approximation of fixed points of multivalued ... · approximation of fixed points of multivalued...

Approximation of Fixed Points of Multivalued Demicontractive andMultivalued Hemicontractive Mappings in Hilbert Spaces

B. G. AkuchuDepartment of Mathematics

University of NigeriaNsukka

e-mail: [email protected]

ABSTRACT: We give control conditions for the approximation of fixed points of multi-valued demicontractive and multivalued hemicontractive maps recently introduced in [1].Many of our conditions are weaker than the conditions used in [1], hence our results im-prove and complement the convergence results in [1] and the references therein.

Keywords and phrases: Proximinal sets, Hilbert spaces, multivalued k-strictlypseudocontractive-type mappings, multivalued pseudocontractive-type mappings,multivalued demicontractive mappings, multivalued hemicontractive mappings.

2000 Mathematical Subject Classification: 47H10; 54H25

1. INTRODUCTION

Let E be a normed space. A subset K of E is called proximinal if for each x ∈ E thereexists k ∈ K such that

||x− k|| = inf{||x− y|| : y ∈ K} = d(x,K).It is known that every closed convex subset of a uniformly convex Banach space is prox-iminal. In fact if K is a closed and convex subset of a uniformly convex Banach space X,then for any x ∈ X there exists a unique point ux ∈ K such that (see for e.g.[24])

||x− ux|| = inf{||x− y|| : y ∈ K} = d(x,K)

We shall denote the family of all nonempty proximinal subsets of X by P (X), the familyof all nonempty closed, convex and bounded subsets of X by CV B(X), the family of allnonempty closed and bounded subsets of X by CB(X) and the family of all nonemptysubsets of X by 2X for a nonempty set X. Let CB(E) be the family of all nonemptyclosed and bounded subsets of a normed space E. Let H be the Hausdorff metric inducedby the metric d on E, that is for every A,B ∈ 2E,

H(A,B) = max{supa∈A d(a,B), supb∈B d(b, A)}.

If A,B ∈ CB(E), then

H(A,B) = inf{ε > 0 : A ⊆ N(ε, B) and B ⊆ N(ε, A)},

where N(ε, C) =⋃c∈C{x ∈ E : d(x, c) ≤ ε}. Let E be a normed space. Let T : D(T ) ⊆

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International Journal of Scientific & Engineering Research, Volume 7, Issue 12, December-2016 ISSN 2229-5518

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E → 2E be a multivalued mapping on E. A point x ∈ D(T ) is called a fixed pointof T if x ∈ Tx. The set F (T ) = {x ∈ D(T ) : x ∈ Tx} is called the fixed point setof T . A point x ∈ D(T ) is called a strict fixed point of T if Tx = {x}. The setFs(T ) = {x ∈ D(T ) : Tx = {x}} is called the strict fixed point set of T . A multivaluedMapping T : D(T ) ⊆ E → 2E is called L − Lipschitzian if there exists L > 0 such thatfor any pair x, y ∈ D(T )

H(Tx, Ty) ≤ L||x− y||. (1)

In (1) if L ∈ [0, 1) T is said to be a contraction while T is nonexpansive if L = 1. T iscalled quasi− nonexpansive if F (T ) = {x ∈ D(T ) : x ∈ Tx} 6= ∅ and for all p ∈ F (T ),

H(Tx, Tp) ≤ ||x− p||. (2)

Clearly every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive.In recent years, much work has been done on the approximation of fixed points of nonlinearmultivalued mappings(see e.g. [4], [5], [6], [8], [9], [13], [16] and references therein), usingthe multivalued equivalence of single valued iterative sequences employed by several au-thors(see e.g. [10], [11], [12]). Different iterative schemes have been introduced by severalauthors to approximate the fixed points of multivalued nonexpansive mappings (see forexample [4], [5], [6]). Sastry and Babu [3] introduced a Mann-type and an Ishikawa-typeiteration schemes as follows:Let T : X → P (X) and p be a fixed point of T . The sequence of Mann iterates is generatedfrom an arbitrary x0 ∈ X by

xn+1 = (1− αn)xn + αnyn, ∀n ≥ 0 (3)

where yn ∈ Txn is such that ||yn − p|| = d(Txn, p) and αn is a real sequence in (0,1)∑∞n=1 αn =∞.

