Existence of fixed points for [Theta]-type contraction and [Theta]-type Suzuki contraction in complete metric spaces

Abstract

The purpose of this paper is to introduce the notions of [Theta]-type contractions and [Theta]-type Suzuki contractions and to establish some new fixed point theorems for these two kinds of mappings in the setting of complete metric spaces. The results presented in the paper are an extension of the Banach contraction principle, the Suzuki contraction theorem, and the Jleli and Samet fixed point theorem. As an application, we utilize our results to study the existence problem of solutions of nonlinear Hammerstein integral equations.

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 DOI 10.1186/s13663-016-0496-5

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Web End = Existence of xed points for -type contraction and -type Suzuki contraction in complete metric spaces

Xin-dong Liu1, Shih-sen Chang2*, Yun Xiao1 and Liang Cai Zhao1

*Correspondence: mailto:[email protected]

Web End [email protected]

2Center for General Education, China Medical University, Taichung, 40402, TaiwanFull list of author information is available at the end of the article

Abstract

The purpose of this paper is to introduce the notions of -type contractions and -type Suzuki contractions and to establish some new xed point theorems for these two kinds of mappings in the setting of complete metric spaces. The results presented in the paper are an extension of the Banach contraction principle, the Suzuki contraction theorem, and the Jleli and Samet xed point theorem. As an application, we utilize our results to study the existence problem of solutions of nonlinear Hammerstein integral equations.

MSC: 47H09; 49M05; 47H10

Keywords: Banach contraction principle; xed point; -type contraction; -type Suzuki contraction

1 Introduction and preliminaries

Let (X, d) be a complete metric space and T : X X be a mapping. If there exists k (, ) such that for all x, y X, d(Tx, Ty) kd(x, y), then T is said to be a contractive mapping.

In , Polish mathematician Banach [] proved a very important result regarding a contraction mapping, known as the Banach contraction principle. It is one of the fundamental results in xed point theory. Due to its importance and simplicity, several authors have obtained many interesting extensions of the Banach contraction principle (see [] and the references therein).

In , Suzuki [] proved the following generalized Banach contraction principle in compact metric spaces.

Theorem . [] Let (X, d) be a compact metric space and T : X X be a mapping. Assume that, for all x, y X with x = y,

d(x, Tx) < d(x, y) d(Tx, Ty) < d(x, y).

Then T has a unique xed point in X.

In , Jleli and Samet [, ] introduced the following notion of a -contraction.

2016 Liu et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License

(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro

vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and

indicate if changes were made.

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 2 of 12

Denition . [] Let (X, d) be a metric space. A mapping T : X X is said to be a

-contraction, if there exists k (, ) such that

x, y X, d(Tx, Ty) = d(Tx, Ty) d(x, y) k, (.)

where : (, ) (, ) is a function satisfying the following conditions:

( ) is non-decreasing; ( ) for each sequence {tn} (, ), limn (tn) = i limn tn = ;

( ) there exist r (, ) and l (, ] such that limt

+ (t)

tr = l;

( ) is continuous.

In the sequel we denote by the set of all functions satisfying the conditions ( )-( ). By using the notion of a -contraction, Jleli et al. [] proved the following xed point theorem.

Theorem . (Jleli et al. []) Let (X, d) be a complete metric space and T : X X be a

-contraction, then T has a unique xed point in X.

Remark . It is obvious that Theorem . is a modied version of the Banach contraction principle. In fact, if T : X X is a Banach contractive mapping with a contractive constant

(, ), i.e.,

d(Tx, Ty) d(x, y), x, y X.

Since (t) = e

t

, t > , by passing it to the above inequality, we arrive at

ed(Tx,Ty) e

d(x,y) = ed(x,y)

= ed(x,y) k, x, y X,

where k = . It follows from Theorem . that T has a unique xed point in X.

From Theorem . it is natural to put forward the following open question.

Open question Could we obtain some xed point theorems for -contractive mappings without the conditions ( ) and ( )?

In order to give an armative answer to this open question, we rst analyze the conditions ( ) and ( ).

It is easy to see that the condition ( ) is so strong that there exist a lot of functions which satisfy the conditions ( ), ( ), and ( ) but they not the condition ( ). For example,

we can prove that the function (t) = ee

tp , p > satises the conditions ( ), ( ), and

( ), but, for any r > ,

lim

t

+

tp tr =

(t) tr =

lim

t

+

ee

tp

tr =

lim

t

+

tr

e

lim

t

+

e

tp = ,

i.e., it does not satisfy the condition ( ).

