Common fixed point and solution of nonlinear functional equations

R E S E A R C H

Open Access

Common fixed point and solution of

nonlinear functional equations

Abdul Rahim Khan

*_{Correspondence:}

[email protected] Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

Abstract

Convergence of a new iterative scheme, containing Mann and Ishikawa iterative schemes, for asymptotically nonexpansive mappings on a 2-uniformly convex hyperbolic space is studied. As application, we find a solution of a system of certain nonlinear functional equations in uniformly convex Banach spaces.

MSC: 47H09; 47H10

Keywords: uniformly convex hyperbolic space; asymptotically nonexpansive mapping; common fixed point; convergence

1 Introduction

The Banach contraction principle asserts that a contraction on a complete metric space has a unique fixed point and its proof hinges on ‘Picard iterations’. This principle is appli-cable to a variety of subjects such as integral equations, partial differential equations and engineering of image processing.

Many important nonlinear problems of mathematics reduce to nonlinear functional equations such as nonlinear integral equations and boundary value problems for non-linear ordinary or partial differential equations which can be translated in terms of a fixed point equationTx=xfor a given nonlinear mappingT on a Banach space or a metric space.

Browder and Petryshyn [] solved the equation

x–Tx=f (.)

for a given elementf ofX(Banach space) and described its relation with the properties of Picard iterates,i.e., the sequence{xn}where

xn+=Txn+f

for an initial valuex.

We know that Picard iterates of nonexpansive mappings fail to converge even on a Ba-nach space. Therefore, Mann [] iterates were introduced to approximate fixed points of nonexpansive mappings. Mann iterates were not adequate for the approximation of fixed points of pseudocontractive mappings and this led to the introduction of Ishikawa iterates [].

Let C be a nonempty subset of a metric space (X,d). A mapping T ofC into itself is (i) asymptotically nonexpansive if there is a sequence {kn} ⊂[,∞) withkn→ as n→ ∞andd(Tn_x,Tn_y)≤k

nd(x,y) for all x,y∈C (whenkn=  for eachn≥, it be-comes nonexpansive); (ii) semi-continuous if for any bounded sequence{xn} inC sat-isfyingd(xn,Txn)→, there exists a subsequence{xni} of{xn} such thatxni →x∈C;

(iii) completely continuous if every bounded sequence{xn}inCimplies that{Txn}has a convergent subsequence.

Nonexpansive mappings (the class of nonlinear mappings containing contractions as a subclass) remain a popular area of research in various fields. The iterative construction of fixed points of these mappings is a fascinating field of research. The fixed point problem for some nonlinear mappings has been studied on linear as well as nonlinear domains [–].

Numerous papers have appeared on the iterative construction of fixed points of asymp-totically nonexpansive and asympasymp-totically quasi-nonexpansive mappings in uniformly convex Banach spaces [, , , –].

The Ishikawa iterative scheme for two asymptotically nonexpansive mappingsSandT is defined as

x=x∈C,

xn+=αnTnyn+ ( –αn)xn, (.)

yn=βnSnxn+ ( –βn)xn, n≥,

whereαn,βn∈I= [, ].

ForS=Tin (.), we have an Ishikawa type iterative scheme for one mapping []

x=x∈C,

xn+=αnTnyn+ ( –αn)xn, (.)

yn=βnTnxn+ ( –βn)xn, n≥.

Whenβn=  in (.), we have the Mann [] type iterative scheme

x=x∈C,

xn+=αnTnxn+ ( –αn)xn, n≥.

(.)

Rhoades [] established Mann and Ishikawa type convergence results as two separate results as follows.

Theorem ([], Theorem ) Let C be a nonempty bounded,closed and convex subset of a uniformly convex Banach space.Let T be a completely continuous asymptotically non-expansive mapping on C with kn≥satisfying∞n=(kn– ) <∞.Define{αn}to satisfy

ε≤αn≤ –εfor all n≥andε> .Then the Mann type iterative scheme{xn}in(.) converges to a fixed point of T.

Theorem ([], Theorem ) Let C be a nonempty bounded,closed and convex subset of a uniformly convex Banach space.Let T be a completely continuous asymptotically nonex-pansive mapping on C with kn≥satisfying

∞

ε≤ –αn,  –βn≤ –εfor all n≥andε> .Then the Ishikawa type iterative scheme

{xn}in(.)converges to a fixed point of T.

