ON A CONJECTURE OF S. REICH

Peter Z. Daffer Department of Mathematics Macon College Macon, Georgia 31297

Hideaki Kaneko and Wu Li Department of Mathematics and Statistics Old Dominion University Norfolk, Virginia 23529-0077

Abstract Simeon Reich [?] 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 [?] 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.

Mathematics Subject Classification (1991): 47H10

1 Introduction

Let (X, d) be a complete metric space. Denote by CB(X) and K(X) the space of all nonempty closed and bounded subsets of X and the space of all nonempty compact subsets of X, respectively. The Hausdorff metric induced by d will be denoted by H. In [?] (p.40), Reich proved that a mapping T: X → K(X) has a fixed point in X if it satisfies H(T x, T y) ≤ k(d(x, y))d(x, y) for all x, y ∈ X with x ≠ y, where k: (0, ∞) → [0, 1) satisfies lim sup_{r→t^{+}} k(r) < 1 for every t ∈ (0, ∞). This result generalizes the fixed point theorem for single-valued mappings that was proved by Boyd and Wong [?]. One of the conjectures made by Reich in [?],[?] asks whether or not the range of T can be relaxed. Specifically the question is whether or not the range of T, K(X), can be replaced by CB(X). In response to Reich’s conjecture, the following theorem was recently proved by Mizoguchi and Takahashi [?], and other proofs have been given by Daffer and Kaneko [?] and Tong-Huei Chang [?];

Theorem 1.1 Let (X, d) be a complete metric space and T: X → CB(X). Assume that T satisfies

H(T x, T y) ≤ k(d(x, y))d(x, y)  (1.1)

for all x, y ∈ X with x ≠ y, where k: (0, ∞) → [0, 1) satisfies lim sup_{r→t^{+}} k(r) < 1 for every t ∈ [0, ∞). Then T has a fixed point in X.

The stronger condition assumed on k in Theorem 1.1, viz., lim sup_{r→t^{+}} k(r) < 1 for 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 (1.1) is a contraction in the sense of Banach over a region for which d(x, y) is sufficiently small. We obtain a theorem which replaces the closed interval [0, ∞) of this result with the open interval (0, ∞) of the classical Boyd-Wong fixed point theorem.

2 The Main Result

In recent papers ([?],[?]) the following class of functions was introduced and studied. Definition 2.1: Let ϕ: R_{+} → R_{+}. ϕ is said to satisfy the condition (Φ) (denoted ϕ ∈ (Φ) ) if (i) ϕ(t) < t for all t ∈ (0, ∞); (ii) ϕ is upper semicontinuous from the right on (0, ∞); and (iii) there exists a positive real number s such that ϕ is nondecreasing on (0, s] and ∑_{n=0}^{∞} ϕ^{n}(t) < ∞ for all t ∈ (0, s].

Chang [?] observed that if k: (0, ∞) → [0, 1) satisfies lim sup_{r→t^{+}} k(r) < 1 for every t ∈ [0, ∞), then there exists a function ϕ ∈ (Φ) such that k(t)t ≤ ϕ(t) for all t ∈ (0, ∞).

Subsequently, Chang proved the following theorem that generalizes Theorem 1.1 above.

Theorem 2.1 Let (X, d) be a complete metric space. Let T: X → CB(X) and suppose that there exists a function ϕ ∈ (Φ) such that

H(T x, T y) ≤ ϕ(max\{d(x, y), d(x, T x), d(y, T y), \frac{d(x, T y) + d(y, T x)}{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 lim sup_{r→t^{+}} k(r) < 1 for every t ∈ (0, ∞) and that belong to (Φ). Utilizing the result of Chang above, we can then obtain a fixed point theorem for multivalued functions that satisfy the conditions required in the conjecture of Reich. The existence of such a class of functions confirms that the conjecture of Reich is still open and that a further investigation toward a complete resolution is required.

