On the second order mixed quasimonotone periodic boundary value systems in ordered Banach spaces

Nonlrneor Analysis, Theory, Printed in Great Britain.

Melhods

& Applrcations,

Vol.

20,

No.

9, pp.

1135-l 144,

0362-546X/93 $6.00+ .OO CC 1993 Pergamon Press Ltd

1993.

ON THE SECOND ORDER MIXED QUASIMONOTONE PERIODIC BOUNDARY VALUE SYSTEMS IN ORDERED BANACH SPACES SEPPO HEIKKILA? and V. LAKSHMIKANTHAM~ TDepartment of Mathematics, University of Oulu, SF-90570 Oulu, Finland; and SDepartment of Applied Mathematics, Florida Institute of Technology, Melbourne, FL 32901-6988, U.S.A. (Received 11 September

1991; received for publication

1 August 1992)

Key words and phrases: Periodic boundary value problems, mixed quasimonotone

systems.

1. INTRODUCTION THE USE

of monotone methods in the study of the periodic boundary value problem

-u”(t)

= f(t, u(t))

for a.e. t E J = [0, T],

u’(0) = u’(T)

u(O) = u(T),

(I)

has recently been quite extensive (see, for example [l-6]). For instance, in [4] the existence of the extremal solutions of this problem is proved as an application of a fixed point theorem for increasing operators in ordered Banach spaces, by assuming that (I) has suitable upper and lower solutions. The function f is assumed to be real valued and decreasing in its second argument. In particular, f need not be continuous, the Lebesgue integrability of f( *, u(a)) for 2.4E C(J, R) will suffice. The above cited result will now be extended as follows. Instead of problem (I) we shall consider the following system of periodic boundary value problems (PBVS) -ui”(f)

= A(t~

ui(“)

=

ui(t),

ui(T),

i”lqitt)t ul(t))

L”lp,Ct), u;(o)

=

u;(T),

for a.e. t E J, i= 13 ---9 n,

(1.1)

with fi: JX En+’ + E, i = 1, . . . . n, where E is an ordered Banach space with regular order cone K and J = [0, T], T > 0. In the notation u = (ui, [u],~, [u],~) for u = (ui, . . ., u,) E E” the term [u],, is formed by pi coordinates of U, different from ui, and [u],~ contains the remaining ones, so that pi + qi = n - 1. We shall prove that the PBVS (1 .l) has coupled extremal quasisolutions if for each i= l,..., n there is a Lebesque integrable function h: J -+ IR, such that the function fi(t, ui, [ulpi, [u],,, v) - hi(t)v is increasing with respect to [ulqi and decreasing with respect to Ui, [u],, and to Z.J,and if fi(*, u(-),u(*)) is strongly measurable on J, when u and v belong to certain order intervals of continuous functions, bounded by later specified coupled generalized upper and lower quasisolutions of (1.1). In the special case when qi = 0 we obtain results concerning the existence of extremal solutions of (1.1). 2. CONVERSION

OF THE PBVS (1.1) TO OPERATOR

EQUATIONS

We say that the functions v = (vi, . . ., u,) and w = (wi, . . ., w,) are coupled quasisolutions of (1.1) if their components have absolutely continuous first derivatives, and - ul’(t) = fi(t, vi(t), [4,,(0, [Wl,,W, G(0) vi(O) = vi(T),

u;(O) = u;(T), 1135

for a.e. t E J, i= l,...,n,

(2.1)

1136

S. HEIKKILA:and V. LAKSHMMANTHAM - wl(t)

= fiCt,

wi(t)v

for a.e. t E J,

[Wlp,(t),[vlqi(t)9wj(t)) l+(O) = W;(T),

w,(O) = w,(T),

i=l

(2.2)

9 .--, n.

