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The structure of fixed-point sets of Lipschitzian type semigroups
Fixed Point Theory and Applications volume 2012, Article number: 163 (2012)
Abstract
The purpose of this paper is to establish some results on the structure of fixed point sets for one-parameter semigroups of nonlinear mappings which are not necessarily Lipschitzian in Banach spaces. Our results improve several known existence and convergence fixed point theorems for semigroups which are not necessarily Lipschitzian.
MSC:47H09, 47H10, 47B20, 54C15.
1 Introduction
Let C be a nonempty subset of a Banach space X and a mapping. We use to denote the set of all fixed points of T. A nonempty closed convex subset D of C is said to satisfy property with respect to mapping T [1] if
(ω) for every ,
where denotes the set of all weak subsequential limits of . Moreover, T is said to satisfy the -fixed point property if T has a fixed point in every nonempty closed convex subset D of C which satisfies property . For a Lipschitzian mapping , we use the symbol to denote the exact Lipschitz constant of S, i.e.,
A mapping is said to be
-
(1)
nonexpansive if ,
-
(2)
asymptotically nonexpansive [2] if for all and ,
-
(3)
uniformly L-Lipschitzian if for all and for some .
In general, the fixed-point set of a nonexpansive mapping need not be convex and can be extremely irregular. Suppose that C is a nonempty closed convex bounded subset of a Banach space X and is a nonexpansive mapping with . Obviously, is a closed set. is convex if X is strictly convex (see [3, 4]).
Nonexpansive retracts have been studied in several contexts (for example, convex geometry [5], extension problems [6], fixed point theory [7], optimal sets [8]). It is well known that if C is a nonempty closed convex bounded subset of a Banach space and if a nonexpansive mapping has a fixed point in every nonempty closed convex subset of C which is invariant under T, then is a nonexpansive retract of C (that is, there exists a nonexpansive mapping such that ) (see [[7], Theorem 2]). The Bruck result was extended by Benavides and Ramirez [1] to the case of asymptotically nonexpansive mappings if the space X was sufficiently regular.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972 and they proved that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping of C has a fixed point. Several authors have studied the existence of fixed points of asymptotically nonexpansive mappings in Banach spaces having rich geometric structure, see [1, 9, 10].
There is a class of mappings which lies strictly between the class of contraction mappings and the class of nonexpansive mappings. The class of pointwise contractions was introduced in Belluce and Kirk [11], and later it was called ‘generalized contractions’ in [12]. Banach’s celebrated contraction principle was extended to this larger class of mappings as follows:
Let C be a nonempty weakly compact convex subset of a Banach space and a pointwise contraction. Then T has a unique fixed point , and converges strongly to for each .
Kirk [13] combined the ideas of pointwise contraction [11] and asymptotic contraction [14] and introduced the concept of an asymptotic pointwise contraction. He announced that an asymptotic pointwise contraction defined on a closed convex bounded subset of a super-reflexive Banach space has a fixed point. Recently, Kirk and Xu [15] gave a simple and elementary proof of the fact that an asymptotic pointwise contraction defined on a weakly compact convex set always has a unique fixed point (with convergence of Picard iterates). They also introduced the concept of pointwise asymptotically nonexpansive mapping and proved that every pointwise asymptotically nonexpansive mapping defined on a closed convex bounded subset of a uniformly convex Banach space has a fixed point.
Every asymptotically nonexpansive mapping is uniformly L-Lipschitzian, and the -fixed point property of uniformly L-Lipschitzian mappings is closely related to the class of nonexpansive and asymptotically nonexpansive mappings. In this connection, a deep result of Casini and Maluta [16] was generalized by Lim and Xu [17] as follows:
Theorem LX (Lim and Xu [[17], Theorem 1])
Let X be a Banach space with a uniform normal structure and let be the normal structure coefficient of X. Let C be a nonempty bounded subset of X and a uniformly L-Lipschitzian mapping with . Then T satisfies the -fixed point property.
The -fixed point property plays a key role in the existence and approximation of solutions of fixed point problems and variational inequality problems, see [17–20].
