Soft ˜n times expansive mapping about fixed point theorem

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J. Math. Comput. Sci. 8 (2018), No. 2, 216-224 https://doi.org/10.28919/jmcs/3579

ISSN: 1927-5307

SOFT n˜ TIMES EXPANSIVE MAPPING ABOUT FIXED POINT THEOREM

SHUNXIN ZHAO∗, MEIMEI SONG

Department of Mathematics, Tianjin University of Technology, Tianjin 300384, China

Copyright c2018 Zhao and Song. This is an open access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.In this paper, we introduce the definitions about soft sets and soft reasonable expansive mapping. Then,

we proved the fixed point theorems about it on complete soft metric spaces. Finally, we present some examples to

validate and illustrate our approach.

Keywords:soft fixed point; complete soft metric spaces; soft contraction mapping.

2010 AMS Subject Classification:47H10, 54H25

1. Introduction

The research about fixed point of expansive mapping was initiated by Machuca(see[1]).

Later, Jungck discussed fixed points for other forms of expansive mapping(see[2]). In the

re-sent, many researchers contribute many structure on soft set theory(see[3-5]). Futher, Das and

Samanta[6] introduced a different notion of soft metric space by using a different concept of

soft point and investigated some basic properties of these spaces.

∗_{Corresponding author}

E-mail address: [email protected]

Received November 23, 2017

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Recently, the study about fixed point theorem for expansive mapping and nonexpansive

map-ping is depply explored and has extended too many other directions. Motivated and inspired by

the works(see[7-10]), in this paper, we define soft n times reasonable expansive mapping and

discuss the existence of fixed point for soft n times reasonable expansive mapping.

2. Preliminaries

In this section, we gather together some definitions and known results which will be used in

section 3. We use those definitions , and we dont make redundant narration. Before we state

our fixed point theorems, we introduce the following basic definitions:

Definition 2.1. Let A be a nonempty subset of E. A soft set (F, A) onU is a set of the form (F, A)={(F(p),p):p∈E}. where F: A→2U is a set-valued map such that F(p)=_∅for allp∈/A. F

is called an approximate function of (F,A), and the collection of all soft set on U will be denoted

byS(U).

Definition 2.2.A soft set (F,A) on U is said to be a null soft set denoted _∅ if for all P∈A, F(p)=_∅.

Definition 2.3.A soft set (F,A) on U is said to be an absolute soft set, if for allP∈A, F(p)=U.

Definition 2.4.A soft set (F,A) on U is said to be a soft point if there is exactly one p∈E, such that F(p)={x}for some x∈U and F(q)=_∅for all q∈E\ {p}. It will be denoted by ˜x_p.

Definition 2.5.Let B (_R) be the collection of all nonempty bounded subsets of _R. Then the mapping F:E →B(_R)is called a soft real set. If (F, E) is a singleton soft set, then identifying

(F, E) with the corresponding soft element, it will be called a soft real number and denoted ˜r, ˜s,

˜

t, ect. If all elements of E are the same singleton soft set. It denoted ¯r, ¯s, ¯t, ect.

For two soft real numbers ˜r, ˜s, we say that:

(N1) ˜r≤˜s, if ˜˜ r(e)≤˜s˜(e), for alle∈A;

(N2) ˜r≥˜s, if ˜˜ r(e)≥˜s˜(e), for alle∈A;

(N3) ˜r<˜s, if ˜˜ r(e)<˜s˜(e), for alle∈A;

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Definition 2.6.A be a nonempty subset of parameters and ˜U be the absolute soft set, i.e.

(f,ϕ)(x_λ) =U for allλ ∈A, where (F,A)= ˜U.

A mappingd: SP(U˜)×SP(U˜)→_R(A)∗is said to be a soft metric on ˜U if for anyUx

λ,U

y

µ,Uγz∈

SP(X˜), the following hold :

(M1)d(Ux

λ,U

y

µ)≥˜¯0 ;

(M2)d(Ux

λ,U

y

µ)=˜¯0, if and only ifU_λx =U

y

µ ;

(M3)d(Ux

λ,U

y

µ)=˜d(U

y

µ,U_λx);

(M4)d(Ux

λ,U

y

µ)≤˜d(U_λx,Uγz) +d(U

y

µ,Uγz).

