Simeon Reich (1974) proved that the fixed point theorem for single-valued mappings proved by Boyd and Wong can be generalized to multivalued mappings which map points into compact sets. He then asked (1983) whether his theorem can be extended to multivalued mappings whose range consists of bounded closed sets. In this note, we provide an affirmative answer for a certain subclass of Boyd-Wong contractive mappings

Daffer, Peter Z.

Kaneko, Hideaki

Li, Wu

English

Old Dominion University

Old Dominion UniversityODU Digital CommonsMathematics & Statistics Faculty Publications Mathematics & Statistics1996On a Conjecture of S. ReichPeter Z. DafferHideaki KanekoOld Dominion UniversityWu LiOld Dominion UniversityFollow this and additional works at: https://digitalcommons.odu.edu/mathstat_fac_pubsPart of the Applied Mathematics CommonsThis Article is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion inMathematics & Statistics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contactdigitalcommons@odu.edu.Repository CitationDaffer, Peter Z.; Kaneko, Hideaki; and Li, Wu, "On a Conjecture of S. Reich" (1996). Mathematics & Statistics Faculty Publications. 145.https://digitalcommons.odu.edu/mathstat_fac_pubs/145Original Publication CitationDaffer, P. Z., Kaneko, H., & Li, W. (1996). On a conjecture of S. Reich. Proceedings of the American Mathematical Society, 124(10),3159-3162. doi:10.1090/s0002-9939-96-03659-3PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 124, Number 10, October 1996ON A CONJECTURE OF S. REICHPETER Z. DAFFER, HIDEAKI KANEKO, AND WU LI(Communicated by Dale Alspach)Abstract. Simeon Reich (1974) proved that the fixed point theorem forsingle-valued mappings proved by Boyd and Wong can be generalized to mul-tivalued mappings which map points into compact sets. He then asked (1983)whether his theorem can be extended to multivalued mappings whose rangeconsists of bounded closed sets. In this note, we provide an affirmative answerfor a certain subclass of Boyd-Wong contractive mappings.1. IntroductionLet (X, d) be a complete metric space. Denote by CB(X) and K(X) the spaceof all nonempty closed and bounded subsets of X and the space of all nonemptycompact subsets of X , respectively. The Hausdorff metric induced by d will bedenoted by H. In [5, p. 40], Reich proved that a mapping T : X → K(X) has afixed point in X if it satisfies H(Tx, Ty) ≤ k(d(x, y))d(x, y) for all x, y ∈ X withx 6= y, where k : (0,∞)→ [0, 1) satisfies lim supr→t+ k(r) < 1 for every t ∈ (0,∞).This result generalizes the fixed point theorem for single-valued mappings that wasproved by Boyd and Wong [1]. One of the conjectures made by Reich in [6, 7] askswhether or not the range of T can be relaxed. Specifically the question is whetheror not the range of T , K(X), can be replaced by CB(X). In response to Reich’sconjecture, the following theorem was recently proved by Mizoguchi and Takahashi[4], and other proofs have been given by Daffer and Kaneko [3] and Tong-HueiChang [2].Theorem 1. Let (X, d) be a complete metric space and T : X → CB(X). Assumethat T satisfiesH(Tx, Ty) ≤ k(d(x, y))d(x, y)(1)for all x, y ∈ X with x 6= y, where k : (0,∞)→ [0, 1) satisfies lim supr→t+ k(r) < 1for every t ∈ [0,∞). Then T has a fixed point in X.The stronger condition assumed on k in Theorem 1, viz., lim supr→t+ k(r) < 1for every t ∈ [0,∞), implies that k(t) < h for some 0 < h < 1 and for small t > 0.Therefore, with this condition, one may conclude that a mapping satisfying (1) isa contraction in the sense of Banach over a region for which d(x, y) is sufficientlysmall. In this paper, we obtain a theorem which replaces the closed interval [0,∞)of this result with the open interval (0,∞) of the classical Boyd-Wong fixed