Some new fixed point results for monotone enriched nonexpansive mappings in ordered Banach spaces

Available online at www.atnaa.org Research Article

Some new xed point results for monotone enriched nonexpansive mappings in ordered Banach spaces

Rahul Shukla^a, Rajendra Pant^a

aDepartment of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa.

Abstract

We study monotone enriched nonexpansive mappings and present some new existence and convergence theorems for these mappings in the setting of ordered Banach spaces. More precisely, we employ the Kras- nosel'ski iterative method to approximate xed points of enriched nonexpansive under dierent conditions.

This way a number of results from the literature have been extended, generalized and complemented.

Keywords: Nonexpansive mapping; Enriched nonexpansive mapping; Banach space.

2010 MSC: 47H10; 47H09.

1. Introduction

Let (B, k.k) be a Banach space and C a nonempty subset of B. A mapping ξ : C → C is said to be nonexpansive if for each pair of elements ϑ, ν ∈ C, we have

kξ(ϑ) − ξ(ν)k ≤ kϑ − νk.

A point ϑ^† ∈ C is said to be a xed point of ξ if ξ(ϑ^†) = ϑ^†. The class of nonexpansive mapping need not have a xed point in the case of general Banach spaces. However, in 1965, Browder [7], Göhde [12] Kirk [13], independently proved the rst xed point result for nonexpansive mappings in Banach spaces having ceratin geometrical properties. After these results, a number of nonlinear mappings have been appeared in the literature to enlarge the class of nonexpansive mappings [11, 15, 9, 25, 18] (see also the references therein).

Email addresses: rshukla.vnit@gmail.com; rshukla@uj.ac.za (Rahul Shukla), pant.rajendra@gmail.com;

rpant@uj.ac.za (Rajendra Pant)

Received :November 20, 2020; Accepted: July 4, 2021; Online: July 6, 2021

Very recently, Berinde [3] introduced a new class of nonexpansive mappings known as enriched nonex- pansive mappings which generalizes nonexpansive mappings. It is shown in [3, 4] that this class of mappings has strong connection with the nonexpansive mappings.

On the other hand, a number of xed point theorems have appeared in the literature where the non- expansive condition on mapping needs to satisfy only for comparable elements in partially ordered spaces.

The motivation behind this approach is to determine the nature of the solution whether it is positive or negative and this approach has fruitful applications. Ran and Reurings obtained the solution of matrix equations through a generalization of the Banach contraction principle, see [19]. Nieto and Rodríguez-López [16] applied similar type of xed point theorem to nd the solution of some dierential equations, for more applications of the xed point theory for monotone mappings, see [8]. Thereafter, many authors developed a metric xed point theory for monotone nonexpansive mappings, see [6, 5, 23, 22, 21, 2, 24].

Motivated by Berinde [3, 4] and others, we extend the class of enriched nonexpansive mappings in the setting of ordered Banach spaces and establish some existence and convergence results for enriched nonexpansive mappings. We employ Krasnosel'ski iterative method to approximate the xed points in ordered Banach spaces under certain assuptions. Our results complement, extend, and generalize certain results from [3, 4, 5, 24, 2].

2. Preliminaries

Let B be a Banach space with a partial order  compatible with the linear structure of B, that is, ϑ  ν implies ϑ + ζ  ν + ζ,

ϑ  ν implies λϑ  λν

for every ϑ, ν, ζ ∈ B and λ ≥ 0. It follows that all order intervals [ϑ, →] = {ζ ∈ B : ϑ  ζ} and [←, ν] = {ζ ∈ B : ζ  ν}are convex. Moreover, we will assume that each [ϑ, →] and [←, ν] is closed. We will say that (B, k · k, ) is an ordered Banach space.

A sequence {ϑn}is said to be an approximate xed point sequence (a.f.p.s. for short) for a mapping ξ if kξ(ϑ_n) − ϑnk → 0 as n → ∞. A sequence {ϑn} is monotone increasing if ϑ1  ϑ₂  ϑ₃  · · · .We shall use the following observation (see [5, Lemma 3.1]). Assume that {ϑn}is a monotone sequence that has a cluster point, i.e., there is a subsequence {ϑnj} that converges to g (with respect to the strong or weak topology).

