Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

and Applied Analysis 3 important, as has been observed by Bruck [18], mainly for the following two reasons. (i) Nonexpansive mappings are intimately connected with the monotonicity methods developed since the early 1960s and constitute one of the first classes of nonlinear mappings for which fixed point theorems were obtained by using the fine geometric properties of the underlying Banach spaces instead of compactness properties. (ii) Nonexpansive mappings appear in applications as the transition operators for initial value problems of differential inclusions of the form 0 ∈ (du/dt) + T(t)u, where the operators {T(t)} are, in general, setvalued and are accretive or dissipative and minimally continuous. The class of strictly pseudocontractive mappings defined in Hilbert spaces which was introduced in 1967 by Browder and Petryshyn [19] is a superclass of the class of nonexpansive mappings and a subclass of the class of Lipschitz pseudocontractions. While pseudocontractive mappings are generally not continuous, the strictly pseudocontractivemappings inherit Lipschitz property from their definitions. The study of fixed point theory for strictly pseudocontractive mappings may help in the study of fixed point theory for nonexpansive mappings and for Lipschitz pseudocontractive mappings. Consequently, the study by several authors of iterative methods for fixed point ofmultivalued nonexpansive mappings has motivated the study in this paper of iterative methods for approximating fixed points of the more general strictly pseudocontractive mappings. Part of the novelty of this paper is that, even in the special case of multivalued nonexpansive mappings, convergence theorems are proved here for the Krasnoselskii-type sequencewhich is known to be superior to the Mann-type and Ishikawa-type sequences so far studied. It is worthmentioning here that iterativemethods for approximating fixed points of nonexpansive mappings constitute the central tools used in signal processing and image restoration (see, e.g., Byrne [20]). LetK be a nonempty subset of a normed space E. The set K is called proximinal (see, e.g., [21–23]) if for each x ∈ E there exists u ∈ K such that d (x, u) = inf {󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 : y ∈ K} = d (x,K) , (13) where d(x, y) = ‖x − y‖ for all x, y ∈ E. Every nonempty, closed, and convex subset of a real Hilbert space is proximinal. Let CB(K) and P(K) denote the families of nonempty, closed, and bounded subsets and of nonempty, proximinal, and bounded subsets ofK, respectively.TheHausdorff metric on CB(K) is defined by