First we shall convert the PBVS (1 .l) to a pair of operator equations. As for the differential and integral calculus of vector valued functions used in the following, see for instance [7,8]. LEMMA 2.1. Let E be a Banach space and fi: J x E”+l -+ E, i = 1, . . . , n. If the Lebesgue integrable functions hi: J + R, are nonvanishing and Hi(t) = Jhhi(s) ds, t E J, then U = (Ur, . . . . un), w = (I+) . . . . w,) are coupled quasisolutions of the PBVS (1.1) if and only if the functions x = (x1, . . . , x,) and y = (yr , . . . , y,), where

xi(t) = (X!(t)9 x:(t)) = (“i(t)9 vit)),

teJ,i=

l,...,

n,

(2.3)

YiCt) = (Y!Ct),Y?Ct))

tEJ,i=

l,...,

n,

(2.4)

= CwiCt)9 wfCt))9

are solutions of the operator equations A(-% Y) = x, the components

A(Y,X)

= Y,

(2.5)

of A = (A,, . . . , A,) being defined by Ai(x* YNt) = (Af(x9Y)(t)9

t E J,

Af(xv Y)Ct)),

(2.6)

where --r 4xdw)

=

T

f e*[xj(s) + x&s)]&

-&

+

e-'

s0

e'[xj(s) + X&Y)]

(2.7)

ds,

s0

and,

Af(x, y)(t) = eeHi(‘)

s’

eHi’“‘[hi(.S)X,?(~) - fits,XiIts),[X11pits)9 [Y ‘lqits)9 &~))I ~

0

e-Hi(t)

+

,fW)

_

1

IT

eH~(“‘[hi(s)x~(s)-

.O

fi( s, xi%)9[x11,;(s), [Y'l,;W,

dw

d&s. (2.8)

Proof. Assume first that v = (v, , . . . , v,) and w = (w, , . . . , w,) are coupled quasisolutions of the PBVS (1.1). Since the functions u; and wl are absolutely continuous and a.e. differentiable, on J, then the functions uf’ and wf’ are Bochner integrable. This implies by (2.1) and (2.2) that the functions Af(x, y), given by (2.8), and also the functions Af(y, x) are defined on J when the components of x and y are given by (2.3), (2.4). Moreover, since vi(O) = u,(T), it follows from (2.3) and (2.7) that

e’&x, y)(t)

= ~_l e =

1

T i0

+[eTUi(T) e

[esui(9 + e”uf(s)] d.s +

-

Vi(O)]

+

1’

[esuj(.s) + esu;(s)] d.s

0

e%,(t) -

U,(O)

=

e’xi(t),

teJ,i=

l,...,

n.

Mixed quasimonotone

1137

systems

From (2.1), (2.3), (2.4) and (2.8) it follows that for all t E J and eHi(‘)Af(x, y)(t) =

st

1, . . . , n,

i=

eHi’“‘[hj(S)Uf(S) - h(S, Ui(S),[Ulp,(s)9 [wlqj(s)t ul(s))l dS

0

1

+ eHiCT)_ 1 0‘eHi”‘[hi(s)Ul(s) - fi(S, Vi(S), [“lpi(s), [wl,,(s)3ul(s))l dS s

=

' [hi(s)enfcs)uf(s)+ eNi%/(

ds

i0

+

eHiw

_

eni%;

=

T

1

lhi

1

-

0

(4 eHi(‘) ui (s)

u;(O) +

+

eHt(s)u/ (,y) J &

eHicTt _ 1 [eH~(T'uf(T) -

u,‘(O)] = eHi%f(t).

Thus, Aj(x, y)(t) = dw,

l,...,

trsJ,i=

n,j=

1,2,

(2.9)

which, together with (2.3), (2.4) and (2.6) implies that A(x, y) = x. Similarly, it can be shown that A(y, x) = y. Conversely, assume that x = (x1, . . . , x,) and y = (yl, . . ., y,) satisfy (2.5). In particular, (2.9) holds. From (2.7) and (2.9) withj = 1 it follows that f

T

s0 for each t E J and

i=

e"[x/(s) + x?(s)] ds,

e'[xi(.s)+ xf(s)] ds +

e'x:(t) = --&

1, . . . . n. This implies by differentiation (xi’)‘(t)

(a)

50

l,...,

tEJ,i=

= x?(t),

that n.

(b)

From (2.8) and (2.9) withj = 2 it follows that eW’x;(t)

=

’ eHWziWx~C$ - fib, 4Y.9, [x11&), b+l,,(s), $(@)I d.~ s0 1 + eK(T) _ 1

Differentiating

T eHi’Wi(~)x~(.s) - fi(s, xii(s), [x11,,(s),[u’l&), .&))I d.s. i0

(c) with respect to

t we

obtain (4

(x?)‘(t) = -At, xi’(t), [xll,i(O, [Y1lqiw’ Xi2(0) foralli=

l,...,

n and for a.e. t E J. Denoting Ui(t) = Xi'(t),

we

+(t)

teJ,i=

= d(t),

l,...,

n,

obtain from (b) that xi” = u;, and from (d) that U:(t) = -At,

ui(t)9

Cc)

[Ulp,(t),[wlq,(t)9uf(t))

for all

i=

1, . . . . n and for a.e. t E J.

