Measure of noncompactness and application to stochastic differential equations

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R E S E A R C H

Open Access

Measure of noncompactness and

application to stochastic differential

equations

Abdelkader Dehici

1,2

_{and Nadjeh Redjel}

1,2*

*_{Correspondence:}

[email protected]

1_{Laboratory of Informatics and}

Mathematics, University of Souk-Ahras, P.O. Box 1553, Souk-Ahras, 41000, Algeria

2_{Department of Mathematics,}

University of Constantine 1, Constantine, 25000, Algeria

Abstract

In this paper, we study the existence and uniqueness of the solution of stochastic diﬀerential equation by means of the properties of the associated condensing nonexpansive random operator. Moreover, by taking account of the results of Diaz and Metcalf, we prove the convergence of Kirk’s process to this solution for small times.

MSC: 47H10; 47H08; 60H10

Keywords: Wiener process; Itô integral; Banach space; ﬁxed point; existence; uniqueness; measure of noncompactness; condensing operators; Kirk’s process

1 Introduction and notations

It has been found over the years that the fixed point theory is a powerful tool for the resolu-tion of nonlinear problems (differential equaresolu-tions, integro-differential equaresolu-tions, . . .). The roots of this theory go back to the famous works of Brouwer () and Banach (), the latter author gave an abstract formulation of the successive approximations method, sys-tematically used by Liouville () in his results. We note that Banach’s work was estab-lished in the case of normed spaces and extended in metric spaces by Caccioppoli (). Since then this theory has become a burgeoning field for several authors who have con-tributed in the elaboration by thousands of papers of the subject. The development of this theory has been heavily linked to that of the functional analysis in the s. The Italian mathematician Darbo has published a result which ensures the existence of fixed point for a type of operators so called condensing operators generalizing the Schauder fixed point and Banach contraction principle. This discovery was the subject of several applications both in linear and nonlinear analysis (integral equations with singular kernels, differen-tial equations defined on unbounded domains, neutral differendifferen-tial equations, differendifferen-tial operators having non-empty essential spectra, boundary value problems in Banach spaces and others). A condensing (or densifying) mapping is a mapping for which the image of any set is in a certain sense more compact than the set itself, the degree of noncompactness of a set is measured by means of functions called measures of noncompactness. Among the application areas of these tools, the theory of probabilistic operators, which is a branch of stochastic analysis which deals with random operators and their properties, is seen as an

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extension of operators theory (determinist case). This axis of research has emerged in the s, thanks to the works of East European school of probabilities whose main purpose was the resolution of stochastic differential equations and stochastic partial differential equations, modeling the trajectories of random phenomena, studied and developed for the first time by Itô in . A stochastic differential equation is an ordinary differential equation perturbed by a white noise (involving the Brownian motion). The history of this direction goes back to the works of the English botanist Brown who described in  this motion as that of an organic fine particle in suspension in a gas or a fluid. In the late  century, scientists (Bachelier, Smoluchowski) addressed the study of this type of motions. Afterwards, and more precisely in , Einstein published a paper in which he showed that the probability density of the Brownian motion satisfies the heat equation. The first rigorous mathematical treatment is due to Wiener in the s of the previous century who has proven the existence of the Brownian motion. For more details of these equa-tions, we can quote for example [–].

In this work, we study the existence and uniqueness of the solution of the following Itô stochastic diﬀerential equation

X(t) =X+

t



aθ,Xf(θ)dθ+

t



aθ,Xf(θ)dw(θ), (.)

whereaandaare Borel measurable functions and ≤f(θ)≤θ.

Recall that this equation models for example the motion of a particle subjected to the infinity of shocks at the timet. Herea is a coefficient of transfer whileais a diffusion

coeﬃcient. In the case wherea anda satisfy the Lipschitz condition with respect to

the second variable, the result forf(θ)≡θwas established by Gikhman and Skorohod [] showing that the associated mapping to (.) is a contraction and the solution was obtained by means of the successive approximations method.

Our goal here is to investigate the problem (.) by imposing general conditions on the functionsa anda, therefore, we show that the associated mappingT is nonexpansive and condensing mapping having a unique ﬁxed point which is the solution of (.). On the other hand, we prove that if this solution satisﬁes the metric property of Diaz and Metcalf, the convergence of Kirk’s process to this solution is ensured.