An extension of a linear version (usually in Banach spaces) of a known result to metric fixed point theory has its own importance. As Mann and Ishikawa iterative schemes in-volve general convex combinations, we need some convex structure in a metric space to investigate their convergence on a nonlinear domain.

Let (X,d) be a metric space. Suppose that there exists a familyof metric segments such that any two pointsx,yinXare endpoints of a unique metric segment [x,y]∈([x,y] is an isometric image of the real line interval [,d(x,y)]). We shall denote byαx⊕( –α)y the unique pointzof [x,y] which satisfies

d(x,z) = ( –α)d(x,y) and d(z,y) =αd(x,y) forα∈I.

Such metric spaces are usually called convex metric spaces []. One can easily deduce x⊕y=y, x⊕y=xandαx⊕( –α)x=xfrom the definition of a convex metric space [–].

A convex metric spaceXis hyperbolic if

dαx⊕( –α)y,αz⊕( –α)w≤αd(x,z) + ( –α)d(y,w)

for allx,y,z,w∈Xandα∈I(see also []).

Forz=w, the hyperbolic inequality reduces to convex structure []

dαx⊕( –α)y,z≤αd(x,z) + ( –α)d(y,z). (.)

A nonempty subsetCof a convex metric spaceXis convex ifαx⊕( –α)y∈Cfor all x,y∈Candα∈I.

Normed spaces and their subsets are linear hyperbolic spaces while Hadamard mani-folds [], the Hilbert open unit ball equipped with the hyperbolic metric [] and the

CAT() spaces qualify for the criteria of nonlinear hyperbolic spaces [, , , ]. A convex metric spaceXis uniformly convex if

δ(r,ε) =inf

 – rd

a, x⊕

 y

:d(a,x)≤r,d(a,y)≤r,d(x,y)≥rε

> 

for anya∈X,r>  andε> .

From now onwards we assume thatXis a uniformly convex hyperbolic space with the property that for everys≥,ε> , there existsη(s,ε) >  depending onsandεsuch that

δ(r,ε) >η(s,ε) >  for anyr>s.

Xu [] extensively used the concept ofp-uniform convexity (see also [, p.]); its nonlinear version forp=  was introduced by Khamsi and Khan [] as follows:

For a fixeda∈X,r> ,ε> , define

(r,ε) =inf

 d(a,x)

₊

d(a,y)

_–d

a, x⊕

 y



where the infimum is taken over all x,y∈ X such that d(a,x)≤r, d(a,y)≤ r and d(x,y)≥rε.

We say thatXis -uniformly convex if

cM=inf

(r,ε)

r_ε :r> ,ε> 

> .

It was shown in [] that anyCAT() space is -uniformly convex withcM=_.

Using the concept of a unique pointαx⊕( –α)yin a metric segment [x,y], we express (.)-(.) in a convex hyperbolic space as follows:

Ishikawa iterative scheme for two mappings

x=x∈C,

xn+=αnTnyn⊕( –αn)xn, (.)

yn=βnSnxn⊕( –βn)xn, n≥,

where ≤αn,βn≤.

Ishikawa iterative scheme for one mapping

x=x∈C,

xn+=αnTnyn⊕( –αn)xn, (.)

yn=βnTnxn⊕( –βn)xn, n≥.

Mann iterative scheme

x=x∈C,

xn+=αnTnxn⊕( –αn)xn, n≥.

(.)

In the sequel, the following results are needed.

Lemma [] Suppose that X is a-uniformly convex hyperbolic space.Then,for anyα∈

(, ),we have that

du,αx⊕( –α)y≤αd(u,x)+ ( –α)d(u,y)– cMmin α, ( –α)

d(x,y)

for any u,x,y∈X.

Lemma [] Let{rn},{sn}and{tn}be nonnegative real sequences and satisfy

rn+≤( +sn)rn+tn for all n≥.

If∞_n=sn<∞and

∞

n=tn<∞,thenlimn→∞rnexists.

2 Convergence in2-uniformly convex hyperbolic spaces We setF(T) ={x∈X:Tx=x}andF=F(S)∩F(T)=∅.

Lemma  Let C be a nonempty convex subset of a hyperbolic space X,and let S,T:C→C be asymptotically nonexpansive mappings with sequence{kn} ⊂[,∞)such that

∞

n=(kn– ) <∞.Then,for the sequence{xn}in(.),limn→∞d(xn,p)exists for all p∈F.

