monotone iterative techniques in ordered banach spaces

NonlinearAnalysis, Theory, Methods &Applications, Vol. 30, No. 8, pp. 5179-5190.1997 Pmt. 2nd World Congress of Ncmhear AM@IS

8 1997 Elsevia Science Ltd

PII: SO362-546X(96)00224-6

Printed in Cheat B&in. All @IIS reserved 0362-546X/97 $17.00 + 0.00

MONOTONE ITERATIVE TECHNIQUES IN ORDERED BANACH SPACES

EDUARDO LIZ* Departamento de Matemhtica Aplicada, E.T.S.I. Telecomunicaci6n, Universidad de Vigo, Spain

Key words and phmaea: Partial order, minimal and maximal fixed points, boundary value problem, lower

and upper solutions, monotone iterative technique.

1. PRELIMINARIES

Let X be a real Banach space and let K be a subset of satisfies

i) K is closed, K # 0, K # (0).

ii) az+&yK,foreveryz,yEKanda,bERwitha

iii) If 5 E K and --z E K, then z = 0.

An order cone permits to define a partial order in X by

x<ywy-XEK

X. K is called an order cone if it

>Oandb>O.

Conversely, let X be a real Banach space with a partial order compatible with the algebraic operations in X, that is,

x 2 0 and X 2 0 implies Xz > 0

xi 5 yi and x2 L y2 implies q + x2 5 yi + y2

Then the positive cone of X is defined by

X+={xEX: 05x)

EXAMPLE 1.1. If X = R” with the partial order defined componentwise,

X+ = {(X*7 227.. .) 2,) 1 Xi 2 0 7 1 5 i 5 n}

DEFINITION 1.1. M c X is said order bounded if there exist u, v E X such that u < x _< V, for all x E M.

Next we recall some definitions about cones

DEFINITION 1.2. The cone K is said normal if there exists a positive constant c satisfying that 11~11 5 cllyll, for all x,y E X such that 0 5 x 5 y.

‘&search partially supported by DGICYT, project PB94-0610

5179

5180 Second World Congress of Nonlinear Analysts

DEFINITION 1.3. The cone K is said regular if every order bounded monotone increasing sequence is convergent.

DEFINITION 1.4. The cone K is said fully regular if every bounded (in norm) increasing sequence is convergent.

EXAMPLE 1.2. If X = C(I), I = [O,l], X+ = {U E C(I) : u(t) >_ 0, V t E I} is normal since

However, Xf is not regular since, for instance, the sequence {un),, defined by un(t) = 1 -t” is increasing and order bounded but does not converge in C(I).

EXAMPLE 1.3. If X = Lp(l), I = [a,b], X+ = {U E Lp(I) : u(t) 2 0 for a. e. t E I} is fully regular.

(6)

(Pz)

P3)

(fi)

Now we give some interesting properties about order cones.

An order interval

is a convex and closed set in X. Moreover, [u, V] is bounded if Xt is normal.

If X+ is normal and M c X is order bounded then M is bounded in norm.

If X+ is normal, an increasing sequence (2,) C X converges to Q, if and only if some subsequence converges to x0.

If X+ is fully regular, then it is also a normal and regular cone.

Increasing Operators

DEFINITION 1.5. An operator F : D(F) c X --+ X is said increasing if z 5 y implies

J’(z) I J’(Y).

DEFINITION 1.6. An increasing operator F is said relatively order compact if {F(z,)}, has a subsequence possessing limit whenever {z,), is increasing (or decreasing) in X.

If the limit of above exists in D(F), then F is called order compact. We shall enunciate here the following key theorem about the existence of fixed points of

increasing operators. (See [3, 5] for more details).

THEOREM 1.1. Let X be a Banach space with cone X+. Let [u, V] be an order interval and F : [u, w] - [u, v] an increasing continuous operator.

Then F has a minimal and a maximal fixed points if one of the following conditions is satisfied:

u) X+ is regular.

b) X+ is normal and F is relatively order compact.

Moreover, the extremal fixed points of F are obtained by

p = ilm F”(u) 7 = ,llw F”(v)

We also observe that {F”(u)} is increasing and {F”(w)} is decreasing.

