Chapter 1

NONEXPANSIVE MAPPINGS: BOUNDARY/INWARDNESS CONDITIONS AND LOCAL THEORY

W. A. Kirk

Department of Mathematics The University of Iowa Iowa City, IA 52242-1419 USA

kirk@math.uiowa.edu

C. H. Morales

Department of Mathematics University of Alabama in Huntsville Huntsville, AL 35899 USA

morales@ultra.math.uah.edu

Keywords: Nonexpansive mappings, pseudocontractive mappings, local theory, inwardness conditions, fixed points.

1. Inwardness conditions 1.1. Introduction

Boundary and inwardness conditions have been particularly useful in extending fixed point theory for nonexpansive mappings to broader classes of mappings, particularly to mappings satisfying local contractive and pseudocontractive assumptions. At the same time these conditions often enable one to relax the assumption that the mapping takes values in its own domain.

The principal inwardness conditions in metric fixed point theory are analogues of one found in the topological theory. For a convex set with nonempty interior the usual inwardness and boundary conditions are stronger than the Leray-Schauder boundary condition, yet weaker than the assumption that the mapping send the boundary of the domain back into the domain. In a variety of circumstances they assure the viability of continuation arguments in much the same way the Leray-Scahuder condition does.

We use standard notation. Throughout X will denote a Banach space and B(x; r) will denote the closed ball centered at x ∈ X with radius r > 0. We use ∂D to denote the boundary of a set D ⊂ X.

Condition 1.1 A mapping f : D → X is said to satisfy the Nagumo boundary condition if

lim_{h→0^{+}} h^{−1} dist (x − hf(x), D) = 0.

For a subset D of X and x ∈ D define the inward set I_{D} (x) as follows:

I_{D} (x) = {x + c (u − x) : u ∈ K and c ≥ 1} .

Similarly we set

\overline{I}_{D} (x) = x + \{y ∈ X : lim inf_{h→0+} h^{−1} dist (x + hy, D) = 0\} .

If D is closed and convex then it is easy to see that

\overline{I}_{D} (x) = x + \overline{\{c (u − x) : u ∈ K and c ≥ 1\}}.

Condition 1.2 A mapping T : D → X is said to be inward if T (x) ∈ I_{D} (x) for each x ∈ D.

Condition 1.3 A mapping T : D → X is said to be weakly inward if T (x) ∈ \overline{I_{D} (x)} for each x ∈ D.

Condition 1.4 A mapping T : D → X is said to be Nagumo inward if T (x) ∈ \overline{I}_{D} (x) for each x ∈ D.

Conditions 1.3 and 1.1 are also related.

Remark 1.5 Let D be a convex subset of X and let f : D → X. Then f satisfies the Nagumo boundary condition for each x ∈ D if and only if the mapping I − f is weakly inward on D.

The following theorem is due to Caristi [3] and implicit in the work of Martin [21].

Theorem 1.6 Let K be a closed convex subset of a Banach space X and suppose T : K → X is a weakly inward contraction mapping. Then T has a (unique) fixed point in K.

Proof. Suppose T has Lipschitz constant k < 1 and let f = I − T. Choose ε > 0 so that k < (1 − ε) / (1 + ε) . By Remark 1.5 f satisfies the Nagumo boundary condition so if x ∈ K with x ≠ T (x) there exists h ∈ (0, 1) such that

dist ((1 − h) x + hT (x) , K) < hε ∥x − T (x)∥ .

Therefore it is possible to select y ∈ K so that

∥(1 − h) x + hT (x) − y∥ < hε ∥x − T (x)∥ .

This gives

∥y − T (y)∥ < ∥x − T (x)∥ + (k − \frac{1 − ε}{1 + ε}) ∥x − y∥ .

A simple calculation now leads to

∥x − y∥ ≤ (\frac{1 − ε}{1 + ε} − k)^{−1} [∥x − T (x)∥ − ∥y − T (y)∥] .

If x ≠ T (x) for each x ∈ K it is now possible to define φ : K → ℝ^{+} by taking

φ (x) = (\frac{1 − ε}{1 + ε} − k)^{−1} ∥x − T (x)∥ .

Upon setting y = g (x) this results in a mapping g : K → K satisfying

∥x − g (x)∥ ≤ φ (x) − φ (g (x)) , x ∈ K.

By Caristi’s Theorem [3] g has a fixed point, contradicting the assumption x ≠ T (x) for each x ∈ K. ■

In the following variant of the above one seeks the weakest condition which will assure the existence of a fixed point. This theorem is due to Martinez Yañez [22].

Theorem 1.7 Let D be a closed subset of a Banach space X and let T : D → X be a contraction mapping which satisfies for each x ∈ D :

T (x) ∈ x + \overline{\{c (y − x) : y ∈ D and c ≥ 1\}}.

Then T has a unique fixed point.

For an extension of the above result to the multivalued case see Deimling [5], Chapter 5, §11; also Yi and Zhao [33] for compact-valued T, Xu [32] for T satisfying the condition that each set T x is proximinal relative to x, and Lim [20] for multivalued contractions merely having closed values.

2. Boundary Conditions 2.1. Introduction

There are three common boundary conditions which frequently arise in metric fixed point theory. The first and most well-known is the so-called Leray-Schauder boundary condition (L-S). This condition is used extensively in conjunction with continuation methods in a wide variety of settings. The remaining two have a more geometric character.

Condition 2.1 A mapping T : D → X satisfies (L-S) if there exists z_{0} ∈ int (D) such that

T (z_{0}) − z_{0} ≠ λ (T (z) − z) for all z ∈ ∂D.

Condition 2.2 A mapping T : D → X is inwardly directed if T (∂D) ⊂ D.

Condition 2.3 A mapping T : D → X satisfies the boundary condition Δ if there exists z_{0} ∈ D such that

∥z_{0} − T (z_{0})∥ < ∥z − T (z)∥ for all z ∈ ∂D.

Condition 2.3 has two obvious variants, although condition Δ in conjunction with inf {∥z − T (z)∥ : z ∈ D} = 0 automatically gives the following stronger version.

Condition 2.4 A mapping T : D → X satisfies the boundary condition Δ_{s} if there exists z_{0} ∈ D such that

∥z_{0} − T (z_{0})∥ < inf_{z∈∂D} ∥z − T (z)∥ .

Condition 2.5 A mapping T : D → X satisfies the boundary condition boundary condition,Δ_{w} Δ_{w} if there exists z_{0} ∈ D such that

∥z_{0} − T (z_{0})∥ ≤ ∥z − T (z)∥ for all z ∈ ∂D.

Two observations are worth noting at the outset.

Remark 2.6 Suppose D is convex and suppose T : D → X is a nonexpansive mapping which satisfies either Condition 2.2 or Condition 2.3. Then T satisfies Condition 2.1.

Remark 2.7 Suppose D is convex and suppose T : D → X is a nonexpansive mapping which satisfies Condition 2.2. Then there exists μ ∈ (0, 1) such that the mapping f_{μ} defined by

f_{μ} := (1 − μ) I + μT

maps D into D (and has the same fixed points as T).

Remark 2.6 is a simple calculation. Remark 2.7 is an observation of F. Browder. See [19] or [29] for more details.

Theorem 2.8 Suppose X is a uniformly convex Banach space and suppose K is a bounded closed convex subset of X. Suppose int (K) ≠ ∅, and suppose T : K → X is nonexpansive and satisfies L-S on K. Then T has a fixed point.

An analogue of the above theorem holds for nonconvex K under the assumption that T satisfies Δ on K. However this is a special case of a corresponding result which holds for locally nonexpansive mappings (see Theorem 3.2 below).