Given a soft metric space(U˜,d), a net{Uxα

λα}α∈Λ of soft point in ˜U will be simply denoted by {Ux

λα}α∈Λ. In particular, a sequence{U

xn

λn}n∈Nof soft points in ˜U will be denoted by{U

x

λn}n.

Definition 2.7. Let{x˜_λ_,n}nbe a sequence of soft points in a soft metric space(U˜,d˜,E). Then

the sequence{x˜_λ_,n}nis said to be convergent in(U˜,d˜,E)if there is a soft point ˜yν∈˜X˜ such that

lim

n→∞

˜

d(x_λ˜_,n,y˜_ν) =¯0

.

Definition 2.8.Let{x˜_λ_,n}nbe a sequence of soft points in a soft metric space (U˜,d˜,E). Then

the sequence{x˜_λ_,n}nis said to be a Cauchy sequence in(U˜,d˜,E)if there is a soft point ˜yν∈˜X˜

such that

lim

i,j→∞

˜

d(x_λ˜_,i,x_λ˜_,_j) =¯0

.

Definition 2.9.A soft metric space(U˜,d˜,E)is called complete if every Cauchy sequence in ˜X converges to some point of ˜X. Throughout this paper,we use Nto denoted the set of soft

natural numbers and ˜[x]to denote the maximum soft integral value that is not large that ˜x.

Definition 2.10.f : ˜U→_U˜ _{is called a soft expansive mapping if there exists a soft constant ˜}_h_≥˜ _¯1 such that

˜

d(f(x˜_λ),f(y˜_µ))≥˜h˜d˜(x˜_λ,y˜˜_µ),for all ˜x_λ,y˜_µ∈˜SP(U˜).

f : ˜U→U˜ is called a soft two times reasonable expansive mapping if there exists a soft constant

˜

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˜

d(x˜_λ,f2(x˜_λ)_ϕ2₍_λ₎)≥˜h˜ d˜(x˜,f(x˜_λ)), for all ˜x_λ∈SP(U˜).

Definition 2.11.f : ˜U →U˜ is called a soft twenty-one type expansive mapping if there exists a soft constant ˜h≥˜¯1 such that

(1) ˜

d(f(x˜_λ),f(y˜µ))≥˜hmin˜ {d˜(x˜λ,y˜µ),d˜(x˜λ,f(x˜λ)),d˜(y˜µ,f(y˜µ)),d˜(x˜λ,f(y˜µ)),d˜(y˜µ,f(x˜λ))},

f or all x˜_λ,y˜_µ∈˜SP(U˜).

f : ˜U→U˜ is called a soft twenty-three type expansive mapping if there exists a soft constant

˜

h≥˜¯1 such that

(2) ˜

d2(f(x˜_λ),f2(y˜µ)ϕ2(µ))≥˜ hmin˜ {d˜2(x˜λ,y˜µ),d˜(x˜λ,y˜µ)·d˜(x˜λ,f(x˜λ)),d˜(x˜λ,f(x˜λ))·d˜(y˜µ,f(y˜µ)),

˜

d2(x˜_λ,f(x˜_λ),d˜(y˜_µ,f(y˜_µ))·d˜(x˜_λ,f(y˜_λ)),d˜(y˜_µ,f(y˜_µ))·d˜(y˜_µ,f(x˜_λ))},f orallx˜_λ,y˜_µ∈˜SP(U˜).

3. Main results

Definition 3.1.Let(U˜,d˜,E)be a complete soft metric space, and let(f,ϕ):(U˜,d˜,E)→(U˜,d˜,E)

is called a soft n times reasonable expansive mapping if there exists a soft constant ˜h>˜¯1 such that

(3) d˜(x˜_λ,fn(x˜_λ)_ϕn

(λ))

˜

≥h˜ d˜(x˜_λ,(f,ϕ)x˜_λ), f or allx˜∈˜ U˜ (n≥2,n∈_N)

is called a softH1-type n times reasonable expansive mapping if there exists a soft constant ˜h≥˜ ¯1 such that

(4) ˜

d(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn−1(y˜_µ) ϕ₍n−1

µ)

