Volume 2013, Article ID 629468,10pages http://dx.doi.org/10.1155/2013/629468
Research Article
Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces
C. E. Chidume,
1C. O. Chidume,
2N. Djitté,
3and M. S. Minjibir
1,41Mathematics Institute, African University of Science and Technology, PMB 681, Garki, Abuja, Nigeria
2Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA
3Universit´e Gaston Berger, 234 Saint Louis, Senegal
4Department of Mathematical Sciences, Bayero University, PMB 3011, Kano, Nigeria
Correspondence should be addressed to C. E. Chidume; [email protected] Received 10 September 2012; Accepted 15 April 2013
Academic Editor: Josef Dibl´ık
Copyright © 2013 C. E. Chidume et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let𝐾be a nonempty, closed, and convex subset of a real Hilbert space𝐻. Suppose that𝑇 : 𝐾 → 2𝐾is a multivalued strictly pseudocontractive mapping such that𝐹(𝑇) ̸= 0. A Krasnoselskii-type iteration sequence {𝑥𝑛}is constructed and shown to be an approximate fixed point sequence of𝑇; that is, lim𝑛 → ∞𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0holds. Convergence theorems are also proved under appropriate additional conditions.
1. Introduction
For several years, the study of fixed point theory of multi- valued nonlinear mappingshas attracted, and continues to attract, the interest of several well-known mathematicians (see, e.g., Brouwer [1], Kakutani [2], Nash [3,4], Geanakoplos [5], Nadler [6], and Downing and Kirk [7]).
Interest in such studies stems, perhaps, mainly from the usefulness of such fixed point theory in real-world applica- tions, such as inGame Theory and Market Economy,and in other areas of mathematics, such as inNonsmooth 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 noncoop- erative 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 amultivalued mappingwhose fixed points coincide with the
equilibrium points of the game. A model example of such an application is theNash equilibrium theorem(see, e.g., [3]).
Consider a game𝐺 = (𝑢𝑛, 𝐾𝑛)with𝑁players denoted by𝑛, 𝑛 = 1, . . . , 𝑁, where𝐾𝑛 ⊂ R𝑚𝑛 is the set of possible strategies of the𝑛th player and is assumed to be nonempty, compact, and convex, and𝑢𝑛: 𝐾 := 𝐾1× 𝐾2⋅ ⋅ ⋅ × 𝐾𝑁 → R is the payoff (or gain function) of the player𝑛and is assumed to be continuous. The player𝑛can takeindividual actions, represented by a vector𝜎𝑛∈ 𝐾𝑛. All players together can take acollective action, which is a combined vector𝜎 = (𝜎1, 𝜎2, . . . , 𝜎𝑁). For each 𝑛,𝜎 ∈ 𝐾 and𝑧𝑛 ∈ 𝐾𝑛, we will use the following standard notations:
𝐾−𝑛:= 𝐾1× ⋅ ⋅ ⋅ × 𝐾𝑛−1× 𝐾𝑛+1× ⋅ ⋅ ⋅ × 𝐾𝑁, 𝜎−𝑛:= (𝜎1, . . . , 𝜎𝑛−1, 𝜎𝑛+1, . . . , 𝜎𝑁) , (𝑧𝑛, 𝜎−𝑛) := (𝜎1, . . . , 𝜎𝑛−1, 𝑧𝑛, 𝜎𝑛+1, . . . , 𝜎𝑁) .
(1)
A strategy 𝜎𝑛 ∈ 𝐾𝑛 permits the 𝑛’th player to maximize his gainunder the conditionthat theremaining playershave chosen their strategies𝜎−𝑛if and only if
𝑢𝑛(𝜎𝑛, 𝜎−𝑛) =max
𝑧𝑛∈𝐾𝑛𝑢𝑛(𝑧𝑛, 𝜎−𝑛) . (2)
Now, let𝑇𝑛: 𝐾−𝑛 → 2𝐾𝑛be the multivalued mapping defined by
𝑇𝑛(𝜎−𝑛) :=Arg max
𝑧𝑛∈𝐾𝑛
𝑢𝑛(𝑧𝑛, 𝜎−𝑛) ∀𝜎−𝑛∈ 𝐾−𝑛. (3) Definition 1. A collective action 𝜎 = (𝜎1, . . . , 𝜎𝑁) ∈ 𝐾 is called aNash equilibrium pointif, for each𝑛,𝜎𝑛is the best response for the𝑛’th player to the action𝜎−𝑛 made by the remaining players. That is, for each𝑛,
𝑢𝑛(𝜎) =max
𝑧𝑛∈𝐾𝑛𝑢𝑛(𝑧𝑛, 𝜎−𝑛) (4) or, equivalently,
𝜎𝑛∈ 𝑇𝑛(𝜎−𝑛) . (5) This is equivalent to that𝜎is a fixed point of the multivalued mapping𝑇 : 𝐾 → 2𝐾defined by
𝑇 (𝜎) := 𝑇1(𝜎−1) × 𝑇2(𝜎−2) × ⋅ ⋅ ⋅ × 𝑇𝑁(𝜎−𝑁) . (6) 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 appli- cations 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 𝑑𝑢
𝑑𝑡 = 𝑓 (𝑡, 𝑢) , a.e. 𝑡 ∈ 𝐼 := [−𝑎, 𝑎] , 𝑢 (0) = 𝑢0. (7) If𝑓 : 𝐼 ×R → Ris discontinuous with bounded jumps, measurable in 𝑡, one looks for solutions in the sense of Filippov [8,9] which are solutions of the differential inclusion
𝑑𝑢
𝑑𝑡 ∈ 𝐹 (𝑡, 𝑢) , a.e. 𝑡 ∈ 𝐼, 𝑢 (0) = 𝑢0, (8) where
𝐹 (𝑡, 𝑥) = [lim inf
𝑦 → 𝑥 𝑓 (𝑡, 𝑦) ,lim sup
𝑦 → 𝑥 𝑓 (𝑡, 𝑦)] . (9) Now set𝐻 := 𝐿2(𝐼)and let𝑁𝐹: 𝐻 → 2𝐻be themultivalued NemyTskii operatordefined by
𝑁𝐹(𝑢) := {V∈ 𝐻 :V(𝑡) ∈ 𝐹 (𝑡, 𝑢 (𝑡)) a.e. 𝑡 ∈ 𝐼} . (10)
Finally, let𝑇 : 𝐻 → 2𝐻be the multivalued mapping defined by𝑇 := 𝑁𝐹𝑜𝐿−1, where𝐿−1 is the inverse of the derivative operator𝐿𝑢 = 𝑢given by
𝐿−1V(𝑡) := 𝑢0+ ∫𝑡
0V(𝑠) 𝑑𝑠. (11) One can see that problem (8) reduces to the fixed point problem:𝑢 ∈ 𝑇𝑢.
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 differen- tial equations or partial differential equations. For more about these topics, one can consult [10–13] and references therein as examples.
We have seen that aNash equilibrium pointis a fixed point 𝜎of amultivalued mapping𝑇 : 𝐾 → 2𝐾, that is, a solution of the inclusion𝑥 ∈ 𝑇𝑥for some nonlinear mapping𝑇. This inclusion can be rewritten as0 ∈ 𝐴𝑥, where𝐴 := 𝐼 − 𝑇and𝐼 is the identity mapping on𝐾.