Theorem 2.9 Suppose X is a uniformly convex Banach space and suppose D is a bounded open subset of X. Suppose T : K → X is nonexpansive and satisfies Δ on K. Then T has a fixed point.

2.2. Preliminary results

The remark below is a straightforward and quite useful observation.

Remark 2.10 Let X and Y be topological spaces, G an open subset of X and let f : X → Y be a mapping for which f(G) is open while f(\overline{G}) is closed. Then ∂f(G) ⊂ f(∂G). Moreover, if f is one-to-one, then ∂f(G) = f(∂G).

Let G be an open subset of a Banach space and let T : G → X be a nonexpansive mapping. We begin with the following basic facts (see e.g., Gatica and Kirk [7]).

Proposition 2.11 (I − tT) (G) is open for t ∈ [0, 1).

Proof. Fix t ∈ (0, 1) and set f_{t} = (I − tT) . Now let x_{0} ∈ G, choose ρ > 0 so that B (x_{0}; ρ) ⊂ G, and let r = (1 − t) ρ. Fix z ∈ B (f_{t} (x_{0}) ; r) . We only need to show that for such z there exists w ∈ B (x_{0}; ρ) such that f_{t} (w) = w − tT (w) = z. To this end define T_{z} : G → x by setting

T_{z} (x) = tT (x) + z, x ∈ G.

Since

∥T_{z} (x) − x_{0}∥ = ∥tT (x) + z − x_{0}∥
= ∥tT (x) + tT (x_{0}) − tT (x_{0}) + z − x_{0}∥
≤ t ∥T (x) + T (x_{0})∥ + ∥z − (x_{0} − tT (x_{0}))∥
≤ t ∥x + x_{0}∥ + ∥z − f_{t} (x_{0})∥
≤ tρ + r = ρ,

it follows that T_{z} : B (x_{0}; ρ) → B (x_{0}; ρ) , and since T_{z} is a contraction mapping it has a unique fixed point, say w ∈ B (x_{0}; ρ) . Thus tT (w) + z = w from which

f_{t} (w) = w − tT (w) = z.
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Proposition 2.12 If G is bounded and if 0 ∈ G then 0 ∈ (I − tT) (G) for sufficiently small t ∈ (0, 1).

Proof. Choose ρ > 0 so that B (0; ρ) ⊂ G. Since T is nonexpansive, G bounded ⇒ T (G) bounded ⇒ tT (G) ⊂ B (0; ρ) for t > 0 sufficiently small. For such t, we have tT : B (0; ρ) → B (0; ρ) . Since tT is a contraction mapping, Banach’s Theorem implies that there exists a unique x_{t} ∈ B (0; ρ) such that tT (x_{t}) = x_{t}. ■

It is obvious that if G is bounded and if 0 ∈ (I − tT) (G) for all t ∈ [0, 1), then

inf {∥x − T (x)∥ : x ∈ G} = 0.

Therefore it is important to study the set

S = {t ∈ (0, 1) : 0 ∈ (I − tT) (G)}

and seek conditions which imply S = (0, 1) . The fact that 0 ∈ int (G) assures that S is nonempty. The rest follows from he boundary condition

0 ∉ (I − tT) (∂G)

for any t ∈ (0, 1) . The key observations needed to see this are the fact that for any t ∈ (0, 1):

(1) (I − tT) (G) is an open set. (Proposition 2.11).

(2) ∂ [(I − tT) (G)] ⊂ (I − tT) (∂G) . (Remark 2.10).

The conclusion then follows upon showing that

t̄= sup {t : t ∈ S} = 1.

The details are routine. Assume t̄ < 1 and choose t_{n} ∈ S with t_{n} ↑ t̄. For each n there exists x_{n} ∈ G such that

t_{n}T (x_{n}) = x_{n}.

This gives

∥x_{n} − x_{m}∥ = ∥t_{n}T (x_{n}) − t_{m}T (x_{m})∥
≤ t_{n} ∥T (x_{n}) − T (x_{m})∥ + |t_{n} − t_{m}| ∥T (x_{m})∥
≤ t_{n} ∥x_{n} − x_{m}∥ + |t_{n} − t_{m}| ∥T (x_{m})∥

from which

(1 − t_{n}) ∥x_{n} − x_{m}∥ ≤ |t_{n} − t_{m}| ∥T (x_{m})∥ .

Letting m, n → ∞ we see that (since G is bounded) {x_{n}} is a Cauchy sequence. Thus x_{n} → x ∈ \overline{G} from which

t_{n}T (x_{n}) → t̄T (x) ,

whence t̄ ∈ S, i.e., 0 ∈ (I − t̄T) (\overline{G}) . Since by assumption 0 ∉ (I − t̄T) (∂G) it follows from (2) that 0 ∈ (I − t̄T) (G) . Since (I − t̄T) (G) is open this in turn implies that 0 ∈ (I − t′T) (G) for t′ > t̄ sufficiently near t̄, contradicting maximality of t̄.

The same argument proves that if T is actually a contraction mapping then 1 ∈ S. Thus we have:

Proposition 2.13 Suppose G is a bounded open subset of a Banach space X with 0 ∈ G, and suppose T : \overline{G} → X is a nonexpansive mapping which satisfies

T (x) ≠ λx for all x ∈ ∂G and λ > 1.

Then inf {∥x − T (x)∥ : x ∈ G} = 0. Moreover if T is a contraction mapping, then T has a fixed point in \overline{G}.

In fact, boundedness of G is not needed for the final conclusion of Proposition 2.13. Also a simple translation allows us to restate the proposition as follows.

Proposition 2.14 Suppose G is a bounded open subset of a Banach space X, and suppose T : \overline{G} → X is a nonexpansive mapping which satisfies for some z_{0} ∈ G :

T (x) − z_{0} ≠ λ (x − z_{0}) for all x ∈ ∂G and λ > 1.  (L-S)

Then inf {∥x − T (x)∥ : x ∈ G} = 0. Moreover if T is a contraction mapping, then T has a fixed point in \overline{G}.

2.3. Nonexpansive and pseudocontractive mappings

An obvious question is whether Proposition 2.14 holds for a wider class of mappings. The key lies in identifying a class of mappings for which Propositions 2.11 and 2.12 hold. We now describe such a class.

Let ⟨·, ·⟩ denote the usual pairing between elements of X and its dual X^{∗} and let J : X → 2^{X^{∗}} denote the normalized duality mapping

J (x) = {j ∈ X^{∗} : ∥j∥^{2} = ∥x∥^{2} = ⟨x, j⟩} .

Definition 2.15 A mapping f : D ⊆ X → X is said to be accretive if for each u, v ∈ D

⟨f (u) − f (v) , j⟩ ≥ 0  (2.1)

for each j ∈ J (u − v) .

Definition 2.16 A mapping f : D ⊆ X → X is said to be strongly accretive if there exists a constant c > 0 such that for each u, v ∈ D

⟨f (u) − f (v) , j⟩ ≥ c ∥u − v∥^{2}  (2.2)

for each j ∈ J (u − v) .

The relationship between accretive mappings and nonexpansive mappings is synergistic. It is known that condition (2.1) has the following equivalent form.

∥u − v∥ ≤ ∥u − v + λ (f (u) − f (v))∥ for all u, v ∈ D, λ > 0.  (2.3)

Therefore if f_{λ} = (I − λf) , then the mapping f is accretive if and only if the resolvent J_{λ} = (f_{λ})^{−1} is nonexpansive on the range of f_{λ}. Replacing f with I − T in (2.3) we have

∥u − v∥ ≤ ∥(1 + λ) u − v − λ (T (u) − T (v))∥ for all u, v ∈ D, λ > 0.  (2.4)

Mappings satisfying (2.4) are said to be pseudocontractive.

