Fixed Points for Pseudocontractive Mappings on Unbounded Domains

Abstract

We give some fixed point results for pseudocontractive mappings on nonbounded domains which allow us to obtain generalizations of recent fixed point theorems of Penot, Isac, and Németh. An application to integral equations is given.

1. Introduction

Let be a nonempty subset of a Banach space with norm . Recall that a mapping is said to be nonexpansive whenever for every . is said to have the fixed point property ((FPP) for short) if every nonexpansive selfmapping of each nonempty bounded closed and convex subset of has a fixed point. It has been known from the outset of the study of this property (around the early sixties of the last century) that it depends strongly on "nice'' geometrical properties of the space. For instance, a celebrated result due to Kirk [1] establishes that those reflexive Banach spaces with normal structure (NS) have the (FPP). In particular, uniformly convex Banach spaces have normal structure (see [2, 3] for more information).

If is a closed convex of a Banach space enjoying the (FPP), in general it is not true that has a fixed point due to the possible unboundedness of (it is enough to consider any translation map, with nonnull vector, in the Banach space ). In 2003 Penot [4] showed that if is a closed convex subset of a uniformly convex Banach space , is a nonexpansive mapping, and for some ,

(1.1)

(in other words if is asymptotically contractive), then has a fixed point.

A celebrated fixed point result due to Altman [5] is the following.

Let be a separable Hilbert space, with inner product and induced norm . Let be a weakly closed mapping where is the closed ball with center and radius . Suppose that maps the sphere into a bounded set in . If the following condition is satisfied:

(1.2)

for all , then has a fixed point in .

In 2006, Isac and Németh [6] gave some fixed point results for nonexpansive nonlinear mappings in Banach spaces inspired by Penot's results where the asymptotically contractiveness was stated in similar terms to condition (1.2).

In this paper we generalize some Penot, Isac, and Németh's fixed point results in several ways. First, we will be concerned with pseudocontractive mappings, a more general class of mappings than the nonexpansive ones. Second, we use an inwardness condition weaker than , and finally our Altmann type assumptions are more general than those required in [4, 6].

We prove our fixed point results as a consequence of some results on the existence of zeroes for accretive operators. Among the problems treated by accretive operators theory, one of the most studied is just this one (see, e.g., Kirk and Schöneberg's paper [7] as well as [3, 8, 9] and the references therein). We obtain here several results of this type, and in particular we give a characterization in the setting of the Banach spaces with (FPP) of those -accretive operators which have zeroes.

2. Preliminaries

Throughout this paper we suppose that is a real Banach space and that is its topological dual. We use to denote the closed ball centered at with radius . We also use the notation , .

If , we will denote by the normalized duality mapping at defined by . We will often use the mapping defined by .

A mapping will be called an operator on . The domain of is denoted by and its range by . It is well known that an operator is accretive if and only if for all .

If, in addition, is for one, hence for all, , precisely , then is called -accretive. We say that satisfies the range condition if for all .

We now recall some important facts regarding accretive operators which will be used in our paper (see, e.g., [10]).

Proposition 2.1.

Let be an operator on . The following conditions are equivalent:

(i) is an accretive operator,

(ii)the inequality holds for all , and for every ,

(iii)for each the resolvent is a single-valued nonexpansive mapping.

Let be a nonempty subset of and let be a mapping. Recall that a sequence of elements of is said to be an a.f.p sequence for whenever . It is well known that if is a nonexpansive mapping which maps a closed convex bounded subset of into itself, then such a mapping always has a.f.p. sequences in .

When the Banach space has the (FPP), Morales [9] gave a characterization of those -accretive operators such that . Let us recall such result.

Theorem 2.2.

Let be a Banach space with the FPP, and let be an -accretive operator. Then if and only if the set is bounded.

