Abstract
Weak and strong convergence theorems are proved in Hilbert spaces for new classes of multivalued demicontractive-type and hemicontractive-type mappings which are related to the class of multivalued pseudocontractive-type mappings studied by Isiogugu (Fixed Point Theory Appl. 2013:61, 2013). Thus our results extend and improve several corresponding results in the contemporary literature.
MSC:47H10, 54H25.
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1 Introduction
Let E be a normed space. A subset K of E is called proximinal if for each there exists such that
It is well known that every closed convex subset of a uniformly convex Banach space is proximinal. For a nonempty set E, we shall denote the family of all nonempty proximinal subsets of E by , the family of all nonempty closed and bounded subsets of E by , the family of all nonempty closed, convex, and bounded subsets of E by , the family of all nonempty closed subsets of E by , the family of all nonempty subsets of E by , the identity on E by I, the weak topology of E by , and the norm (or strong) topology of E by .
Let H denote the Hausdorff metric induced by the metric d on E, that is, for every ,
If , then
where . Let E be a normed space. Let be a multivalued mapping on E. A point is called a fixed point of T if . The set is called the fixed point set of T. A point is called a strict fixed point of T if . The set is called the strict fixed point set of T. A multivalued mapping is called L-Lipschitzian if there exists such that for all
In (1.1) if , T is said to be a contraction, while T is nonexpansive if . T is called quasi-nonexpansive if and for all ,
Clearly every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive.
Several authors have studied various classes of multivalued mappings. In [1], Shahzad and Zegeye studied certain classes of multivalued nonself mappings in Banach spaces and constructed an appropriate net which converges strongly to a fixed point of the classes of the mappings. Recently, Isiogugu [2] introduced new classes of multivalued mappings as follows.
Definition 1.1 ([2])
Let X be a normed space. A multivalued mapping is said to be k-strictly pseudocontractive-type in the sense of Browder and Petryshyn [3] if there exists such that given any and , there exists satisfying and
If in (1.3) T is said to be a pseudocontractive-type mapping. T is called nonexpansive-type if . Clearly, every multivalued nonexpansive mapping is nonexpansive-type mapping.
From the definitions, it is clear that every multivalued nonexpansive-type mapping is k-strictly pseudocontractive-type and every k-strictly pseudocontractive-type mapping is pseudocontractive-type. Examples to show that the class of nonexpansive-type mappings is properly contained in the class of k-strictly pseudocontractive-type mappings and that the class of k-strictly pseudocontractive-type mappings is properly contained in the class of pseudocontractive-type mappings were given in [2]. The following theorems were also proved in [2].
Theorem 1.1 Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that is a k-strictly pseudocontractive-type mapping from K into the family of all proximinal subsets of K with such that and for all . Suppose is weakly demiclosed at zero. Then the Mann-type sequence defined by
converges weakly to , where with and is a real sequence in satisfying: (i) ; (ii) ; (iii) .
Theorem 1.2 Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose that is an L-Lipschitzian pseudocontractive-type mapping from K into the family of all proximinal subsets of K such that and for all . Suppose for any pair and with , there exists with satisfying the conditions of Definition 1.1. Suppose T satisfies condition (1) (i.e., if there exists a nondecreasing function with and for all such that , . Then the Ishikawa sequence defined by
converges strongly to , where with , with satisfying the conditions in Definition 1.1 and and are real sequences satisfying: (i) ; (ii) ; (iii) .
In [4], Chidume et al. also considered a class of multivalued k-strictly pseudocontractive mappings defined as follows.
Let H be a real Hilbert space. A multivalued mapping is said to be k-strictly pseudocontractive if there exists such that for all one has
If , T is said to be pseudocontractive mapping. They constructed a Mann-type iteration scheme which is an approximate fixed point sequence and obtain some strong convergence theorems for the class of k-strictly pseudocontractive mappings.
The following example shows that the class of multivalued pseudocontractive-type mappings considered by Isiogugu [2] is not a subclass of the multivalued pseudocontractive mappings considered by Chidume et al. [4].
Example 1.1 Let (the reals with usual metric). Define by
It was shown in [2] that T is k-strictly pseudocontractive-type mapping hence pseudocontractive-type. However, for , if we choose and then and . Consequently,
which implies that T is not pseudocontractive and hence not k-strictly pseudocontractive mapping in the sense of Chidume et al. [4].
