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Fixed point results for generalized mappings
Fixed Point Theory and Applications volume 2014, Article number: 217 (2014)
Abstract
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 CAT(0) spaces. Our results improve and extend some corresponding known results in the literature.
MSC:47H09, 47H10, 54H25.
1 Introduction
The notion of asymptotic pointwise contraction was introduced by Kirk [1] as follows.
Let (M,d) be a metric space. A mapping T:M\to M is called an asymptotic pointwise contraction if there exists a function \alpha :M\to [0,1) such that, for each integer n\ge 1,
where {\alpha}_{n}\to \alpha pointwise on M.
Moreover, Kirk and Xu [2] proved that if C be a weakly compact convex subset of a Banach space E and T:C\to C an asymptotic pointwise contraction, then T has a unique fixed point v\in C and for each x\in C the sequence of Picard iterates \{{T}^{n}x\} converges in norm to v.
Rakotch [3] proved that if M be a complete metric space and f:M\to M satisfies d(f(x),f(y))\le \alpha (d(x,y))d(x,y), for all x,y\in M, where \alpha :[0,\mathrm{\infty})\to [0,1) is monotonically decreasing, then f has a unique fixed point z and \{{f}^{n}(x)\} converges to z, for each x\in M.
Boyd and Wong [4] proved that if M be a complete metric space and f:M\to M satisfies d(f(x),f(y))\le \psi (d(x,y))d(x,y), for all x,y\in M, where \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is upper semicontinuous from the right and satisfies 0\le \psi (t)<t for t>0, then f has a unique fixed point z and \{{f}^{n}(x)\} converges to z, for each x\in M.
Using the diameter of an orbit, Walter [5] obtained a result that may be stated as follows:
Let (M,d) be a complete metric space and let T:M\to M be a mapping with bounded orbits. If there exists a continuous, increasing function \phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+} for which \phi (r)<r for every r>0 and
where {O}_{T}(x,y)=\{{T}^{n}x\}\cup \{{T}^{n}y\}, then T has a unique fixed point {x}_{0}. Moreover, \{{T}^{n}x\} converges to {x}_{0}, for each x\in M.
In [6], the authors proved coincidence results for asymptotic pointwise nonexpansive mappings. Kirk [7], proved that if M be a complete metric space and T:M\to M satisfies d({T}^{n}x,{T}^{n}y)\le {\varphi}_{n}(d(x,y)), for all x,y\in M, where {\varphi}_{n}:[0,\mathrm{\infty})\to [0,\mathrm{\infty}), {\varphi}_{n}\to \varphi uniformly on the range of d and ϕ is continuous with \varphi (s)<s for all s>0, then T has a unique fixed point z and \{{T}^{n}(x)\} converges to z, for each x\in M.
References [3, 4, 6–11] present simple and elegant proofs of fixed point results for pointwise contraction, asymptotic pointwise contraction, and asymptotic nonexpansive mappings.
Very recently, Saeidi [11] introduced the concept of (weak) asymptotic pointwise contraction type mappings.
Let (M,d) be a metric space. A mapping T:M\to M is said to be of asymptotic pointwise contraction type (resp. of weak asymptotic pointwise contraction type) if {T}^{N} is continuous for some integer N\ge 1 and there exists a function \alpha :M\to [0,1) such that, for each x in M,
where {\alpha}_{n}\to \alpha pointwise on M. Taking
it can easily be seen from (1.1) (resp. (1.2)) that
for all x\in M and
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 [11]. Furthermore, if C is a nonempty weakly compact subset of a Banach space E and T:C\to C a mapping of weak asymptotic pointwise contraction type, then T has a unique fixed point v\in C and, for each x\in C, the sequence of Picard iterates \{{T}^{n}x\} converges in norm to v (see [11]). Golkarmanesh and Saeidi [12] 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 [1] and Saeidi [11].
2 Preliminaries
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 \mathcal{A}(M), 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:
{r}_{x}(A)=sup\{d(x,y):y\in A\};
R(A)=inf\{{r}_{x}(A):x\in A\};
diam(A)=sup\{d(x,y):x,y\in A\};
{C}_{A}(A)=\{x\in A:{r}_{x}(A)=R(A)\};
cov(A)=\bigcap \{B:B\text{is a closed ball and}B\supseteq A\}.
diam(A) is called the diameter of A, R(A) is called the Chebyshev radius of A, {C}_{A}(A) is called the Chebyshev center of A and cov(A) is called the cover of A.
