Composite Implicit Random Iterations for Approximating Common Random Fixed Point for a Finite Family of Strictly Pseudocontractive

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Composite Implicit Random Iterations for Approximating Common Random Fixed Point for a Finite Family of Strictly Pseudocontractive

Random Operators

Shrabani Banerjee¹ and Binayak S. Choudhury²

1Department of Mathematics

Bengal Engineering and Science University, Shibpur Howrah-711103, India

E-mail: [email protected]

2Department of Mathematics

Bengal Engineering and Science University, Shibpur Howrah-711103, India.

E-mail: [email protected]

(Received: 21-4-11/Accepted:24-6-11) Abstract

In the present work we construct a composite implicit random iterative process for a finite family of strictly pseudocontractive random operators and study the necessary and sufficient condition for the convergence of this process to a common random fixed point of a finite family strictly pseudocontractive random operators.

Keywords: Banach spaces, Condition(B), Composite implicit random it- erative process, measurable spaces, strictly pseudocontractive random opera- tors.

2000 MSC No: 47H10.

1 Introduction

Random fixed point theory was initiated by the Prague school of probabilists in the works of Hans [14] and Spacek [20] as stochastic generalization of de- terministic fixed point theory. After that till recent times a large number of research papers focussing on various aspects of random fixed point theory have appeared in recent literatures. Some of these references are noted in [1], [9]

and references therein. Fixed point iteration schemes for nonlinear operators on Banach and Hilbert spaces have been developed and studied by many au- thors in recent times. The development of random fixed point iterations was initiated by Choudhury in [8] where random Ishikawa iteration scheme was defined and its strong convergence to a random fixed point in Hilbert spaces was discussed. After that several authors have worked on random fixed point iterations some of which are noted in [2], [10], [11], [12], [13], [17] and [18].

In 2005 Beg and Abbas [2] constructed and studied different random iterative algorithms for weakly contractive and asymptotically nonexpansive random operators on arbitrary Banach spaces. They also established the convergence of an implicit random iteration process to a common random fixed point for a finite family of asymptotically quasi-nonexpansive random operators. Very recently Plubtieng et al. [17] constructed and established the convergence of an implicit random iteration process with errors for a common random fixed point of a finite family of asymptotically quasi-nonexpansive random operators in the setting of uniformly convex Banach spaces. Very recently Kumam et al.

[16] have established the convergence of an implicit random iteration process to a common random fixed point for a finite family of strictly pseudocontrac- tive random operators in real separable Banach spaces.

The purpose of this paper is to construct a composite implicit random itera- tive scheme for a finite family of strictly pseudocontractive random operators and to study the convergence of this iterative process in real separable Banach spaces. Our results extend and improve some recent results in the existing literature.

2 Preliminaries

Throughout this paper, (Ω,Σ) denotes a measurable space and X stands for a real Banach space. For any function T : Ω×X → X we denote the n-th iterateT(t, T(t, ..., T(t, x)))) ofT byTⁿ(t, x). A functionf : Ω→X is said to be measurable if f⁻¹(B) ∈ Σ for every Borel subset B of X. A single-valued operator T : Ω×X → X is called a random operator if T(·, x) : Ω → X is measurable, for every x ∈ X. The letter I denotes the identity random operator I : Ω× X → X defined by I(t, x) = x and T⁰ = I. A random operator T : Ω×X → X is continuous if T(t,·) : X → X is continuous for

each t ∈ Ω. A measurable function p : Ω → X is said to be a random fixed point of the random operatorT : Ω×X →X ifT(t, p(t)) =p(t),∀t∈Ω. The set of all random fixed points ofT is denoted by RF(T).

A mapping J fromX to 2^X? defined by

J(x) = {x^? ∈X^? :hx, x^?i=kxk.kx^?k,kx^?k=kxk}

where h., .i denotes the generalized duality pairing is called the normalized duality mapping.

Definition 2.1 [16] LetC be a nonempty subset of a real separable Banach space X and T : Ω×C →C be a random operator. Then T is said to be (i)strictly pseudocontractive random operator if for some measurable mapping λ: Ω→(0,1)and for any x, y ∈C, there exists j(x−y)∈J(x−y) such that for each t∈Ω,

hT(t, x)−T(t, y), j(x−y)i ≤ kx−yk²−λ(t)kx−y−(T(t, x)−T(t, y))k² (ii) a L-Lipschitzian random operator if for any x, y ∈C and for each t∈Ω,

kT(t, x)−T(t, y)k ≤Lkx−yk where L is a positive constant.