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 3 of 12

Furthermore, the condition ( ) can be replaced by an equivalent but a more simple condition inft(,) (t) = . This fact can be seen from the following lemmas.

Lemma . [] If {tk}k is a bounded sequence of real numbers such that all its convergent subsequences have the same limit l, then {tk}k is convergent and limk tk = l.

Lemma . Let : (, ) (, ) be a non-decreasing and continuous function with

inft(,) (t) = and {tk}k be a sequence in (, ). Then the following conclusion holds:

lim

k

(tk) = ,

which is a contradiction with (M) > . Therefore {tk} is bounded. Hence there exists a subsequence {tkn} {tk} such that limn tkn = (some nonnegative number). Clearly

.If > , then there exists n

N such that tkn ( , ) for all n n. As is nondecreasing, we deduce that ( ) limn (tkn) = which contradicts with ( ) > . Consequently = . By Lemma ., we know that limk tk = .

() (Suciency) Since inft(,) (t) = , if tk , then for any given > , there is > such that () (, + ) and there exists k

N such that tk < for all k > k. Therefore < (tk) () < + , for k > k. This shows that (tk) .

The conclusion of Lemma . is proved.

In the sequel, we denote by

the set of functions : (, ) (, ) satisfying the following conditions:

( ) is non-decreasing and continuous; ( ) inft(,) (t) = .

Examples of functions belonging to

It is obvious that the following are examples of the functions belonging to

:

tp , p > ; (t) := e

t, t > ;(t) := + t, t > ; (t) := arctan( t ), < < , t > .

such that

x, y X, d(Tx, Ty) > d(Tx, Ty) M(x, y) k. (.)

k tk = .

Proof () (Necessity) If limk (tk) = , then we claim that the sequence {tk} is bounded. Indeed, if the sequence is unbounded, we may assume that tk , then for every M > , there is k

N such that tk > M for any k > k. Hence we have (M) (tk), and so

(M) lim

k

(tk) = lim

(.)

Based on the above argument, now we are in a position to give the following denition.

Denition . Let (X, d) be a complete metric space and T : X X be a mapping.

() T is said to be a -type contraction, if there exist k (, ) and

(t) := ee

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 4 of 12

() T is said to be a -type Suzuki contraction, if there exist k (, ) and

that for all x, y X with Tx = Ty,

d(x, Tx) < d(x, y)

. (.)

The purpose of this paper is to prove some existence theorems of xed points for -type contraction and -type Suzuki contraction in the setting of complete metric spaces. The results presented in the paper improve and extend the corresponding results in Banach [], Suzuki [], Jleli and Samet [, ]. As an application, we shall utilize our results to study the existence problem of solutions for a class of nonlinear Hammerstein integral equations.

2 Existence theorems of xed point for -type Suzuki contractions and -type contractions

In this section, we are going to give some existence theorems of xed point for -type Suzuki contractions and -type contractions.

Theorem . Let (X, d) be a complete metric space and T : X X be a -type Suzuki contraction, i.e., there exist

and k (, ) such that for all x, y X with Tx = Ty,

d(x, Tx) < d(x, y)

where

M(x, y) := max d(x, y), d(x, Tx), d(y, Ty),

d(x, Ty), d(y, Tx)

such

d(Tx, Ty) M(x, y) k, (.)

where

M(x, y) = max d(x, y), d(x, Tx), d(y, Ty),

d(x, Ty), d(y, Tx)

d(Tx, Ty) M(x, y) k, (.)

, (.)

then T has a unique xed point z X and for each x X the sequence {Tnx} converges to z.

Proof Let x be an arbitrary point in X. If for some positive integer p such that Tpx = Tpx,

then Tpx will be a xed point of T. So, without loss of generality, we can assume that d(Tnx, Tnx) > for all n .

Therefore,

d Tnx, Tnx < d Tnx, Tnx , n . (.)

Hence from (.), for all n , we have

d TTnx, TTnx = d Tnx, Tn+x M Tnx, Tnx k, (.)

where

M Tnx, Tnx = max d Tnx, Tnx , d Tnx, Tnx , d Tnx, Tn+x ,

d Tnx, Tn+x , d Tnx, Tnx

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 5 of 12

= max d Tnx, Tnx , d Tnx, Tn+x ,

= max d Tnx, Tnx , d Tnx, Tn+x . (.)

