Characterizations of Fixed Points of Multi-Valued

Maps by Metric Projections

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

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

Abstract The purpose of this paper is twofold. First a theorem from [2] is generalized by assuming compactness of a certain set of metric projection. Second, we examine a theorem of Dugundji and Granas [5] concerning fixed points of weakly contractive point-valued maps. We prove that the theorem of Dugundji and Granas can be extended to multi-valued maps provided that the corresponding metric projections satisfy the weak contractive condition.

Mathematics Subject Classification (1991): 47H10

1 Introduction

Let (X, d) be a metric space, f a continuous map of X into X and T a map of X into 2^{X} \{∅}. A subset K of X is called proximinal if, for each x ∈ X, there exists an element k ∈ K such that d(x, k) = d(x, K) where d(x, K) = inf{d(x, y) : y ∈ K}. The family of all bounded proximinal subsets of X is denoted by P (X). We denote the family of all nonempty closed subsets of X and that of all nonempty closed and bounded subsets of X by F (X) and CB(X) respectively. A map ϕ: X × X → [0, ∞) is called compactly positive if inf{ϕ(x, y) : a ≤ d(x, y) ≤ b} > 0 for each finite interval [a, b] ⊆ (0, ∞). A multi-valued map T: X → F (X) is called weakly contractive if there exists a compactly positive mapping ϕ such that

H(T (x), T (y)) ≤ d(x, y) − ϕ(x, y)

for each x, y ∈ X where H denotes the Hausdorff metric on CB(X) induced by d. T: X → F (X) is called an k f-contraction if there exists k ∈ [0, 1) such that

H(T x, T y) ≤ kd(fx, fy)

for each x, y ∈ X. Moreover, T: X → F (X) is called f-contractive if

H(T x, T y) < d(fx, fy) whenever fx ≠ fy

for each x, y ∈ X. A point x ∈ X is called a coincidence point of f and T if fx ∈ T x. Of course, a k f-contraction generalizes the classical Banach contraction, whereas the idea of f-contractive maps generalizes that of contractive multifunctions that was studied by Smithson in [9]. Recently, it was demonstrated in [3] that fixed point theorem for k f-contractions can be used to prove the existence of solutions of certain classes of nonlinear integral equations. The study of various conditions that guarantee the existence of fixed points of multi-valued k f-contractions and those of f-contractive maps were made in [6,2]. A coincidence point plays a critical role in proving the existence of fixed points of such maps, (see e.g. theorem 1 [2]). Anf-orbit of x ∈ X under T is a sequence {x_{n} : x_{n} ∈ X and fx_{n} ∈ T x_{n−1}, x_{0} = x}. An forbit of x under T is called regular provided that for each n, d(fx_{n}, fx_{n+1}) ≤ H(T x_{n−1}, T x_{n}) and d(fx_{n}, fx_{n+1}) ≤ d(fx_{n−1}, fx_{n}). An f-orbit of x under T is called strongly regular if T: X → P (X) and for each n, d(fx_{n+1}, fx_{n}) = d(fx_{n}, T x_{n}). In a recent paper [2], the following theorem was proved.

Theorem 1.1 [2] Let X be a connected metric space, T: X → F (X) an f-contractive map, where f: X → X is continuous. Suppose that T (X) ⊆ f(X) and that f ∘ T (x) ⊆ T ∘ f(x) for every x ∈ X. If for some x ∈ X an f-orbit of x is regular and contains a subsequence {x_{n_{k}}} such that fx_{n_{k}} → t_{0} and fx_{n_{k}+1} → t_{1}, then t_{0} = t_{1} and ft_{0} ∈ T t_{0}.

Our first task in this paper is to re-examine Theorem 1.1. We shall develop a general topological framework that can be used to prove a similar theorem concerning a coincidence point of f and T. The connectivity of X is dropped from assumptions. This will be done in Section 2.

Our second task is to re-examine a theorem of Dugundji and Granas. In [5], they proved that a single-valued weakly contractive map of a complete metric space into itself has a unique fixed point. It is still an open question whether the theorem of Dugundji and Granas can be extended to multi-valued maps. Partial results were provided in theorem 2 [7] and in theorem 2.3 [4]. In Section 3, we prove a theorem that guarantees a fixed point of multi-valued maps provided that their metric projections are weakly contractive.

