ON SOLVABILITY OF MIXED MONOTONE OPEI : TOI : t EQUATIONS WITH APPLICATIONS TO MIXED QUASIMONOTONE DIFFEINTIAL SYSTEMS INVOLVING DISCONTINUITIES

In this paper we shall first study solvability of mixed monotone systems of operator equations in an ordered normed space by using a generalized iteration method. The obtained results are then applied to prove existence of coupled extremal quasisolutions of the systems of first and second order mixed quasimonotone differential equations with discontinuous right hand sides. Most of the results deal with systems in a Banach space ordered by a regular order cone.


INTRODUCTION
In [6] existence of coupled minimal and maximal quasisolutions of the system u(t) = fi(t, ui(t),[u]pi(t),[U]qi(t)) t e [0, T], ui(O = Uol, i = 1,...,n (1) is studied by assuming that the functions fi are continuous, mixed quasimonotone and satisfy one-sided Lipschitz condition with respect to u i.The last assumption is replaced in [5] by g- monotonicity of the functions fi" In this paper we shall extend the above mentioned results to the case when the functions fi are allowed to be discontinuous in all of their variables.These extensions will be 1Peceived: July, 1991. Revised: September, 1991.element of Y. Thus, F has by Theorem 3.1 of [2] the least fixed point u = (v,w) in Y.
Moreover, u = sup{Fu a [a E A}, where (ua) a e A is the longest transfinite sequence, indexed by ordinals, which satisfies u 0 = a, and if 0 < c E A then ua = sup < aFu and u a < Fu a.
The above results imply by (a) and (b) that (v, w) is a solution of (2.1), and that v = sup{A(v, wa) a A} and w = inf{A(wa, va) la A}, where (va) A and (wa) a A are the longest transfinite sequences which satisfy (I) If 0 < a e A then v a = sup < aA(v, w#), w a = inf# < aA(w/, v#), and the inequalities v a < A(va, wa) A(wa, va) < w a hold, at least one of them being strict.
If (, @) is a solution of (2.1), it follows from (2.1) and (b) that (, ) is a fixed point of F in Y.Because (v,w) is the least one, then (v,w)< (V,@), i.e. v < and @_<w.In particular, we can choose V = w and @ = v, whence v _< w.Since (@, ) is also a solution of (2.1), it follows that V, @ e Iv, w], so that (i) holds.ttemark 2.1: If A in Theorem 2.1 is continuous, then also F is continuous, whence the sequence (ua) a e h is reduced to a finite or infinite sequence of ordinary iterations Fie, j e .In particular, v = lira vj and w = lira w:, where (vj)= o and (wj)=o are defined by vj + 1 = A (vj, wj), wj + = A(wj, vj), j e N.
If v = w = u, we say that u is a lower, an upper or an ordinary solution of (3.1), respectively.
In the following, we shall equip C(J,En) with the maximum norm and pointwise ordering, and Lfspaces with p-norms and a.e.pointwise orderings.
g-monotone and mixed quasimonotone real systems.
Consider first the case when E = R.Given fi:J x Rn--*R and Uoi R, i = 1,...,n, assume that (fl) (f2) (3.1) has coupled lower and upper quasisolutions Vo, w o such that v o <_ w o. fi( .,u(. ))is Lebesgue measurable on J and whenever u e [Vo, W o] is absolutely continuous.
and for a.e.t J.
(hi x converges by the Monotone Convergence Theorem to a function Thus, the sequence ,._j,j 1 h L 1 (J, i), and Defining ui:J--,R by li.mj,/ h(x)dx = / hi(x)dx, tJ.This and (c) imply that the sequence (Ai(vj, wj))j= converges uniformly on J to u i.Thus, denoting u = (u,..., Un) then A(vj, wj)--,u in X = C(J,Rn) with respect to the uniform norm.Moreover, it is ey to see that u Z, whence (A(vj, wj))= converges in Z.This convergence can be proved similarly also in the ce (vj)= is a decreeing and (wj)j = x an increing sequence in Z.Thus, A satisfies the condition (A2) of Theorem 2.1.Noting aho that v 0 is the let and w 0 is the greatest element of Z, it follows from Theorem 2.1 that (2.1) has a solution (v, w)such that (i)holds.
