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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

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 Boyd-Wong 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 Meir-Keeler 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 Boyd-Wong contractions, all Matkowski contractions, all Meir-Keeler 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 Boyd-Wong contractions is very large. Therefore Matkowski contractions and Boyd-Wong contractions are essentially different. Considering the appearance of the statements, we might have considered that the difference between Boyd-Wong contractions and Matkowski contractions was small and the difference between Boyd-Wong and Meir-Keeler 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 Grant-in-Aid 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, 1-68 (1975). MR Meir,A,Keeler,E:Atheoremoncontractionmappings.J.Math.Anal.Appl.28, (1969). MR Ćirić, LB: A new fixed-point theorem for contractive mappings. Publ. Inst. Math. (Belgr.) 30, (1981).MR Jachymski, J: Equivalent conditions and the Meir-Keeler 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 fixed-point 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

### On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces

Bachar and Khamsi Fixed Point Theory and Applications (215) 215:16 DOI 1.1186/s13663-15-45-3 R E S E A R C H Open Access On common approximate fixed points of monotone nonexpansive semigroups in Banach

### Quasi Contraction and Fixed Points

Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa-00168, 6 Pages doi:10.5899/2012/jnaa-00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady

### COMMON FIXED POINTS OF FOUR MAPS USING GENERALIZED WEAK CONTRACTIVITY AND WELL-POSEDNESS

Int. J. Nonlinear Anal. Appl. (011) No.1, 73 81 ISSN: 008-68 (electronic) http://www.ijnaa.com COMMON FIXED POINTS OF FOUR MAPS USING GENERALIZED WEAK CONTRACTIVITY AND WELL-POSEDNESS MOHAMED AKKOUCHI

### Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s13661-015-0508-0 R E S E A R C H Open Access Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations

Wang and Agarwal Advances in Difference Equations (2015) 2015:312 DOI 10.1186/s13662-015-0650-0 R E S E A R C H Open Access Relatively dense sets, corrected uniformly almost periodic functions on time

### ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska

Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. Gacovska-Barandovska Smirnov (1949) derived four

### TRIPLE FIXED POINTS IN ORDERED METRIC SPACES

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197-207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON

### How many miles to βx? d miles, or just one foot

How many miles to βx? d miles, or just one foot Masaru Kada Kazuo Tomoyasu Yasuo Yoshinobu Abstract It is known that the Stone Čech compactification βx of a metrizable space X is approximated by the collection

### Math 317 HW #5 Solutions

Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction

italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University

### On Some Properties of I 2 -Convergence and I 2 -Cauchy of Double Sequences

Gen. Math. Notes, Vol. 7, No. 1, November 2011, pp.1-12 ISSN 2219-7184; Copyright c ICSRS Publication, 2011 www.i-csrs.org Available free online at http://www.geman.in On Some Properties of I 2 -Convergence

### Infinite product spaces

Chapter 7 Infinite product spaces So far we have talked a lot about infinite sequences of independent random variables but we have never shown how to construct such sequences within the framework of probability/measure

### 1 Fixed Point Iteration and Contraction Mapping Theorem

1 Fixed Point Iteration and Contraction Mapping Theorem Notation: For two sets A,B we write A B iff x A = x B. So A A is true. Some people use the notation instead. 1.1 Introduction Consider a function

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

### On Statistically Convergent Sequences in Intuitionistic Fuzzy 2-Normed Spaces

International Mathematical Forum, 5, 010, no. 0, 967-978 On Statistically Convergent Sequences in Intuitionistic Fuzzy -Normed Spaces Abdullah M. Alotaibi King Abdulaziz university, P.O. Box 8003 Jeddah

### CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOM-VALUED VECTOR FUNCTIONS

C O L L O Q U I U M M A T H E M A T I C U M VOL. 97 2003 NO. 1 CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOM-VALUED VECTOR FUNCTIONS BY RAFAŁ KAPICA (Katowice) Abstract. Given a probability space (Ω,

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh

Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp -. TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### SOME TYPES OF CONVERGENCE OF SEQUENCES OF REAL VALUED FUNCTIONS