If M(Tnx, Tnx) = d(Tnx, Tn+x), then it follows from (.) that

d Tnx, Tn+x d Tnx, Tn+x k.

This implies that

ln d Tnx, Tn+x k ln d Tnx, Tn+x ,

which is a contradiction with k (, ). Hence, from (.) we have M(Tnx, Tnx) = d(Tnx, Tnx). This together with inequality (.) yields

d Tnx, Tn+x d Tnx, Tnx k d Tnx, Tnx k

n

Therefore we have limn (d(Tnx, Tn+x)) = . This together with ( ) and Lemma . gives

lim

n d

N. (.)

Tpnx, Tqnx = . (.)

From (.) and (.), we can choose a positive integer n such that

d Tpnx, TTpnx <

d Tnx, Tn+x

d(x, Tx) k

. (.)

Since : (, ) (, ), it follows from (.) that

lim

n

d Tnx, Tn+x lim

n

d(x, Tx) k n

= .

Tnx, Tn+x = . (.)

Now, we claim that {Tnx}n= is a Cauchy sequence. Arguing by contradiction, we assume that there exist > and a sequence {pn}n= and {qn}n= of natural numbers such that

pn > qn > n, d Tpnx, Tqnx , d Tpnx, Tqnx < , n

So, we have

d Tpnx, Tqnx d Tpnx, Tpnx + d Tpnx, Tqnx

d Tpnx, Tpnx + .

It follows from (.) and the above inequality that

lim

n d

< d Tpnx, Tqnx , n n.

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 6 of 12

So, from the assumption of the theorem, we get

d Tpn+x, Tqn+x M Tpnx, Tqnx k, n n, (.)

where

M Tpnx, Tqnx = max d Tpnx, Tqnx , d Tpnx, Tpn+x , d Tqnx, Tqn+x ,

d Tpnx, Tqn+x , d Tqnx, Tpn+x

max d Tpnx, Tpn+x , d Tqnx, Tqn+x ,

d Tpnx, Tpn+x + d Tpn+x, Tqn+x + d Tqn+x, Tqnx . (.)

Substituting (.) into (.), then letting n and by using the condition ( ) , (.), (.), we get

lim

n

d Tpn+x, Tqn+x lim

n

d Tpn+x, Tqn+x k.

This is a contradiction. Therefore {Tnx}n= is a Cauchy sequence. By the completeness of (X, d), without loss of generality, we can assume that {Tnx}n= converges to some point z X, i.e.,

lim

n d

Tnx, z = . (.)

Now, we claim that

d Tn+x, Tn+x < d Tn+x, z , n

N. (.)

Suppose to the contrary that (.) is not true. Therefore the following inequality is also not true:

d Tnx, Tn+x < d Tnx, z , n

N. (.)

It follows from (.) and (.) that there exists an m

N such that

d Tmx, Tm+x d Tmx, z and

d Tm+x, Tm+x d Tm+x, z . (.)

Therefore,

d Tmx, z d Tmx, Tm+x d Tmx, z + d z, Tm+x .

This implies that

d Tmx, z d z, Tm+x . (.)

This together with (.) shows that

d Tmx, z d z, Tm+x

d Tm+x, Tm+x . (.)

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 7 of 12

Since d(Tmx, Tm+x) < d(Tmx, Tm+x), by the assumption of the theorem, we get

d Tm+x, Tm+x M Tmx, Tm+x k, (.)

where

M Tmx, Tm+x = max d Tmx, Tm+x , d Tmx, TTmx , d Tm+x, TTm+x ,

d Tmx, TTm+x , d Tm+x, TTmx

= max d Tmx, Tm+x , d Tmx, Tm+x , d Tm+x, Tm+x ,

d Tmx, Tm+x + d Tm+x, Tm+x

= max d Tmx, Tm+x , d Tm+x, Tm+x .

If M(Tmx, Tm+x) = d(Tm+x, Tm+x), then from (.) we have

d Tm+x, Tm+x d Tm+x, Tm+x k < d Tm+x, Tm+x .

This is a contradiction. Therefore we have

M Tmx, Tm+x = d Tmx, Tm+x .