Since the order intervals are (weakly) closed, we have g ∈ [ϑn, →) for each n, that is, g is an upper bound for {ϑn}. If g1 is another upper bound for {ϑn}, then ϑn ∈ (←, g₁]for each n, and hence g  g1.It follows that {ϑn}converges to g = sup{ϑn}.If {ϑn}is a monotone increasing (resp. monotone decreasing) sequence which converges to p, then ϑn p(resp. p  ϑn). We say that ϑ, ν ∈ B are comparable whenever ϑ  ν or ν  ϑ.

Denition 2.1. [10]. A Banach space B is said to be uniformly convex if for every ε ∈ (0, 2] there is some δ > 0so that, for any ϑ, ν ∈ B with kϑk = kνk = 1, the condition kϑ − νk ≥ ε implies that ^ϑ+ν₂

≤ 1 − δ.

Denition 2.2. Let (B, k · k, ) be an ordered Banach space.

• [17]. A space B satises weak-Opial property if, for every weakly convergent sequence {ϑn} with weak limit ϑ ∈ B it holds:

lim inf

n→∞ kϑ_n− ϑk < lim inf

n→∞ kϑ_n− νk for all ν ∈ B with ϑ 6= ν.

• [1] A space B satises the monotone weak-Opial property if, for every monotone weakly convergent sequence {ϑn} with weak limit ϑ ∈ B it holds:

lim inf

n→∞ kϑ_n− ϑk < lim inf

n→∞ kϑ_n− νk

for all ν ∈ B and ν is greater or less than all the elements of the sequence {ϑn}.

All nite dimensional Banach spaces, all Hilbert spaces and `^p (1 ≤ p < ∞) satisfy the weak-Opial property. But Lp([0, 1]),for p > 1 do not have the Opial property [10]. In [1], it is proved that Lp([0, 1]),for p > 1,satisfy monotone weak-Opial property.

Denition 2.3. [10]. A mapping ξ : C → C is said to be quasi-nonexpansive if for all ϑ ∈ C and ϑ^†∈ F (ξ) 6=

∅,

kξ(ϑ) − ϑ^†k ≤ kϑ − ϑ^†k.

where F (ξ) is the set of all xed points of ξ.

It is well known that a nonexpansive mapping with a xed point is quasi-nonexpansive. However the converse need not to be true.

Denition 2.4. [20]. The mapping ξ : C → C with F (ξ) 6= ∅ satises Condition (I) if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0, f(r) > 0 for r ∈ (0, ∞) such that kϑ − ξ(ϑ)k ≥ f(d(ϑ, F (ξ))) for all ϑ ∈ C, where d(ϑ, F (ξ)) = inf{kϑ − yk : ν ∈ F (ξ)}.

Let C be a convex subset of a Banach space B and ξ : C → C a monotone nonexpansive mapping. The following iteration process is known as the Krasnosel'ski-Mann iteration process (see [14]):

(ϑ₁∈ C

ϑn+1= αnϑn+ (1 − αn)ξ(ϑn) (1)

where {αn}is a sequence in [a, b] with a, b ∈ (0, 1).

Lemma 2.5. [5]. Let C be a nonempty bounded closed convex subset of an ordered Banach space (B, k · k, ) and ξ : C → C a monotone nonexpansive mapping. Suppose that {ϑn} is a sequence dened by (1) and ϑ1  ξ(ϑ₁). Then lim

n→∞kϑ_n− ξ(ϑ_n)k = 0.

Lemma 2.6. [5]. Let C be a nonempty closed convex subset of an ordered convex space (B, k · k, ) and ξ : C → C a monotone mapping. Suppose that {ϑn} is a sequence dened by (1) and ϑ1  ξ(ϑ₁). Then

ϑ_n ϑ_n+1 ξ(ϑ_n) for all n ∈ N.

Lemma 2.7. Let C a nonempty convex subset a Banach space B. Let ξ : C → C be a mapping, dene S : C → C as follows:

S(ϑ) = (1 − λ)ϑ + λξ(ϑ) for all ϑ ∈ C and λ ∈ (0, 1). Then F (S) = F (ξ).

Denition 2.8. Let C be a nonempty subset of a Banach space B.