Second World Congress of Nonlinear Analysts 5181

2. ABSTRACT MONOTONE METHOD

Our aim consists in describing an abstract scheme in order to obtain some results of existence and approximation of solutions for some different problems involving ordinary and functional differential equations. For it we shall consider an abstract equation

L(x) = N(x), 2 E X)

where X is a Banach space partially ordered with cone Xt,

(2.1)

L : D(L) c x --t x x Y and N: D(N) c x -+ x x Y

are two operators with D(L) f~ D(N) # 0, Y b ein another Banach space partially ordered. g In many applications L will be a linear operator, but we shall assume only the following

weaker condition:

L(x) - L(y) < L(x - y) , for every x,y E D(L). vfl)

We shall show an application in which L is not linear but it satisfies condition (Hi). We shall assume the operators L and N taking values in X x Y in order to include the

boundary conditions or other facts like impulse effects. In X x Y we shall consider the partial order defined by

(“1,Yl) I (227Y2) onXxYifandonlyifxr<x2onXandyi<yzonY.

We are in a position to introduce the concepts of lower and upper solutions for (2.1).

DEFINITION 2.1. We shall say that o E D(L) II D(N) is a lower solution of (2.1) if L(Q) L: N(cr) on X X Y.

DEFINITION 2.2. A function /3 E D(L) rl D(N) is an upper solution of (2.1) if L(p) 2 N(P) on X X Y.

One of the key assumptions to develop the monotone method is the fact that the operator L satisfy a maximum principle. In order to establish such a condition, we shall consider a linear positive operator J : X - X x Y, that is, x 2 0 on I implies J(x) 2 0 on X x Y. Thus, our second main hypothesis is the following:

(H2) There exists a positive operator J : X - X x Y and a real number X > 0 for which the operator

(L + AJ)-’ : X x Y --f X

exists and L + XJ is inverse positive in D(L), that is, (L + XJ)(x) 2 0 implies x > 0.

For the operator N we shall assume the following growth condition:

(H3) There exists m 2 0 such that

N(u) - N(v) 2 -m J(u - u),


whenever (Y < v 5 u 5 p.

Here cy and ,8 will be respectively a lower and an upper solution of (2.1) with cr 5 /?. Assume that [a, p] C D(N) and (Hz) is s.atisfied. Thus we can define the operator

F=(L+XJ)-‘o(N+X.J):[a,/3]-X.

For this operator we obtain the following result:

THEOREM 2.1. Let a and ,f3 be respectively a lower and an upper solution of (2.1) with (Y 5 /J’ and [a,P] c D(N). Assume that the conditions (H,),(Hz) and (Hs) are satisfied for some m 5 X.

Then F is increasing and F( [a, /!I]) c [a, /3].

Proof Let ~1~72 E [~,fi]. We shall show that

(i) F(Q~) E b,Pl. (ii) 71 5112 implies F(m) 5 F(q2).

To prove that (Y 5 F(ql), let v = F(Q) - a. Then, using (HI) and definition 2.1, we obtain:

[L + X4(v) 1 [(L + XJ) o F](Q) - [L + xJ](a) L [N + ~J](Q) - [N + xJ](a) = N(Q*) - N(a) + XJ(V1 - a) 2 (-m t X) J(V1 - (Y).

Since m 5 A, (Y 5 r)l and J is positive, we have that [L + U](v) 2 0 on X x Y and hence (Hz) implies that v 2 0 on X, that is, c~ < F(ql).

In an analogous way, we can prove that F(ql) 5 /?. Now, to prove (ii), set w = F(q2) - F(ql). Then

[L t V(w) 2 [(L t X.4 0 F](712) - [(L + W 0 J’](cJI)

= IN + V(pI2) - [N t Mh) 1 (A - m> 472 - n) L 0.

Condition (Hz) again implies that w > 0 and then F(v1) 5 F(r]2).

Observe that in the conditions of Theorem 2.1, z E [(Y, /3] is a solution of (2.1) if and only if it is a fixed point of F. Thus Theorem 2.1 permits us to assure the existence of the minimal and maximal solutions of (2.1) on [a,p] ‘f 1 one of the following conditions holds:

u) X+ is regular and F is continuous.

b) X+ is normal and F is continuous and relatively order compact.

We also note that F is continuous if, for instance, (L + XJ)-’ and N are continuous. Moreover, in the conditions of Theorem 2.1, one can see that if (L + XJ)-’ is relatively order compact, then so is F.