Real Analysis Exchange Vol. 28(2), 2002/2003, pp. 43 59 Ruchi Das, Department of Mathematics, Faculty of Science, The M.S. University Of Baroda, Vadodara - 390002, India. email: ruchid99@yahoo.com Nikolaos

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### Some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative variables

Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 6 (013), 74 85 Research Article Some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative

### Compactness in metric spaces

MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### Computing divisors and common multiples of quasi-linear ordinary differential equations

Computing divisors and common multiples of quasi-linear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univ-lille1.fr

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### 1 Introduction, Notation and Statement of the main results

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matemática Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1 6) Skew-product maps with base having closed set of periodic points. Juan

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### A NOTE ON THE RATES OF CONVERGENCE OF DOUBLE SEQUENCES

SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22) (2014), 87 92 DOI: 10.5644/SJM.10.1.11 A NOTE ON THE RATES OF CONVERGENCE OF DOUBLE SEQUENCES METIN BAŞARIR Abstract. In this paper, we define the rates of convergence

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### Cylinder Maps and the Schwarzian

Cylinder Maps and the Schwarzian John Milnor Stony Brook University (www.math.sunysb.edu) Bremen June 16, 2008 Cylinder Maps 1 work with Araceli Bonifant Let C denote the cylinder (R/Z) I. (.5, 1) fixed

### Weak topologies. David Lecomte. May 23, 2006

Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such

### Math 3000 Running Glossary

Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

### A Simple Proof of Duality Theorem for Monge-Kantorovich Problem

A Simple Proof of Duality Theorem for Monge-Kantorovich Problem Toshio Mikami Hokkaido University December 14, 2004 Abstract We give a simple proof of the duality theorem for the Monge- Kantorovich problem

### Advanced results on variational inequality formulation in oligopolistic market equilibrium problem

Filomat 26:5 (202), 935 947 DOI 0.2298/FIL205935B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Advanced results on variational

### Double Sequences and Double Series

Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions,

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam

Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad

### 9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### Practice Problems Solutions

Practice Problems Solutions The problems below have been carefully selected to illustrate common situations and the techniques and tricks to deal with these. Try to master them all; it is well worth it!

### Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend

### The square-free kernel of x 2n a 2n

ACTA ARITHMETICA 101.2 (2002) The square-free kernel of x 2n a 2n by Paulo Ribenboim (Kingston, Ont.) Dedicated to my long-time friend and collaborator Wayne McDaniel, at the occasion of his retirement

### Lattice Points in Large Borel Sets and Successive Minima. 1 Introduction and Statement of Results

Lattice Points in Large Borel Sets and Successive Minima Iskander Aliev and Peter M. Gruber Abstract. Let B be a Borel set in E d with volume V (B) =. It is shown that almost all lattices L in E d contain

### The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line

The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,

### Math Real Analysis I

Math 431 - Real Analysis I Solutions to Homework due October 1 In class, we learned of the concept of an open cover of a set S R n as a collection F of open sets such that S A. A F We used this concept

### 3.1. Sequences and Their Limits Definition (3.1.1). A sequence of real numbers (or a sequence in R) is a function from N into R.

CHAPTER 3 Sequences and Series 3.. Sequences and Their Limits Definition (3..). A sequence of real numbers (or a sequence in R) is a function from N into R. Notation. () The values of X : N! R are denoted

### n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

### Math 421, Homework #5 Solutions

Math 421, Homework #5 Solutions (1) (8.3.6) Suppose that E R n and C is a subset of E. (a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as defined in Definition

### Minimal R 1, minimal regular and minimal presober topologies

Revista Notas de Matemática Vol.5(1), No. 275, 2009, pp.73-84 http://www.saber.ula.ve/notasdematematica/ Comisión de Publicaciones Departamento de Matemáticas Facultad de Ciencias Universidad de Los Andes

### ON TOPOLOGICAL SEQUENCE ENTROPY AND CHAOTIC MAPS ON INVERSE LIMIT SPACES. 1. Introduction

Acta Math. Univ. Comenianae Vol. LXVIII, 2(1999), pp. 205 211 205 ON TOPOLOGICAL SEQUENCE ENTROPY AND CHAOTIC MAPS ON INVERSE LIMIT SPACES J. S. CANOVAS Abstract. The aim of this paper is to prove the

