HYPERBOLIC MONOTONICITY IN THE HILBERT BALL

Monotone operator theory has been intensively developed with many applications to Convex and Nonlinear Analysis, Partial Differential Equations, and Optimization. In this note we intend to apply the concept of (hyperbolic) monotonicity to Complex Analysis. As we will see, this application involves the generation theory of one-parameter continuous semigroups of holomorphic mappings. Let (H ,〈·,·〉) be a complex Hilbert space with inner product 〈·,·〉 and norm | · |, and let B := {x ∈H : |x| < 1} be its open unit ball. The hyperbolic metric ρ on B×B [5, page 98] is defined by


Introduction
Monotone operator theory has been intensively developed with many applications to Convex and Nonlinear Analysis, Partial Differential Equations, and Optimization. In this note we intend to apply the concept of (hyperbolic) monotonicity to Complex Analysis. As we will see, this application involves the generation theory of one-parameter continuous semigroups of holomorphic mappings.
Let (H, ·, · ) be a complex Hilbert space with inner product ·, · and norm | · |, and let B := {x ∈ H : |x| < 1} be its open unit ball. The hyperbolic metric ρ on B × B [5, page 98] is defined by ρ(x, y) := argtanh 1 − σ(x, y) 1/2 , (1.1) Recall that if C is a subset of H, then a (single-valued) mapping f : C → H is said to be monotone if Re x − y, f (x) − f (y) ≥ 0, x, y ∈ C. (1.4) Equivalently, f is monotone if Re x, f (x) + y, f (y) ≥ Re y, f (x) + x, f (y) , x, y ∈ C. (1.5) It is also not difficult to see that f is monotone if and only if |x − y| ≤ x + r f (x) − y + r f (y) , x, y ∈ C, (1.6) for all (small enough) r > 0. Let I denote the identity operator. A mapping f : C → H is said to satisfy the range condition if (I + r f )(C) ⊃ C, r > 0. (1.7) If f is monotone and satisfies the range condition, then the mapping J r : C → C, welldefined for positive r by J r := (I + r f ) −1 , is called a (nonlinear) resolvent of f . It is clearly nonexpansive, that is, 1-Lipschitz: J r x − J r y ≤ |x − y|, x, y ∈ C. (1.8) As a matter of fact, this resolvent is even firmly nonexpansive: J r x − J r y ≤ J r x − J r y + s x − J r x − y − J r y (1.9) for all x and y in C and for all positive s. This is a direct consequence of (1.6) because x − J r x = r f (J r x) and y − J r y = r f (J r y) for all x and y in C. We remark in passing that, conversely, each firmly nonexpansive mapping is a resolvent of a (possibly set-valued) monotone operator. To see this, let T : C → C be firmly nonexpansive. Then the operator is monotone because T satisfies (1.9). We now turn to the concept of hyperbolic monotonicity which was introduced in [19, page 244]; there it was called ρ-monotonicity. In the present paper we will use both terms interchangeably.
We say that a mapping f : B → H is ρ-monotone on B if for each pair of points (x, y) ∈ ρ(x, y) ≤ ρ x + r f (x), y + r f (y) (1.11) for all r > 0 such that the points x + r f (x) and y + r f (y) belong to B.
We say that f : B → H satisfies the range condition if If a ρ-monotone f satisfies the range condition (1.12), then for each r > 0, the resolvent J r := (I + r f ) −1 is a single-valued, ρ-nonexpansive self-mapping of B. As a matter of fact, this resolvent is firmly nonexpansive of the second kind in the sense of [5, page 129] (see Lemma 4.2 below). We remark in passing that this resolvent is different from the one introduced in [17] which is firmly nonexpansive of the first kind [5, page 124]. Our first aim in this note is to establish the following characterization of ρ-monotone mappings. Recall that a subset of B is said to lie strictly inside B if its distance from the boundary of B (the unit sphere of H) is positive.
This result shows that in some cases the hyperbolic monotonicity of f : B → H already implies the range condition (1.12). This is in analogy with the Euclidean Hilbert space case, where it is known that if f : H → H is continuous and monotone, then the range R(I + r f ) = H for all r > 0. To see this, we may first note that a continuous and monotone f : H → H is maximal monotone and then invoke Minty's classical theorem [11] to conclude that R(I + r f ) is indeed all of H for all positive r.
