ON NONLINEAR BOUNDARY VALUE PROBLEMS WITH DEVIATING ARGUMENTS AND DISCONTINUOUS RIGHT ttAND SIDE

In this paper we shall study the existence of the extremal 
solutions of a nonlinear boundary value problem of a second order 
differential equation with general Dirichlet/Neumann form boundary 
conditions. The right hand side of the differential equation is assumed to 
contain a deviating argument, and it is allowed to possess discontinuities 
in all the variables. The proof is based on a generalized iteration 
method.

A function x C(J) is said to be a lower solution of the BVP (1.1)-(1.2) if its first derivative exists and is absolutely continuous on I, and if Lx(t) < f(t, z(t), x((t)) a.e. on I, and Bjx(t) < aj(t), C I j, j = 0, 1.

If
In the case when f is zero function, the problem (1.1)-(1.2) has a unique solution which we denote by z.In the special case when f is a function of alone, it can be shown that the problem (1.1)-(1.2) has at most one solution in the class AC(I).The existence of the solution is guaranteed by the above regularity and boundary conditions, and can be extended via boundary conditions to a unique solution of (1.1)-(1.2) in the sense defined above.
The existence and uniqueness of the solution to problem (1.1)-(1.2) is discussed in [1, 2] by using classical comparison and iteration methods, and assuming that f is a continuous function in all its three arguments.Recently, the existence of extremal solutions to the BVP (1.1)-(1.2) is studied in [3] in the special case without deviating arguments, but allowing discontinuous nonlinearity for f, by using a generalized iteration method.In the present paper we shall study the existence of the extremal solutions of the general problem (1.1)-(1.2) between given lower and upper solutions via a generalized iteration method developed in [3,4], and a comparison method.The function f is required to be continuous only at the points (tj, x,y), j = O, 1.