(iii) a semi-compact random operator if for a sequence of measurable mappings {ξn}from ΩtoC withlimn→∞kξn(t)−T(t, ξn(t))k= 0, for all t∈Ω, we have a subsequence {ξ_nk} of {ξ_n} such that ξ_nk(t) →ξ(t), for all t∈ Ω, where ξ is a measurable mapping from Ωto C.

Every strictly pseudocontractive random operator is aL-Lipschitzian random operator.

Definition 2.2 Condition(B) A finite family {T_i :i∈ {1,2, ..., N}} of N continuous random operators from Ω×C → C with F =^TN_i=1RF(T_i) 6= ∅ is said to satisfy Condition(B) if there is a nondecreasing functionf : [0,∞)→ [0,∞) with f(0) = 0 and f(r)>0 for all r ∈(0,∞) such that for all t∈Ω

f(d(x(t), F))≤ max

1≤i≤N{kx(t)−T_i(t, x(t))k} for all x where x: Ω→C is a measurable function.

Lemma 2.3 ([15]) Let (Ω,Σ) be a measurable space, X be a separable Ba- nach space and T : Ω×X → X be a continuous random operator. Then for any measurable functionx: Ω→X, the functiont→T(t, x(t))is measurable.

We define the composite implicit random iterative process in the following:

Definition 2.4 Composite implicit random iterative scheme Let{T_i :i∈ {1,2, ...., N}}be a finite family ofN continuous random operators fromΩ×C toCwhere Cbe a nonempty closed convex subset of a real separable Banach spaceX. Let ξ₀ : Ω→C be any measurable function. Then Composite implicit random iteration scheme is defined as follows:

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ξ₁(t) = α₁ξ₀(t) + (1−α₁)T₁(t, β₁ξ₁(t) + (1−β₁)T₁(t, ξ₁(t))) ξ₂(t) = α₂ξ₁(t) + (1−α₂)T₂(t, β₂ξ₂(t) + (1−β₂)T₂(t, ξ₂(t)))

·

·

ξ_N(t) = α_Nξ_N−1(t) + (1−α_N)T_N(t, β_Nξ_N(t) + (1−β_N)T_N(t, ξ_N(t)))

ξ_N+1(t) = α_N+1ξ_N(t) + (1−α_N+1)T₁(t, β_N+1ξ_N+1(t) + (1−β_N+1)T₁(t, ξ_N+1(t))

·

·

ξ_2N(t) =α_2Nξ2N−1(t) + (1−α_2N)T_N(t, β_2Nξ_2N(t) + (1−β_2N)T_N(t, ξ_2N(t)))

ξ_2N+1(t) =α_2N+1ξ_2N(t) + (1−α_2N+1)T₁(t, β_2N+1ξ_2N+1(t) + (1−β_2N+1)T₁(t, ξ_2N+1(t))

·

·

which can be written in the compact form as

( ξ_n(t) =α_nξn−1(t) + (1−α_n)T_n(t, η_n(t))

η_n(t) =β_nξ_n(t) + (1−β_n)T_n(t, ξ_n(t)), n≥1,∀t ∈Ω (1) where T_n =T_nmodN and {α_n},{β_n} are sequences in [0,1].

Remark 2.5 By Lemma 2.3 the sequence{ξn} defined in (1) is a sequence of measurable functions.

Lemma 2.6 ([19], Lemma1) Let {a_n},{b_n} and {δ_n} be sequences of non- negative real numbers satisfying the inequality

a_n+1≤(1 +δ_n)a_n+b_n,∀n≥1.

If ^P∞_n=1δ_n <∞ and ^P∞_n=1b_n <∞, then (i) lim_n→∞a_n exists,

(ii) limn→∞a_n = 0 whenever lim infn→∞a_n= 0.

Lemma 2.7 ([5], Lemma2.1)LetX be a real Banach space and letJ :X → 2^X? be the normalized duality mapping. Then for any given x, y ∈ X, we have

kx+yk² ≤ kxk²+ 2< y, j >,∀j ∈J(x+y)

3 Main Results

Lemma 3.1 Let X be a real separable Banach space and C be a nonempty closed convex subset of X. Let {T_i :i∈ {1,2, .., N}} be N strictly pseudocon- tractive continuous random operators and F =^TN_i=1RF(T_i) 6=∅. Let {ξ_n} be the implicit random iterative sequence defined by (1) satisfying the following conditions:

(i)0<lim infn→∞α_n ≤lim sup_n→∞α_n <1, (ii)^P∞_n=1(1−αn)(1−βn)<∞.