Lemma 2.2 If φ(t) = t − at^{b}, where a > 0, then (i) if b ≥ 2, then ∑_{n=0}^{∞} φ^{n}(t) = +∞, for all t ∈ R except for t = 0 and t = a^{− \frac{1}{b−1}} ; (ii) if 1 < b < 2, then

∑_{n=0}^{∞} φ^{n}(t) = \{ \begin{matrix}< +∞, & for 0 ≤ t ≤ a^{− \frac{1}{b−1}} , \\ = +∞, & for t < 0 or a^{− \frac{1}{b−1}} < t \end{matrix}

Here φ^{n} = φ ∘ ⋯ ∘ φ denotes the n-fold composition.

Proof: A sketch shows that ∑_{n=0}^{∞} φ^{n}(t) = +∞ for t < 0 or a^{− \frac{1}{b−1}} < t, for any value of b greater than 1. Also, it is clear that the series converges for t = 0 and t = a^{− \frac{1}{b−1}} . We thus restrict attention in what follows to t ∈ (0, a^{− \frac{1}{b−1}} ).

Let b ≥ 2 and choose k > a, k ∈ N. Then \frac{1}{n} − \frac{a}{n^{b}} ≥ \frac{1}{n} − \frac{a}{n^{2}} ≥ \frac{1}{n+k}, for all n ≥ \frac{ak}{k−a} = n_{0}. Thus φ(\frac{1}{n}) ≥ \frac{1}{n+k} for n ≥ n_{0}, for some n_{0}. On the interval (0, (ab)^{− \frac{1}{b−1}} ), φ is increasing, and so for t in this interval and N > max\{[t^{−1}] + 1, n_{0}\} we have φ(t) ≥ \frac{1}{n+k}. It follows that φ^{n}(t) ≥ \frac{1}{N+kn}. It follows that φ^{n}(t) ≥ \frac{1}{N+kn}. Indeed, this is true for n = n_{0}, and we see that, for n ≥ n_{0},

φ^{n+1}(t) = φ(φ^{n}(t)) ≥ φ(\frac{1}{N + kn}) ≥ \frac{1}{N + kn + k} = \frac{1}{N + k(n + 1)},

completing the induction. Hence,

∑_{n=0}^{∞} φ^{n}(t) ≥ ∑_{n=N}^{∞} φ^{n}(t) ≥ ∑_{n=N}^{∞} \frac{1}{N + kn} = +∞.

Now let 1 < b < 2. For 1 < c < \frac{1}{b−1}, we have

n^{bc}(\frac{1}{n^{c}} − \frac{1}{(n + 1)^{c}}) = \frac{n^{bc}((n + 1)^{c} − n^{c})}{n^{c}(n + 1)^{c}} = \frac{n^{bc−c−1}}{(1 + \frac{1}{n})^{c}} · \frac{(1 + \frac{1}{n}) − 1}{\frac{1}{n}} → 0

as n → ∞. Thus, there is n_{0}, which we can choose so that n_{0}^{−c} < (ab)^{− \frac{1}{b−1}} (so that _{\frac{1}{n_{0}^{−c}}} is in the interval where φ is increasing) such that \frac{1}{n^{c}} − \frac{1}{(n+1)^{c}} ≤ \frac{1}{n^{bc}}, for all n ≥ n_{0}. Thus we have φ(\frac{1}{n^{c}}) ≤ \frac{1}{(n+1)^{c}} for all n ≥ n_{0}. For every t ∈ [0, a^{− \frac{1}{b−1}} ], lim_{n→∞} φ^{n}(t) = 0, and so for some k_{0} ∈ N we have φ^{k_{0}} ≤ \frac{1}{n_{0}^{c}}. This implies that φ^{n}(t) ≤ \frac{1}{(n_{0}+n−k_{0})^{c}}, for all n ≥ k_{0}. Indeed, this is true for n = k_{0}, and we have that,