Proof Letp∈F. By (.) and (.), we have

d(xn+,p) =d

αnTnyn⊕( –αn)xn,p

≤αnd

Tnyn,p

+ ( –αn)d(xn,p)

≤αnknd(yn,p) + ( –αn)d(xn,p)

=αnknd

βnSnxn⊕( –βn)xn,p

+ ( –αn)d(xn,p)

≤αnkn

βnd

Snxn,p

+ ( –βn)d(xn,p)

+ ( –αn)d(xn,p)

≤αnβnknd(xn,p) +αn( –βn)knd(xn,p) + ( –αn)d(xn,p)

≤αnβnknd(xn,p) +αn( –βn)knd(xn,p) + ( –αn)knd(xn,p)

=k_nd(xn,p).

That is,

d(xn+,p)≤knd(xn,p). (.)

Since{kn}is bounded, therefore

d(xn+,p)≤

 +M(kn– )

d(xn,p),

whereM=sup_n≥(kn+ ). AsM

∞

n=(kn– ) <∞, so by Lemma ,limn→∞d(xn,p) exists.

Lemma  Let C be a nonempty convex subset of a hyperbolic space X,and let S,T:C→C be asymptotically nonexpansive mappings with sequence{kn} ⊂[,∞)such that

∞

n=(kn– ) <∞.Then,for the sequence{xn}in(.),we have that

d(xn,p)≤sd(xn,p)

for all n>n≥,p∈F and some s> .

Proof With the help of inequalityx≤ex–_{forx≥ and (.), we have}

d(xn,p)≤kn–d(xn–,p)

≤e(kn––)d(xn–,p) ≤ · · ·

≤e

n–

j=n(kj–)_d(xn ,p)

≤e

_∞ j=(kj–)_d(x

n,p)

=sd(xn,p) wheres=e

∞_j=(kj–)_.

Theorem  Let C be a nonempty closed and convex subset of a complete hyperbolic space X,and let S,T :C→C be asymptotically nonexpansive mappings with sequence{kn} ⊂ [,∞)such that∞_n=(kn– ) <∞.Then{xn}in(.)converges to a point in F if and only iflim infn→∞d(xn,F) = ,where d(x,F) =inf{d(x,p) :p∈F}.

Proof We only prove sufficiency. Suppose thatlim inf_n→∞d(xn,F) = . It has been shown in the proof of Lemma  thatd(xn+,p)≤knd(xn,p). By the properties ofinf, we have that d(xn+,F)≤knd(xn,F) and hence, by Lemma ,limn→∞d(xn,F) exists. Therefore the hy-pothesislim inf_n→∞d(xn,F) =  gives thatlimn→∞d(xn,F) = . Next we show that{xn} is a Cauchy sequence. Letε> . Sincelimn→∞d(xn,F) = , there existsn≥ such that

d(xn,F) <

ε

s. Hence there must existq∈Fsuch thatd(xn,q) <

ε

s.

Now, for anyn>m≥n, we have from the estimate in the proof of Lemma 

d(xn+m,xn)≤d(xn+m,p) +d(xn,p)

≤sd(xn,p) <ε.

This proves that{xn}is a Cauchy sequence. SinceXis complete andCis its closed subset, thereforelimn→∞xn=q∈C. Nowlimn→∞d(xn,F) =  gives thatd(q,F) = . AsFis closed,

soq∈F.

Lemma  Let C be a nonempty convex subset of a-uniformly convex hyperbolic space X, and let S,T:C→C be asymptotically nonexpansive mappings with sequence{kn} ⊂[,∞) such that∞_n=(kn– ) <∞.Define{αn}and{βn}to satisfy <ε≤αn,βn≤ –εfor all n≥.Then,for the sequence{xn}in(.),limn→∞d(Sxn,xn) =  =limn→∞d(Txn,xn).

