Abstract
We prove weak and strong convergence theorems and the demiclosedness property for classes of multivalued mappings T such that is not nonexpansive, where . Thus our results extend and improve the results on multivalued and single-valued mappings 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 known that every closed convex subset of a uniformly convex Banach space is proximinal. In fact, if K is a closed and convex subset of a uniformly convex Banach space X, then for any there exists a unique point such that (see, e.g., [1, 2])
We will denote the family of all nonempty proximinal subsets of X by , the family of all nonempty closed, convex and bounded subsets of X by , the family of all nonempty closed and bounded subsets of X by and the family of all nonempty subsets of X by for a nonempty set X. Let be the family of all nonempty closed and bounded subsets of a normed space E. Let H be 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 a fixed point set of T. A point is called a strict fixed point of T if . The set is called a strict fixed point set of T. A multivalued mapping is called L-Lipschitzian if there exists such that, for any pair ,
In (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.
In recent years, several works have been done on the approximation of fixed points of multivalued nonexpansive mappings by many authors (see, for example, [3–5] and references therein). Different iterative schemes have been introduced by several authors to approximate the fixed points of nonexpansive mappings (see, for example, [3–5]). Among the iterative schemes, Sastry and Babu [3] introduced Mann and Ishikawa iteration as follows.
Let and p be a fixed point of T. The sequence of Mann iterates is given for by
where is such that and is a real sequence in .
The sequence of Ishikawa iterates is given by
where , are such , and , are real sequences satisfying: (i) ; (ii) ; (iii) .
Using the above iterative schemes, Panyanak [4] generalized the result proved by [3].
Nadler [6] made the following useful remark.
Lemma 1 Let and . If , then there exists such that
Using Lemma 1, Song and Wang [5] modified the iteration process due to Panyanak [4] and improved the results therein. They made the important observation that generating the Mann and Ishikawa sequences in [3] is in some sense dependent on the knowledge of the fixed point. They gave their iteration scheme as follows.
Let K be a nonempty convex subset of X, let and such that . Choose , . Let
Choose such that and
Choose such that and
Choose such that and
Inductively, we have
where , satisfy , and , are real sequences in satisfying , .
Using the above iteration, they then proved the following theorem.
Theorem 1 (Theorem 1, [5])
Let K be a nonempty compact convex subset of a uniformly convex Banach space X. Suppose that is a multivalued nonexpansive mapping such that and for all . Then the Ishikawa sequence defined as above converges strongly to a fixed point of T.
Shahzad and Zegeye [7] observed that if X is a normed space and is any multivalued mapping, then the mapping defined for each x by
has the property that for all . Using this idea, they removed the strong condition ‘ for all ’ introduced by Song and Wang [5].
Recently, Khan and Yildirim [8] introduced a new iteration scheme for multivalued nonexpansive mappings using the idea of the iteration scheme for a single-valued nearly asymptotically nonexpansive mapping introduced by Agarwal et al. [9] as follows:
where , and . Also, using a lemma in Schu [10], the idea of removal of the condition ‘ for all ’ introduced by Shahzad and Zegeye [7] and the method of direct construction of a Cauchy sequence as indicated by Song and Cho [11], they stated the following theorems.
Theorem 2 (Theorem 1, [8])
Let X be a uniformly convex Banach space satisfying Opial’s condition and K be a nonempty closed convex subset of X. Let be a multivalued mapping such that and is a nonexpansive mapping. Let be the sequence as defined in (7). Let be demiclosed with respect to zero. Then converges weakly to a point of .