• A mapping ξ : C → C is said to be compact if ξ(C) has a compact closure.

• A mapping ξ : C → C is said to be weakly compact if ξ(C) has a weakly compact closure.

Lemma 2.9. [26] For given r > 0. A Banach space B is uniformly convex if and only if there exists a continuous strictly increasing function ϕ : [0, ∞) → [0, ∞), ϕ(0) = 0, such that

kλϑ + (1 − λ)νk² ≤ λkϑk²+ (1 − λ)kνk²− λ(1 − λ)ϕ(kϑ − νk) (2) for all λ ∈ [0, 1] and ϑ, ν ∈ M with kϑk ≤ r, kνk ≤ r.

3. Main Results

Recently, Berinde [3] introduced the following class of nonlinear mappings.

Denition 3.1. Let (B, k.k) be a Banach space. A mapping ξ : B → B is said to be b-enriched nonexpansive mapping if there exists b ∈ [0, ∞) such that for all ϑ, ν ∈ B

kb(ϑ − ν) + ξ(ϑ) − ξ(ν)k ≤ (b + 1)kϑ − νk. (3)

It is shown that every nonexpansive mapping ξ is a 0-enriched mapping. The classes of b-enriched nonexpansive mappings and that of quasi-nonexpansive mappings are independent in nature. The following two examples illustrate this fact.

Example 3.2. [3]. Let C = ¹₂, 2 ⊂ R and ξ : C → C be a mapping dened as ξ(ϑ) = _ϑ¹. Then F (ξ) = {1}

and ξ is a ³₂-enriched nonexpansive mapping. On the other hand at ϑ = ¹₂

|ξ(ϑ) − 1| = |2 − 1| = 1 > 1

2 = |ϑ − 1|.

Thus ξ is not a quasi-nonexpansive mapping.

Example 3.3. Let C = [0, 4] ⊂ R and ξ : C → C be a mapping dened as

ξ(ϑ) =

(0, if ϑ 6= 4 3, if ϑ = 4.

Then F (ξ) = {0} and ξ is a quasi-nonexpansive mapping. On the other hand at ϑ = 3 and ν = 4, ξ is not a b-enriched nonexpansive mapping for any b ∈ [0, ∞).

The following useful denition is due to [6]:

Denition 3.4. Let (B, k.k, ) be an ordered Banach space and C a nonempty subset of B. A mapping ξ : C → C is said to be monotone if

ϑ  ν implies ξ(ϑ)  ξ(ν), where ϑ, ν ∈ C.

Now, we extend Denition 3.1 in the setting of partially ordered Banach spaces as follows:

Denition 3.5. Let (B, k.k, ) be an ordered Banach space and C a nonempty subset of B. A mapping ξ : C → C is said to be monotone b-enriched nonexpansive mapping if ξ is monotone and there exists b ∈ [0, ∞)such that

kb(ϑ − ν) + ξ(ϑ) − ξ(ν)k ≤ (b + 1)kϑ − νk (4)

for all ϑ, ν ∈ C with ϑ and ν are comparable.

It can be seen that every monotone nonexpansive mapping ξ is a monotone 0-enriched mapping.

Theorem 3.6. Let (B, k.k, ) be an ordered uniformly convex Banach space and C a nonempty bounded closed convex subset of B. Let ξ : C → C be a monotone b-enriched nonexpansive mapping. Suppose that there exists a point ϑ1 in C such that ϑ1 and ξ(ϑ1) are comparable. Then F (ξ) 6= ∅.

Moreover, for given λ ∈

0,_b+1¹  the sequence {ϑn} dened by (Krasnosel'ski iterative method)

ϑn+1= (1 − λ)ϑn+ λξ(ϑn) (5)

converges weakly to a point in F (ξ).

Proof. By the denition of monotone b-enriched nonexpansive mapping, we have

kb(ϑ − ν) + ξ(ϑ) − ξ(ν)k ≤ (b + 1)kϑ − νk (6)

for all ϑ  ν. Take µ = _b+1¹ ∈ (0, 1)and put b = ^1−µ_µ in (6) then the above inequality is equivalent to k(1 − µ)(ϑ − ν) + µ(ξ(ϑ) − ξ(ν))k ≤ kϑ − νk. (7) Dene the mapping S as follows:

S(ϑ) = (1 − µ)ϑ + µξ(ϑ)for all ϑ ∈ C.