1138

S. HEIKKILKand V. LAKSHMIKANTHAM

From (a) it follows that U,(T) Thus, (2.1) holds. From (c) it it can be shown that (2.2) u = (Ui ) . . .) u,) and w = (wi ,

= Vi(O), and from (c) that v;(T) = u;(O), for each i = 1, . . . . n. follows, moreover, that vi is absolutely continuous. Similarly, holds and that each W; is absolutely continuous, whence . . . , w,) are coupled quasisolutions of the PBVS (1.1).

3. HYPOTHESES

AND

AUXILIARY

RESULTS

Assume now that E is an ordered Banach space with regular order cone K. Given a positive natural number m, define a partial ordering and norm in C(J, Em) by xryifandonlyify,(t)-X,(t)EK,foralltEJandi=

l,...,m,

and [(X1( = SUp(((Xi(t)I(

(t

E J,

1, . . . . ml.

i =

Denote by L’(J, E) the space of all Bochner integrable functions x: J + E, and by L’(J, R,) the set of all Lebesgue integrable functions h: J + R, . The following integral inequality, which is proved, for instance in [9], is frequently applied in the sequel. LEMMA

3.1.

If x, y E L’(J, E) and 0 5 j:yw

x(t)

I

for a.e. t E J, then for all t E J

y(t)

- j:x(sm

5 j;y@)&

- j:*0&.

Denoting P = (u: J + E” 1u’ is absolutely continuous

and a.e. differentiable

on J),

let us impose the following hypotheses on the functions fi: J x E”+I -+ E, i = 1, . . . , n. (f0) There exist i, = (co, . . . . fi,), G = (tii, . . . . ti,) E P with (G, 6’) I (c, G’), and I$‘, cry E L’(J, E) such that for each i = 1, . . . , n and for a.e. t E J -G/(t)

I

fi(t,

-G/(t)

1 fi(t,

B,(t),

[G]pi(t),

[G]qi(t),

c/(t))

-

C:(t),

iri(T)

I

fii(O)y

and

(f 1) For each Hi(t)

=

Sb hi(S)

i = dS,

iG;i(t)y [*lpi(t),

[G]q,(t),

i?:(t))

1, . . . , n there exists a nonvanishing

C:(t),

@j(T)

2

Gi(O)-

function hi E L’(J, R,) such that for

t E J, r

T q(T)

+

-

C/(O)

eWHUT’cy(s)

5 i

&

+

t

eHiCs’c/(s)

ds,

t E

J,

10

and T

eHiWW)C,y(S)

l?;(o) - G;(r) I

& +

st

t eHi@)cly(s)ds,

t E J.

30

(f2) fi(*, u(a), Vi(.)) is strongly measurable = 1, ..,,n.

on J whenever U, u E C(J, E”), I? 5 u 5 d,

I?,’ I u I 8’andi (f3)

The

function

h(t,

ui,

[ul,, [ul,, , vi)

-

hi(t)ui

is decreasing

with respect

to Ui on

hi], to [u!,,on [[C$(t), [u]Jt)] and to Uion [G:(t), D;(t)], and increasing with respect to [ulqi on [[C&(t), [u],,(t)], for a.e. t E J and for each i = 1, . . . . n.

[ai(

Mixed quasimonotone

systems

1139

LEMMA 3.2. Let fi: J x En+’ + E, i = 1, . . . . n satisfy conditions (U1, . . . , fi,) and ti = (WI, . . . , W,), where

pi

= (;i(t), ~f(t))l

(fO)-(f3).

Denoting

t E J,

#i(t) = (G.;i(f), G/(t)),

b= (3.1)

then A(B, W) I 6 and W I A(ti, t?). Proof. Hypotheses (fO)-(f3) imply that for all u = (ur , . .., un), u = (q, . . ., u,) in C(J, E”), satisfying (I?, 9) % (u, V) 5 (i?, c’), for all i = 1, . . . . n, and for a.e. t E J am

+

Cr(t) + hi(t)tii(t)I hi(t)ui(t) -fi(t, u(t), Vi(t)) I

;I(t) - Cy(t) + hi(t)iri(t).