The sequence of Ishikawa iterates is given by{yn = (1− βn)xn + βnznxn+1 = (1− αn)xn + αnun,

(4)

where zn ∈ Txn, un ∈ Tyn are such ||zn− p|| = d(p, Txn), ||un− p|| = d(Tyn, p) and {αn},{βn} are real sequences satisfying:(i) 0 ≤ αn, βn < 1; (ii) lim

n→∞βn = 0; (iii)

∑∞n=1 αnβ =∞.

Using the above iterative schemes, Panyanak [4] generalized the results proved in [3].Nadler [6] made the following useful Remark, presented as a lemma:

Lemma [6]: let A,B ∈ CB(X) and a ∈ A. If γ > 0 then there exists b ∈ B such that

d(a, b) ≤ H(A,B) + γ. (5)

Using lemma 1, Song and Wang [5] modified the iteration process due to Panyanak [4]and improved the results therein. They made the important observation that generatingthe Mann and Ishikawa sequences in [3] is in some sense dependent on the knowledge ofthe fixed point. They gave their iteration scheme as follows:Let K be a nonempty convex subset of X, Let αn, βn ∈ [0, 1] and γn ∈ (0,∞) such thatlimn→∞

γn = 0. Choose x1 ∈ K, z1 ∈ Tx1. Let

y1 = (1− β1)x1 + β1z1.

Choose u1 ∈ Ty1 such that ||z1 − u1|| ≤ H(Tx1, T y1) + γ1 and

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x2 = (1− α1)x1 + α1u1.

Choose z2 ∈ Tx2 such that ||z2 − u1|| ≤ H(Tx2, T y1) + γ2 and

y2 = (1− β2)x2 + β2z2.

Choose u2 ∈ Ty2 such that ||z2 − u2|| ≤ H(Tx2, T y2) + γ2 and

x3 = (1− α2)x2 + α1u2

Inductively, we have {yn = (1− βn)xn + βnznxn+1 = (1− αn)xn + αnun,

(6)

where zn ∈ Txn, un ∈ Tyn satisfy ||zn − un|| ≤ H(Txn, T yn) + γn, ||zn+1 − un|| ≤H(Txn+1, T yn) + γn and {αn},{βn} are real sequences in [0, 1) satisfying lim

n→∞βn = 0,∑∞

n=1 αnβn =∞.Using the above iteration scheme, they proved the following theorem:

Theorem 1(Theorem 1 [5]): Let K be a nonempty compact convex subset of a uni-formly convex Banach space X. Suppose that T : K → CB(K) is a multivalued nonex-pansive mapping such that F (T ) 6= ∅ and T (p) = {p} for all p ∈ F (T ). Then the Ishikawasequence defined as above converges strongly to a fixed point of T .Shahzad and Zegeye [16] observed that if X is a normed space and T : D(T ) ⊆ E → P (X)is a multivalued mapping then the mapping PT : D(T )→ P (X) defined for each x by

PT (x) = {y ∈ Tx : d(x, Tx) = ‖x− y‖}

has the property that PT (q) = {q} for all q ∈ F (T ). Using this idea they removed thestrong condition “T (p) = {p} for all p ∈ F (T )” introduced by Song and Wang [5].Khan and Yildirim [17] introduced a new iteration scheme for multivalued nonexpansivemappings using the idea of the iteration scheme for single valued nearly asymptoticallynonexpansive mapping introduced by Agarwal et.al [15] as follows:

x1 ∈ K,xn+1 = (1− λ)vn + λunyn = (1− η)xn + ηvn,∀n ∈ N.