Deﬁnition . A probability space (,F,P) is a triplet for whichis a non-empty set,F is aσ-algebra ofandPis a probability measure deﬁned onF(P() = ).

Notice that some results concerning the existence of ﬁxed point theorems involving probabilistic metric spaces can be found for example in [–].

A real random variableXis aP-measurable function deﬁned onwith values in R. A family of random variablesXt(ω) (t≥) (denoted also byX(t,ω) or simplyXt) is called a stochastic process. Forω∈, the functiont−→X(t,ω) is the path of the stochastic processX(t,ω).

The mean value or expectationE(X) of the random variableXis deﬁned as the integral

E(X) =

X(ω)dP(ω),

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Two random variablesXandYare said to be independent if, for anya,b∈R,

Pω∈|X(ω) <aandY(ω) <b=Pω∈|X(ω) <a×Pω∈|Y(ω) <b.

Deﬁnition . A Wiener process (also called Brownian motion){wt}t≥is a stochastic process with the following properties:

(a) w= ;

(b) for <t<· · ·<tnthe random variableswt–wt,wt–wt, . . . ,wtn–wtn–are

independent;

(c) the random variableswt+s–wthave a normal distribution with zero expectation and as a variance.

Remark . Ifwis a random variable with zero expectation and varianceσ=E(w_),

thenE|w|=



πσ. Thus, we obtain

E|wtj+–wtj|=

 π

tj+–tj

and hence the series_jE|wtn

j+–wtjn|diverges witht n

j+–tjn−→, where  <tn <tn<· · ·< tn

n=T.

For a pair (w(t),X(t)) of a Wiener processw(t) and random processX(t), we deﬁne the Itô integral as follows:

I(X) =

T



X(t)dw(t).

The Itô integral is not a classical integral, this is due to the nonsmoothness of the paths

w(t) and the divergence of the series_jE|wtn_j_+–wt_jn|.

With a Wiener process, we can associate a ﬁltrationFt (Ft⊂F), ≤t≤T, which is a family ofσ-algebra generated by the Brownian paths up to timet, in other words,

Ft=σ

X(s) : ≤s≤t≤T.

It is easy to show that the familyFtis nondecreasing (with respect to the inclusion).

Deﬁnition . A random variableY is said to beFt-measurable if knowledge ofY de-pends only on the information known up to timet.

Deﬁnition . A sequence of real random variablesXnonconverges to the random variableXin probability, written

Xn p

−→X,

if for every> 

lim

n−→+∞P

ω∈|Xn(ω) –X(ω)<

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LetM([,T]) denote the set of all functionsX(t,ω) deﬁned and jointly measurable in t∈[,T] andω∈which are also measurable with respect toFt for allt∈[,T] and such that

P

ω∈

T



_X(t,ω)dt<∞

= .

In the sequel, without loss of generality,X(t,·) will be denoted byX(t). In the case whereX(t) =X(tk) fort∈[tk,tk+) ( =t<t<t<· · ·<tn=T).

T

 X(t)dw(t)

is given by the formula

T



X(t)dw(t) = n–

k=

X(tk)(wtk+–wtk).

In the general case whereX(t) is an arbitrary element ofM([,T]), then there exists a

sequence of step functionsXn(t) such that

lim

n−→+∞

T



_X_n(t) –X(t)dt=  (in probability)

and the sequence_TXn(t)dw(t) converges in probability to some limitξ, which is called the Itô stochastic integral ofX(t) denoted by_TX(t)dw(t).

Some properties of the Itô stochastic integral(see[]):

(i) Itô integral is linear; (ii) if_TE(|f(t)|₎_dt_<_∞_{, then}

E

T



f(t)dw(t)

= , (.)

and

E

sup ≤s≤μ

_sf(t)dw(t)

 ≤

μ



Ef(t)dt (≤μ≤T). (.)

In the sequel, we assume that in (.), the initial data which is the random variableXis F-measurable.

Deﬁnition . The processX(t) is called a strong solution of (.) if the following three conditions are satisﬁed:

(i) X(t)isFt-measurable;

(ii) the integrals in (.) exist;

(iii) P( ) = where is the set ofω∈such that (.) holds for allt∈[,T].

2 Main results

We denote byXTthe vector space of measurable random functionsξ(t,ω) with respect to theσ-algebraFtfor anyt∈[,T] such thatP({ω∈|t−→ξ(t,ω) is continuous}) = .