Thus we have the following: A mapping f : D ⊂ X → X is accretive if and only if the mapping T = I − f is pseudocontractive.

The study of accretive operators seems partly motivated by its close connection to the theory of partial differential equations, in addition to the fact all nonexpansive mappings are pseudocontractive, while all contraction mappings are strongly pseudocontractive.

At this point two things are clear. First, if one wishes to study the existence of zeros of accretive mappings, then the focus should be on the fixed point theory for pseudocontractive mappings rather than nonexpansive mappings. Second, the nonexpansive theory is intimately intertwined with the pseudocontractive theory via nonexpansiveness of the resolvent operators. Our next two results amply illustrate this fact.

We begin with the basic existence result for pseudo-contractive mappings which was discovered independently by Caristi [3] and Vidossich [29] using entirely different methods.

Theorem 2.17 Suppose K is a closed convex subset of a Banach space X and suppose K has the fixed point property for nonexpansive mappings. Suppose T : K → X is a continuous pseudocontractive mapping which is and weakly inward on K. Then T has a fixed point.

Proof. We consider only the lipschitzian case here. This is Caristi’s original approach. The general case may be deduced from the approach in [8], Chapter 13.

If T is lipschitzian it is possible to choose r > 0 so small that kT is a contraction mapping for k = r/ (1 + r) . Set H = [I + r (I − T)]^{−1} . Since T is pseudocontractive H is a nonexpansive mapping on its domain D = [I + r (I − T)] (K) . Also, H (D) = K. We will show that D ⊃ K and thus conclude that H |_{K}: K → K.

Fix z ∈ K and define T̃ : K → X by setting T̃ (x) = kT (x)+(1 − k) z. The choice of k assures that T̃ is a contraction mapping. Also, for each x ∈ K, I_{K} (x) is a convex set which contains K, and since T̃ (x) is a convex combination of elements of the closure of this set it follows that T̃ is also weakly inward on K. By Theorem 1.6 T̃ has a fixed point, say x^{∗} ∈ K. Thus

x^{∗} = kT (x^{∗}) + (1 − k) z = r (r + 1)^{−1} T (x^{∗}) + (r + 1)^{−1} z.

This implies z = (r + 1) x^{∗} − rT (x^{∗}) = x^{∗} + r (I − T) (x^{∗}) . Therefore z is in the range of I + r (I − T) and thus the domain of H contains K. We now conclude that there exists x ∈ K such that H (x) = x = x + r (x − T (x)) from which T (x) = x. ■

We now replace the inwardness assumption with a boundary condition. This will pave the way for the development of the local theory. Indeed, a complete local analog of Theorem 2.19 below is given later in the paper (see Theorem 4.18). A key tool in the development here and elsewhere is the following Domain Invariance Theorem due to Deimling [4].

Theorem 2.18 If D is an open subset of a Banach space X, and if T : D → X is continuous and strongly accretive, then T (D) is open.

Theorem 2.19 Suppose D is a bounded open subset of a Banach space X and suppose T : \overline{D} → X is a continuous pseudocontractive mapping. Suppose also that there exists z ∈ D such that

∥z − T (z)∥ < ∥x − T (x)∥ for all x ∈ ∂D.

Then inf\{∥x − T (x)∥ : x ∈ \overline{D}\} = 0. Moreover, if it is the case that \overline{D} has the fixed point property with respect to nonexpansive mappings, then T has a fixed point in \overline{D}.

Proof. The proof of Theorem 2.19 can be broken into three steps.

Step (I). inf \{∥x − T (x)∥ : x ∈ \overline{D}\} = 0.

Proof. Since T is pseudocontractive, for fixed r ∈ (0, 1) and u, v ∈ \overline{D} :

(1 − r) ∥u − v∥ ≤ ∥(I − rT) (u) − (I − rT) (v)∥ .  (2.5)

Thus the mapping U = (1 − r) (I − rT)^{−1} is defined and nonexpansive on

B = (I − rT) (\overline{D}).

Using the duality between pseudocontractivity and accretivity, there exists j ∈ J (u − v) such that

⟨ (I − rT) (u) − (I − rT) (v) , j⟩ ≥ (1 − r) ∥u − v∥^{2} .

It follows that the mapping I − rT is strongly accretive and one may apply Theorem 2.18 to conclude that (I − rT) (D) is open. On the other hand (2.5) implies that (I − rT) (\overline{D}) is closed. Thus ∂B ⊂ (I − rT) (∂D) .

Next let T_{r} (x) = x − rT (x) , x ∈ \overline{D}. Then T_{r} (x) ∈ ∂B ⇒ x ∈ ∂D, and

∥T_{r} (z) − U (T_{r} (z))∥ < ∥T_{r} (x) − U (T_{r} (x))∥ .

Straightforward calculations in conjunction with the fact that U is nonexpansive lead to the conclusion:

U (T_{r} (x)) − T_{r} (z) ≠ λ (T_{r} (x) − T_{r} (z)) for all T_{r} (x) ∈ ∂B and λ > 1.

It follows from this that for each t ∈ (0, 1) the contraction mapping U_{t} : B → X defined by

U_{t}(T_{r} (x)) = (1 − t) T_{r} (z) + tU (T_{r} (x))

satisfies the Leray-Schauder boundary condition

U_{t} (T_{r} (x)) − T_{r} (z) ≠ λ (T_{r} (x) − T_{r} (z)) for all T_{r} (x) ∈ ∂B and λ > 1.

By Proposition 2.13, U_{t} has a fixed point T_{r} (x_{t}) for x_{t} ∈ \overline{D} (i.e., for T_{r} (x_{t}) ∈ B). Thus

∥T_{r} (x_{t}) − U (T_{r} (x_{t}))∥ = ∥(1 − t) T_{r} (z) + tU (T_{r} (x_{t})) − U(T_{r} (x_{t}))∥
≤ (1 − t) [ ∥T_{r} (z)∥ + ∥U (T_{r} (x_{t}))∥ ].

But U maps B into (1−r)(\overline{D}) with D bounded, so it must be the case that \{U(T_{r}(x_{t}))\} is bounded. Letting t → 1,

inf \{ ∥T_{r} (x) − U (T_{r} (x))∥ : T_{r} (x) ∈ B\} = inf \{r ∥x − T (x)∥ : x ∈ \overline{D}\} = 0.

This proves Step (I).

In view of Step (I) there exists z ∈ D such that

∥z − T (z)∥ < inf { ∥x − T (x)∥ : x ∈ ∂D}.  (2.6)

Since D is bounded (2.6) implies that α ∈ (0, 1) may be chosen so near 1 that for all y ∈ \overline{D},

α ∥z − T (z)∥ + (1 − α) ∥z − y∥ < inf {α ∥x − T (x)∥ − (1 − α) ∥x − y∥ : x ∈ ∂D}.

Now define U_{α,y} : \overline{D} → X by setting

U_{α,y} (x) = (1 − α) y + αT (x) , x ∈ \overline{D}.

Step (II). U_{α,y} has a fixed point F_{α} (y) for each y ∈ \overline{D}.

Proof. By the choice of α

∥z − U_{α,y} (z)∥ < inf { ∥x − U_{α,y} (x)∥ : x ∈ ∂D}.  (2.7)

We now fix y ∈ \overline{D} and r ∈ (0, 1) , and let S = I − rU_{r,α}. Then for u, v ∈ \overline{D} and suitable j ∈ J (u − v) we have by (2.2)

⟨S (u) − S (v) , j⟩ = ∥u − v∥^{2} − rα⟨T (u) − T (v) , j⟩
≥ (1 − αr) ∥u − v∥^{2} .