However, as pointed out on [14, page 393], Minty's theorem is equivalent to the Kirszbraun-Valentine extension theorem which is no longer valid, generally speaking, outside Hilbert space, or for the Hilbert ball of dimension larger than 1 [8,9]. On the other hand, it is known [10] that if E is any Banach space and f : E → E is continuous and accretive, then f is m-accretive, that is, R(I + r f ) = E for all r > 0.
Our proof of Theorem 1.1 uses finite dimensional projections. The separable case is due to Itai Shafrir (see [19,Theorem 2.3]). This proof is presented in Section 3, which also contains a discussion of continuous semigroups of holomorphic mappings and their (infinitesimal) generators (see Corollary 3.2). It is preceded by three preliminary results in Section 2. In Section 4, the last section of our note, we study the asymptotic behavior of compositions and convex combinations of resolvents of ρ-monotone mappings (see Theorems 4.14 and 4.15). Theorem 4.14, in particular, provides two methods for finding a common null point of finitely many (continuous) ρ-monotone mappings.

Preliminaries
We precede the proof of Theorem 1.1 with the following three preliminary results.
Given z ∈ B, let {u α : α ∈ Ꮽ} be a complete orthonormal system in H which contains z/|z| if z = 0. Let Γ be the set of all finite dimensional subspaces of H which contain z and are spanned by vectors from {u α : α ∈ Ꮽ}, ordered by containment. For each F ∈ Γ, let P F : H → F be the orthogonal projection of H onto F.
Note that (2.2) is the hyperbolic analog of the Euclidean (1.5). Finally, we recall a fixed point theorem which will be used in the proof of Theorem 1.1. Let C be a subset of a vector space E and let the point x belong to C. Recall that the inward set I C (x) of x with respect to C is defined by If E is a topological vector space, then a mapping f : C → E is said to be weakly inward if f (x) belongs to the closure of I C (x) for each x ∈ C. Theorem 2.3. Let C be a nonempty, compact and convex subset of a locally convex, Hausdorff topological vector space E. If a continuous f : C → E is weakly inward, then it has a fixed point. This theorem is due to Halpern and Bergman [6]. A simple proof can be found in [13].

The range condition
We begin this section with the proof of Theorem 1.1.
Proof of Theorem 1.1. One direction is clear: if J r is ρ-nonexpansive, and the points x, y, x + r f (x), y + r f (y) all belong to B, then Thus, it is enough to prove that if f is ρ-monotone, then for each z ∈ B and r > 0, there exists a solution x ∈ B to the equation x + r f (x) = z. Fix z ∈ B and consider the corresponding directed set Γ of finite dimensional subspaces of H.
, and f F is seen to be ρ-monotone by the characterization (2.2). Now we want to show that there is a point w F ∈ B F such that Indeed, consider the mapping h : Proposition 2] (alternatively, it satisfies the Leray-Schauder condition on {x ∈ F : |x| = s}) and therefore has a fixed point by Theorem 2.3. This fixed point Our next claim is that |v| = t. To see this, note first that Writing (2.2) with x := v and y := v E , we see that Re (3.8) Thus, Taking limits, we get Why is it important to know that in certain cases a ρ-monotone mapping already satisfies the range condition? To answer this question, let D be a domain (open, connected subset) in a complex Banach space X, and recall that a holomorphic mapping f : D → X is said to be a semi-complete vector field on D if the Cauchy problem It is known (see, e.g., [1,18]) that if a holomorphic f : D → X is semi-complete, then the family is a one-parameter (nonlinear) semigroup (semiflow) of holomorphic self-mappings of D, that is, where I denotes the restriction of the identity operator on X to D. In addition, uniformly on each ball which is strictly inside D.
A semigroup {F t } t≥0 is said to be generated if, for each z ∈ D, there exists the strong limit This mapping f is called the (infinitesimal) generator of the semigroup. It is, of course, a semi-complete vector field. Analogous definitions apply to (continuous) semigroups of ρ-nonexpansive mappings, where ρ is a pseudometric assigned to D by a Schwarz-Pick system [5, page 91].
