TRANSFINITE METHODS IN METRIC FIXED-POINT THEORY

This is a brief survey of the use of transfinite induction in metric fixed-point theory. Among the results discussed in some detail is the author’s 1989 result on directionally nonexpansive mappings (which is somewhat sharpened), a result of Kulesza and Lim giving conditions when countable compactness implies compactness, a recent inwardness result for contractions due to Lim, and a recent extension of Caristi’s theorem due to Saliga and the author. In each instance, transfinite methods seem necessary.

result of [23], and a recent extension of Caristi's theorem given in [20].The paper concludes with some historical comments.
This paper is largely expository although Theorem 2.4 is a new formulation, and in this case we give a detailed proof.

"Directional" contractions
We begin by looking back at the transfinite method of our 1989 Marseille-Luminy paper [15].That approach arose in an attempt to sharpen results about "weak directional contractions." For points x, y of a metric space (M,d), we denote (x, y) = z ∈ M : d(x,z) + d(z, y) = d(x, y) and x = z = y . (2.1) The metric space M is said to be metrically convex if (x, y) = ∅ whenever x, y ∈ M, x = y.
The mapping T : M → M is said to be pointwise Lipschitzian on M with constant k if T is continuous, and for each x ∈ M, T is called a pointwise contraction if k ∈ [0,1).The mapping T : M → M is said to be almost directionally Lipschitzian on M with constant k if for each x, y ∈ M, inf z∈(x,y) d T(x),T(z) d(x,z) ≤ k. (2. 3) The mapping T : M → M is said to be weakly directionally Lipschitzian on M with constant k if T is continuous, and for each x ∈ M, T is a weak directional contraction if k < 1.In [6], Clarke proved that every weak directional contraction, defined on a complete metric space, has a fixed point.He asked whether pointwise contractions on a complete and convex metric spaces are global contractions.The following answers this in the affirmative.
(An example of a weak directional contraction that is not a contraction is given in [6].)

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Theorem 2.1 (Kirk-Ray [19]).Let M and N be complete metric spaces with M metrically convex, and suppose T : M → N is almost directionally Lipschitzian with constant k.Suppose, in addition, that T is a closed mapping (i.e., has a closed graph).Then T is Lipschitzian with global Lipschitz constant k.
A Banach space version of the above result was given in an earlier paper [18].In both instances, transfinite induction arguments were used although it is possible to give proofs of these results based on Zorn's lemma.
The above considerations motivated the approach of [15].Now we assume that K is a bounded closed convex subset of a Banach space, and here we use S(x, y) to denote the set {(1 − α)x + αy : α ∈ (0,1)}.Definition 2.2.A mapping T : K → K is said to be weakly directionally nonexpansive on K if T is continuous, and for each for all z ∈ S(x,T(x)) sufficiently near x.
Theorem 2.3 (Kirk [15]).Let K be a bounded closed and convex subset of a Banach space and suppose T : Now we show how the method of [15] can be used to prove a minor variant of the above result.The proof serves to illustrate the delicate nature of the transfinite argument.Here we assume that the mapping is locally nonexpansive.By this we mean that each point has a neighborhood such that the restriction of the mapping to that neighborhood is nonexpansive.The domain is not assumed to be convex otherwise the mapping would be globally nonexpansive.The condition on T is an "inwardness" type assumption-see Section 4.
Theorem 2.4.Let D be a bounded closed subset of a Banach space X and let T : D → X be a locally nonexpansive mapping.Suppose also that (2.6) We will apply the following lemma, which is Lemma 2.1 of [15].This is a transfinite version of a 1976 result basically due to Ishikawa [13].

The Kulesza-Lim theorem
Now we describe the result of Kulesza and Lim [21], a result is motivated by the following question.Are there normal structure type conditions, weaker than hyperconvexity, yet strong enough to assure that the intersection of any descending chain of nonempty admissible sets in a metric space is nonempty (and admissible)?The Kulesza-Lim result shows that if the underlying metric space is complete, the answer is yes.This is an instance where the very nature of the problem calls for transfinite methods.We need some definitions and notation.Let (M,d) be a bounded metric space and We use Ꮽ(M) to denote the family of all admissible subsets of M. Thus, where x i ∈ M, r i > 0, and i ∈ I (some index set).For D ∈ Ꮽ(M), define Definition 3.2.The family Ꮽ(M) is said to be countably compact (resp., compact) if every descending sequence (resp., chain) of nonempty sets in A(M) has nonempty intersection.
The starting point is the following, which can be proved using elementary methods.
Passing from countable compactness to compactness requires an escalation of the method of argument.The key idea in the proof of Theorem 3.3 is the following fact.Any uncountable chain of real numbers which is either strictly decreasing or strictly increasing is eventually constant.The following routine observation is needed for the proof.(3.4) The set of all ordinals for which the conclusion of Lemma 3.5 holds is nonempty, and thus by well-ordering, there is a smallest such ordinal which we again call Γ.A chain {D α } that satisfies the conclusions of Lemma 3.5 for Γ is called a Γ-chain.
Given a Γ-chain (3.5) Finally, we say that a Γ-chain {J α } is a refinement of a Γ-chain {D α } if J α ⊆ D α for each α < Γ.This leads to the following lemma.Lemma 3.6.Under the hypothesis of Lemma 3.5, there exists a Γ-chain {D α } with the property: if We mention that a result similar in spirit to the above is found in [16].Transfinite methods are used there as well.The result of [16] is formulated for topologies defined by a collection of sets which may properly contain the admissible sets.A convexity structure Σ on a metric space M is a family of subsets of M that contains M, contains the closed balls of M, and is closed under intersections.A proximinal set in Σ is a set (in Σ) which lies on the boundary of some closed ball in D.