φ^{n+1}(t) = φ(φ^{n}(t)) ≤ φ(\frac{1}{(n_{0} + n − k_{0} + 1)^{c}}) ≤ \frac{1}{(n_{0} + n − k_{0} + 1)^{c}} ,

completing the induction. Hence

∑_{n=0}^{∞} φ^{n}(t) = ∑_{n=0}^{k_{0}−1} φ^{n}(t) + ∑_{n=k_{0}}^{∞} φ^{n}(t) = ∑_{n=0}^{k_{0}−1} φ^{n}(t) + ∑_{n=k_{0}}^{∞} \frac{1}{n_{0} + n − k_{0})^{c}} < +∞

since c > 1. □

Theorem 2.3 Let (X, d) be a complete metric space and T: X → CB(X). If there exists a function φ: R_{+} → R_{+} such that φ(t) < t for all t > 0 and φ(t) ≤ t − at^{b}, a > 0, for some 1 < b < 2 on some interval [0, s], s > 0, such that

H(T x, T y) ≤ φ(d(x, y))

for all x, y ∈ X, then T has a fixed point in X.

Proof: Writing ψ(t) = t − at^{b}, we have that φ^{n}(t) ≤ ψ^{n}(t) for every n ∈ N and t ∈ [0, (ab)^{\frac{1}{b−1}} ] an interval upon which ψ is increasing). Indeed, this holds for n = 0, 1 and if it holds for n, then (since ψ^{n}(t) ∈ [0, (ab)^{\frac{1}{b−1}} ] for every n,

ψ^{n+1}(t) = ψ(ψ^{n}(t)) ≥ ψ(φ^{n}(t)) ≥ φ(φ^{n}(t)) = φ^{n+1}(t)

completing the induction. By Lemma 2.2, ψ ∈ (Φ); hence we also have φ ∈ (Φ) and the conclusion follows from Theorem 2.1. □

We remark that Theorem 2.3 can be formulated using the contractive condition

H(T x, T y) ≤ φ(max\{d(x, y), d(x, T x), d(y, T y), \frac{d(x, T y) + d(y, T x)}{2}\})

because of Theorem 2.1 above. However, because of a more relevance to the conjecture of Reich, we chose to describe our theorem using the contractive condition H(T x, T y) ≤ φ(d(x, y)).

Theorem 2.3 reveals that with k(t) = 1 − at^{b−1} we have a class of functions that satisfy all the conditions in the conjecture of Reich. Hence we obtain the following;

Theorem 2.4 Let (X, d) be a complete metric space and T: X → CB(X). Assume that T satisfies

H(T x, T y) ≤ 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 − at^{b−1}, a > 0, for some b ∈ (1, 2) on some interval [0, s], s > 0, 0 < s < a^{− \frac{1}{b−1}} . Then T has a fixed point in X.

References

[1] D. W. Boyd and J. S. W. Wong, On Nonlinear Contractions, Proc. A.M.S. 89(1968), 458-464.

[2] T. H. Chang, Common Fixed Point Theorems for Multi-Valued Mappings, Math. Japonica (in press).

[3] P. Z. Daffer and H. Kaneko, Fixed Points of Generalized Contractive Multi-Valued Mappings, Jl. Math. Anal. Appl. (in press).

[4] N. Mizoguchi and W. Takahashi, Fixed Points Theorems for Multi-Valued Mappings on Complete Metric Spaces, Jl. Math. Anal. Appl. 141(1989) 177-188.

[5] S. Reich, Fixed Points of Contractive Functions, Boll. Un. Mat. Ital., (4) 5 (1972), 26-42.

[6] S. Reich, Some Fixed Point Problems, Atti. Acad. Naz. Lincei, 57(1974), 194-198.

[7] S. Reich, Some Problems and Results in Fixed Point Theory, Contemporay Math. A.M.S. 21(1983), 179-187.

[8] K. Sastry, S.Naidu and J. Prasad, Common Fixed Points for Mutimaps in a Metric Space, Nonlinear Anal. Thy. Methods & Appl. 13(3) (1989) 221-229.