The mapping theory for accretive mappings is closely related to the fixed point theory of pseudo-contractive mappings. Recently, applications of the semigroup result on the existence of solutions to certain partial differential equations have been explored in Hester and Morales [21]. They proved that the semigroup result directly implies the existence of unique global solutions to time evolution equations of the form , where A is a combination of derivatives. In many applications, semigroups are not necessarily Lipschitzian. It is an interesting problem to extend fixed point existence results, namely Theorem LX, for semigroups of nonlinear mappings which are not necessarily Lipschitzian.
Motivated by the results above, in this paper we establish some results on the structure of fixed point sets for one-parameter semigroups of nonlinear mappings which are not necessarily Lipschitzian in Banach spaces. Our theorems significantly extend Theorem LX to more general Banach spaces and to a more general class of operators. We obtain a general convergence theorem for semigroups of non-Lipschitzian pseudo-contractive mappings. Our results improve several known fixed point problems and variational inequality problems for semigroups which are not necessarily Lipschitzian.
2 Preliminaries
Let denote the set of nonnegative real numbers, and let denote the set of nonnegative integers. Throughout this paper, G denotes an unbounded set of such that for all and for all with (often or ).
2.1 Lipschitzian type mappings
Let C be a nonempty subset of a Banach space X and a mapping. Then T is called
-
(i)
pointwise contractive [11] if there exists a function such that for all ;
-
(ii)
asymptotic pointwise contractive [13] if for each , there exists a function such that for all , where pointwise on C;
-
(iii)
pointwise asymptotically nonexpansive [15] if for each integer , for all , where pointwise;
-
(iv)
asymptotically nonexpansive in the intermediate sense [22] provided T is uniformly continuous and
(2.1) -
(v)
mapping of asymptotically nonexpansive type [23] if
Fix a sequence in with . A mapping is said to be nearly Lipschitzian with respect to [24] if for each , there exists a constant such that
for all . The infimum of constants in (2.2) is called nearly Lipschitz constant and is denoted by . A nearly Lipschitzian mapping T with the sequence is called
-
(i)
nearly contractive if for all ,
-
(ii)
nearly uniformly L-Lipschitzian if for all ,
-
(iii)
nearly uniformly k-contractive if for all ,
-
(iv)
nearly nonexpansive if for all ,
-
(v)
nearly asymptotically nonexpansive if for all with .
The mapping T is said to be demicontinuous if, whenever a sequence in C converges strongly to , then converges weakly to Tx. The mapping T is said to be weakly contractive if
where is a continuous and nondecreasing function such that , for and .
Let C be a convex subset of a Banach space X and D a nonempty subset of C. Then a continuous mapping P from C onto D is called a retraction if for all , i.e., . A retraction P is said to be sunny if for each and with . If the sunny retraction P is also nonexpansive, then D is said to be a sunny nonexpansive retract of C.
In what follows, we shall make use of the following lemmas:
Lemma 2.1 [3]
Let C be a nonempty closed convex subset of a Banach space X and a continuous strongly pseudo-contractive mapping. Then T has a unique fixed point in C.
Lemma 2.2 (Goebel and Reich [[4], Lemma 13.1])
Let C be a convex subset of a smooth Banach space X, D a nonempty subset of C and P a retraction from C onto D. Then the following are equivalent:
-
(a)
P is sunny and nonexpansive.
-
(b)
for all , .
-
(c)
for all .
2.2 Semigroups
Let C be a nonempty subset of a Banach space X. The one-parameter family is said to be a strongly continuous semigroup of mappings from C into itself if
-
(I)
for all ;
-
(II)
for all ;
-
(III)
for each , the mapping from G into C is continuous.
We denote by the set of all common fixed points of ℱ, i.e., . For a Lipschitzian semigroup ℱ, we write
If ℱ satisfies (I)-(III) and
then ℱ is called asymptotically regular on C. If ℱ satisfies (I)-(III) and
then ℱ is called uniformly asymptotically regular on C.