2 Fixed Points of f-Contractions

Throughout this section, for A_{n}, A ∈ F (X), A_{n} ⇒ A denotes the convergence of A_{n} to A in the Hausdorff metric. For each x ∈ X, we denote the metric projection of x onto A by P_{A}(x), namely P_{A}(x) = {a ∈ A : d(x, a) = d(x, A)}. We also let A^{(ϵ)} = {x ∈ X : d(x, A) < ϵ} and U_{ϵ}(x) = {y : d(y, x) < ϵ}. We are now ready to prove a sequence of lemmas that will be used to establish the main result of this section.

Lemma 2.1 Let C_{n} and C ∈ F (X). Moreover assume that C is compact. If C_{n} ⊆ C^{(ϵ)} for all sufficiently large n and for each ϵ > 0, and if x_{n} ∈ C_{n} for each n ∈ N(=natural numbers), then there exists a subsequence {x_{n_{k}}} such that x_{n_{k}} → x for some x ∈ C as k → ∞.

Proof: By passing to a subsequence which we again denote by {C_{n}}, we can assume that C_{n} ⊆ C^{( \frac{1}{n})} for all n. If no subsequence of \{x_{n}\} converges to any point in C, then to each x ∈ C, there is ϵ = ϵ(x) such that U_{ϵ}(x)∩{x_{n}} = ∅. Now {U_{ϵ(x)}(x): x ∈ C} is an open cover of C. Since C is compact, there is a finite subcover {U_{ϵ(x_{i})}(x_{i}): i = 1, …, N}. For large enough n, we have C^{( \frac{1}{n})} ⊆ ∪_{i=1}^{N}U_{ϵ(x_{i})}(x_{i}). Moreover, since \{x_{n}\} ∩ ∪_{i=1}^{N}U_{ϵ(x_{i})}(x_{i}) = ∅, we have C^{( \frac{1}{n})} ∩ \{x_{n}\} = ∅. This contradiction proves the Lemma. □

Now let A ∈ P (X), x ∈ X, C = P_{A}(x) and C̃^{(ϵ)} = {z ∈ A: d(z, x) < d(x, C) + ϵ} for ϵ > 0. Also we denote the complement of A by A^{c}.

Lemma 2.2 C = ∩_{ϵ>0}C̃^{(ϵ)}.

Proof: It is clear that C ⊆ ∩_{ϵ>0}C̃^{(ϵ)}. If z ∈ C^{c} ∩ A, then for some ϵ > 0, d(z, x) ≥ d(x, C) + ϵ. Thus z ∈ (∩_{ϵ>0}C̃^{(ϵ)})^{c} = ∪_{ϵ>0}(C̃^{(ϵ)})^{c}. □

Lemma 2.3 inf_{z∈(_{C^{(ϵ)}})^{c}∩A} d(z, x) > d(x, C).

Proof: If this is not a case, then there is a sequence {z_{n}}, z_{n} ∈ (C̃^{(ϵ)})^{c} ∩ A for each n such that d(x, C) ≤ d(z_{n}, x) < d(x, C) + \frac{1}{n}. This yields that z_{n} ∈ C̃^{(ϵ)} whenever n > [\frac{1}{ϵ}] which is a contradiction. □

Lemma 2.4 Let x_{n}, x ∈ X and A_{n}, A ∈ P (X). Define C_{n} = P_{A_{n}}(x_{n}) and C = P_{A}(x). Suppose that x_{n} → x and A_{n} ⇒ A. Then for every ϵ > 0, we have C_{n} ⊆ C^{(ϵ)} for all sufficiently large n.

Proof: If the conclusion is not true, then for some ϵ > 0, there is a subsequence {C_{n_{k}}} and z_{n_{k}} ∈ C_{n_{k}} ∩(C^{(ϵ)})^{c} for each k. Now d(z_{n_{k}}, x_{n_{k}}) = d(C_{n_{k}}, x_{n_{k}}) = d(A_{n_{k}}, x_{n_{k}}) → d(A, x) = d(C, x). Since x_{n_{k}} → x,

lim sup_{k→∞} d(z_{n_{k}}, x) ≤ lim sup_{k→∞} d(z_{n_{k}}, x_{n_{k}}) + lim sup_{k→∞} d(x_{n_{k}}, x)
= d(x, C).