We have the following consequence of Theorems 1.1 and 2.1,

THEOREM 2.2. In the conditions of Theorem 2.1, assume that F is continuous and one of the following assumptions holds:


u) X+ is regular.


Then there exist monotone sequences {a,}, and {p,}, such that (~0 = o 5 cy, < 13, < /IO = p for every n E N which converge to the minimal solution p and the maximal solution y of (2.1) on [(r,/3j respectively, that is, if r is any solution of L(r) = N(s) with z E [c-1,0], then

r E hY1. Proof. The sequences {cr,}, and {,&}, are obtained by

a, = F((Y,,_~) = F"(o), n E N

P, = F(Pn-1) = F”(P), n E N .

Now we shall consider the situation when the lower solution is over the upper solution. Following the same arguments employed for the case N < ,B, one can easily prove the following results.

THEOREM 2.3. Let a and /3 be respectively a lower and an upper solution of (2.1) with (Y 2 ,B and [P,o] c D(N). Assume that (HI) holds and the following conditions are satisfied for some m 5 -X:

(H4) There exists X < 0 for which the operator

G = (L + XJ)-r : X x Y --, X

exists and L + XJ is inverse negative in D(L), that is, [L + XJ](x) 10 implies 2 _< 0.

(Hs) There exists m 2 0 such that

N(u) - N(v) 5 m J(u - v) ,

whenever p 5 v 5 u 5 cr.

Then the operator F = (L + XJ)-’ o (N + XJ) : [p, a] 4 X is increasing and F( [,B, a]) C

P, 4’

THEOREM 2.4. In the conditions of Theorem 2.3, assume that F is continuous and one of the following assumptions holds:

a) X+ is regular.


Then there exist monotone sequences {/3,} and {cr,,} such that PO = p 5 /?I,, < CY, 5 ~0 = a for every n E N which converge to the minimal solution and the maximal solution of (2.1) on [,B,a] respectively.


3. APPLICATIONS

Periodic Boundary Value Problem for Ordinary Differential Equations of n-th Order Consider the following boundary value problem of n-th order with n 2 2:

u(“)(t) = f(t,u(t)), for a. e. t E Z = [a, b]

&)(a) = u(‘)(b) , i = 0 1 (3.1) 1 ,“‘, n-l,

where f : I x R -+ R is a Carathkodory function, that is, f (s, z) is measurable for all 2 E R, f(t, 0) is continuous for a.e. t E I and there exists hR E L’(Z) such that

IfWl I h(t) f or a.e. t E I and for all zr E R such that \]zj\ L: R.

We shall consider the Banach space X = L’(Z), with cone

X+ = (2~ E L’(Z) : u(t) 2 0 for a. e. t E I}.

We recall that X+ is a fully regular cone and hence it is also a regular and normal cone. Next we introduce the sets of functions

D(L) = {U E IV’(Z) : u(‘)(a) = Ji)(b), i = O,,l,. . . ,n - 2}

and

D(N) = {u E L’(Z) : f(.,u(.)) E L’(Z)}

We note that L”(Z) C D(N) since f is a CarathQdory function. Now we define the operators L : D(L) - L’(Z) x R and N : D(N) ---+ L’(Z) x R by

L(U) = (~(~1, u(“-l)(a) - u (“-l)(b)) and N(u) = (f(., u(a)) , 0) respectively. Since f is a Carathkdory function, it follows that N is well-defined and continuous (see [l]).

We shall also consider the positive linear operator J : L’(Z) + L’(Z) x R defined by J(u) = (U) 0).

Problem (3.1) is equivalent to the abstract equation

L(u) = N(u), u E D(L).

The concepts of lower and upper solutions for (3.1) are defined by L(a) < N(o) and L(p) 2 N(P) respectively, that is, (Y E W”J(Z) is a lower solution of (3.1) if it satisfies

c&*)(t) 5 f(t,a(t)) , for a. e. t E Z

cx(i)(a) = afi)(b), i = 0 1 7 ,.“, n-2 (3.2) cx(‘+l)(a) _< ‘y(“-l)(b)

Analogously, an upper solution ,Z3 is defined by reversing the inequalities (3.2). In order to apply the results of section 2, we shall investigate when conditions (Hi)-(ZZ,)

hold for the problem (3.1).