pointtheorem.Received by the editors April 5, 1995.1991 Mathematics Subject Classification. Primary 47H10.c©1996 American Mathematical Society3159License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use3160 P. Z. DAFFER, HIDEAKI KANEKO, AND WU LI2. The main resultIn recent papers ([2, 8]) the following class of functions was introduced andstudied.Definition 2. Let φ : R+ → R+. The function φ is said to satisfy the condition (Φ)(denoted φ ∈ (Φ) ) if (i) φ(t) < t for all t ∈ (0,∞); (ii) φ is upper semicontinuousfrom the right on (0,∞); and (iii) there exists a positive real number s such that φis nondecreasing on (0, s] and∑∞n=0 φn(t) <∞ for all t ∈ (0, s].Chang [2] observed that if k : (0,∞) → [0, 1) satisfies lim supr→t+ k(r) < 1 forevery t ∈ [0,∞), then there exists a function φ ∈ (Φ) such that k(t)t ≤ φ(t) for allt ∈ (0,∞).Subsequently, Chang proved the following theorem that generalizes Theorem 1above.Theorem 3. Let (X, d) be a complete metric space. Let T : X → CB(X) andsuppose that there exists a function φ ∈ (Φ) such thatH(Tx, Ty) ≤ φ(max{d(x, y), d(x, Tx), d(y, T y),d(x, Ty) + d(y, Tx)2})for all x, y ∈ X. Then T has a fixed point in X.The purpose of this note is to establish a class of functions that satisfy the in-equality lim supr→t+ k(r) < 1 for every t ∈ (0,∞) and that belong to (Φ). Utilizingthe result of Chang above, we can then obtain a fixed point theorem for multival-ued functions that satisfy the conditions required in the conjecture of Reich. Theexistence of such a class of functions confirms that the conjecture of Reich is stillopen and that a further investigation toward a complete resolution is required.Lemma 4. Suppose ϕ(t) = t− atb, where a > 0.(i) If b ≥ 2, then∑∞n=0 ϕn(t) = +∞, for all real numbers t except for t = 0 andt = a−1b−1 ;(ii) If 1 < b < 2, then the series∑∞n=0 ϕn(t) converges if 0 ≤ t ≤ a− 1b−1 andit diverges if t < 0 or t > a−1b−1 , where ϕn = ϕ ◦ · · · ◦ ϕ denotes the n-foldcomposition.Proof. A sketch shows that∑∞n=0 ϕn(t) = +∞ for t < 0 or a− 1b−1 < t, for anyvalue of b greater than 1. Also, it is clear that the series converges for t = 0 andt = a−1b−1 . Thus we restrict our attention to the case when t ∈ (0, a− 1b−1 ).For t ∈ (0, a− 1b−1 ), 0 < ϕ(t) < t. Since {ϕn(t) : n = 1, 2, . . .} is a strictlymonotone decreasing sequence and bounded below by 0, it has a limit z. Since ϕis a continuous function, we have ϕ(z) = limn→∞ ϕ (ϕn(t)) = z. As a consequence,z = 0. That is,limn→∞ϕn(t) = 0 for 0 < t < a−1b−1 .(2)Let b ≥ 2 and choose an integer k > a. Then 1n −anb≥ 1n −an2 ≥1n+k , for alln ≥ akk−a =: m0. Thus ϕ(1n ) ≥1n+k for n ≥ m0. On the interval(0, (ab)−1b−1), ϕis increasing. Let ε = min{1m0, (ab)−1b−1}. Thenϕ(t) ≥ ϕ(1mt)≥ 1mt + kfor 0 < t < ε,(3)License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-useA CONJECTURE OF S. REICH 3161where mt :=[1t + 1]. ([x] denotes the largest integer less than or equal to x.)Now let t ∈ (0, a− 1b−1 ). By (2), there exists a positive integer n0 such that0 < ϕn(t) < ε for n ≥ n0. Let r := 1ϕn0(t) . Then it follows thatϕn(t) ≥ 1r + k(n− n0)for n ≥ n0.(4)Indeed, this is true for n = n0, and we see that, for n ≥ n0,ϕn+1(t) = ϕ(ϕn(t))≥ ϕ(1r + k(n− n0))≥ 1r + k(n− n0) + k=1r + k(n+ 1− n0),where we use the induction assumption for the first inequality. The inductionmethod proves (4). Hence,∞∑n=1ϕn(t) ≥∞∑n=n0ϕn(t) ≥∞∑n=n01r + k(n− n0)= +∞.Now let 1 < b < 2. For 1 < c < 1b−1 , we havenbc(1nc− 1(n+ 1)c)=nbc((n+ 1)c − nc)nc(n+ 1)c=nbc−c−1(1 + 1n )c·(1 + 1n )c − 11n→ 0as n→∞. Thus, we can choose n0 such that n−c0 < (ab)−1b−1 (so that 1nc0is in theinterval where ϕ is increasing) and 1nc −1(n+1)c ≤anbc, for all n ≥ n0. As a result,we have ϕ( 1nc ) ≤1(n+1)c for all n ≥ n0. For every t ∈ [0, a− 1b−1 ], limn→∞ ϕn(t) = 