A mapping is said to be pseudocontractive if for every , and for all positive , . Pseudocontractive mappings are easily seen to be more general than nonexpansive mappings ones. The interest in these mappings also stems from the fact that they are firmly connected to the well-known class of accretive mappings. Specifically is pseudocontractive if and only if is accretive where is the identity mapping.

We say that a mapping is demiclosed at if for any sequence in weakly convergent to with norm convergent to one has that . It is well known that if is weakly compact and convex, is nonexpansive, and is demiclosed at , then has a fixed point in .

We say that the mapping is weakly inward on if for all . Such condition is always weaker than the assumption of mapping the boundary of into . Recall that if is a continuous accretive mapping, is convex and closed, and is weakly inward on , then has the range condition (see [11]).

We say that a semi-inner-product is defined on , if to any there corresponds a real number denoted by satisfying the following properties:

(s1) for ,

(s2) for , and ,

(s3) for ,

(s4) .

It is known (see [12, 13]) that a semi-inner-product space is a normed linear space with the norm and that every Banach space can be endowed with a semi-inner-product (and in general in infinitely many different ways, but a Hilbert space in a unique way).

In [6] the authors considered several fixed point results for nonexpansive mappings with unbounded domains satisfying additional asymptotic contractive-type conditions in terms of a function under the following assumptions:

(G1) for any and ,

(G2) for any ,

(G3) for any ,

(G4) there exists an such that for every .

3. Zeroes for Accretive Operators

We begin with the definition of a certain kind of functions on which we will be concerned. This class is more general than the corresponding one considered in [6]. Let be a real Banach space and a mapping which satisfies the following conditions:

(g1) for any and ,

(g2) there exists such that for any with ,

(g3) for any ,

(g4) for each , there exists (depending on ), such that if , then .

Notice that if we consider either or , then satisfies (g1)–(g4).

Let be a Banach space with the (FPP). If is an -accretive operator such that its domain is a bounded set, then it is well known that (see, e.g., [7, 9]). If is not bounded, then we give the following result.

Theorem 3.1.

Let be a Banach space with the (FPP). Let be a mapping satisfying (g1) and (g2). If is an -accretive operator such that there exists with

(3.1)

whenever , then, .

Proof.

Since is -accretive and has the (FPP), by Theorem 2.2 we know that if and only if the set is bounded.

In order to get a contradiction we assume that is an unbounded set. This fact means that for each there exists such that .

Since , then there exist and such that . This means that .

Consequently, for every , we have

(3.2)

which is a contradiction.

In the following theorem we are going to give a characterization in terms of a particular function , (in the framework of the Banach spaces with the (FPP)), of those -accretive operators which have zeroes.

Theorem 3.2.

Let be a Banach space with the (FPP). Let be the mapping

(3.3)

If is an -accretive operator, then the following conditions are equivalent:

(1)there exists such that whenever and ;

(2).

Proof.

()() It is clear that satisfies conditions (g1) and (g2), thus by Theorem 3.1 we obtain that .

()() In order to get a contradiction, assume that for each there exists with such that

(3.4)

The above inequality implies that for each , there exist and such that .

By definition of , we have that , and thus .

From the above fact, we derive that for each ,

(3.5)

that is, is unbounded. By Theorem 2.2, it follows that if is unbounded, then ; therefore, we have a contradiction.

As a consequence of the above characterization it is easy to capture the following result which is related to [7, Theorems  2 and  3].

Corollary 3.3.

Let be a real Banach space with the (FPP). Suppose that is an -accretive operator for which there exist and such that

(3.6)

for all with . Then .

Proof.

Without loss of generality we may assume that . Otherwise, we work with the operator defined by .

If we take as in Theorem 3.2, to obtain the conclusion it is enough to see that

(3.7)

whenever .

In order to get a contradiction, assume that for each there exists with such that

(3.8)

The above inequality implies that for each , there exist and such that .

By definition of , we have that , and thus .

By hypothesis, we know that the inequality holds for every .