We put ξ XT=

E(sup_≤_s_≤_T|ξ(s,ω)|)_{. It is easy to show that}_·

XTdeﬁnes a norm onXT.

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Proof It suﬃces to prove thatXTis complete with respect to the norm · XT. Letξnbe a Cauchy sequence inXT, then we can extract a subsequenceξnkwhich converges almost surely for allt∈[,T]. From the set of indices{nk}(k≥), we choose a subset of integers

By multiplying by k_{, we obtain}

E(sup_≤_s_≤_tξn(s) –ξn(s)

Using Chebyshev’s inequality, it follows that

P

Since the series+∞_k_=–k_{converges, the Borel-Cantelli lemma gives}

P

It follows that the partial sums

ξm(t) +

converge uniformly in [,T] and letξ(t) its limit (for this topology). This gives

sup

Sinceξnis a Cauchy sequence inXT and contains a subsequenceξmk which converges

toξ,ξnconverges toξinXT, and by this we achieve the proof.

Hereafter, the principal goal is to transform equation (.) to a ﬁxed point problem. To this aim, we associate it to the following mappingCgiven by

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It is easy to observe thatCis deﬁned onXT, but so that it takes its values inXT, we need to add the following additional conditions on the functionsaanda, which is the goal of

the following proposition.

Proposition . Assume that the following assumptions (called growth polynomial as-sumptions)are satisﬁed.

a(s,u)≤M|u|+  and a(s,u)≤M|u|+  (M> ). (.)

Using the Cauchy-Schwarz inequality, it follows that

(CX)(s)≤s

By passing to the sup and using the monotonicity of the expectation, we obtain

Esup

Using (.) and the commutativity between the expectation and the integral, it follows that

Esup

By the assumptions given above, we get

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The fact that ≤f(θ)≤θ(θ∈[,T]) and the last inequality give

(CX)_X η≤M

η+ η X _X_η+ . (.)

Thus (CX) Xηis ﬁnite if X Xηis ﬁnite, which completes the proof.

If X(t) is a solution of the equation (.) on the interval [,s], ≤s≤T, we denote ϕ(s) = X _Xs.

Lemma . The functionϕis bounded on the interval[,T].

Proof From the inequality (.), by changingηbys, we infer that

Esup

≤r≤s

₍_CX₎₍_r₎ ≤M

s

 X 

Xf(θ)dθ

+s+ 

s

 X 

Xf(θ)dθ+ s

≤M

T

s

 X 

Xf(θ)dθ

+T+ 

s

 X 

Xf(θ)dθ+ T

.

Using the fact thatX(t) is a solution of the equation (.) on the interval [,s] and ≤

f(s)≤s, it follows that

ϕ(s)≤K+K s



ϕ(r)dr,

whereK= M(T_{+ }_T_{) and}_K_{= }_M₍_T_{+ ).}

Now, Gronwall’s lemma implies that

ϕ(s)≤KeKT,

which gives the result.

We introduce here the concept of measure of noncompactness of Hausdorﬀ which is a real positive function measuring the degree of noncompactness of sets.

LetXbe a complex Banach space and letP(X) be the set of all subsets ofX, we denote byB(x,r) andB(x,r), respectively, the open and closed ball of centerxand radiusr> .

Definition . The Hausdorff measure of noncompactnessα(A) ofA∈P(X) is defined as the infimum of the numbers>  such thatAhas a finite-net inX. Recall that a set

S⊆Xis called an-net ofAifA⊆S+B(, ) ={s+b:s∈S,b∈B(, )}.

The Hausdorﬀ measure of noncompactnessαenjoys the following properties:

(a) regularity:α(A) = if and only ifAis totally bounded;

(b) nonsingularity:αis equal to zero on every one-element set;

(c) monotonicity:A⊆Aimpliesα(A)≤α(A);

(d) semi-additivity:α(A∪A) =max{α(A),α(A)};

(e) Lipschitzianity:|α(A) –α(A)| ≤ρ(A,A); hereρdenotes the Hausdorﬀ

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(f ) continuity: for anyA∈P(X)and any, there existsδ> such that |α(A) –α(A)|<for allAsatisfyingρ(A,A) <δ;

(g) semi-homogeneity:α(tA) =|t|α(A)for any numbert;

(h) algebraic semi-additivity:α(A+A)≤α(A) +α(A);

(i) invariance under translations:α(A+x) =α(A)for anyx∈X.