Then for each t∈Ω,

(i) limn→∞kξ_n(t)−ξ(t)k exists, (ii) limn→∞d(ξn(t), F) exists,

(iii) limn→∞kξ_n(t)−T_l(t, ξ_n(t))k= 0, for alll = 1,2, ...., N.

Proof: Lett ∈Ω. Letξ ∈F. Then for alln ≥1, kξ_n(t)−ξ(t)k² =hξ_n(t)−ξ(t), j(ξ_n(t)−ξ(t))i

= α_nhξn−1(t)−ξ(t), j(ξ_n(t)−ξ(t))i+ (1−α_n)hT_n(t, η_n(t))−ξ(t), j(ξ_n(t)−ξ(t))i

= α_nkξn−1(t)−ξ(t)k.kξ_n(t)−ξ(t)k+ (1−α_n)hT_n(t, ξ_n(t))−ξ(t), j(ξ_n(t)−ξ(t))i +(1−α_n)hT_n(t, η_n(t))−T_n(t, ξ_n(t)), j(ξ_n(t)−ξ(t))i

≤ α_n

2 [kξn−1(t)−ξ(t)k²+kξn(t)−ξ(t)k²] + (1−αn)[kξn(t)−ξ(t)k²−

λ(t)kξ_n(t)−T_n(t, ξ_n(t))k²] + (1−α_n)kT_n(t, η_n(t))−T_n(t, ξ_n(t))k.kξ_n(t)−ξ(t)k

≤ α_n

2 kξn−1(t)−ξ(t)k²+ α_n

2 kξ_n(t)−ξ(t)k²+ (1−α_n)kξ_n(t)−ξ(t)k²

−(1−αn)λ(t)kξn(t)−Tn(t, ξn(t))k²+ (1−αn)Lkηn(t)−ξn(t)k.kξn(t)−ξ(t)k (2) Now

kη_n(t)−ξ_n(t)k = (1−β_n)kT_n(t, ξ_n(t))−ξ_n(t)k

≤ (1−β_n)[kT_n(t, ξ_n(t))−ξ(t)k+kξ_n(t)−ξ(t)k]

≤ (1−β_n)(L+ 1)kξ_n(t)−ξ(t)k (3) From (2) and (3) it follows that for all t∈Ω and for all n≥1,

kξ_n(t)−ξ(t)k²

≤ α_n

2 kξn−1(t)−ξ(t)k²+ α_n

2 kξn(t)−ξ(t)k²+ (1−αn)kξn(t)−ξ(t)k²

−(1−α_n)λ(t)kξ_n(t)−T_n(t, ξ_n(t))k²+ (1−α_n)(1−β_n)(L+ 1)Lkξ_n(t)−ξ(t)k²

= α_n

2 kξn−1(t)−ξ(t)k²+ [α_n

2 + 1−α_n+ (1−α_n)(1−β_n)(L+ 1)L]kξ_n(t)−ξ(t)k²

−(1−α_n)λ(t)kξ_n(t)−T_n(t, ξ_n(t))k²

which implies that

α_nkξ_n(t)−ξ(t)k² ≤ α_nkξn−1(t)−ξ(t)k²+ 2(1−α_n)(1−β_n)(L+ 1)Lkξ_n(t)−ξ(t)k²

−2(1−α_n)λ(t)kξ_n(t)−T_n(t, ξ_n(t))k²

= α_nkξn−1(t)−ξ(t)k²+µ_nkξ_n(t)−ξ(t)k²−

2(1−α_n)λ(t)kξ_n(t)−T_n(t, ξ_n(t))k² (4) ( whereµn= 2(1−αn)(1−βn)(L+ 1)L)

Since 0 <lim infn→∞α_n ≤ lim sup_n→∞α_n <1, there exists a positive integer n₀ and η, η⁰ ∈ (0,1) such that 0 < η < α_n < η⁰ < 1 for all n ≥ n₀. Therefore from (4) it follows that for all n≥n0,