This means that there exists such that . Therefore

(3.9)

On the other hand, since is an accretive operator, it is clear that

(3.10)

which is a contradiction.

The above corollary allows us to recapture the following well-known result.

Corollary 3.4.

Let be a real Banach space with the (FPP). Suppose that is an -accretive operator; if

(3.11)

then .

Corollary 3.5.

Let be a Banach space with the (FPP). Let be a mapping satisfying (g1) and (g2). If is an -accretive operator such that

(3.12)

then .

Proof.

It is clear that condition (3.12) implies assumption (3.1).

Corollary 3.6.

Let be a real Hilbert space. Let be a convex proper lower semicontinuous mapping with effective domain . Suppose that for some there exists such that for all with . Then has an absolute minimum on .

Proof.

Consider the subdifferential associated to , that is

(3.13)

It is well known that is an -accretive operator on (see [14]). Now, we consider defined as in Theorem 3.2.

In order to get a contradiction, suppose that given there is with such that

(3.14)

By definition of we have that there exists such that

(3.15)

This means that , hence . Consequently

(3.16)

By hypothesis, when , we obtain the following contradiction:

(3.17)

This contradiction allows us to conclude that there exists such that if with then

(3.18)

Since has the (FPP), from Theorem 3.1 we conclude that ; that is, there exists such that

(3.19)

and therefore is an absolute minimum of .

If has the (FPP), is an accretive operator with the range condition, and is convex and bounded, then, ; see [8]. For the case that is not bounded we have the following result.

Theorem 3.7.

Let be a Banach space. Suppose that is a mapping satisfying conditions (g1)–(g4).

If has the (FPP), is an accretive operator with the range condition, is convex, and satisfies condition (3.1), then .

Proof.

Since is accretive with the range condition, then the following two conditions hold:

(i)

(ii) is a nonexpansive mapping.

Fix . For each positive integer , from (i) there exist and such that

(3.20)

Hence, . It follows that

(3.21)

We claim that is a bounded sequence. Indeed, otherwise we can assume that there exists a subsequence of such that . Without loss of generality we may assume that , , where the constants and are given in the definitions of conditions (g2) and (g4), respectively.

Therefore, we have

(3.22)

Consequently,

(3.23)

This is a contradiction which proves our claim.

Since is a bounded sequence, it is clear that goes to as goes to infinity.

Now we claim that has a bounded a.f.p. sequence. Indeed, consider for each positive integer , . It is not difficult to see that because . In this case, we obtain

(3.24)

Finally, if we call , we obtain that the following set

(3.25)

is bounded closed convex and -invariant. Thus, since enjoys the (FPP), there exists such that and then .

Remark 3.8.

If we check the proof of Theorem 3.7, we may notice that such theorem still holds if we omit conditions (g3) and (g4) but we add .

Corollary 3.9.

Let be a Banach space. Suppose that is a mapping satisfying conditions (g1)–(g4).

If has the (FPP), is an accretive operator with the range condition, is convex, and satisfies condition (3.12), then .

4. Fixed Point Results

Theorem 4.1.

Let be a Banach space with the (FPP). Suppose that is a mapping satisfying conditions (g1) and (g2). Let be a closed convex and unbounded subset of with . Let be a continuous pseudocontractive mapping. Assume that the following conditions are satisfied.

(a) is weakly inward on .

(b)There exists such that for every with the inequality

(4.1)

holds.

Then has a fixed point in .

Proof.

Since is a continuous, pseudocontractive mappings weakly inward on , then is an accretive operator with the range condition (see [11, 15]).

Let us see that condition (3.1) is satisfied. Indeed, if with ,

(4.2)

The above equality along with (4.1) allows us conclude that condition (3.1) holds.

On the other hand, since , by Remark 3.8 and following the same argument developed in the proof of Theorem 3.7, it is not difficult to see that

(4.3)

has a bounded a.f.p. sequence , and thus, if we call , we obtain that the set

(4.4)

is bounded closed convex and -invariant. Thus, since enjoys the (FPP), there exists such that and then .