However, we observe that there are many multivalued mappings T for which neither T nor is nonexpansive. Based on the above observation, it is our purpose in this paper to firstly introduce the new classes of multivalued nonexpansive-type, k-strictly pseudocontractive-type and pseudocontractive-type mappings which are more general than the class of multivalued nonexpansive mappings. Secondly, we prove that if H is a real Hilbert space and K is a nonempty weakly closed subset of H, is a multivalued mapping from K into the family of all nonempty proximinal subsets of H. Suppose that is a k-strictly pseudocontractive-type mapping. Then is demiclosed at zero (i.e., the graph of is closed at zero in or weakly demiclosed at zero), where I denotes the identity on E, the weak topology, the norm (or strong) topology and . Lastly, we prove weak and strong convergence theorems for these classes of multivalued mappings without the compactness condition on the domain of the mappings using Mann and Ishikawa iteration schemes. Thus our results extend and improve the results on single-valued and multivalued mappings in the contemporary literature.
In the sequel, we will need the following definitions and lemmas.
Definition 1 (See, e.g., [12])
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 (See, e.g., [12, 13])
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 3 Let E be a Banach space. Let be a multivalued mapping. A point is called an asymptotic fixed point of T if there exists a sequence such that converges weakly to p and a sequence with for all such that converges strongly to zero. We denote the set of asymptotic fixed points of T by .
Definition 4 (See, e.g., [12, 13])
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 5 ([14])
Let H be a real Hilbert space. Let be a multivalued mapping. T is said to be monotone if given and , there exists such that
Definition 6 ([5])
A multivalued mapping is said to satisfy condition (1) (see, for example, [5]) if there exists a nondecreasing function with and for all such that
Definition 7 Let E be a real Banach space. Let be a multivalued mapping. is said to be monotone in the sense of [14] if given any pair and , there exists such that
Lemma 2 ([15])
Let , and be sequences of nonnegative real numbers satisfying the following relation:
where is a nonnegative integer. If , , then exists.
Lemma 3 ([11])
Let K be a normed space. Let be a multivalued mapping and . Then the following are equivalent:
-
(1)
;
-
(2)
;
-
(3)
.
Moreover, .
Lemma 4 Let H be a real Hilbert space. Then the following well-known result holds: if is a sequence in H which converges weakly to , then
2 Main results
Definition 8 Let X be a normed space. A multivalued mapping is said to be k-strictly pseudocontractive-type in the sense of Browder and Petryshyn [16] if there exists such that given any and , there exists satisfying and
If in (9), T is said to be a pseudocontractive-type mapping. T is called nonexpansive-type if . Clearly, every multivalued nonexpansive mapping is a 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. The following examples show that the class of nonexpansive-type mappings is properly contained in the class of k-strictly pseudocontractive-type mappings and the class of k-strictly pseudocontractive-type mappings is properly contained in the class of pseudocontractive-type mappings.
Example 1 Let (the reals with usual metric). Define by
Then for all , hence it is not nonexpansive.
Also, for each , , , choose . Then
and
Hence
Consequently, T is k-strictly pseudocontractive-type with . It then follows that T is pseudocontractive-type. Observe that T is not nonexpansive-type so that the class of multivalued nonexpansive-type mappings is properly contained in the class of multivalued k-strictly pseudocontractive-type mappings. Next, we show that the class of multivalued pseudocontractive-type mappings properly contains the class of multivalued k-strictly pseudocontractive-type mappings.
Example 2 Let (the reals with usual metric). Define by
Then which is not nonexpansive. Also,
Furthermore, for each , , , choose . Then
and
It then follows that
Now, for any pair , . It then follows that for any the corresponding . In particular, for ,
It then follows that T is a pseudocontractive-type mapping but not a k-strictly pseudocontractive-type mapping.
Using the multivalued version of the method of the proof used in [17], we then prove the following.
Proposition 1 Let E be a real Banach space. Suppose is a pseudocontractive-type mapping. Then is monotone.
Proof Since T is pseudocontractive-type, for any pair and , there exists such that
Now,
Hence, . Hence, is monotone. □
Proposition 2 Let E be a normed space. And let be a k-strictly pseudocontractive-type mapping. Then T is a L-Lipschitzian mapping.