This proves that S is strongly accretive, so again by domain invariance S (D) is open. Hence S (z) ∈ D, and since S (\overline{D}) is closed we have ∂S (D) ⊂ S (∂D) .

Now let H = (1 − r) S^{−1}. It is now possible to show that:

(i) H is a contraction mapping.

(ii) H satisfies the Leray-Schauder boundary condition:

H \binom{}{x̄} − z̄ ≠ λ (x̄ − z̄) for all x̄ ∈ ∂G and λ > 1

where G = S (D) and z̄ = S (z) .

Having established (i) and (ii) it follows that the mapping H has a fixed point w̄ ∈ G by Proposition 2.13. From the definition of H,

(1 − r) (I − rU_{α,y})^{−1} \binom{}{w̄} = w̄;

hence

U_{α,y} (\frac{w̄}{1 − r} ) = \frac{w̄}{1 − r} .

This proves Step (II).

It is now possible to define a mapping F_{α} : \overline{D} → \overline{D} by taking F_{α} (y) to be the fixed point of U_{α,y}. Thus

F_{α} (y) = (1 − α) y + αT (F_{α} (y)) , y ∈ \overline{D}.

Step (III). F_{α} is nonexpansive.

Proof. Let u, v ∈ \overline{D}. Then

F_{α} (u) − F_{α} (v) = α (T (F_{α} (u)) − T (F_{α} (v))) + (1 − α) (u − v)

and so for some j ∈ J (F_{α} (u) − F_{α} (v)) ,

⟨F_{α} (u) − F_{α} (v) , j⟩ = α⟨T (F_{α} (u)) − T (F_{α} (v)) , j⟩ + (1 − α) ⟨(u − v) , j⟩.

Therefore

∥F_{α} (u) − F_{α} (v)∥^{2} ≤ α ∥F_{α} (u) − F_{α} (v)∥ + (1 − α) ∥u − v∥^{2}

from which

∥F_{α} (u) − F_{α} (v)∥ ≤ ∥u − v∥ .

It is now possible to invoke the assumption that \overline{D} has the fixed point property for nonexpansive mappings and complete the proof of the theorem:

F_{α} (y) = y ⇒ y = (1 − α) y + αT (y) ⇒ T (y) = y.
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3. Locally nonexpansive mappings

Boundary conditions are particularly suited to the study of the local theory. A mapping T : G → X, where G is an open subset of X, is said to be locally k-lipschitzian if given any x ∈ G there exists r_{x} > 0 such that T restricted to B(x; r_{x}) ∩ G satisfies

∥T (u) − T (v)∥ ≤ k ∥u − v∥ for each u, v ∈ B(x; r_{x}) ∩ G.

If k < 1, then T is called a local contraction, and if k = 1 T is said to be locally nonexpansive. T is said to be a local radial contraction if for each x ∈ G there exists α (x) ∈ (0, 1) and r_{x} > 0 such that

∥T (u) − T (x)∥ ≤ α (x) ∥u − x∥ for each u ∈ B(x; r_{x}) ∩ G.

The study of locally lipschitzian mappings is motivated by the well-known fact that if T : G → X has a continuous Gâteaux derivative T′_{x} at each x ∈ G and if ∥T′_{x}∥ ≤ k, then the restriction of T to any convex neighborhood of a point in G is lipschitzian with Lipschitz constant k. (See, for example, the discussion in [1].)

Remark 3.1 Suppose D is an open subset of a Banach space and suppose T : \overline{D} → X is a continuous mapping which is a local contraction on D. Suppose also that T satisfies the boundary condition Δ_{w} on D. Then T has a fixed point.

The above is a trivial consequence of the fact that if

∥z_{0} − T (z_{0})∥ ≤ ∥x − T (x)∥ for all x ∈ ∂D

then {T^{n} (z_{0})} is a Cauchy sequence which necessarily converges to a fixed point of T.

Fixed point theory for locally nonexpansive mappings and, more generally, locally pseudocontractive mappings has been studied rather extensively. See, [15] for a survey of early results and, e.g., [25], [27], [30], for more recent developments. The standard underlying assumption in the nonexpansive case is that G is an open subset of a Banach space X with T : \overline{G} → X continuous on \overline{G} and locally nonexpansive on G. Various conditions on X (e.g., uniform convexity) in conjunction with certain boundary conditions assure that such a mapping always has at least one fixed point.

3.1. Existence theorems

The following result is proved in [13].

Theorem 3.2 Let G be a bounded open subset of a uniformly convex Banach space X. Suppose T : \overline{G} → X is continuous on \overline{G} and locally nonexpansive on G. Suppose also that there exists z ∈ G such that

∥z − T (z)∥ < ∥x − T (x)∥ for all x ∈ ∂G.  (3.1)

Then T has a fixed point in G.

Another fact is found in Morales [25].

Theorem 3.3 Let X be a Banach space which is uniformly smooth, let G be a bounded open subset of X, and let T : \overline{G} → X be continuous and locally nonexpansive on G. Let y ∈ G satisfy

∥y − T (y)∥ < ∥x − T (x)∥ for all x ∈ ∂G.

Then for each y ∈ G there is a unique path t ↦ y_{t} ∈ G, 0 ≤ t < 1, satisfying

y_{t} = (1 − t)y + tT (y_{t})

and for which lim_{t→1^{−}} y_{t} ∈ F (T).

As noted in [13] and [25] respectively , Theorems 2.19 and 3.2 actually hold for the more general class of locally pseudocontractive mappings. Also, Theorem 3.3 extends a result of [2] where it is assumed that X is both uniformly convex and uniformly smooth.

4. Locally Pseudocontractive Mappings 4.1. Introduction

Using the Leray-Schauder boundary condition as well as the Conditions Δ and Δ_{w} it is possible to extend some of the previous results to mappings characterized by their local behavior. The continuity of the path induced by the eigenvectors of pseudocontractive mappings is a useful tool in this approach. However, these local extensions require a few global results, which we discuss first.

As noted earlier, the accretive operator theory is intimately related to the pseudocontractive one. This fact is one of the motivations for the study of more general classes of operators as can be seen below.

Let α : [0, ∞) → [0, ∞) be a function for which α(0) = 0 and the lim inf_{r→r_{0}} α(r) > 0 for every r_{0} > 0.

Definition 4.1 A mapping A : D ⊂ X → X is said to be α-strongly accretive if for u, v ∈ D there exists j ∈ J(u − v) such that

(A(u) − A(v), j⟩ ≥ α(∥ u − v ∥) ∥ u − v ∥ .  (4.1)

Analogously, a mapping T is said to be α-strongly pseudocontractive if I − T is αstrongly accretive.

A mapping A satisfying (4.1) locally, i.e., if each x ∈ D has a neighborhood U such that the restriction of A to U is globally α-strongly accretive, is said to be locally αstrongly accretive. The notion of locally α-strongly pseudocontractive can be defined similarly.

If α(r) = cr for some c > 0, then A is strongly accretive (with constant c) since in this case (4.1) becomes

⟨A(u) − A(v), j⟩ ≥ c ∥ u − v ∥^{2},

while for T = I − A and c ∈ (0, 1), T is strongly pseudocontractive (with constant 1 − c) and (4.1) takes the form of

⟨T (u) − T (v), j⟩ ≤ (1 − c) ∥ u − v ∥^{2} .  (4.2)

These definitions have obvious local formulations. Of course if c = 0, then the operator A is accretive, while T is pseudocontractive. In the latter case a result of Kato [12] shows that (4.2) and (2.4) are actually equivalent.

The observation below can be easily derived from the definition.