If , are coupled quisolutions of (3.1) in Ivo, Wo], then they belong to Z, whence the above equivalences imply that (, ) is a solution of (2.1) when the components of A are dnd by ().This and the result (i) of Theorem 2.1 imply that , q [v, w].
As a consequence of Theorem 3.1 we obtain, Proition 3.1: Given the funclions gi (R, R + and hi: J x n, i = 1,..., n, assume lhal for each i = 1,...,n, gi has positive essenlial infimum, that h is Borel measurable, that hi(t ) is increasing on R n for a.e.t J, nd that lhere is m x(J,R +) such that hi(t, u) <_ mi(t) for all u R n and for a.e.t e J.
'" "' Uoi 0 From (3.4), (a) and (b) we obtain (b) Voi(t <_ ui(t < Woi(t for all t e J. Thus, v 0 _< u _< w0, so that v is the least and w is the greatest of all the solutions of (3.5).
Next we shall give an extension of Theorem 1.4.1 of [6] to the case when f = (fl,'",fn) is a mapping from J x E n to En, where E is an ordered Bausch space, by allowing f also to possess discontinuities.Theorem 3.2: Let E be an ordered Banach space with regular order cone, and let fi:J En--.E and Uoi E, i-1,..., n satisfy the conditions (fl), (f3), (fS) fit',u(')) is strongly measurable whenever u:J--.E n is absolutely continuous and a.e.differentiable, and (f6) there is Pi Lx(J,R+ such that ui-,fi(t,u,...,ui,...,Un)+ Pi(t)ui is increasing on [Voi(t), Woi(t)] for all u = (Ul,... Un) e [v0(t), w0(t)] and for a.e.t e J.
Then the results of Theorem 3.1 hold.

Proof:
Denote Z = {z e Iv0' w0]lz is absolutely differentiable}.The given hypotheses imply that the equation continuous and a.e.Ai(v,w)(t) = fi(t, vi(t),[v]pi(t),[w]qi(t))-t-pi(t)vi(t), t e S (a) defines for each i = 1,...,n a mapping i:Z x Z--,L(J,E), which is increasing in the first argument and decreasing in the second one.Thus, the initial value problem u(t) + ti(t)ui(t) = i(v, w)(t) for a.e.t E J, ui(O) = Uoi has for each = 1,...,n and v,w E Z a unique absolutely continuous solution u = Ai(v,w) given by Ai(v w)(t) = exp( f Ii(x)dz)(Uoi + ezp( f li(s)ds)Ai(v, w)(x)dx), t E J. (c) 0 0 0 The so obtained functions Ai:Z x Z--.C(J,E) have absolutely continuous and a.e.differentiable values, and they possess the mixed monotonicity conditions stated for the functions i above.
The above results imply that A-(A,...,An) is a mapping from Z x Z to Z, and satisfies condition (A1) of Theorem 2.1.
is an increasing and (wj)= 1 a decreasing sequence in g.Assume now that (vj)j 1 The.sequences (i(vj, wj)(t))= 1, t e J are increasing and are contained in the order interval [Ai(Vo, Wo)(t),)q(Wo, Vo)(t)], whence they converge, because the order cone g of E is regular.Denote h(t) = ti, (v, o)(t), t e J.
Since K is also normal, there is a positive constant c such that I I a(,,o)(t)II <_ ( + )( II (,o, Oo)(t)II + I I (Oo, Vo)(t)II) for .. t e y.
Thus, hi6 LI(J,E) by the Dominated Convergence Theorem for Bochner Integrals (cf. [4]),  which implies by (d) and the normality of K that the convergence in (J') is uniform on J.
Thus, v and w are coupled quasisolutions of (3.1) and v o < v < w _< w 0.
If , are coupled quasisolutions of (3.1) in Iv0' w0], then they belong to Z, whence the above equivalences imply that (, ) is a solution of (2.1) when the components of A are defined by (c).Thus, the result (i) of Theorem 2.1 ensures that , e Iv, w].