(7)

where vn ∈ PT (xn), un ∈ PT (yn) and λ ∈ [0, 1). Also, using a lemma in Schu [18], the ideaof removal of the condition “T (p) = {p} for all p ∈ F (T )” introduced by Shahzad andZegeye [16], and the method of direct construction of a Cauchy sequence as indicated bySong and Cho [14], they proved the following theorems:

Theorem 2(Theorem 2 [17]: Let X be a uniformly convex Banach space satisfyingOpial’s condition and K a nonempty closed convex subset of X. Let T : K → P (K) be amultivalued mapping such that F (T ) 6= ∅ and PT is a nonexpansive mapping. Let {xn}be the sequence as defined in (7). Let (I − PT ) be demiclosed with respect to zero. Then{xn} converges weakly to a point of F (T ).

Recently, Isiogugu [2], introduced the new classes of multivalued nonexpansive-type, mul-tivalued k-strictly pseudocontractive-type and multivalued pseudocontractive-type map-pings which are more general than the class of multivalued nonexpansive mappings, where:Definitions 1: Let X be a normed space. A multivalued mapping T : D(T ) ⊆ X → 2X

is said to be k-strictly pseudocontractive-type in the sense of Browder and Petryshyn [25]

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if there exist k ∈ [0, 1) such that given any x, y ∈ D(T ) and u ∈ Tx, there exits v ∈ Tysatisfying, ‖u− v‖ ≤ H(Tx, Ty) and

H2(Tx, Ty) ≤ ||x− y||2 + k‖x− u− (y − v)‖2 (8)

If k = 1 in (8) T is said to be a pseudocontractive-type mapping. T is called multivaluednonexpansive-type if k = 0. Clearly, every multivalued nonexpansive mapping is bothmultivalued k-strictly pseudocontractive-type and multivalued pseudocontractive-type.Using the above definitions, the author proved weak and strong convergence theorems forthese classes of multivalued mappings without compactness condition on the domain ofthe mappings using Mann and Ishikawa iteration schemes.

More recently, Isiogugu and Osilike [1], introduced the new classes of multivalued demicontractive-type and multivalued hemicontractive-type mappings which are more general than the classof multivalued quasi-nonexpansive mappings and which are also related to: the classesof multivalued k-strictly pseudocontractive-type and multivalued pseudocontractive -typemappings introduced in [2], single valued mappings of Browder and Petryshyn [25], Hicksand Kubicek [20]. The authors gave the following definitions:

Definition 2 [1]: Let X be a normed space. A multivalued mapping T : D(T ) ⊆X → 2X is said to be demicontractive in the terminology of Hicks and Kubicek [20] ifF (T ) 6= ∅ and for all p ∈ F (T ), x ∈ D(T ) there exists k ∈ [0, 1) such that

H2(Tx, Ty) ≤ ||x− p||2 + kd2(x, Tx),

where H2(Tx, Ty) = [H(Tx, Ty)]2 and d2(x, p) = [d(x, p)]2. If k = 1, then T is calledhemicontractive.Using the above definitions, the authors stated and proved the following results:

Theorem 3 [1]: Let K be a nonempty closed and convex subset of a real Hilbert spaceH. Suppose that T : K → P (K) is a demicontractive mapping from K into the familyof all proximinal subsets of K with k ∈ (0, 1) and T (p) = {p} for all p ∈ F (T ). Suppose(I − T ) is weakly demiclosed at zero. Then the Mann type sequence defined by

xn+1 = (1− αn)xn + αnyn

converges weakly to q ∈ F (T ), where yn ∈ Txn and αn is a real sequence in (0, 1) satisfy-ing: (i) αn → α < 1− k; (ii) α > 0;

Theorem 4 [1]: Let K be a nonempty closed and convex subset of a real Hilbert spaceH. Suppose that T : K → P (K) is an L-Lipschitzian hemicontractive mapping from Kinto the family of all proximinal subsets of K such that T (p) = {p} for all p ∈ F (T ).Suppose T satisfies condition (1). Then the Ishikawa sequence defined by{

yn = (1− βn)xn + βnunxn+1 = (1− αn)xn + αnwn.