Many problems in applications can be modeled in the form0 ∈ 𝐴𝑥, where, for example,𝐴 : 𝐻 → 2𝐻is amonotone operator, that is, ⟨𝑢 −V, 𝑥 − 𝑦⟩ ≥ 0 for all 𝑢 ∈ 𝐴𝑥,V∈ 𝐴𝑦, 𝑥, 𝑦 ∈ 𝐻. Typical examples include the equilibrium state of evolution equationsand critical points of some functionals defined on Hilbert spaces 𝐻. Let 𝑓 : 𝐻 → (−∞, +∞]
be a proper, lower-semicontinuous convex function; then it is known (see, e.g., Rockafellar [14] or Minty [15]) that the multivalued mapping 𝑇 := 𝜕𝑓, thesubdifferential of 𝑓, is maximal monotone, where for𝑤 ∈ 𝐻,
𝑤 ∈ 𝜕𝑓 (𝑥) ⇐⇒ 𝑓 (𝑦) − 𝑓 (𝑥) ≥ ⟨𝑦 − 𝑥, 𝑤⟩ ∀𝑦 ∈ 𝐻
⇐⇒ 𝑥 ∈Arg min(𝑓 − ⟨⋅, 𝑤⟩) .
(12) In this case, the solutions of the inclusion0 ∈ 𝜕𝑓(𝑥), if any, correspond to the critical points of𝑓, which are exactly its minimizer points.
Also, theproximal point algorithm of Martinet [16] and Rockafellar [17] studied also by a host of authors is connected with iterative algorithm for approximating a solution of0 ∈ 𝐴𝑥where𝐴is a maximal monotone operator on a Hilbert space.
In studying the equation𝐴𝑢 = 0, Browder introduced an operator𝑇defined by𝑇 : 𝐼−𝐴where𝐼is the identity mapping on𝐻. He called such an operator𝑇pseudocontractive. It is clear that solutions of 𝐴𝑢 = 0 now correspond to fixed points of 𝑇. 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, mappings𝑇 : 𝐾 → 𝐾such that‖𝑇𝑥 − 𝑇𝑦‖ ≤ ‖𝑥 − 𝑦‖for all 𝑥, 𝑦 ∈ 𝐾. 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 compact- ness 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, set- valued and areaccretiveordissipativeandminimally 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 nonex- pansive mappings and a subclass of the class of Lipschitz pseudocontractions. While pseudocontractive mappings are generally not continuous, the strictly pseudocontractive map- pings 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 mappingshas 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 theKrasnoselskii-type sequencewhich 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 thecentral toolsused insignal processingandimage restoration(see, e.g., Byrne [20]).
Let𝐾be a nonempty subset of a normed space𝐸. The set 𝐾is calledproximinal(see, e.g., [21–23]) if for each 𝑥 ∈ 𝐸 there exists𝑢 ∈ 𝐾such that
𝑑 (𝑥, 𝑢) =inf{𝑥 − 𝑦 : 𝑦 ∈ 𝐾} = 𝑑(𝑥,𝐾), (13) where𝑑(𝑥, 𝑦) = ‖𝑥 − 𝑦‖for all𝑥, 𝑦 ∈ 𝐸. Every nonempty, closed, and convex subset of a real Hilbert space is proxim- inal. Let𝐶𝐵(𝐾)and𝑃(𝐾)denote the families of nonempty, closed, and bounded subsets and of nonempty, proximinal, and bounded subsets of𝐾, respectively. TheHausdorff metric on𝐶𝐵(𝐾)is defined by
𝐷 (𝐴, 𝐵) =max{sup
𝑎∈𝐴𝑑 (𝑎, 𝐵) ,sup
𝑏∈𝐵𝑑 (𝑏, 𝐴)} (14) for all𝐴, 𝐵 ∈ 𝐶𝐵(𝐾). Let𝑇 : 𝐷(𝑇) ⊆ 𝐸 → 𝐶𝐵(𝐸)be a multivalued mappingon𝐸. A point𝑥 ∈ 𝐷(𝑇)is called afixed point of 𝑇if𝑥 ∈ 𝑇𝑥. The fixed point set of𝑇is denoted by 𝐹(𝑇) := {𝑥 ∈ 𝐷(𝑇) : 𝑥 ∈ 𝑇𝑥}.
A multivalued mapping𝑇 : 𝐷(𝑇) ⊆ 𝐸 → 𝐶𝐵(𝐸)is called 𝐿-Lipschitzianif there exists𝐿 > 0such that
𝐷 (𝑇𝑥, 𝑇𝑦) ≤ 𝐿 𝑥 − 𝑦 ∀𝑥,𝑦 ∈ 𝐷(𝑇). (15) When𝐿 ∈ (0, 1)in (15), we say that𝑇is acontraction, and𝑇 is callednonexpansiveif𝐿 = 1.
Several papers deal with the problem of approximating fixed points ofmultivalued nonexpansivemappings (see, e.g., [21–26] and the references therein) and their generalizations (see, e.g., [27,28]).
Sastry and Babu [21] introduced the following iterative schemes. Let𝑇 : 𝐸 → 𝑃(𝐸)be a multivalued mapping, and let𝑥∗be a fixed point of𝑇. Define iteratively the sequence {𝑥𝑛}𝑛∈Nfrom𝑥0∈ 𝐸by
𝑥𝑛+1= (1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑦𝑛, 𝑦𝑛∈ 𝑇𝑥𝑛,
𝑦𝑛− 𝑥∗ = 𝑑(𝑇𝑥𝑛, 𝑥∗) , (16) where𝛼𝑛 is a real sequence in (0,1) satisfying the following conditions:
(i)∑∞𝑛=1𝛼𝑛= ∞, (ii) lim𝛼𝑛 = 0.
They also introduced the following scheme:
𝑦𝑛= (1 − 𝛽𝑛) 𝑥𝑛+ 𝛽𝑛𝑧𝑛, 𝑧𝑛∈ 𝑇𝑥𝑛,
𝑧𝑛− 𝑥∗ = 𝑑(𝑥∗, 𝑇𝑥𝑛) , 𝑥𝑛+1= (1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑢𝑛, 𝑢𝑛∈ 𝑇𝑦𝑛,
𝑢𝑛− 𝑥∗ = 𝑑(𝑇𝑦𝑛, 𝑥∗) ,
(17)
where{𝛼𝑛}and{𝛽𝑛}are sequences of real numbers satisfying the following conditions:
(i)0 ≤ 𝛼𝑛, 𝛽𝑛 < 1, (ii) lim𝑛 → ∞𝛽𝑛= 0, (iii)∑∞𝑛=1𝛼𝑛𝛽𝑛= ∞.
Sastry and Babu called a process defined by (16) a Mann iteration process and a process defined by (17) where the iteration parameters𝛼𝑛and𝛽𝑛satisfy conditions (i), (ii), and (iii) an Ishikawa iteration process. They proved in [21] that the Mann and Ishikawa iteration schemes for a multivalued mapping𝑇with fixed point 𝑝converge to a fixed point of 𝑇under certain conditions. More precisely, they proved the following result for a multivalued nonexpansive mapping with compact domain.
Theorem SB (Sastry and Babu [21]). Let 𝐻be real Hilbert space, let𝐾be a nonempty, compact, and convex subset of𝐻, and let𝑇 : 𝐾 → 𝐶𝐵(𝐾)be a multivalued nonexpansive map- ping with a fixed point𝑝. Assume that (i)0 ≤ 𝛼𝑛,𝛽𝑛 < 1, (ii)𝛽𝑛 → 0, and (iii)∑ 𝛼𝑛𝛽𝑛 = ∞. Then, the sequence{𝑥𝑛} defined by(17)converges strongly to a fixed point of𝑇.