Since ξ is monotone, for all ϑ  ν

S(ϑ) = (1 − µ)ϑ + µξ(ϑ)  (1 − µ)ϑ + µξ(ν)  (1 − µ)ν + µξ(ν) = S(ν) and S is monotone. Then from (7), we get

kS(ϑ) − S(ν)k ≤ kϑ − νk

for all ϑ  ν. Thus S is a monotone nonexpansive mapping. Since ϑ1  ξ(ϑ₁) ϑ1 = (1 − µ)ϑ1+ µϑ1  (1 − µ)ϑ₁+ µξ(ϑ1) = S(ϑ1).

Thus all the assumptions of [6, Theorem 4.1] are satised and S has a xed point in C. From Lemma 2.7, F (S) = F (ξ) 6= ∅.

Next, for given ϑ1∈ C and any λ ∈ (0, 1), consider the sequence

ϑ_n+1= (1 − λ)ϑ_n+ λS(ϑ_n). (8)

From Lemma 2.6 (with λ = αn for all n ∈ N)

ϑn ϑ_n+1 S(ϑ_n) for all n ∈ N. Again from Lemma 2.5

n→∞lim kϑ_n− S(ϑ_n)k = 0.

Therefore {ϑn}is an a.f.p.s. for a monotone nonexpansive mapping S and all the assumptions of [24, Theorem 1] are fullled. Hence {ϑn} converges weakly to a xed point of S. But F (S) = F (ξ) and

(1 − λ)ϑ + λS(ϑ) = (1 − λµ)ϑ + λµξ(ϑ) for all ϑ ∈ C. Since λ ∈ (0, 1) and µ = _b+1¹ .This implies that λµ ∈

0,_b+1¹ 

.Therefore for any λ ∈

0,_b+1¹ , the sequence {ϑn}dened by (5) converges weakly to a point in F (ξ).

Theorem 3.7. Let (B, k.k, ) be an ordered uniformly convex Banach space and C a nonempty bounded closed convex subset of B. Let ξ : C → C be a monotone b-enriched nonexpansive mapping. Suppose that there exists a point ϑ1 in C such that ϑ1 and ξ(ϑ1) are comparable. Then F (ξ) 6= ∅.

Moreover, the sequence {ϑn} dened by

ϑn+1 =



1 − 1 b + 1



ϑn+ 1

b + 1ξ(ϑn) converges weakly to a point in F (ξ).

Proof. Following the same proof technique as in Theorem 3.6, we can dene a mapping S : C → C as follows:

S(ϑ) =



1 − 1 b + 1



ϑ + 1

b + 1ξ(ϑ) for all ϑ ∈ C

and S is a monotone nonexpansive mapping with ϑ1  S(ϑ₁). Then all the assumptions of [24, Theorem 5]

are satised, hence {Sⁿ(ϑ1)}converges weakly to a xed point of S. But F (S) = F (ξ) and Sⁿ(ϑ₁) =



1 − 1 b + 1



ϑ_n+ 1

b + 1ξ(ϑ_n) for all n ∈ N. This completes the proof.

Remark 3.8. In Theorem 3.7, we extend the value of λ to _b+1¹ . In [4, Theorem 3.2], the value of λ lies in



0,_b+1¹ .

Theorem 3.9. Let (B, k.k, ) be an ordered Banach space having the weak-Opial property and C a nonempty weakly compact convex subset of B. Let ξ : C → C be a monotone b-enriched nonexpansive mapping. Suppose that there exists a point ϑ1 in C such that ϑ1 and ξ(ϑ1) are comparable. Then F (ξ) 6= ∅.

Moreover, for given λ ∈

0,_b+1¹  the sequence {ϑn} dened by (Krasnosel'ski iterative method) ϑ_n+1= (1 − λ)ϑ_n+ λξ(ϑ_n)

converges weakly to a point in F (ξ).