Consequently, from (.) we have

d Tm+x, Tm+x d Tmx, Tm+x k. (.)

Since k (, ), we have

d Tmx, Tm+x k < d Tmx, Tm+x .

Hence from condition ( ) and (.) we have

d Tm+x, Tm+x < d Tmx, Tm+x . (.)

This together with (.) shows that

d Tm+x, Tm+x < d Tmx, Tm+x

d Tmx, z + d z, Tm+x

d Tm+x, Tm+x +

d Tm+x, Tm+x

= d Tm+x, Tm+x , (.)

which is a contradiction. Therefore the inequality (.) is proved.

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 8 of 12

N,

d TTn+x, Tz M Tn+x, z k. (.)

On the other hand, from (.) we know that Tnx z, hence we have

M Tn+x, z = max d Tn+x, z , d Tn+x, TTn+x , d(z, Tz),

d Tn+x, Tz , d Tz, TTn+x

d(z, Tz) (as n ). (.)

Now, we claim that d(z, Tz) = .

In fact, if d(z, Tz) > . Letting n in (.), and by using (.), (.), and the condition ( ) , we obtain

d(z, Tz) = lim

n

lim

n

This together with (.) shows that

d(z, u) = d(Tz, Tu) d(z, u) k < d(z, u) , (.)

which is a contraction. Hence we have u = v.

This completes the proof of Theorem ..

Remark . Theorem . is a generalization and improvement of the main results in

Suzuki [].

It follows from Denition . that if T : X X is a -type contraction, then T : X X is a -type Suzuki contraction. Hence from Theorem . we can obtain the following existence theorem of xed point for -type contractions.

By the assumption of Theorem . and (.) we have, for every n

d TTn+x, Tz

M Tn+x, z k

= d(z, Tz) k

< d(z, Tz) .

This is a contradiction. Hence, z = Tz, i.e., z is a xed point of T.

Now we prove that z is the unique xed point of T in X. In fact, if z, u X are two distinct xed points of T, that is Tz = z = u = Tu, then d(z, u) = d(Tz, Tu) > . Since = d(z, Tz) < d(z, u), as follows from the assumption of the theorem, we obtain

d(z, u) = d(Tz, Tu) M(z, u) k, (.)

where

M(z, u) = max d(z, u), d(z, Tz), d(u, Tu),

d(z, Tu), d(u, Tz)

= d(z, u).

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 9 of 12

Theorem . Let (X, d) be a complete metric space and T : X X be a -type contractive mapping, i.e., there exist

and k (, ) such that

x, y X, d(Tx, Ty) = d(Tx, Ty) M(x, y) k, (.)

where

M(x, y) := max d(x, y), d(x, Tx), d(y, Ty),

d(x, Ty), d(y, Tx)

, (.)

then T has a unique xed point z X and for any given x X, the sequence {Tnx} converges to z.

Proof Denote by (t) := e

t : (, ) (, ). It is easy to check that

. Hence from

This implies that T is a -type contractive mapping with k = . Therefore the conclusion of Corollary . can be obtained from Theorem . immediately.

The following corollary can be obtained from Corollary . immediately.

Corollary . Let (X, d) be a complete metric space and T : X X be a mapping. Suppose that there exist , , , , with + + + + < such that

d(Tx, Ty) d(x, y) + d(x, Tx) + d(y, Ty) +

xp

, (.)

then T has a unique xed point z X, and for each x X the sequence {Tnx} converges to z.

Remark . Theorem . is a generalization and improvement of the Banach contraction principle [] and some recent results in Jleli and Samet [, ].

3 Some consequencesCorollary . Let (X, d) be a complete metric space and T : X X be a mapping. If there exists (, ) such that

d(Tx, Ty) M(x, y), x, y X, (.)

where

M(x, y) = max d(x, y), d(x, Tx), d(y, Ty),

d(x, Ty), d(y, Tx)

(.) we have

ed(Tx,Ty) e

M(x,y)

= eM(x,y)

.

d(x, Ty) +

d(y, Tx), x, y X.

Then T has a unique xed point z and, for each x X, the sequence {Tnx} converges to z.

We note that if p > , then (t) = ee

. Hence from Theorem . we can obtain the

following corollary.