αnkξn(t)−ξ(t)k² ≤ αnkξn−1(t)−ξ(t)k²+µn.α_n

η kξn(t)−ξ(t)k²− 2(1−α_n)λ(t)kξ_n(t)−T_n(t, ξ_n(t))k²

which implies that for all n≥n₀, kξ_n(t)−ξ(t)k² ≤ kξn−1(t)−ξ(t)k²+µ_n

η kξ_n(t)−ξ(t)k²−21−α_n

α_n λ(t)kξ_n(t)−T_n(t, ξ_n(t))k² which implies that for all n≥n₀,

kξ_n(t)−ξ(t)k²

≤ η

η−µ_nkξn−1(t)−ξ(t)k²−2 η

η−µ_n.1−α_n

α_n .λ(t)kξn(t)−Tn(t, ξn(t))k²

= (1 + µ_n

η−µ_n)kξn−1(t)−ξ(t)k²−2 η

η−µ_n.1−α_n

α_n .λ(t)kξ_n(t)−T_n(t, ξ_n(t))k² (5) By the condition of the theorem we get that^P∞_n=1µ_n<∞. Therefore limn→∞µ_n= 0. So there existsn1(> n0)∈N such that µn < ^η₂ for all n≥n1. Thus for all n≥n₁, from (5) it follows that

kξ_n(t)−ξ(t)k²

≤ (1 + 2

ηµ_n)kξn−1(t)−ξ(t)k²−2 η η−µn

.1−α_n αn

.λ(t)kξ_n(t)−T_n(t, ξ_n(t))k²

= (1 +σ_n)kξn−1(t)−ξ(t)k²−2 η

η−µ_n.1−αn

α_n .λ(t)kξ_n(t)−T_n(t, ξ_n(t))k² ( whereσ_n = 2

ηµ_n) (6)

From (6) it follows that for alln ≥n₁,

kξ_n(t)−ξ(t)k² ≤(1 +σ_n)kξn−1(t)−ξ(t)k² (7)

Since^P∞_n=1µ_n <∞, we have ^P∞_n=1σ_n<∞. Thus by Lemma2.6 we have that limn→∞kξ_n(t)−ξ(t)k exists for all ξ∈F and for eacht ∈Ω. From (7) we get that for all t∈Ω and for all n≥n1,

kξ_n(t)−ξ(t)k ≤(1 +σ_n)¹²kξ_n−1(t)−ξ(t)k ≤(1 + 1

2σ_n)kξ_n−1(t)−ξ(t)k (8) (as√

1 +x≤1 + ¹₂x)

Taking infimum over all ξ∈F, from (8) it follows that for each t∈Ω, d(ξ_n(t), F)≤(1 + 1

2σ_n)d(ξn−1(t), F) (9) Hence by Lemma 2.6 we have lim_n→∞d(ξ_n(t), F) exists for each t ∈Ω. Since fort∈Ω, limn→∞kξ_n(t)−ξ(t)kexists, there existsM(t)>0 such thatkξ_n(t)−

ξ(t)k ≤M(t). Therefore from (6) it follows that for all n ≥n₁ and t∈Ω, 21−η⁰

η⁰ .λ(t)kξ_n(t)−T_n(t, ξ_n(t))k²

≤ 2 η

η−µ_n.1−αn

α_n .λ(t)kξ_n(t)−T_n(t, ξ_n(t))k²

≤ kξn−1(t)−ξ(t)k²− kξn(t)−ξ(t)k²+σn(M(t))² (10) From (10) it follows that for eacht ∈Ω,

∞

X

n=n1

kξ_n(t)−T_n(t, ξ_n(t))k² <∞ which implies that

n→∞lim kξ_n(t)−T_n(t, ξ_n(t))k= 0 (11) Now for each t∈Ω,

kη_n(t)−ξ_n(t)k= (1−β_n)kT_n(t, ξ_n(t))−ξ_n(t)k →0 as n → ∞(by(11)) (12) Now for each t∈Ω,

kξn(t)−ξn−1(t)k = (1−αn)kTn(t, ηn(t))−ξn−1(t)k

≤ (1−α_n)kT_n(t, η_n(t))−T_n(t, ξ_n(t))k+

(1−α_n)kT_n(t, ξ_n(t))−ξ_n(t)k+ (1−α_n)kξ_n(t)−ξn−1(t)k which implies that

kξn(t)−ξn−1(t)k ≤ 1−αn

α_n Lkηn(t)−ξn(t)k+1−αn

α_n kTn(t, ξn(t))−ξn(t)k(13)

