Fixed point theorems for a new nonlinear mapping in Hilbert Spaces

Abstract: This study presents a new class of nonlinear mappings in Hilbert space. We establish demiclosed princi-

ples and fixed point theorems for this nonlinear mapping. Some works of literature are improved on and expanded

upon by the results that this study presents.

Key-Words:Nonlinear mappings, fixed point, convergence, nonexpansive mapping.

AMS Subject Classification: 47H09; 47H10.

Received: January 23, 2023. Revised: October 26, 2023. Accepted: November 29, 2023. Published: December 31, 2023.

1 Introduction and Preliminaries

Throughout this paper, we denote by Nthe set of pos-

itive integers and by Rthe set of real numbers. Let

Hbe a real Hilbert space with inner product ⟨., .⟩and

norm ∥.∥, respectively.

Let be Hbe a real Hilbert space and Mbe a

nonempty closed convex subset of H, and F:M →

Mbe a mapping. A point z∈ M is called a fixed

point of F:M → M if z=Fz. We denote F ix(F)

the set of a fixed points of F.

A mapping F:M→Mis called nonexpansive

∥Fu− Fv∥ ≤ ∥u−v∥

for all u, v ∈ M.Fis called quasi-nonexpansive if

F ix(F)=∅and

∥Fu−z∥ ≤ ∥u−z∥

for all u∈ M and z∈F ix(F). If F:M→Mis

nonexpansive mapping and the set F ix(F)=∅, then

Fis quasi-nonexpansive. It is well-known that the

set F ix(F)of fixed points of a quasi-nonexpansive

mapping Fis closed and convex, [1], [2].

The following demiclosed principle for nonexpan-

sive mappings in Hilbert spaces was provided in 1965

by [3].

Theorem 1.1 Let Mbe a nonempty closed convex

subset of a real Hilbert space H. Let Fbe a non-

expansive mapping of Minto itself, and let {un}be

a sequence in M. If un→zand ∥un− Fun∥= 0,

then Fz=z.

The following fixed point theorem for nonexpansive

mappings in Hilbert spaces was provided in 1971 by

[4].

Theorem 1.2 Let Mbe a nonempty closed convex

subset of a real Hilbert space H. Let Tbe a non-

expansive mapping of Minto itself, Then {Fnu}is a

bounded sequence for some u∈ M iff F ix(F)=∅.

In a real Hilbert space, the following result was pro-

vided in 1980 by [5].

Theorem 1.3 Let Mbe a nonempty closed convex

subset of a real Hilbert space H.Then the following

conditions are equivalent.

(1) Every nonexpansive mapping of Minto itself has

a fixed point in M;

(2) Mis bounded.

A firmly nonexpansive mapping is an essential exam-

ple of a nonexpansive mapping in a Hilbert space. A

mapping F:M→Mis said to be firmly nonex-

pansive if

∥Fu− Fv∥2≤ ⟨u−v, Fu− Fv⟩

for all u, v ∈ M, [3], [6], [7].

Nonspreading mapping was first introduced in

2008 by [8]. They also obtained a common fixed

point theorem for a commutative family of non-

spreading mappings in Banach spaces, as well as a

fixed point theorem for a single nonspreading map-

ping. A mapping F:M → M is called nonspread-

ing, [8], if

2∥Fu− Fv∥2≤ ∥Fu−v∥2+∥Fv−u∥2

for all u, v ∈ M.

An extension of Theorem 1.2 for nonspreading

mapping in Hilbert spaces was made by [8]. An ex-

tension of Ray’s type theorem for nonspreading map-

ping in Hilbert spaces was made in 2010 by [9].