The goal of the following theorem is to address other properties in view of their impor-tance.

Theorem .([], Theorem ..) The Hausdorﬀ measure of noncompactness is invariant under passage to the closure and to the convex hull:α(A) =α(A) =α(coA).

We note that the measure of noncompactness has many applications in mathematics. On this topic, we refer to [, –].

Deﬁnition . LetXbe a Banach space. A functionψ deﬁned onP(X) with values in some partially ordered set (,≤) is called a measure of noncompactness in the general sense ifψ(A) =ψ(coA) for allA∈P(X).

Deﬁnition . Let (X, · ) be a normed space andϑone of measure of noncompactness given above. A continuous mappingG:X−→Xis said to be densifying or condensing, if, for every bounded subset ofXsuch thatϑ(A) > , we haveϑ(G(A)) <ϑ(A).

LetM([,T]) be the vector space of scalar functions deﬁned on [,T]; it is partially ordered by the usual order≤. Letγ:P(XT)−→M([,T]) deﬁned by

γ :P(XT)−→M([,T]); −→γ().

Here

γ() : [,T]−→R;

t−→γt(t),

wheret={Xt=X|[,t]:X∈} ⊂Xtandγt is the measure of noncompactness of Haus-dorﬀ on the spaceXt.

Lemma . The functionγ deﬁnes a measure of noncompactnesss in the general sense on XTwhich is additively nonsingular(i.e.,γ(A∪ {X}) =γ(A)for all A⊂XTand X∈XT).

Proof LetA⊂XT, we haveA⊂co(A), this givesAt⊂(co(A))tfor allt∈[,T], the property of the monotonicity of the Hausdorﬀ measure of noncompactness implies thatγt(At)≤ γt(co(A))tfor allt∈[,T], this givesγ(A)≤γ(co(A)). On the other hand

co(A)_t=X|[,t]:X∈co(A)

⊂co(At).

Again, the monotonicity and the invariance by closure of the Hausdorﬀ measure of non-compactness leads to

γt

co(A)_t≤γt

co(At)

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Hence

γco(A)(t)≤γ(A)(t), ≤t≤T.

It follows thatγ(co(A))≤γ(A), by which we achieve the proof for the ﬁrst assertion. The

fact thatγ is additively nonsingular is trivial.

Deﬁnition . Let (X, · ) be a normed space andKa nonempty bounded subset ofX. A selfmappingTonKis called a nonexpansive mapping if T(x) –T(y) ≤ x–y for all

x,y∈K.

Now, let us introduce the following conditions:

(H) |a(s,u) –a(s,u)|≤h(s)g( u–u ),

whereh: [,T]−→[, +∞[ integrable and_Th(s)ds≤  T+ and

g: [, +∞[−→[ , +∞[ is nondecreasing and concave function withg(s)≤s

for alls∈[, +∞[.

(H) For allA> , the inequality

h(s)≤A s



h(θ)ghf(θ)dθ, ≤s≤T

cannot admit nontrivial solutions.

(H) λ(f–(B))−→whenλ(B)−→; hereλis the Lebesgue measure andB⊂[,t](≤ t≤T).

Remark . We note that if we takeh(t) =α (α> ),g(u) =u,f(x) =x, andT = – +

 +__α, then the assumptions given in (H), (H), and (H) are satisﬁed.

Proposition . Under the assumption(H),the mapping C deﬁned by(.)is nonexpan-sive on every Xt(≤t≤T).

Proof We have

CX(s) –CY(s)=

s



aθ,Xf(θ)–aθ,Yf(θ)dθ

+

s



a

θ,Xf(θ)–a

θ,Yf(θ)dω(θ).

By using the inequality (x+x)_≤__x

+ x, we obtain

CX(s) –CY(s)≤

s



aθ,Xf(θ)–aθ,Yf(θ)dθ



+ 

s



a

θ,Xf(θ)–a

θ,Yf(θ)dω(θ)



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The Cauchy-Schwarz inequality enables us to write

Passing to the sup on [,T] and using the monotonicity of the expectation, it follows that

Esup

The stochastic inequality (.) gives

Esup

The commutativity between the expectation and the integral together with the fact thatg

is concave gives

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Remark . We note that nonexpansive selfmappings on bounded subsets of Banach spaces do not necessarily have ﬁxed points, we can refer for example to the famous work of Alspach [] who gave an example of a weakly compact convex subsetMof the space

L([, ]) and a ﬁxed point free isometry onM.