So for eacht ∈Ω and for all n ≥n₀, from (13) it follows that kξ_n(t)−ξn−1(t)k ≤ 1−η

η Lkη_n(t)−ξ_n(t)k+1−η

η kT_n(t, ξ_n(t))−ξ_n(t)k

→ 0 as n → ∞(by(11),(12)) (14) So we have

n→∞lim kξ_n(t)−ξ_n+l(t)k= 0 for all l∈ {1,2, ...., N} and for eacht ∈Ω. (15) Now for each t∈Ω and for all l ∈ {1,2, ...., N},

kξ_n(t)−T_n+l(t, ξ_n(t))k ≤ kξ_n(t)−ξ_n+l(t)k+kξ_n+l(t)−T_n+l(t, ξ_n+l(t))k+ kTn+l(t, ξn+l(t))−Tn+l(t, ξn(t))k

≤ (1 +L)kξ_n(t)−ξ_n+l(t)k+kξ_n+l(t)−T_n+l(t, ξ_n+l(t))k

→ 0 as n → ∞(by(11),(15)) Consequently we have

n→∞lim kξ_n(t)−T_l(t, ξ_n(t))k= 0 for alll ∈ {1,2, ...., N} and for each t∈Ω.(16) Theorem 3.2 LetXbe a real separable Banach space and Cbe a nonempty closed convex subset of X. Let {T_i : i ∈ {1,2, ...., N}} be N strictly pseudo- contractive continuous random operators and F =^TN_i=1RF(T_i)6=∅. Let {ξ_n} be the implicit random iterative sequence defined by(1) satisfying the following conditions:

(i)0<lim infn→∞α_n ≤lim sup_n→∞α_n <1, (ii)^P∞_n=1(1−α_n)(1−β_n)<∞.

Then {ξ_n} converges strongly to a common random fixed point of the random operators{T_i, i∈ {1,2, ...., N}}if and only if for allt∈Ω,lim infn→∞d(ξ_n(t), F) = 0, where d(ξ_n(t), F) = inf{kξ_n(t)−ξ(t)k:ξ ∈F}.

Proof: The necessary part is obvious. We only prove the sufficiency part. Let ξ∈F. Now from (8) we have that for each t∈Ω and for all n ≥n₁,

kξ_n(t)−ξ(t)k ≤(1 + 1

2σ_n)kξn−1(t)−ξ(t)k= (1 +λ_n)kξn−1(t)−ξ(t)k (17) where λ_n = ¹₂σ_n. Now from (17) we have that for each t ∈ Ω and for all m, n≥n₁,

kξ_n+m(t)−ξ(t)k ≤ [1 +λ_n+m]kξn+m−1(t)−ξ(t)k ≤e^λn+mkξn+m−1(t)−ξ(t)k

≤ e^{λn+m+λn+m−1}kξn+m−2−ξ(t)k

≤ · · · ·

≤ e^P

n+m

i=n+1λikξ_n(t)−ξ(t)k ≤Rkξ_n(t)−ξ(t)k (18)

where R = e^P∞n=1λn < ∞. Therefore for any ξ ∈ F and for t ∈ Ω, we have that

kξ_n+m(t)−ξ_n(t)k ≤ Rkξ_n(t)−ξ(t)k+kξ_n(t)−ξ(t)k

= (R+ 1)kξ_n(t)−ξ(t)k (19) Since limn→∞d(ξ_n(t), F) = 0, there exists n₂(≥ n₁) ∈ N such that for all n ≥ n₂ we have d(ξ_n(t), F) < _R+1 . So there exists q ∈ F such that kξ_n(t)− q(t)k< _R+1 for all n ≥n₂. Therefore from (19) we have that for all n ≥n₂,

kξ_n+m(t)−ξ_n(t)k ≤ (R+ 1) R+ 1 =

which in turn implies that{ξ_n(t)}is a cauchy sequence for each t∈Ω. There- foreξ_n(t)→p(t) asn→ ∞for eacht ∈Ω where the functionp: Ω→X, being the limit of the sequence of measurable functions is also measurable. Now we prove thatp∈F. Since for eacht∈Ω,ξ_n(t)→p(t) asn→ ∞there existsn₃ ∈ N such thatkξ_n(t)−p(t)k< _2(1+L) for alln ≥n₃. Since lim_n→∞d(ξ_n(t), F) = 0 for each t ∈ Ω, there exists n4 ∈ N such that d(ξn(t), F) < _2(1+L) for all n ≥ n4. So there exists ξ^∗ ∈ F such that kξn(t)−ξ^∗(t)k ≤ _2(1+L) for all n ≥ n₄, where L is the Lipschitz constant. Letn₅ = max {n₃, n₄}. Now for alll ∈ {1,2, ...., N}, t∈Ω and for all n≥n₅