Remark 4.2 If T is pseudocontractive, then tT is strongly pseudocontractive (with constant t) for every t ∈ (0, 1). On the other hand, if A is accretive, then tI + A is strongly accretive for t > 0.

4.2. Preliminary Results

Proposition 4.3 Let K be a closed convex subset of a Banach space X. Suppose A : K → X is a continuous and α-strongly accretive mapping, where

lim inf_{r→∞} α(r) >∥ A(x_{0}) ∥

for some x_{0} ∈ K. Suppose also that

lim_{h→0} \frac{1}{h} dist(x − hA(x); K) = 0  (4.3)

for every x ∈ K. Then A has a unique zero in K.

Proof. As usual we may assume that x_{0} = 0. By a result of Martin [21], K ⊂ (I + A)(K), and since I + A is invertible, the mapping g = (I + A)^{−1} is a nonexpansive self-mapping on K. Because the fixed points of g are precisely the zeros of A, it is sufficient to show that g has a fixed point in K. To this end, one may first show that the set

E = {x ∈ K : Ax = tx for some t < 0}

is bounded. To see this, let x ∈ E. Then Ax = tx for some t < 0 and,

⟨A(x) − A(0), j⟩ ≥ α(∥ x ∥) ∥ x ∥,

for some j ∈ J(x). This implies,

α(∥ x ∥) ∥ x ∥≤ t ∥ x ∥^{2} + ∥ A(0) ∥ ∥ x ∥ .

Since t < 0, α(||x||) ≤ ||A(0)||, and this implies that E is bounded. As a consequence of this one may easily show that the set

F = {y ∈ K : g(y) = λy for some λ > 1}

is also bounded. Next it is shown that (I−g)(K) is a closed set of X (this fact in general does not hold for arbitrary nonexpansive mappings g). Suppose {y_{n}} is a sequence in K so that y_{n} − g(y_{n}) → u, for some u ∈ X. Let g(y_{n}) = x_{n}. Then y_{n} − g(y_{n}) = Ax_{n} → u. On the other hand,

⟨A(x_{n}) − A(x_{m}), j⟩ ≥ a(∥ x_{n} − x_{m} ∥)||x_{n} − x_{m}||,

which implies that α(||x_{n} −x_{m}||) ≤ ||Ax_{n} −Ax_{m}||, and thus {x_{n}} is a Cauchy sequence. This means x_{n} → x for some x ∈ K, and since I + A is continuous, y_{n} → y for some y ∈ K. This implies u = (I − g)(y).

Finally, let t_{n} ∈ (0, 1) so that t_{n} → 1^{−} as n → ∞. Then t_{n}g(y_{n}) = y_{n} for some y_{n} ∈ K, which implies that

y_{n} − g(y_{n}) = (1 − t_{n}^{−1})y_{n}.

Since {y_{n}} ⊂ F and F is bounded, y_{n} − g(y_{n}) → 0 ∈ (I − g)(K). ■

Proposition 4.3 and Remark 1.5 yield the following facts.

Corollary 4.4 Let K be a closed convex subset of a Banach space X. Suppose that T : K → X is a continuous and α-strongly pseudocontractive mapping which is weakly inward on K. Then T has a unique fixed point.

Corollary 4.5 Let K and X be as in Proposition 4.3. Suppose T : K → K is a continuous and α-strongly pseudocontractive mapping. Then T has a unique fixed point.

Proposition 4.6 Let T : D ⊂ X → X be α-strongly pseudocontractive mapping. Suppose there exists a bounded sequence {x_{n}} in D and a convergent sequence {t_{n}} in (0, 1] such that x_{n} −t_{n}T (x_{n}) = u_{n} for a Cauchy sequence {u_{n}}. Then {x_{n}} is a Cauchy sequence.

Proof. Since T is strongly pseudocontractive (with constant k) there exists j in J(x_{n} − x_{m}) such that

⟨t_{n}^{−1}x_{n} − t_{m}^{−1}x_{m}, j⟩ ≤ [1 − \frac{α(r_{n,m})}{r_{n,m}}] · r_{n,m}^{2} + ⟨u_{n} − u_{m}, j⟩,

where r_{n,m} = ||x_{n} − x_{m}||. This implies

⟨t_{n}^{−1}x_{n} − t_{n}^{−1}x_{m}, j⟩ + ⟨t_{n}^{−1}x_{m} − t_{m}^{−1}x_{m}, j⟩ ≤ [1 − \frac{α(r_{n,m})}{r_{n,m}}] · r_{n,m}^{2} + ||u_{n} − u_{m}|| ||j||.

Therefore,

(t_{n}^{−1} − 1)||x_{n} − x_{m}|| + α(||x_{n} − x_{m}||) ≤ ||x_{m}|||t_{n}^{−1} − t_{m}^{−1}| + ||u_{n} − u_{m}||,

which completes the proof. ■

4.3. Global Results

The fact that pseudocontractive mappings are a much wider family than the nonexpansive mappings leads to the question of whether some of the results obtained earlier for nonexpansive mappings actually extend to pseudocontractive ones. The same question can be posed for the strong pseudocontractions relative to contractions.

Theorem 4.7 Let X be a Banach space and let D be an open subset of X. Suppose T : \overline{D} → X is a continuous α- strongly pseudocontractive mapping which satisfies for some z_{0} ∈ D, lim inf_{r→∞} α(r) > ||z_{0} − T (z_{0})|| and

T (x) − z_{0} ≠ λ(x − z_{0}) for all x ∈ ∂D and λ > 1.  (L-S)

Then T has a unique fixed point in \overline{D}.

Proof. By replacing T (x) with T (x + z_{0}) − z_{0} and D by D − z_{0}, one may select z_{0} = 0 in (L-S). Consider now the set

E = {x ∈ D : tT (x) = x for some t ∈ [0, 1]}.

Then, as in the proof of Proposition 4.3, it can be shown that this set E is also bounded. On the other hand, let

E = {t ∈ [0, 1] : tT (x) = x for some x ∈ D}.

Then E ≠ ϕ (since 0 ∈ E). The proof is complete by establishing 1 ∈ E. To accomplish this, we first observe that due to the continuity of T at 0, we may choose \overline{t} ∈ (0, 1) and δ > 0 such that tT maps the closed ball B(0; δ) into itself for each t ∈ (0, \overline{t}]. Hence, as a consequence of Corollary 4.5, t_{0} := sup E > 0.

Next, let {t_{n}} be a sequence in E such that t_{n} → t as n → ∞. Then t_{n}T (x_{n}) = x_{n} for some x_{n} ∈ D. Since T is α-strongly pseudocontractive and {x_{n}} is bounded, Proposition 4.6 implies that {x_{n}} is a Cauchy sequence, and hence it converges, say to x. Consequently, the continuity of T implies that tT (x) = x, and because of the (L-S) condition, x ∈ D. Therefore, sup E ∈ E. To complete the proof, suppose sup E < 1. Then we may choose a sequence {t_{n}} ∈

[0, 1] such that t_{n} → t_{0}^{+}. Let t_{0}T (x_{0}) = x_{0} for some x_{0} ∈ D, and let B be an open ball centered at x_{0} (in D). For t ∈ [0, 1], define h_{t}(x) = x − tT (x). Then for each n ∈ ℕ,

y_{n} := h_{t_{n}}(x_{0}) ∈ h_{t_{n}}(B),

while 0 ∉ h_{t_{n}}(B). Choose u_{n} ∈ seg[0, y_{n}] ∩ ∂h_{t_{n}}(B). Since h_{t_{n}} is strongly accretive, h_{t_{n}}(B) is open (by Theorem 2.18), while h_{t_{n}}(\overline{B}) is closed. Hence, by Remark 2.10, ∂h_{t_{n}}(B) ⊂ h_{t_{n}}(∂B). This means, there exists x_{n} ∈ ∂B such that h_{t_{n}}(x_{n}) = u_{n}. Since y_{n} → 0 as n → ∞ and u_{n} ∈ seg[0, y_{n}], u_{n} → 0. Therefore, by Proposition 4.6 again, {x_{n}} is a Cauchy sequence, which converges to x_{1} ∈ ∂B. Continuity of T implies that h_{t_{0}}(x_{1}) = 0, and since x_{0} ≠ x_{1}, this contradicts the one-to-oneness of h_{t_{0}}. ■

The connection between the boundary conditions Δ and L-S for nonexpansive mappings was noted earlier. A similar connection for α-strongly pseudocontractive mapping gives the following result.