0(cf. (2)), and so for some integer k0 we have ϕk0(t) ≤ 1nc0 .We claim that ϕn(t) ≤ 1(n0+n−k0)c , for all n ≥ k0. Indeed, this is true for n = k0,and we have thatϕn+1(t) = ϕ (ϕn(t)) ≤ ϕ(1(n0 + n− k0 + 1)c)≤ 1(n0 + n+ 1− k0)c,where the induction assumption is used for the first inequality. This completes theproof of our claim. Hence∞∑n=0ϕn(t) =k0−1∑n=0ϕn(t) +∞∑n=k0ϕn(t) =k0−1∑n=0ϕn(t) +∞∑n=k01(n0 + n− k0)c< +∞since c > 1.Theorem 5. Let (X, d) be a complete metric space and T : X → CB(X). If thereexists an upper right semi-continuous function ϕ : R+ → R+ such that (i) ϕ(t) < tfor all t > 0, (ii) ϕ(t) ≤ t− atb, a > 0, for some 1 < b < 2 on some interval [0, s],s > 0, and (iii)H(Tx, Ty) ≤ ϕ(d(x, y))for all x, y ∈ X, then T has a fixed point in X.Proof. Writing ψ(t) = t−atb and ε = min{s, (ab)1b−1}, we have that ϕn(t) ≤ ψn(t)for every positive integer n and t ∈ [0, ε] (an interval upon which ψ is increasing).Indeed, this holds for n = 1 and if it holds for n, thenψn+1(t) = ψ(ψn(t)) ≥ ψ(ϕn(t)) ≥ ϕ(ϕn(t)) = ϕn+1(t),License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use3162 P. Z. DAFFER, HIDEAKI KANEKO, AND WU LIsince ψn(t), ϕn(t) ∈ [0, ε] for every n. This completes the induction proof of ϕn(t) ≤ψn(t). By Lemma 4, ψ ∈ (Φ); hence we also have ϕ ∈ (Φ) and the conclusion followsfrom Theorem 3.We remark that Theorem 5 can be formulated using the contractive conditionH(Tx, Ty) ≤ ϕ(max{d(x, y), d(x, Tx), d(y, T y),d(x, Ty) + d(y, Tx)2}),because of Theorem 3 above. However, because of a more relevance to the con-jecture of Reich, we chose to describe our theorem using the contractive conditionH(Tx, Ty) ≤ ϕ(d(x, y)).Theorem 5 reveals that with k(t) = 1− atb−1 we have a class of functions thatsatisfy all the conditions in the conjecture of Reich. Hence we obtain the following:Corollary 6. Let (X, d) be a complete metric space and T : X → CB(X). Assumethat T satisfiesH(Tx, Ty) ≤ k(d(x, y))d(x, y)for all x, y ∈ X, where k : R+ → [0, 1] with k(t) < 1 for t > 0, and k(t) ≤ 1−atb−1,a > 0, for some b ∈ (1, 2) on some interval [0, s], 0 < s < a− 1b−1 . Then T has afixed point in X.References1. D. W. Boyd and J. S. W. Wong, On Nonlinear Contractions, Proc. A.M.S. 89(1968), 458-464.MR 39:9162. T. H. Chang, Common Fixed Point Theorems for Multi-Valued Mappings, Math. Japonica 41(1995), 311–320. CMP 1995:113. P. Z. Daffer and H. Kaneko, Fixed Points of Generalized Contractive Multi-Valued Mappings, J.Math. Anal. Appl. 192 (1995), 655–666. CMP 1995:124. N. Mizoguchi and W. Takahashi, Fixed Points Theorems for Multi-Valued Mappings on Com-plete Metric Spaces, J. Math. Anal. Appl. 141(1989) 177-188. MR 90f:470865. S. Reich, Fixed Points of Contractive Functions, Boll. Un. Mat. Ital., (4) 5 (1972), 26-42. MR46:82066. S. Reich, Some Fixed Point Problems, Atti. Accad. Naz. Lincei, 57(1974), 194-198. MR 53:13467. S. Reich, Some Problems and Results in Fixed Point Theory, Contemporary Math. A.M.S.21(1983), 179–187. MR 85e:470828. K. Sastry, S. Naidu and J. Prasad, Common Fixed Points for Mutimaps in a MetricSpace, Nonlinear Anal. 13(3) (1989) 221-229. MR 90b:54034Department of Mathematics, Macon College, Macon, Georgia 31297E-mail address: pdaffer@cennet.mc.peachnet.eduDepartment of Mathematics and Statistics, Old Dominion University, Norfolk, Vir-ginia 23529-0077E-mail address: kaneko@math.odu.eduDepartment of Mathematics and Statistics, Old Dominion University, Norfolk, Vir-ginia 23529-0077E-mail address: wuli@math.odu.eduLicense or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

On a Conjecture of S. Reich

https://digitalcommons.odu.edu/cgi/viewcontent.cgi?article=1145&context=mathstat_fac_pubs

On a Conjecture of S. Reich

Abstract

Similar works

Full text

Available Versions

Old Dominion University