Corollary 4.2.

Let be a Banach space with the (FPP). Suppose that is a mapping satisfying conditions (g1) and (g2). Let be a closed convex and unbounded subset of with . If is a continuous pseudocontractive mapping weakly inward on and

(4.5)

then has a fixed point in .

Proof.

Clearly inequality (4.5) implies condition (4.1).

Corollary 4.3.

Let be a Banach space with the (FPP). Suppose that is a mapping satisfying conditions (g1)–(g4). Let be a closed convex and unbounded subset of . If is a continuous pseudocontractive mapping weakly inward on and satisfies condition (4.1), then has a fixed point in .

Proof.

From the above theorem, we know that is an accretive operator with the range condition and with condition (3.1). Therefore by Theorem 3.7 we obtain the result.

Remark 4.4.

In order to give an alternative proof of Corollary 4.3, it is enough to see that condition (4.5) implies that has an a.f.p. sequence and thus, using [16, Theorem  4.3], we obtain the same conclusion. In this case, if we assume that is a reflexive Banach space and is demiclosed at zero, then we can remove the assumption on the (FPP) for the space . Nevertheless, it is well known that there exist nonreflexive Banach spaces with the FPP (see [13]). On the other hand, if is a reflexive Banach space such that for every nonexpansive mapping, say , the mapping is demiclosed at , then the Banach space has the FPP.

Remark 4.5 (Theorem  3.2 in [6] reads).

Let be a reflexive Banach space. Suppose that satisfies conditions (G1), (G2), (G3), and (G4). Let be a nonempty unbounded closed convex set. If is a nonexpansive mapping such that , is demiclosed and

(4.6)

for some , then has a fixed point in .

Notice that Corollary 4.3 generalizes this theorem in several senses.

(i)Our assumptions (g1)–(g4) on mapping are weaker than the corresponding in that theorem.

(ii)Every nonexpansive mapping is in fact continuous and pseudocontractive.

(iii)The inwardness condition is more general than the assumption .

(iv)Condition (4.6) implies that

(4.7)

Interchanging the roles of and we can conclude that

(4.8)

for every . Therefore, there exists such that if , then

(4.9)

which is just condition (4.1) of Theorem 4.1.

In the same sense, Theorem 4.1 is a generalization of Theorem  3.1 of [6].

Corollary 4.6.

Let be a Banach space with the (FPP). Let be a closed convex and unbounded subset of such that . Let be a continuous pseudocontractive mapping. Assume that the following conditions are satisfied.

(a) is weakly inward on .

(b)There exists such that for every and for every ,.

Then has a fixed point in .

Proof.

It is enough to apply Theorem 4.1, where is defined by

(4.10)

and if , then .

Remark 4.7.

Notice that the above condition (b) is similar to the well-known Leray-Schauder boundary condition. Some results of this type can be found in [17–19].

Corollary 4.8.

Let be a Banach space with the (FPP). Let be a closed convex and unbounded subset of . If is a continuous pseudocontractive mapping weakly inward on and for every and large enough

(4.11)

for some , then has a fixed point in .

Proof.

Let be the function defined by . It is clear that satisfies conditions (g1) and (g3) . Moreover,

(4.12)

for some . Therefore,

(4.13)

Since is a fix element of , clearly there exists such that whenever . This means that satisfies (g2).

To see that satisfies condition (g4) we argue as follows.

Given a fix , we know that .

Since as , we can find such that for every .

Then,

(4.14)

Now, we will see that satisfies inequality (4.1) in Corollary 4.3. Indeed, if , we have, for some , that

(4.15)

Thus the conclusion follows from Corollary 4.3.

Remark 4.9.

In the case that for all , , then the mapping is said to have as a center; see [20], where some fixed point theorems are given for this class of mappings.