In the sequel, we will need the following two lemmas; the ﬁrst lemma is one of the clas-sical results in measure theory.

Lemma . Let> and letφ: [,T]−→Ra monotone function,then the set of discon-tinuity points ofφhaving a magnitude≥is a ﬁnite set in[,T].

Lemma . For any⊂XT,the function

γ() : [,T]−→[, +∞[;

t−→γt(t),

is a nondecreasing bounded function on[,T].

Proof Lett,t∈[,T] such thatt≤t; thent⊂t. The monotonicity of the

mea-sure of noncompactness of Hausdorﬀ givesγt(t)≤γt(t) and implies thatγ()(t)≤

γ()(t), which proves the ﬁrst assertion. The second assertion follows directly from the ﬁrst one, indeed by the monotonicity, we deduce that γ()(t)≤γ()(T) for all

t∈[,T].

Theorem . Under the assumptions(H), (H), (H)together with the growth polyno-mial conditions given in(.),C is a condensing mapping with respect to the measure of noncompactnessγ on Xtfor all t∈[,T].

Proof We show that if there exists A⊂Xt such that γ(A)≤γ(C(A)), then we obtain necessarily γ(A) = . Let t>  and > , by using Lemmas . and ., we denote by

{tj}mj= the set of points in [,t] for whichγ(A)(tj+ ) –γ(A)(tj– )≥. It is easy to de-duce that there existsδsuﬃciently small such thatinft,t∈]tj–δ,tj+δ[|γ(A)(t) –γ(A)(t)| ≥

for allj= , . . . ,m. Letting= [,t]\ m

j=]tj–δ,tj+δ[, we observe that= m+

k= Ik where each Ik is a closed bounded interval with Ik ∩Ij=∅ fork=j. On Ik, the func-tiont−→γ(A)(t) is uniformly continuous, this implies that the existence ofδ> 

suf-ﬁciently small such that for alls,s∈Ik:|s–s|<δ, then |γ(A)(s) –γ(A)(s)|<. On

the other hand, in Ik, we choose a ﬁnite set {bks}rks= for which δ <bks –bks–< _δ.

Now, for all ≤s≤rk (k= , . . . ,m+ ), let {Mi,Mi,Mi, . . . ,Mis} a (γ(A)(bks) +

)-net of the setAb_ks. Thus, we can construct a family of paths{Gl:l= , . . . ,h} such that P{ω∈|t−→Gl(t)(ω) is continuous (l= , . . . ,h)}=  as follows:Gl≡Mif on the inter-valsJks= [bks–+δ_,bks–δ_] for all ≤f ≤s(k= , . . . ,m+ ) (l= , . . . ,h) and linear on the

complementary intervals. On the other hand, sinceγ(A)≤γ(C(A)),γ(A)(θ)≤γ(C(A))(θ) for allθ∈[,t]. LetZ∈(C(A))θ={Y|[,θ]/Y∈C(A)}, this shows the existence ofV∈Asuch

thatY=C(V). Moreover, we haveV|[,b_ks]∈Ab_ks (≤s≤rk) (k= , . . . ,m+ ), it follows that there exists ≤f≤sfor which

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SinceGl≡MifonJks, it follows that forθ∈Jks, we have

By using the same techniques as Proposition ., we get

E sup

Sincegis concave, it follows that

(CV) – (CGl)_X

The fact thatgis nondecreasing and (.) lead to

I≤κ

τ



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The continuity of the integral shows that there existsη>  such that, for all measurable setB⊂[,τ] withλ(B) <η, we have

B

h(θ)gEVf(θ)–Gl

f(θ)dθ< κ.

Moreover, the condition (H) implies that it is always possible to chooseδgiven above

suﬃciently small such that

I<.

Consequently, we can write

(CV) – (CGl)



Xτ≤+κ

t



h(θ)gγ(A)f(θ)+ dθ. (.)

The deﬁnition of the measure of noncompactness of Hausdorﬀ together with (.) leads to

γ(A)(t)≤γC(A)(t)≤(CV) – (CGl)_X τ

≤+κ

t



h(θ)gγ(A)f(θ)+ dθ.

By using the assumption (H), we obtainγ(A)≡, which gives the result.