(1)

Also since z_{n_{k}} ∈ C_{n_{k}} ⊆ A_{n_{k}} and A_{n_{k}} ⇒ A, we have

d(z_{n_{k}}, (C^{(ϵ)})^{c} ∩ A) → 0.  (2)

Since C = ∩_{ϵ>0}C^{(ϵ)} (C is closed) and by Lemma 2.2, C = ∩_{ϵ>0}C̃^{(ϵ)}, using C̃^{(ϵ)} ⊆ A, we have that for some ϵ′, 0 < ϵ′ ≤ ϵ, C̃^{(ϵ′)} ⊆ C^{(ϵ)}. Hence z_{n_{k}} ∉ C̃^{(ϵ′)}. There are two cases for z_{n_{k}}. If z_{n_{k}} ∈ A, then this implies that

d(z_{n_{k}}, x) ≥ d(x, C) + ϵ′  (3)

by definition of C̃^{(ϵ′)}. If z_{n_{k}} ∉ A, then d(z_{n_{k}}, A) < \frac{ϵ′}{2} for sufficiently large k so that by (2), for k large enough, there is z′ ∈ A ∩ (C^{(ϵ)})^{c} with d(z_{n_{k}}, z′) < \frac{ϵ′}{2} . In this case we have, since z′ ∉ C̃^{(ϵ′)},

d(x, C) + ϵ′ ≤ d(z′, x)
≤ d(z′, z_{n_{k}}) + d(z_{n_{k}}, x)
< \frac{ϵ′}{2} + d(z_{n_{k}}, x),

i.e.,

d(x, C) + \frac{ϵ′}{2} < d(z_{n_{k}}, x).  (4)

In either case, (3) or (4), we get using (1)

d(x, C) + \frac{ϵ′}{2} < lim sup_{k→∞} d(z_{n_{k}}, x) ≤ d(x, C).

This contradiction shows that there is no subsequence and for every ϵ > 0, C_{n} ⊆ C^{(ϵ)} for all sufficiently large n. □

We are now ready to prove our first theorem. The connectivity of metric space (X, d) is dropped from Theorem 1.1 in exchange for an assumption that a certain metric projection is compact.

Theorem 2.5 Let (X, d) be a metric space and T: X → P (X) a continuous f-contractive map with T (X) ⊆ f(X). Let f be continuous and one-to-one with f^{−1} continuous on f(X). If for some x ∈ X, a strongly regular f-orbit of x under T has a cluster point x^{∗}, and if P_{T x^{∗}}(fx^{∗}) is a compact set, then x^{∗} is a coincidence point of f and T. Proof: Let {x_{n}}, fx_{n+1} ∈ T x_{n}, for each n ≥ 0, x_{0} = x, be a strongly regular f-orbit of x under T and put c_{n} = d(fx_{n+1}, fx_{n}). If c_{n} = 0 for some n, then 0 = d(fx_{n+1}, fx_{n}) = d(fx_{n}, T x_{n}) by the strong regularity of the f-orbit. Hence x_{n} is a coincidence point of f and T and the proof is complete. We thus assume that c_{n} > 0 for all n. First notice that {c_{n}} is monotonically strictly decreasing. To see this, c_{n+1} = d(fx_{n+2}, fx_{n+1}) = d(T x_{n+1}, fx_{n+1}) ≤ H(T x_{n+1}, T x_{n}) < d(fx_{n+1}, fx_{n}) = c_{n}. Thus we assume that lim_{n→∞} c_{n} = c for some c ≥ 0.

By hypothesis there is a subsequence {x_{n_{k}}} of the f-orbit that converges to x^{∗}. By the continuity of T and f we have

c = lim_{k→∞} c_{n_{k}} = lim_{k→∞} d(fx_{n_{k}+1}, fx_{n_{k}})
= lim_{k→∞} d(T x_{n_{k}}, fx_{n_{k}})
= d(T x^{∗}, fx^{∗}).