When is a mapping f : D → X a generator? An answer to this question is provided by the following result [19, page 239]. Recall that if D is a convex domain, then all the pseudometrics assigned to D by Schwarz-Pick systems coincide. If D is also bounded, then this common pseudometric is, in fact, a metric, which we call the hyperbolic metric of D. If, in the setting of this theorem, f : D → X is a generator of a ρ-nonexpansive semigroup {F t } t≥0 , then the following exponential formula holds: (3.17) Combining Theorems 1.1 and 3.1, we obtain the following corollary. If follows from the Cauchy inequalities that this corollary applies, in particular, to holomorphic mappings which are bounded on each ρ-ball.
Note that all the mappings of the form f = I − T, where I is the identity operator and T : B → B is ρ-nonexpansive (in particular, holomorphic), are generators of semigroups of ρ-nonexpansive (resp., holomorphic) mappings. More applications of hyperbolic monotonicity and, in particular, of the characterizations provided by Proposition 2.2 and Corollary 3.2, can be found in [2].

Asymptotic behavior
In this section we study the asymptotic behavior of compositions and convex combinations of resolvents of ρ-monotone mappings.
Consider the function ψ : [0,δ] → [0,∞) defined by  1). Then the following are equivalent: Let D be a subset of the Hilbert ball B. Recall that a mapping T : D → B is said to be firmly nonexpansive of the second kind [5, page 129] if the function ϕ : is decreasing for all points x and y in D.
We denote the family of firmly nonexpansive mappings of the second kind by FN 2 . Indeed, since f is ρ-monotone, x − J r x = r f (J r x), and y − J r y = r f (J r y), we know that, by (1.11), ρ J r x,J r y ≤ ρ J r x + s f J r x ,J r y + s f J r x = ρ J r x + (s/r) x − J r x ,J r y + (s/r) y − J r y (4.4) for all 0 ≤ s ≤ r. In other words, for all 0 ≤ t ≤ 1, as required.
To define this concept for fixed point free mappings, we first recall two notations.  E(e,r), r > 0, are invariant under T. We say that such a mapping is strongly nonexpansive if for any sequence {x n : n = 1,2,...} ⊂ B such that {ϕ e (x n )} is bounded, the condition ϕ e (x n ) − ϕ e (Tx n ) → 0 implies that x n − Tx n → 0.
Proofs of the following two lemmas can be found in [15].  Our interest in strongly nonexpansive mappings stems from the following two facts.
Lemma 4.5. If a mapping T ∈ FN 2 has a fixed point, then it is strongly nonexpansive.
Proof. Suppose that the sequence {x n } is ρ-bounded, y ∈ F(T), and ρ(x n , y) − ρ(Tx n , y) → 0. In order to prove that ρ(x n ,Tx n ) → 0, we may assume without loss of generality that lim n→∞ ρ(x n , y) = lim n→∞ ρ(Tx n , y) = d > 0. Since T ∈ FN 2 , we also have ρ Tx n , y ≤ ρ x n + Tx n /2, y ≤ ρ x n , y . Hence lim n→∞ ρ((x n + Tx n )/2, y) = d, too. Now we can invoke Lemma 4.3 to conclude that x n − Tx n → 0. Since {x n } is ρ-bounded, it follows that ρ(x n ,Tx n ) → 0 as well. Next, we recall [16] the following weak convergence result. In view of Lemma 4.5, this result applies, in particular, to all those mappings T : B → B in FN 2 which have a fixed point.
It follows from [8,9] that in the setting of Proposition 4.7, strong convergence does not hold in general. However, our next result shows that when a strongly nonexpansive mapping is fixed point free, its iterates do converge strongly. Proof. Let e be the sink point of T and denote T n x by x n , n = 1,2,.... Since ϕ e (Tx) ≤ ϕ e (x) for all x ∈ B, the sequences {ϕ e (x n )} and {ϕ e (Tx n )} decrease to the same limit M. Since T is strongly nonexpansive, it follows that x n − Tx n → 0. Since T is fixed point free, this implies that {x n } cannot have a ρ-bounded subsequence. Thus lim n→∞ |x n | = 1, x n ,e → 1, and x n → e, as asserted. Now we consider compositions and convex combinations of strongly nonexpansive mappings.
