BROWDER & G ¨OHDE FIXED POINT THEOREM FOR G-NONEXPANSIVE MAPPINGS
M. R. ALFURAIDAN & S. A. SHUKRI
Abstract. In this paper, we prove the analog to Browder & G¨ohde fixed point theorem for G-nonexpansive mappings in complete hyperbolic metric spaces uniformly convex. In the linear case, this result is refined. Indeed, we prove that ifXis a Banach space uniformly convex in every direction endowed with a graphG, then any G-nonexpansive mapping T : C → C, where C is a nonempty weakly compact convex subset of X, has a fixed point provided that there existsx0∈C such thatT(x0) andx0 areG-connected.
1. Introduction
The theory of fixed points is one of the most influential tools of modern math- ematics. The flourishing field of fixed point theory started in the early days of topology (the work of Poincare, Lefschetz-Hopf, and Leray-Schauder). For exam- ple, the existence problems are usually translated into a fixed point problem like the existence of solutions to elliptic partial differential equations, or the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming.
Following the publication of Ran and Reurings paper [26], there was a huge interest to the new theory of monotone mappings which are Lipschitzian on com- parable elements. Later on, Nieto and Rodr´ıguez-L´opez [25] extended the new fixed point result discovered in [26] and used such extension when trying to prove the existence of periodic solutions once a lower or upper solutions exist. In [17]
Jachymski was the first to recognize the power behind using graphs instead of partial orders (see also the recent papers [1, 2, 3, 4, 5, 6, 7]).
2010Mathematics Subject Classification. Primary 47H09, Secondary 46B20, 47H10, 47E10.
Key words and phrases. Directed graph, Fixed point, uniformly convex space, hyperbolic metric space,G-nonexpansive mapping, Mann iteration.
1
Nonexpansive mappings are those maps which have Lipschitz constant equal to one. They are a natural extension of contractive mappings. However, the fixed point problem for nonexpansive mappings differ sharply from that of the contractive mappings. Indeed, the existence of the fixed points of nonexpansive mappings requires restrictive conditions on the domain. This explains why it took more than four decades to prove the first fixed point results for nonexpan- sive mappings in Banach spaces following the publication in 1965 of the works of Browder [10], G¨ohde [13], and Kirk [19].
In this paper, we examine the existence of fixed points of G-nonexpansive map- pings defined in either a uniformly convex hyperbolic metric space or a uniformly convex in every direction Banach space. This is the analog to Browder and G¨ohde’s fixed point theorem for nonexpansive mappings in uniformly convex Ba- nach spaces. Such a result gives a good example of the recent bridge between the graph theory and the metric fixed point theory. This work is inspired by [9].
2. Preliminaries
Let us start by giving the basic definitions and properties of graph theory and hyperbolic metric spaces which will be used all through.
A graph G is an nonempty set V(G) of elements called vertices together with a possibly empty subset E(G) of V(G) × V(G) called edges. We assume in this paper, that all graphs are reflexive, i.e., (x, x) ∈ E(G) for each x ∈ V(G).
Moreover, we assume that there exists a distance function d defined on the set of vertices V(G). We could treat Gas a weighted graph by giving each edge the metric distance between its vertices. The conversion graph G−1 is obtained by reversing the direction of E(G). In this case, we have
E(G−1) ={(y, x)| (x, y)∈E(G)}.
An oriented graphGis a digraph such that whenever (x, y)∈E(G), then (y, x)∈/ E(G). The graphGe is obtained from G by removing the direction of edges, i.e., (x, y)∈Ge if (x, y)∈G or (y, x)∈G.
Letxandybe inV(G). A (directed) path fromxtoyis a finite sequence (xi)i=Ni=1 of vertices such that x0 =x, xN =y and (xn−1, xn) ∈E(G) for i= 1, ..., N. In
this case, the length of the path (xi)i=Ni=1 is N + 1. Two vertices x and y are G-connected if (x, y)∈E(G). The graphe Gis said to be connected if there exists a path between any two vertices. The graphG is said to be weakly connected if Ge is connected.