Also, the demiclosed principles were extended for

nonspreading mappings by [10]. In Hilbert spaces,

the following two nonlinear mappings introduced by

[11], are said to be T J −1,T J −2; see also, [12]. A

mapping F:M → M is called a T J −1mapping,

[11], if

2∥Fu− Fv∥2≤ ∥u−v∥2+∥Fu−v∥2

SEYİT TEMİR

Department of Mathematics, Faculty of Arts and Sciences,

Adıyaman University, 02040, Adıyaman

TURKEY

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for all u, v ∈ M. A mapping F:M → M is called

aT J −2, [11], mapping if

3∥Fu− Fv∥2≤2∥Fu−v∥2+∥Fv−u∥2

for all u, v ∈ M. Similar results to the above theo-

rems were also obtained for T J −1and T J −2map-

pings, [11]. Motivated by the above works, we intro-

duce a new nonlinear mappings in Hilbert spaces.

Definition 1.4 Let Mbe a nonempty closed convex

subset of a Hilbert space H. We say F:M → M

is a new nonlinear mapping if there exists α, β ∈ R

with 0≤α+β≤2such that

2∥Fu− Fv∥2≤α∥Fu−v∥2+β∥Fv−u∥2

+ (2 −α−β)∥u−v∥2

for all u, v ∈ M.

Remark 1.5 Especially with special choices of αand

β, the nonlinear mapping as defined in Definition 1.4

becomes nonspreading mapping, T J −1mapping,

T J −2mapping and nonexpansive mapping. Indeed,

in Definition 1.4, we know that if we choose

(1) α=β= 1 for all u, v ∈ M , then Fis a non-

spreading mapping;

(2) α= 1, β = 0 for all u, v ∈ M, then Fis a

T J −1mapping;

(3) α=4

3, β =2

3for all u, v ∈ M, then Fis a

T J −2mapping.

(4) α= 0, β = 0 for all u, v ∈ M, then Fis a

nonexpansive mapping.

We can also show that if u=Fu, then for any v∈

2∥u− Fv∥2≤α∥u−v∥2+β∥Fv−u∥2

+ (2 −α−β)∥u−v∥2

(2 −β)∥u− Fv∥2≤(2 −β)∥u−v∥2

∥u− Fv∥2≤ ∥u−v∥2.

This means that the nonlinear mapping as de-

fined in Definition 1.4 with a fixed point is quasi-

nonexpansive mapping.

Let ℓ∞be the Banach space of bounded sequences

with the supremum norm. A linear functional µon

ℓ∞is called a mean if µ(e) = ||µ|| = 1, where e=

(1,1,1, ....). For u= (u1, u2, u3, ....), the value µis

also denoted by µn(un). A Banach limit on ℓ∞is an

invariant mean, that is, µn(un) = µn(un+1). If µis

a Banach limit on ℓ∞, then for u= (u1, u2, u3, ...)∈

ℓ∞,liminfn→∞ un≤µnun≤limsupn→∞ un. In

particular, if u= (u1, u2, u3, ...)∈ℓ∞and un→

a∈ R, then we have µn(un) = µn(un+1) = a. For

details, [7].

Proposition 1.6 Let Mbe a nonempty closed convex

subset of a Hilbert space H. Let α, β be the same as

in Definition 1.4. Then, F:M → M is a nonlinear

mapping if and only if

∥Fu− Fv∥2≤α−β

2−β∥Fu−u∥2+∥u−v∥2

+2⟨Fu−u, α(u−v) + β(Fv−u)⟩

2−β

for all u, v ∈ M.

Proof. We have that for u, v ∈ M,

2∥Fu− Fv∥2≤α∥Fu−v∥2+β∥Fv−u∥2

+ (2 −α−β)∥u−v∥2

=α∥Fu−u∥2+2α⟨Fu−u, u−v⟩+α∥u−v∥2

+β∥Fv− Fu∥2+ 2β⟨Fv− Fu, Fu−u⟩

+β∥Fu−u∥2+ (2 −α−β)∥u−v∥2

= (α+β)∥Fu−u∥2+β∥Fu− Fv∥2

+(2 −β)∥u−v∥2

+2α⟨Fu−u, u−v⟩+2β⟨Fv−Fu, Fu−u⟩

= (α+β)∥Fu−u∥2+β∥Fv− Fu∥2

+(2 −β)∥u−v∥2+ 2α⟨Fu−u, u −v⟩

+2β⟨Fv−u, u−Fu⟩−2β⟨Fu−u, u−Fu⟩

= (α−β)∥Fu−u∥2+β∥Fu− Fv∥2

+(2 −β)∥u−v∥2

+2⟨Fu−u, α(u−v) + β(Fv−u)⟩.