This and the normality of the order cone of E imply that each fi( *, u(e), vi( *)) is pointwise a.e. norm bounded by a function of L’(J, R,). This and (f2) ensure that Aj(V, a) and A$(#, 0) are defined for all i = 1, . . ., n and j = 1,2. Condition (f0) implies that for each t E J and i = 1, . . . , n t

T

e’Af(fi, a)(t) = &

e"[iTi(s)+ fif(S)]d.S +

e"[&(s) + G/(S)]d.9

s0

0

=

&[e’Ci(T) - a,(O)] +e’Gi(t) e

5

Ci(O) + ddi(t) - pi

=

- S,(O)

e’;!(t).

Applying lemma 3.1 and conditions (f0) and (fl) we have for each t E J and i = 1, . . . , n eHi”)Af(fi, W)(t) = 1 'P""'[h,(S)O~(S) .i0

+ ef4(T) _ 1 T

1 s

eWT)

-fi(S, ci(S),[i)]pi(S)9 [+]qi(S),fif(S))] ~

_

1

s’ +

ihi

6)

e

ffi’“‘fi;(s)

+

effiW

[h,(s) eHi(“)i7f(s) + eHi’“‘(d/(s)

+

@I’(s)

-

cy(.s))]

d.s

0

- c;(s))] ds

0

1

=

e%(T)

_

1

T

t

eHJ(T)$i(T)

_

eHi(‘)fi:(t) - cl(O) -

fii(())

_

enics)c#) ds

I0

I

eHi’“)c/(s) ds

0 eWT)

=

eHg(‘)fif(t) +

eHi(T) _

- i&f(O)]

[a;(T)

T

1 eEi,(T)

1

_

1

f eHi(‘)ctY(s)

&

-

eHi’“‘q?(s)

&

5

eHi(‘)fii(t).

0 s0 Thus, Af(ti, W) 5 pi and Af(6, 6) I 6: for i = 1, . . . . n, whence A(fi, W) 5 0. The proof that I? I A(#, 0) is similar.

1140

S.

HEIKKILK and V. LAKSHMIKANTHAM

LEMMA3.3.Letfi:JXE”+‘-tE,i=

l,...,

n satisfy conditions (fO)-(f3). Denoting

Z = (x = (x1,x2) E C(J, E”) x C(J, E”) 1(ti, 3) I (x1,x2) 5 (6, i?‘)], thenforallx,Y,zEZ,i=

l,...,

nandj= and

A_i(x, z) I &(Y, z) Proof.

(3.2)

1,2 whenever x 5 y.

A{(z, y) I Aj(z, x)

(3.3)

The given hypotheses imply that for each t E J and i = 1, . . . , n T

e’A,‘(x, z)(t) = --&

0

[x!(s) + x&Z>] ds +

[yi’(s) + v%)l ds +

f [xi’(s) + x:(S)] d.s i0

s

1 fyi’@) + y?(s)1 ds = e’A:(y, z)(t),

and eHi@‘Af(x, z)(t) = 1 + eW(r) _ 1 07‘eH,(S)[hi(.S)xf(s)- fi(s* x!(s), [X’lpi(s), [Z’lq,(S)9 X?(S)>1dS i I

’ eHi’“‘[hi(S)yi2(S) - fi(s9 Yi’(S), [Y’lp,(S)9 [Z11q,@)9Y,26))1 dS j0

1 + eWT) _ 1

TeH,(r)[hi(s)yF(s) - fi(s,Yi’(s)9 [Y11pi(s)9[Z’lqi(s), Y?(s))l dS i0

ZZeHi”)Af(y, z)(t). Thus, the first inequalities of (3.3) hold. The latter inequalities can be proved similarly.

LEMMA3.4. Let fi: J x En+’ -+ E satisfy conditions Then equations (2.6)-(2.8) define a mapping A: Z x (Al) A(*, z) is increasing and A(z, *) is decreasing (A2) If (x&‘= 1and (Yk)T=, are sequences in Z, one then (A(xk , yk))F= 1 converges in Z.