Corollary 4.8 Let X be a Banach space and let D be an open subset of X. Suppose T : \overline{D} → X is a continuous α-strongly pseudocontractive mapping which satisfies for some z_{0} ∈ D, lim inf_{r→∞} α(r) > ||z_{0} − T (z_{0})|| and

||z_{0} − T (z_{0})|| ≤ ||x − T (x)|| for all x ∈ ∂D.  (Δ_{w})

Then T has a unique fixed point in \overline{D}.

Proof. In view of Theorem 4.7 it is sufficient to show that (Δ_{w}) implies (L-S). To see this, suppose there exists x ∈ ∂D and λ > 1 such that T (x) − z_{0} = λ(x − z_{0}). Then there exists j ∈ J(x − z_{0}) such that

⟨λ(x − z_{0}) + z_{0} − T (z_{0}), j⟩ ≤ ||x − z_{0}||^{2} − α(||x − z_{0}||)||x − z_{0}||,

which implies

(λ − 1)||x − z_{0}|| ≤||z_{0} − T (z_{0})|| − α(||x − z_{0}||)
≤||x − T (x)|| − α(||x − z_{0}||)
=(λ − 1)||x − z_{0}|| − α(||x − z_{0}||).

Hence, x = z_{0} which is a contradiction. ■

Theorem 4.7 yields an extension of the first conclusion of Proposition 2.14.

Corollary 4.9 Let X be a Banach space and let D be a bounded open subset of X. Suppose T : \overline{D} → X is a continuous pseudocontractive mapping which satisfies the Leray-Schauder condition. Then inf{||x − T (x)|| : x ∈ D} = 0.

Proof. For each t ∈ [0, 1], let T_{t}(x) = (1 − t)z_{0} + tT (x). Then T_{t} satisfies the Condition (L-S), and hence, by Theorem 11, T_{t} has a fixed point x_{t} ∈ \overline{D}, such that tT (x_{t}) + (1 − t)z_{0} = x_{t}. Since D is bounded, it follows that x_{t} − T (x_{t}) → 0 as t → 1^{−}.

■

In view of the proof of Corollary 4.8, one might wonder whether the Leray-Schauder condition would imply the Condition Δ_{w}. Indeed, the proposition below answers this question.

Proposition 4.10 Suppose D, X and T are as in Corollary 4.8. Suppose also that T satisfies the boundary condition (L-S) and is fixed-point free on ∂D. Then T satisfies the boundary condition Δ_{w}.

Proof. We first prove that

δ := inf{||x − T (x)|| : x ∈ ∂D} > 0.

Suppose the contrary. Then there exists a sequence {x_{n}} in ∂D such that x_{n} − T (x_{n}) → 0 as n → ∞. However, according to Proposition 4.6 {x_{n}} is Cauchy, and consequently, it converges to some x ∈ ∂D. This contradicts one of the assumptions on T. On the other hand, Corollary 4.9 implies that inf{||x − T (x)|| : x ∈ D} = 0. This means that there exists w_{0} ∈ D such that ||w_{0} − T (w_{0})|| < δ, and hence, the condition △_{w} holds. ■

It is unknown whether Theorem 2.8 holds for pseudocontractive mappings. In fact, this seems to be one of the few results for nonexpansive mappings for which the continuous pseudocontractive case remains in doubt. At the same time, the following is true.

Theorem 4.11 Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let D be a bounded open subset of X. Suppose T : \overline{D} → X is a continuous pseudocontractive mapping which satisfies the Leray-Schauder condition with respect to some z_{0} ∈ D. Then T has a fixed point in \overline{D}.

Proof. By the proof of Corollary 4.9, there exists x_{n} := x_{t_{n}} ∈ \overline{D} such that

t_{n}T (x_{n}) + (1 − t_{n})z_{0} = x_{n}  (4.4)

for each t_{n} ∈ [0, 1] with t_{n} → 1^{−}. Since X is reflexive, we may assume without loss of generality that {x_{n}} converges weakly to some x ∈ X. Then from (4.4) and the fact that T is pseudocontractive, we get

⟨(t_{n}^{−1} − 1)x_{n} − (t_{m}^{−1} − 1)x_{m}, J(x_{n} − x_{m})⟩ ≤ (t_{n}^{−1} − t_{m}^{−1})⟩z_{0}, J(x_{n} − x_{m})⟩.

By letting m → ∞, we obtain

⟨x_{n}, J(x_{n} − x)⟩ ≤ ⟨z_{0}, J(x_{n} − x)⟩

and thus

||x_{n} − x||^{2} ≤ ⟨z_{0} − x, J(x_{n} − x)⟩.

Therefore, x_{n} → x as n → ∞, and hence the continuity of T implies that T (x) = x ■

4.4. Local Existence Results

The boundary conditions introduced in Section 2.1 will now be used for locally kpseudocontractive operators to prove the existence of fixed points. The next result is proven in [23].

Theorem 4.12 Let X be a Banach space and let D be an open subset of X. Suppose T : \overline{D} → X is a continuous and locally strongly pseudocontractive mapping on D which satisfies for some z_{0} ∈ D:

T (x) − z_{0} ≠ λ(x − z_{0}) for all x ∈ ∂D and λ > 1.

Then T has a fixed point in \overline{D}.

The following facts will be used in the proof of Theorem 4.12.

Proposition 4.13 Let X be a Banach space, D an open subset of X, and T : \overline{D} → X a continuous mapping which is locally k-pseudocontractive on D (with k ≤ 1). Suppose t_{0}T (x_{0}) = x_{0} for some x_{0} ∈ D and t_{0} ∈ (0, 1], and suppose for σ_{0} > 0, B(x_{0}; σ_{0}) ⊂ D. Then:

(a) If k < 1 and for all t ∈ (0, 1] satisfying

|t − t_{0}| ≤ \frac{t_{0}}{||x_{0}||}σ_{0}(1 − k),  (4.5)

there exists a unique point x_{t} ∈ B(x_{0}; σ_{0}) such that tT (x_{t}) = x_{t}. Moreover, this point satisfies

||x_{t} − x_{0}|| ≤ \frac{||x_{0}||}{t_{0}(1 − tk)}|t − t_{0}|.  (4.6)

(b) If k = 1 and t ∈ (0, \frac{1 + t_{0})}{2}] which satisfies

|t − t_{0}| ≤ \frac{t_{0}(1 − t_{0})}{2||x_{0}||}σ_{0},

there exists a unique point x_{t} ∈ B(x_{0}; σ_{0}) such that tT (x_{t}) = x_{t}. Moreover, this point satisfies

||x_{t} − x_{0}|| ≤ \frac{||x_{t}||}{t(1 − t_{0})}|t − t_{0}|.