Panyanak [22] extended the above result of Sastry and Babu [21] to uniformly convex real Banach spaces. He proved the following result.
Theorem P 1 (Panyanak [22]). Let𝐸be a uniformly convex real Banach space, and let 𝐾 be a nonempty, compact, and convex subset of 𝐸 and𝑇 : 𝐾 → 𝐶𝐵(𝐾) a multivalued nonexpansive mapping with a fixed point𝑝. Assume that (i) 0 ≤ 𝛼𝑛,𝛽𝑛< 1, (ii)𝛽𝑛 → 0, and (iii)∑ 𝛼𝑛𝛽𝑛 = ∞. Then, the sequence{𝑥𝑛}defined by(17)converges strongly to a fixed point of𝑇.
Panyanak [22] also modified the iteration schemes of Sastry and Babu [21]. Let𝐾be a nonempty, closed, and convex subset of a real Banach space, and let𝑇 : 𝐾 → 𝑃(𝐾)be a multivalued mapping such that𝐹(𝑇)is a nonempty proximi- nal subset of𝐾.
The sequence of Mann iterates is defined by𝑥0∈ 𝐾, 𝑥𝑛+1= (1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑦𝑛, 𝛼𝑛∈ [𝑎, 𝑏] , 0 < 𝑎 < 𝑏 < 1,
(18) where𝑦𝑛∈ 𝑇𝑥𝑛is such that‖𝑦𝑛− 𝑢𝑛‖ = 𝑑(𝑢𝑛, 𝑇𝑥𝑛)and𝑢𝑛 ∈ 𝐹(𝑇)is such that‖𝑥𝑛− 𝑢𝑛‖ = 𝑑(𝑥𝑛, 𝐹(𝑇)).
The sequence of Ishikawa iterates is defined by𝑥0∈ 𝐾, 𝑦𝑛= (1 − 𝛽𝑛) 𝑥𝑛+ 𝛽𝑛𝑧𝑛, 𝛽𝑛∈ [𝑎, 𝑏] , 0 < 𝑎 < 𝑏 < 1,
(19) where𝑧𝑛 ∈ 𝑇𝑥𝑛is such that‖𝑧𝑛− 𝑢𝑛‖ = 𝑑(𝑢𝑛, 𝑇𝑥𝑛)and𝑢𝑛 ∈ 𝐹(𝑇)is such that‖𝑥𝑛− 𝑢𝑛‖ = 𝑑(𝑥𝑛, 𝐹(𝑇)). The sequence is defined iteratively by the following way
𝑥𝑛+1= (1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑧𝑛, 𝛼𝑛∈ [𝑎, 𝑏] , 0 < 𝑎 < 𝑏 < 1, (20) where𝑧𝑛 ∈ 𝑇𝑦𝑛 is such that‖𝑧𝑛 −V𝑛‖ = 𝑑(V𝑛, 𝑇𝑦𝑛) and V𝑛∈ 𝐹(𝑇)is such that‖𝑦𝑛−V𝑛‖ = 𝑑(𝑦𝑛, 𝐹(𝑇)). Before we state his theorem, we need the following definition.
Definition 2. A mapping𝑇 : 𝐾 → 𝐶𝐵(𝐾)is said to satisfy condition (I)if there exists a strictly increasing function𝑓 : [0, ∞) → [0, ∞)with𝑓(0) = 0,𝑓(𝑟) > 0for all𝑟 ∈ (0, ∞) such that
𝑑 (𝑥, 𝑇 (𝑥)) ≥ 𝑓 (𝑑 (𝑥, 𝐹 (𝑇)) ∀𝑥 ∈ 𝐷. (21) Theorem P 2 (Panyanak [22]). Let𝐸be a uniformly convex real Banach space, let𝐾be a nonempty, closed, bounded, and convex subset of𝐸, and let𝑇 : 𝐾 → 𝑃(𝐾)be a multivalued nonexpansive mapping that satisfies condition (I). Assume that (i)0 ≤ 𝛼𝑛 < 1and (ii)∑ 𝛼𝑛 = ∞. Suppose that𝐹(𝑇)is a nonempty proximinal subset of𝐾. Then, the sequence{𝑥𝑛} defined by(18)converges strongly to a fixed point of𝑇.
Panyanak [22] then asked the following question.
Question (P).Is Theorem P2 true for the Ishikawa iteration defined by (19) and (20)?
For multivalued mappings, the following lemma is a con- sequence of the definition of Hausdorff metric, as remarked by Nadler [6].
Lemma 3. Let𝐴, 𝐵 ∈ 𝐶𝐵(𝑋)and𝑎 ∈ 𝐴. For every𝛾 > 0, there exists𝑏 ∈ 𝐵such that
𝑑 (𝑎, 𝑏) ≤ 𝐷 (𝐴, 𝐵) + 𝛾. (22) 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.
Let𝐾be a nonempty, closed, and convex subset of a real Banach space, and let𝑇 : 𝐾 → 𝐶𝐵(𝐾)be a multivalued mapping. Let𝛼𝑛, 𝛽𝑛 ∈ [0, 1]and 𝛾𝑛 ∈ (0, ∞)be such that lim𝑛 → ∞𝛾𝑛 = 0. Choose𝑥0∈ 𝐾,
𝑦𝑛= (1 − 𝛽𝑛) 𝑥𝑛+ 𝛽𝑛𝑧𝑛,
𝑥𝑛+1= (1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑢𝑛, (23) where𝑧𝑛∈ 𝑇𝑥𝑛and𝑢𝑛∈ 𝑇𝑦𝑛are such that
𝑧𝑛− 𝑢𝑛 ≤ 𝐷(𝑇𝑥𝑛, 𝑇𝑦𝑛) + 𝛾𝑛,
𝑧𝑛+1− 𝑢𝑛 ≤ 𝐷(𝑇𝑥𝑛+1, 𝑇𝑦𝑛) + 𝛾𝑛. (24) They then proved the following result.
Theorem SW (Song and Wang [23]). Let𝐾be a nonempty, compact and convex subset of a uniformly convex real Banach space𝐸. Let𝑇 : 𝐾 → 𝐶𝐵(𝐾)be a multivalued nonexpansive mapping with𝐹(𝑇) ̸= 0satisfying𝑇(𝑝) = {𝑝}for all𝑝 ∈ 𝐹(𝑇).
Assume that (i) 0 ≤ 𝛼𝑛,𝛽𝑛 < 1, (ii) 𝛽𝑛 → 0, and (iii)
∑ 𝛼𝑛𝛽𝑛 = ∞. Then, the Ishikawa sequence defined by (23) converges strongly to a fixed point of𝑇.
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 restriction𝑇𝑝 = {𝑝}for all𝑝 ∈ 𝐹(𝑇)in Theorem SW, they introduced a new iteration scheme as follows.