A Lipschitzian semigroup ℱ is called a
-
(i)
uniformly L-Lipschitzian semigroup if ;
-
(ii)
nonexpansive semigroup if for all ;
-
(iii)
asymptotically nonexpansive semigroup if for all and .
2.3 Asymptotic center
Throughout the paper, is a Banach space which is assumed not to be Schur. That is, X has weakly convergent sequences that are not norm convergent. Let C be a nonempty closed convex subset of a Banach space X and a bounded set in X. Consider the functional defined by
The infimum of over C is said to be the asymptotic radius of with respect to C and is denoted by . A point is said to be an asymptotic center of with respect to C if
The set of all asymptotic centers of with respect to C is denoted by . A number is called an asymptotic diameter of . It is well known that if X is reflexive, then is nonempty closed convex and bounded, and if X is uniformly convex, then consists only of a single point, , i.e., is the unique point which minimizes the functional
over .
2.4 Normal structure
Normal structure plays a key role in some problems of metric fixed point theory. Let C be a nonempty bounded subset of a Banach space X. We denote by
the diameter of C. Put
This nonnegative real number is called the Chebyshev radius of C relative to itself. The normal structure coefficient of a Banach space X is defined [25] by
The space X is said to have the uniformly normal structure if . It is well known that, for every uniformly convex Banach space X, . A weakly convergent sequence coefficient of X is defined (see [25]) by
It is proved in [[26], Theorem 1] that
where . It is readily seen that
The space X is said to have the weak uniformly normal structure if . If X is a reflexive Banach space with modulus of convexity , then
Thus, if X is a uniformly convex Banach space, then and also the equation
has a unique solution . A general formula for in an arbitrary Banach space is not known. In particular, it has been calculated that for a Hilbert space H,
for (),
and for ,
Remark 2.3 is an example of a reflexive Banach space such that and are different. Indeed,
A Banach space X is said to satisfy the Opial condition, if whenever a sequence in X converges weakly to x, then
The Opial modulus of X is defined by
where and the infimum is taken over all with and all sequences in X such that and . For any Banach space X, we have the following inequality:
3 Nonemptiness of common fixed-point sets
First, we introduce some wider classes of semigroups.
Definition 3.1 Let C be a nonempty subset of a normed space X and a strongly continuous semigroup of mappings from C into itself. The semigroup ℱ is said to be nearly Lipschitzian if there exist a function with and a function such that
For a nearly Lipschitzian semigroup ℱ, we write
We say ℱ is
-
(a)
pointwise nearly Lipschitzian if for each , there exist a function with and a function such that
-
(b)
pointwise nearly uniformly -Lipschitzian if there exist a function with and a function such that
-
(c)
asymptotic pointwise nearly Lipschitzian if for each , there exist a function with and two functions with pointwise such that
We say that an asymptotic pointwise nearly Lipschitzian semigroup ℱ is pointwise nearly asymptotically nonexpansive (pointwise asymptotically nonexpansive) if for all and pointwise ( and for all and pointwise). Further, we say that an asymptotic pointwise nearly Lipschitzian semigroup ℱ is asymptotic pointwise nearly contractive if pointwise and for all . The semigroup ℱ is said to be nearly uniformly L-Lipschitzian if there exist a constant and a function with such that
The nearly uniformly L-Lipschitzian semigroup will be called nearly nonexpansive semigroup.
Before presenting the main result of this section, we give another definition:
Definition 3.2 Let C be a nonempty weakly compact convex subset of Banach space X, a strongly continuous semigroup of mappings from C into itself. A nonempty closed convex subset D of C is said to satisfy property with respect to semigroup ℱ if
(ω) for every ,
where denotes the set of all weak limits of as .
The semigroup ℱ is said to satisfy the -fixed point property if ℱ has a common fixed point in every nonempty closed convex subset D of C which satisfies property .
We now establish that a semigroup ℱ of a certain class of Lipschitzian type mappings satisfies the -fixed point property.
