Fixed point of asymptotic pointwise nonexpansive semigroups in metric spaces
© Al-Mezel and Khamsi; licensee Springer. 2013
Received: 15 February 2013
Accepted: 18 June 2013
Published: 28 August 2013
Let C be a bounded, closed, convex subset of a uniformly convex metric space . In this paper, we introduce the concept of asymptotic pointwise nonexpansive semigroups of nonlinear mappings , i.e., a family such that , , and , where for every . Then we investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups. The proof is based on the concept of types extended to one parameter family of points.
MSC:47H09, 46B20, 47H10, 47E10.
The purpose of this paper is to prove the existence of common fixed points for semigroups of nonlinear mappings acting in metric spaces. Recently, Khamsi and Kozlowski presented a series of fixed point results for pointwise contractions, asymptotic pointwise contractions, pointwise nonexpansive and asymptotic pointwise nonexpansive mappings acting in modular functions spaces [1, 2].
Let us recall that a family of mappings forms a semigroup if , and . Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the vector function space would define the state space, and the mapping would represent the evolution function of a dynamical system. The question about the existence of common fixed points, and about the structure of the set of common fixed points, can be interpreted as a question whether there exist points that are fixed during the state space transformation at any given point of time t, and if yes - what does the structure of a set of such points may look like. In the setting of this paper, the state space is a nonlinear metric space.
The existence of common fixed points for families of contractions and nonexpansive mappings in Banach spaces has been the subject of the intensive research since the early 1960s, as investigated by Belluce and Kirk [3, 4], Browder , Bruck , DeMarr , and Lim . The asymptotic approach for finding common fixed points of semigroups of Lipschitzian (but not pointwise Lipschitzian) mappings has also been investigated, see, e.g., Tan and Xu . It is worthwhile mentioning the recent studies on the special case, when the parameter set for the semigroup is equal to , and , the n th iterate of an asymptotic pointwise nonexpansive mapping. Kirk and Xu  proved the existence of fixed points for asymptotic pointwise contractions and asymptotic pointwise nonexpansive mappings in Banach spaces, while Hussain and Khamsi  extended this result to metric spaces, and Khamsi and Kozlowski to modular function spaces [1, 2]. In the context of modular function spaces, Khamsi discussed in  the existence of nonlinear semigroups in Musielak-Orlicz spaces and considered some applications to differential equations.
2 Uniform convexity in metric spaces
for all p, x, y in M, then M is said to be a hyperbolic metric space (see ).
Obviously, normed linear spaces are hyperbolic spaces. As nonlinear examples, one can consider the Hadamard manifolds , the Hilbert open unit ball equipped with the hyperbolic metric , and the spaces [17–19] (see Example 2.1). We will say that a subset C of a hyperbolic metric space M is convex if , whenever x, y are in C.
The definition of uniform convexity finds its origin in Banach spaces . To the best of our knowledge, the first attempt to generalize this concept to metric spaces was made in . The reader may also consult [14, 16, 22].
Most of the results in this section may be found in .
Let us observe that , and is an increasing function of ε for every fixed r.
- (ii)For there holds
- (iii)If is uniformly convex, then is strictly convex, i.e., whenever
for any , then we must have .
Let be such that . Then is a Cauchy sequence.
Recall that a hyperbolic metric space is said to have the property (R) if any nonincreasing sequence of nonempty, convex, bounded and closed sets, has a nonempty intersection .
Our next result deals with the existence and the uniqueness of the best approximants of convex, closed and bounded sets in a uniformly convex metric space. This result is of interest by itself as uniform convexity implies the property (R), which reduces to reflexivity in the linear case.
The following result gives the analogue of the well known theorem that states any uniformly convex Banach space is reflexive (see Theorem 2.1 in ).
If is complete and uniformly convex, then has the property (R).
Note that any hyperbolic metric space M, which satisfies the property (R), is complete. The following technical lemma will be needed.
Example 2.1 Let be a metric space. A geodesic from x to y in X is a mapping c from a closed interval to X such that , , and for all . In particular, c is an isometry and . The image α of c is called a geodesic (or metric) segment joining x and y. The space is said to be a geodesic space if every two points of X are joined by a geodesic and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each , which we will be denoted by , and called the segment joining x to y.
A geodesic triangle in a geodesic metric space , consisting of three points in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for geodesic triangle in is a triangle in such that for . Such a triangle always exists (see ).
A geodesic metric space is said to be a space if all geodesic triangles of appropriate size satisfy the following comparison axiom:
One may also find the modulus of uniform convexity via similar triangles.
Assume that is complete and uniformly convex. Let C be any a nonempty, closed, convex and bounded subset of M. Let τ be a type defined on C. Then any minimizing sequence of τ is convergent. Its limit is independent of the minimizing sequence.
3 Asymptotic Pointwise Nonexpansive Semigroups
for any . If for any , then T is called asymptotic pointwise nonexpansive. A point is called a fixed point of T if . The set of fixed points of T will be denoted by .
This definition is now extended to a one parameter family of mappings.
for and ;
- (iii)for each , is an asymptotic pointwise nonexpansive mapping, i.e., there exists a function such that(3.1)
for each , the mapping is strong continuous.
Therefore, we will throughout this work assume that , for any and , and .
The concept of type functionals is a powerful technical, tool which is used in the proofs of many fixed point results. The definition of a type is based on a given sequence. In this work, we generalize this definition to a one-parameter family of mappings.
A typical method of proof for the fixed point theorems in Banach and metric spaces is to construct a fixed point by finding an element, on which a specific type function attains its minimum. To be able to proceed with this method, one has to know that such an element indeed exists.
The next lemma is the generalization of the minimizing sequence property for types defined by sequences in Lemma 4.3 in  to the one-parameter case in modular function spaces.
If , then .
Moreover any minimizing sequence of τ is convergent. Moreover the limit of is independent of the minimizing sequence.
Contradiction. Therefore, we must have .
for any . If we let , we get . This contradiction implies that , which completes the proof. □
4 Main result
Using the Lemma 3.1, we are ready to prove the main fixed point result for asymptotic pointwise nonexpansive semigroup in metric spaces.
Theorem 4.1 Let be a uniformly convex metric space. Let C be a closed bounded convex nonempty subset of M. Let be an asymptotically pointwise nonexpansive semigroup on C. Then ℱ has a common fixed point and the set of common fixed points is closed and convex.
and letting , we get for any , i.e., . □
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