We have

∥Fu−Fv∥2≤α−β

2−β∥Fu−u∥2+∥u−v∥2

+2⟨Fu−u, α(u−v)+β(Fv−u)⟩

2−β.

Hence, the proof is completed.

Remark 1.7 If α=β= 1 , then Proposition 1.6 is

reduced to Lemma 3.2 in [10]. In the sequel, we need

the following lemmas as tools.

Following the similar argument as in the proof of The-

orem 3.1.5, [7], we get the following result.

Lemma 1.8 Let Mbe a nonempty closed convex

subset of a real Hilbert space H, and let µbe a Ba-

nach limit. Let {un}be a sequence with un⇀ z.

If u=z, then µn∥un−z∥< µn∥un−u∥and

µn∥un−z∥2< µn∥un−u∥2.

The following fixed point theorem was proven using

Banach limits.

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Theorem 1.9 , [11]. Let Hbe a Hilbert space, let

Mbe a nonempty closed convex subset of H, and let

Fbe a nonlinear mapping of Minto itself. Suppose

that there exists an element u∈ M such that Fnuis

bounded and

µn∥Fnu− Fv∥2≤µn∥Fnu−v∥2,∀v∈ M

for some Banach limit µ. Then , Fhas a fixed

point in M.

2 Main results

In this section, we study the fixed point theorems,

demiclosed principles for nonlinear mappings in

Hilbert spaces.

2.1 Fixed point theorems

Theorem 2.1 Let Mbe a nonempty closed convex

subset of a real Hilbert space H,Fbe a nonlinear

mappings defined in Definition 1.4 .Then, F ix(F)=

∅if and only if {Fnz}is bounded for some z∈ M.

Proof. Since F:M→Mis a nonlinear mapping if

there exist α, β ∈ R with 0≤α+β≤2such that

2∥Fu− Fv∥2≤α∥Fu−v∥2+β∥Fv−u∥2+

(2 −α−β)∥u−v∥2

for all u, v ∈ M. If F ix(F)=∅, then Fnz=z

for z∈F ix(F). So {Fnz}is bounded. We show

reverse. Take z∈ M such that {Fnz}is bounded.

Let µbe a Banach limit. Then for any v∈ M and

n∈ N ∪ {0}, we have

2∥Fn+1z− Fv∥2≤α∥Fn+1z−v∥2

+β∥Fv− Fnz∥2

+(2 −α−β)∥Fnz−v∥2

for any v∈ M. Since {Fnz}is bounded, we can ap-

ply a Banach limit µto both sides of inequality. Then

we have

µn(2∥Fn+1z− Fv∥2)≤µn(α∥Fn+1z−v∥2

+β∥Fv− Fnz∥2

+ (2−α−β)∥Fnz−v∥2)

µn(2∥Fn+1z−Fv∥2)≤µn(α∥Fn+1z−v∥2)

+µn(β∥Fv− Fnz∥2)

+µn((2−α−β)∥Fnz−v∥2

)

(2 −β)µn∥Fnz− Fv∥2≤αµn∥Fnz−v∥2

+(2−α−β)µn∥Fnz−v∥2

(2 −β)µn∥Fnz−Fv∥2≤(2 −β)µn∥Fnz−v∥2

for all v∈ M. By Theorem 1.9, then we have that

F ix(F)is nonempty.