)≥˜ hmin˜ {d˜(x˜_λ,y˜_µ),d˜(x˜_λ,(f,ϕ)(x˜_λ)),d˜(fn−2(y˜µ)_ϕn−2 (µ)

,

fn−1(y˜_µ) ϕ₍n−1

µ)

),d˜(x˜_λ,fn−1(y˜_µ) ϕ₍n−1

µ)

)}

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such that

(5) ˜

d2(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn−1(y˜_µ) ϕ₍n−1

µ)

)≥˜ hmin˜ {d˜2(x˜_λ,y˜_µ),d˜(x˜_λ,y˜_µ)·d˜(x˜_λ,(f,ϕ)(x˜_λ)),d˜(x˜_λ,

(f,ϕ)(x˜_λ))·d˜(fn−2(y˜µ)_ϕn−2 (µ)

,fn−1(y˜_µ) ϕ₍n−1

µ)

),d˜2(x˜_λ,(f,ϕ)(x˜_λ)),

˜

d(fn−2(y˜_µ) ϕ₍n−2

µ)

,fn−1(y˜_µ) ϕ₍n−1

µ)

)·d˜(x˜_λ,fn−1(y˜_µ) ϕ₍n−1

µ)

)}

Lemma 3.4. Let(U˜,d˜,E)be a complete soft metric space, let ˜U₁be a subset of ˜U, and left the soft mappings(f,ϕ1),(g,ϕ2) : ˜U1→U˜ satisfy the following conditions:

(i)(f,ϕ1)is a surjective soft mapping(f,ϕ)(U˜1) = (U˜);

(ii) There is a functionalφ : ˜U→_R+ which is lower semicontinuous bounded from below such

that ˜d((f,ϕ1)(x˜λ1),(g,ϕ2)(xλ˜2))≤˜ φ((f,ϕ1)(xλ˜1))−φ((g,ϕ2)(xλ˜2))for all ˜xλi∈˜ U˜1.

Then,(f,ϕ1)and(g,ϕ2)have a coincidence point, that is, there exists at least an ˜xλ ∈˜ U˜1such

that(f,ϕ1)(x˜λ) = (g,ϕ2)(x˜λ).

Theorem 3.5. Let(U˜,d˜,E)be a complete soft metric space and(f,ϕ)be a continuous and soft

surjective mapping if there exists a constant ˜h>˜ ¯1 such that

(6) d˜(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎)≥˜ h˜ d˜(x˜_λ,(f,ϕ)(x˜_λ)) f or all x˜_λ ∈˜ SP(U˜)

Then,(f,ϕ)has a fixed point in(U˜,d˜,E).

Proof.By (2.4) we have

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˜

d(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎)−d˜(x˜_λ,(f,ϕ)(x˜_λ))≥˜ h˜d˜(x˜_λ,(f,ϕ)(x˜_λ))−d˜(x˜_λ,(f,ϕ)(x˜_λ))

Thus,

(8) _d˜₍_x_˜

λ,(f,ϕ)(x˜λ))≤˜

¯1 ¯

h−¯1[d˜(f

n−1₍_x_˜

λ)ϕn−1(λ),fn(x˜λ)ϕn(λ))−d˜(x˜λ,(f,ϕ)(x˜λ))]

Letφ(x˜_λ) = _¯¯1

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So we can getφ((f,ϕ)(x˜

λ)). Then we have ˜d(x˜λ,(f,ϕ)(x˜λ)≤˜ φ((f,ϕ)(x˜λ))−φ(x˜λ).

From the continuity of ˜d, we know that φ(x˜_λ) is continuous. Thus φ(x˜_λ) is lower

semicon-tinuous bounded from below. Therefore the conclusion follows immediately from the above

Lemma 2.4. This completes the proof.

Theorem 3.6.Let(U˜,d˜,E)be a complete soft metric space and(f,ϕ)be a continuous and soft

surjective ˜n(n˜≥˜2,n˜∈˜ _N˜)times reasonable expansive mapping. Assume that either (i)or(ii)holds:

(i)(f,ϕ)is anH1−typen˜times reasonable expansive mapping; (ii)(f,ϕ)is anH2−typen˜times reasonable expansive mapping. Then,(f,ϕ)has a fixed point in ˜U.

