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Vol. 19 (2018), No. 1, pp. 365–383 DOI: 10.18514/MMN.2018.2268

INFINITE SYSTEMS OF INTEGRAL EQUATIONS IN FR ´ECHET SPACES USING THE NOTION OF L-FUNCTIONS

MOHSEN HOSSEINZADEH MOGHADDAM, REZA ALLAHYARI, MOHSEN ERFANIAN OMIDVAR, AND MAHMOUD HASSANI

Received 09 March, 2017

Abstract. The aim of this paper is to prove some fixed point theorems for L-functions with the help of measure of noncompactness and the Tychonoff fixed point theorem. Also, we prove some existence theorems for a general infinite system of integral equations. As an application, we study the problem of the existence of solutions for infinite systems of nonlinear integral equations of Hammerstein type in two variables. The results obtained extend several ones. Finally, an example is presented to guarantee our results.

2010Mathematics Subject Classification: 47H09; 47H10; 54H25

Keywords: measure of noncompactness, Frechet spaces, Tychonoff fixed point theorem, infiniteK systems of equations

1. INTRODUCTION AND PRELIMINARIES

There are three major branches of fixed point theory in functional analysis, and each branch has its celebrated theorems. In the topological branch of fixed point the- ory, the main three theorems are Brouwer’s fixed point theorem, its infinite dimen- sional version, Schauder’s fixed point theorem, and Tychonoff fixed point theorem on locally convex spaces which in each of them compactness plays an essential role.

Theorem 1 (Schauder [1]). Let C be a closed and convex subset of a Banach space E. Then every compact and continuous mapF WC !C has at least one fixed point.

Theorem 2(Tychonoff fixed point theorem [1]). LetEbe a Hausdorff locally con- vex linear topological space,C a convex subset ofEandF WC !E a continuous mapping such that

F .C /AC withAcompact. ThenF has at least one fixed point.

The concept of measure of noncompactness together with the well-known Darbo’s fixed point theorem have played a basic role in nonlinear functional analysis, espe- cially in topological fixed point theory. Up to now, several papers have been published

c 2018 Miskolc University Press

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on the existence and behavior of solutions of nonlinear differential and integral equa- tions, using the technique of measure of noncompactness together with the Darbo’s fixed point theorem. Some of these works are noted in [3,4,7,8,10,12–16,18,19].

Recently, the theory of infinite systems of integral or differential equations can also be used in solving some problems for differential and integral equations (see [5,11, 22–25]).

On the other hand, In 1969, Meir and Keeler [21] proved the following very interest- ing fixed-point theorem, which is a generalization of the Banach contraction principle [6].

Definition 1([21]). Let.X; d /be a metric space. Then a mapping T on X is said to be a Meir-Keeler contraction (MKC, for short) if for any" > 0, there existsı > 0 such that

"d.x; y/ < "CıH)d.T x; T y/ < "; (1.1) for allx; y2X.

Theorem 3 (Meir and Keeler [21]). Let .X; d / be a complete metric space. If T WX !X is a Meir-Keeler contraction, then T has a unique fixed point.

Next, Lim [20] and Suzuki [26] introduced the notion of L-functions and char- acterized Meir-Keeler contractions in metric spaces. Moreover, Aghajani et al. [2]

introduce the notion of Meir-Keleer condensing operator on a Banach space, a char- acterization using strictly L-functions and provide a few generalizations of Darbo’s fixed point theorem. In this paper, we state and prove some fixed point theorems for L-functions on Frechet spaces with the help of measure of noncompactness and theK Tychonoff fixed point theorem, which is an extension of the results [21], Lim [20], Suzuki [26] and Darbo [13]. Then, we prove some existence theorems for a general infinite system of integral equations. Finally, using the obtained results, we are go- ing to study the existence of continuous solutions of the infinite system of nonlinear integral equations of Hammerstein type

xn.t; s/ (1.2)

Dfn

t; s; x1.t; s/ : : : ; xn.t; s/;

Z 1 0

Z 1 0

kn.t; s; v; w/hn.v; w; .xj.v; w//j1D1/dvdw

; t; s2RC, n2N. The functionsfn, kn andhn (n2N) are continuous and sat- isfy some certain conditions, specified later. Furthermore, an example is presented to guarantee our results.