Proof Let T be a k-strictly pseudocontractive-type mapping. Then there exists such that, for all and , there exists satisfying and
We then have that
It then follows that
Hence
□
Proposition 3 Let H be a real Hilbert space. Let K be a nonempty weakly closed subset of H. Let be a multivalued mapping from K into the family of all nonempty proximinal subsets of H. Suppose that is a k-strictly pseudocontractive-type mapping. Then is demiclosed at zero (i.e., the graph of is closed at zero in or weakly demiclosed at zero).
Proof Let be such that converges weakly to p and a sequence with for all such that converges strongly to 0. We prove that (i.e., for some ). Since converges weakly, it is bounded. Let be arbitrary. From the definition of k-strictly pseudocontractive-type, for each , there exists such that
and
for all . Also, since , . Thus we have
For each , define by
Then, from Lemma 4, we obtain
Thus
Therefore,
Observe also that
Hence it follows from (14) and (15) that . Therefore . □
Theorem 3 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 (I-T) 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) .
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 that . Also, since K is closed and with bounded, there exists a subsequence such that converges weakly to some . Also, implies that . Since is weakly demiclosed at zero, we have that . Since H satisfies Opial’s condition [18], we have that converges weakly to . □
Corollary 1 Let H be a real Hilbert space and K be a nonempty closed and convex subset of H. Let be a multivalued mapping from K into the family of all proximinal subsets of K with . Suppose is a k-strictly pseudocontractive-type mapping with . Then the Mann sequence defined in Theorem 3 converges weakly to a point of .
Proof The proof follows easily from Lemma 3, Proposition 3 and Theorem 3. □
Remark 1 It is easy to see that Examples 1 and 2 satisfy the condition ‘given any pair and with , there exists with satisfying the conditions of Definition 8’. Also, if T is a multivalued mapping such that is a pseudocontractive-type mapping, then given any pair and with the corresponding satisfying the conditions of Definition 8, it is the case that and .
Based on Lemma 3 and Remark 1 above, we will first prove weak and strong convergence for the new class of pseudocontractive-type mappings with the following two conditions: (i) given any pair and with , there exists with satisfying the conditions of Definition 8; (ii) for all , then obtain the case for an arbitrary multivalued mapping T such that is a pseudocontractive-type mapping without the two conditions on as corollary.
Theorem 4 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 8. Suppose T satisfies condition (1). Then the Ishikawa sequence defined by
converges strongly to , where with , with satisfying the conditions in Definition 8 and and are real sequences satisfying: (i) ; (ii) ; (iii) .
Proof
Also,
(17) and (18) imply that
(19) and (20) imply that
It then follows from Lemma 2 that exists. Hence is bounded, so are and . We then have from (21), (ii) and (iii) that
It then follows that . Since , we have that as . Since T satisfies condition (1), . Thus there exists a subsequence of such that for some . From (21)
We now show that is a Cauchy sequence in .
Therefore is a Cauchy sequence and converges to some because K is closed. Now,
Hence as ,
Hence, and converges strongly to q. Since exists, we have that converges strongly to . □
Corollary 2 Let H be a real Hilbert space and K be 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 pseudocontractive-type mapping. If T satisfies condition (1), then the Ishikawa sequence defined in (16) converges strongly to .
Proof The proof follows easily from Lemma 3, Remark 1 and Theorem 4. □
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Acknowledgements
This work was carried out at the University of Kwazulu Natal, South Africa when the author visited under the OWSDW [formally TWOWS], Abdus Salam International Centre for Theoretical Physics (ICTP) Trieste, Italy, Postgraduate Training Fellowship. She is grateful to OWSDW for the Fellowship and to University of Kwazulu Natal for making their facilities available and for hospitality.
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Isiogugu, F.O. Demiclosedness principle and approximation theorems for certain classes of multivalued mappings in Hilbert spaces. Fixed Point Theory Appl 2013, 61 (2013). https://doi.org/10.1186/1687-1812-2013-61
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DOI: https://doi.org/10.1186/1687-1812-2013-61