Let𝐾be a nonempty, closed, and convex subset of a real Banach space, and let𝑇 : 𝐾 → 𝑃(𝐾) be a multivalued mapping and𝑃𝑇𝑥 := {𝑦 ∈ 𝑇𝑥 : ‖𝑥 − 𝑦‖ = 𝑑(𝑥, 𝑇𝑥)}. Let 𝛼𝑛, 𝛽𝑛 ∈ [0, 1]. Choose𝑥0∈ 𝐾, and define{𝑥𝑛}as follows:
𝑦𝑛= (1 − 𝛽𝑛) 𝑥𝑛+ 𝛽𝑛𝑧𝑛,
𝑥𝑛+1= (1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑢𝑛, (25) where𝑧𝑛 ∈ 𝑃𝑇𝑥𝑛 and𝑢𝑛 ∈ 𝑃𝑇𝑦𝑛. They then proved the fol- lowing result.
Theorem SZ (Shahzad and Zegeye [29]). Let 𝑋 be a uni- formly convex real Banach space, let 𝐾 be a nonempty, closed, and convex subset of 𝑋, and let𝑇 : 𝐾 → 𝑃(𝐾) be a multivalued mapping with 𝐹(𝑇) ̸= 0 such that 𝑃𝑇 is nonexpansive. Let{𝑥𝑛}be the Ishikawa iterates defined by(25).
Assume that𝑇satisfies condition (I) and𝛼𝑛, 𝛽𝑛 ∈ [𝑎, 𝑏] ⊂ (0, 1). Then,{𝑥𝑛}converges strongly to a fixed point of𝑇.
Remark 4. In recursion formula (16), the authors take𝑦𝑛 ∈ 𝑇(𝑥𝑛)such that‖𝑦𝑛− 𝑥∗‖ = 𝑑(𝑥∗, 𝑇𝑥𝑛). The existence of𝑦𝑛 satisfying this condition is guaranteed by the assumption that
𝑇𝑥𝑛is proximinal. In general such a𝑦𝑛is extremely difficult to pick. If𝑇𝑥𝑛is proximinal, it is not difficult to prove that it is closed. If, in addition, it is a convex subset of a real Hilbert space, then𝑦𝑛isuniqueand is characterized by
⟨𝑥∗− 𝑦𝑛, 𝑦𝑛− 𝑢𝑛⟩ ≥ 0 ∀𝑢𝑛∈ 𝑇𝑥𝑛. (26) One can see from this inequality that it is not easy to pick 𝑦𝑛∈ 𝑇𝑥𝑛satisfying
𝑦𝑛− 𝑥∗ = 𝑑(𝑥∗, 𝑇𝑥𝑛) (27) 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{𝑢𝑛}and{𝑧𝑛}are not known precisely. Only their existenceis guaranteed by Lemma 3. Unlike as in the case of formula (16), characteri- zations of{𝑢𝑛}and{𝑧𝑛}guaranteed byLemma 3are not even known. So, recursion formulas (23) are not really useable.
It is our purpose in this paper to first introduce the impor- tant class ofmultivalued strictly pseudocontractive mappings which is more general than the class ofmultivalued nonexpan- sive mappings. Then, we prove strong convergence theorems for this class of mappings. The recursion formula used in our more general setting is of theKrasnoselskii 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 inLemma 7.
2. Preliminaries
In the sequel, we will need the following definitions and results.
Definition 5. Let𝐻be a real Hilbert space and let 𝑇be a multivalued mapping. The multivalued mapping(𝐼 − 𝑇) is said to bestrongly demiclosedat 0 (see, e.g., [27]) if for any sequence{𝑥𝑛} ⊆ 𝐷(𝑇)such that{𝑥𝑛}converges strongly to𝑥∗ and𝑑(𝑥𝑛, 𝑇𝑥𝑛)converges strongly to 0, then𝑑(𝑥∗, 𝑇𝑥∗) = 0.
Definition 6. Let𝐻be a real Hilbert space. A multivalued mapping𝑇 : 𝐷(𝑇) ⊆ 𝐻 → 𝐶𝐵(𝐻)is said to be𝑘-strictly pseudocontractiveif there exist𝑘 ∈ (0, 1) such that for all 𝑥, 𝑦 ∈ 𝐷(𝑇)one has
(𝐷 (𝑇𝑥, 𝑇𝑦))2
≤ 𝑥 − 𝑦2+ 𝑘(𝑥 − 𝑢) − (𝑦 −V)2 ∀𝑢 ∈ 𝑇𝑥, V∈ 𝑇𝑦.
(28) If𝑘 = 1in (28), the mapping𝑇is said to bepseudocontractive.
We now prove the following lemma which will play a central role in the sequel.
Lemma 7. Let𝐸be a reflexive real Banach space and let𝐴, 𝐵 ∈ 𝐶𝐵(𝑋). Assume that𝐵is weakly closed. Then, for every𝑎 ∈ 𝐴, there exists𝑏 ∈ 𝐵such that
‖𝑎 − 𝑏‖ ≤ 𝐷 (𝐴, 𝐵) . (29)
Proof. Let𝑎 ∈ 𝐴and let{𝜆𝑛}be a sequence of positive real numbers such that𝜆𝑛 → 0as𝑛 → ∞. FromLemma 3, for each𝑛 ≥ 1, there exists𝑏𝑛∈ 𝐵such that
𝑎 − 𝑏𝑛 ≤ 𝐷(𝐴,𝐵) + 𝜆𝑛. (30) It then follows that the sequence{𝑏𝑛}is bounded. Since𝐸is reflexive and𝐵is weakly closed, there exists a subsequence {𝑏𝑛𝑘}of{𝑏𝑛}that converges weakly to some𝑏 ∈ 𝐵. Now, using inequality (30), the fact that{𝑎−𝑏𝑛𝑘}converges weakly to𝑎−𝑏 and𝜆𝑛𝑘 → 0, as𝑘 → ∞, it follows that
‖𝑎 − 𝑏‖ ≤lim inf𝑎 − 𝑏𝑛𝑘 ≤ 𝐷 (𝐴, 𝐵) . (31) This proves the lemma.
Proposition 8. Let𝐾be a nonempty subset of a real Hilbert space𝐻and let𝑇 : 𝐾 → 𝐶𝐵(𝐾)be a multivalued𝑘-strictly pseudocontractive mapping. Assume that for every𝑥 ∈ 𝐾, the set𝑇𝑥is weakly closed. Then,𝑇is Lipschitzian.
Proof. Let𝑥, 𝑦 ∈ 𝐷(𝑇)and𝑢 ∈ 𝑇𝑥. FromLemma 7, there existsV∈ 𝑇𝑦such that
‖𝑢 −V‖ ≤ 𝐷 (𝑇𝑥, 𝑇𝑦) . (32) Using the fact that 𝑇 is 𝑘-strictly pseudocontractive, and inequality (32), we obtain the following estimates:
(𝐷 (𝑇𝑥, 𝑇𝑦))2≤ 𝑥 − 𝑦2+ 𝑘(𝑥 − 𝑢) − (𝑦 −V)2
≤ (𝑥 − 𝑦 + √𝑘𝑥 − 𝑢 − (𝑦 −V))2, (33) so that
𝐷 (𝑇𝑥, 𝑇𝑦) ≤ 𝑥 − 𝑦 + √𝑘(𝑥 − 𝑦 + ‖𝑢 −V‖)
≤ 𝑥 − 𝑦 + √𝑘(𝑥 − 𝑦 + 𝐷(𝑇𝑥,𝑇𝑦)).
(34) Hence,
𝐷 (𝑇𝑥, 𝑇𝑦) ≤ (1 + √𝑘
1 − √𝑘) 𝑥 − 𝑦. (35) Therefore,𝑇is𝐿-Lipschitzian with𝐿 =: (1+√𝑘)/(1−√𝑘).