Now, we recall some basic facts concerning measures of noncompactness. Denote by Rthe set of real numbers and putRCDŒ0;C1/. Let.E;kk/be a real Banach space with zero element0. LetB.x; r/denote the closed ball centered atxwith radiusr.

The symbolBr stands for the ballB.0; r/. ForX, a nonempty subset ofE, we de- note by X and C onvX the closure and the closed convex hull ofX, respectively.

Moreover, let us denote byME the family of nonempty bounded subsets ofE and byNE its subfamily consisting of all relatively compact sets.

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A topological vector space (TVS) is a vector spaceX over the fieldRwhich is en- dowed with a topology such that the maps .x; y/ !xCy and .˛; x/! ˛x are continuous fromXX andRX toX. A topological vector space is called locally convex if there is a basic for the topology consisting of convex sets (that is, setsA such that ifx; y2AthentxC.1 t /y2Afor0 < t < 1).

Definition 2(Definition 1.11 in [9]). A Frechet space is a complete linear metricK space or equivalently, a complete total paranormed space. In other words, a locally convex space is called a Frechet space if it is metrizable and the underlying metricK space is complete.

Definition 3([5]). LetMbe a class of subsets of a Frechet space E, we sayK Mis an admissible set ifNE\M¤¿and ifX 2M, thenC onv.X /; X2M.

Definition 4 ([5]). LetMbe an admissible subset of a FreKchet space E, we say thatWM !RCbe a measure of noncompactness on Frechet space E if it satisfiesK the following conditions:

.1ı/ The familykerD fX2MW.X /D0gis nonempty andkerNE. .2ı/ XY H).X /.Y /.

.3ı/ .X /D.X /.

.4ı/ .C onvX /D.X /.

.5ı/ .XC.1 /Y /.X /C.1 /.Y /for2Œ0; 1.

.6ı/ IffXngis a sequence of closed sets fromMsuch thatXnC1XnfornD 1; 2; , and if lim

n!1.Xn/D0, thenX1D \1nD1Xn¤¿.

Definition 5(Lim [20]). A function' fromRCinto itself is called an L-function if'.0/D0,'.s/ > 0fors2.0;C1/, and for everys2.0;C1/there existsı > 0 such that'.t /s, for allt2Œs; sCı.

Definition 6([2]). A function fromRCinto itself is called a strictly L-function if.0/D0,.s/ > 0fors2.0;C1/, and for everys2.0;C1/there existsı > 0 such that.t / < s;for allt 2Œs; sCı.

2. FIXED POINT THEOREMS

In this section, we present some fixed point theorems on a Frechet space.K

Definition 7. Let˝be a nonempty subset of a Frechet spaceK E,Man admissible set such that˝2MandWM !RCbe a measure of noncompactness onE. We say that an operatorF W˝ !˝ is a Meir-Keeler condensing operator if for any

" > 0, there existsı > 0such that

".X / < "CıH).FX / < "; (2.1) andF .X /2Mfor any nonempty subsetX2M.

Now, we are ready to state our first main theorem for this section.

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Theorem 4. Let˝ be a nonempty, closed and convex subset of a Frechet spaceK E,Man admissible set such that˝2MandWM !RCbe a measure of non- compactness onE. LetF W˝ !˝be a continuous and Meir-Keeler condensing operator, thenF has at least one fixed point.

Proof. By induction, we obtain a sequence f˝ngsuch that ˝0D˝ and ˝nD C onv.F ˝n 1/,n1. It is obvious that˝n2Mand˝nC1˝n for alln2N.

If there exists an integerN 0such that.˝N/D0, then ˝N is compact. Thus, Tychonoff fixed point theorem implies thatF has a fixed point. Now assume that .˝n/¤0forn0. Define"nD.˝n/. Since ˝nC1˝n and by.2ı/so we have f"ngis a positive decreasing sequence of real numbers and there exists 0 such that"n! asn! 1. We claim that D0. Suppose, on the contrary, that > 0, then there existsn0such thatn > n0implies"n< Cı. /, therefore by the definition of Meir-Keeler condensing operator,"nC1< which is a contradiction.

Therefore, D0, that is,"n!0asn! 1. Since the sequence.˝n/is nested, in view of axiom.6ı/of Definition4we deduce that the set˝1D

1

\

nD1

˝nis nonempty, closed and convex subset of the set˝. Moreover, the set˝1 is invariant under the operatorF and belongs toKer. Thus, applying Tychonoff fixed point theorem,F

has a fixed point.