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Theorem 3.8 ( [16]).Let (M,d) be a bounded metric space, let Σ be a countably compact convexity structure which contains its closed r-neighbors, and let τ be the topology on M generated by Σ as a subbase for closed sets.Then if the proximinal sets in Σ are either separable or quasinormal, M is τ-compact.

Lim's inwardness result
Now we turn to Lim's recent inwardness result.This too is a theorem for which the standard mode of proof seems to fail.It may be too much to say that it requires a transfinite induction argument, but it does seem clear that a straightforward application of Caristi's theorem is not adequate.We will illustrate this in detail.
Theorem 4.1.Let D be a nonempty closed subset of a Banach space X and let T : D → 2 X \{∅} be a multivalued contraction with closed values which is weakly inward on D. Then T has a fixed point.
The above result was proved by Martínez-Yañez in 1991 [24] for single-valued T, by Yi and Zhao in 1994 [35] for compact-valued T, and by Xu in 2001 [34] for T satisfying the condition that each set Tx is proximinal relative to x.In each of these instances, it was possible to apply Caristi's theorem directly.Reich [29] also uses Caristi's theorem to give an extension of Lim's result in [22] to certain inward maps.(See [33] for another exposition on the ideas of this section.) For a closer look at this result, we define the terms.Let T be a nonempty closed subset of a Banach space X and T : D → 2 X \{∅}, a multivalued contraction mapping with closed values.Thus, there exists k ∈ (0,1) such that for all x, y ∈ D, where H denotes the (extended) Hausdorff metric on the nonempty closed subsets of X.Thus, where 3) The condition on T assures that if some value of T is bounded, then all are.It is well known that in this case, T takes values in the complete metric space of all bounded nonempty closed subsets of X endowed with the Hausdorff metric.On the other hand, if Tx is unbounded for some x ∈ D, then the space of all nonempty closed subsets of X, having finite Hausdorff distance from Tx, is also a complete metric space.
The inward set of D relative to x ∈ D is the set T is said to be weakly inward on D if for each x ∈ D, Note that Therefore, the inward set of D relative to x consists of D along with those points w ∈ X \D which have the property that some point z ∈ D with z = x lies on the segment joining x and w.
T is said to be weakly inward on D if for each x ∈ D, T lies in the closure of the inward set of D relative to x, that is, Note that in Deimling [9] and elsewhere, T : where Lim has observed that it is always the case that In fact, for convex D, the two concepts coincide (see [5,28]) but this is not true in general (see [9,Example 11.1]).First we approach the proof of Theorem 4.1 with a view of applying Caristi's theorem which we now state.Theorem 4.2 (Caristi).Let (M,d) be a complete metric space and suppose g : M → M is an arbitrary mapping which satisfies for all x ∈ M where ϕ : M → R is a lower semicontinuous mapping which is bounded below.Then g has a fixed point in M.
Now we assume that T satisfies the assumptions of Theorem 4.1.Let ∈ (k,1) where k is the Lipschitz constant of T and choose ε ∈ (0,1) so that (4.12)

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Let x ∈ D. In order to apply Caristi's theorem, we would like to show that if T is fixed point free, then it is possible to choose g(x) ∈ D so that Toward this end, let ε > 0 and choose y ∈ Tx so that Since T is fixed point free, dist(x,Tx) > 0, so by the weak inwardness of T, there exist g(x) ∈ D and λ ≥ 1 such that Since ε ∈ (0,1), this in particular implies g(x It can be shown that this implies dist g(x),Tg(x) ≤ x − g(x) + dist(x,Tx) and thus At this point it is tempting to say that since ε > 0 is arbitrary, it follows that where ϕ(x) = b −1 dist(x,Tx), whence Caristi's theorem applies.Unfortunately, however, the choice of y, and hence g(x), depends on ε .On the other hand, with the assumption that Tx is proximinal relative to x, it is not necessary even to introduce ε .We can merely take y in the above argument to be the point of Tx which is nearest x.This is, in fact, precisely the observation of Xu [34] who does assume that the sets Tx are all proximinal relative to x.Now we briefly describe how Lim proceeds to obtain the full result using transfinite induction.
Proof of Theorem 4.1 (Outline).Let ∈ (k, 1) where k is the Lipschitz constant of T and choose ε ∈ (0,1) so that Let x 0 ∈ D and choose y 0 ∈ Tx 0 .We assume T is fixed point free and proceed by transfinite induction.Let Ω denote the first uncountable ordinal, let γ ∈ Ω, and suppose y α ,x α have been defined for all α < β < γ so that We proceed to define y γ ,x γ so that (i), (ii), and (iii) remain valid for all α < β ≤ γ.Case 1. Suppose γ = µ + 1.Since y µ ∈ Tx µ and T is fixed point free, we have x µ − y µ > 0. By the weak inwardness of T, there exist x µ+1 ∈ D and λ µ+1 ≥ 1 such that Since y µ ∈ Tx µ and there exists y µ+1 ∈ Tx µ+1 such that y µ+1 − y µ ≤ x µ+1 − x µ .(Note: y µ+1 depends on both y µ and the contractive condition.It is not the point of Tx µ+1 which nearest x µ+1 .)Case 2. Suppose γ is a limit ordinal.This case is fairly straightforward by passing to limits.