Interest in such studies stems, perhaps, mainly from the usefulness of such fixed point theory in real-world applications, such as in Game Theory and Market Economy, and in other areas of mathematics, such as in Nonsmooth Differential Equations.We describe briefly the connection of fixed point theory of multivalued mappings and these applications.
Game Theory and Market Economy.In game theory and market economy, the existence of equilibrium was uniformly obtained by the application of a fixed point theorem.In fact, Nash [3,4] showed the existence of equilibria for noncooperative static games as a direct consequence of Brouwer [1] or Kakutani [2] fixed point theorem.More precisely, under some regularity conditions, given a game, there always exists a multivalued mapping whose fixed points coincide with the equilibrium points of the game.A model example of such an application is the Nash equilibrium theorem (see, e.g., [3]).
Consider a game  = (  ,   ) with  players denoted by ,  = 1, . . ., , where   ⊂ R   is the set of possible strategies of the th player and is assumed to be nonempty, compact, and convex, and   :  :=  1 ×  2 ⋅ ⋅ ⋅ ×   → R is the payoff (or gain function) of the player  and is assumed to be continuous.The player  can take individual actions, represented by a vector   ∈   .All players together can take a collective action, which is a combined vector  = ( 1 ,  2 , . . .,   ).For each ,  ∈  and   ∈   , we will use the following standard notations: − := ( 1 , . . .,  −1 ,  +1 , . . .,   ) , (  ,  − ) := ( 1 , . . .,  −1 ,   ,  +1 , . . .,   ) . (1) A strategy   ∈   permits the 'th player to maximize his gain under the condition that the remaining players have chosen their strategies  − if and only if   (  ,  − ) = max   ∈    (  ,  − ) . ( From the point of view of social recognition, game theory is perhaps the most successful area of application of fixed point theory of multivalued mappings.However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a nonequilibrium point and convergent to equilibrium solution.This is part of the problem that is being addressed by iterative methods for fixed point of multivalued mappings. Nonsmooth Differential Equations.The mainstream of applications of fixed point theory for multivalued mappings has been initially motivated by the problem of differential equations (DEs) with discontinuous right-hand sides which gave birth to the existence theory of differential inclusion (DIs).Here is a simple model for this type of application.Consider the initial value problem If  :  × R → R is discontinuous with bounded jumps, measurable in , one looks for solutions in the sense of Filippov [8,9] which are solutions of the differential inclusion where Now set  :=  2 () and let   :  → 2  be the multivalued NemyTskii operator defined by   () := {V ∈  : V () ∈  (,  ()) a.e. ∈ } .
Finally, let  :  → 2  be the multivalued mapping defined by  :=    −1 , where  −1 is the inverse of the derivative operator  =   given by One can see that problem (8) reduces to the fixed point problem:  ∈ .
Finally, a variety of fixed point theorems for multivalued mappings with nonempty and convex values is available to conclude the existence of solution.We used a first-order differential equation as a model for simplicity of presentation, but this approach is most commonly used with respect to second-order boundary value problems for ordinary differential equations or partial differential equations.For more about these topics, one can consult [10][11][12][13] and references therein as examples.
We have seen that a Nash equilibrium point is a fixed point  of a multivalued mapping  :  → 2  , that is, a solution of the inclusion  ∈  for some nonlinear mapping .This inclusion can be rewritten as 0 ∈ , where  :=  −  and  is the identity mapping on .
In this case, the solutions of the inclusion 0 ∈ (), if any, correspond to the critical points of , which are exactly its minimizer points.Also, the proximal point algorithm of Martinet [16] and Rockafellar [17] studied also by a host of authors is connected with iterative algorithm for approximating a solution of 0 ∈  where  is a maximal monotone operator on a Hilbert space.
In studying the equation  = 0, Browder introduced an operator  defined by  : − where  is the identity mapping on .He called such an operator  pseudocontractive.It is clear that solutions of  = 0 now correspond to fixed points of .In general, pseudocontraactive mappings are not continuous.However, in studying fixed point theory for pseudocontractive mappings, some continuity condition (e.g., Lipschitz condition) is imposed on the operator.An important subclass of the class of Lipschitz pseudocontractive mappings is the class of nonexpansive mappings, that is, mappings  :  →  such that ‖ − ‖ ≤ ‖ − ‖ for all ,  ∈ .Apart from being an obvious generalization of the contraction mappings, nonexpansive mappings are important, as has been observed by Bruck [18], mainly for the following two reasons.
(i) Nonexpansive mappings are intimately connected with the monotonicity methods developed since the early 1960s and constitute one of the first classes of nonlinear mappings for which fixed point theorems were obtained by using the fine geometric properties of the underlying Banach spaces instead of compactness properties.
(ii) Nonexpansive mappings appear in applications as the transition operators for initial value problems of differential inclusions of the form 0 ∈ (/) + (), where the operators {()} are, in general, setvalued and are accretive or dissipative and minimally continuous.
The class of strictly pseudocontractive mappings defined in Hilbert spaces which was introduced in 1967 by Browder and Petryshyn [19] is a superclass of the class of nonexpansive mappings and a subclass of the class of Lipschitz pseudocontractions.While pseudocontractive mappings are generally not continuous, the strictly pseudocontractive mappings inherit Lipschitz property from their definitions.The study of fixed point theory for strictly pseudocontractive mappings may help in the study of fixed point theory for nonexpansive mappings and for Lipschitz pseudocontractive mappings.Consequently, the study by several authors of iterative methods for fixed point of multivalued nonexpansive mappings has motivated the study in this paper of iterative methods for approximating fixed points of the more general strictly pseudocontractive mappings.Part of the novelty of this paper is that, even in the special case of multivalued nonexpansive mappings, convergence theorems are proved here for the Krasnoselskii-type sequence which is known to be superior to the Mann-type and Ishikawa-type sequences so far studied.It is worth mentioning here that iterative methods for approximating fixed points of nonexpansive mappings constitute the central tools used in signal processing and image restoration (see, e.g., Byrne [20]).
Sastry and Babu [21] introduced the following iterative schemes.Let  :  → () be a multivalued mapping, and let  * be a fixed point of .Define iteratively the sequence where   is a real sequence in (0,1) satisfying the following conditions: They also introduced the following scheme: where {  } and {  } are sequences of real numbers satisfying the following conditions: Sastry and Babu called a process defined by ( 16) a Mann iteration process and a process defined by (17) where the iteration parameters   and   satisfy conditions (i), (ii), and (iii) an Ishikawa iteration process.They proved in [21] that the Mann and Ishikawa iteration schemes for a multivalued mapping  with fixed point  converge to a fixed point of  under certain conditions.More precisely, they proved the following result for a multivalued nonexpansive mapping with compact domain.
Panyanak [22] extended the above result of Sastry and Babu [21] to uniformly convex real Banach spaces.He proved the following result.
Panyanak [22] also modified the iteration schemes of Sastry and Babu [21].Let  be a nonempty, closed, and convex subset of a real Banach space, and let  :  → () be a multivalued mapping such that () is a nonempty proximinal subset of .
Recently, Song and Wang [23] modified the iteration process due to Panyanak [22] and improved the results therein.They gave their iteration scheme as follows.
More recently, Shahzad and Zegeye [29] extended and improved the results of Sastry and Babu [21], Panyanak [22], and Son and Wang [23] to multivalued quasi-nonexpansive mappings.Also, in an attempt to remove the restriction  = {} for all  ∈ () in Theorem SW, they introduced a new iteration scheme as follows.
Theorem SZ (Shahzad and Zegeye [29]).Let  be a uniformly convex real Banach space, let  be a nonempty, closed, and convex subset of , and let  :  → () be a multivalued mapping with () ̸ = 0 such that   is nonexpansive.Let {  } be the Ishikawa iterates defined by (25).Assume that  satisfies condition (I) and   ,   ∈ [, ] ⊂ (0, 1).Then, {  } converges strongly to a fixed point of .Remark 4. In recursion formula (16), 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   ∈   satisfying       −  *     =  ( * ,   ) at every step of the iteration process.So, recursion formula ( 16) is not convenient to use in any possible application.Also, the recursion formula defined in (23) is not convenient to use in any possible application.The sequences {  } and {  } are not known precisely.Only their existence is guaranteed by Lemma 3. Unlike as in the case of formula ( 16), characterizations of {  } and {  } guaranteed by Lemma 3 are not even known.So, recursion formulas (23) are not really useable.
It is our purpose in this paper to first introduce the important class of multivalued strictly pseudocontractive mappings which is more general than the class of multivalued nonexpansive mappings.Then, we prove strong convergence theorems for this class of mappings.The recursion formula used in our more general setting is of the Krasnoselskii type [30] which is known to be superior (see, e.g., Remark 20) to the recursion formula of Mann [31] or Ishikawa [32].We achieve these results by means of an incisive result similar to the result of Nadler [6] which we prove in Lemma 7.

