Fixed point results for generalized mappings
© Golkarmanesh et al.; licensee Springer. 2014
Received: 28 April 2014
Accepted: 8 October 2014
Published: 22 October 2014
In this paper, first we establish fixed point results for weak asymptotic pointwise contraction type mappings in metric spaces. Then we study the existence of fixed points for weak asymptotic pointwise nonexpansive type mappings in spaces. Our results improve and extend some corresponding known results in the literature.
MSC:47H09, 47H10, 54H25.
The notion of asymptotic pointwise contraction was introduced by Kirk  as follows.
where pointwise on M.
Moreover, Kirk and Xu  proved that if C be a weakly compact convex subset of a Banach space E and an asymptotic pointwise contraction, then T has a unique fixed point and for each the sequence of Picard iterates converges in norm to v.
Rakotch  proved that if M be a complete metric space and satisfies , for all , where is monotonically decreasing, then f has a unique fixed point z and converges to z, for each .
Boyd and Wong  proved that if M be a complete metric space and satisfies , for all , where is upper semicontinuous from the right and satisfies for , then f has a unique fixed point z and converges to z, for each .
Using the diameter of an orbit, Walter  obtained a result that may be stated as follows:
where , then T has a unique fixed point . Moreover, converges to , for each .
In , the authors proved coincidence results for asymptotic pointwise nonexpansive mappings. Kirk , proved that if M be a complete metric space and satisfies , for all , where , uniformly on the range of d and ϕ is continuous with for all , then T has a unique fixed point z and converges to z, for each .
Very recently, Saeidi  introduced the concept of (weak) asymptotic pointwise contraction type mappings.
It is easy to see that the class of asymptotic pointwise contraction type mappings contains the class of an asymptotic pointwise contraction mappings, but the converse is not true . Furthermore, if C is a nonempty weakly compact subset of a Banach space E and a mapping of weak asymptotic pointwise contraction type, then T has a unique fixed point and, for each , the sequence of Picard iterates converges in norm to v (see ). Golkarmanesh and Saeidi  obtained some results for this mappings in modular spaces.
In this paper, motivated by [1, 9–11], we combine the above results and obtain fixed point results for classes of mappings that extend the notions of asymptotic contraction and asymptotic pointwise contraction introduced by Kirk  and Saeidi .
Let M be a metric space and ℱ a family of subsets of M. We say that ℱ defines a convexity structure of M if it contains the closed ball and is stable by intersection. For instance , the class of the admissable subsets of M, defines a convexity structure on any metric space M. Recall that a subset of M is admissable if it is a nonempty intersection of closed balls.
At this point we introduce some notation which will be used throughout the remainder of this work. For a subset A of a metric space M, set:
is called the diameter of A, is called the Chebyshev radius of A, is called the Chebyshev center of A and is called the cover of A.
Definition 1 ()
We will say that ℱ is compact if any family of elements of ℱ, has a nonempty intersection provided for any finite subset .
We will say that ℱ is normal if for any , not reduced to one point, we have .
We will say that ℱ is uniformly normal if there exists such that, for any , not reduced to one point, we have . It is easy to check that .
where is a bounded sequence in M. Types are very useful in the study of the geometry of Banach spaces and the existence of fixed point of mappings. We will say that a convexity structure ℱ on M is T-stable if types are ℱ-convex. We have the following lemma.
Lemma 1 ()
Theorem 1 ()
Let M be a bounded metric space. Assume that there exists a convexity structure ℱ which is compact and T-stable. be an asymptotic pointwise contraction. Then T has a unique fixed point . Moreover, the orbit converges to for each .
Theorem 2 ()
where for each and the sequence converges pointwise to a function . Then T has a unique fixed point . Moreover, the orbit converges to , for each .
3 Fixed point results for asymptotic pointwise contractive type mappings in metric spaces
Theorem 3 Let be a bounded metric space. Assume that there exists a convexity structure ℱ which is compact and T-stable. Let be a weak asymptotic pointwise contraction type mapping. Then T has a unique fixed point . Moreover, the orbit converges to for each .
Since the choice of is arbitrary, we get .
Taking lim inf in the above inequality, it follows that . Since , we immediately get . □
In the following, we present an example in a bounded metric space which shows that a mapping of asymptotic pointwise contraction type is not necessary an asymptotic pointwise contraction.
where is some discontinuous function with . We deduce that T is discontinuous, and then it would not be an asymptotic pointwise contraction. But we see that for all , and so T is of asymptotic pointwise contraction type.
where for each and the sequence converges pointwise to a function . Then T has a unique fixed point z. Moreover, the orbit converges to z for each .
Since is arbitrary, we get .
Since , we immediately get . □
4 Fixed point results for weak asymptotic pointwise nonexpansive type mappings in metric spaces
In this section we introduce weak asymptotic pointwise nonexpansive type mappings in metric spaces and we extend the results found of .
A metric space is said to be a length space if each two points of M are joined by a rectifiable path (that is, a path of finite length) and the distance between any two points is taken to be the infimum of the lengh of all rectifiable paths joining them. In this case, d is said to be a length metric (otherwise, known an inner metric or intrinsic metric). In the case that there is no rectifiable path joins two points of the space, the distance between them is said to be ∞.
A geodesic path joining to (or, more briefly, a geodesic from x to y) is a map c from a closed interval to M 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. is said to be a geodesic space if every two points of are joined by a geodesic. M is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each , which we will denote by , called the segment joining x to y.
A geodesic metric space in a geodesic metric space consists of three points in M (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle in is a triangle in such that for . If it is further assumed that perimeter of is less than , where denotes the diameter of . Such a triangle always exists.
A geodesic metric space is said to be a space if all geodesic triangles of appropriate size satisfy the following comparison axiom.
Complete spaces are often called Hadamard spaces. These spaces are of particular relevance to this study.
for any .
Theorem 5 Let M be a complete metric space. Let C be a bounded closed nonempty convex subset of M. If is a weak asymptotic pointwise nonexpansive type, then the fixed point set is closed and convex.
for any . Since T is of weak asymptotic pointwise nonexpansive type, we get , which implies that , i.e., . □
where C is a closed convex subset which contains the bounded sequence and .
Theorem 6 Let M be a complete metric space. Let C be a bounded closed nonempty convex subset of M. Let be a weak asymptotic pointwise nonexpansive type. Let be an approximate fixed point sequence, i.e., and . Then we have .
Letting , we will get . The rest of the proof is similar to the one used for Theorem 3. □
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks the DSR for technical and financial support.
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