Proof. Following largely the proof of Theorem 3.6, we can dene a monotone nonexpansive mapping S with ϑ₁  S(ϑ₁). Thus all the assumptions of [5, Theorem 3.3] are satised and it is guaranteed that S has at least one xed point. From Lemma 2.7, F (S) = F (ξ) 6= ∅. For given ϑ1 ∈ C and for any λ ∈ (0, 1), consider a sequence

ϑ_n+1= (1 − λ)ϑ_n+ λS(ϑ_n). (9)

From [5, Theorem 3.3], {ϑn} converges weakly to a xed point of S. But F (S) = F (ξ), the rest of proof directly follows from Theorem 3.6.

Theorem 3.10. Let (B, k.k, ) be an ordered Banach space having the monotone weak-Opial property and C a nonempty bounded closed convex subset of B. Let ξ : C → C be a weakly compact monotone b-enriched nonexpansive mapping. Suppose that there exists a point ϑ1 in C such that ϑ1 and ξ(ϑ1) are comparable.

Then F (ξ) 6= ∅.

Moreover, for given λ ∈

0,_b+1¹  the sequence {ϑn} dened by (Krasnosel'ski iterative method) ϑn+1= (1 − λ)ϑn+ λξ(ϑn)

converges weakly to a point in F (ξ).

Proof. From the proof of Theorem 3.6, we can dene a monotone nonexpansive mapping S with ϑ1  S(ϑ₁).

For given ϑ1 ∈ C and for any λ ∈ (0, 1), consider a sequence

ϑ_n+1= (1 − λ)ϑ_n+ λS(ϑ_n). (10)

From Lemma 2.6 (with λ = αn for all n ∈ N)

ϑ_n ϑ_n+1 S(ϑ_n) for all n ∈ N. Again from Lemma 2.5

n→∞lim kϑ_n− S(ϑ_n)k = 0

and

n→∞lim kϑ_n− ξ(ϑ_n)k = 0. (11)

Since the range of C under ξ is contained in a weakly compact set, there exists a subsequence {ξ(ϑnj)} of {ξ(ϑ_n)} converges weakly to ϑ^† ∈ C. By (11), the subsequence {ϑnj} converges weakly to ϑ^†. Since {ϑn} is monotone increasing, the sequences {ϑn} and {S(ϑn)} converge weakly to ϑ^†. Thus for each n ∈ N, ϑ_n  S(ϑ_n)  ϑ^†. By the monotonicity of S, for each n ∈ N, S(ϑn)  S(ϑ^†). Suppose that S(ϑ^†) 6= ϑ^†, by the monotone weak-Opial property, we get

lim inf

n→∞ kϑ_n− ϑ^†k < lim inf

n→∞ kϑ_n− S(ϑ^†)k. (12)

By the triangle inequality and using the fact that S is monotone nonexpansive mapping, kϑ_n− S(ϑ^†)k ≤ kϑn− S(ϑ_n)k + kS(ϑn) − S(ϑ^†)k ≤ kϑn− S(ϑ_n)k + kϑn− ϑ^†k and

lim inf

n→∞ kϑ_n− S(ϑ^†)k ≤ lim inf

n→∞ kϑ_n− ϑ^†k

a contradiction from (12). Thus S(ϑ^†) = ϑ^†,and the rest of proof directly follows from Theorem 3.6.

Theorem 3.11. Let (B, k.k, ) be an ordered uniformly convex Banach space and C a nonempty closed convex subset of B. Let ξ : C → C be a monotone b-enriched nonexpansive mapping and ξ satises Condition (I). Suppose that there exists a point ϑ1 in C such that ϑ1  ξ(ϑ₁), F (ξ) 6= ∅ and ϑ1  ζ for all ζ ∈ F (ξ).

For given λ ∈

0,_b+1¹  the sequence {ϑn} dened by (Krasnosel'ski iterative method) ϑn+1= (1 − λ)ϑn+ λξ(ϑn)

converges strongly to a point in F (ξ).

Proof. Following largely the proof of Theorem 3.6, we can dene a monotone nonexpansive mapping S with ϑ1  S(ϑ₁). Let λ ∈ (0, 1) and dene

ϑ_n+1= (1 − λ)ϑ_n+ λS(ϑ_n). (13)

Since ϑ1  ζ for all ζ ∈ F (ξ) = F (S) and S is monotone mapping, S(ϑ1)  S(ζ) = ζ and ϑ₂ = (1 − λ)ϑ₁+ λS(ϑ₁)  (1 − λ)ϑ₁+ λζ  (1 − λ)ζ + λζ = ζ similarly, it can be seen that ϑn ζ for all ζ ∈ F (S) and n ∈ N.

