1. Introduction
Many of the real world known problems that scientists are looking to solve are nonlinear. Therefore, translating linear version of such problems into their equivalent nonlinear version has a great importance. Mathematicians have tried to transfer the structure of covexity to spaces that are not linear spaces. Takahashi [
1], Kirk [
2,
3], and Penot [
4], for example, presented this notion in metric spaces. Takahashi [
1] introduced the following notion of convexity in metric spaces:
Definition 1. ([
1])
Let be a metric space and . A mapping is said to be a convex structure on X if for each and all ,A metric space together with a convex structure W is called a convex metric space and is denoted by .
A subset C of X is called convex if , for all and all .
Example 1. Let . For any and
and , we define the mapping byand the metric by Then is a convex metric space.
Example 2. Let with the metricfor any and define the mapping byfor each and . Then is a convex metric space. Example 3. Let be the metric space with the metric and define by , for all and . Then is a convex metric space.
This notion of convex structure is a generalization of convexity in normed spaces and allows us to obtain results that seem to be possible only in linear spaces. One of its useful applications is the iterative approximation of fixed points in metric spaces. All of the sequences that are used in fixed point problems require linearity or convexity of the space. So, this concept of convexity helps us to define various iteration schemes and to solve fixed point problems in metric spaces. In recent years, many authors have established several results on the covergence of some iterative schemes using different contractive conditions in convex metric spaces. For more details, refer to [
5,
6,
7,
8,
9,
10,
11,
12,
13,
14].
Now, let us recall some definitions and concepts that will be needed to state our results:
Definition 2. ([
15])
Let be a metric. A subset D is called proximinal if for each there exists an element such that , where . We denote the family nonempty proximinal and bounded subsets of D by and the family of all nonempty closed and bounded subsets of X by .
For two bounded subsets
A and
B of a metric space
, the Pompeiu–Hausdorff metric between
A and
B is defined by
Definition 3. ([
16])
Let be a metric space. A multi-valued mapping is said to be nonexpansive if , for all .An element is called a fixed point of T if . The set of all fixed points of T are denoted by .
Definition 4. ([
17])
Let be a metric space and D be a nonempty subset of X. A multi-valued mapping is called −nonexpansive if for all and with , there exists with such thatIt is clear that if T is a −nonexpansive map, then is a nonexpansive map, where for is defined byfor all . Definition 5. ([
16])
Let be a metric space. A multi-valued mapping is said to satisfy condition (I) if there is a nondecreasing function with , for such that , for all . First of all, Moudafi [
18] introduced the viscosity approximation method for approximating the fixed point of nonexpansive mappings in Hilbert spaces. Since then, many authors have been extending and generalizing this result by using different contractive conditions on several spaces. For some new works in these fields, we can refer to [
19,
20,
21,
22,
23,
24,
25,
26,
27]. Inspired and motivated by the research work going on in these fields, in this paper we investigate the convergence of some viscosity approximation processes for
−nonexpansive multi-valued mappings in a complete convex metric spaces. The convergence theorems for finite and infinite family of such mappings are also presented. Our results can improve and extend the corresponding main theorems in the literature.
2. Main Results
At first, we present two lemmas that are used to prove our main result. Since the idea is similar to the one given in Lemmas
and
in [
28], we only state the results without the proof:
Lemma 1. Let and be sequences in a convex metric space and be a sequence in such that . Set Let for all . Suppose thatand .Thenfor all . Lemma 2. Let and be bounded sequences in a convex metric space and be a sequence in with . Suppose that and Then
Now, we state and prove the main theorem of this paper:
Theorem 1. Let D be a nonempty, closed and convex subset of a complete convex metric space and be a −nonexpansive multi-valued mapping with , such that T satisfies condition (). Suppose that such that and such that . Let be the Mann type iterative scheme defined bywhere for . Then converges to a fixed point of T. Proof. Take
. Then
and we have
Hence,
is a decreasing and bounded below sequence and thus
exists for any
. Therefore
is bounded and so
is bounded. On the other hand,
Thus
Applying Lemma 2, we get
Hence, we have
. Since
T satisfies condition (
), we conclude that
Next, we show that
is a Cauchy sequence. Since
, thus for
, there exists
such that for all
Thus, there exists
such that for all
,
It follows that
for all
. Therefore
is a Cauchy sequence and hence it is convergent. Let
. We will show that
is a fixed point of
T.
Since
, thus for given
, there exists
such that for all
,
Moreover,
implies that there exists a natural number
such that for all
,
and thus there exists
such that for all
,
Therefore
Thus,
and therefore
is a fixed point of
T. □
As a result of Theorem 1, Corollaries 1 and 2 are obtained:
Corollary 1. Let D be a nonempty, closed and convex subset of a complete convex metric space , be −nonexpansive multi-valued mapping with such that T satisfies condition () and be a contractive mapping with a contractive constant . Then the iterative sequence defined bywhere and , converges to a fixed point of T. Corollary 2. Let D be a nonempty, closed, and convex subset of a complete convex metric space and be −nonexpansive multi-valued mapping with . Let be the Ishikawa type iterative scheme defined bywhere , , and . Then converges to a fixed point of T if and only if . The above result can be generalized to the finite and infinite family of −nonexpansive multi-valued mappings:
Theorem 2. Let D be a nonempty, closed, and convex subset of a complete convex metric space and be a finite family of −nonexpansive multi-valued mappings such that . Consider the iterative process defined bywhere and , for all and . Then converges to a point in F if and only if . Proof. The necessity of conditions is obvious and we will only prove the sufficiency. Let
. we have
Therefore, is a decreasing sequence and so , for all . As in the proof of Theorem 1, is a Cauchy sequence and thus exists and equals to some . Again, with a similar process as in the proof of Theorem 1, we conclude that for all . Hence and this completes the proof of theorem. □
Theorem 3. Let D be a nonempty, closed, and convex subset of a complete convex metric space and be an infinite family of −nonexpansive multi-valued mappings such that . Consider the iterative process defined bywhere , and . Then converges to a point in F if and only if .