kT_l(t, p(t))−p(t)k ≤ kT_l(t, p(t))−ξ^∗(t)k+kξ^∗(t)−p(t)k

≤ kT_l(t, p(t))−T_l(t, ξ^∗(t))k+kξ^∗(t)−p(t)k

≤ Lkξ^∗(t)−p(t)k+kξ^∗(t)−p(t)k

= (1 +L)kξ^∗(t)−p(t)k

≤ (1 +L)kξ^∗(t)−ξ_n(t)k+ (1 +L)kξ_n(t)−p(t)k

< (1 +L)

2(1 +L) + (1 +L)

2(1 +L) =

which implies thatT_l(t, p(t)) =p(t) for all l∈ {1,2, ...., N}and for eacht∈Ω.

Therefore p ∈ F. Thus {ξ_n} converges strongly to a common random fixed point of{T_i, i∈ {1,2, ...., N}}.

Theorem 3.3 LetXbe a real separable Banach space and Cbe a nonempty closed convex subset of X. Let {T_i : i ∈ {1,2, ...., N}} be N strictly pseudo- contractive continuous random operators and F =^TN_i=1RF(T_i)6=∅. Let {ξ_n} be the implicit random iterative sequence defined by(1) satisfying the following conditions:

(i)0<lim infn→∞α_n ≤lim sup_n→∞α_n <1, (ii)^P∞_n=1(1−α_n)(1−β_n)<∞.

If the family {T_i : i ∈ {1,2, ...., N}} satisfies Condition(B), for each t ∈ Ω, then {ξ_n} converges strongly to a common random fixed point of {T_i, i ∈ {1,2, ...., N}}.

Proof: By the proof of Lemma 3.1 we have limn→∞d(ξ_n(t), F) exists for each t ∈Ω. Again by Lemma 3.1 and Condition(B), we have that for each t ∈Ω, limn→∞f(d(ξn(t), F)) = 0. Since f : [0,∞) → [0,∞) is a nondecreasing function with f(0) = 0 so we have limn→∞d(ξ_n(t), F) = 0. Hence the result follows by Theorem 3.2.

Theorem 3.4 LetXbe a real separable Banach space and Cbe a nonempty closed convex subset of X. Let {T_i : i ∈ {1,2, ...., N}} be N strictly pseudo- contractive continuous random operators and F =^TN_i=1RF(T_i)6=∅. Let {ξ_n} be the implicit random iterative sequence defined by(1) satisfying the following conditions:

(i)0<lim infn→∞α_n ≤lim sup_n→∞α_n <1, (ii)^P∞_n=1(1−α_n)(1−β_n)<∞.

Let one member of the family {T_i : i∈ {1,2, ...., N}} to be semi-compact ran- dom operator, then {ξ_n} converges strongly to a common random fixed point of {T_i, i∈ {1,2, ...., N}}.

Proof: From Lemma 3.1 we get that limn→∞kξn(t)−Tl(t, ξn(t))k= 0 for each t ∈ Ω and for each l ∈ {1,2, ...., N}. Let us assume that T₁ is semi-compact random operator. So there exists a subsequence{ξ_nk(t)} of {ξ_n(t)} such that ξn_k(t)→ ξ(t) for each t ∈Ω, where ξ is a measurable mapping from Ω to C.

Now for each t∈Ω and for each l ∈ {1,2, ...., N}, we have kξ(t)−T_l(t, ξ(t))k= lim

k→∞kξ_nk(t)−T_l(t, ξ_nk(t))k= 0

From above it follows that ξ ∈ F. Since {ξ_n(t)} has a subsequence {ξ_nk(t)}

such thatξ_nk(t)→ξ(t) for eacht ∈Ω, we have that lim infn→∞d(ξ_n(t), F) = 0.

Hence the result follows by Theorem 3.2.

Remark 3.5 Our results in this paper extend and improve the correspond- ing results of [16].

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