(9)

converges strongly to p ∈ F (T ), where un ∈ Txn and wn ∈ Tyn satisfying the conditionsof lemma 3.1 and {αn} and {βn} are real sequences satisfying (i) 0 ≤ αn ≤ βn < 1;(ii)lim infn→∞

αn = α > 0; (iii) supn≥1

βn ≤ β ≤ 1√1+L2+1

.

Corollary 1 [1]: Let H be a real Hilbert space and K be a nonempty closed and con-vex subset of H. Suppose that T : K → P (K) is a k-strictly pseudocontractive-type

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mapping from K into the family of all proximinal subsets of K with k ∈ (0, 1) such thatF (T ) 6= ∅ and Tp = {p} for all p ∈ F (T ). Suppose (I − T ) is weakly demiclosed at zero.Then the Mann sequence {xn} defined in theorem 3.1 converges weakly to a point of F (T ).

Corollary 2 [1]: Let H be a real Hilbert space and K a nonempty closed and convexsubset of H. Let T : K → P (K) be a multivalued mapping from K into the family of allproximinal subsets of K. Suppose PT is an demicontractive mapping with k ∈ (0, 1) and(I − PT ) is weakly demiclosed at zero. Then the Mann sequence {xn} defined in theorem3.1 converges weakly to a point of F (T ).

Corollary 3 [1]: Let H be a real Hilbert space and K be a nonempty closed and convexsubset of H. Suppose that T : K → P (K) is an L-Lipschitzian pseudocontractive-typemapping from K into the family of all proximinal subsets of K such that F (T ) 6= ∅ andTp = {p} for all p ∈ F (T ). Suppose T satisfies condition (1). Then the Ishikawa sequence{xn} defined in (3.10) converges strongly to p ∈ F (T ).

Corollary 4 [1]: Let H be a real Hilbert space and K a nonempty closed and convexsubset of H. Let T : K → P (K) be a multivalued mapping from K into the family of allproximinal subsets of K such that F (T ) 6= ∅. Suppose PT is an L-Lipschitzian hemicon-tractive mapping and (I − PT ). If T satisfies condition (1). Then the Ishikawa sequence{xn} defined in (3.10) converges strongly to p ∈ F (T ).

We observe that many of the conditions on the control sequences for which the aboveresults are proven are strong. For example, the condition lim

n→∞αn → α imposed on {αn}

in theorem 3 is strong. This condition disenfranchises many non-convergent sequencesin (0, 1), for which the results can be proven. Furthermore, the condition sup

n≥1βn ≤ β in

theorem 4 can be dropped and replaced with a milder condition.

It is our purpose in this article to give different sets of control conditions, which are milderthan the above condition, under which the results still hold. Furthermore, we drop thecondition sup

n≥1βn ≤ β in theorem 4. Some of our computations and analyses pass through

the subsequence of a subsequence arguments

Before we proceed to the main results, we state some lemmas and give some definitionswhich will be useful in the sequel.

Lemma 1 [26]: Let {an}, {bn} and {cn} be sequences of nonnegative real numberssatisfying

an+1 ≤ (1 + bn)an + cn, n ≥ n0,

where n0 is a nonnegative integer. If∑bn <∞,

∑cn <∞ . Then lim

n→∞an exists.

Lemma 2 [14]: Let K be a normed space. Let T : K → P (K) be a multivalued mappingand PT (x) = {y ∈ Tx : ‖x− y‖ = d(x, Tx)}. Then the following are equivalent:(1) x ∈ Tx;(2) PTx = {x};(3) x ∈ F (PT ).Moreover, F (T ) = F (PT ).