Since L is linear, (HI) is trivially satisfied. We shall assume that there exist a lower solution Q and an upper solution ,B of (3.1) an we shall distinguish two cases. d

Let us observe that [qp] c L”(I) since X+ is normal and therefore [a, ,0] c D(N). Concerning to condition (Hz), Lemma 2.2 in [2] proves that for A E (0, A(n)), the operator

(L + XJ) is inverse positive in D(L), being

n ! A(n) = (j; 0) _ ayn rn _ l),-1 ’

where d, is the integer part of n/2. Moreover, (L + U)-’ exists and it is continuous. As a consequence,

F=(L+XJ)-‘o(N+XJ):[a,p]+X

is continuous since (L + XJ)-’ and N are continuous. Condition (Hs) may be reformulated as follows:

There exists m 2 0 such that

f(t, U) - f(t, V) 2 -m(u - V) , for a. e. t E I, a(t) 2 2) 5 21 i P(t) . (3.3)

Since X+ is regular and F is continuous, an application of Theorem 2.2 provides the monotone iterative method for (3.1):

THEOREM 3.1. Let (Y and /? be respectively a lower and an upper solution of (3.1) with Q 5 p. Assume that f is a CarathQdory function satisfying (3.3) for some m E (0, A(n)).

Then there exist monotone sequences {cr,,} and {a} such that a0 = o < on < pn 5 /?,-, = /3 for every n E N which converge uniformly to the extremal solutions of (3.1) on [o,,B].

In this case, Lemma 2.2 in [2] again proves that the operator (L+XJ) is inverse negative in D(L) for X E (-A(n),O). Th us, using the same reasonings employed in case one and applying Theorem 2.4, we obtain the following analogous to Theorem 3.1:

THEOREM 3.2. Let (Y and /? be respectively a lower and an upper solution of (3.1) with p 5 CK Assume that f is a Carathkodory function such that

f(t, u) - f(t, V) 5 m(u - v), for a. e. t E I, j?(t) 5 2) 5 u 5 a(t).

for some m E (O,A(n)). Then there exist monotone sequences {/?,,} and {cY~} such that cro = (Y < cr, < ,9,, <

PO = p for every n E N which converge uniformly to the extremal solutions of (3.1) on [/3, a].

REMARK 3.1 When f is continuous and X = C(I), one can obtain analogous results to The- orems 3.1 and 3.2 since X+ is a normal cone and it is not difficult to prove that the operator F is relatively order compact (in fact, F is a compact operator [12]).


Periodic Boundary Value Problem for First Order Impulsive Diflerential Equations Consider the following impulsive first order differential equation with periodic boundary

conditions:

u’(t) = f(t,u(t)) t E I, t # tk, k = 1,2,. . . ,p

u(t;) = Ik(u(tk)), k = I, 2,. . . ,p

u(0) = u(T) (3.4)

where I = [O,T],O = ta < ti < t2 < 1.. < t, < t,+i = T,f : I x R 4 R is a continuous function for t # tk and I k : R -+ R is a continuous function for each k = 1,2,. . . ,p.

The validity of the monotone method for impulsive first order differential equations has been studied in [6] for the periodic case and more general boundary conditions were considered in [7].

In this section we obtain a new maximum principle that permits to apply the results of section 2 to problem (3.4).

In order to get the operatorial form of (3.4), we shall introduce the following space of functions.

X={u:I+R: u is continuous for t # tk ,

there exist u(t:), u(ti) and n(ti) = u(tk) , k = 1,2,,. . . ,p}

X can be identified with the Banach space fi C[t t kr k+l and thus X is also a Banach space ] k=O

with the norm

Moreover, we can define a partial order in X by

u 5 n if and only if u(t) < v(t), t E I.

The positive cone X+ = {u E x : u(t) 2 0, t E I} is a closed normal cone in X. Now define

D(L) = {u E X : Ul(t&+l) E c’(tk,tk+l), k = 0,1,-,P}

A solution of (3.4) will be a function u E D(L) satisfying (3.4). Let Y = RP x R. In Y we shall consider the partial order defined componentwise. Now we introduce the operators L : D(L) -t X x Y and N : X + X x Y by

L(u) = (u’, MWL 7 40 - u(T))

and N(u) = (f(.,u(.)) , {Ik(+k)));=1, 0) .

We shall also consider the positive linear operator J : X --+ X x Y defined by J(u) =

(u, om:).