Lemma 1. Let' WRC !RC be an increasing and right continuous function.

Then the following conditions are equivalent:

.a/ '.t / < t for anyt > 0and'.0/D0.

.b/ 'is an L-function.

.c/ 'is a strictly L-function.

Proof. Let ' satisfy condition (a). Since '.t / < t for any t > 0 and' a right continuous function, so for anys > 0there existıs> 0such that

j'.t / '.s/j< s '.s/

for allt2Œs; sCıs. Thus,'.t / < sfor allt2Œs; sCısand'satisfies condition (b).

Now, assume that ' satisfies condition (b). Thus, for anys > 0there exists ı > 0 such that'.t /sfor allt2Œs; sCı. Since'is an increasing function, so'.t / < s for allt2Œs; sCı2and' satisfies condition (c).

It remains to be shown that condition (c) implies condition (a). Because'is a strictly L-function, it implies that'.s/ < s for alls > 0and therefore the proof is obvious.

Now, we formulate and prove a fixed point theorem using strictly L-functions as an application of Theorem4.

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Theorem 5. Let˝be a nonempty, closed and convex subset of a Frechet space E,K Man admissible set such that˝2MandWM !RCa measure of noncompact- ness onE. ThenT is a Meir-Keeler condensing operator if and only if there exists a (increasing, right continuous) strictly L-function#such that

.TX /#..X //; (2.2)

for eachX ˝andX 2M.

Proof. Let" > 0be given. By the assumption, there existsı > 0such that#.t / < "

if"t < "Cı. IfX is a subset of˝such that

".X / < "Cı."/;

thus,

.T .X //#..X // < "

andT is a Meir-Keeler condensing operator. For the necessity part, assume that T is a Meir-Keeler condensing. From the definition of Meir-Keeler condensing, we can define a function˛W.0;1/!.0;1/, such that

".X / < "C2˛."/H).T .X // < "; (2.3) for "2.0;1/. Using such ˛, we define a nondecreasing function ˇW.0;1/! Œ0;1/, by

ˇ.t /Dinff"Wt"C˛."/g

for t2.0;1/. Sincet tC˛.t /, we note thatˇ.t /t fort 2.0;1/. Define a function1fromŒ0;1/into itself by

1.t /D 8

<

:

0 if tD0;

ˇ.t / if t > 0 andminf" > 0Wt"C˛."/gexi st s;

ˇ .t /Ct

2 ot herwi se:

(2.4) Similar to Proposition 1 in [20],1be an L-function. Now, we define2,3 and# fromŒ0;1/into itself by

2.t /Dsupf1.s/Wstg; 3.t /Dinff2.s/Ws > tg and

#.t /D 3.t /Ct 2 fort2Œ0;1/. Then we have

01.t /2.t /3.t /#.t /t

for all t 2.0;1/, 2 is a nondecreasing L-function, 3 is a nondecreasing, right continuous L-function and#is an increasing, right continuous L-function. Therefore, by Lemma1we have# is a increasing and right continuous strictly L-function. Fix

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X 2Msuch that.X /¤0. From the definition ofˇ, there exists".X /such that".X /"C˛."/. Thus,

.TX / < "ˇ..X //#..X //

holds.

Remark1. If there exists a strictly L-function#such thatT satisfies in condition (2.2) then there exists an increasing and right continuous strictly L-function#0such thatT satisfies in condition (2.2).

3. INFINITE SYSTEMS OF CONDENSING OPERATORS

In this section, we prove some existence theorems for a general infinite system of equations involving condensing operators.

Let.Ei; di/be a Frechet space for allK i 2N,d.x; y/Dsup˚1

2iminf1; di.xi; yi/g W i 2N , xD.x1; x2; : : :/; yD.y1; y2; : : :/2 Y

i2N

Ei. Then .Y

i2N

Ei; d / is a FrechetK space.

Theorem 6(Tychonoff’s theorem[17]). Letf.Xi; i/Wi 2Ngbe any family of to- pological spaces. ThenY

i2N

.Xi; i/is compact if and only if each.Xi; i/is compact.

Remark2. We use the notationR!which denotes the countable Cartesian product ofRCwith itself, andl1consists of all bounded sequences of scalars.