The following result is proved in [16]. Here is an analog of this result for the fixed point free case. Proof. Let T 1 and T 2 be two fixed point free and strongly nonexpansive mappings with a common sink point e = e(T 1 ) = e(T 2 ). We first note that the composition T = T 2 T 1 is also fixed point free. Indeed, let x ∈ B and consider the iterates x n = T n x, n = 1,2,.... Since the decreasing sequence {ϕ e (x n )} converges, we see that 0 ≤ ϕ e x n − ϕ e T 1 x n ≤ ϕ e x n − ϕ e Tx n −→ 0, (4.10) and therefore x n − T 1 x n → 0. If {x n } were ρ-bounded, then its asymptotic center [5, page 116] would be a fixed point of T 1 . Hence {x n } is ρ-unbounded and T is fixed point free, as claimed. Thus e = e(T) is also the sink point of T. To show that T is strongly nonexpansive, let {x n } ⊂ B be a sequence such that {ϕ e (x n )} is bounded and ϕ e (x n ) − ϕ e (Tx n ) → 0. Then 0 ≤ ϕ e x n − ϕ e T 1 x n ≤ ϕ e x n − ϕ e T 2 T 1 x n , 0 ≤ ϕ e T 1 x n − ϕ e T 2 T 1 x n ≤ ϕ e x n − ϕ e T 2 T 1 x n . and so lim n→∞ (x n − T 2 T 1 x n ) = 0, too. The proof can now be completed by using induction on m.
Turning to convex combinations, we first note the following fact. It is a consequence of [4, Theorem 9.5 (ii)].
Lemma 4.11. Let the mappings T j : B → B, 1 ≤ j ≤ m, be strongly nonexpansive, and let T = m j=1 λ j T j , where 0 < λ j < 1 and m j=1 λ j = 1. If is not empty, then F = F(T) and T is also strongly nonexpansive.
We now formulate an analog of this fact for the fixed point free case. Proof. Once again, let T 1 and T 2 be two fixed point free and strongly nonexpansive mappings with a common sink point e = e(T 1 ) = e(T 2 ). We claim that the convex combination T = λ 1 T 1 + λ 2 T 2 , where 0 < λ 1 , λ 2 < 1 and λ 1 + λ 2 = 1, is also fixed point free. To see this, let x ∈ B and consider the iterates x n = T n x, n = 1,2,.... Note that ϕ e (x n ) − ϕ e (Tx n ) → 0 because the decreasing sequence {ϕ e (x n )} is convergent. Assume that {x n } has a ρ-bounded subsequence. Passing to a further subsequence and relabeling, if necessary, we may assume without loss of generality that ϕ e T 1 x n = max ϕ e T 1 x n ,ϕ e T 2 x n . (4.14) Since all the ellipsoids E(e,r) are convex, it follows that ϕ e (Tx n ) ≤ ϕ e (T 1 x n ) and therefore 0 ≤ ϕ x n − ϕ e T 1 x n ≤ ϕ e x n − ϕ e Tx n −→ 0.  We continue with a known fact [7]. We are now ready to formulate and prove the main result of this section.
Theorem 4.14. For each 1 ≤ j ≤ m, let f j : B → H be a continuous ρ-monotone mapping which is bounded on each ρ-ball. Let r j be positive and denote the resolvent (I + r j f j ) −1 of f j by R j . Furthermore, let 0 < λ j < 1 satisfy m j=1 λ j = 1. If the common null point set  The composition R m R m−1 ··· R 1 and the convex combination m j=1 λ j R j are also strongly nonexpansive by Lemmas 4.10 and 4.12, respectively. The existence of the strong lim n→∞ (R m R m−1 ··· R 1 ) n x and the strong lim n→∞ ( m j=1 λ j R j ) n x is now seen to follow from Proposition 4.8.

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:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009