PRELIMINARIES
Let C + (J) denote the space of all nonnegative-valued functions of C(J).We assume that the space C(J) is endowed with the norm 1[ [[ 0 and with the partial ordering < defined by II x II 0 max x(t) and x < y if and only if x(t) < y(t) for all t E J. (2.1) We shall impose the following assumptions on the function f: I x 2---[R: (f0) The BVP (1.1)-(1.2) has a lower solution a and an upper solution b such that a<b.
(2.4) Hence, the equation 1 Fy(t) = /k(t,s)g(s,y(s),y((s)))ds, t I, to where k(t,s) is the Green's function associated with the operator and the homogeneous boundary conditions ajFy(t:i 1)JbjFy(t:i O, j = O, 1, defines a function Fy C(I) for each y e [a, b].Since q + h q C + (R), it follows from the maximum principle that k(t, s) is nonnegative on I x I.
Employing the Green's function and superposition principle for ordinary linear differential equations, one can show that the BVP (2.2) has for each y [a, b] a unique solution which can be expressed in the form x(t) = Gy(t) = z(t) + Fy(t) b1 0, tIj, bj#O,j=0,1 t E I j, b1 = 0, j = 0, 1.The operator G, defined on [a,b] by (2.7), is nondecreasing and satisfies a < Ga and Gb < b.
Since a, is by (fl) a lower solution of the BVP (1.1)-(1.2), it is also a lower solution of the BVP (2.2) with y = a.By using this, the maximum principle (cf.[5]) and the definition of G, it can be shown that a g Ga.Similarly, since b, as an upper solution of (1.1)-(1.2), is also an upper solution of (2.2) with y b, it follows that Gb <_ b.
The proof of our main result will be based on the following lemma, which can be proved by a generalized iteration method (cf.[3,4]).If the hypotheses (i), (ii) and (f0)-(f4) hold, then the B VP (1.1)-(1.2) has the least and the greatest solution in the order interval [a,b]. Proof: Let a,b E C(J) be as in condition (f0).From Lemma 2.1 it follows that (2.7) defines a nondecreasing mapping G: [a, b]--[a, b].In view of (2.3), (2.5) and (2.7) we have for each y E [a, b] Gy(t)--Gy(T)I <_ z(t)-z(T) / w(t)w(T)l, y e [a,b], t," E J, where o(t) = 1 f k(t,s)M(s)ds, ::vp(aj(tbi to})( Fa(tj then (GYn(t))= o is for each t J a monotone sequence in [a(t),b(t)].Thus z(t)-lrnooGYn(t exist in [a(t),b(t)] for each t E J.
(3.3) From (3.1) it follows that the sequence (GYn) n=o is equicontinuous, whence the convergence in (3.3) is uniform, in particular, the limit functions z of (Gy,)= o belongs to C(J).
The above proof shows that the hypotheses of Lemma 2.2 hold when X = C(J), endowed with the norm and the partial ordering defined in (2.1), whence G has the least fixed point z. and the greatest fixed point :*.In view of the definition of G, both these fixed points are solutions of the BVP (2.2) with y = z.This and (2.3) imply that z. and z* are solutions of the BVP (1.1) 2), then it satisfies the BVP (2.2) with y = z, whence z is a fixed point of G. Since z. and z" are the least and the greatest fixed points of G, then z. _< z < z*.Thus, z. and z* are the least and the greatest solutions of the BVP (1.1)-(1.2) in [a,b].
The existence of the least and the greatest of all the solutions of the BVP (1.1)-(1.2) is ensured if the condition (f0) is replaced by (f6) f(t,z,Y) <-Pl(t) x + P2(t) Y for all t E I\Z and for all x,y E R, where Z is a null set in I, Pi E LI+ (I), i= 1,2, and r(Q)< 1, where r(Q) is the spectral radius of the operator Q: C(J)--,C(J), defined by It is easy to see that (3.6) defines a bounded, linear and nondecreasing operator Q: C(d)--,C(J).Thus the norm of Q has the following representations: e where e denotes the constant function e(t)--_ 1.Each iterate Qn of Q is also bounded, linear and nondecreasing, whence the spectral radius of Q can be given by r(Q) = lira (maxlQne(t)) = inf rata IQne(t)).
n.--*oo" E n N In particular, r(Q) < 1 if and only if there exists n E N such that Qne(t) < 1 for each t E J. Lw(t) = pl(t)w(t) + pz(t))w(o(t)) a.e. on I, BiT(t) = cj(t) l, t e I,/, j = 0, 1. (3.7)If conditions (fl)-(f3) hold with these a, b, then the BVP (1.1)-(1.2) has the least and the greatest solutions, and they belong to the order interval[-w, w]. Proof: Denoting by y the solution of (3.4) with H 0, the condition r(Q)< 1 ensures that the Neumann series n0 converges in C+ (J) uniformly, and that the sum function w is the solution of the operator equation w = y + Qw. (3.9) From the choice of y and from the definition of Q it follows that w is the solution of the BVP (3.7).Thus condition (fh) holds with H(t,u,v)= pl(t)u + p2(t)v, which implies by Proposition 3.1 that (f0) holds with a=-w and b-w.Hence, if (fl)-(f3) hold, then the BVP (1.1)-(1.2) has by Proposition 3.1 the extremal solutions in [--w, w].
Moreover, g(z) < 2lz for each z E R..

Lemma 2. 2 :
Let [a,b] be a nonempty order interval in an ordered Banach space X, and G: [a, b]--[a, b] a nondecreasing mapping.If each monotone sequence inGin, b]   converges, then G has the least fixed point x. and the greatest fixed point x*.Moreover, x, = min{y [a, bl Gy <_ y}, max{vy [a,b] y <_ Gy}.(2.s)3.EXISTENCE OF THE EXTREMAL SOLUTIONSWe are now ready to prove the following result concerning the existence of the extremal solutions of the BVP (1.1)-(1.2) in [a,b].

Proposition 3 . 2 :
If f: I R2R satisfies condition (f6), then condition (f0) holds for a w, b = w, where w 6' + (J) is the solution o the B VP