Remark 9. We note that for a single-valued mapping𝑇, for each𝑥 ∈ 𝐷(𝑇), the set𝑇𝑥is always weakly closed.
We now prove the following lemma which will also be crucial in what follows.
Lemma 10. Let𝐾be a nonempty and closed subset of a real Hilbert space 𝐻and let 𝑇 : 𝐾 → 𝑃(𝐾) be a 𝑘-strictly pseudocontractive mapping. Assume that for every𝑥 ∈ 𝐾, the set𝑇𝑥is weakly closed. Then,(𝐼 − 𝑇)is strongly demiclosed at zero.
Proof. Let{𝑥𝑛} ⊆ 𝐾be such that𝑥𝑛 → 𝑥and𝑑(𝑥𝑛, 𝑇𝑥𝑛) → 0as𝑛 → ∞. Since𝐾is closed, we have that𝑥 ∈ 𝐾. Since, for every𝑛,𝑇𝑥𝑛is proximinal, let𝑦𝑛 ∈ 𝑇𝑥nsuch that‖𝑥𝑛− 𝑦𝑛‖ = 𝑑(𝑥𝑛, 𝑇𝑥𝑛). UsingLemma 7, for each𝑛, there exists𝑧𝑛 ∈ 𝑇𝑥 such that
𝑦𝑛− 𝑧𝑛 ≤ 𝐷(𝑇𝑥𝑛, 𝑇𝑥) . (36) We then have
𝑥 − 𝑧𝑛 ≤ 𝑥 − 𝑥𝑛 + 𝑥𝑛− 𝑦𝑛 + 𝑦𝑛− 𝑧𝑛
≤ 𝑥 − 𝑥𝑛 + 𝑥𝑛− 𝑦𝑛 + 𝐷(𝑇𝑥𝑛, 𝑇𝑥)
≤ 𝑥 − 𝑥𝑛 + 𝑥𝑛− 𝑦𝑛 + (1 + √𝑘)
(1 − √𝑘)𝑥𝑛− 𝑥 . (37) Observing that𝑑(𝑥, 𝑇𝑥) ≤ ‖𝑥 − 𝑧𝑛‖, it then follows that
𝑑 (𝑥, 𝑇𝑥) ≤ 𝑥 − 𝑥𝑛 + 𝑥𝑛− 𝑦𝑛 + (1 +√𝑘
1 − √𝑘) 𝑥𝑛− 𝑥 . (38) Taking the limit as𝑛 → ∞, we have that𝑑(𝑥, 𝑇𝑥) = 0. There- fore,𝑥 ∈ 𝑇𝑥, completing the proof.
3. Main Results
We prove the following theorem.
Theorem 11. Let𝐾be a nonempty, closed, and convex subset of a real Hilbert space𝐻. Suppose that𝑇 : 𝐾 → 𝐶𝐵(𝐾)is a multivalued𝑘-strictly pseudocontractive mapping such that 𝐹(𝑇) ̸= 0. Assume that𝑇𝑝 = {𝑝}for all𝑝 ∈ 𝐹(𝑇). Let{𝑥𝑛}be a sequence defined by𝑥0∈ 𝐾,
𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (39) where𝑦𝑛∈ 𝑇𝑥𝑛and𝜆 ∈ (0, 1−𝑘). Then,lim𝑛 → ∞𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0.
Proof. Let 𝑝 ∈ 𝐹(𝑇). We have the following well-known identity:
𝑡𝑥 + (1 − 𝑡)𝑦2
= 𝑡‖𝑥‖2+ (1 − 𝑡) 𝑦2− 𝑡 (1 − 𝑡) 𝑥 − 𝑦2, (40) which holds for all𝑥, 𝑦 ∈ 𝐻 and for all𝑡 ∈ [0, 1]. Using inequality (28) and the assumption that 𝑇𝑝 = {𝑝} for all 𝑝 ∈ 𝐹(𝑇), we obtain the following estimates:
𝑥𝑛+1− 𝑝2= (1 − 𝜆) (𝑥𝑛− 𝑝) + 𝜆 (𝑦𝑛− 𝑝)2
= (1 − 𝜆) 𝑥𝑛− 𝑝2+ 𝜆𝑦𝑛− 𝑝2
− 𝜆 (1 − 𝜆) 𝑥𝑛− 𝑦𝑛2
≤ (1 − 𝜆) 𝑥𝑛− 𝑝2+ 𝜆(𝐷 (𝑇𝑥𝑛, 𝑇𝑝))2
− 𝜆 (1 − 𝜆) 𝑥𝑛− 𝑦𝑛2
≤ (1 − 𝜆) 𝑥𝑛− 𝑝2
+ 𝜆 (𝑥𝑛− 𝑝2+ 𝑘𝑥𝑛− 𝑦𝑛2)
− 𝜆 (1 − 𝜆) 𝑥𝑛− 𝑦𝑛2
= 𝑥𝑛− 𝑝2+ 𝜆𝑘𝑥𝑛− 𝑦𝑛2
− 𝜆 (1 − 𝜆) 𝑥𝑛− 𝑦𝑛2
(41)
= 𝑥𝑛− 𝑝2− 𝜆 (1 − 𝑘 − 𝜆) 𝑥𝑛− 𝑦𝑛2. (42) It then follows that
𝜆 (1 − 𝑘 − 𝜆)∑∞
𝑛=1𝑥𝑛− 𝑦𝑛2≤ 𝑥0− 𝑝2 (43) which implies that
∑∞
𝑛=1𝑥𝑛− 𝑦𝑛2< ∞. (44) Hence, lim𝑛 → ∞‖𝑥𝑛− 𝑦𝑛‖ = 0. Since𝑦𝑛 ∈ 𝑇𝑥𝑛, we have that lim𝑛 → ∞𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0.
A mapping𝑇 : 𝐾 → 𝐶𝐵(𝐾)is called hemicompactif, for any sequence{𝑥𝑛}in𝐾such that𝑑(𝑥𝑛, 𝑇𝑥𝑛) → 0as𝑛 →
∞, there exists a subsequence{𝑥𝑛𝑘}of{𝑥𝑛}such that𝑥𝑛𝑘 → 𝑝 ∈ 𝐾. We note that if𝐾is compact, then every multivalued mapping𝑇 : 𝐾 → 𝐶𝐵(𝐾)is hemicompact.
We now prove the following corollaries ofTheorem 11.
Corollary 12. Let𝐾be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇 : 𝐾 → 𝐶𝐵(𝐾) be a multivalued 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇) ̸= 0such that𝑇𝑝 = {𝑝}for all𝑝 ∈ 𝐹(𝑇). Suppose that𝑇 is hemicompact and continuous. Let{𝑥𝑛}be a sequence defined by𝑥0∈ 𝐾,
𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (45) where𝑦𝑛 ∈ 𝑇𝑥𝑛and𝜆 ∈ (0, 1 − 𝑘). Then, the sequence{𝑥𝑛} converges strongly to a fixed point of𝑇.
Proof. FromTheorem 11, we have that lim𝑛 → ∞𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0. Since𝑇is hemicompact, there exists a subsequence{𝑥𝑛𝑘} of{𝑥𝑛} such that𝑥𝑛𝑘 → 𝑞as 𝑘 → ∞for some𝑞 ∈ 𝐾.