An extension of Caristi's theorem
It is well known that if T : M → M is a contraction mapping with Lipschitz constant k ∈ (0,1), then where ϕ : It is also well known that T : M → M has a unique fixed point if for some integer p > 1, the mapping T p is a contraction mapping.This latter assumption leads to the inequality where ).On the other hand, the assumption that T p is a contraction mapping also leads to the inequality with ϕ as above.This raises the obvious question of whether it is possible to replace condition (5.1) with condition (5.3).The answer is "yes" provided that {ϕ(T n (x))} is decreasing, but it is not obvious that this fact follows from either Caristi's theorem or the Brézis-Browder order principle.
Theorem 5.1 ( [20]).Let (M,d) be a complete metric space and suppose T : M → M is an arbitrary mapping which satisfies for all x ∈ M, where p ∈ N is fixed and ϕ : M → R is lower semicontinuous and bounded below.Suppose also that ϕ(T(x)) ≤ ϕ(x) for each x ∈ M. Then T has a fixed point in M.
5.1.Remarks.One of the main objectives of [20] was to show, by using the Brézis-Browder order principle, that the lower semicontinuity condition in Theorem 4.2 can be weakened so that the resulting theorem contains the extension of Caristi's theorem given in [10].In particular, it suffices to assume that ϕ : M → R is lower semicontinuous from above.This means that given any sequence {x n } in M, the conditions lim n x n = x and ϕ(x n ) ↓ r ⇒ ϕ(x) ≤ r.In fact, an inspection of the proof of [20,Theorem 2.1] shows that an even weaker assumption suffices; namely, it is enough to assume that lim n x n = x and x n x n+1 ⇒ ϕ(x) ≤ r, where is defined by: x y ⇐⇒ d(x, y) ≤ ϕ(x) − ϕ(y).The same reasoning applies to Theorem 5.1.(Actually, this general idea seems to have arisen earlier in [11], where Gajek and Zagrodny use the notion of lower semicontinuous from above, which they call decreasingly lower semicontinuous to establish an extension of Ekeland's principle.) We might also wonder whether it is possible to allow p in Theorem 5.1 to depend on x.However, this weaker assumption does not even imply that the orbits of T are bounded.

Historical comments
In some sense, the very origins of metric fixed-point theory are rooted in transfinite induction.Indeed, Brodskiȋ and Mil'man, in their seminal paper [3], used transfinite induction to show that if a subset K of a Banach space has "normal structure and is compact in some topology τ for which the normed closed balls are τ-closed (e.g., the weak or weak * topology), then K contains a uniquely determined point (called the center of K) which is fixed under every isometry of K onto itself.Other early uses of transfinite induction include the work of Sadovskii on condensing operators ( [30,31]; although in the latter instance, an elementary proof, without transfinite induction, has been given by Reich [27]; (see also [25]).Altman [1] makes heavy use of transfinite methods in his study of contractors and contractor directions.Here again, however, other methods sometimes suffice (e.g., see [7,10]).In [17], the theory of ultranets is used to define a transfinite iteration process.As a consequence, it is noted that given any weakly compact set K and any contractive mapping T of K into K (i.e., T(x) − T(y) < x − y for x, y ∈ K, x = y), there is a unique point z ∈ K such that T Ω (x) = z for each x ∈ K.In [26], conditions, under which this point z is W. A. Kirk 323 actually a fixed point of T, are discussed.Finally, we also mention that Wong subsequently gave a simpler transfinite induction proof of Caristi's theorem in [32].

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:
in fact for α sufficiently large, d α ≡ d).Call d the diameter of {D α } and write d = diam D α .

Lemmas 3 .
5 and 3.6 are then used to prove the following result which, in conjunction with Lemma 3.4, gives Theorem 3.3.Theorem 3.7 ([21]).Suppose (M,d) is a bounded metric space for which Ꮽ(M) is countably compact and normal.Then Ꮽ(M) is compact.