Definition 2.1. The graph G is said to be transitive whenever (x, z) ∈ E(G) provided (x, y) ∈ E(G) and (y, z) ∈ E(G), for any x, y, z ∈ V(G). In another words, G is transitive if for any two vertices x and y that are connected by a directed finite path, we have (x, y)∈E(G).
Next, we introduce the concept of a hyperbolic metric space. Indeed, let (X, d) be a metric space. For any x, y ∈X, the subset [x, y] is called a metric segment if [x, y] is an isometric image of the real line interval [0, d(x, y)]. Denote by F the family of all metric segments inX. If any two pointsx, y inX are endpoints of a unique metric segment, (X, d) is called a convex metric space [24]. In this case, the unique point z of [x, y] which satisfies
d(x, z) = (1−β)d(x, y), and d(z, y) =βd(x, y),
where β ∈ [0,1], is denoted by βx⊕ (1−β)y. Furthermore, if the following inequality holds
d
αp⊕(1−α)x, αq⊕(1−α)y
≤αd(p, q) + (1−α)d(x, y),
for all p, q, x, y in X, and α ∈ [0,1], then X is called a hyperbolic metric space (see [27]).
Normed linear spaces are obvious examples of hyperbolic spaces. For nonlinear examples, we can take the Hilbert open unit ball equipped with the hyperbolic metric [16], the Hadamard manifolds [11] and the CAT(0) spaces [20, 21, 22].
A subset C of a hyperbolic metric space X is said to be convex if [x, y] ⊂ C whenever x, y are in C.
Definition 2.2. Let (X, d) be a hyperbolic metric space. We say that X is uniformly convex (in short, UC) if for any a∈X, for every r >0, and for each
>0
δ(r, ε) = infn 1−1
r d1 2x⊕ 1
2y, a
;d(x, a)≤r, d(y, a)≤r, d(x, y)≥rεo
>0.
From now onwards we assume thatX is a hyperbolic metric space and if (X, d) is uniformly convex, then for every s≥0, ε >0, there exists η(s, ε)>0 depending ons and such that
δ(r, ε)> η(s, ε)>0 f or any r > s.
Property 1. [18] Let (X, d) be a uniformly convex hyperbolic metric space and {Cn} be a decreasing sequence of nonempty bounded convex closed subsets of X.
Then T
n≥1
Cn 6=∅.
The following lemma will be used throughout.
Lemma 2.1. [9]LetC be a nonempty closed convex subset of a uniformly convex hyperbolic metric space(X, d). Letτ :C →[0,+∞)be a type function, i.e., there exists a bounded sequence {xn} ∈X such that
τ(x) = lim sup
n→+∞
d(xn, x),
for any x ∈ C. Then τ is continuous. Since X is hyperbolic, then τ is convex, i.e., the subset {x ∈ C; τ(x) ≤ r} is convex for any r ≥ 0. Moreover, there exists a unique minimum point z ∈C such that
τ(z) = inf{τ(x); x∈C}.
3. G-nonexpansive mappings in Metric Spaces
Throughout this section, we assume that (X, d) is a hyperbolic metric space endowed with a graph G. Let C be a nonempty, closed, convex and bounded subset of X not reduced to one point. Assume that G is transitive and G- intervals are convex and closed. In this work, G-intervals are any of the subsets [a,→) = {x ∈ C; (a, x) ∈ E(G)} and (←, b] = {x ∈ C; (x, b) ∈ E(G)}, for any a, b∈C.
Definition 3.1. Let C be a nonempty subset of X. A mapping T : C → C is called
(i) G-monotone if for anyx, y ∈C such that(x, y)∈E(G), we have(T(x), T(y))∈ E(G).
(ii) G-nonexpansive if T isG-monotone and d(T(x), T(y))≤d(x, y), for any x, y ∈C such that (x, y)∈E(G).