As a direct consequence of Theorem 2.1, we have

the following

Theorem 2.2 Let Mbe nonempty bounded closed

convex subset of a Hilbert space Hand let Fbe a

nonlinear mapping from Mto itself. Then Thas a

fixed point.

Using Theorem 2.1, we can also prove the following

well-known fixed point theorems. We first prove a

fixed point theorem for nonexpansive mappings in a

Hilbert space.

Corollary 2.3 Let Mbe a nonempty closed convex

subset of a real Hilbert space H,Fbe a nonexpansive

mappings. Then, F ix(F)=∅if and only if {Fnz}is

bounded for some z∈ M.

Proof. If α= 0, β = 0 in Theorem 2.1, (0,0)-

nonlinear mapping of Minto itself is nonexpansive.

By Theorem 2.1, Fhas a fixed in M.

The following is a fixed point theorem for non-

spreading mappings in a Hilbert space.

Corollary 2.4 Let Mbe a nonempty closed convex

subset of a real Hilbert space H,Fbe a nonspreading

mappings .Then, F ix(F)=∅if and only if {Fnz}is

bounded for some z∈ M.

Proof. If α= 1, β = 1 in Theorem 2.1, (1,1)-

nonlinear mapping of Minto itself is nonspreading.

By Theorem 2.1, Fhas a fixed in M.

The following is a fixed point theorem for T J −1

mappings in a Hilbert space.

Corollary 2.5 Let Mbe a nonempty closed convex

subset of a real Hilbert space H,Fbe a T J −1

mappings .Then, F ix(F)=∅if and only if {Fnz}is

bounded for some z∈ M.

Proof. If α= 1, β = 0 in Theorem 2.1, (1,0)-

nonlinear mapping of Minto itself is T J −1. By

Theorem 2.1, Fhas a fixed in M.

The following is a fixed point theorem for T J −2

mappings in a Hilbert space.

Corollary 2.6 Let Mbe a nonempty closed convex

subset of a real Hilbert space H,Fbe a T J −2

mappings .Then, F ix(F)=∅if and only if {Fnz}is

bounded for some z∈ M.

Proof. If α=4

3, β =2

3in Theorem 2.1, (4

3,2

3)-

nonlinear mapping of Minto itself is T J −2. By

Theorem 2.1, Fhas a fixed in M.

2.2 Demiclosed principles

Theorem 2.7 Let Hbe a Hilbert space and let M

be a nonempty closed convex subset of H. Let F

be a nonlinear mapping defined in Definition 1.4 of

Minto itself such that F ix(F)=∅. Then Fis

demiclosed,i.e.,un⇀ z and un− Fun→0imply

z∈F ix(F).

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Proof. Let unbe a sequence in K with un⇀ z and

un− Fun→0as n→ ∞. Then unand Funare

bounded. Suppose z=Fz. From Opial’s condition,

[13], we have

lim inf

n→∞ ∥un−z∥2<lim inf

n→∞ ∥un− Fz∥2

=lim inf

n→∞ (∥un− Fun∥2

+∥Fun−Fz∥2+ 2⟨un−Fun,Fun−Fz⟩)

≤lim inf

n→∞ (∥un− Fun∥2

+α−β

2−β∥Fun−un∥2+∥un−z∥2

2⟨un− Fun, α(un−z) + β(Fz−un)⟩

2−β

+2⟨un− Fun,Fun− Fz⟩)

=lim inf

n→∞ ∥un−z∥2.

This is a contradiction. Hence we get the conclusion.

We have the following results as special of Theo-

rem 2.7.

Corollary 2.8 , [10]. Let Hbe a Hilbert space and

let Mbe a nonempty closed convex subset of H. Let

Fbe a nonspreading mapping of Minto itself such

that F ix(F)=∅. Then Fis demiclosed,i.e.,un⇀ z

and un− Fun→0imply z∈F ix(F).

Proof. If α= 1, β = 1 ,(1,1)-nonlinear mapping

of Minto itself becomes nonspreading. By Theorem

2.7, we get the conclusion.