Proof. To prove (a), we use Theorem 4.7. Since T is globally strongly pseudocontractive on B := B(x_{0}; σ_{0}), it is enough to show that tT satisfies the Leray-Schauder condition on ∂B for t that satisfies (4.5). To see this, suppose for some λ > 0 and x ∈ ∂B,

tT (x) − x_{0} = λ(x − x_{0}).

Then by the strong pseudocontractiveness of T on B, there exists j ∈ J(x − x_{0}) such that

⟨t^{−1}λ(x − x_{0}) + t^{−1}x_{0} − t_{0}^{−1}x_{0}, j⟩ ≤ k||x − x_{0}||^{2}.

It follows that

(λ − tk)||x − x_{0}|| ≤ \frac{|t − t_{0}|}{t_{0}}||x_{0}||,

which implies that λ ≤ 1. Therefore, tT has a fixed point x_{t} ∈ B. The uniqueness follows directly from the fact that tT satisfies (4.2) on B. To complete the proof of (a), observe that for some j ∈ J(x_{t} − x_{0}),

⟨t^{−1}x_{t} − t_{0}^{−1}x_{0}, j⟩ ≤ k||x_{t} − x_{0}||^{2},

which implies

⟨t^{−1}(x_{t} − x_{0}) + t^{−1}x_{0} − t_{0}^{−1}x_{0}, j⟩ ≤ k||x_{t} − x_{0}||^{2}.

Thus

(1 − tk)||x_{t} − x_{0}||^{2} ≤ (tt_{0}^{−1} − 1)⟨x_{0}, j⟩,

and hence

||x_{t} − x_{0}|| ≤ \frac{||x_{0}||}{t_{0}(1 − tk)}|t − t_{0}|.

The proof of (b) follows the same path as in the proof of (a) with k = 1. ■

Proposition 4.14 Let X, D and T be as in Proposition 4.13. For H ⊂ D, set

E_{H} = {t ∈ [0, 1] : tT (x) = x for some x ∈ H}

and let

E = {x ∈ D : tT (x) = x for some t ∈ [0, 1]}.

Then

(i) the set E is the union of disjoint nontrivial components, each of which is a continuous image of a subinterval of [0, 1].In addition, if F is any component of E, then

(ii) the function h : E_{F} → ℝ defined by h(t) = ||x_{t} − T (x_{t})|| where x_{t} ∈ F and tT (x_{t}) = x_{t} is nondecreasing;

(iii) if k < 1 and t_{0} ∈ E_{F}, then the set S = {x ∈ F : tT (x) = x for some t ∈ E_{F} ∩[t_{0}, 1]} is bounded, and

(iv) if t_{n}T (x_{t_{n}}) = x_{t_{n}} with t_{n} → t^{−} ∈ (0, 1] and {x_{t_{n}}} ⊂ F (with k < 1), then {x_{t_{n}}} is a Cauchy sequence.

Proof. (i) is an immediate consequence of Proposition 4.13.

(ii) Let t ∈ E_{F} and x_{t} ∈ F such that tT (x_{t}) = x_{t}. Then, by assumption, there exists σ > 0 such that T is k-pseudocontractive on B(x_{t}; σ) ⊂ D. Suppose sT (x_{s}) = x_{s} with x_{s} ∈ B(x_{t}; σ) ∩ F and 0 ≤ t < s ≤ 1. then by (4.6) we have

||x_{s} − x_{t}|| ≤ \frac{||x_{t}||}{t(1 − sk)}|t − s|  (4.7)

which yields

||x_{s} − T (x_{s})|| =\frac{1 − s}{s}||x_{s}|| ≤ \frac{1 − s}{s} [||x_{s} − x_{t}|| + ||x_{t}||]
≤ (\frac{1 − s}{s}) [\frac{s − t}{t(1 − sk)} + 1] ||x_{t}||
≤\frac{1 − t}{t}||x_{t}|| = ||x_{t} − T (x_{t})||.

(iii) Suppose x_{t_{0}} ∈ F with t_{0}T (x_{t_{0}}) = x_{t_{0}} and select s, x_{s} as in (ii). If t_{0} > 0, then by (4.7)

||x_{s}|| ≤ [\frac{s − t_{0}}{t_{0}(1 − sk)} + 1] ||x_{t_{0}}||
=\frac{s(1 − t_{0}k)}{t_{0}(1 − sk)}||x_{t_{0}}||
≤\frac{1 − t_{0}k}{t_{0}(1 − k)}||x_{t_{0}}||.

If t_{0} = 0, then x_{t_{0}} = 0 and ||x_{s}|| ≤ \frac{1}{1−k}||T (0)|| for x_{s} ∈ B(0; δ) with an appropriate δ. Therefore, S is bounded.

(iv) For this part of the proof, we follow the argument given by Kirk and Morales [16]. If t_{m} < t_{n}, then by Proposition 4.13, the segment |t_{m} − t_{n}| can be covered by finitely many overlapping subintervals {I_{i}}_{i=1}^{r} which have the property that for each i and s, t ∈ I_{i}, the eigenvectors x_{s}, x_{t} corresponding to s and t satisfy

||x_{s} − x_{t}|| ≤ \frac{M}{t_{0}(1 − k)}|t − s|  (4.8)

where M = sup{||x_{t}|| : x_{t} ∈ F, t_{0} ≤ t ≤ 1} with t_{0} = inf{t_{n}}.

We may now select s_{i} ∈ I_{i} ∩ I_{i+1} such that t_{m} = s_{0} < s_{i} < ⋯ < s_{r} = t_{n}. Then by (4.8),

||x_{s_{i−1}} − x_{s_{i}}|| ≤ \frac{M}{t_{0}(1 − k)}|s_{i−1} − s_{i}|, i = 1, …, r,

and thus

||x_{t_{m}} − x_{t_{n}}|| ≤ ∑_{i=1}^{r} ||x_{s_{i}} − x_{s_{i+1}}|| ≤ \frac{M}{t_{0}(1 − k)} ∑_{i−1}^{r} |s_{i−1} − s_{i}|.

Therefore, {x_{t_{n}}} is a Cauchy sequence. ■

Proof of Theorem 4.12 By replacing T (x) with T (x + z_{0}) − z_{0} one may take z_{0} = 0 in (L-S). Define E as in Proposition 4.14. Since 0 ∈ D and T is continuous, there exists t_{0} ∈ (0, 1) and σ > 0 such that tT maps the closed ball B(0; σ) into itself for each t ∈ (0, t_{0}). Hence, by Corollary 4.5, tT (x) = x for some x ∈ D. This means, [0, t_{0}) ⊂ E. Now consider the \overline{t} = sup E. Then, by Proposition 4.14 (iv), \overline{t} ∈ E, and according to Proposition 4.13 (a), \overline{t} = 1. Therefore, T has a fixed point in \overline{D}. ■

Corollary 4.15 Let D be an open subset of a Banach space X and let T : \overline{D} → X be continuous and locally strongly pseudocontractive on D, which satisfies for some z_{0} ∈ D

||z_{0} − T (z_{0})|| ≤ ||x − T (x)|| for x ∈ ∂D  (4.9)

Then T has a fixed point in \overline{D}.

Proof. As before, without loss of generality, one may take z_{0} = 0 in (4.9). Let E be as in Proposition 4.14, and let F_{0} be the component of E for which 0 ∈ F_{0}. Let B be a closed ball centered at 0 such that T restricted to B is strongly pseudocontractive, and let x ∈ B ∩ F_{0} with tT (x) = x for some t ∈ (0, 1). Then by (4.2) there exists j ∈ J(x) such that

⟨t^{−1}x − T (0), j⟩ ≤ k||x||^{2},

which implies that (t^{−1} − k)||x|| ≤ ||T (0)||, and thus ||x − T (x)|| ≤ ||T (0)||. However, Propositions 4.13 and 4.14 (ii) imply that ||x − T (x)|| ≤ ||T (0)|| for all x ∈ F_{0}. Hence, F_{0} ⊂ D, and since this set is a component of E, there exists an open set G in X such that G ∩ E = F_{0}, which implies that T satisfies the L-S condition on ∂G. The conclusion now follows from Theorem 4.12. ■

We are now in the position of obtaining a local version of Theorem 2.19. We begin with some preliminaries.