Now, we show that lim

n→∞d(ϑ_n, F (S)) = 0.For any ζ ∈ F (S),

kS(ϑ_n) − ζk ≤ kϑn− ζkfor all n ≥ 1. (14)

Thus

kϑ_n+1− ζk ≤ (1 − λ)kϑ_n− ζk + λkS(ϑ_n) − ζk ≤ kϑ_n− ζk.

Hence the sequences {kϑn−ζk}and {d(ϑn, F (S))}are monotone nonincreasing and lim

n→∞kϑ_n−ζk, lim

n→∞d(ϑn, F (S)) exist. Again

kϑ_n+1− ζk² = k(1 − λ)(ϑ_n− ζ) + λ(S(ϑ_n) − ζ)k²

≤ (1 − λ)kϑ_n− ζk²+ λkS(ϑ_n) − ζ)k²− λ(1 − λ)ϕ(kϑ_n− S(ϑ_n)k)

≤ kϑ_n− ζk²− λ(1 − λ)ϕ(kϑ_n− S(ϑ_n)k).

Thus

λ(1 − λ)ϕ(kϑn− S(ϑ_n)k) ≤ kϑn+1− ζk²− kϑ_n− ζk² → 0as n → ∞ and

kϑ_n− S(ϑ_n)k → 0 as n → ∞. (15)

Since S(ϑ) = (1 − µ)ϑ + µξ(ϑ) for all ϑ ∈ C,

ϑ − S(ϑ) = µ(ϑ − ξ(ϑ)) for all ϑ ∈ C. (16)

Since ξ satises Condition (I), and (16), we obtain kϑ_n− S(ϑ_n)k

µ = kϑn− ξ(ϑ_n)k ≥ f (d(ϑn, F (ξ))) = f (d(ϑn, F (S))).

From (15), lim

n→∞f (d(ϑ_n, F (S))) = 0and

n→∞lim d(ϑ_n, F (S)) = 0. (17)

Now, it can be seen that the sequence {ϑn}is Cauchy. For the sake of completeness we include the argument.

For given ε > 0, in view of (17), there exists a n0 ∈ N such that for all n ≥ n0

d(ϑn, F (S)) < ε 4. In particular,

inf{kϑn0 − ζk : ζ ∈ F (S)} < ε 4, and there exists ζ ∈ F (S) such that

kϑ_n0 − ζk < ε 2. Therefore, for all m, n ≥ n0,

kϑ_n+m− ϑ_nk ≤ kϑ_n+m− ζk + kζ − ϑ_nk ≤ kϑ_n− ζk < 2ε 2 = ε,

and the sequence {ϑn} is Cauchy. Since C is a closed subset of B, so {ϑn}converges to a point ϑ^†∈ C and ϑ_n ϑ^†for all n ∈ N.

kϑ^†− S(ϑ^†)k ≤ kϑ^†− ϑ_nk + kϑ_n− S(ϑ_n)k + kS(ϑ_n) − S(ϑ^†)k

≤ 2kϑ^†− ϑ_nk + kϑ_n− S(ϑ_n)k

from (15), ϑ^†= S(ϑ^†). Hence, the sequence {ϑn} converges strongly to a point in F (ξ).

Theorem 3.12. Let (B, k.k, ) be an ordered Banach space and C a nonempty bounded closed convex subset of B.Let ξ : C → C be a compact monotone b-enriched nonexpansive mapping. Suppose that there exists a point ϑ₁ in C such that ϑ1 ξ(ϑ₁). For given λ ∈

0,_b+1¹  the sequence {ϑn} dened by (Krasnosel'ski iterative method)

ϑn+1= (1 − λ)ϑn+ λξ(ϑn) converges strongly to a point in F (ξ).

Acknowledgement

The authors are thankful to the reviewers and the editor for their constructive comments. The rst author acknowledges the support from the GES 4.0 fellowship, University of Johannesburg, South Africa.

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