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 10 of 12

Corollary . Let (X, d) be a complete metric space and T : X X be a mapping. Suppose that there exist p > , k (, ) such that

ee

[d(Tx,Ty)]p

s, x(s) ds, (.)

where the unknown function x(t) takes real values.

Let X = C([, E]) be the space of all real continuous functions dened on [, E]. It is well known that C([, E]) endowed with the metric

d(x, y) = x y = max

t[,E]

x(t) y(t)

is a complete metric space. Dene a mapping T : X X by

T(x)(t) = h(t) +

Assumption .

(I) f C([, E] (, +)), h X, and K C([, E]) ([, E]) such that K(t, s) ;(II) f (t, ) : (, +) (, +) is increasing for all t [, E];(III) there exists [, +) such that

f (t, x) f (t, y) e Q(x, y), x, y X, t [, E],

where Q(x, y) := max{|x y|, |x Tx|, |y Ty|, |x Ty|, |y Tx|};(IV) maxt,s[,E] |K(t, s)| .

For x X, we dened x = maxt[,E] |x(t)|et, where is chosen arbitrarily. It is easy to check that is a norm equivalent to the maximum norm in X, and X endowed with the metric d dened by

d (x, y) = x y = max

t[,E]

x(t) y(t) et , x, y X,

is a complete metric space.

ee

[M(x,y)]p k, x, y X, Tx = Ty,

where M(x, y) is given by (.). Then T has a unique xed point z and, for each x X, the sequence {Tnx} converges to z.

4 Application to nonlinear Hammerstein integral equations

As an application, in this section, we shall use the xed point theorems proved in Section to study the existence and uniqueness problem of solutions for some kind of nonlinear Hammerstein integral equations.

Let us consider the following nonlinear Hammerstein integral equation:

x(t) = h(t) +

t K(t, s)f

t K(t, s)f

s, x(s) ds, t [, E]. (.)

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 11 of 12

Theorem . Let X = C([, E]), (X, d ), T, f , K(t, s) be the same as above. If Assumption . is satised, then the nonlinear Hammerstein integral equation (.) has a unique solution x C([, E]), and for each x C([, E]) the iterative sequence {xn = Tnx} converges to the unique solution x X of equation (.).

Proof We rst show that the mapping T : X X dened by (.) is a -type contraction. Indeed, from the conditions (III) and (IV), for each x, y C([, E]), t [, E], we have

Tx(t) Ty(t) =

t

K(t, s)

f s, x(s) f s, y(s) ds

t

K(t, s) f s, x(s) f s, y(s) ds

t

f s, x(s) f s, y(s) ds

t

e Q x(s), y(s) ds

max x(s) y(s) es , x(s) Tx(s) es ,

y(s) Ty(s) es ,

= e

t es

x(s) Ty(s) es , y(s) Tx(s) es

ds

e

t

es

max d (x, y), d (x, Tx),

d (y, Ty),

d (x, Ty), d (y, Tx)

ds

= e M(x, y)

t es ds

= e M(x, y)et

= e(t)M(x, y),

where

M(x, y) = max d (x, y), d (x, Tx), d (y, Ty),

d (x, Ty), d (y, Tx)

.

This implies that |Tx(t) Ty(t)|et e M(x, y). Hence we have

d (Tx, Ty) = max

t[,E]

Tx(t) Ty(t) et e M(x, y).

Since (t) = e

t

, t > , we have

ed

eM(x,y) = eM(x,y) k, x, y X, (.)

where k = e . Since , k (, ). Therefore the mapping T is a -type contraction. By Theorem ., T has a unique xed point x X, i.e., x is the unique solution of the nonlinear Hammerstein integral equation (.) and, for each x X, the sequence {xn =

Tnx} converges uniformly to x.

(Tx,Ty)

e

Liu et al. Fixed Point Theory and Applications (2016) 2016:8 Page 12 of 12

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Author details

1Institute of Mathematics, Yibin University, Yibin, Sichuan 644000, China. 2Center for General Education, China Medical University, Taichung, 40402, Taiwan.

Acknowledgements

The authors would like to express their thanks to the editor and the referees for their helpful comments and advices. This work was supported by Scientic Research Fund of SiChuan Provincial Education Department (No. 14ZA0272). This work was also supported by the National Natural Science Foundation of China (Grant No. 11361070) and the Natural Science Foundation of China Medical University, Taiwan.

Received: 29 May 2015 Accepted: 4 January 2016

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