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Weak and Strong Convergence Theorems of Modified Projection-Type Ishikawa Iteration

Scheme for Lipschitz α-Hemicontractive Mappings

Imo Kalu Agwu∗, Donatus Ikechi Igbokwe
Department of Mathematics, Micheal Okpara University of Agriculture, Umudike, Umuahia Abia State,

Nigeria
agwuimo@gmail.com, igbokwedi@yahoo.com
∗Correspondence: agwuimo@gmail.com

Abstract. In this paper, we establish weak and strong convergence theorems of a two-step modifiedprojection-type Ishikawa iterative scheme to the fixed point of α-hemicontractive mappings withoutany compactness assumption on the operator or the space. Our results extend, improve and generalizeseveral previously known results of the existing literature.

1. Introduction
Let H be a real Hilbert space with inner product 〈, .,〉 and induced norm ‖, .,‖, K a nonemptyconvex and closed subset of H and T : K −→ K a selfmap on K. We use F (T ) to denote the setof fixed point of T , N to denote the set of natural numbers and xn → x (respectively xn ⇀ x) todenote the strong (weak) convergence of the sequence {xn}∞n=0 to the point x.

Definition 1.1. Let T : K −→ K be a maaping. Then

I. T is said to be L-Lipschitizian if there exists L > 0 such that

‖Ts −Tz‖≤‖s −z‖,∀s,z ∈ K. (1.1)
From the definition, it easy to observe that every nonexpansive mapping is Lipschitizian
with L = 1.

II. T is called k-strictly pseudocontraction (see, for example, [9]) if there exists k ∈ (0, 1] such
that for all s,z ∈ K, the inequality

‖Ts −Tz‖2 ≤‖s −z‖2 + k‖(I −T )s − (I −T )z‖2 (1.2)
hods. Note that if k = 1 in (1.2), then T is a pseudocontraction. It well-known that in
real Hilbert spaces, the class of nonexpansive mapping is a proper subclass of the class of

Received: 1 Nov 2021.
Key words and phrases. strong convergence; modified Ishikawa iterative scheme; weak convergence; α-hemicontractive operator; fixed point; real Hilbert space. 1

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 2
k-strictly pseudocontive mapping. Also, the class of k-strictly pseudocontive mapping is a
proper subclass of the class of pseudocontive mapping.

III. T is called demicontractive mapping (see, for example, [?]) if F (T ) = {x ∈ K : x = Tx} 6= ∅
and ∀(s ×q) ∈ (K ×F (T )), there exists k ∈ [0, 1) such that the inequality

‖Ts −Tq‖2 ≤‖s −q‖2 + k‖s −Ts‖2 (1.3)
hods.

IV. T is said to satisfy condition A (see, for example [?]) F (T ) = {x ∈ K : x = Tx} 6= ∅ and
there exists λ > 0 such that

〈s −TS,s −q〉≥ λ‖s −Ts‖2,∀(s ×q) ∈ (K ×F (T )). (1.4)
It is worthy to mention that the class of k-strictly pseudocontions with a nonempty fixed
point set is a proper subclass of the class demicontractions. T is called hemicontraction (see,
for example, [17]) if k = 1 in (1.3). The class of pseudocontractive maps is a proper subclass
of the class of hemicontractive maps. Again, the class of demicontractive maps is a proper
subclass of the class of hemicontractive maps (see, for example, [?]). These two classes of
mappings have been studied extensively by many researchers (see, for example, [?], [13], [17]
and the references therein).

V. T is called α-demicontraction (see, for examole, [13] ) if F (T ) = {x ∈ K : x = Tx} 6= ∅
and ∀(s ×q) ∈ (K ×F (T )), there exist λ > 0 and α ≥ 1 such that the inequality

〈s −TS,s −αq〉≥ λ‖s −Ts‖2,∀(s ×q) ∈ (K ×F (T )). (1.5)
holds. Clearly, (1.5) is equivalent to

‖Ts −αq‖2 ≤‖s −αq‖2 + k‖s −Ts‖2, (1.6)
where k = 1 − 2λ ∈ [0, 1).