Definition 3 [5]: A multivalued mapping T : K → P (K) is said to satisfy condition

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(1) (see for example [4] if there exists a nondecreasing function f : [0,∞) → [0,∞) withf(0) = 0 and f(r) > 0 for all r ∈ (0,∞) such that

d(x, Tx) ≥ f(d(x, F (T )), ∀x ∈ K

Definition 4(see e.g. [19], [23]: Let E be a Banach space. Let T : D(T ) ⊆ E → 2E

be a multivalued mapping. I − T is said to be weakly demiclosed at zero if for anysequence {xn}∞n=1 ⊆ D(T ) such that {xn} converges weakly to p and a sequence {yn} withyn ∈ Txn for all n ∈ N such that {xn− yn} converges strongly to zero. Then p ∈ Tp (i.e.,0 ∈ (I − T )p).

Main ResultsTheorem 5: Let K be a nonempty closed and convex subset of a real Hilbert space H.Suppose that T : K → P (K) is a multivalued demicontractive mapping from K into thefamily of all proximinal subsets of K with k ∈ (0, 1) such that T (p) = {p} for all p ∈ F (T ).Suppose (I-T) is weakly demiclosed at zero. Then the Mann type sequence defined by


converges weakly to q ∈ F (T ), where yn ∈ Txn and αn is a real sequence in (0,1) satisfying:(i) 0 < a1 ≤ αn ≤ a2 < 1− k, for some a1, a2 ∈ (0, 1).Proof. Using the well known identity

||tx+ (1− t)y||2 = t||x||2 + (1− t)||y||2 − t(1− t)||x− y||2

which holds for all x, y ∈ H and for all t ∈ [0, 1], p ∈ F (T ) we have

||xn+1 − p||2 = ||(1− αn)xn + αnyn − p||2

= ||(1− αn)(xn − p) + αn(yn − p)||2

= (1− αn)||xn − p||2 + αn||yn − p||2 − αn(1− αn)||xn − yn||2

≤ (1− αn)||xn − p||2 + αnH2(Txn, Tp)− αn(1− αn)||xn − yn||2

≤ (1− αn)||xn − p||2 + αn

[‖xn − p‖2 + k||xn − yn||2

]−αn(1− αn)||xn − yn||2

= ||xn − p||2 + αnk||xn − yn||2 − αn(1− αn)||xn − yn||2

= ||xn − p||2 − αn(1− (αn + k))||xn − yn||2

It then follows from (i) and lemma 1 that limn→∞

‖xn − p‖ exists. Hence {xn} is bounded.

Also,

∞∑n=1

a1(1− (a2 + k))||xn − yn||2 ≤∞∑n=1

αn(1− (αn + k))||xn − yn||2

≤ ||x0 − p||2 <∞.

This yields limn→∞

||xn− yn|| = 0. Also since K is closed and {xn} ⊆ K with {xn} bounded,

there exist a subsequence {xnt} of {xn} such that {xnt} converges weakly to some q ∈ K.Also lim

n→∞||xn − yn|| = 0 implies that lim

t→∞||xnt − ynt || = 0. Since (I − T ) is weakly demi-

closed at zero we have that q ∈ Tq. Since H satisfies Opial’s condition (see [22]), we havethat {xn} converges weakly to q ∈ F (T ). 2

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Theorem 6: Let K be a nonempty closed and convex subset of a real Hilbert space H.Suppose that T : K → P (K) is a multivalued demicontractive mapping from K into thefamily of all proximinal subsets of K with k ∈ (0, 1) such that T (p) = {p} for all p ∈ F (T ).Suppose (I-T) is weakly demiclosed at zero. Then the Mann type sequence defined by


converges weakly to q ∈ F (T ), where yn ∈ Txn and αn is a real sequence in (0,1) satisfyingthe condition (i) 0 < αn < 1− k, (ii) 0 < α = lim inf αn

Proof: Using the well known identity

||tx+ (1− t)y||2 = t||x||2 + (1− t)||y||2 − t(1− t)||x− y||2

which holds for all x, y ∈ H and for all t ∈ [0, 1], we compute as in theorem 5 to have

||xn+1 − p||2 ≤ ||xn − p||2 − αn(1− (αn + k))||xn − yn||2

It then follows from (i) and lemma 1 that limn→∞

‖xn − p‖ exists. Hence {xn} is bounded.