Now we are ready to state and prove the main results of this section.

Theorem 7. Supposei be a measure of noncompactness on Frechet spacesK Ei

for all i 2N. Moreover assume that the function F Wl1 !RC is convex, non- decreasing andF ..xi/1iD1/D0if and only ifxi D0for alli 2N. If we define

MD fX

1

Y

iD1

EiWsup

i fi.i.X //g<1g; wherei.X /denotes the natural projections of

1

Y

iD1

Ei intoEi and WM !RCbe defined by

.X /DF

i.i.X //1

iD1

; (3.1)

thenMis an admissible set andis a measure of noncompactness on Frechet spaceK ED

1

Y

iD1

Ei.

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Proof. It can be easily verified thatMis an admissible set. Now, we proveis a measure of noncompactness onED

1

Y

iD1

Ei. For proving the condition.2ı/, sup- pose thatX Y. Sincei be a measure of noncompactness, we havei.i.X //

i.i.Y //for alli 2N, and usingF is nondecreasing we imply that .X /DF

i.i.X //1

iD1

.Y /DF

i.i.Y //1

iD1

;

which shows that the condition.2ı/ is valid. The properties .3ı/-.5ı/ are simple consequences of

i.UC.1 /V /Di.U /C.1 /i.V /; (3.2) i.C onvX /DC onvi.X /;

i.X /i.X /i.X /:

Now we show.1ı/. If .X /D0forX 2Mthen i.i.X //D0for each i 2N.

Hence, by virtue of.1ı/of Definition4for measure of noncompactnessi,i.X /is relatively compact for alli2Nand by Theorem6,

1

Y

iD1

i.X /is relatively compact.

Thus,X

1

Y

iD1

i.X /is relatively compact. Finally, it suffices to show.6ı/. suppose thatfXngis a sequence of closed sets fromMsuch thatXnC1Xnforn2Nand

nlim!1.Xn/D0. So we have

nlim!1F

i.i.Xn//1

iD1

D0: (3.3)

SinceXnC1Xn,i.i.XnC1//i.i.Xn//andi.i.Xn//0for alli2N.

Thus, there is anri 0so that

nlim!1i.i.Xn//Dri: BecauseF is continuous, then

nlim!1F

i.i.Xn//1 iD1

DF

nlim!1 i.i.Xn//1 iD1

DF .ri/1iD1 : On the other hand, using (3.3) we haveF .ri/1iD1

D0. By assumption of the the- orem we imply thatri D0for alli 2N, and so lim

n!1i.i.Xn//D0. By (6ı) of definition of measure of noncompactness on En we have Xi1 D

1

\

nD1

i.Xn/¤¿ for alli 2N. Therefore we get

1

Y

iD1

Xi1 X1 andX1¤¿. This completes the

proof.

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Example1. Letn.n2N/be measures of noncompactness on Frechet spacesK En, respectively. ConsideringF1..xn/1nD1/D sup

n2N

bnxn andF2..xn/1nD1/D

1

X

nD1

anxn

(the functionsF 1; F 2are defined onl1) such thatan; bn2RC,

1

X

nD1

an<1andfbng be a bounded sequence, then all the conditions of Theorem7are satisfied. Therefore, e1WD sup

n2N

bn.Xn/ande2WD

1

X

nD1

an.Xn// define measures of noncompactness in the FreKchet spaceED

1

Y

iD1

Ei whereXn.n2N/, denotes the natural projections of X intoEn.

Theorem 8. Let˝i.i2N/be a nonempty, convex and closed subset of a FrechetK spaceEi,i an arbitrary measure of noncompactness onEi andsup

i fii/g<1. LetFiW

1

Y

iD1

˝ii .iD1; 2; : : :/be a continuous operator such that

aii.Fi.

1

Y

jD1

Xj//'.sup

j fajj.Xj/g/ (3.4) for any subsetXi of˝i(i2N) where'WRC !RCis a strictly L-function andfaig is a bounded sequence of positive real numbers. Then there exist.xj/j1D12

1

Y

jD1

˝j

such that

Fi..xj/j1D1/Dxi (3.5) for alli2N.

Proof. Let us considerFeW

1

Y

iD1

˝i !