Definition 1 ([9])
Let ℱ be a convexity structure on M.

(i)
We will say that ℱ is compact if any family {({A}_{\alpha})}_{\alpha \in \mathrm{\Gamma}} of elements of ℱ, has a nonempty intersection provided {\bigcap}_{\alpha \in F}{A}_{\alpha}\ne \mathrm{\varnothing} for any finite subset F\subset \mathrm{\Gamma}.

(ii)
We will say that ℱ is normal if for any A\in \mathcal{F}, not reduced to one point, we have R(A)<diam(A).

(iii)
We will say that ℱ is uniformly normal if there exists c\in (0,1) such that, for any A\in \mathcal{F}, not reduced to one point, we have R(A)\le c(diam(A)). It is easy to check that c\ge 1/2.
Let M be a metric space and ℱ a convexity structure. We will say that a function \mathrm{\Phi}:M\to M is ℱconvex if \{x:\mathrm{\Phi}(x)\le r\}\in \mathcal{F} for any r\ge 0. Also we define a type to be a function \mathrm{\Phi}:M\to [0,\mathrm{\infty}] such that
where ({x}_{n}) 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 Tstable if types are ℱconvex. We have the following lemma.
Lemma 1 ([9])
Let M be a metric space and ℱ a compact convexity structure on M which is Tstable. Then, for any type Φ, there exists {x}_{0}\in M such that
Hussain and Khamsi [9] and Nicolae [10] proved the following results in metric spaces.
Theorem 1 ([9])
Let M be a bounded metric space. Assume that there exists a convexity structure ℱ which is compact and Tstable. T:M\to M be an asymptotic pointwise contraction. Then T has a unique fixed point {x}_{0}. Moreover, the orbit \{{T}^{n}x\} converges to {x}_{0} for each x\in M.
Theorem 2 ([10])
Let M be a bounded metric space, T:M\to M and suppose that there exists a convexity structure ℱ which is compact and Tstable. Assume that
where {\alpha}_{n}:M\to \mathbb{R} for each n\in \mathbb{N} and the sequence {\{{\alpha}_{n}\}}_{n\in \mathbb{N}} converges pointwise to a function \alpha :M\to [0,1). Then T has a unique fixed point {x}_{0}. Moreover, the orbit \{{T}^{n}x\} converges to {x}_{0}, for each x\in M.
3 Fixed point results for asymptotic pointwise contractive type mappings in metric spaces
In this section, we generalize the results obtained by Hussain and Khamsi [9] and Nicolae [10] for the wider class of weak asymptotic pointwise contraction type mappings.
Theorem 3 Let (M,d) be a bounded metric space. Assume that there exists a convexity structure ℱ which is compact and Tstable. Let T:M\to M be a weak asymptotic pointwise contraction type mapping. Then T has a unique fixed point {x}_{0}. Moreover, the orbit \{{T}^{n}x\} converges to {x}_{0} for each x\in M.
Proof Fix x\in M and define a function f by
Since ℱ is compact and Tstable, there exists {x}_{0}\in M such that
Let us show that f({x}_{0})=0. Indeed, for any m\ge 1 we have
which implies
Now, by (1.3) and (3.1), we obtain
which forces f({x}_{0})=0 as \alpha ({x}_{0})<1. Hence, d({T}^{n}x,{x}_{0})\to 0 as n\to \mathrm{\infty}. From this and the continuity of {T}^{N}, for some N\ge 1, it follows that
namely, {x}_{0} is a fixed point of {T}^{N}. Now, repeating the above proof for {x}_{0} instead of x, we deduce that \{{T}^{n}{x}_{0}\} is convergent to an element of M. But {T}^{kN}{x}_{0}={x}_{0} for all k\ge 1. Hence, {T}^{n}{x}_{0}\to {x}_{0}. We show that T{x}_{0}={x}_{0}. For this purpose, consider an arbitrary \u03f5>0. Then there exists a {k}_{0}>0 such that d({T}^{n}{x}_{0},{x}_{0})<\u03f5 for all n>{k}_{0}. So, choosing a natural number k>{k}_{0}/N, we obtain
Since the choice of \u03f5>0 is arbitrary, we get T{x}_{0}={x}_{0}.
It is easy to verify that T has only one fixed point. Indeed, if a,b\in M are two fixed points of T, then we have
Taking lim inf in the above inequality, it follows that d(a,b)\le \alpha (a)d(a,b). Since \alpha (a)<1, we immediately get a=b. □
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.