V. T is called α-hemicontraction (see, for examole, [17] ) if F (T ) = {x ∈ K : x = Tx} 6= ∅
and ∀(s ×q) ∈ (K ×F (T )), there exists α ≥ 1 such that the inequality

‖Ts −αq‖2 ≤‖s −αq‖2 + ‖s −Ts‖2 (1.7)
holds. Observe that (1.7) is equivalent to

〈s −Ts,s −αq〉≥ 0,∀(s ×q) ∈ (K ×F (T )). (1.8)
In [ [17], Example 2.2], Osilike and Onah gave an example of α-hemicontractive mapping with
α > 1 which is not hemicontractive mapping, and also showed that there are hemicontractive
(1-hemicontractive) mappings which are not α-hemicontraction for α > 1(see [ [17], Example
2.1] for details). Again, Osilike and Onah [17] presented an example of a mapping which
is hemicontractive (1-hemicontractive) and alpha-hemicontractive mapping for α > 1 but

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 3
neither demicontractive (1-demicontractive) nor α-demicontractive mapping for α > 1(see
[17], Example 2.3 for details). For further cheracterisation of α-hemicontractive mapping,
interested reader should consult [17].

A mapping T : H −→ H is called ν-strongly monotone if there exists ν > 0 such that

〈s −Ts,s −z〉≥ ν‖s −z‖2,∀s,z ∈ H.. (1.9)
Iterative method for approximating fixed point of L-Lipschitz pseudocontractive mapping has beenan active area of investigation in recent times (see, for example, [?], [?], [20], [14], [26], [27] and thereferences contained in them). In [24], Voluhan introduced the modified projection-type Ishikawaiterative method in the following way: Let H be a Hilbert space, K nonempty, closed and convexsubset of H and T : K −→ K be an L-Lipshitz pseudocontractive mapping. For an arbitrary

x0 ∈ K, define the sequence {xn}∞n=0 iteratively as follows. xn+1 = PK[(1 −αn −γn)xn + γnTyn]
yn = (1 −βn)xn + βnTxn,n ≥ 1,

(1.10)
where {αn}∞n=0,{βn}∞n=0,{γn}∞n=0 ∈ (0, 1) and PK is a projection map from H onto K. Using(1.10), she proved the following theorem.
Theorem 1.1. Let H be a Hilbert space, D a nonempty closed convex subset of H and T : D −→ D
an L-Lipschitz pseudocontractive mapping such that F (T ) 6= ∅. For any given x0 ∈ H, let {xn}∞n=0
be the sequence defined by (1.10). Assume the sequences {αn}∞n=0,{βn}∞n=0,{γn}∞n=0 ∈ (0, 1)
satisfy

(1) βn(1 −αn) > γn,∀n ≥ 1;(2) limn→∞αn = 0 and ∑∞n=0αn = ∞;(3) 0 < α ≤ γn ≤ βn ≤ β < 1√
1 + L2 + 1

,∀n ≥ 1.

Then, the sequence {xn}∞n=0 strongly converges to the fixed point of T .

Remark 1.1. If αn = 0,∀n ≥ 1, and PK is an identity, (1.10) reduces to the well-known Ishikawa
iteration method  xn+1 = (1 −γn)xn + γnTyn

yn = (1 −βn)xn + βnTxn,n ≥ 1,
(1.11)

which has been used by several researchers to approximate the fixed points of different operators
or operator equations in different spaces.

Motivated and inspired by the works in [17], [24] and some ongoing research in this direction, itis our purpose in this paper to extend the results in [24] and other related results from Lipschitzpseudocontractive mapping to the more general α-hemicontractive mapping. Our results is more

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 4
general and also more applicable because fewer and simpler conditions are required to attainconvergence.

2. Preliminary
The following definitionS and lemmas will be needed to prove our main results.

Definition 2.1. (see [27]) Let H and K be as defined above. For each x ∈ H, there exists a unique
nearest point of K, denoted by PKx, such that

‖x −PKx‖≤‖x −y‖,∀y ∈ K.

Such a PK is called metric projection from H onto K. It is well-known that PK is firmly nonexpansive
mapping from H onto K; that is,

‖PKx −PKy‖2 ≤〈PKx −PKy,x −y〉,∀x,y ∈ H.

Also, for any x ∈ H and z ∈ K,z = PKx if and only if

〈x −z,z −y〉≥ 0,∀y ∈ K.

Definition 2.2. The Banach space Z is said to have Opial property, if for each weakly convergent
sequence {zn}∞n=0with weak limit z ∈ Z, the following inequality holds:

lim sup
n→∞

‖zn −z‖ < ‖zn −y‖,∀y ∈ Zwithz 6= y.