Since𝑇is continuous, we also have𝑑(𝑥𝑛𝑘, 𝑇𝑥𝑛𝑘) → 𝑑(𝑞, 𝑇𝑞) as𝑘 → ∞. Therefore,𝑑(𝑞, 𝑇𝑞) = 0and so𝑞 ∈ 𝐹(𝑇). Setting 𝑝 = 𝑞in the proof ofTheorem 11, it follows from inequality (42) that lim𝑛 → ∞‖ 𝑥𝑛−𝑞 ‖exists. So,{𝑥𝑛}converges strongly to𝑞. This completes the proof.
Corollary 13. Let 𝐾 be a nonempty, compact, and convex subset of a real Hilbert space𝐻, and let𝑇 : 𝐾 → 𝐶𝐵(𝐾) be a multivalued 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇) ̸= 0such that𝑇𝑝 = {𝑝}for all𝑝 ∈ 𝐹(𝑇). Suppose that 𝑇is continuous. Let{𝑥𝑛}be a sequence defined by𝑥0∈ 𝐾,
𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (46) where𝑦𝑛 ∈ 𝑇𝑥𝑛and𝜆 ∈ (0, 1 − 𝑘). Then, the sequence{𝑥𝑛} converges strongly to a fixed point of𝑇.
Proof. Observing that if𝐾 is compact, every mapping 𝑇 : 𝐾 → 𝐶𝐵(𝐾) is hemicompact, the proof follows from Corollary 12.
Corollary 14. Let𝐾be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇 : 𝐾 → 𝐶𝐵(𝐾)be a multivalued nonexpansive mapping such that𝑇𝑝 = {𝑝} for all𝑝 ∈ 𝐹(𝑇). Suppose that𝑇is hemicompact. Let{𝑥𝑛}be a sequence defined by𝑥0∈ 𝐾,
𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (47) where𝑦𝑛 ∈ 𝑇𝑥𝑛and𝜆 ∈ (0, 1). Then, the sequence{𝑥𝑛}con- verges strongly to a fixed point of𝑇.
Proof. Since 𝑇is nonexpansive and hemicompact, then it is strictly pseudocontractive, hemicompact, and continuous.
So, the proof follows fromCorollary 12.
Remark 15. InCorollary 12, the continuity assumption on𝑇 can be dispensed with if we assume that for every𝑥 ∈ 𝐾,𝑇𝑥is proximinal and weakly closed. In fact, we have the following result.
Corollary 16. Let𝐾be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇 : 𝐾 → 𝑃(𝐾) be a multivalued𝑘-strictly pseudocontractive mapping with𝐹(𝑇) ̸=
0such that for every𝑥 ∈ 𝐾,𝑇𝑥is weakly closed and𝑇𝑝 = {𝑝}
for all𝑝 ∈ 𝐹(𝑇). Suppose that𝑇is hemicompact. Let{𝑥𝑛}be a sequence defined by𝑥0∈ 𝐾,
𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (48) where𝑦𝑛 ∈ 𝑇𝑥𝑛and𝜆 ∈ (0, 1 − 𝑘). Then, the sequence{𝑥𝑛} converges strongly to a fixed point of𝑇.
Proof. Following the same arguments as in the proof of Corollary 12, we have𝑥𝑛𝑘 → 𝑞and lim𝑛 → ∞𝑑(𝑥𝑛𝑘, 𝑇𝑥𝑛𝑘) = 0.
Furthermore, fromLemma 10,(𝐼 − 𝑇)is strongly demiclosed at zero. It then follows that 𝑞 ∈ 𝑇𝑞. Setting 𝑝 = 𝑞 and following the same computations as in the proof of Theorem 11, we have from inequality (42) that lim‖𝑥𝑛 − 𝑞‖
exists. Since{𝑥𝑛𝑘}converges strongly to𝑞, it follows that{𝑥𝑛} converges strongly to𝑞 ∈ 𝐹(𝑇), completing the proof.
Corollary 17. Let𝐾be a nonempty, closed, and convex subset of a real Hilbert space𝐻, and let𝑇 : 𝐾 → 𝑃(𝐾)be a multi- valued 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇) ̸= 0 such that for every𝑥 ∈ 𝐾,𝑇𝑥is weakly closed and𝑇𝑝 = {𝑝}
for all𝑝 ∈ 𝐹(𝑇). Suppose that𝑇satisfies condition (I). Let{𝑥𝑛} be a sequence defined by𝑥0∈ 𝐾,
𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (49) where𝑦𝑛 ∈ 𝑇𝑥𝑛and𝜆 ∈ (0, 1 − 𝑘). Then, the sequence{𝑥𝑛} converges strongly to a fixed point of𝑇.
Proof. FromTheorem 11, we have that lim𝑛 → ∞𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0. Using the fact that𝑇satisfies condition (I), it follows that
lim𝑛 → ∞𝑓(𝑑(𝑥𝑛, 𝐹(𝑇))) = 0. Thus there exist a subsequence {𝑥𝑛𝑘}of{𝑥𝑛}and a sequence{𝑝𝑘} ⊂ 𝐹(𝑇)such that
𝑥𝑛𝑘− 𝑝𝑘 < 1
2𝑘 ∀𝑘. (50)
By setting𝑝 = 𝑝𝑘and following the same arguments as in the proof ofTheorem 11, we obtain from inequality (42) that
𝑥𝑛𝑘+1− 𝑝𝑘 ≤𝑥𝑛𝑘− 𝑝𝑘 < 1
2𝑘. (51)
We now show that{𝑝𝑘}is a Cauchy sequence in𝐾. Notice that
𝑝𝑘+1− 𝑝𝑘 ≤ 𝑝𝑘+1− 𝑥𝑛𝑘+1 +𝑥𝑛𝑘+1− 𝑝𝑘
< 1 2𝑘+1+ 1
2𝑘
< 1 2𝑘−1.
(52)
This shows that{𝑝𝑘}is a Cauchy sequence in𝐾and thus con- verges strongly to some𝑞 ∈ 𝐾. Using the fact that𝑇is𝐿- Lipschitzian and𝑝𝑘 → 𝑞, we have
𝑑 (𝑝𝑘, 𝑇𝑞) ≤ 𝐷 (𝑇𝑝𝑘, 𝑇𝑞)
≤ 𝐿 𝑝𝑘− 𝑞 , (53) so that𝑑(𝑞, 𝑇𝑞) = 0and thus𝑞 ∈ 𝑇𝑞. Therefore,𝑞 ∈ 𝐹(𝑇)and {𝑥𝑛𝑘}converges strongly to𝑞. Setting𝑝 = 𝑞in the proof of Theorem 11, it follows from inequality (42) that lim𝑛 → ∞‖𝑥𝑛− 𝑞‖exists. So,{𝑥𝑛}converges strongly to𝑞. This completes the proof.
Corollary 18. Let𝐾be a nonempty compact convex subset of a real Hilbert space𝐻, and let𝑇 : 𝐾 → 𝑃(𝐾)be a multivalued 𝑘-strictly pseudocontractive mapping with𝐹(𝑇) ̸= 0such that for every𝑥 ∈ 𝐾,𝑇𝑥is weakly closed and𝑇𝑝 = {𝑝}for all 𝑝 ∈ 𝐹(𝑇). Let{𝑥𝑛}be a sequence defined by𝑥0∈ 𝐾,
𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (54) where𝑦𝑛 ∈ 𝑇𝑥𝑛and𝜆 ∈ (0, 1 − 𝑘). Then, the sequence{𝑥𝑛} converges strongly to a fixed point of𝑇.