Proof.In the case of (i),taking ˜yµ = (f,ϕ)(x˜λ)in (2.2),we have

(9) ˜

d(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎)≥˜ h min˜ {d˜(x˜_λ,(f,ϕ)(x˜_λ)), d˜(x˜_λ,(f,ϕ)(x˜_λ)),

˜

d(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎), d˜(x˜_λ,fn(x˜_λ)_ϕn₍_λ₎)}

=h min˜ {d˜(x˜_λ,(f,ϕ)(x˜_λ)),d˜(fn−1(x˜_λ)_ϕn−1₍_λ₎,

fn(x˜_λ)_ϕn₍_λ₎), d˜(x˜_λ,fn(x˜_λ)_ϕn₍_λ₎)}

Because(f,ϕ)is an ˜ntimes reasonable expansive mapping, we have

(10) d˜(x˜_λ,fn(x˜_λ)_ϕn₍_λ₎)≥˜ h˜ d˜(x˜_λ,(f,ϕ)(x˜_λ))>˜ d˜(x˜_λ,(f,ϕ)(x˜_λ))

Thus,we obtain

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˜

d(fn−1(x˜_λ)_ϕn−1₍

λ),f

n₍

˜

x_λ)_ϕn₍_λ₎)≥˜ h min˜ {d˜(x˜_λ,(f,ϕ)(x˜_λ)),d˜(fn−1(x˜_λ)_ϕn−1₍

λ),f

n₍

˜

x_λ)_ϕn₍_λ₎)}

If ˜d(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎) = min{d˜(x˜_λ,(f,ϕ)(x˜_λ)),d˜(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎)},

then ˜d(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎)≥˜ h˜ {d˜(fn−1(x˜_λ)

ϕn−1(λ),f

n₍_x_˜

λ)ϕn(λ))}. Hence,

˜

d(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎) = 0. (otherwise, ˜d(fn−1(x˜_λ)

ϕn−1(λ),fn(x˜λ)ϕn(λ))≥˜

˜

d(fn−1(x˜_λ)_ϕn−1₍

λ),f

n₍_x_˜

λ)ϕn(λ)), which is a contradiction). Therefore, f

n−1₍_x_˜

λ)ϕn−1(λ)= f

n₍_x_˜

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that is fn−1(x˜_λ)_ϕn−1₍_λ₎ = f(fn−1(x˜_λ)_ϕn−1₍_λ₎), which implies that fn−1(x˜_λ)_ϕn−1₍_λ₎ is a fixed

point of(f,ϕ)in ˜U.

If ˜d(x˜_λ,(f,ϕ)(x˜_λ)) = min{d˜(x˜_λ,(f,ϕ)(x˜_λ)),d˜(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎)}, then

˜

d(fn−1(x˜_λ)_ϕn−1₍_λ₎,fn(x˜_λ)_ϕn₍_λ₎)≥˜ hd˜(x˜_λ,(f,ϕ)(x˜_λ)).

By Theorem 2.5, we obtain that (f,ϕ) has a fixed point in ˜U. In the case of (ii),taking

˜

y_µ = (f,ϕ)(x˜_λ)in (2.3),we have

(12) ˜

d2(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ) ϕ₍n−1

λ)

)≥˜ hmin˜ {d˜2(x˜_λ,(f,ϕ)(x˜_λ)),d˜(x˜_λ,(f,ϕ)(x˜_λ))·d˜(x˜_λ,(f,ϕ)(x˜_λ)),

˜

d(x˜_λ,(f,ϕ)(x˜_λ))·d˜(fn−1(x˜_λ)_ϕn−1 (λ)

,fn(x˜_λ)_ϕn

(λ)),

˜

d2(x˜_λ,(f,ϕ)(x˜_λ)),

˜

d(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)_ϕn

(λ))·

˜

d(x˜_λ,fn(x˜_λ)_ϕn

(λ))}

= h min{d˜2(x˜_λ,(f,ϕ)(x˜_λ)),d˜(x˜_λ,(f,ϕ)(x˜_λ))·d˜(fn−1(x˜_λ)_ϕn−1 (λ)

