Convergence Theorems for Fixed Points of Multivalued Mappings in Hilbert Spaces

Let H be a real Hilbert space and K a nonempty closed convex subset of H. Suppose T : K → CB(K) is a multivalued Lipschitz pseudocontractive mapping such that F(T) ̸ = 0. An Ishikawa-type iterative algorithm is constructed and it is shown that, for the corresponding sequence {x n }, under appropriate conditions on the iteration parameters, lim inf n→∞ d (x n , Tx n ) = 0 holds. Finally, convergence theorems are proved under approximate additional conditions. Our theorems are significant improvement on important recent results of Panyanak (2007) and Sastry and Babu (2005).

Existence theorem for fixed point of multivalued contractions and nonexpansive mappings using the Hausdorff metric have been proved by several authors (see, e.g., Nadler Jr. [5], Markin [6], and Lim [7]).Later, an interesting and rich fixed point theory for such maps and more general maps was developed which has applications in control theory, convex optimization, differential inclusion, and economics (see Gorniewicz [8] and references cited therein).
Sastry and Babu [2] introduced the following iterative scheme.Let  :  → () be a multivalued mapping and let  * be a fixed point of .The sequence of iterates is given for  0 ∈  by where   is a real sequence in (0,1) satisfying the following conditions: They also introduced the following sequence: where {  }, {  } are real sequences satisfying the following conditions: Sastry and Babu called the process defined by (4) a Mann iteration process and the process defined by (5) where the iteration parameters   ,   satisfy conditions (i), (ii), and (iii) an Ishikawa iteration process.They proved in [2] that the Mann and Ishikawa iteration schemes for a multivalued map  with fixed point  converge to a fixed point of  under certain conditions.More precisely, they proved the following result for a multivalued nonexpansive map with compact domain.
Panyanak [1] extended the above result of Sastry and Babu [2] to uniformly convex real Banach spaces.He proved the following result.
Panyanak [1] also modified the iteration schemes of Sastry and Babu [2].Let  be a nonempty closed convex subset of a real Banach space and let  :  → () be a multivalued map with () a nonempty proximinal subset of .

Question (P).
Is Theorem P2 true for the Ishikawa iterates defined by (7) and (8)?As remarked by Nadler Jr. [5], the definition of the Hausdorff metric on CB() gives the following useful result.Lemma 1.Let ,  ∈ CB() and  ∈ .For every  > 0, there exists  ∈  such that  (, ) ≤  (, ) + . ( Song and Wang [3,4] modified the iteration process by Panyanak [1] and improved the results therein.They gave their iteration scheme as follows. Let  be a nonempty closed convex subset of a real Banach space and let  :  → CB() be a multivalued map. where They then proved the following result.

Shahzad and Zegeye
Let  be a nonempty closed convex subset of a real Banach space,  :  → () a multivalued map, and Choose  0 ∈ , and define {  } as follows: where   ∈     ,   ∈     .They then proved the following result.
Remark 2. In recursion formula (4), the authors take   ∈ (  ) such that ‖  −  * ‖ = ( * ,   ).The existence of   satisfying this condition is guaranteed by the assumption that   is proximinal.In general such a   is extremely difficult to pick.If   is proximinal, it is not difficult to prove that it is closed.If, in addition, it is a convex subset of a real Hilbert space, then   is unique and is characterized by One can see from this inequality that it is not easy to pick at every step of the iteration process.So, recursion formula (4) is not convenient to use in any possible application.Also, the recursion formulas defined in ( 7) and ( 8) are not convenient to use in any possible application.The sequences {  } and {   } are not known precisely.The restrictions   ∈   , ‖  −  ‖ = (  ,   ),   ∈ (), and    ∈   , ‖   − V  ‖ = (V  ,   ), V  ∈ (), make them difficult to use.These restrictions on   and    depend on (), the fixed points set.So, the recursions formulas (7) and (8) If  = 1 in ( 16), the map  is said to be pseudocontractive.
Browder and Petryshyn [14] introduced and studied the class of strictly pseudocontractive maps as an important generalization of the class of nonexpansive maps (mappings  :  →  satisfying ‖ − ‖ ≤ ‖ − ‖ ∀,  ∈ ).It is trivial to see that every nonexpansive map is strictly pseudocontractive.
Motivated by this, Chidume et al. [15] introduced the class of multivalued strictly pseudocontractive maps defined on a real Hilbert space  as follows.
We observe from ( 17) that every nonexpansive mapping is strict pseudocontractive and hence the class of pseudocontractive mappings is a more general class of mappings.
Then, they proved strong convergence theorems for this class of mappings.The recursion formula used is of the Krasnoselskii-type [16].
Remark 5. We note that, for the more general situation of approximating a fixed point of a multivalued Lipschitz pseudocontractive map in a real Hilbert space, an example of Chidume and Mutangadura [17] shows that, even in the single-valued case, the Mann iteration method does not always converge in the setting of Theorem CA2.We now give an example of multivalued pseudocontractive map (Definition 4) which is not nonexpansive.
It is our purpose in this paper to prove strong convergence theorems for the class of multivalued Lipschitz pseudocontractive maps in real Hilbert spaces.We use the recursion formula (11), dispensing with the second restriction on the sequences {  } and {  }: ‖ +1 −  −1 ‖ ≤ ( +1 ,   ) +   ,  ≥ 1.This class of maps is much more larger than that of multivalued nonexpansive maps used in Theorem SW.So, in the setting of real Hilbert spaces, our theorem improves and extends the result of Song and Wang [3,4].

Preliminaries
In the sequel, we will need the following results.

Main Results
We use the following iteration scheme.