Proposition 3.2: Given an ordered Banach space E with regular order cone, assume that the functions fi: J x EnE satisfy the hypotheses of Lemma 3.1, that fi( u( )) is strongly measurable whenever u: J---E n is absolutely continuous and a.e.differentiable, and that there is Ii Lx(J,R + ) such that u1'(t, u) + ui(t)u is increasing for a.e.t J and for each ., .., E the least and the i= 1,.. n.Then the system (3.9) has for each choice of uol, UOn greatest solution. Proof: Given u01,..., Uon E, it is easy to see that hypotheses of Theorem 3.2 hold with v0, w o defined by (3.7)-(3.8).Thus, the system (3.9) has the least solution v and the greatest solution w between v o and w o.
If u = (Ul,...,un) is any solution of (3.9), it follows from (3.6), (3.7) and (3.9) that r = u Voi satisfies for each = 1,..., n which implies that 1 r(t)ai(t)ri(t > 0 for a.e.t e J, ri(O) = O, Thus, v 0 < u. ri(t) >_ ri(O)exp( / a](x)dx) = 0 for each t e J. 0 Similarly it can be shown that u _< w0, whence all the solutions of (3.9) are between v 0 and w 0. Consequently, v is the least and w the greatest of all the solutions of the system (3.9).lmarlm 3.1: Condition (f2) holds, for instance when fi is a "standard function" in the sense defined in [8], and in particular, when fi is Borel measurable.If each h in Proposition 3.1 is standard, then the assumption on the Borel measurability of h is not needed.
Condition (f5) of Theorem 3.2 holds, if E is separable and each fi is a standard function (cf.IS]).
If in Theorems 3.1 and 3.2 the functions gi are continuous and each fi is a Carathodory function, it follows from lemark 2.1 that the asserted coupled quasisolutions v and w of (3.1) are obtained as the uniform limits of the successive approximations given by (2.2).Moreover, if the functions fi and Pi are continuous then v, w are coupled extremal quasisolutions of the system (1) given in Introduction.Similar special cases are obtained also for Propositions 3.1 and 3.2.Theorem 3 in [9] contains the special case of Theorem 3.2 where E = R and the functions fi are mixed monotone, i.e. each Pi is the zero function in the condition (f6).
If the equalities hold in (4.2) and in (4.3), we say that v, w are coupled quasisolutions of (4.1).In particular, if v = w = u, we say that u is a lower, an upper or an ordinary solution of (4.1), respectively.
Our considerations are based on Theorem 2.1 and the following Lemma, which is a consequence of Example 2.1 and Lemma 6.1 of [3].If p LI(J,i and C LI(J,E then the IVP u"(t) + p(t)u'(t) = C(t) for a.e.t J, u(O) = Uo, u'(O) = u I (4.4) has for each choice of Uo, u 1 6 E the unique solution z 0 z 0 0 0 Moreover, V v, w are lower and upper soluions of (4.4) then v u w and v' S u' S w'.
Threm 4.1: Let E be an ordered Banach space with regular order cone K.If the functions fi:J x E n +IE satisfy the hypotheses (fa)-(fd), then the system (4.1) has coupled quasisolutions v, w satisfying (Vo, v) (v, v') (w, w') (To, w), such that V, [p, w] and ', ' Iv', w'] whenever V, [Vo, To] are coupled quasisolutions of (4.1) such t v', e [v, %1 Proof: Denote Z = {u 6 Iv0' w0] u' is absolutely continuous a.e.differentiable and u' e Ivy, w]}, and define a norm and a partial ordering in X = C(J,En) by I I u !1 = p{ I I (t)!1 + I I '(t)II It J} u < v if and only if u(t) < v(t) and u'(t) <_ v'(t) for all t e J.
The given hypotheses imply that the equation defines for each i= 1,...,n a mapping Ci:ZxZL(J,E), which is increasing in the first argument and decreasing in the second one.Thus, the initial value problem 7(t) + ,(t)(t) = c(, o)(t) fo .. t e , u(0) = Uo, u(0) = has by Lemma 4.1 for each i = 1,...,n and v,w Z a unique absolutely continuous solution, u = Ai(v, w), given by x Ai(v w)(t) = Uoi + ezp( f #i(Y)dy)ulidz + ezp( f #i(s)ds)Ci(v, w)(y)dydz.The so obtained functions Ai(v w):JE have absolutely continuous and a.e.differentiable first derivatives, and they possess the mixed monotonicity conditions stated for the functions C above.Moreover, from (4.2), (a) and (b) it follows when v = v 0 and w = w 0 that r = Voi -u is for each i = 1,..., n a lower solution of e'(t) + z,(t)e(t) = 0 fo a.. t J, (0) = e(0) = 0.