Example 1 Let M={\prod}_{i=1}^{n}{I}_{i} ({I}_{i}=[0,1]), equipped with the Euclidean norm. Then M is a bounded metric space. For each ({x}_{1},{x}_{2},\dots ,{x}_{n})\in M, define
where f:[0,1]\to [0,1] is some discontinuous function with f(0)=0. We deduce that T is discontinuous, and then it would not be an asymptotic pointwise contraction. But we see that {T}^{n}x=0 for all x\in M, and so T is of asymptotic pointwise contraction type.
Theorem 4 Let (M,d) be a bounded metric space, T:M\to M and suppose there exists a convexity structure ℱ which is compact and Tstable and {T}^{N} is continuous for some integer N\ge 1. Assume
where {\alpha}_{n}:M\to \mathbb{R} for each n\in N and the sequence \{{\alpha}_{n}\} converges pointwise to a function \alpha :M\to [0,1). Then T has a unique fixed point z. Moreover, the orbit \{{T}^{n}x\} converges to z for each x\in M.
Proof Taking
it can easily be seen that
for all x\in M, and
Fix x\in M and define a function f by
Since ℱ is compact and Tstable, there exists z\in M such that
Let us prove that f(z)=0. Indeed, for any m\ge 1 we have
which implies
By (3.2) we have {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\gamma}_{m}(z)\le 0, thus, for the subsequence \{{\gamma}_{{m}_{k}}(z)\} of \{{\gamma}_{m}(z)\}, we have
Now, by (3.4) and (3.5) we obtain
which forces f(z)=0 as \alpha (z)<1. Hence, d({T}^{n}x,z)\to 0 as n\to \mathrm{\infty}. From this and the continuity of {T}^{N}, for some N\ge 1, it follows that
namely, z is a fixed point of {T}^{N}. Now, repeating the above proof for z instead of x, we deduce that {T}^{n}z is convergent to a member of M. But {T}^{kN}z=z for all k\ge 1. Hence, {T}^{n}z\to z. We show that Tz=z; for this purpose, consider an arbitrary \u03f5>0. Then there exists a {k}_{0}>0 such that d({T}^{n}z,z)<\u03f5 for all n>{k}_{0}. So, by choosing a natural number k>{k}_{0}/N, we obtain
Since \u03f5>0 is arbitrary, we get Tz=z.
Assume that T has two fixed points a,b\in M, then, for each n\in \mathbb{N},
Taking the lim inf in the above inequality, it follows that
Since \alpha (a)<1, we immediately get a=b. □
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 [9].
Definition 2 Let (M,d) be a metric space. A mapping T:M\to M is said to be of asymptotic pointwise nonexpansive type (resp. weak asymptotic pointwise nonexpansive type) if {T}^{N} is continuous for some integer N\ge 1 and there exists a sequence {\alpha}_{n}:M\to [0,+\mathrm{\infty}) such that
where {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(x)\le 1. Taking
it can easily be seen from (4.1) (resp. (4.2)) that
for all x\in M and
A metric space (M,d) 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 x\in M to y\in M (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]\subset \mathbb{R} to M such that c(0)=x, c(l)=y, and d(c(t),c({t}^{\prime}))=t{t}^{\prime} for all t,{t}^{\prime}\in [0,l]. In particular, c is an isometry and d(x,y)=l. The image α of c is called a geodesic (or metric) segment joining x and y. (M,d) is said to be a geodesic space if every two points of (M,d) are joined by a geodesic. M is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,y\in M, which we will denote by [x,y], called the segment joining x to y.
A geodesic metric space \Delta ({x}_{1},{x}_{2},{x}_{3}) in a geodesic metric space (M,d) 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 \Delta ({x}_{1},{x}_{2},{x}_{3}) in (M,d) is a triangle \overline{\Delta}({x}_{1},{x}_{2},{x}_{3}):=\Delta ({\overline{x}}_{1},{\overline{x}}_{2},{\overline{x}}_{3}) in {M}_{k}^{2} such that {d}_{{\mathbb{R}}^{2}}({\overline{x}}_{i},{\overline{x}}_{j})=d({x}_{i},{x}_{j}) for i,j\in \{1,2,3\}. If k>0 it is further assumed that perimeter of \Delta ({x}_{1},{x}_{2},{x}_{3}) is less than 2{D}_{k}, where {D}_{k} denotes the diameter of {M}_{k}^{2}. Such a triangle always exists.
A geodesic metric space is said to be a CAT(k) space if all geodesic triangles of appropriate size satisfy the following CAT(k) comparison axiom.