It is our purpose in this work to introduce and study new classes of multivalued demicontractive-type and hemicontractive-type mappings which are more general than the class of multivalued quasi-nonexpansive mappings and are also related to the multivalued k-strictly pseudocontractive-type and pseudocontractive-type mappings of Isiogugu [2], single-valued mappings of Browder and Petryshyn [3], Hicks and Kubicek [5] and Naimpally and Singh [6]. We also prove weak and strong convergence theorems for approximation of fixed points of our classes of mappings.
2 Preliminaries
We shall need the following definitions and lemmas.
Definition 2.1 (see, e.g., [7])
Let E be a Banach space. Let be a multivalued mapping. is said to be strongly demiclosed at zero if for any sequence such that converges strongly to p and a sequence with for all such that converges strongly to zero, then (i.e., ).
Observe that if T is a multivalued Lipschitzian mapping, then is strongly demiclosed.
Definition 2.2 (see, e.g., [7, 8])
Let E be a Banach space. Let be a multivalued mapping. is said to be weakly demiclosed at zero if for any sequence such that converges weakly to p and a sequence with for all such that converges strongly to zero. Then (i.e., ).
Definition 2.3 (see, e.g., [7, 8])
Let E be a Banach space. Let be a multivalued mapping. The graph of is said to be closed in (i.e., is weakly demiclosed or demiclosed) if for any sequence such that converges weakly to p and a sequence with for all such that converges strongly to y. Then (i.e., for some ).
Definition 2.4 A Banach X is said to satisfy Opial’s condition if whenever a sequence converges weakly to then it is the case that
for all , .
Definition 2.5 ([9])
A multivalued mapping is said to satisfy condition (1) (see for example [9]) if there exists a nondecreasing function with and for all such that
Lemma 2.1 ([10])
Let , , and be sequences of nonnegative real numbers satisfying the following relation:
where is a nonnegative integer. If , , then exists.
Lemma 2.2 ([11])
Let K be a normed space. Let be a multivalued mapping and . Then the following are equivalent:
-
(1)
;
-
(2)
;
-
(3)
.
Moreover, .
Lemma 2.3 ([12])
Let and . If , then there exists such that
3 Main results
We now introduce the new classes of multivalued demicontractive-type and hemicontractive-type mappings and prove some convergence theorems for these classes of mappings.
Definition 3.1 Let X be a real normed space. A mapping is said to be demicontractive in the terminology of Hicks and Kubicek [5] if and for all , there exists such that
where and .
If in (3.1) then T is called a hemicontractive mapping.
The following are some examples of demicontractive mappings.
Example 3.1 Every multivalued quasi-nonexpansive mapping is demicontractive.
Example 3.2 Let X be a normed space. Suppose that T is a multivalued mapping such that and that is a k-strictly pseudocontractive-type mapping; then is demicontractive.
Example 3.3 Let X be a normed space. Let be a multivalued k-strictly pseudocontractive-type with a nonempty fixed point set. Suppose for all ; then for any , and with we have
therefore, T is demicontractive-type.
Example 3.4 Let (the reals with usual metric). Define by
Then . For each ,
Therefore,
Consequently, T is demicontractive-type with . It then follows that T is hemicontractive. Observe that T is not quasi-nonexpansive so that the class of multivalued quasi-nonexpansive mappings is properly contained in the class of multivalued demicontractive-type mappings.
Next is an example of a multivalued mapping T with , for all for which is a demicontractive-type but not a k-strictly pseudocontractive-type mapping.
Example 3.5 Let (the reals with usual metric). Define by
Then . For each ,
which is demicontractive-type but not k-strictly pseudocontractive-type (see for example [5]).
The following example shows that the class of demicontractive mapping is properly contained in the class of hemicontractive mappings.
Example 3.6 Let (the reals with the usual metric). Define by
Then . For each ,
Therefore,
and . Therefore, T is hemicontractive but not demicontractive.
Other examples of hemicontractive mappings include the following.
Example 3.7 Let X be a normed space. Suppose T is a multivalued mapping such that and is pseudocontractive-type mapping; then is hemicontractive.
Example 3.8 Let X be a normed space. Let be a multivalued pseudocontractive-type with a nonempty fixed point set. Suppose for all ; then for any , and with we have
The following lemma shows that Lemma 2.3 is also valid for all and .
Lemma 3.1 Let E be a metric space. If and , then it is a simple consequence of the Hausdorff metric H that there exists such that
Proof Let E be a metric space and be the family of all nonempty proximinal subsets of E. Let and . Since B is proximinal, there exists such that
Observe that
Hence the result follows. □
Remark 3.1 Lemma 3.1 holds if E is a reflexive real Banach space and is replaced with with B weakly closed (see for example [4]).