Theorem 3.3 Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X and a strongly continuous semigroup of demicontinuous mappings from C into itself. Suppose that for each , there exist a function with and two functions with pointwise and such that
Also suppose that there exists a nonempty closed convex subset M of C which satisfies property (ω) with respect to ℱ. Then:
-
(a)
For arbitrary , there exist a sequence in G with and an iterative sequence in M defined by
(3.1) -
(b)
If ℱ is asymptotically regular on C, then there exists an element such that converges strongly to .
Proof (a) Since one can easily construct a nonempty closed convex separable subset of C which is invariant under each (i.e., for ), we may assume that C itself is separable.
The separability of makes it possible to select a sequence of such that
and
For any , consider a sequence in C. Suppose . Using property (ω), we obtain that . Now we can construct a sequence in M in the following way:
-
(b)
The weak asymptotic regularity of ℱ ensures that , . We now show that converges strongly to a common fixed point of ℱ. Set
and
for all . By the property of , we have
By the asymptotic regularity of ℱ and the w-l.s.c. of the norm , we have
On the other hand, by the asymptotic regularity of ℱ, we have
Set . From (3.2), we obtain
For any , one can see that
so it follows that is a Cauchy sequence in M. Let . Observe that
Taking the limit superior as on both sides, we get
Hence . Let . Note , so it follows from the demicontinuity of that . Observe that . By the uniqueness of the weak limit of , we have . Therefore, . □
Theorem 3.3 generalizes the result due to Górnicki [27] in the context of the (ω)-fixed point property for a wider class of mappings. Theorem 3.3 also extends corresponding results of Sahu, Agarwal and O’Regan [18], Sahu, Liu and Kang [28] and Sahu, Petruşel and Yao [29] for asymptotic pointwise nearly Lipschitzian semigroups. As , and there are Banach spaces for which while , the following result is an improvement on Casini and Maluta [16] and Lim and Xu [[17], Theorem 1].
Corollary 3.4 Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X and a strongly continuous semigroup of demicontinuous nearly uniformly L-Lipschitzian of mappings from C into itself. Suppose that and that there exists a nonempty closed convex subset M of C which satisfies property (ω) with respect to ℱ. Then:
-
(a)
For arbitrary , there exist a sequence in G with and an iterative sequence in M defined by
-
(b)
If ℱ is asymptotically regular on C, then there exists an element such that converges strongly to .
Corollary 3.5 Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X and a strongly continuous semigroup of demicontinuous nearly uniformly L-Lipschitzian asymptotically regular mappings from C into itself such that . Then ℱ has a common fixed point in C.
4 Common fixed-point sets as Lipschitzian retracts
Theorem 4.1 Let X be a uniformly Banach space with the Opial condition, C a nonempty closed convex bounded subset of X and a strongly continuous semigroup of demicontinuous mappings from C into itself. Suppose that there exists a function with such that
where with . Also suppose that ℱ is asymptotically regular on C. Then and is -Lipschitzian retract of C.
Proof Using similar arguments as in the proof of Theorem 3.3(a), we may select a sequence of such that and
Let denote a mapping which associates with a given a unique , that is, . Since for all , it follows from the lower weak semi-continuity of the norm that
i.e., A is -Lipschitzian mapping. It follows that A is uniformly continuous.
For any , consider a sequence in C. Suppose . Now we can construct a sequence in C in the following way:
From (4.1), we have
Set , , and for all . From (3.3), we have
for and . It follows that
Thus, the sequence converges uniformly to a function Q defined by
For , we have
Taking the limit superior as , we get
Hence . Let . From the demicontinuity of , we obtain that . One can see that for all . Thus, for all and . Therefore, Q is a retraction of C onto . □
Corollary 4.2 Let X be a uniformly Banach space with the Opial condition, C a nonempty closed convex bounded subset of X and a demicontinuous asymptotically regular nearly Lipschitzian mapping such that . Then and is a -Lipschitzian retract of C.
One sees from Theorem 4.1 that if , then is a nonexpansive retract of C. In the next section, we show that is a sunny nonexpansive retract of C when a strongly continuous semigroup of asymptotically pseudo-contractive mappings (see Theorem 5.6).