1

Y

iD1

˝i in the following way:

F ..xe j/j1D1/D.F1..xj/j1D1/; F2..xj/j1D1/; : : : ; Fi..xj/j1D1/; : : :/

for all.xj/j1D12

1

Y

iD1

˝i. It is obvious thatFeis continuous. It suffices to show that the hypothesis (2.2) of Theorem5holds whereis defined by Example1. Take an

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arbitrary nonempty subsetX of

1

Y

iD1

˝i. Now, by.2ı/and (3.4) we obtain

.eF .X //.

1

Y

iD1

Fi.

1

Y

jD1

j.X ///

Dsup

i

aii.Fi..

1

Y

jD1

j.X ////

sup

i

'.sup

j

ajj.j.X ///

sup

i

'..X //

'..X //: (3.6)

Thus, using Theorem 5, eF is a continuous and Meir-Keeler condensing operator.

Now applying Theorem4,Fehas a fixed point and there exist.xj/j1D12

1

Y

jD1

˝j such that

.xj/j1D1DF ..xe j/j1D1/D.F1..xj/j1D1/; F2..xj/j1D1/; : : : ; Fj..xj/j1D1/; : : :/

and that (3.5) holds.

4. EXISTENCE OF A SOLUTION FOR A INFINITE SYSTEM OF INTEGRAL EQUATIONS

In the following section, we will work in the classical Banach space BC.RC RC/ consisting of all real functions defined, bounded and continuous onRCRC equipped with the standard norm

kxk Dsupfjx.t; s/j Wt; s0g:

Now, we present the definition of a special measure of noncompactness inBC.RC RC/which will be needed in the sequel.

To do this, letX be a fixed nonempty and bounded subset ofBC.RCRC/and fix a positive numberT. Forx2X and > 0, denote by!T.x; /the modulus of the continuity of functionxon the intervalŒ0; T ;i.e.,

!T.x; /Dsupfjx.t; s/ x.u; v/j Wt; s; u; v2Œ0; T ;jt uj ;js vj g: Further, let us put

!T.X; /Dsupf!T.x; /Wx2Xg;

!0T.X /D lim

!0!T.X; /

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and

!0.X /D lim

T!1!0T.X /:

Moreover, for two fixed numbers t; s2RC let us the define the function on the familyMBC.RCRC/by the following formula

.X /D!0.X /C lim sup

k.t;s/k!1

d i amX.t; s/;

wherek.t; s/k Dmax.t; s/andX.t; s/D fx.t; s/Wx2Xg. Similar to [8], it can be shown that the function is a measure of noncompactness in the spaceBC.RC RC/.

As an application of Theorem8 we prove the existence of solutions for the infinite system of integral equations of Hammerstein type in two variables (1.2).

Theorem 9. Assume that the following conditions are satisfied:

.a1/ fnWRCRCRnR !R.n2N/is continuous. Moreover, there exists a nondecreasing, right continuity and concave strictly L-function'such that

jfn.t; s; x1; : : : ; xn; u/ fn.t; s; y1; : : : ; yn; v/j '. max

1injxi yij/C ju vjI (4.1)

.a2/ M WDsupfjfn.t; s; 0; : : : ; 0/j Wt; s2RC; n2Ng<1;

.a3/ knWRCRCRCRC !Rare continuous functions for alln2N;

.a4/ hnWRCRCR! !R.n2N/is continuous and there exist a continuous functionanWRCRC !RCand a continuous and nondecreasing function bnWRC !RCsuch that

jhn.t; s; .xj/j1D1/j an.t; s/bn. sup

1j <1jxjj/ for allt; s2RCand.xj/j1D12R!with sup

1j <1jxjj<1. Also the function .v; w/ !an.v; w/kn.v; w; t; s/ is integrable over RCRC for any fixed t; s2RCandn2N.

.a5/ there exists a positive constantDsuch that DDsupf

Z 1

0

Z 1

0

an.v; w/jkn.t; s; v; w/jdvdwWt; s2RC; n2Ng<1; and

kt;slimk!1

Z 1

0

Z 1

0

an.v; w/jkn.t; s; v; w/jdvdwD0I (4.2) .a6/ the following equalities are hold:

Tlim!1

supf

Z 1

T

Z T 0

an.v; w/jkn.t; s; v; w/jdvdwWt; s2RCg

D0; (4.3)

Tlim!1

supf

Z 1

0

Z 1

T

an.v; w/jkn.t; s; v; w/jdvdwWt; s2RCg

D0;

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for alln2N;

.a7/ there exists a positive solutionr0of the inequality '.r/CMCDbn.r/r for alln2N.