Corollary 2.9 , [11]. Let Hbe a Hilbert space and

let Mbe a nonempty closed convex subset of H. Let

Fbe a T J −1mapping of Minto itself such that

F ix(F)=∅. Then Fis demiclosed,i.e.,un⇀ z and

un− Fun→0imply z∈F ix(F).

Proof. If α= 1, β = 0 ,(1,0)-nonlinear mapping of

Minto itself becomes T J −1. By Theorem 2.7, we

get the conclusion.

Corollary 2.10 , [11]. Let Hbe a Hilbert space and

let Mbe a nonempty closed convex subset of H. Let

Fbe a T J −2mapping of Minto itself such that

F ix(F)=∅. Then Fis demiclosed,i.e.,un⇀ z and

un− Fun→0imply z∈F ix(F).

Proof. If α=4

3, β =2

3,(4

3,2

3)-nonlinear mapping of

Minto itself becomes T J −2. By Theorem 2.7, we

get the conclusion.

2.3 Weak convergence theorem

Theorem 2.11 Let Hbe a Hilbert space and let M

be a nonempty closed convex subset of H. Let Fbe

a nonlinear mapping defined in Definition 1.4 of M

into itself such that F ix(F)=∅. Let {ςn}be a real

sequence in (0,1). Let {un}be defined by

u∈ M chosen arbitrary ,

un+1 = (1 −ςn)un+ςnFun,∀n∈ N ,

Assume that lim inf

n→∞ ςn(1 −ςn)>0, then un⇀ z for

z∈F ix(F).

Proof. For any z∈F ix(F)and all u∈ M, we

consider Fnonlinear mapping defined Definition 1.4,

i.e., 2∥Fu−Fz∥2≤α∥Fu−z∥2+β∥Fz−u∥2+(2−

α−β)∥u−z∥2, then we have ∥Fu−z∥2≤ ∥u−z∥2.

Therefore we get for each n∈ N ,∥Fun−z∥ ≤

∥un−z∥. Now

∥un+1 −z∥2=∥((1 −ςn)un+ςnFun)−z∥2

= (1−ςn)∥un−z∥2+ςn∥Fun−z∥2

−ςn(1−ςn)∥Fun−un∥2

≤(1 −ςn)∥un−z∥2+ςn∥un−z∥2

−ςn(1 −ςn)∥Fun−un∥2

=∥un−z∥2−ςn(1−ςn)∥Fun−un∥2.

Hence {∥un−z∥} is a nonincreasing sequence and

lim

n→∞ ∥un−z∥exists. Besides, we know that

ςn(1−ςn)∥Fun−un∥2≤ ∥un−z∥2− ∥un+1 −z∥2.

This implies that lim

n→∞ ∥Fun−un∥= 0. Now, it

is enough to show that {un}has a unique weak sub-

sequential limit in F ix(F). Let {unj}and {unk}be

two subsequences of {un}, converge weakly to zand

w, respectively. Then lim

n→∞ ∥Fun−un∥= 0 and

I−Fis demiclosed at zero by Theorem 2.7, where I

is identity mapping. This implies that (I−F)z= 0.

That is, F(z) = z; similarly F(w) = w. Next we

prove the uniqueness. Suppose that z=w. Then by

the Opial’s condition, [13], we have

lim

n→∞ ∥un−z∥=lim

j→∞

∥unj−z∥<lim

j→∞

∥unj−w∥

=lim

n→∞ ∥un−w∥=lim

k→∞

∥unk−w∥

<lim

k→∞

∥unk−z∥=lim

n→∞ ∥un−z∥.

This leads to a contradiction. So, z=w. Therefore

un⇀ z. This completes the proof.

2.4 Example for nonlinear mapping as

defined in Definition 1.4.

Example 2.12 The following example shows that F

is a nonlinear mapping as defined in Definition 1.4.

