- Research Article
- Open Access
The structure of fixed-point sets of Lipschitzian type semigroups
© Sahu et al.; licensee Springer 2012
Received: 15 March 2012
Accepted: 22 August 2012
Published: 25 September 2012
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.
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  if
(ω) for every ,
nonexpansive if ,
asymptotically nonexpansive  if for all and ,
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 , extension problems , fixed point theory , optimal sets ). 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 [, Theorem 2]). The Bruck result was extended by Benavides and Ramirez  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  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 , and later it was called ‘generalized contractions’ in . 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  combined the ideas of pointwise contraction  and asymptotic contraction  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  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  was generalized by Lim and Xu  as follows:
Theorem LX (Lim and Xu [, 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 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 . 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.
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
pointwise contractive  if there exists a function such that for all ;
asymptotic pointwise contractive  if for each , there exists a function such that for all , where pointwise on C;
pointwise asymptotically nonexpansive  if for each integer , for all , where pointwise;
- (iv)asymptotically nonexpansive in the intermediate sense  provided T is uniformly continuous and(2.1)
- (v)mapping of asymptotically nonexpansive type  if
nearly contractive if for all ,
nearly uniformly L-Lipschitzian if for all ,
nearly uniformly k-contractive if for all ,
nearly nonexpansive if for all ,
nearly asymptotically nonexpansive if for all with .
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 
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 [, Lemma 13.1])
P is sunny and nonexpansive.
for all , .
for all .
for all ;
for all ;
for each , the mapping from G into C is continuous.
then ℱ is called uniformly asymptotically regular on C.
uniformly L-Lipschitzian semigroup if ;
nonexpansive semigroup if for all ;
asymptotically nonexpansive semigroup if for all and .
2.3 Asymptotic center
2.4 Normal structure
3 Nonemptiness of common fixed-point sets
First, we introduce some wider classes of semigroups.
- (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
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.
- (a)For arbitrary , there exist a sequence in G with and an iterative sequence in M defined by(3.1)
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.
- (b)The weak asymptotic regularity of ℱ ensures that , . We now show that converges strongly to a common fixed point of ℱ. Set
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  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 , Sahu, Liu and Kang  and Sahu, Petruşel and Yao  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  and Lim and Xu [, Theorem 1].
- (a)For arbitrary , there exist a sequence in G with and an iterative sequence in M defined by
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
where with . Also suppose that ℱ is asymptotically regular on C. Then and is -Lipschitzian retract of C.
i.e., A is -Lipschitzian mapping. It follows that A is uniformly continuous.
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
In , Schu introduced the concept of asymptotically pseudo-contractive mapping as follows:
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.
Note that T is a pseudo-contractive mapping which is not Lipschitzian (see ). Since T is not Lipschitzian, it is not asymptotically nonexpansive. It is shown in  that T is an asymptotically pseudo-contractive mapping with sequence .
- (i)The semigroup ℱ is pseudo-contractive if and only if the following holds:
Every nonexpansive semigroup must be a continuously pseudo-contractive semigroup.
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.
Since T is uniformly continuous, we have as for fixed .
Therefore, for all . □
The following result extends the celebrated convergence theorem of Browder  and many results concerning Browder’s convergence theorem to a semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings.
- (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)
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 .
Since , we get that . Hence the set is sequentially compact.
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), .
Therefore, by Lemma 2.2, we conclude that Q is sunny nonexpansive. □
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 [, 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  from nonexpansive mappings to a semigroup of pseudo-contractive mappings.
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 .
- (b)For each , the mapping defined by
Thus, . Therefore, . □
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