Note that all finite dimensional Banach spaces, all Hilbert spaces and `p(0 ≤ p < ∞) satisfy the
Opial property. But Lp(1 < p < ∞.p 6= 2) do not satisfies the Opial property.

Definition 2.3. (see [27]) Let E be a real Banach space. A mapping T, with domain D(T ) ∈ E,
is said to be demiclosed at 0 if for any sequence zn ⊂ E,zn � q ∈ D(T ) and ‖zn−Tzn‖→ 0, then
Tq = q.

Lemma 2.1. (see [27]) Let H be a real Hilbert space. Then, the following inequality holds:

‖λx + (1 −λ)y‖2 ≤ λ‖x‖2 + (1 −λ)‖y‖2 −λ(1 −λ)‖x −y‖,∀λ ∈ [0, 1],∀x,y ∈ H.

Lemma 2.2. (see [27]) Let {sn}n∈N be a sequence of nonnegative real numbers satisfying the
inequality:

sn+1 ≤ (1 −γn)sn + δn,∀n ≥ 1,

where {γn}n∈N and {δn}n∈N satisfy the following conditions:(i) {γn}n∈N ⊂ (0, 1);(ii) ∑∞n=1γn = ∞.
Suppose

∑∞
n=1δn < ∞, then,limn→∞ sn = 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 5
Lemma 2.3. (see [4]) Let E be a real Hilbert space. Then, for all x,y ∈ H, the following inequalities
hold:

I. ‖x −y‖2 ≤‖x‖2 − 2〈y, (x + y)〉 + ‖y‖2;

II. ‖x −y‖2 ≤‖x‖2 − 2〈y, (x + y)〉.

Lemma 2.4. (see [?]) Let D be a sunset of a real Hilbert space, T : D −→ H be a nonexpansive
mapping and z a weak cluster point of the sequence {yn}∞n=0. If ‖Tyn −yn‖→ 0, then z ∈ F (T )

Proposition 2.5. (see [27]) Let D be a nonempty subset of a real Hilbert space amd Γ : D −→ D
an α-demicontractive mapping. Assume that x ∈ D and α ≥ 1. Then, Γ is Lipschitizian.

Theorem 2.6. (see [4]) A Banach space E is reflexive if and only if every (normed) bounded sequence
in E has a subsequence which converges weakly to an element of E.

3. Convergence Results
Now, we prove our main results.

Theorem 3.1. Let H be a real Hilbert space, K a nonempty closed convex subset of H and T :
K −→ K an L-Lipschitz α-hemicontractive mapping. For any arbitrary x0 ∈ H, define the sequence
{xn}∞n=0 iteratively as follows: xn+1 = PK[(1 −αn −γn)xn + γnTyn]

yn = (1 −βn)xn + βnTxn,n ≥ 1,
(3.1)

where the sequences {δn}∞n=0,{γn}
∞
n=0,{βn}

∞
n=0 ∈ (0, 1) satisfy the following conditions:

(i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤
1 −δ

1 + L2
;

(ii) limn→∞δn = 0 and
∑∞
n=0δn = ∞.

Then, the sequence {xn}∞n=0 generated by (3.1) weakly and strongly converges to the fixed point
of T .

Proof. Since F (T ) is nonempty, let αq ∈ F (T ) and x ∈ K. Using (3.1), Lemma 2.1 and the factthat T is L-Lipschitizian, we estimate as follows:
‖xn+1 −αq‖2 = ‖PK[(1 −δn −γn)xn + γnTyn] −αq‖

≤ ‖(1 −δn −γn)xn + γnTyn −αq‖

= ‖(1 −δn −γn)(xn −αq) + γn(Tyn −αq) −δnαq‖

≤ ‖(1 −δn −γn)(xn −αq) + γn(Tyn −αq)‖ + δn‖αq‖. (3.2)
Set Qn = ‖(1 −δn −γn)(xn −αq) + γn(Tyn −αq)‖2 and observe that

Qn = ‖(1 −δn)(xn −αq) − (1 −γn)(xn −αq) + γn(Tyn −αq)‖2. (3.3)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 6
Since

(1 −δn)(xn −αq) = (1 −δn)(1 −γn)(xn −αq) + γn(1 −δn))(xn −αq) (3.4)
and

γn(Tyn −αq) = γn(1 −δn)(Tyn −αq) + γnδn(Tyn −αq), (3.5)
it follows from (3.3) that