Then the infinite system of equations (1.2) has at least one solution in the space .BC.RCRC//!.

Proof. Let us fix arbitrarilyn2N. DefineHnW.BC.RCRC//! !BC.RC RC/by

Hn..xj/j1D1/.t; s/Dfn.t; s; x1.t; s/ : : : ; xn.t; s/;

Z 1 0

Z 1 0

kn.t; s; v; w/hn.v; w; .xj.v; w//j1D1/dvdw/: (4.4)

In light of (4.4) and assumptions.a1/-.a7/,fnis continuous,knis continuous,hn

is continuous andxi fori 2Nare continuous. On the other hand, integral of con- tinuous function is continuous. Therefore, we infer that the functionHn..xj/j1D1/is continuous for arbitrarily.xj/j1D12.BC.RCRC//!because it is the composition of continuous functions. Moreover, in view of our assumptions, for arbitrarily fixed .xj/j1D12.BC.RCRC//!andt; s2RC, we obtain

jHn..xj/j1D1/.t; s/j (4.5)

ˇ ˇ ˇfn

t; s; x1.t; s/ : : : ; xn.t; s/;

Z 1

0

Z 1

0

kn.t; s; v; w/hn.v; w; .xj.v; w//j1D1/dvdw

fn.t; s; 0; : : : ; 0/ˇ ˇ

ˇC jfn.t; s; 0; : : : ; 0/j '.max

1injxi.t; s/j/C jfn.t; s; 0; : : : ; 0/j C

Z 1

0

Z 1

0 jkn.t; s; v; w/jjan.v; w/jbn.j.xj.v; w//j1D1j/ '.max

1injxi.t; s/j/CMCDbn. sup

1j <1

jxj.v; w/j/

'.max

1inkxik/CMCDbn. sup

1j <1kxjk/:

Thus,

kHn.x/k '. max

1inkxik/CMCDbn. sup

1j <1kxjk/ (4.6) andHn..xj/j1D1/2BC.RCRC/for any.xj/j1D12.BC.RCRC//! . By (4.6) and using.a7/, the functionHnmaps.BNr0/!intoBNr0.

Now we claim thatHnis a continuous function on.BNr0/! for alln2N. To do this, let us fix0 < " <21n and take arbitraryxD..xj/j1D1/; yD..yj/j1D1/2.BC.RC

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RC//!such thatd.x; y/Dsup˚1

2i minf1;kxi yikg Wi2N < ". Then, fort; s; v; w2 RC, we get

ˇ ˇ

ˇHn..xj/j1D1/.t; s/ Hn..yj/j1D1/.t; s/

ˇ ˇ ˇ

ˇ ˇ ˇfn

t; s; x1.t; s/; : : : ; xn.t; s/;

Z 1

0

Z 1

0

kn.t; s; v; w/hn.v; w; .xj.v; w//j1D1/dvdw

fn

t; s; y1.t; s/; : : : ; yn.t; s/;

Z 1

0

Z 1

0

kn.t; s; v; w/hn.v; w; .yj.v; w//j1D1/dvdwˇ ˇ ˇ '. max

1injxi.t; s/ yi.t; s/j/

Cˇ ˇ ˇ

Z 1

0

Z 1

0

kn.t; s; v; w/Œhn.v; w; .xj.v; w//j1D1/ hn.v; w; .yj.v; w//j1D1/dvdwˇ ˇ ˇ:

On the other hand, assumption .a5/ensure that there exists a positive numberT such that for max.t; s/ > T we have

ˇ ˇ

ˇHn..xj/j1D1/.t; s/ Hn..yj/j1D1/.t; s/

ˇ ˇ

ˇ'. max

1injxi.t; s/ yi.t; s/j/

C2bn.r0/ Z 1

0

Z 1

0 jkn.t; s; v; w/jan.v; w/dvdw '.2n"/Cbn.r0/":

