Demiclosedness principle and approximation theorems for certain classes of multivalued mappings in Hilbert spaces
© Isiogugu; licensee Springer 2013
Received: 19 July 2012
Accepted: 24 February 2013
Published: 18 March 2013
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.
Keywordsproximinal sets Hilbert spaces nonexpansive-type mappings k-strictly pseudocontractive-type mappings pseudocontractive-type mappings
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  introduced Mann and Ishikawa iteration as follows.
where is such that and is a real sequence in .
where , are such , and , are real sequences satisfying: (i) ; (ii) ; (iii) .
Nadler  made the following useful remark.
Using Lemma 1, Song and Wang  modified the iteration process due to Panyanak  and improved the results therein. They made the important observation that generating the Mann and Ishikawa sequences in  is in some sense dependent on the knowledge of the fixed point. They gave their iteration scheme as follows.
where , satisfy , and , are real sequences in satisfying , .
Using the above iteration, they then proved the following theorem.
Theorem 1 (Theorem 1, )
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.
has the property that for all . Using this idea, they removed the strong condition ‘ for all ’ introduced by Song and Wang .
where , and . Also, using a lemma in Schu , the idea of removal of the condition ‘ for all ’ introduced by Shahzad and Zegeye  and the method of direct construction of a Cauchy sequence as indicated by Song and Cho , they stated the following theorems.
Theorem 2 (Theorem 1, )
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., )
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.
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 .
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 ()
Definition 6 ()
Lemma 2 ()
where is a nonnegative integer. If , , then exists.
Lemma 3 ()
2 Main results
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.
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.
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 , we then prove the following.
Proposition 1 Let E be a real Banach space. Suppose is a pseudocontractive-type mapping. Then is monotone.
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.
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).
Hence it follows from (14) and (15) that . Therefore . □
converges weakly to , where with and is a real sequence in satisfying: (i) ; (ii) ; (iii) .
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 , 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.
converges strongly to , where with , with satisfying the conditions in Definition 8 and and are real sequences satisfying: (i) ; (ii) ; (iii) .
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. □
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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