Qn = ‖(1 −δn)(1 −γn)(xn −αq) + γn(1 −δn))(xn −αq) − (1 −γn)(xn −αq)

+γn(1 −δn)(Tyn −αq) + γnδn(Tyn −αq)‖2

= ‖(1 −δn)[(1 −γn)(xn −αq) + γn(Tyn −αq)] + δnγn(Tyn −xn)‖2. (3.6)
(3.6) and Lemma 2.1 imply that

Qn = (1 −δn)‖(1 −γn)(xn −αq) + γn(Tyn −αq)‖2 + δn‖γn(Tyn −xn)‖2

−δn(1 −δn)‖xn −αq‖2. (3.7)
If we denote Vn = ‖(1 −γn)(xn −αq) + γn(Tyn −αq)‖2 and use similar technique as above, thenwe get

Vn = (1 −γn)‖xn −αq‖2 + γn‖Tyn −αq‖2 −γn(1 −γn)‖xn −Tyn‖2. (3.8)
(3.7) and (3.8) imply
Qn = (1 −δn)[(1 −γn)‖xn −αq‖2 + γn‖Tyn −αq‖2 −γn(1 −γn)‖xn −Tyn‖2]

+δnγ
2
n‖Tyn −xn‖

2 −δn(1 −δn)‖xn −αq‖2

= (1 −δn)(1 −γn)‖xn −αq‖2 + (1 −δn)γn‖Tyn −αq‖2 −γn(1 −γn)(1 −δn)‖xn −Tyn‖2

+δnγ
2
n‖Tyn −xn‖

2 −δn(1 −δn)‖xn −αq‖2

≤ (1 −δn)(1 −γn)‖xn −αq‖2 + (1 −δn)γnL2‖yn −αq‖2

−(γn −δnγn −γ2n + γ
2
nδn)‖xn −Tyn‖

2 + δnγ
2
n‖Tyn −xn‖

2 −δn(1 −δn)‖xn −αq‖2

= (1 −δn)(1 −γn)‖xn −αq‖2 + γnL2‖yn −αq‖2 −δnγnL2‖yn −αq‖2

−(γn −δnγn −γ2n)‖xn −Tyn‖
2 −δn(1 −δn)‖xn −αq‖2. (3.9)

Observr that
|xn −Tyn‖ ≤ (‖xn −αq‖ + L‖yn −αq‖)2

= ‖xn −αq‖2 + L(2‖xn −αq‖‖yn −αq‖) + L2‖yn −αq‖2

≤ ‖xn −αq‖2 + L‖xn −αq‖2 + L‖yn −αq‖2 + L2‖yn −αq‖2

= (1 + L)‖xn −αq‖2 + L(1 + L)‖yn −αq‖2. (3.10)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 7
(3.9) and (3.10) imply
Qn ≤ (1 −δn)(1 −γn)‖xn −αq‖2 + γnL2‖yn −αq‖2 −δnγnL2‖yn −αq‖2

−(γn −δnγn −γ2n)[(1 + L)‖xn −αq‖
2 + L(1 + L)‖yn −αq‖2] −δn(1 −δn)‖xn −αq‖2

= (1 −δn)(1 −γn)‖xn −αq‖2 − (1 + L)(γn −δnγn −γ2n)‖xn −αq‖

−[(γn −δnγn −γ2n)L−L
2γ2n]‖yn −αq‖

2 −δn(1 −δn)‖xn −αq‖2 (3.11)
Again, from (3.1), we get

‖yn −αq‖2 = ‖(1 −βn)(xn −αq) + βn(Txn −αq)‖2 (3.12)
Since T is α-hemicontractive mapping, it follows from (3.12) and Lemma 2.1 that
‖yn −αq‖2 ≤ (1 −βn)‖xn −αq‖2 + βn[‖xn −αq‖2‖2 + ‖xn −Txn‖2] −βn(1 −βn)‖xn −Txn‖2

= (1 −βn)‖xn −αq‖2 + β2n‖xn −Txn‖
2. (3.13)

Putting (3.13) into (3.11), we have
Qn ≤ (1 −δn)(1 −γn)‖xn −αq‖2 − (1 + L)(γn −δnγn −γ2n)‖xn −αq‖