We finish the proof by showing that c = 0. Suppose c > 0. Let C ≡ P_{T x^{∗}}(fx^{∗}). Then C is compact by hypothesis. For every z ∈ C, we have c = d(T x^{∗}, fx^{∗}) = d(z, fx^{∗}). Recall that T (X) ⊆ f(X). Since f^{−1} is continuous, f^{−1}(C) is compact. For every y ∈ f^{−1}(C), we have c = d(T x^{∗}, fx^{∗}) = d(fy, fx^{∗}) > H(T y, T x^{∗}). Let φ(y) ≡ H(T y, T x^{∗}). Then φ: f^{−1}(C) → R_{+} is continuous and since f^{−1}(C) is compact, sup_{y∈f^{−1}(C)} φ(y) ≡ r < d(fy, fx^{∗}) = c.

Now put C_{k} ≡ P_{T x_{n_{k}}} (fx_{n_{k}}). Then by the strong regularity of the f-orbit, fx_{n_{k}+1} ∈ C_{k}. Using the continuity of T and f, we have fx_{n_{k}} → fx^{∗} and T x_{n_{k}} → T x^{∗}. By Lemma 4, for every ϵ > 0, C_{k} ⊆ C^{(ϵ)} for all sufficiently large k. Now fx_{n_{k}+1} ∈ C_{k} for every k. Lemma 1 yields a subsequence of {fx_{n_{k}+1}}, which we denote by the same notation, which converges to some point z ∈ C. Then since f is continuous and one-to-one, x_{n_{k}+1} = f^{−1}fx_{n_{k}+1} → f^{−1}z ≡ y ∈ f^{−1}(C). It now follows that

lim_{k→∞} H(T x_{n_{k}+1}, T x_{n_{k}}) = H(T y, T x^{∗}) ≤ r < c.  (5)

But by the strong regularity of the f-orbit,

H(T x_{n_{k}+1}, T x_{n_{k}}) ≥ d(T x_{n_{k}+1}, fx_{n_{k}+1}) = d(fx_{n_{k}+1}, fx_{n_{k}+2}) = c_{n_{k}+2}.

Since lim_{n→∞} c_{n} = c, lim_{k→∞} c_{n_{k}+2} = c and hence lim inf_{k→∞} H(T x_{n_{k}+1}, T x_{n_{k}}) ≥ c, which contradicts (5). Thus we conclude that c = 0 and this yields d(T x^{∗}, fx^{∗}) = 0, implying fx^{∗} ∈ T x^{∗}, i.e., x^{∗} is a coincidence point. □

Theorem 2.6 Suppose all the hypotheses in Theorem 2.5 are satisfied. Suppose further that f ∘ T (x) ⊆ T ∘ f(x) for every coincidence point x^{∗} of f and T. If {f^{n}x^{∗}} converges, then f and T possess a common fixed point .

Proof: Since f ∘T ⊆ T ∘f for every coincidence point of f and T, we have f^{n} ∘T x^{∗} ⊆ T ∘f^{n}x^{∗}. If f^{n}x^{∗} → x, we get from fx^{∗} ∈ T x^{∗} that f^{n+1}x^{∗} ∈ T ∘ f^{n}x^{∗} and letting n → ∞, we obtain x ∈ T x, i.e., x is a fixed point of T. □

REMARK 1: If (X, d) is a connected metric space, then an f contractive map T: X → F (X) is continuous [2] and this need not be assumed. REMARK 2: If T: X → C̆(X) or if T: X → K(X), where C̆(X) stands for the Chebyshev subsets [8] and K(X) the nonempty compact subsets of X, then P_{T x^{∗}}(fx^{∗}) is compact and this assumption can be dropped.

3 Fixed Points of Weak Contractions

Throughout this section, we assume that (X, d) is a complete metric space. In this section, we study metric projections that are weakly contractive. We let δ(A, B) ≡ sup{d(x, y): x ∈ A, y ∈ B} for A, B ⊆ X. The following lemma generalizes lemma 1.3 of Dugundji and Granas [5].