5 Common fixed-point sets as sunny nonexpansive retracts
Let C be a nonempty subset of a Banach space X and a semigroup of mappings from C into itself. A sequence in C is said to an approximating fixed point sequence of ℱ if for all . The family is demiclosed at zero if is a sequence in C weakly converging to and for all imply for all . Following [18], we say that ℱ has property () if for every bounded set in C, we have
In [30], Schu introduced the concept of asymptotically pseudo-contractive mapping as follows:
Let H be a real Hilbert space whose inner product and norm are denoted by and respectively. Let C be a nonempty subset of H and a mapping. Then T is called an asymptotically pseudo-contractive mapping if there exists a sequence in with such that
The class of asymptotically pseudo-contractive mappings contain properly the class of asymptotically nonexpansive mappings. The following example shows that a continuous asymptotically pseudo-contractive mapping is not necessarily asymptotically nonexpansive.
Example 5.1 Let and . Define by
Note that T is a pseudo-contractive mapping which is not Lipschitzian (see [31]). Since T is not Lipschitzian, it is not asymptotically nonexpansive. It is shown in [30] that T is an asymptotically pseudo-contractive mapping with sequence .
Let be a semigroup of mappings from C into itself. Then ℱ is said to be pseudo-contractive if
Remark 5.2
-
(i)
The semigroup ℱ is pseudo-contractive if and only if the following holds:
-
(ii)
Every nonexpansive semigroup must be a continuously pseudo-contractive semigroup.
We say ℱ is asymptotically pseudo-contractive if there exists a function with such that
Example 5.3 Let , , and . For , define by
and define
Set and . Note that
and
For and , we have , and hence
Thus,
Therefore, is an asymptotically pseudo-contractive semigroup with function . Moreover, for each , is discontinuous at and hence ℱ is not a Lipschitzian semigroup.
We begin with the following:
Theorem 5.4 (Demiclosedness Principle)
Let C be a nonempty closed convex bounded subset of a real Hilbert space H. Let be a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Then the family is demiclosed at zero.
Proof Assume that is a sequence in C weakly converging to z and for all . Let with and let be a sequence in G defined by for all . Notice that . Fix and define
Since T is uniformly continuous, we have as for fixed .
Indeed, for fixed , we have
for all . Since ℱ is a uniformly continuous semigroup, it follows that for each fixed . Noticing that ℱ is an asymptotically pseudo-contractive semigroup, for fixed , we have
Since and , it follows from (5.1) that
Note that
which implies that
Letting in (5.2), we obtain that . It follows from the continuity of that
Therefore, for all . □
The following result extends the celebrated convergence theorem of Browder [32] and many results concerning Browder’s convergence theorem to a semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings.
Theorem 5.5 Let C be a nonempty closed convex bounded subset of a real Hilbert space H and a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Let be a sequence in and a sequence in such that for all , and . Then:
-
(a)
There exists a sequence in C defined by
(5.3) -
(b)
If ℱ has property (), then and converges strongly to such that
(5.4)
Proof (a) Let be a strongly continuous semigroup of asymptotically pseudo-contractive mappings with a net . Set . Note for all , it follows that and hence for all . Then, for each , the mapping defined by
is continuous and strongly pseudo-contractive. Indeed, for x, y in C, we have
Therefore, by Lemma 2.1, there exists a sequence in C described by (5.3).
-
(b)
Assume that ℱ has property (). From (5.3), we have as . The property () of ℱ gives that as for all . Since is bounded, we can assume that a subsequence of such that for some . By Theorem 5.4, we have .
For , we have
From (5.3), we have
so it follows that
Since and C is bounded, it follows from (5.5) that
We claim that the set is sequentially compact. For , we have
which implies that
By the weak compactness of C, there exists a weakly convergent subsequence . Suppose that as . Since is an approximating fixed point sequence of ℱ, we infer from Theorem 5.4 that . In (5.7), interchange v and to obtain that
Since , we get that . Hence the set is sequentially compact.