The above results imply that A-(A1,...,An) is a mapping from Z xZ to Z, and satisfies condition (A1) of Theorem 2.1.
Assume now that (vj)=x is an increasing and (wj)j=l a decreasing sequence in Z.The sequences (Ci(vj, wj)(t))% 1, t e J are increasing and are contained in the order interval [Ci(vo, Wo)(t),Ci(wo, v0)(t)] whence they converge, because the order cone g of E is regular.Denote hi(t) = li.mCi(vj, wj)(t), t E J.
o o These inequalities, (d) and the normality of K imply that the convergence in (e) is uniform on J. Thus, denoting t, = (t,1,...,n) then A(vj, wj)r, in C(J, En).Obviously, t, Z, whence (A(vj, wj))jo= converges in Z.This convergence in the case when (vj)j= is decreasing and (wj)j__ is increasing is proved similarly.Thus, A satisfies the condition (A2) of Theorem 2.1, whence (2.1) has a solution (v, w) such that (i) holds.
Proposition 4.2: Given an ordered Banach space E with regular order cone and f i: J En +I''*E, i = 1,...,n, assume that the hypotheses of Lemma 4.2 hold, that f(.,u(.),v(.)) is strongly measurable whenever u' and v are absolutely continuous and a.e.differentiable on J, and that there is i Lx(J,R+) such that (u,v)fi(t,u,v)+ti(t)v is increasing for a.e.t J and for each i= 1,...,n.Then for each choice of Uoi, uxi E, i= 1,...,n the system (4.8) has solutions v and w such that (v,v')< (u,u')< (w,w') for all solutions u of (4.8). Proof: Let Uoi ui fi E, i= 1,...,n be given.It is easy to show that the hypotheses of Theorem 4.1 hold with v0, w 0 given as in the proof of Lemma 4.2.If u = (ux,...,un) is any solution of (4.8), it follows from (4.6) that u is for each i= 1,...,n an upper solution of (4.7) when j = 1, and a lower solution when j = 2.This implies by Lemma 6.1 of [3] that (v0, v) _< (u, u') _< (w0, w)).This holds for each solution u of (4.8), so that the assertions follow from the results of Theorem 4.1.
Remark 4.1: If E is separable and each fi is standard then the condition (fb)   and the corresponding hypothesis of Proposition 4.2 hold (cf.[8]).These assumptions hold also without separability of E in the special cases when each fi is a Carath6odory function, or all the functions in question are continuous.In these special cases we obtain analogous consequences for systems (4.1), (4.8) and (2) as stated in Remarks 3.1 for first order systems.
The regularity of the order cone K of E is essentially used in the proofs of Theorems 3.2 and 4.1.This holds if, for instance, E is finite-dimensional, or E is a real Hilbert space, and (z[y)>_0 for all z, yE K, or E is reflexive and K is normal.In particular, the nonnegative elements form a regular cone in the LP-spaces of real-valued functions, defined on any measured space , if 1 _< p < oo.More generally, if K is a regular order cone in E, then the cone LP(f,K) of a.e.K-valued functions of LP(f,E) is regular in LP(f,E).The nonnegative sequences form a regular order cone in/P-spaces with 1 <_ p < oo, and also in c 0.
The above examples of ordered Banach spaces with regular order cone imply that the results of Theorems 3.2 and 4.1 can be applied to finite and infinite systems of first and second order initial value problems, as well as to finite systems of first and second order stochastic initial value problems. [1] [4] [6] I.V. Shragin, "On the Carath6odory conditions", Russian Math.Surveys 34, 3 (1979),   pp.220-221.Y. Sun, "A fLxed point theorem for mixed monotone operators with applications", J.
holds if and only if