The point x∈C is called a fixed point of T if T(x) =x. The set of fixed points of T will be denoted by F ix(T).
Let T :C → C be G-nonexpansive mapping. Fix λ ∈(0,1) and x0 ∈C. Define the iteration sequence {xn}in C by
(MS) xn+1 = (1−λ)xn⊕λT(xn), n ≥0.
Such sequence is known as Mann iteration sequence [23]. Assume that x0 and T(x0) areG-connected. Without loss of generality, we assume that (x0, T(x0))∈ E(G). SinceG-intervals are convex and T isG-monotone, we have
(x0, x1),(x1, T(x0)),(T(x0), T(x1))∈E(G).
By induction, we have
(xn, xn+1),(xn+1, T(xn),(T(xn), T(xn+1))∈E(G), for any n≥1, which implies, since T is G-nonexpansive,
d(T(xn+1), T(xn))≤d(xn+1, xn).
In order to proceed further, we will need the following fundamental result [14, 15].
Proposition 3.1. Under the above assumptions, we have (GK) (1 +nλ) d(T(xi), xi) ≤ d(T(xi+n), xi)+
(1−λ)−n
d(T(xi), xi)−d(T(xi+n), xi+n)
,
for any i, n∈N. This inequality implies
n→+∞lim d(xn, T(xn)) = 0, i.e., {xn} is an approximate fixed point sequence ofT. Next, we give the main theorem of this section.
Theorem 3.1. Let the triplet(X, d, G)be as described above. Assume that(X, d) is uniformly convex. Let C be a nonempty closed convex and bounded subset of X not reduced to one point. Let T : C → C be a G-nonexpansive mapping.
Then T has a fixed point provided there existsx0 ∈C such that x0 andT(x0)are G-connected.
Proof. Consider the Mann iteration sequence {xn} generated by (M S) starting at x0 with λ ∈ (0,1). WLOG, we can assume that (x0, T(x0)) ∈ E(G). Using the properties of {xn} and the transitivity of G, the subsets [xn,→), n ≥ 0, are nonempty, non-increasing, convex and closed. Since X is uniformly convex, Property 1 implies that
C∞= \
n≥0
[xn,→)∩C = \
n≥0
{x∈C; (xn, x)∈E(G)} 6=∅.
Let x ∈ C∞, then (xn, x) ∈ E(G) for any n ≥ 0. Since T is G-monotone we have (T(xn), T(x)) ∈ E(G). As (xn+1, T(xn)) ∈ E(G), then by transitivity of G, we get (xn+1, T(x)) ∈ E(G) for any n ≥ 0, i.e., T(C∞) ⊂C∞. Consider the type function τ :C∞ →[0,+∞) generated by{xn}, i.e.,τ(x) = lim sup
n→+∞
d(xn, x).
Since lim
n→+∞d(xn, T(xn)) = 0, we get τ(x) = lim sup
n→+∞
d(T(xn), x), for any x ∈ C∞. Lemma 2.1 implies the existence of a unique z ∈ C∞ such that τ(z) = inf{τ(x); x∈C∞}. Sincez∈C∞, we have (xn, z)∈E(G), for anyn ≥1, which implies
τ(T(z)) = lim sup
n→+∞
d(T(xn), T(z))≤lim sup
n→+∞
d(xn, z) = τ(z).
The uniqueness of the minimum point implies that z = T(z), i.e., z is a fixed
point of T.
In the next section, we will show how to weaken the uniform convexity property when we assume thatX is a linear space.
4. G-nonexpansive mappings in Banach Spaces
Let (E,k.k) be a Banach space. We say that E is uniformly convex in the direction z ∈E, with kzk= 1, if δ(ε, z)>0, where
δ(ε, z) = inf
1−
x+y 2
;kxk ≤1, kyk ≤1, x−y=α z, andkx−yk ≥ε
,
for anyε∈(0,2]. Uniform convexity in every direction was introduced by Garkavi [12] in connection with his study of Chebyshev centers. Zizler [29] proved that any separable Banach space has an equivalent norm which is uniformly convex in every direction. It is also known that uniformly convex Banach spaces are super-reflexive [8] which shows that the class of uniformly convex is a lot smaller than the class of uniformly convex in every direction.