Define

F: [0,10] →[0,10]

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by Fu=0,0≤u < 1

0.005, u ≥1

Indeed, Fis a nonlinear mapping as defined in Def-

inition 1.4. Especially with special choices of αand

β, the nonlinear mapping as defined in Definition 1.4

becomes nonspreading mapping, T J −1mapping,

T J −2mapping and nonexpansive mapping. If we

choose

(1) α=β= 1 for all u, v ∈ M , then Fis a non-

spreading mapping;

(2) α= 1, β = 0 for all u, v ∈ M , then Fis a

T J −1mapping;

(3) α=4

3, β =2

3for all u, v ∈ M , then Fis a

T J −2mapping;

(4) α= 0, β = 0 for all u, v ∈ M , then Fis a

nonexpansive mapping.

Table 1: To verify that Fis a nonlinear mapping de-

fined in Definition 1.4, consider the following cases:

Cases α, β u, v Definition 1.4

111

10 ,9

10 0.3,10 0.00005≤110.07832

211

10 ,9

10 10,0.5 0.00005≤90.26952

100 ,199

100 1.2,0.2 0.00005 ≤2.85698

200 ,99

100 0.9,1.1 0.00005 ≤0.83926

5 0 ,2 0.9,1.1 0.00005 ≤1.60205

6 2 ,0 0.9,1.1 0.00005 ≤2.42000

200 ,99

100 1,0.2 0.00005 ≤1.63339

8 2 ,0 1,0.2 0.00005 ≤0.07605

9 0 ,2 1,0.2 0.00005 ≤2

10 1

200 ,99

100 0.9,10 0.00005≤84.51706

11 1

200 ,99

100 0.99,9.99 0.00005 ≤82.86452

12 1

200 ,99

100 1.01,0.99 0.00005 ≤1.01515

Table 2: To verify that Fis a nonspreading mapping

for α= 1, β = 1, consider the following cases:

Cases α, β u, v Nonspreading

1 1,1 1.01 ,0.99 0.00005 ≤1.99032

2 1,1 0.99 ,1.01 0.00005 ≤1.99032

3 1,1 1.2,0.2 0.00005 ≤1.47802

Table 3: To verify that Fis a T J −1mapping for

α= 1, β = 0, consider the following cases:

Cases α, β u, v T J −1

1 1,0 1.01 ,0.99 0.00005 ≤0.97062

2 1,0 1.2,0.2 0.00005 ≤1.03802

3 1,0 0.99 ,1.01 0.00005 ≤1.02050

Table 4: To verify that Fis a T J −2mapping for

α=4

3, β =2

3, consider the following cases:

Cases α, β u, v T J −2

3,2

31.01 ,0.99 0.000075 ≤2.96055

3,2

31.2,0.2 0.000075 ≤1.51605

3,2

30.99 ,1.01 0.000075 ≤3.01042

Table 5: To verify that Fis a nonexpansive mapping

for α= 0, β = 0, consider the following cases:

Cases α, β u, v Nonexpansive

1 0,0 1.01 ,0.99 0.005000 ≤0.02000

2 0,0 1.2,0.2 0.005000 ≤1.00000

3 0,0 0.99 ,1.01 0.005000 ≤0.02000

3 Conclusion

In this paper we introduce a new class of nonlinear

mappings in Hilbert space. We prove fixed point the-

orems and demiclosed principles for this nonlinear

mapping. Furthermore, we have given an example of

a nonlinear mapping as defined in Definition 1.4. It

is shown that for various αand βvalues, this map-

ping satisfies the inequality given in Definition 1.4 as

mentioned in Table 1. Especially with special choices

of αand β, the nonlinear mapping as defined in Def-

inition 1.4 becomes nonspreading mapping(Table 2),

T J −1mapping(Table 3), T J −2mapping(Table 4)

and nonexpansive mapping(Table 5).

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EQUATIONS

DOI: 10.37394/232021.2023.3.20

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E-ISSN: 2732-9976

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Volume 3, 2023