Lemma 3.1 Let T: X → P (X) satisfy

H(P_{T x}(x), P_{T y}(y)) ≤ d(x, y) − φ(x, y), for all x, y ∈ X

where φ: X × X → R_{+} is compactly positive. For r > 0, write Λ(\frac{r}{2}, r) = inf\{φ(x, y): \frac{r}{2} ≤ d(x, y) < r\}. Let x_{0} ∈ X. If d(x_{0}, P_{T x_{0}}(x_{0})) < min\{\frac{r}{2}, Λ(\frac{r}{2}, r)\}, then P_{T y}(y) ∩ U_{r}(x_{0}) ≠ ∅ for every y ∈ U_{r}(x_{0}).

Proof: Let y ∈ U_{r}(x_{0}). Then

d(x_{0}, P_{T y}(y)) ≤ d(x_{0}, P_{T x_{0}}(x_{0})) + H(P_{T x_{0}}(x_{0}), P_{T y}(y))
< min\{\frac{r}{2}, Λ(\frac{r}{2}, r)\} + d(x_{0}, y) − φ(x_{0}, y).

Two cases arise: (i) d(x_{0}, y) < \frac{r}{2}. Here d(x_{0}, P_{T y}(y)) < \frac{r}{2} + \frac{r}{2} = r. (ii) \frac{r}{2} ≤ d(x_{0}, y) ≤ r. Here d(x_{0}, P_{T y}(y)) < Λ(\frac{r}{2}, r)+r −Λ(\frac{r}{2}, r) = r. Thus in all cases d(x_{0}, P_{T y}(y)) < r. Hence there exists z ∈ P_{T y}(y) with d(x_{0}, z) < r. □

Theorem 3.2 Let (X, d) be a complete metric space and T: X → P (X) be continuous. Suppose that T satisfies

H(P_{T x}(x), P_{T y}(y)) ≤ d(x, y) − φ(x, y) for all x, y ∈ X

where φ is compactly positive. Then T has a unique fixed point.

Proof: Let x_{0} ∈ X and let {x_{n}} be a strongly regular orbit of x_{0} under T. Then

d(x_{n}, x_{n+1}) = d(x_{n}, P_{T x_{n}}(x_{n})) ≤ H(P_{T x_{n−1}}(x_{n−1}), P_{T x_{n}}(x_{n}))
≤ d(x_{n−1}, x_{n}) − φ(x_{n−1}, x_{n})
≤ d(x_{n−2}, x_{n−1}) − φ(x_{n−2}, x_{n−1}) − φ(x_{n−1}, x_{n})
…
≤ d(x_{0}, x_{1}) − ∑_{i=0}^{n−1} φ(x_{i}, x_{i+1}).

Thus ∑_{i=0}^{n−1} φ(x_{i}, x_{i+1}) ≤ d(x_{0}, x_{1}) and ∑_{i=0}^{∞} φ(x_{i}, x_{i+1}) < ∞. Write c(φ, r) = min\{\frac{r}{2}, Λ(\frac{r}{2}, r)\}, with Λ(\frac{r}{2}, r) as in Lemma 3.1. To every k ∈ N, there is n_{k} ∈ N such that

φ(x_{n}, x_{n+1}) < inf\{φ(ξ, η): c(φ, \frac{1}{k}) ≤ d(ξ, η) ≤ d(x_{0}, x_{1})\}

for all n ≥ n_{k}. Hence for n ≥ n_{k}, we have

d(x_{n}, x_{n+1}) = d(x_{n}, P_{T x_{n}}(x_{n})) < c(φ, \frac{1}{k}).

We may choose the sequence \{n_{k}\} to be increasing. By Lemma 3.1, P_{T x_{n_{k}}} (x_{n_{k}}) ∩ U_{\frac{1}{k}} (x_{n_{k}}) ≠ ∅, and so x_{n_{k}+1} can be chosen in this intersection. Then x_{n_{k}+1} ∈ U_{\frac{1}{k}} (x_{n_{k}}) and again by Lemma 3.1 x_{n_{k}+2} can be chosen in P_{T x_{n_{k}+1}}(x_{n_{k}+1})∩U_{\frac{1}{k}} (x_{n_{k}}) ≠ ∅. Inductively, we get that x_{n_{k}+j} ∈ U_{\frac{1}{k}} (x_{n_{k}}) and so x_{n_{k}+j+1} can be chosen in P_{T x_{n_{k}+j}}(x_{n_{k}+j}) ∩ U_{\frac{1}{k}} (x_{n_{k}}) for all j ≥ 0.