−[(γn −δnγn −γ2n)L−L
2γ2n]{(1 −βn)‖xn −αq‖

2 + β2n‖xn −Txn‖
2}

−δn(1 −δn)‖xn −αq‖2

≤ (1 −δn)(1 −γn)‖xn −αq‖2 − [(γn −δnγn −γ2n)(1 + L) + δn(1 −δn) −L
2γ2n]‖xn −αq‖

2

−β2n[(γn −δnγn −γ
2
n)L−L

2γ2n]‖xn −Txn‖
2. (3.14)

Since from condition (i), (γn −δnγn −γ2n) −L2γ2n ≥ 0, it follows from (3.14) that
Qn ≤ (1 −δn)2‖xn −αq‖2 (3.15)

(3.2) and (3.15) imply
|xn+1 −αq‖ ≤ (1 −δn)‖xn −αq‖2 + δn‖αq‖

≤ max{‖xn −αq‖2,‖αq‖},∀n ∈N.

It is easy to see, using mathematical induction, that
|xn+1 −αq‖ ≤ max{‖xn −αq‖2,‖αq‖}

= ‖x0 −αq‖2. (3.16)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 8
Hence, {xn}∞n=0 is bounded.Furthermore, since from (3.1),

‖xn+1 −αq‖2 = ‖PK[(1 −δn −γn)xn + γnTyn] −αq‖2

≤ ‖(1 −δn −γn)xn + γnTyn −αq‖2

= ‖xn −αq −γn(xn −Tyn) −δnxn‖2,

it follows from Lemma 2.3(i) that
‖xn+1 −αq‖2 ≤ ‖xn −αq −γn(xn −Tyn)‖2 − 2δn〈xn,xn+1 −αq〉. (3.17)

Since
‖xn −αq −γn(xn −Tyn)‖2 = ‖(1 −γn)(xn −αq) + γn(αq −Tyn)‖2

= (1 −γn)‖xn −αq‖2 + γn‖αq −Tyn‖2 −γn(1 −γn)‖Tyn −xn‖2

≤ (1 −γn)‖xn −αq‖2 + γnL2‖yn −αq‖2

−γn(1 −γn)‖Tyn −xn‖2, (3.18)
it follows from (3.10) that
‖xn −αq −γn(xn −Tyn)‖2 ≤ (1 −γn)‖xn −αq‖2 + γnL2‖yn −αq‖2

−γn(1 −γn){(1 + L)‖xn −αq‖2 + L(1 + L)‖yn −αq‖2}

= (1 −γn)‖xn −αq‖2 + γnL2‖yn −αq‖2

−γn(1 −γn)(1 + L)‖xn −αq‖2 −γn(1 −γn)L‖yn −αq‖2

−γnL2‖yn −αq‖2 + γ2nL
2‖yn −αq‖2

= (1 −γn)‖xn −αq‖2 −γn(1 −γn)(1 + L)‖xn −αq‖2

−[γn(1 −γn)L−L2γ2n]‖yn −αq‖
2. (3.19)

(3.13) and (3.19) imply
‖xn −αq −γn(xn −Tyn)‖2 ≤ (1 −γn)‖xn −αq‖2 −γn(1 −γn)(1 + L)‖xn −αq‖2

−[γn(1 −γn)L−L2γ2n]{(1 −βn)‖xn −αq‖
2 + β2n‖xn −Txn‖

2}

≤ (1 −γn)‖xn −αq‖2 −γnL[1 −γn −γnL]{(1 −βn)‖xn −αq‖2

+β2n‖xn −Txn‖
2}. (3.20)

By condition (i), 1 −γn −γnL > 0,∀n ≥ 0. Consequently,
‖xn −αq −γn(xn −Tyn)‖2 ≤ ‖xn −αq‖2

−(1 −γn −γnL)β2nγnL‖xn −Txn‖
2. (3.21)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 9
(3.17)and (3.21) imply

‖xn+1 −αq‖2 ≤ ‖xn −αq‖2 − (1 −γn −γnL)β2nγnL‖xn −Txn‖
2

−2δn〈xn,xn+1 −αq〉.

Since {xn} is bounded, there exists a constant B > 0 such that −2〈xn,xn+1 −αq〉≤ B. Thus,
‖xn+1 −αq‖2 ≤ ‖xn −αq‖2 − (1 −γn −γnL)β2nγnL‖xn −Txn‖

2

δnB.