Proof Letp∈F. Then, by Lemma , we have

d(xn+,p) =d

αnTnyn⊕( –αn)xn,p



≤αnd

Tnyn,p



+ ( –αn)d(xn,p)

– cMmin αn, ( –αn)

dTnyn,xn



≤αnd

Tnyn,p



+ ( –αn)d(xn,p)

– cMαn( –αn)d

Tnyn,xn



≤αnknd(yn,p)+ ( –αn)d(xn,p)

– cMεd

Tnyn,xn



=αnknd

βnSnxn⊕( –βn)xn,p



+ ( –αn)d(xn,p)

– cMα( –α)d

Tnyn,xn

≤αnknβnd

Snxn,p



+αnkn( –βn)d(xn,p)

– cMαnknmin βn, ( –βn)

dTnyn,xn



+ ( –αn)d(xn,p)– cMεd

Tnyn,xn

≤αnβnknd(xn,p)+αn( –βn)knd(xn,p)

+ ( –αn)d(xn,p)

– cMknεd

Tnxn,xn

– cMεd

Tnyn,xn

≤αnβnknd(xn,p)+αn( –βn)knd(xn,p)

+ ( –αn)knd(xn,p)

– cMknεd

Tnxn,xn

– cMεd

Tnyn,xn

=k_nd(xn,p)– cMknεd

Snxn,xn

– cMεd

Tnyn,xn

=d(xn,p)+

k_n– d(xn,p)– cMknεd

Snxn,xn

– cMεd

Tnyn,xn

.

Sincelimn→∞d(xn,p) exists, therefore we have

d(xn+,p)≤d(xn,p)– cMεd

Tnxn,xn

– cMεd

Tnyn,xn

+k_n– M

for someM> .

This inequality implies the following two important inequalities:

cMεd

Tnyn,xn

≤d(xn,p)–d(xn+,p)+

k_n– M (.)

and

cMεd

Snxn,xn

≤d(xn,p)–d(xn+,p)+

k_n– M. (.)

Letmbe any positive integer. Summing up the terms from  tomon both sides in in-equality (.), we have

cMε m

n=

dTnyn,xn

≤d(x,p)–d(xm+,p)+

m

n=

k_n– M

≤d(x,p)+

m

n=

k_n– M.

Whenm→ ∞in the above inequality, we get that

cMε ∞

n=

dTnyn,xn

<∞,

and hence

lim

n→∞d

Tnyn,xn

Adapting a similar procedure for inequality (.), we get that

lim

n→∞d

Snxn,xn

= . (.)

Therefore the inequality

dTnxn,xn

≤dTnxn,Tnyn

+dTnyn,xn

≤knd(xn,yn) +d

Tnyn,xn

=knd

xn,βnSnxn⊕( –βn)xn

+dTnyn,xn

=knβnd

xn,Snxn

+dTnyn,xn

≤knd

xn,Snxn

+dTnyn,xn

together with (.) and (.) gives that

lim

n→∞d

Tnxn,xn

= .

Next we prove that

lim

n→∞d(Sxn,xn) =  =nlim→∞d(Txn,xn).

Note that

d(xn+,xn) =d

αnTnyn⊕( –αn)xn,xn

=αnd

Tnyn,xn

≤( –ε)dTnyn,xn

→ asn→ ∞.

Finally,

d(xn+,Sxn+)≤d

xn+,Sn+xn+

+dSxn+,Sn+xn+

≤dxn+,Sn+xn+

+kd

xn+,Snxn+

≤k

d(xn+,xn) +d

xn,Snxn

+dSnxn,Snxn+

+dxn+,Sn+xn+

≤dxn+,Sn+xn+

+kd

xn,Snxn

+k( +kn)d(xn+,xn)

gives that

lim

n→∞d(Sxn,xn) = .

Similarly,

lim

n→∞d(Txn,xn) = .

That is,

lim

The following concept is needed to proceed further.

Letf be nondecreasing on [,∞) withf() =  andf(t) >  for allt∈(,∞). Then the mappingsS,T:C→CwithF=∅satisfy Condition (A) if

d(x,Tx)≥fd(x,F) or d(x,Sx)≥fd(x,F) forx∈C.

Using Condition (A) and Theorem , we prove a convergence theorem in complete -uniformly convex spaces as follows.

Theorem  Let C be a nonempty convex subset of a complete-uniformly convex hyper-bolic space X.Let S,T:C→C be asymptotically nonexpansive mappings with sequence

{kn} ⊂[,∞)such that

_∞

n=(kn– ) <∞and satisfy Condition(A).Define{αn}and{βn}to satisfy <ε≤αn,βn≤ –εfor n≥.Then the sequence{xn}in(.)converges to a point in F.

Proof By Lemma ,limn→∞d(Sxn,xn) =  =limn→∞d(Txn,xn). Using Condition (A), we get thatlimn→∞d(xn,F) = . Now Theorem  gives that{xn}converges to a point inF.