Proof. FromTheorem 11, we have that lim𝑛 → ∞𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0. Since{𝑥𝑛} ⊆ 𝐾and𝐾is compact,{𝑥𝑛}has a subsequence {𝑥𝑛𝑘}that converges strongly to some𝑞 ∈ 𝐾. Furthermore, fromLemma 10, (𝐼 − 𝑇) is strongly demiclosed at zero. It then follows that𝑞 ∈ 𝑇𝑞. Setting𝑝 = 𝑞and following the same arguments as in the proof ofTheorem 11, we have from inequality (42) that lim‖𝑥𝑛− 𝑞‖exists. Since{𝑥𝑛𝑘}converges strongly to𝑞, it follows that{𝑥𝑛}converges strongly to𝑞 ∈ 𝐹(𝑇). This completes the proof.
Corollary 19. Let 𝐾 be a nonempty, compact, and convex subset of a real Hilbert space𝐸, and let𝑇 : 𝐾 → 𝑃(𝐾)be a multivalued nonexpansive mapping. Assume that𝑇𝑝 = {𝑝}
for all𝑝 ∈ 𝐹(𝑇). Let{𝑥𝑛}be a sequence defined by𝑥0∈ 𝐾, 𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (55) where𝑦𝑛 ∈ 𝑇𝑥𝑛and𝜆 ∈ (0, 1). Then, the sequence{𝑥𝑛}con- verges strongly to a fixed point of𝑇.
Remark 20. Recursion formula (39) of Theorem 11 is the Krasnoselskii type (see, e.g., [30]) and is known to be superior than the recursion formula of the Mann algorithm (see, e.g., Mann [31]) in the following sense.
(i) Recursion formula (39) requires less computation time than the Mann algorithm because the parameter 𝜆in formula (39) is fixed in(0, 1 − 𝑘), whereas in the algorithm of Mann,𝜆is replaced by a sequence{𝑐𝑛}in (0, 1)satisfying the following conditions:∑∞𝑛=1𝑐𝑛= ∞ and lim𝑐𝑛 = 0. The𝑐𝑛must be computed at each step of the iteration process.
(ii) The Krasnoselskii-type algorithm usually yields rate of convergence as fast as that of a geometric progres- sion, whereas the Mann algorithm usually has order of convergence of the form𝑜(1/𝑛).
Remark 21. Any consideration of the Ishikawa iterative algo- rithm (see, e.g., [32]) involvingtwoparameters (two sequen- ces in(0, 1)) for the above problem is completely undesirable.
Moreover, the rate of convergence of the Ishikawa-type algorithm is generally of the form𝑜(1/√𝑛)and the algorithm requires a lot more computation than even the Mann process.
Consequently, the question asked in [22],Question (P)above, whether an Ishikawa-type algorithm will converge (when it was already known that a Mann-type process converges) has no merit.
Remark 22. Our theorem and corollaries improve conver- gence theorems for multivalued nonexpansive mappings in [21–23,25,26,28] in the following sense.
(i) In our algorithm,𝑦𝑛 ∈T𝑥𝑛is arbitrary and does not have to satisfy the very restrictive condition‖𝑦𝑛 − 𝑥∗‖ = 𝑑(𝑥∗, 𝑇𝑥𝑛) in recursion formula (16), and similar restrictions in recursion formula (17). These restrictions on𝑦𝑛depend on𝑥∗, a fixed point that is being approximated.
(ii) The algorithms used in our theorem and corol- laries which are proved for the much larger class of multivalued strict pseudocontractions are of the Krasnoselskii type.
Remark 23. In [29], the authors replace the condition𝑇𝑝 = {𝑝}for all𝑝 ∈ 𝐹(𝑇)with the following two restrictions: (i) on the sequence{𝑦𝑛}:𝑦𝑛 ∈ 𝑃𝑇𝑥𝑛, for example,𝑦𝑛 ∈ 𝑇𝑥𝑛 and
‖𝑦𝑛 − 𝑥𝑛‖ = 𝑑(𝑥𝑛, 𝑇𝑥𝑛). We observe that if𝑇𝑥𝑛 is a closed convex subset of a real Hilbert space, then𝑦𝑛is unique and is characterized by
⟨𝑥𝑛− 𝑦𝑛, 𝑦𝑛− 𝑢𝑛⟩ ≥ 0 ∀𝑢𝑛 ∈ 𝑇𝑥𝑛; (56) (ii) on𝑃𝑇: the authors demand that𝑃𝑇be nonexpansive. So, the first restriction makes the recursion formula difficult to use in any possible application, while the second restriction reduces the class of mappings to which the results are applicable. This is the price to pay for removing the condition 𝑇𝑝 = {𝑝}for all𝑝 ∈ 𝐹(𝑇).
Remark 24. Corollary 12 is an extension of Theorem 12 of Browder and Petryshyn [19] from single-valued to multival- ued strictly pseudocontractive mappings.
Remark 25. A careful examination of our proofs in this paper reveals that all our results have carried over to the class of multivalued quasinonexpansive mappings.
Remark 26. The addition ofboundederror terms to the recur- sion formula (39) leads to no generalization.
We conclude this paper with examples where for each𝑥 ∈ 𝐾,𝑇𝑥is proximinal and weakly closed.
Example 27. Let 𝑓 : R → R be an increasing function.
Define𝑇 :R → 2Rby
𝑇𝑥 = [𝑓 (𝑥−) , 𝑓 (𝑥+)] ∀𝑥 ∈R, (57) where𝑓(𝑥−) := lim𝑦 → 𝑥−𝑓(𝑦)and𝑓(𝑥+) := lim𝑦 → 𝑥+𝑓(𝑦).
For every𝑥 ∈ R, 𝑇𝑥is either a singleton or a closed and bounded interval. Therefore,𝑇𝑥is always weakly closed and convex. Hence, for every𝑥 ∈R, the set𝑇𝑥is proximinal and weakly closed.
Example 28. Let𝐻be a real Hilbert space, and let𝑓 : 𝐻 → R be a convex continuous function. Let𝑇 : 𝐻 → 2𝐻be the multivalued mapping defined by
𝑇𝑥 = 𝜕𝑓 (𝑥) ∀𝑥 ∈ 𝐻, (58)
where𝜕𝑓(𝑥)is the subdifferential of𝑓at𝑥which is defined by
𝜕𝑓 (𝑥) = {𝑧 ∈ 𝐻 : ⟨𝑧, 𝑦 − 𝑥⟩ ≤ 𝑓 (𝑦) − 𝑓 (𝑥) ∀𝑦 ∈ 𝐻} . (59) It is well known that for every𝑥 ∈ 𝐻, 𝜕𝑓(𝑥)is nonempty, weakly closed, and convex. Therefore, since𝐻is a real Hilbert space, it then follows that for every𝑥 ∈ 𝐻, the set𝑇𝑥 is proximinal and weakly closed.
The condition𝑇𝑝 = {𝑝}for all𝑝 ∈ 𝐹(𝑇)which is imposed in all our theorems of this paper is not crucial. Our emphasis in this paper is to show that a Krasnoselskii-type sequence converges. It is easy to construct trivial examples for which this condition is satisfied. We do not do this. Instead, we show how this condition can be replaced with another condition which does not assume that the multivalued mapping is single-valued on the nonempty fixed point set. This can be found in the paper by Shahzad and Zegeye [29].