(fO)-(f3), and let Z be defined by (3.2). Z --t Z which satisfies for each z E Z. being increasing and the other decreasing,

Proof. From the results of lemmas 3.2 and 3.3 it follows that the equations (2.6)-(2.8) define a mapping A: Z x Z + Z which satisfies condition (Al). To prove (A2), let (x&= 1 be an increasing and (Yk)z= 1 a decreasing sequence in Z. The sequence (A(xk, yk))T= 1 is by (Al) increasing in Z. In particular, the sequences (&(x,, yk)(t))T= 1 are increasing and order bounded sequences in E for all t E J, i = 1, . . . , n and j = 1,2, whence they converge, since the order cone K of E is regular. By applying lemma 3.1, definitions (2.7), (2.8) and hypothesis (f3) itiseasytoseethatforalltEJandi= l,...,n,

0 5 e’[&x,,

_Mf)

- &x,,

YdOl

5 eT[Af(xk, yd(T)

- A&L,

Y,)G?I,

and 0 5; eHi%4f(xkI yk)W -

Af
5 eHi’T’[&k,

Y~V)

- &(x,,

Y,K?I

Mixed quasimonotone

1141

systems

whenever m I k. Since K is also normal, there exists c > 0 such that for all t E J, i = 1, . . . , n andj = 1,2 II&x, 9Y/c)(f) - &,

9umw)ll

5 Cll&k,

Y/&n

- ~hl,

Y*>VIll

whenever m I k. Thus, the sequence (A(xk, yk))r= 1 converges uniformly on J, whence its limit function z is continuous. Since Z has the least and the greatest element, it is easy to see that z E Z. Similar reasoning shows that (A(xk, yk))F= 1 converges in Z also when (xk)F= 1 is decreasing and (JJ~):=, is increasing. 4. MAIN

RESULTS

As an application of theorem 2.1 in [lo], lemmas 2.1 and 3.4 we obtain the following existence theorem, which is the main result of this paper. THEOREM 4.1. Let E be an ordered Banach space with regular order cone fi: En+’ -+E, i= l,..., n, satisfy conditions (fO)-(f3), then the PBVS quasisolutions U, w satisfying (G, r?‘) I (w, w’) 5 (u, u’) I (C, 8’) such and 9, 9 E [w’, u’] whenever 6, i7 E [G, li] are coupled quasisolutions 9, it’ E [l?, 91.

K. If the functions (1 .I) has coupled that r?,,B E [w, v] of (1.1) such that

Proof. Let Z be defined by (3.2), and A : Z x Z --t Z by equations (2.6)-(2.8). From theorem 2.1 in [lo] and lemma 3.4 it follows that system (2.5) has a solution (x,y) such that (i) y z~ x, and ,i?,J E [y, x] for any other solution (i, j) of (2.5). Lemma 2.1 implies that the functions w = (wr , . . . . w,) and v = (v,, . . . . on), where q(t)

= x,!(t),

wi(t) = .Yi’(t)9

i= I,...,

n,t~J,

6-O

are coupled quasisolutions of the PBVS (1.1). Since ti 5 y I x 5 0, it follows from (a), (2.3), (2.4) and (3..1) that (G, G’) I (w, w’) 5 (u, u’) I (~7,C’). Assume now that G, 6 E [G, G]are coupled quasisolutions of (1 .I) such that 9, 9 E [G’, $1. By lemma 2.1 the functions ,? = (2, , . . . , ZJ and j = (Jr, . . . , J,), where cfi(t) = (17,‘(t), Z;(t))

= (C,(t), iif(t

tEJ,i=

l,...,n

@I

tEJ,i=

I,...,n

(c)

_ijiCt)= (Yf(t)7_F?(t)) = (ti,i(f)9 +lCt)),

are solutions of the operator equations (2.5). Obviously, _C,j E Z. This, together with (i), (b) and (c) implies that G, fi E [w, u] and G’, fi’ E [w’, u’], which completes the proof. The functions B = (d,, . .., i;,), G = (Gl, . . . . I?,,) E P are said to be coupled upper and lower quasisolutions of (1. l), if -G!(t)

5 J;:(t, ai(t)9 [dIpi

Ci(T) I fii(O),

[t;tlq,(t)tcl(t))

C;(r) 5 a;(o),

for a.e. t E J, i= 1, **.9 n,

(4.1)

and -$/(t)

2 fi(t, Gi(t)7 [Glp,(t), [6lq,(t)* +itt)) G+(T) 2 r&(O),

@j(T) 2 r?;(o),

for a.e. t E J, i= 19 a.*, n.