We now replace the original strongly regular orbit {x_{n}} by a new one {x̃_{n}}. For all n ≤ n_{1}, we put x̃_{n} = x_{n}. For k = 1 and n > n_{1}, we replace {x_{n}} by a new sequence {x_{n}^{(1)}} formed as described above. We thus have x_{n}^{(1)} ∈ U_{1}(x_{n_{1}}) for all n ≥ n_{1}. For k = 2 and n > n_{2}, we replace the terms of the sequence {x_{n}^{(1)}} by new ones {x_{n}^{(2)}} using the procedure described above. Thus x_{n}^{(2)} ∈ U_{\frac{1}{2}} (x_{n_{2}}) for all n ≥ n_{2}. In general, having arrived at \{x_{n}^{(k−1)}\} for all n > n_{k} replace x_{n}^{(k−1)} by x_{n}^{(k)} ∈ U_{\frac{1}{k}} (x_{n_{k}}). This generates a new sequence \{x̃_{n}\}, which is also a strongly regular orbit of x_{0}. This sequence has the property that x̃_{n} ∈ U_{\frac{1}{k}} (x̃_{n_{k}}) for all n ≥ n_{k} and for every k ∈ N. We claim that the sequence thus constructed is Cauchy. To prove this claim, let ϵ > 0 and choose k > \frac{1}{ϵ}. Then for all m, n ≥ n_{k} we have x̃_{m}, x̃_{n} ∈ U_{\frac{1}{k}} (x̃_{n_{k}}) and so d(x̃_{m}, x̃_{n}) ≤ \frac{2}{k} < 2ϵ. Now since X is complete, {x̃_{n}} converges, say x̃_{n} → z. Since T is continuous, we have T x̃_{n} → T z, and an appeal to lemma 2 of Assad and Kirk [1] now yields z ∈ T z, i.e., T has a fixed point. Uniqueness of fixed point is clear. □ REMARK 3: In Theorem 3.2, the continuity condition assumed on T was necessitated because of the appeal to lemma 2 of Assad and Kirk [1] that we made at the end of the proof. The following lemma shows that Theorem 3.2 can be improved by relaxing the continuity assumption on T. First we recall the following [8]; T: X → CB(X) is said to be upper Kuratowski semicontinuous (u.K.s.c) at x_{0}, respectively lower Kuratowski semi-continuous (l.K.s.c.) at x_{0}, if lim_{n→∞}x_{n} = x_{0}, y_{n} ∈ T (x_{n}), n ∈ N, lim_{n→∞} y_{n} = y_{0} imply y_{0} ∈ T (x_{0}), respectively if lim_{n→∞} x_{n} = x_{0}, y_{0} ∈ T (x_{0}) imply lim d(y_{0}, T (x_{n})) = 0. T: X → CB(X) is said to be upper semi-continuous (u.s.c.) at x_{0}, respectively lower semi-continuous (l.s.c.) at x_{0} if for every open set U ⊂ X such that T (x_{0}) ⊂ U, respectively such that T (x_{0}) ∩ U ≠ ∅, there exists in X an open neighborhood V of x_{0} such that T (x) ⊂ U for all x ∈ V, respectively such that T (x) ∩ U ≠ ∅ for all x ∈ V. Finally, T: X → CB(X) is said to be upper Hausdorff semi-continuous (u.H.s.c.) at x_{0}, respectively lower Hausdorff semi-continuous (l.H.s.c.) at x_{0}, if lim_{n→∞} x_{n} = x_{0} implies lim_{n→∞} sup_{g∈T (x_{n})} d(g, T (x_{0})) = 0, respectively lim_{n→∞} sup_{g∈T (x_{0})} d(g, T (x_{n})) = 0.

Lemma 3.3 Let (X, d) be a metric space and let T: X → CB(X) and let {x_{n}} be a sequence such that x_{n+1} ∈ T (x_{n}), for n ∈ N. Suppose that x_{n} → x_{0} and that T is u.s.c., u.K.s.c. or u.H.s.c., then x_{0} ∈ T (x_{0}), i.e., x_{0} is a fixed point of T.