,

fn(x˜_λ)ϕ₍n λ)),

˜

d(fn−1(x˜_λ) ϕ₍n_λ−₎1,f

n₍_x_˜

λ)ϕ₍n λ))·

˜

d(x˜_λ,fn(x˜_λ)ϕ₍n λ))}

Because ˜U is an ˜ntimes reasonable expansive mapping, we have

(13) d˜(x˜_λ,fn(x˜_λ)_ϕn₍_λ₎)≥˜ h˜ d˜(x˜_λ,(f,ϕ)(x˜_λ))>˜ d˜(x˜_λ,(f,ϕ)(x˜_λ))

Hence, ˜d(x˜_λ,fn(x˜_λ)_ϕn₍_λ₎)·d˜(fn−1(x˜_λ)

ϕ₍n−1 λ)

,fn(x˜_λ)_ϕn

(λ))>˜

˜

d(x˜_λ,(f,ϕ)(x˜_λ))

·d˜(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)_ϕn

(λ)).

Therefore, we have

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˜

d2(fn−1(x˜_λ) ϕ₍n_λ−₎1,f

n₍_x_˜

λ)_ϕn−1

(λ) )

˜

≥hmin˜ {d˜2(x˜_λ,(f,ϕ)(x˜_λ)),d˜(x˜_λ,(f,ϕ)(x˜_λ))

·d˜(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)_ϕn

(λ))}

If ˜d2(x˜_λ,(f,ϕ)(x˜_λ)) = min{d˜2(x˜_λ,(f,ϕ)(x˜_λ)),d˜(x˜_λ,(f,ϕ)(x˜_λ))·d˜(fn−1(x˜_λ)

ϕ₍n−1 λ)

,fn(x˜_λ)_ϕn

(λ))},

then

(15) d˜2(fn−1(x˜_λ) ϕ₍n−1

λ)

(8)

that is, ˜d(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)_ϕn

(λ))

˜

≥√˜hd˜(x˜_λ,(f,ϕ)(x˜_λ)).

Because√˜h≥˜ 1, by Theorem 2.5, we obtain that ˜U has a fixed point in ˜U.

If ˜d(x˜_λ,(f,ϕ)(x˜_λ))·d˜(fn−1(x˜_λ)_ϕn−1 (λ)

,fn(x˜_λ)_ϕn

(λ)) = min{

˜

d2(x˜_λ,(f,ϕ)(x˜_λ)),d˜(x˜_λ,(f,ϕ)(x˜_λ))·

˜

d(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)_ϕn

(λ))}, then ˜d

2₍_fn−1₍_x_˜

λ)_ϕn−1

(λ)

λ)

)≥˜ h˜d˜(x˜_λ,(f,ϕ)(x˜_λ))

·d˜(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)_ϕn

(λ)),that is

(16) d˜(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)ϕ₍n λ)) · (

˜

d(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)ϕ₍n λ))−h

˜

d(x˜_λ,(f,ϕ)(x˜_λ)))≥˜ ˜0.

If ˜d(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)_ϕn

(λ)) = ˜0, then f n−1₍_x_˜

λ)_ϕn−1

(λ)

= fn(x˜_λ)_ϕn

(λ), that is f n−1₍_x_˜

λ)_ϕn−1

(λ)

=

f(fn−1(x˜_λ) ϕ₍n−1

λ)

), which implies that fn−1(x˜_λ) ϕ₍n−1

λ) is a fixed point of

(f,ϕ)in ˜U.

If ˜d(fn−1(x˜_λ) ϕ₍n−1

λ)

,fn(x˜_λ)_ϕn

(λ)) 6= ˜0, then ˜d(f n−1₍_x_˜

λ)_ϕn−1

(λ)

,fn(x˜_λ)_ϕn

(λ))≥˜ h

˜

d(x˜_λ,(f,ϕ)(x˜_λ)))≥˜ ˜0.

By Theorem 2.5, we obtain that(f,ϕ)has a fixed point in ˜U.

Therefore,by induction we derive that(f,ϕ)has a fixed point in ˜U.

This completes the proof.

Corollary 3.7.Let(U˜,d˜,E)be a complete soft metric space and(f,ϕ)be a continuous and soft

surjective twenty-three type expansive mapping and(f,ϕ)is a two times reasonable expansive

mapping, then(f,ϕ)has a fixed point in ˜U.

Conflict of Interests

The authors declare that there is no conflict of interests.

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