The following lemma is the analog to Lemma 2.1 in Banach spaces that are uniformly convex in every direction.
Lemma 4.1. [9] Let (X,k.k) be a Banach space which is uniformly convex in every direction. Let C be a weakly compact nonempty convex subset of X. Let τ :C →[0,+∞) be a type function. Then there exists a unique minimum point z ∈C such that
τ(z) = inf{τ(x); x∈C}.
The following proposition is an analog to Proposition 3.1 to Banach spaces as they are hyperbolic metric spaces.
Proposition 4.1. Let the triple (X,k.k, G) be a Banach space endowed with a directed graph G. Let C be a nonempty, convex and bounded subset of X not reduced to one point such thatV(G) = C. Assume that Gis reflexive and transi- tive and G-intervals are convex and closed. Let T :C →C be a G-nonexpansive mapping. Fix λ ∈ (0,1) and x0 ∈ C such that x0 and T(x0) are G-connected.
Consider the sequence {xn} in C defined by (MS). Hence (GK) (1 +nλ) kT(xi)−xik ≤ kT(xi+n)−xik+
(1−λ)−n
kT(xi)−xik − kT(xi+n)−xi+nk ,
for any i, n∈N. Then we have lim
n→+∞kxn−T(xn)k= 0, i.e., {xn} is an approx- imate fixed point sequence of T.
Property 2. [17] The triple (E,k.k, G) has property (P) if and only if for any sequence {xn}n∈N in C if xn → x and (xn, xn+1) ∈E(G), for n ∈N, then there is a subsequence {xkn} with (xkn, x)∈E(G), for n∈N.
Using Lemma 4.1 together with the ideas of the proof of Theorem 3.1, we get the following fixed point result.
Theorem 4.1. Let (X,k.k, G) be a Banach space endowed with a directed re- flexive and transitive graph G such that Property (P) is satisfied and G-intervals are convex and closed. Assume that X is uniformly convex in every direction.
Let C be a nonempty weakly compact convex subset of X. Let T : C → C be a G-nonexpansive mapping. Then T has a fixed point provided that there exists x0 ∈C such that x0 and T(x0) are G-connected.
Let us finish this paper with the following two examples.
Example 4.1. Consider the Hilbert space `2 defined by
`2 =n
(xn)∈RN; X
n∈N
|xn|2 <+∞o .
Define the digraph G on `2 by:
(x, y)∈E(G) if and only if xn ≤yn, n≥2,
where x = (xn) and y = (yn) are in `2. Then G is transitive. Moreover, it is easy to check that G-intervals are convex and closed. Let x, y ∈`2 defined by
x= (1,0,0,· · ·) and y = (2,0,0,· · ·).
Then, we have (x, y)∈E(G) and(y, x)∈E(G), i.e., G contains a cycle. There- fore the graph G will not be generated by a partial order. Such example enforces the idea that replacing the partial order by a graph is worthy of consideration.
Inspired by an example in [28], we have the following:
Example 4.2. Let C = [0,1]. Define the graph G on C by (x, y)∈E(G)⇐⇒x+y≤1and |x−y| ≤ 3
8.
It is easy to show that G is convex. Now let T :C→C be defined by T(x) =x2.
We can easily show that T is G-nonexpansive. However, it is not nonexpansive because kT x−T yk>kx−yk where x= 12 and y= 1.
Acknowledgement
The authors would like to acknowledge the support provided by the Deanship of Scientific Research at King Fahd University of Petroleum & Minerals for funding this work through project No. IP142-MATH-111.
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Monther Rashed Alfuraidan, Department of Mathematics & Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.
E-mail address: [email protected]
Sami Atif Shukri, Department of Mathematics & Statistics, King Fahd Uni- versity of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.
E-mail address: [email protected]
