) and k(tx)=k(x) 1,we have ρ(tx, z)=2 2l(Tx)+4 2 2l(Tx)+3 k(tx) =2 2l(x)+4 2 2l(x)+4 k(x) = ϕ ( 2 2l(x)+4 2 2l(x)+3 k(x)) = ϕ ( ρ(x, z) ). We have shown ρ(tx, Tz)=ρ(Tx, z) ϕ(ρ(x, z)) for all x X \{z}.usingthisandthestrict monotony of ϕ on the range of ρ,weobtain ρ(tx, Ty) max { ρ(tx, z), ρ(ty, z) } max { ϕ ( ρ(x, z) ), ϕ ( ρ(y, z) )} = ϕ ( max { ρ(x, z), ρ(y, z) }) = ϕ ( ρ(x, y) ). Therefore T BWC(X, ρ). From Theorem 9, we obtain the following. Corollary 10 Let (X, d) be a complete metric space and let T be a mapping on X. Then (A), (E), (F), and (G) are equivalent. (E) There exists a complete metric ρ on X such that the topology of (X, ρ) is stronger than that of (X, d) and T BWC(X, ρ). (F) There exists a complete metric ρ on X such that the topology of (X, ρ) is stronger than that of (X, d) and T MKC(X, ρ). (G) There exists a complete metric ρ on X such that the topology of (X, ρ) is stronger than that of (X, d) and T CJMC(X, ρ). Proof It is obvious that (E) (F) (G). The proof of (G) (A) is almost the same as that of (D) (A).Since (E) is weaker than (D),we have (A) (E) by Theorem 9. We note that (E) is much simpler than (B) and (C). 4 Additional results Intheprevioussection,wehaveshowedthat(D)isequivalentto(A).Inthissection,we will show that the condition (H) on contractions is not equivalent to (A). Theorem 11 Let (X, d) be a complete metric space and let T be a mapping on X. Then (H) (A) holds.
7 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 7 of 10 (H) There exists a complete metric ρ on X such that the topology of (X, ρ) is stronger than that of (X, d) and T Cont(X, ρ). Proof The proof of (D) (A) works. The following example tells that (A) (H) does not hold. Example 12 ([4]) Let A be the set of all real sequences {a n } such that a n (0, )forn N, {a n } is strictly decreasing, and {a n } converges to 0. Let H be a Hilbert space consisting of all the functions x from A into R satisfying a A x(a) 2 < with inner product x, y = a A x(a)y(a) for all x, y H. Putd(x, y)= x y, x y 1/2.DefineacompletesubsetX of H by ( ) X = {0} {a n e a : n N}, a A where e a H is defined by e a (a)=1ande a (b)=0forb A \{a}. Define a mapping T on X by T0=0 and T(a n e a )=a n+1 e a. Then (A) holds. However, (H) does not hold. Proof It is obvious that (A) holds. Arguing by contradiction, we assume (H). That is, there exist a metric ρ on X and r [0, 1) such that the topology of (X, ρ) is stronger than that of (X, d), (X, ρ) iscompleteandρ(tx, Ty) rρ(x, y) for all x, y X. Since the topology of (X, ρ) is stronger than that of (X, d), inf { ρ(0, y):d(0, y)>t } >0 for every t > 0. So, there exists a strictly increasing sequence {κ n } in N such that r κ n < inf { ρ(0, y):d(0, y)>1/n }. Then ρ(0, x) r κ n d(0, x) 1/n holds. We choose α A such that α 2κn +1 >1/n. Fixν N with r κ ν ρ(0, α 1 e α ) 1. Then we have ρ(0, α 2κν +1e α )=ρ ( T 2κ ν 0, T 2κ ν (α 1 e α ) ) r 2κ ν ρ(0, α 1 e α ) r κ ν and hence 1/ν < α 2κν +1 = d(0, α 2κν +1e α ) 1/ν. This is a contradiction.