As an immediate consequence of theorem 4.1 we obtain the following proposition.

(4.2)

1142

S.

PROPOSITION 4.1. The results c/(t)

=

hi(t)ai,

HEIKKILA and V. LAKSHMIKANTHAM

of theorem

c:(t)

=

hi(t)bi

4.1 hold,

if in condition

with ai, bi E K, and if in condition

G;(T) - S;(O) 5 a,(1 - e-Hi(T) ) and G:(O) - G;(T) In particular, quasisolutions

(f0)

conditions (f0) and (fl) hold if 6 and of (1.1) such that (G, 6’) 5 (6, 5’).

5 bi(l

tit are

(fl)

- e-Hl(T)).

coupled

upper

and

lower

Proof. Routine calculations show that condition (fl) holds with the modifications given in the first assertion. Choosing cr = cy = 0 we obtain the second assertion. Inthecasewhenq,=Oforeachi= l,..., n, we shall rewrite the system (1) in the form -U/(t)

= fi(t, 24(t), U/(t))

uj(“) = ui(T),

for a.e. t E J,

i= 1 , --*, n.

uf(0) = u;(T),

(4.3)

We say that u E P is an upper solution (resp. a lower sohtion) of (4.3), if u satisfies (4.3) with all the equality signs replaced by 5 (resp. 2). If u and w are upper and lower solutions of (4.3) such that (w, w’) 5 (u, u’), the solutions u and ii of (4.3) are said to be extremal solutions of (4.3) between w and u, if (w, w’) 5 (u, u’) 5 (ii, ti’) I (u, u’), and if (u, u’) I (u, u’) I (U, U’) whenever u is a solution of (4.3) such that (w, w’) 5 (u, u’) I (u, u’). As a consequence of proposition 4.1 we obtain the following corollary. an ordered Banach space E with regular order cone and the functions n, assume that the system (4.3) has a lower solution G and an upper solution i, such that (G, 8’) cr (6, fi’), and that for each i = 1, . . ., n there is hi E L’(J, IR,) such that (x,Y) ++ fi(t, X, y) - hi(t)y is decreasing on [G(t), C(t)] x [G:(t), i?;(t)] for a.e. t E J. If condition (f2) also holds, then system (4.3) has extremal solutions between G and 6. COROLLARY4.1. Given

fi:JxE”+’

-+E, i= l,...,

Condition (f3) holds, in particular, if f is also decreasing in its last argument. Moreover, condition (f2) holds, if f satisfies Caratheodory conditions, whence we have corollary 4.2. COROLLARY 4.2. Let E be an ordered Banach space with regular order cone, and let J;;: Jx En+’ --t E, i = 1, . . . . n, be such that f;:( -, x, y) is strongly measurable for all x E E”, y E E, and that A(t, . , *) is continuous and decreasing for a.e. t E J. If the PBVS (4.3) has upper and lower solutions i?, 6, satisfying (G, 6’) I (6, i?‘), then (4.3) has extremal solutions between G and B. As a special case of corollary

4.1 we obtain

corollary

4.3.

COROLLARY4.3. Given an ordered Banach space E with regular order cone, gi : J x E” -+ E and pi E L’(J, R), assume that for each i = 1, . . . . n the function gi(t, *) is decreasing for a.e. t E J and that gi( *, u( *)) is strongly measurable on J for each u E C(J, E”). If the PBVS -Ul (t) + Pi(t)uf(r) Uj(o) = Ui(T)* has upper and lower solutions between G and d.

= giCt, U(t)) U;(o) = u/(T),

for a.e. t E J,

i= l,...,n

fi, G with (G, 6’) 5 (6, B’ ), then (4.4) has extremal

(4.4) solutions

Mixed quasimonotone

Proof.

x

It is easy to see that the functions fi: J A (t, x, Y) = gi (t, x) - Pi (QY,

1143

systems

-+ E, defined by

En”

tEJ,xEE”,yEE,

satisfy the hypotheses of corollary 4.1 with hi(t) = Ipi(t)l,

t E J.