We now prove the following theorems.
Theorem 3.1 Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that is a demicontractive mapping from K into the family of all proximinal subsets of K with and for all . Suppose is weakly demiclosed at zero. Then the Mann type sequence defined by
converges weakly to , where and is a real sequence in satisfying: (i) ; (ii) .
Proof Using the well-known identity:
which holds for all and for all , we obtain
It then follows that exists; hence is bounded. Also,
Since from (ii), we have . Thus . Also since K is closed and with bounded, there exist a subsequence such that converges weakly to some . Also implies that . Since is weakly demiclosed at zero we have . Since H satisfies Opial’s condition [13] we find that converges weakly to . □
Corollary 3.1 Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that is k-strictly pseudocontractive-type mapping from K into the family of all proximinal subsets of K with such that and for all . Suppose is weakly demiclosed at zero. Then the Mann sequence defined in Theorem 3.1 converges weakly to a point of .
Proof The proof follows easily from Example 3.3 and Theorem 3.1. □
Corollary 3.2 Let H be a real Hilbert space and K a nonempty closed and convex subset of H. Let be a multivalued mapping from K into the family of all proximinal subsets of K. Suppose is a demicontractive mapping with and is weakly demiclosed at zero. Then the Mann sequence defined in Theorem 3.1 converges weakly to a point of .
Proof The proof follows easily from Lemma 2.2 and Theorem 3.1. □
Remark 3.2 Since the choice of in the Mann-type iteration scheme is independent of , we can also replace with in Theorem 3.1 and its corollaries. Furthermore, since , one can impose standard conditions on T or K which guarantee strong convergence.
Theorem 3.2 Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose that is an L-Lipschitzian hemicontractive mapping from K into the family of all proximinal subsets of K and for all . Suppose T satisfies condition (1). Then the Ishikawa sequence defined by
converges strongly to , where , satisfying the conditions of Lemma 3.1 and and are real sequences satisfying: (i) ; (ii) ; (iii) .
Proof
Equations (3.11) and (3.12) imply that
Equations (3.13) and (3.14) imply that
It then follows from Lemma 2.1 that exists. Hence is bounded so and also are. We then have from (3.15), (ii), and (iii)
It then follows that . Since we have as . Since T satisfies condition (1), . Thus there exists a subsequence of such that for some . From (3.10)
We now show that is a Cauchy sequence in . We have
Therefore is a Cauchy sequence and converges to some because K is closed. Now,
Hence as . We have
Hence, and converges strongly to q. Since exists we see that converges strongly to . □
Corollary 3.3 Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose that is an L-Lipschitzian pseudocontractive-type mapping from K into the family of all proximinal subsets of K such that and for all . Suppose T satisfies condition (1). Then the Ishikawa sequence defined in (3.10) converges strongly to .
Proof The proof follows easily from Example 3.8, Lemma 3.1, and Theorem 3.2. □
Corollary 3.4 Let H be a real Hilbert space and K a nonempty closed and convex subset of H. Let be a multivalued mapping from K into the family of all proximinal subsets of K such that . Suppose is an L-Lipschitzian hemicontractive mapping. If T satisfies condition (1). Then the Ishikawa sequence defined in (3.10) converges strongly to .
Proof The proof follows easily from Lemma 2.2 and Theorem 3.2. □
Remark 3.3 In Theorem 3.2 and its corollaries we can replace with with additional condition that T is weakly closed for all in order to ensure that and satisfy Lemma 3.1 as indicated in Remark 3.1. Furthermore, since , the additional requirement that is weakly demiclosed at zero in Theorem 3.2 yields weak convergence without condition (1).
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Acknowledgements
The second author is grateful to the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy for their facilities and for hospitality. He contributed to the work during his visits to the Centre as a regular associate. The work was completed while the first author was visiting the University of Kwazulu Natal, South Africa under the OWSD (formally TWOWS) Postgraduate Training Fellowship. She is grateful to OWSD (formally TWOWS) for the Fellowship and to University of Kwazulu Natal for making facilities available and for hospitality.
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Isiogugu, F.O., Osilike, M.O. Convergence theorems for new classes of multivalued hemicontractive-type mappings. Fixed Point Theory Appl 2014, 93 (2014). https://doi.org/10.1186/1687-1812-2014-93
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DOI: https://doi.org/10.1186/1687-1812-2014-93