Also,∞∑n=1

αn(1− (αn + k))||xn − yn||2 ≤ ||x0 − p||2 <∞.

This implies limαn(1−(αn+k))||xn−yn||2 = 0. Using (ii), this yields lim inf ||xn−yn||2 = 0.That is lim inf ||xn − yn|| = 0. This implies there exists a subsequence {xnk

− ynk} of

{xn − yn} such that lim ||xnk− ynk

|| = 0. Also since K is closed and {xn} ⊆ K with{xn} bounded, then {xnk

} is also bounded. Then there exist a subsequence {xnkj} of

{xnk} such that {xnkj

} converges weakly to some q ∈ K. Since lim ||xnk− ynk

|| = 0,

then limj→∞||xnkj

− ynkj|| = 0. Since (I − T ) is weakly demiclosed at zero we have that

q ∈ Tq. Since H satisfies Opial’s condition (see [22]), we have that {xn} converges weaklyto q ∈ F (T ). 2

Prototype: A prototype of the sequence {αn} which satisfies the conditions of theorems5 and 6 is

{αn} =

12, if n is even

13, if n is odd

Remark: Observe that all the conditions in our theorems do not require that {αn} be aconvergent sequence as in [1].

Theorem 7: Let K be a nonempty closed and convex subset of a real Hilbert space H.Let T : K → P (K) be an L-Lipschitzian hemicontractive-type mapping from K into thefamily of all proximinal subsets of K and T (p) = {p} for all p ∈ F (T ). Suppose T satisfiescondition (1). Then the Ishikawa sequence defined by

yn = (1− βn)xn + βnun

xn+1 = (1− αn)xn + αnwn,

converges strongly to p ∈ F (T ), where un ∈ Txn, wn ∈ Tyn and {αn}, {βn} are realsequences satisfying (i) 0 ≤ αn ≤ βn ≤ 1√

1+L2+1(ii) 0 < α = lim inf αn = lim inf βn

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Proof: Using the well known identity||tx+ (1− t)y||2 = t||x||2 + (1− t)||y||2 − t(1− t)||x− y||2which holds for all x, y ∈ H and for all t ∈ [0, 1], p ∈ F (T ) we have

||xn+1 − p||2 = ||(1− αn)xn + αnwn − p||2

= ||(1− αn)(xn − p) + αn(wn − p)||2

= (1− αn)||xn − p||2 + αn||wn − p||2

−αn(1− αn)||xn − wn||2

≤ (1− αn)||xn − p||2 + αnH2(Tyn, Tp)

−αn(1− αn)||xn − wn||2

≤ (1− αn)||xn − p||2 + αn

[||yn − p||2

+||yn − wn||2]− αn(1− αn)||xn − wn||2

= (1− αn)||xn − p||2 + αn||yn − p||2 + αn||yn − wn||2

−αn(1− αn)||xn − wn||2 (10)Also,

||yn − wn||2 = ||(1− βn)xn + βnun − wn||2

= ||(1− βn)(xn − wn) + βn(un − wn)||2

= (1− βn)||xn − wn||2 + βn||un − wn||2

−βn(1− βn)||xn − un||2 (11)(10) and (11) imply that

||xn+1 − p||2 ≤ (1− αn)||xn − p||2 + αn||yn − p||2

+αn

[(1− βn)||xn − wn||2 + βn||un − wn||2

−βn(1− βn)||xn − un||2]

−αn(1− αn)||xn − wn||2 (12)

||yn − p||2 = ||(1− βn)xn + βnun − p||2

= ||(1− βn)(xn − p) + βn(un − p)||2

= (1− βn)||xn − p||2 + βn||un − p||2

−βn(1− βn)||xn − un||2

≤ (1− βn)||xn − p||2 + βnH2(Txn, Tp)

−βn(1− βn)||xn − un||2

≤ (1− βn)||xn − p||2 + βn

[||xn − p||2 + ||xn − un||2

]−βn(1− βn)||xn − un||2

= (1− βn)||xn − p||2 + βn||xn − p||2 + βn||xn − un||2

−βn(1− βn)||xn − un||2

= ||xn − p||2 + β2n||xn − un||2. (13)