The last inequality implies that
‖xn+1 −αq‖2 −‖xn −αq‖2 + (1 −γn −γnL)β2nγnL‖xn −Txn‖

2 ≤ δnB. (3.22)
Now, we consider the following two cases:Case A: Suppose there exists n0 ∈N such that {‖xn −αq‖} is non-increasing. Then, {‖xn −αq‖}is convergent. Clearly, ‖xn+1−αq‖−‖xn−αq‖→ 0. In view, of condition (ii) and (3.22), we have
‖xn−Txn‖→ 0. By Lemma 2.4, it is obvious that ωω(xn) ⊂ F (T ), where ωω(xn){x : ∃xnk ⇀ αx?}is the weak limit set of {xn}. This implies that the sequence {xn} converges weakly to a fixed point
αx? of T .Suppose there exists some subsequences {xnk}∞k=0 ⊂{xn}∞n=0 such that xnk ⇀ αy? weakly and
αy? 6= αx?. Since limn→∞‖xn −αv‖ exists for αv ∈ F (T ), by virtue of Opial condition on H, wehave

lim
n→∞

‖xn −αx?‖ = lim
n→∞

‖xnj −αx
?‖ < lim

n→∞
‖xnj −αy

?‖ = lim
n→∞

‖xnk −αy
?‖

< lim
n→∞

‖xnk −αx
?‖ = lim

n→∞
‖xnj −αy

?‖,

which is a contradiction. Consequently, αy? = αx?. This implies that {xnj}∞j=0 converges wealy toa common fixed point of T.Next, we prove that {xn}∞n=0 converges strongly to x?/ Let ξn = γnTyn + (1−γnxn). Then, from(3.1), we obtain xn+1 = PK[ξn −δnxn],n ≥ 0. This implies that
xn+1 = PK[ξn + δnξn + δnξn −δnxn

= PK[(1 −δn)ξn + δn(ξn −xn)]. (3.23)
Observe that

‖ξn −αx?‖2 = ‖xn −αx? −γn(xn −Tyn)‖2. (3.24)By using the same argument as in (3.20), with αx? = αq, we get, from (3.24), that
‖ξn −αx?‖ = ‖xn −αx?‖. (3.25)

Again, from (3.1), we obtain
‖yn −xn‖ = βn‖xn −Txn‖→ 0 as n →∞,βn ∈ (0, 1). (3.26)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 10
In addition, since T is Lipschitz, it follows that

‖ξn −xn‖ = ‖γn[(Tyn −Txn) − (xn −Txn)]‖

≤ γn‖Tyn −Txn‖−γn‖xn −Txn‖

≤ γnL‖yn −xn‖ + γn‖xn −Txn‖→ 0 as n →∞. (3.27)
Now, using (3.23), we get

‖xn+1 −αx?‖2 ≤ ‖(1 −δn)ξn + δn(ξn −xn) −αx?‖2

= ‖(1 −δn)(ξn −αx?) + δn(ξn −xn) −δnαx?‖2,

which by Lemma 2.3 yields
‖xn+1 −αx?‖2 ≤ ‖(1 −δn)(ξn −αx?) + δn(ξn −xn)‖2 − 2δn〈αx?,xn+1 −αx?〉

= (1 −δn)‖ξn −αx?‖2 + δn‖ξn −xn‖2 −δn(1 −δn)‖xn −αx?‖2

−2δn〈αx?,xn+1 −αx?〉

≤ (1 −δn)‖ξn −αx?‖2 + ‖ξn −xn‖2 − 2δn〈αx?,xn+1 −αx?〉

= (1 −δn)‖ξn −αx?‖2 − 2δn〈αx?,xn+1 −αx?〉 ( by (3.27)) (3.28)
≤ (1 −δn)‖ξn −αx?‖2 (3.29)

(3.29) and Lemma 2.2 imply that xn → αx? as n →∞.Case B: Assume that {‖xn − αq‖}∞n=0 is not a monotonically increasing sequence. Set Vn =
‖xn −αq‖2 and let τ : N−→N be a mapping defined by

τn = max{k ∈N : k ≤ n,Vn ≤ Vn+1},∀n ≥ n0,

for some n0 large enough. Obviously, {τn}∞n=0 is a nondecreasing sequence given that τn →
∞ as n →∞ and Vτn ≤ Vτn+1 for all n ≥ n0. From (3.22),