On the other hand, in [21, Corollary  1.6, page 54] one can read a similar condition, where the domain of the mapping is required to be bounded.

If is asymptotically contractive in the sense due to Penot, then it is easy to see that

(4.16)

which implies condition (4.11) of Corollary 4.8 for , and therefore Penot's fixed point theorem is a consequence of Corollary 4.8.

Example 4.10.

Next, we are concerned with the solvability of the following Hammerstein's integral equation:

(4.17)

in . Here , is a bounded domain of such that its Lebesgue's measure , and . Suppose that and satisfy the following conditions:

(1) is a Carathéodory function,

(2), where and ,

(3),

(4)the function is strongly measurable and whenever ,

(5)there exists a function , belonging to such that for all ,

(6) and .

Proposition 4.11.

Assume that conditions (1)–(6) are satisfied, then problem (4.17) has at least one solution in .

Proof.

First notice that (4.17) may be written in the form where is given by

(4.18)

Let us see that satisfies the conditions of Corollary 4.8. In this sense, we are going to prove that is a nonexpansive mapping. Indeed,

(4.19)

Since , by Holder's inequality, we obtain that

(4.20)

Finally, we are going to show that there exists such that if , then . Indeed, we know that

(4.21)

hence,

(4.22)

Applying again Holder's inequality, we derive that

(4.23)

Moreover, it is clear that

(4.24)

therefore there exists such that if , then as we claimed.

Notice that if then, Corollary in [4] does not apply because under this condition we cannot guarantee that is asymptotically contractive on .

Let be a closed convex subset of a Banach space . A family of mappings is called a one-parametric strongly continuous semigroup of nonexpansive mappings (nonexpansive semigroup, for short) on if the following assumptions are satisfied:

(1) for all ,

(2)for each , the mapping from into is continuous,

(3)for each , is a nonexpansive mapping.

In the next result we study when a nonexpansive semigroup has a common fixed point.

Theorem 4.12.

Let be a Banach space with the (FPP). Suppose that is a mapping satisfying conditions (g1)–(g4). Let be a closed convex and unbounded subset of . If is a nonexpansive semigroup such that there exist with satisfying that

(4.25)

whenever large enough, then the semigroup has a least one common fixed point.

Proof.

By Theorem of [22] in order to get the conclusion it is enough to show that, given , the mapping defined by

(4.26)

has a fixed point.

By hypotheses we know that there exists such that for every with the inequality

(4.27)

holds. Since satisfies conditions (g1)–(g4), we have

(4.28)

The above inequality means that satisfies the conditions of Corollary 4.3 and therefore has a fixed point, which implies by Theorem of [22] that the semigroup has a common fixed point.

Corollary 4.13.

Let be a Banach space with the (FPP). Let be a closed convex and unbounded subset of . If is a nonexpansive semigroup such that there exist , with satisfying that

(4.29)

whenever large enough, then the semigroup has a least one common fixed point.

Proof.

It is enough to apply the above theorem with (see the proof of Corollary 4.8).

We conclude this section by presenting a corollary of Theorem 4.1 which guarantees the existence of positive eigenvalues.

Corollary 4.14.

Let be a Banach space with the (FPP). Suppose that is a mapping satisfying conditions (g1) and (g2). Let be a closed convex and unbounded subset of with . Let be a continuous pseudocontractive mapping. Assume that the following conditions are satisfied.

(a).

(b)There exists such that for every with the inequality

(4.30)

holds for some .

Then any is an eigenvalue of associated to an eigenvector in .

Proof.

Consider a fixed . Let us see that is a continuous pseudocontractive mapping such that . Indeed, since , is convex, , and , then .

To see that is a pseudocontractive mapping, it is enough to prove that is an accretive mapping:

(4.31)

The above inequality holds since is a pseudocontractive mapping and therefore .

Finally, if with , we have

(4.32)

The above facts show that is under the assumption of Theorem 4.1 and hence there exists such that .