8 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 8 of 10 The Matkowski contraction version of (H) is equivalent to (H) itself. Theorem 13 Let (X, d) be a complete metric space and let T be a mapping on X. Then (H) (I) holds. (I) There exists a complete metric ρ on X such that the topology of (X, ρ) is stronger than that of (X, d) and T MC(X, ρ). Proof (H) (I): Obvious. (I) (H): Assume (I). Then there exists a function ψ satisfying (i)(iii) of Theorem 4 with replacing d := ρ.defineafunctionϕ from [0, )intoitselfby ϕ(t)=max { ψ(t), t/2, [ t] 1 }. Then from Lemma 8, ϕ satisfies (i)(iii) of Theorem 4 with replacing d := ρ and ϕ := ψ. We note 0 < ϕ(t) fort (0, ). Also we note k ϕ(t) ift satisfies k < t k +1forsome k N. We define a sequence {t n } n Z.Putt 0 =1andt n = ϕ n (1) for n N.Fork N,wecan choose ν(k) N satisfying ϕ ν(k) (k +1)=k. Soforn N, thereexistsκ(n) N such that κ(n) 1 j=1 ν(j)<n κ(n) j=1 ν(j), where 0 j=1 ν(j)=0.weputt κ(n) n = ϕ j=1 ν(j) n (κ(n) + 1). Then thefollowingareobvious: t n+1 = ϕ(t n ) for every n Z, {t n } n Z is a strictly decreasing sequence, {t n } n Z converges to 0 as n tends to, {t n } n Z converges to as n tends to. Let z X be a unique fixed point of T and fix r (0, 1). Define a function f from X \{z} into (0, )by f (x)=r n if t n+1 < ρ(z, x) t n for some n Z. Then we have f (Tx) rf (x) provided Tx z. Indeed t n+1 < ρ(z, x) t n implies f (x)=r n and ρ(z, Tx) ψ ( ρ(z, x) ) ϕ ( ρ(z, x) ) ϕ(t n )=t n+1 and hence f (Tx) r n+1 = rf (x). We denote by q a complete metric defined by (1) with replacing q := ρ.let{x n } be a sequence in X such that x n are all different and {x n } converges to some x X in (X, q). From the definition of q, x = z holds. Without loss of generality, we may assume x n z.sincelim n f (x n )=lim n q(z, x n )=0,wehavelim n ρ(z, x n )=0.Thus, {x n } converges to z in (X, ρ). Therefore the topology of (X, q) is stronger than that of (X, ρ), which is stronger than that of (X, d). Let x and y be two distinct elements of X \{z}.inthe case where Tx = z,wehave q(tx, Tz)=q(Tx, z)=0 rq(x, z).
9 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 9 of 10 In the other case, where Tx z,wehave q(tx, Tz)=q(Tx, z)=f (Tx) rf (x)=q(x, z). We obtain q(tx, Ty) max { q(tx, z), q(ty, z) } max { rq(x, z), rq(y, z) } = r max { q(x, z), q(y, z) } = rq(x, y). Therefore T Cont(X, q). The following result due to Bessaga [15] (seealso[16]) shows that the topological condition appearing in condition (H) cannot be removed, because otherwise the convergence of iterates in the metric space (X, d) cannot be ensured. Theorem 14 (Bessaga [15]) Let X be a set and let T be a mapping on X. Then (J) and (K) are equivalent. (J) There exists a complete metric ρ on X such that T Cont(X, ρ). (K) There exists a unique fixed point z of T and the set of periodic points of T is {z}. If X is a metric space, then (J) is strictly weaker than (A) because (K) is strictly weaker than (A). In conclusion, we obtain (H) (I) (A) (E) (F) (G) (J) under the assumption that (X, d) is a complete metric space. We can tell that, from this point of view, the difference between contractions and Matkowski contractions is small and the difference between contractions and BoydWong contractions is very large. Therefore Matkowski contractions and BoydWong contractions are essentially different. Considering the appearance of the statements, we might have considered that the difference between BoydWong contractions and Matkowski contractions was small and the difference between BoydWong and MeirKeeler contractions was large. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. ( HiCi). The authors, therefore, acknowledge technical and financial support of KAU. The first author is supported in part by GrantinAid for Scientific Research from Japan Society for the Promotion of Science. Received: 20 January 2015 Accepted: 24 March 2015 References 1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, (1922) 2. Caccioppoli, R: Un teorema generale sull esistenza di elementi uniti in una transformazione funzionale. Rend. Accad. Naz. Lincei 11, (1930)
10 Suzuki and Alamri Fixed Point Theory and Applications (2015) 2015:59 Page 10 of Suzuki, T: A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point. Proc. Am. Math. Soc. 136, (2008). MR Suzuki, T: Convergence of the sequence of successive approximations to a fixed point. Fixed Point Theory Appl. 2010, Article ID (2010). MR Leader, S: Equivalent Cauchy sequences and contractive fixed points in metric spaces. Stud. Math. 76, (1983). MR Suzuki, T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 253, (2001). MR Jachymski, J: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, (1997). MR Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, (1969). MR Matkowski, J: Integrable solutions of functional equations. Diss. Math. 127, 168 (1975). MR Meir,A,Keeler,E:Atheoremoncontractionmappings.J.Math.Anal.Appl.28, (1969). MR Ćirić, LB: A new fixedpoint theorem for contractive mappings. Publ. Inst. Math. (Belgr.) 30, (1981).MR Jachymski, J: Equivalent conditions and the MeirKeeler type theorems. J. Math. Anal. Appl. 194, (1995). MR Kuczma, M, Choczewski, B, Ger, R: Iterative Functional Equations. Encyclopedia of Mathematics and Its Applications, vol. 32. Cambridge University Press, Cambridge (1990). MR Matkowski, J: Fixed point theorems for contractive mappings in metric spaces. Čas. Pěst. Mat. 105, (1980). MR Bessaga, C: On the converse of the Banach fixedpoint principle. Colloq. Math. 7, (1959). MR Jachymski, J: A short proof of the converse to the contraction principle and some related results. Topol. Methods Nonlinear Anal. 15, (2000).MR
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