Convergence theorems for new classes of multivalued hemicontractive-type mappings
© Isiogugu and Osilike; licensee Springer. 2014
Received: 27 September 2013
Accepted: 13 February 2014
Published: 9 April 2014
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.
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 .
Clearly every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive.
Several authors have studied various classes of multivalued mappings. In , 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  introduced new classes of multivalued mappings as follows.
Definition 1.1 ()
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 . The following theorems were also proved in .
converges weakly to , where with and is a real sequence in satisfying: (i) ; (ii) ; (iii) .
converges strongly to , where with , with satisfying the conditions in Definition 1.1 and and are real sequences satisfying: (i) ; (ii) ; (iii) .
In , Chidume et al. also considered a class of multivalued k-strictly pseudocontractive mappings defined as follows.
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  is not a subclass of the multivalued pseudocontractive mappings considered by Chidume et al. .
which implies that T is not pseudocontractive and hence not k-strictly pseudocontractive mapping in the sense of Chidume et al. .
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 , single-valued mappings of Browder and Petryshyn , Hicks and Kubicek  and Naimpally and Singh . We also prove weak and strong convergence theorems for approximation of fixed points of our classes of mappings.
We shall need the following definitions and lemmas.
Definition 2.1 (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., ).
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 ).
for all , .
Definition 2.5 ()
Lemma 2.1 ()
where is a nonnegative integer. If , , then exists.
Lemma 2.2 ()
Lemma 2.3 ()
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.
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.
therefore, T is demicontractive-type.
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.
which is demicontractive-type but not k-strictly pseudocontractive-type (see for example ).
The following example shows that the class of demicontractive mapping is properly contained in the class of hemicontractive mappings.
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.
The following lemma shows that Lemma 2.3 is also valid for all and .
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 ).
We now prove the following theorems.
converges weakly to , where and is a real sequence in satisfying: (i) ; (ii) .
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  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.
converges strongly to , where , satisfying the conditions of Lemma 3.1 and and are real sequences satisfying: (i) ; (ii) ; (iii) .
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).
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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