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(12) and (13) imply that

||xn+1 − p||2 ≤ (1− αn)||xn − p||2

+αn

[||xn − p||2 + β2

n||xn − un||2]

+αn

[(1− βn)||xn − wn||2 + βn||un − wn||2

−βn(1− βn)||xn − un||2]

−αn(1− αn)||xn − wn||2

= (1− αn)||xn − p||2 + αn||xn − p||2 + αnβ2n||xn − un||2

+αn(1− βn)||xn − wn||2 + αnβn||un − wn||2

−αnβn(1− βn)||xn − un||2 − αn(1− αn)||xn − wn||2

≤ ||xn − p||2 + αnβ2n||xn − un||2 + αnβnH

2(Txn, T yn)

−αn(βn − αn)||xn − wn||2

−αnβn(1− βn)||xn − un||2

≤ ||xn − p||2 + αnβ2n||xn − un||2 + αnβ

3nL

2‖xn − un‖2

−αnβn(1− βn)||xn − un||2


= ||xn − p||2 − αnβn[1− 2βn − L2β2n]||xn − un||2


= ||xn − p||2 − αnβn[1− 2βn − L2β2n]||xn − un||2 (14)

It then follows from condition (i) and Lemma 1 that limn→∞

‖xn − p‖ exists. Hence {xn} is

bounded, so also are {un} and {wn}. We have from (14) that

∞∑n=0

αnβn[1− 2βn − L2β2n]||xn − un||2 ≤

∞∑n=0

[||xn − p||2 − ||xn+1 − p||2]

≤ ||x0 − p||2 +D <∞

This implies that limαnβn[1 − 2βn − L2β2n]||xn − un||2 = 0. It then follows from (ii) that

lim inf ||xn − un|| = 0. Then there exists a subsequence {xnj− unj

} of {xn − un} suchthat lim ||xnj

− unj|| = 0. Since unj

∈ Txnjwe have that d(xnj

, Txnj) ≤ ||xnj

− unj|| → 0

as j →∞. Since T satisfies condition (1) then limj→∞

d(xnj, F (T )) = 0. Thus there exists a

subsequence {xnjk} of {xnj

} such that ‖xnjk− pk‖ ≤ 1

2kfor some {pk} ⊆ F (T ). From (14)

‖xnjk+1− pk‖ ≤ ‖xnjk

− pk‖.

We now show that {pk} is a Cauchy sequence in F (T ).

‖pk+1 − pk‖ ≤ ‖pk+1 − xnjk+1‖+ ‖xnjk+1

− pk‖

≤ 1

2k+1+

1

2k

≤ 1

2k−1

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Therefore {pk} is a Cauchy sequence and converges to some q ∈ K since K is closed. Now,

‖xnjk− q‖ ≤ ‖xnjk

− pk‖+ ‖pk − q‖

Hence xnjk→ q as k →∞.

d(q, T q) ≤ ‖q − pk‖+ ‖pk − xnjk‖+ d(xnk

, Txnk) +H(Txnk

, T q)

≤ ‖q − pk‖+ ‖pk − xnjk‖+ d(xnjk

, Txnjk) + L‖xnjk

− q‖

Hence q ∈ Tq. Since lim ‖xn − q‖ exists and {xnjk} converges strongly to q, we have that

{xn} converges strongly to q ∈ F (T ).

Corollary 5: Let H be a real Hilbert space and K a nonempty closed and convex subsetof H. Let T : K → P (K) be a multivalued mapping from K into the family of all proxim-inal subsets of K such that F (T ) 6= ∅. Suppose PT is an L-Lipschitzian hemicontractivemapping. If T satisfies condition (1). Then the Ishikawa sequence {xn} defined in theorem7 converges strongly to p ∈ F (T ).

Proof: The proof follows easily from Lemma 2 and Theorem 7. 2

Authors’ Contribution: All authors contributed equally to the research work embodiedherein.

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