Suppose thatt; s2Œ0; T . By applying the assumptions, we infer that

ˇ ˇ

ˇHn..xj/j1D1/.t; s/ Hn..yj/j1D1/.t; s/

ˇ ˇ ˇ '. max

1injxi.t; s/ yi.t; s/j/ C

ˇ ˇ ˇ

Z 1

0

Z 1

0 kn.t; s; v; w/Œhn.v; w; .xj.v; w//j1D1/ hn.v; w; .yj.v; w//j1D1/dvdw ˇ ˇ ˇ

'.2n"/C (Z 1

0

Z T

0 jkn.t; s; v; w/jjhn.v; w; .xj.v; w//j1D1/ hn.v; w; .yj.v; w//j1D1/jdv C

Z 1

T jkn.t; s; v; w/jŒjhn.v; w; .xj.v; w//j1D1/j Chn.v; w; .yj.v; w//j1D1/jdv

dw )

'.2n"/C Z T

0

Z T

0 jkn.t; s; v; w/jjhn.v; w; .xj.v; w//j1D1/ hn.v; w; .yj.v; w//j1D1/jdvdw C

Z T 0

Z 1

T jkn.t; s; v; w/jŒjhn.v; w; .xj.v; w//j1D1/j C jhn.v; w; .yj.v; w//j1D1/jdvdw C

Z 1

T

Z 1

0 jkn.t; s; v; w/jŒjhn.v; w; .xj.v; w//j1D1/j C jhn.v; w; .yj.v; w//j1D1/jdvdw '.2n"/CKTn!rT0.hn; "/C2bn.r0/

Z T 0

Z 1

T

an.v; w/jkn.t; s; v; w/jdvdw C2bn.r0/

Z 1

T

Z 1

0

an.v; w/jkn.t; s; v; w/jdvdw;

where

KTnDsupfkn.t; s; v; w/Wt; s; v; w2Œ0; T g

(13)

!rT0.hn; "/Dsupfjhn.v; w; .xj/j1D1/ hn.v; w; .yj/j1D1/j Wv; w2Œ0; T ; xi; yi 2Œ r0; r0; jxi yij "g:

From the continuity of the functionhnon the compact setŒ0; T Œ0; T Œ r0; r0! (by using Tychonoff’s theorem), we have!rT

0.hn; "/ !0as" !0and in view of assumption.a6/we can chooseT in such a way that three last terms of the above estimate are sufficiently small. Thus Hn is a continuous function on .BC.RC RC//!.

Now we assert thatHnsatisfies all the conditions of Theorem8. LetXibe nonempty and bounded subsets ofBNr0 for alli2Nsuch that supi..Xi// <1, and assume thatT > 0and" > 0are arbitrary constants. Lett1; t2; s1; s22Œ0; T such thatjt2

t1j ",js2 s1j "andxi2Xi for alli2N. Then, by the assumptions we have:

ˇ ˇ

ˇHn..xi/1iD1/.t2; s2/ Hn..xi/1iD1/.t1; s1/ ˇ ˇ ˇ

ˇ ˇ ˇfn

t2; s2; x1.t2; s2/; : : : ; xn.t2; s2/;

Z 1

0

Z 1

0

kn.t2; s2; v; w/hn.v; w; .xi.v; w//1iD1/dvdw

fn

t1; s1; x1.t1; s1/; : : : ; xn.t1; s1/;

Z 1

0

Z 1

0

kn.t1; s1; v; w/hn.v; w; .xi.v; w//1iD1/dvdwˇ ˇ ˇ

ˇ ˇ ˇ ˇ

fn

t2; s2; x1.t2; s2/; : : : ; xn.t2; s2/;

Z 1

0

Z 1

0

kn.t2; s2; v; w/hn.v; w; .xi.v; w//j1D1/dvdw

fn

t2; s2; x1.t1; s1/; : : : ; xn.t1; s1/;

Z 1

0

Z 1

0

kn.t2; s2; v; w/hn.v; w; .xi.v; w//1iD1/dvdw ˇ

ˇ ˇ ˇ

C ˇ ˇ ˇ ˇ

fn

t2; s2; x1.t1; s1/; : : : ; xn.t1; s1/;

Z 1

0

Z 1

0

kn.t2; s2; v; w/hn.v; w; .xi.v; w//1iD1/dvdw

fn

t1; s1; x1.t1; s1/; : : : ; xn.t1; s1/;

Z 1

0

Z 1

0

kn.t2; s2; v; w/hn.v; w; .xi.v; w//1iD1/dvdw ˇ

ˇ ˇ ˇ

C ˇ ˇ ˇ ˇ

fn

t1; s1; x1.t1; s1/; : : : ; xn.t1; s1/;