Theorem .([], p.) Let A:K−→K be a mapping deﬁned on a closed,bounded, con-vex subset K of a Banach space X.Assume that A is condensing with respect to the additively nonsingular measure of noncompactness in the general sense.Then A has at least one ﬁxed point in K.

Theorem . The mapping C:XT−→XTdeﬁned by(.)has a unique ﬁxed point in XT.

Proof The existence follows from Theorem ., more precisely, forτ belonging to the

interval [, – +

 +__M_HH_+], the inequality (.) shows that the random mappingC:

Xτ−→XτleavesBXT(,

√

H) (the ball of center  and radius√H) invariant, which implies

the existence of the solutionX(t) inXτ(τ∈[, – +

 +  M

H

H+]), the result inXTfollows

from an extension to the whole interval [,T].

For the uniqueness, we proceed as follows: Assume thatX=X(t)t∈[,T]andY=Y(t)t∈[,T]

are two strong solutions of equation (.) such thatX() =Y() = . In other words

X(s) =

s

 a

θ,Xf(θ)dθ+

s

 a

θ,Xf(θ)dw(θ) (.)

and

Y(s) =

s



aθ,Yf(θ)dθ+

s



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It follows that

Passing to the sup and expectation and using the assumption (H) together with the

stochastic inequality (.), we obtain

Esup

By the monotonicity of the expectation and the functiong, we infer that

Esup

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3 Application to the convergence of Kirk’s iterative process Let us recall the following theorem due to Diaz and Metcalf [].

Theorem . Let B be a continuous selfmapping on a metric space(X,d)such that

(i) Fix(B)=∅(Fix(B)is the set of ﬁxed point ofB);

(ii) for eachy∈Xsuch thaty∈/Fix(B),and for eachz∈Fix(B)we have

dB(y),z<d(y,z).

Then one,and only one,of the following properties holds:

(a) for eachx∈Xthe Picard sequence{Bn₍_x₎_}_{contains no convergent subsequences}_;

(b) for eachx∈Xthe sequence{Bn₍_x₎_}_{converges to a point belonging to}_F₍_B₎_.

Kirk iteration([]): Let (X, · ) be a normed space andKa closed, convex, and bounded subset ofX. LetAbe a selfmapping onK. For eachx∈K, the sequence{Sn₍_x₎_}_{deﬁned by}

S:K−→K, where

S=λI+λA+· · ·+λnAn, λi≥,λ> ,

n

i=

λi= ,

is said to be Kirk’s iterative process.

Theorem .[] Let K be a convex subset of a Banach space and A:K−→K be a non-expansive mapping.Then S(x) =x if and only if A(x) =x.

LetT> , andH=KeKTa real positive which is an upper bound of the functionϕ(t) =

X 

Xt (≤t≤T). We denote byBXt(,

√

H) the closed ball of center  and radius√H

inXt.

Theorem . If there exists τ∈[, – +

 +__M_HH_+] such that C :BXτ(,

√ H)−→

BXτ(,

√

H)satisﬁes Diaz-Metcalf ’s condition,then,for each X∈BXτ(,

√

H),Kirk’s pro-cess{Sn₍_X₎_}₍_{associated to the mapping C}₎_{converges to the unique ﬁxed point of T}_.

Proof Following Theorem ., it follows that the mappingChas a unique ﬁxed pointX_. Moreover,Sis a densifying mapping. Indeed, ifKis a bounded subset ofBXτ(,

√ H) such thatγ(K) > , then

S(K)⊆λK+λC(K) +· · ·+λnCn(K).

Hence, by the monotonicity, semi-additivity, and homogeneity properties, we get

αS(K)≤λα(K) +λα

C(K)+· · ·+λnα

Cn(K).

The fact thatCis densifying shows that

αC(K)<α(K), (.)

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γCn(K)≤γCn–(K)≤ · · · ≤γC(K)<γ(K) (.)

and therefore

γS(K)< (λ+· · ·+λn)γ(K) =γ(K). (.)

Also, by use of Theorem . together with Theorem .,Fix(S) =Fix(C) ={X_}_. ForX∈K, let

K=

+∞

n= Sn_{_X_}_.

We haveS(K) =+∞_n_=Sn_{_X_{} ⊂}_K_{and since}_K₌_{_X_{} ∪}_S₍_K_{), this gives}

γ(K) =maxγ{X},αS(K)=max,γS(K)=γS(K).