Next, we show that . Suppose, for contradiction, that is another subsequence of such that . It is easy to see that . Observe that
Since , we get
From (5.6), we obtain
Similarly, we have
Adding inequalities (5.8) and (5.9) yields
a contradiction. In a similar way it can be shown that each cluster point of the sequence is equal to . Therefore, the entire sequence converges strongly to . It is easy to see, from (5.6), that the inequality (5.4) holds. □
Theorem 5.6 Let C be a nonempty closed convex bounded subset of a real Hilbert space H and a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Suppose that ℱ has property (). Then and is a sunny nonexpansive retract of C.
Proof Assume that is a semigroup of asymptotically pseudo-contractive mappings from C into itself with a function with . Without loss of generality, we may assume that in and in such that for all , and . Then, for an arbitrarily fixed element , there exists a sequence in C defined by (5.3). By Theorem 5.5(b), .
By Theorem 5.5(b), converges strongly to an element such that the inequality (5.4) holds. Define a mapping by
In view of (5.4), we have
Therefore, by Lemma 2.2, we conclude that Q is sunny nonexpansive. □
6 Application
Let C be a nonempty convex subset of a real Hilbert space H and D a nonempty subset of C. For a nonlinear mapping , the variational inequality problem over D is to find a point such that
It is important to note that the theory of variational inequalities has played an important role in the study of many diverse disciplines, for example, partial differential equations, optimal control, optimization, mathematical programming, mechanics, finance, etc.; see, for example, [33, 34] and references therein.
We now turn our attention to dealing with the problem of the existence of solutions of by sunny nonexpansive retractions.
Following Wong, Sahu and Yao [[35], Proposition 4.6], one can show that the variational inequality problem with is equivalent to the fixed point problem. Indeed,
Proposition 6.1 Let C be a nonempty convex subset of a smooth Banach space X and a strongly continuous semigroup of mappings from C into itself with . Let be a mapping with and let Q be the sunny nonexpansive retraction from C onto . Then is a solution of variational inequality problem over if and only if is a fixed point of Qf.
The following result improves the so-called viscosity approximation method which was first introduced by Moudafi [36] from nonexpansive mappings to a semigroup of pseudo-contractive mappings.
Theorem 6.2 Let C be a nonempty closed convex bounded subset of a real Hilbert space H, a weakly contractive mapping with function ψ and a strongly continuous semigroup of uniformly continuous pseudo-contractive mappings from C into itself. Suppose that ℱ has property () and ℱ is nearly nonexpansive with function . Let be a sequence in and a sequence in such that and . Then, we have the following:
-
(a)
The variational inequality problem over has a unique solution in .
-
(b)
There exists a sequence in C defined by
(6.1)
such that converges strongly to the unique solution of the variational inequality problem .
Proof (a) By Theorem 5.6, there is a sunny nonexpansive retraction Q from C onto . Since Qf is a weakly contractive mapping from C into itself, it follows from Rhoades [[39], Theorem 1] that there exists a unique element such that . Note is an element of . It follows from Proposition 6.1 that is the unique solution of the variational inequality problem over .
-
(b)
For each , the mapping defined by
is continuous and strongly pseudo-contractive. In fact, for all and , we have
Hence each is continuous -strongly pseudo-contractive. Therefore, by [37, 38], there exists a sequence in C described by (6.1). As in Theorem 5.5(a), we may define a sequence in C by
By Theorem 5.5, we have that . Observe that
It follows that
Thus, . Therefore, . □
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DRS designed of the study, performed the nonlinear analysis and also wrote the article. DO participated in the design of the study, carried out the materials and helped to check the manuscript. RA conceived of the study, participated in its design and also helped to draft the manuscript. All authors read and approved the final manuscript.
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Sahu, D., Agarwal, R. & O’Regan, D. The structure of fixed-point sets of Lipschitzian type semigroups. Fixed Point Theory Appl 2012, 163 (2012). https://doi.org/10.1186/1687-1812-2012-163
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DOI: https://doi.org/10.1186/1687-1812-2012-163