Z 1

0

Z 1

0

kn.t2; s2; v; w/hn.v; w; .xi.v; w//1iD1/dvdw

fn

t1; s1; x1.t1; s1/; : : : ; xn.t1; s1/;

Z 1

0

Z 1

0

kn.t1; s1; v; w/hn.v; w; .xi.v; w//1iD1/dvdw ˇ

ˇ ˇ ˇ '. max

1jnjxj.t2; s2/ xj.t1; s1/j/C!Tr0;D1.fn; "/

C j Z 1

0

Z 1

0 Œkn.t2; s2; v; w/ kn.t1; s1; v; w/hn.v; w; .xi.v; w//1iD1/dvdwj '. max

1jnjxj.t2; s2/ xj.t1; s1/j/C!Tr0;D1.fn; "/

C Z T

0

Z T

0 jkn.t2; s2; v; w/ kn.t1; s1; v; w/jjhn.v; w; .xi.v; w//1iD1/jdvdw C

Z T 0

Z 1

T jkn.t2; s2; v; w/ kn.t1; s1; v; w/jjhn.v; w; .xi.v; w//1iD1/jdvdw C

Z 1

T

Z 1

0 jkn.t2; s2; v; w/ kn.t1; s1; v; w/jjhn.v; w; .xi.v; w//1iD1/jdvdw

(14)

'. max

1jn!T.xj; "//C!Tr0;D1.fn; "/CT2UrT0!T.kn; "/

Cbn.r0/ Z T

0

Z 1

T Œjkn.t2; s2; v; w/j C jkn.t1; s1; v; w/jan.v; w/dvdw Cbn.r0/

Z 1

T

Z 1

0 Œjkn.t2; s2; v; w/j C jkn.t1; s1; v; w/jan.v; w/dvdw (4.7)

where

D1Dbn.r0/D.see assumpt i on .a5//

!rT0;D1.fn; "/Dsupfjfn.t2; s2; x1; ; xn; y/ fn.t1; s1; x1; ; xn; y/j Wt1; s1; t2; s22Œ0; T ;

jt2 t1j ";js2 s1j ";jxij r0;jyj D1g;

!T.xi; "//Dsupfjxi.t2; s2/ xi.t1; s1/j Wt1; s1; t2; s22Œ0; T ;jt2 t1j ";js2 s1j "g;

!T.kn; "/Dsupfjkn.t2; s2; v; w/ kn.t1; s1; v; w/j W

t1; s1; t2; s2; v; w2Œ0; T ;jt2 t1j ";js2 s1j "g;

UrT0Dsupfjhn.v; w; .xj/j1D1/j Wv; w2Œ0; T ; xi2Œ r0; r0g:

Sincexi was an arbitrary element ofXi for alli2Nin (4.7), we obtain

!T.Hn.

1

Y

iD1

Xi//'. max

1in!T.xi; "//C!rT

0;D1.fn; "/CT2UrT0!T.kn; "/

Cbn.r0/ Z T

0

Z 1

T

Œjkn.t2; s2; v; w/j C jkn.t1; s1; v; w/jan.v; w/dvdw Cbn.r0/

Z 1

T

Z 1

0

Œjkn.t2; s2; v; w/j C jkn.t1; s1; v; w/jan.v; w/dvdw;

and by the uniform continuity of fn and kn on the compact sets Œ0; T Œ0; T  Œ r0; r0nŒ D1; D1andŒ0; T Œ0; T Œ0; T Œ0; T respectively, we have

!rT

0;D1.fn; "/ !0and!T.kn; "/ !0as" !0. Therefore, we obtain

!T.Hn.

1

Y

iD1

Xi//'. max

1in!T.xi; "//

Cbn.r0/ Z T

0

Z 1

T

Œjkn.t2; s2; v; w/j C jkn.t1; s1; v; w/jan.v; w/dvdw Cbn.r0/

Z 1

T

Z 1

0

Œjkn.t2; s2; v; w/j C jkn.t1; s1; v; w/jan.v; w/dvdw:

Now takingT ! 1and by using assumption.a6/we get

!0.Hn.

1

Y

iD1

Xi//'. max

1in!0.Xi//: (4.8)

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