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### Andrew McLennan January 19, Winter Lecture 5. A. Two of the most fundamental notions of the dierential calculus (recall that

Andrew McLennan January 19, 1999 Economics 5113 Introduction to Mathematical Economics Winter 1999 Lecture 5 Convergence, Continuity, Compactness I. Introduction A. Two of the most fundamental notions

### Continuous first-order model theory for metric structures Lecture 2 (of 3)

Continuous first-order model theory for metric structures Lecture 2 (of 3) C. Ward Henson University of Illinois Visiting Scholar at UC Berkeley October 21, 2013 Hausdorff Institute for Mathematics, Bonn

### On the greatest and least prime factors of n! + 1, II

Publ. Math. Debrecen Manuscript May 7, 004 On the greatest and least prime factors of n! + 1, II By C.L. Stewart In memory of Béla Brindza Abstract. Let ε be a positive real number. We prove that 145 1

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### ARTICLE IN PRESS Nonlinear Analysis ( )

Nonlinear Analysis ) Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A question concerning a polynomial inequality and an answer M.A. Qazi a,,

### More Mathematical Induction. October 27, 2016

More Mathematical Induction October 7, 016 In these slides... Review of ordinary induction. Remark about exponential and polynomial growth. Example a second proof that P(A) = A. Strong induction. Least

### Solutions of Equations in One Variable. Fixed-Point Iteration II

Solutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### Multiplicative Relaxation with respect to Thompson s Metric

Revista Colombiana de Matemáticas Volumen 48(2042, páginas 2-27 Multiplicative Relaxation with respect to Thompson s Metric Relajamiento multiplicativo con respecto a la métrica de Thompson Gerd Herzog

### Available online at ScienceDirect. Procedia Computer Science 74 (2015 ) 47 52

Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 74 (05 ) 47 5 International Conference on Graph Theory and Information Security Fractional Metric Dimension of Tree and

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### Coupled best proximity point theorems for proximally g-meir-keelertypemappings in partially ordered metric spaces

Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 DOI 10.1186/s13663-015-0355-9 R E S E A R C H Open Access Couple best proximity point theorems for proximally g-meir-keelertypemappings in

### Chapter 6 SEQUENCES. sn 1+1,s 1 =1 is again an abbreviation for the sequence { 1, 1 2, 1 } n N ε,d (s n,s) <ε.

Chapter 6 SEQUENCES 6.1 Sequences A sequence is a function whose domain is the set of natural numbers N. If s is a sequence, we usually denote its value at n by s n instead of s (n). We may refer to the

### On the Union of Arithmetic Progressions

On the Union of Arithmetic Progressions Shoni Gilboa Rom Pinchasi August, 04 Abstract We show that for any integer n and real ɛ > 0, the union of n arithmetic progressions with pairwise distinct differences,

### LINEAR SEARCH FOR A BROWNIAN TARGET MOTION 1

23,23B(3):321-327 LINEAR SEARCH FOR A BROWNIAN TARGET MOTION 1 A. B. El-Rayes 1 Abd El-Moneim A. Mohamed 1 Hamdy M. Abou Gabal 2 1.Military Technical College, Cairo, Egypt 2.Department of Mathematics,

### A Simple Observation Concerning Contraction Mappings

Revista Colombiana de Matemáticas Volumen 46202)2, páginas 229-233 A Simple Observation Concerning Contraction Mappings Una simple observación acerca de las contracciones German Lozada-Cruz a Universidade

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients

DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca

### Concentration of points on two and three dimensional modular hyperbolas and applications

Concentration of points on two and three dimensional modular hyperbolas and applications J. Cilleruelo Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas Universidad

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Measure Theory. 1 Measurable Spaces

1 Measurable Spaces Measure Theory A measurable space is a set S, together with a nonempty collection, S, of subsets of S, satisfying the following two conditions: 1. For any A, B in the collection S,

### Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Rate of convergence towards Hartree dynamics

Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and