CAT(k): Let Δ be a geodesic triangle in M and \overline{\Delta}\subset {M}_{k}^{2} be a comparison triangle for Δ. Then Δ is said to satisfy the CAT(k) inequality if for all x,y\in \Delta and all comparison points \overline{x},\overline{y}\in \overline{\Delta},
Complete CAT(0) spaces are often called Hadamard spaces. These spaces are of particular relevance to this study.
Finally, we observe that if x, {y}_{1}, {y}_{2} are points of a CAT(0) space and if {y}_{0} is the midpoint of the segment [{y}_{1},{y}_{2}], which we denote by \frac{{y}_{1}\oplus {y}_{2}}{2}, then the CAT(0) inequality implies
Let M be a complete CAT(0) space and x,y\in M, then for any \alpha \in [0,1] there exists a unique point \alpha x\oplus (1\alpha )y\in [x,y] such that
for any z\in M.
Let M be a complete CAT(0) space. A subset C\subset M is convex if for any x,y\in C we have [x,y]\subset C. Any type function achieves its infimum, i.e., for any bounded sequence \{{x}_{n}\} in a CAT(0) space M, there exists \omega \in M such that f(\omega )=inf\{f(x):x\in M\}, where
Theorem 5 Let M be a complete CAT(0) metric space. Let C be a bounded closed nonempty convex subset of M. If T:C\to C is a weak asymptotic pointwise nonexpansive type, then the fixed point set Fix(T) is closed and convex.
Proof Fix x\in C and define a function f by
Let {x}_{0}\in C be such that f({x}_{0})=inf\{f(x);x\in M\}={f}_{0}. According to the proof of Theorem 3 we have f({T}^{n}{x}_{0})\le {\alpha}_{n}({x}_{0}){f}_{0}+{r}_{n}({x}_{0}), for any n\ge 1. The CAT(0) inequality implies
If we let n go to infinity, we get
which implies
Since T is of weak asymptotic pointwise nonexpansive type, we get
which implies \{{T}^{n}{x}_{0}\} is a Cauchy sequence. Let v={lim}_{n\to \mathrm{\infty}}{T}^{n}{x}_{0}. By the proof of Theorem 3 Tv=v and Fix(T) is closed. In order to prove that Fix(T) is convex, it is enough to prove \frac{x\oplus y}{2}\in Fix(T), whenever x,y\in Fix(T). Let z=\frac{x\oplus y}{2}. The CAT(0) inequality implies
for any n\ge 1. Since
and
we get
for any n\ge 1. Since T is of weak asymptotic pointwise nonexpansive type, we get {lim}_{n\to \mathrm{\infty}}{T}^{n}z=z, which implies that T(z)=z, i.e., z\in Fix(T). □
Before we state the next and final result of this work, we need the following notation:
where C is a closed convex subset which contains the bounded sequence \{{x}_{n}\} and f(x)={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({x}_{n},x).
Theorem 6 Let M be a complete CAT(0) metric space. Let C be a bounded closed nonempty convex subset of M. Let T:C\to C be a weak asymptotic pointwise nonexpansive type. Let \{{x}_{n}\}\in C be an approximate fixed point sequence, i.e., {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0 and \{{x}_{n}\}\rightharpoonup z. Then we have Tz=z.
Proof Since \{{x}_{n}\} is an approximate fixed point sequence, we have
for any m\ge 1. Hence f({T}^{m}x)\le {\alpha}_{m}(x)f(x)+{r}_{m}(x) (see the proof of Theorem 3). In particular, {lim}_{n\to \mathrm{\infty}}f({T}^{m}z)=f(z). The CAT(0) inequality implies
for any m,n\ge 1. If n\to \mathrm{\infty}, we will get
for any m\ge 1. The definition of z implies
for any m\ge 1, or
Letting m\to \mathrm{\infty}, we will get {lim}_{n\to \mathrm{\infty}}d(z,{T}^{m}z)=0. The rest of the proof is similar to the one used for Theorem 3. □
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Acknowledgements
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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Golkarmanesh, F., AlMazrooei, A.E., Parvaneh, V. et al. Fixed point results for generalized mappings. Fixed Point Theory Appl 2014, 217 (2014). https://doi.org/10.1186/168718122014217
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DOI: https://doi.org/10.1186/168718122014217
Keywords
 asymptotic center
 asymptotic pointwise contraction type
 convexity structure
 Tstable
 weak asymptotic pointwise nonexpansive type
 CAT(0) space