Remark 4.16 (cf. Kirk[13]). Let D be an open subset of a Banach space X, and suppose T : D → X is a continuous and locally strongly accretive mapping. Let u = T (x) for x ∈ D, and let S = seg[u, v] ⊂ T (D) for some v ∈ X. Then there exists a unique path (up to parametrization) whose image Γ begins at x, ends at w ∈ D with T (w) = v, and T (Γ) = S. Moreover, the inverse of the restriction of T to Γ is a Lipschitz mapping of S onto Γ.

Remark 4.16 leads to a proof of the existence of an inverse for a locally strongly accretive operator.

Proposition 4.17 (Morales-Mutangadura [26]) Let X be a Banach space, let D be a connected open subset of X, and let T : D → X be continuous and locally strongly accretive. Then T is globally one-to-one on D.

We now give a result which is identical to Theorem 2.19, except that the mapping T is assumed to be locally pseudocontractive rather than globally pseudocontractive. However the proofs in the two cases are markedly different.

Theorem 4.18 Let X be a Banach space, D a bounded open subset of X, and T : \overline{D} → X a continuous mapping which is locally pseudocontractive on D. Suppose there exists z ∈ D such that

||z − T (z)|| < ||x − T (x)|| for all x ∈ ∂D.  (4.10)

Then inf\{||x − T (x)|| : x ∈ D\} = 0. Moreover, if it is the case that \overline{D} has the fixed point property with respect to nonexpansive selfmappings, then T has a fixed point in D.

Proof. As before we may take z = 0 in (4.10). Since T is continuous at 0, there exists t ∈ (0, 1) such that t ∈ E_{D}. Let t_{0} = sup E_{F_{0}}, where F_{0} is the component of E (as in Proposition 4.14) which contains 0. If t_{0} < 1, then by Proposition 4.14 (iv), t_{0} ∈ E\overline{D}. This means, t_{0}T (x_{0}) = x_{0} for some x_{0} ∈ \overline{D}. However, as we observed earlier, ||x_{0} − T (x_{0})|| ≤ ||T (0)||, which due to (4.10) implies that x_{0} ∈ D. This means, according to Proposition 4.13, there exists t > t_{0} with t ∈ E_{F_{0}}, which contradicts the maximality of t_{0}, and hence, t_{0} = 1. Therefore, inf{||x − T (x)|| : x ∈ D} = 0.

To prove the existence of a fixed point for T under the additional assumption on \overline{D}, we use the first part of this theorem to redefine the element z in (4.10) so that

||z − T (z)|| < inf {||x − T (x)|| : x ∈ ∂D|}.

Since D is bounded, we may choose ρ > 0 large enough and r ∈ (0, 1) such that D ⊂ B(z; ρ) and

r(||z − y|| + ||y − x||) + ||z − T (z)|| < ||x − T (x)||  (4.11)

for all y ∈ B(z; ρ) and for all x ∈ ∂D.

For each fixed y ∈ B(z; ρ), define U : \overline{D} → X by setting

U(x) = T (x) + r(y − x), x ∈ \overline{D}.

If x ∈ ∂D, then (4.11) implies

||z − U(z)|| =||z − T (z) + r(z − y)||
≤||z − T (z)|| + r||z − y||
<||x − T (x)|| − r||y − x||
≤||x − U(x)||.

Since U is continuous on \overline{D} and locally strongly pseudocontractive on D, Corollary 4.15 implies that U has a fixed point x ∈ D. Hence (1 + r)x − T (x) = ry, and thus, f(x) := (1 + s)x − sT (x) = y for s = r^{−1}. Therefore, B(z; ρ) ⊂ f(D). Since, without loss of generality, D may be assumed to be connected, Proposition 4.17 implies that f is globally one-to-one, and hence the mapping g := f^{−1} : B(z; ρ) → D is locally nonexpansive, and consequently, is also globally nonexpansive. Therefore, the restriction of g to \overline{D} has a fixed point and the proof is complete. ■

As a consequence of the proof of Theorem 4.18, the following result can be obtained. See [25] for the details.

Theorem 4.19 Let X be a Banach space for which the closed unit ball has the fixed point property for nonexpansive selfmappings, let D be a bounded open subset of X, and suppose T : \overline{D} → X is continuous mapping which is locally pseudocontractive on D. Suppose there exists z ∈ D such that

||z − T (z)|| < ||x − T (x)||  for all x ∈ ∂D.  (4.12)

Then T has a fixed point in \overline{D}.

As a corollary of Theorem 4.19 we obtain an extension of Theorem 3.2 for locally pseudocontractive mappings.

Corollary 4.20 Let X be a uniformly convex Banach space, let D be a bounded open subset of X, and let T : \overline{D} → X be a continuous mapping which is locally pseudocontractive on D. Suppose there exists z ∈ D such that

||z − T (z)|| < ||x − T (x)|| for all x ∈ ∂D.

Then T has a fixed point in \overline{D}.

As mentioned earlier, Condition Δ provides a way to approximate fixed points for locally pseudocontractive mappings. The following result is proved in [25]. See also [2], [24], and [26] for related results.

Theorem 4.21 Let X be a uniformly smooth Banach space, D a bounded open subset of X, and T : \overline{D} → X a continuous mapping which is locally pseudocontractive on D. Suppose there exists z ∈ D such that

||z − T (z)|| < ||x − T (x)|| for all x ∈ ∂D.

Then there exists a unique path t → z_{t} ∈ D, t ∈ [0, 1), satisfying

z_{t} = tT (z_{t}) + (1 − t)z,  (4.13)

where the strong lim_{t→1^{−}} z_{t} exists, and this limit is a fixed point T.

5. Remarks

As we have remarked earlier, the locally lipschitzian mappings mapping,locally lipschitzianarise in a natural way from differentiability assumptions. Whether locally pseudocontractive mappings arise in a similar natural way is less clear. Thus many of the results of the previous section might seem to be more interesting when viewed as results about locally nonexpansive mappings. On the other hand, it seems to be the case that it is just as easy (or difficult) to obtain results for locally pseudocontractive mappings as it is for locally nonexpansive mappings.

There are also two observations in [18] that might be worth mentioning in connection with our discussion of the local theory. If D is an open subset of a metric space X and if T is a mapping of D into a convex metric space Y, then T is said to be locally expansive on D if for each u_{0} ∈ D there exists r > 0 such that B (u_{0}; r) ⊆ D and d (T (u) , T (v)) ≥ d (u, v) for all u, v ∈ D. In [18] it is shown that if X and Y are complete with Y metrically convex, and if T is a closed mapping which maps open subsets of D onto open subsets of Y, then an element y ∈ Y belongs to T (D) if and only if there exists x_{0} ∈ D such that d (T (x_{0}) , y) ≤ d (T (x) , y) for all x ∈ X\D.

It is also shown in [18] that if a continuous locally accretive mapping defined on a convex open subset of a Banach space is actually accretive. In view of this, for a convex domain D Theorem 4.18 coincides with Theorem 2.19. Also, in both cases, the domain D is assumed to have the fixed point property for nonexpansive mappings. While the class of such sets has not been completely classified, it is known to include nonconvex sets (e.g, [9]).

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