4.1. If A = (Ai, . . . . A,) with Ais given by (2.6)-(2.8) in the proof of theorem 4.1, it follows from the proof of theorem 2.1 in [lo] that the solution (x, y) of (2.5) is obtained as x = inf(A(x,, y,) 1a E A) and y = sup(A(y,, x,) 1a E A), where (x,>,.* and (y&E* are the longest transfinite sequences, whose first elements are x,, = D,y0 = W, and which satisfy the following condition. If 0 < cy E A then x, = infi3,, A(x~, ys), y, = supO,.,A(yB, x,), and the inequalities A(x,, y,) 5 x,, y, I A(y,, x,) hold, at least one being strict. If fis are Caratheodory functions, then A is continuous. In this case x and y are the uniform limits of the successive approximations given by

Remark

xj+l

= A(xj9 Yj),

Yj+l

= A(Yj9

xj),

jE tN.

Moreover, if the functionsfi are continuous, then the differential equations in (2.1) and (2.2) hold for all t E J. If E is separable then condition (f2) holds for instance when eachA is a “standard function” in the sense defined in [l I], and in particular when the functions fi are Bore1 measurable. The hypothesis of corollary 4.3 on strong measurability of gi( *, u(a)) for each u E C(J, E”) also holds when E is separable and gi is standard. The method described above also applies to second order initial value problems, as well as to first order initial value and periodic boundary value problems involving discontinuous nonlinearities (cf. [9, 10,12-141). The regularity of the order cone K of E is essentially used in the proof of theorem 4.1. This holds if, for instance, E is finite-dimensional, or E is a real Hilbert space, and (x )y) z 0 for all x, y E K, or E is reflexive and K is normal. In particular, the nonnegative elements form a regular order cone in the LP-spaces of real-valued functions, defined on any measured space a, if 1 s p < 03. More generally, if K is a regular order cone in E, then the cone Lp(Q, K) of a.e. K-valued functions of LP(Q E) is regular in L*(Q, E). The nonnegative sequences form a regular order cone in lP-spaces with 1 I p < CO, and also in c, . The above examples of ordered Banach spaces with regular order cone imply that the results derived in this paper can be applied to finite and infinite systems of second order periodic boundary value problems, as well as the second order stochastic periodic boundary value systems. REFERENCES 1. CABADA A. 81 NIETO J. J., A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problem, J. mafh. Analysis Applic. (to appear). 2. LADDE G. S., LAKSHMIKANTHAMV. C VATSALAA. S., Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman Publishing, Boston (1985). 3. LAKSHMIKANTHAM V., Periodic boundary value problems of first and second order differential equations,

J. Appl. math Simulation 2(3), 131-138 (1989). 4. LAKSHMIKANTHAMV., NIETO J. J. & SUN Y., An existence result about periodic boundary value problems of second order differential equations, Applicable Analysis 40(l), l-10 (1991). 5. NIETO J. J., Nonlinear second order periodic boundary value problem, J. math. Analysis Applic. 130, 22-29

(1988).

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second order periodic boundary value problem with Caratheodory functions, Applicable Analysis 34(1 + 2), 111-128 (1989). 7. HILLE E., Methods in Classical and Functional Analysis. Addison-Wesley, Reading, Massachusetts (1972). 8. HILLE E. & PHILLIPS R. S., Functional Analysis and Semigroups. American Mathematical Society Coll. 6. NIETO J. J., Nonlinear

Publications, Vol. 31 (1957). 9. HEIKKIL.&S., On the Cauchy problem for ordinary differential equations in ordered Banach spaces, Report ISBN 9.51-42-2820-0, Mathematics, University of Oulu, 29 pp (1989). 10. HEIKKILK S., KUMPULAINEN M. & LAKSHMIKANTHAM V., On solvability of mixed monotone operator equations with applications to mixed quasimonotone differential systems involving discontinuities, J. Appl. math. Stochastic Analysis 5(l), 1-17 (1992). 11. SHRAGIN I. V., On the Caratheodory conditions, Russ. math. Survs 34(3), 220-221 (1979). 12. HEIKKILA;S., LAKSHMIKANTHAMV. & LEELA S., Applications of monotone techniques to differential equations with discontinuous right hand side, Diff. Integral Eqns l(3), 287-297 (1988). 13. HEIKKIL~~S., LAKSHMIKANTHAMV. & SUN Y., Fixed points results in ordered normed spaces with applications to abstract and differential equations, J. math. Analysis Applic 163(2), 422-437 (1992). 14. HEIKKILK S. & LEELA S., On the solvability of the second order initial value problems in Banach spaces, in Dynamic Systems and Applications, Vol. 1, pp. 141-170 (1992).