Let𝐾be a nonempty, closed, and convex subset of a real Hilbert space, let𝑇 : 𝐾 → 𝑃(𝐾)be a multivalued mapping, and let𝑃𝑇: 𝐾 → 𝐶𝐵(𝐾)be defined by
𝑃𝑇(𝑥) := {𝑦 ∈ 𝑇𝑥 : 𝑥 − 𝑦 = 𝑑(𝑥,𝑇𝑥)}. (60) We will need the following result.
Lemma 29 (Song and Cho [33]). Let𝐾be a nonempty subset of a real Banach space, and let𝑇 : 𝐾 → 𝑃(𝐾)be a multivalued mapping. Then, the following are equivalent:
(i)𝑥∗∈ 𝐹(𝑇);
(ii)𝑃𝑇(𝑥∗) = {𝑥∗};
(iii)𝑥∗∈ 𝐹(𝑃𝑇).
Moreover,𝐹(𝑇) = 𝐹(𝑃𝑇).
Remark 30. We observe from Lemma 29that if𝑇 : 𝐾 → 𝑃(𝐾) is any multivalued mapping with 𝐹(𝑇) ̸= 0, then the corresponding multivalued mapping𝑃𝑇satisfies𝑃𝑇(𝑝) = {𝑝}
for all𝑝 ∈ 𝐹(𝑃𝑇), condition imposed in all our theorems and corollaries. Consequently, examples of multivalued mappings 𝑇 : 𝐾 → 𝐶𝐵(𝐾)satisfying the condition𝑇𝑝 = {𝑝}for all 𝑝 ∈ 𝐹(𝑇)abound.
Furthermore, we now prove the following theorem where we dispense with the condition𝑇𝑝 = {𝑝}for all𝑝 ∈ 𝐹(𝑇).
Theorem 31. Let𝐾be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇 : 𝐾 → 𝑃(𝐾) be a multivalued mapping such that𝐹(𝑇) ̸= 0. Assume that𝑃𝑇 is 𝑘-strictly pseudocontractive. Let {𝑥𝑛} be a sequence defined iteratively from arbitrary point𝑥1∈ 𝐾by
𝑥𝑛+1= (1 − 𝜆) 𝑥𝑛+ 𝜆𝑦𝑛, (61) where𝑦𝑛 ∈ 𝑃𝑇(𝑥𝑛)and𝜆 ∈ (0, 1 − 𝑘). Then,lim𝑛 → ∞𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0.
Proof. Let 𝑝 ∈ 𝐹(𝑇). We have the following well-known identity:
𝑡𝑥 + (1 − 𝑡)𝑦2
= 𝑡‖𝑥‖2+ (1 − 𝑡) 𝑦2− 𝑡 (1 − 𝑡) 𝑥 − 𝑦2, (62) which holds for all𝑥, 𝑦 ∈ 𝐻 and for all𝑡 ∈ [0, 1]. Using recursion formula (61), the identity (62), the fact that𝑃𝑇 is 𝑘-strictly pseudocontractive, and Lemma 29, we obtain the following estimates:
𝑥𝑛+1− 𝑝2= (1 − 𝜆) (𝑥𝑛− 𝑝) + 𝜆 (𝑦𝑛− 𝑝)2
= (1 − 𝜆) 𝑥𝑛− 𝑝2+ 𝜆𝑦𝑛− 𝑝2
− 𝜆 (1 − 𝜆) 𝑥𝑛− 𝑦𝑛2
≤ (1 − 𝜆) 𝑥𝑛− 𝑝2+ 𝜆[𝐷 (𝑃𝑇(𝑥𝑛) , 𝑃𝑇(𝑝))]2
− 𝜆 (1 − 𝜆) 𝑥𝑛− 𝑦𝑛2
≤ (1 − 𝜆) 𝑥𝑛− 𝑝2
+ 𝜆 (𝑥𝑛− 𝑝2+ 𝑘𝑥𝑛− 𝑦𝑛2)
− 𝜆 (1 − 𝜆) 𝑥𝑛− 𝑦𝑛2
= 𝑥𝑛− 𝑝2+ 𝜆𝑘𝑥𝑛− 𝑦𝑛2
− 𝜆 (1 − 𝜆) 𝑥𝑛− 𝑦𝑛2
= 𝑥𝑛− 𝑝2− 𝜆 (1 − 𝑘 − 𝜆) 𝑥𝑛− 𝑦𝑛2.
(63)
It then follows that 𝜆 (1 − 𝑘 − 𝜆)∑∞
𝑛=1𝑥𝑛− 𝑦𝑛2≤ 𝑥0− 𝑝2, (64) which implies that
∑∞
𝑛=1𝑥𝑛− 𝑦𝑛2< ∞. (65) Hence, lim𝑛 → ∞‖ 𝑥𝑛−𝑦𝑛‖= 0. Since𝑦𝑛∈ 𝑃𝑇(𝑥𝑛)(and hence, 𝑦𝑛 ∈ 𝑇𝑥𝑛), we have that lim𝑛 → ∞𝑑(𝑥𝑛, 𝑇𝑥𝑛) = 0, completing the proof.
We conclude this paper with examples of multivalued mappings𝑇for which𝑃𝑇is strictly pseudocontractive, a con- dition assumed inTheorem 31. Trivially, every nonexpansive mapping is strictly pseudocontractive.
Example 32. Let𝐻 =R, with the usual metric and𝑇 :R → 𝐶𝐵(R)be the multivalued mapping defined by
𝑇𝑥 = {{ {{ {{ {
[0,𝑥
2] , 𝑥 ∈ (0, ∞) , [𝑥
2, 0] , 𝑥 ∈ (−∞, 0] .
(66)
Then𝑃𝑇is strictly pseudocontractive. In fact,𝑃𝑇𝑥 = {𝑥/2}for all𝑥 ∈R.
Example 33. The following example is given in Shahzad and Zegeye [29]. Let𝐾be nonempty subset of a normed space 𝐸. A multivalued mapping 𝑇 : 𝐾 → 𝐶𝐵(𝐸) is called
∗-nonexpansive (see, e.g., [34]) if for all𝑥, 𝑦 ∈ 𝐾and𝑢𝑥∈ 𝑇𝑥 with‖𝑥−𝑢𝑥‖ = 𝑑(𝑥, 𝑇𝑥), there exists𝑢𝑦∈ 𝑇𝑦with‖𝑦−𝑢𝑦‖ = 𝑑(𝑦, 𝑇𝑦)such that
𝑢𝑥− 𝑢𝑦 ≤𝑥 − 𝑦. (67) It is clear that if𝑇is∗-nonexpansive, then𝑃𝑇is nonexpansive and hence, strictly pseudocontractive. We also note that
∗-nonexpansiveness is different from nonexpansiveness for multivalued mappings. Let K =[0, +∞), and let𝑇be defined by𝑇𝑥 = [𝑥, 2𝑥]for𝑥 ∈ 𝐾. Then,𝑃𝑇(𝑥) = {𝑥}for𝑥 ∈ 𝐾and thus it is nonexpansive and hence strictly pseudocontractive.
Note also that𝑇is∗-nonexpansive but is not nonexpansive (see [35]).
Acknowledgment
The authors thank the referees for their comments and remarks that helped to improve the presentation of this paper.
References
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