Preliminaries
In the sequel, we will need the following definitions and results.
Definition 5. Let  be a real Hilbert space and let  be a multivalued mapping.The multivalued mapping ( − ) is said to be strongly demiclosed at 0 (see, e.g., [27]) if for any sequence {  } ⊆ () such that {  } converges strongly to  * and (  ,   ) converges strongly to 0, then ( * ,  * ) = 0. Definition 6.Let  be a real Hilbert space.A multivalued mapping  : () ⊆  → () is said to be -strictly pseudocontractive if there exist  ∈ (0, 1) such that for all ,  ∈ () one has If  = 1 in (28), the mapping  is said to be pseudocontractive.We now prove the following lemma which will play a central role in the sequel.
Remark 9. We note that for a single-valued mapping , for each  ∈ (), the set  is always weakly closed.
We now prove the following lemma which will also be crucial in what follows.
Lemma 10.Let  be a nonempty and closed subset of a real Hilbert space  and let  :  → () be a -strictly pseudocontractive mapping.Assume that for every  ∈ , the set  is weakly closed.Then, ( − ) is strongly demiclosed at zero.

Main Results
We prove the following theorem.
We now prove the following corollaries of Theorem 11.

Corollary 12.
Let  be a nonempty, closed, and convex subset of a real Hilbert space , and let  :  → () be a multivalued -strictly pseudocontractive mapping with () ̸ = 0 such that  = {} for all  ∈ ().Suppose that  is hemicompact and continuous.Let {  } be a sequence defined by  0 ∈ , where   ∈   and  ∈ (0, 1 − ).Then, the sequence {  } converges strongly to a fixed point of .
Proof.From Theorem where   ∈   and  ∈ (0, 1).Then, the sequence {  } converges strongly to a fixed point of .

)
Proof.Let  ∈  and let {  } be a sequence of positive real numbers such that   → 0 as  → ∞.From Lemma 3, for each  ≥ 1, there exists   ∈  such that      −       ≤  (, ) +   .(30) It then follows that the sequence {  } is bounded.Since  is reflexive and  is weakly closed, there exists a subsequence {   } of {  } that converges weakly to some  ∈ .Now, using inequality (30), the fact that {−   } converges weakly to − and    → 0, as  → ∞, it follows that ‖ − ‖ ≤ lim inf       −         ≤  (, ) .Let  be a nonempty subset of a real Hilbert space  and let  :  → () be a multivalued -strictly pseudocontractive mapping.Assume that for every  ∈ , the set  is weakly closed.Then,  is Lipschitzian.