‖xτ(n) −Txτ(n)‖
2 ≤

δτ(n)B

(1 −γτ(n) −γτ(n)L)β2τ(n)γτ(n)L
→ 0 as n →∞. (3.30)

Therefore, limn→∞‖xτ(n) − Txτ(n)‖ = 0. Using similar argument as Case A above, we concludethat {xτ(n)}→ αx? →∞.From (3.28), we have
0 ≤‖xτ(n)+1 −αx

?‖2 −‖xτ(n) −αx
?‖2 ≤ δτ(n)[2〈αx

? −xτ(n)+1 −‖xτ(n) −αx
?‖2], (3.31)

for δτ(n) ∈ (0, 1). Hence, limn→∞‖xτ(n) − αx?‖2 = 0. This implies that limn→∞Vτ(n) =
limn→∞Vτ(n)+1 = 0. In addition, for n ≥ n0, it is easy to see that Vτ(n) = Vτ(n)+1 if n 6= τ(n)(i.e., τ(n) < n) because Vj > Vj+1, f or τ(n) + 1 ≤ n. Consequently. we obtain, for all n ≥ n0,
0 ≤ Vτ(n)max{Vτ(n),Vτ(n)+1} = Vτ(n)+1. Hence, limn→∞Vn = 0. That is, {xn}∞n=0 converges

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 11
strongly to αx?, and this completes the proof.

�

The following corollaries are immediate consequence of Theorem 3.1.
Corollary 3.2. Let H be a real Hilbert space, K a nonempty closed convex subset of H and
T : K −→ K an L-Lipschitz hemicontractive mapping. For any arbitrary x0 ∈ H, define the
sequence {xn}∞n=0 iteratively as follows: xn+1 = PK[(1 −αn −γn)xn + γnTyn]

yn = (1 −βn)xn + βnTxn,n ≥ 1,
(3.32)

where the sequences {δn}∞n=0,{γn}
∞
n=0,{βn}

∞
n=0 ∈ (0, 1) satisfy the following conditions:

(i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤
1 −δ

1 + L2
;

(ii) limn→∞δn = 0 and
∑∞
n=0δn = ∞.

Then, the sequence {xn}∞n=0 generated by (3.32) weakly and strongly converges to the fixed point
of T .

Corollary 3.3. Let H be a real Hilbert space, K a nonempty closed convex subset of H and
T : K −→ K is α-demicontractive mapping. For any arbitrary x0 ∈ H, define the sequence
{xn}∞n=0 iteratively as follows: xn+1 = PK[(1 −αn −γn)xn + γnTyn]

yn = (1 −βn)xn + βnTxn,n ≥ 1,
(3.33)

where the sequences {δn}∞n=0,{γn}
∞
n=0,{βn}

∞
n=0 ∈ (0, 1) satisfy the following conditions:

(i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤
1 −δ

1 + L2
;

(ii) limn→∞δn = 0 and
∑∞
n=0δn = ∞.

Then, the sequence {xn}∞n=0 generated by (3.33) weakly and strongly converges to the fixed point
of T .

Corollary 3.4. Let H be a real Hilbert space, K a nonempty closed convex subset of H and
T : K −→ K is demicontractive mapping. For any arbitrary x0 ∈ H, define the sequence {xn}∞n=0
iteratively as follows:  xn+1 = PK[(1 −αn −γn)xn + γnTyn]

yn = (1 −βn)xn + βnTxn,n ≥ 1,
(3.34)

where the sequences {δn}∞n=0,{γn}
∞
n=0,{βn}

∞
n=0 ∈ (0, 1) satisfy the following conditions:

(i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤
1 −δ

1 + L2
;

(ii) limn→∞δn = 0 and
∑∞
n=0δn = ∞.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.10 12
Then, the sequence {xn}∞n=0 generated by (3.34) weakly and strongly converges to the fixed point
of T .

Competing Interest. The authors declare that there is no conflict of interest.
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https://doi.org/10.28924/ada/ma.2.10
https://doi.org/10.1006/jmaa.1993.1309
https://doi.org/10.1016/j.na.2009.03.075
https://doi.org/10.1016/j.jmaa.2008.01.045
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https://doi.org/10.1090/S0002-9939-00-05573-8
https://doi.org/10.28919/afpt/4442

	1. Introduction
	2. Preliminary
	3. Convergence Results 
	Competing Interest

	References