SinceSis densifying, we establish thatγ(K) = , by the property of regularity, we establish thatKis totally bounded, and consequentlyKis compact (sinceXτis a Banach space),

therefore the sequence{Sn₍_X₎_}_{contains a convergent subsequence.}

On the other hand, the fact thatCsatisﬁes the Diaz-Metcalf condition shows that, for allX∈BXτ(,

√

H)\{X_}_{, we have}

C(X) –X_<_X_–_X_.

This implies that

Ck(X) –X≤X–X. (.)

Thus,

S(X) –X=

n

k=

λkCk(X) –X

=

n

k=

λkCk(X) – n

k=

λkX

≤

n

k=

λkCk(X) –X

< n

k=

λkX–X=X–X.

Then Ssatisﬁes also the assumptions of Theorem ., this implies that limn−→+∞Sn(X)

exists and is equal toX_.

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point ofA (see for example [–]) with the additional condition that the spaceX is uniformly convex or strictly convex. Unfortunately, in our case the Banach spaceXTis not strictly convex or uniformly convex, indeed, it suﬃces to take its subspace of functions ζ(t) independent ofωequipped with its norm sup, to see that this does not have these properties.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors would like to thank the editor and the anonymous referees for their remarks and valuable comments, which helped to improve this paper.

Received: 3 August 2015 Accepted: 10 January 2016

References

1. Akhmerov, RR, Kamenskii, MI, Potapov, AS, Rodkina, AE, Sadovskii, BN: Measures of Noncompactness and Condensing Operators. Birkhäuser, Basel (1992)

2. Yamada, T, Watanabe, S: On the uniqueness of solutions of stochastic diﬀerential equations. I. J. Math. Kyoto Univ. 11(1), 155-167 (1971)

3. Yamada, T, Watanabe, S: On the uniqueness of solutions of stochastic diﬀerential equations. II. J. Math. Kyoto Univ. 11(3), 533-563 (1971)

4. Rodkina, A: Solubility of stochastic differential equations with perturbed argument. Ukr. Math. J.37(1), 98-103 (1985) 5. Veretennikov, AY: On strong solutions of stochastic differential equations. Teor. Veroâtn. Primen.24(2), 348-360 (1979) 6. Gikhman, II, Skorohod, AV: Introduction to the Theory of Random Processes. Nauka, Moscow (1965) (in Russian) 7. Pap, E, Hadzic, O, Mesiar, R: A fixed point theorem in probabilistic metric spaces and applications. J. Math. Anal. Appl.

202(2), 431-440 (1996)

8. Sehgal, VM, Bharucha-Reid, AT: Fixed points of contraction mappings on probabilistic metric spaces. Theory Comput. Syst.6(1), 97-102 (1972)

9. Sen, MD, Karapınar, E: Some results on best proximity points of cyclic contractions in probabilistic metric spaces. J. Funct. Spaces2015, Article ID 470574 (2015)

10. Aghajani, A, Pourhadi, E: Application of measure of noncompactness tol1solvability of inﬁnite systems of second

order diﬀerential equations. Bull. Belg. Math. Soc. Simon Stevin22(1), 105-118 (2015)

11. Ezzinbi, K, Taoudi, MA: Sadovskii-Krasnosel’skii type ﬁxed point theorems in Banach spaces with application to evolution equations. J. Appl. Math. Comput. (2014). doi:10.1007/s12190-014-0836-8

12. Losada, J, Nieto, JJ, Pourhadi, E: On the attractivity of solutions for a class of multi-term fractional functional diﬀerential equations. J. Comput. Appl. Math. (Available online 23 July 2015)

13. Mursaleen, M, Noman, A: Hausdorﬀ measure of noncompactness of certain matrix operators on the sequence of generalized means. J. Math. Inequal. Appl.417, 96-111 (2014)

14. Alspach, DE: A ﬁxed point free nonexpansive map. Proc. Am. Math. Soc.3, 423-424 (1981)

15. Diaz, JB, Metcalf, FT: On the set of subsequential limit points of successive approximations. Trans. Am. Math. Soc.135, 459-485 (1969)

16. Kirk, WA: On successive approximations for nonexpansive mappings in Banach spaces. Glasg. Math. J.12(1), 6-9 (1971)

17. Barbuti, U, Guerra, S: Un Teorema costruttivo di punto ﬁsso negli spazi di Banach. Rend. Ist. Mat. Univ. Trieste4, 115-122 (1972)