Proof: Let T be u.s.c. and suppose that x_{0} ∉ T (x_{0}). Let 0 < ϵ < \frac{1}{3}d(x_{0}, T (x_{0})) be given. Define V ≡ {y ∈ X|d(y, T (x_{0})) < ϵ}. Since T is u.s.c., there is δ > 0 such that d(x, x_{0}) < δ implies T (x) ⊂ V. We choose δ < \frac{1}{3}d(x_{0}, T (x_{0})). Now choose n_{0} such that n ≥ n_{0} implies d(x_{n}, x_{0}) < δ. Then T (x_{n}) ⊂ V. But since x_{n+1} ∈ T (x_{n}), we have x_{n+1} ∈ V. Hence, for n ≥ n_{0} + 1, we have d(x_{n}, x_{0}) > \frac{1}{3}d(x_{0}, T (x_{0})). This contradiction proves this lemma.

If T is u.K.s.c., then by the definition of u.K.s.c. the result follows immediately.

If T is u.H.s.c., then x_{n} → x_{0} implies that sup_{x∈T (x_{n})} d(x, T (x_{0})) → 0. Since d(x_{0}, T (x_{0})) ≤ d(x_{0}, x_{n}) + d(x_{n}, T (x_{0})) ≤ d(x_{0}, x_{n}) + sup_{x∈T (x_{n−1})} d(x, T (x_{0})) and since the last two terms tend to 0 as n → ∞, we conclude that d(x_{0}, T (x_{0})) = 0 and x_{0} ∈ T (x_{0}). (We note that this last result in fact takes care of u.s.c. case, since u.s.c. implies u.H.s.c. [8;p.56]) □

Theorem 3.2 can now be improved vastly using the lemma 3.3.

Theorem 3.4 Let (X, d) be a complete metric space and T: X → P (X) be u.s.c., u.K.s.c. or u.H.s.c. Suppose further that T satisfies

H(P_{T x}(x), P_{T y}(y)) ≤ d(x, y) − φ(x, y) for all x, y ∈ X

where φ is compactly positive. Then T has a unique fixed point.

REMARK 4: Lemma 3.3 does not hold when T is l.s.c., l.K.s.c. or l.H.s.c. This can be seen by the following example. Let T: R → CB(R) be defined by T (x) = [0, 1] for x ≠ 0 and T (0) = {1}. Then T is l.s.c., l.K.s.c., and l.H.s.c. But it is not true that if x_{n} → x_{0} and x_{n+1} ∈ T (x_{n}), then x_{0} ∈ T (x_{0}). Let {x_{n}} be a sequence in (0, 1] which converges to 0. Then x_{n+1} ∈ T (x_{n}) = [0, 1] for every n, but x = 0 ∉ T (0) = {1}. Of course this example does not provide a counterexample to Theorem 3.2. It is interesting to investigate whether or not the continuity conditions on T can be further weakened. REMARK 5: The condition of Theorem 3.2 can be replaced by the following stronger condition.

H(P_{T x}(x), P_{T y}(y)) ≤ Φ(d(x, y)),

where Φ: R_{+} → R_{+} satisfies Φ(t) < t and lim sup_{s→t} Φ(s) < t, for all t > 0. Indeed, defining φ(t) = t − Φ(t), we have H(P_{T x}(x), P_{T y}(y)) ≤ d(x, y) − φ(d(x, y)). To see that φ is compactly positive, consider the interval [a, b], 0 < a ≤ b < ∞. If inf{φ(s)|s ∈ [a, b]} = 0, then there is a sequence {s_{n}} in [a, b] such that lim_{n→∞} φ(s_{n}) = 0. If s_{n} → s for some s ∈ [a, b], then lim_{n→∞}(s_{n} − Φ(s_{n})) = s − lim_{n→∞} Φ(s_{n}) = 0 so that lim_{n→∞} Φ(s_{n}) = s. This contradiction proves that φ is compactly positive. REMARK 6: It is interesting to ask whether or not the condition lim sup_{s→t} Φ(s) < t, for all t > 0 in REMARK 5 can be replaced by the condition lim sup_{s→t^{+}} Φ(s) < t, for all t > 0

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