Another convergence theorem is established in the following result under any of Con-ditions (ii)-(iii) without requiring the completeness of the spaceX.

Theorem  Let C be a nonempty convex subset of a-uniformly convex hyperbolic space X. Let S,T :C→C be asymptotically nonexpansive mappings with sequence{kn} ⊂[,∞) such that∞_n=(kn– ) <∞and either S or T is semi-compact.Define{αn}and{βn}to satisfy <ε≤αn,βn≤ –εfor all n≥.Then the sequence{xn}in(.)converges to a point in F.

Proof Lemma  gives thatlimn→∞d(Sxn,xn) =  =limn→∞d(Txn,xn). Suppose thatT is semi-compact. Sincelim_n→∞d(xn,p) exists, therefore{xn}is bounded. Aslimn→∞d(Txn, xn) =  and T is semi-compact, so there is a subsequence {xni} of {xn} such that xni→q∈C, and henceTxni→Tq andSxni→Sq. Therefore limi→∞d(Sxni,xni) =  = lim_i→∞d(Txni,xni) implies thatd(Sq,q) =  =d(Tq,q). That is,q∈F. Aslimn→∞d(xn,p)

exists andxni→q, thereforexn→q.

Let{Ti:i= , , . . . ,k}be a family of mappings onC. The multi-step iteration scheme of Khanet al.[] may be adapted in a convex hyperbolic space as follows:

xn+= ( –αkn)xn⊕αknTkny(k–)n,

y(k–)n= ( –α(k–)n)xn⊕α(k–)nTkn–y(k–)n,

y(k–)n= ( –α(k–)n)xn⊕α(k–)nTkn–y(k–)n,

· · ·

yn= ( –αn)xn⊕αnTnyn,

yn= ( –αn)xn⊕αnTnyn,

(.)

Following the line of action of the proofs of Theorem  and Lemma , we can easily prove the following results.

Theorem  Let C be a nonempty closed and convex subset of a complete-uniformly convex hyperbolic space X,and let{Ti:i= , , . . . ,k}be a family of asymptotically quasi-nonexpansive self-mappings of C,i.e.,d(Tn

ix,pi)≤uind(x,pi)for all x∈C and pi∈F(Ti), i= , , . . . ,k.Suppose that F=

k

i=F(Ti)=∅,x∈C and

∞

n=(uin– ) <∞for all i.Then the iterative sequence{xn},defined by(.),converges to a common fixed point of the family of mappings if and only iflim infn→∞d(xn,F) = .

Theorem  Let C be a nonempty closed and convex subset of a-uniformly convex hy-perbolic space X,and let{Ti:i= , , , . . . ,k}be a family of asymptotically nonexpansive mappings of C,i.e.,d(T_inx,pi)≤uind(x,pi)for all x∈C and pi∈F(Ti),where{uin}are se-quences in[,∞)with∞_n=(uin– ) <∞for each i∈ {, , , . . . ,k}.Assume that F=∅and

the sequence{xn}is in(.)withαin∈[δ,  –δ]for someδ∈(,_).If for some i, ≤i≤k, Tiis semi-compact,then{xn}converges to a point in F.

Remark  () Theorem  extends (unifies) Theorem  of Khan and Takahashi [] (The-orems -) in the setting of -uniformly convex hyperbolic spaces.

() Theorem  establishes Theorem  by Qihou [] together with its Corollaries  and , which are themselves extensions of the results of Ghosh and Debnath [] and Petryshyn and Williamson [], for two asymptotically nonexpansive mappings on a -uniformly convex hyperbolic space.

() All the results of this paper, in particular, hold inCAT() spaces.

Remark  In a uniformly convex Banach spaceB, iterative scheme (.) for nonexpansive mappings becomes the scheme (∗) of Kuhfittig ([], p.) which he applied to solve the system of equations of the type

x–Six=fi fori= , , , . . . ,m,

where eachSiis a nonexpansive self-mapping onXand eachfiis a given element ofX. Following Kuhfittig [], we can apply our iteration scheme (.) to find a solution of the system of equations of the type

x–Sn_ix=fi fori= , , , . . . ,m (.)

for a family{Si}of asymptotically nonexpansive mappings onB.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author is grateful to King Fahd University of Petroleum & Minerals for supporting the research project IN 121037.

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