It is obvious that problem (3.4) is equivalent to the abstract equation L(U) = N(U), u E D(L). Hence the concepts of lower and upper solutions for (3.4) are defined by L(o) 5 N(o) and L(p) > N(P) respectively, that is, o E D(L) is a lower solution of (3.4) if it satisfies

a’(l) I f(t,a(t)) , t E I, t # tk, k = 1,2,. . . ,p

a(Q) L Q(b)), ic = 1,2,. . . ,p (3.5)

o(0) L o(T)

Analogously, an upper solution p is defined by reversing the inequalities (3.5). If (Y and p are respectively a lower and an upper solution of (3.4), we shall consider the

following hypothesis for the function f:

(A,) There exist lim,,,; f(t, z(t)), lim,,,- f(t, s(t)> and linq_+t; f(t, r(t)) = f(k Z(&)), for every k= l,..., p, whenever x E X ind Q 5 x 5 /3.

Obviously L is linear and hence (Hi) is true. Next we shall descry when conditions (Hz) and (Ha) hold for the problem (3.4).

In [8] we prove that condition (Hz) is satisfied for an arbitrary X > 0. On the other hand, let us observe that in the present case, condition (Ha) is equivalent

to the following two conditions:

(A,) There exists m > 0 such that

f(t, u) - f(t, V) 2 -m(u - u) ,for a(t) < 2,s 21 5 P(t), t E I.

(Aa) Ik is a nondecreasing function for every k = 1,2,. . . ,p.

Thus we have obtained the following consequence of Theorem 2.1

LEMMA 3.1. Let (Y and ,8 be respectively a lower and an upper solution of (3.4) with a < ,8. Assume that f(t, ) u is a continuous function for t # tk and condition (A,) is satisfied, Ik is a continuous function verifying (AZ) for each k = 1,2,. . . ,p, and there exists m > 0 such that hypothesis (AZ) holds.

Then the operator A = (L t XJ)-’ o (N t XJ) : [a, /!I] --+ X is increasing and A( [a, /?I) c b> PI.

Now, by using the open map theorem, we can assure that the operator (L $ XJ)-’ is continuous. Since IV is also continuous, we have easily that A is continuous.

Finally, in [8] it is proved that A is relatively order compact. Moreover, A is a compact operator.

As a consequence, we obtain the following corollary of Theorem 2.2

THEOREM 3.3. Assume that the conditions of lemma 3.1 are satisfied. Then there exist monotone sequences {cr,} and {a} such that cro = Q _< on 5 ,B, 2

/3o = ,0 for every n E N which converge piecewise uniformly to the minimal solution p and the maximal solution y of (3.4) on [o,p] respectively, that is, if u is any solution of (3.4) with

u E [o, PI, then u E [P, rl.


Periodic Periodic Boundary Value Problem for Diflerential Equations with Maxima Consider the following boundary value problem for a nonlinear differential equation with

maximum:

u’(t) = f (6 4% ,_y&“(“)) : t E P7Tl

u(o) = u(T) = u(t) - t E [-h,O] (3.6)

where h > 0 and j E C((O,T] x R2,R). It is worth mentioning that equations with maxima find wide application in the theory

of automatic regulation (see [ll]). Problem (3.6) has been studied in [13]. We shall show that our scheme is adequate for

it despite the equations with maxima generate functionals not having the property of linearity even if the equations are linear. Indeed, let 4 : C( [0, T]) --+ C( [0, T]) the operator defined by

(3.7)

First of all, observe that finding solutions of (3.6) in C([-h, T]) flC’([O, 2’1) is equivalent to find solutions in C’([O, T]) of the problem

u’(t) = f (4 u(t), [44(t)), t E [07 Tl 1 40) = u(T) (3.8) and to extend them to functions of C([-h, T]) n C’( [0, T]) by

u(t) = u(O), Vt E [-h,O] .

Thus we shall consider the Banach space X = C([O,T]) with X+ the cone of nonnegative functions.

For 6 > 0 consider the operator

L : D(L) = C’([O,T]) - C([O,T]) x R

L(u) = (4) + w4(*h 40) - u(T)). The operator N : C( [0, T]) --+ C( [0, T]) x R is defined by

N(u) = (f (.? 4.L kJ4(.)) + a4(.)7 0) * Thus, solving problem (3.6) is equivalent to find a function u E D(L) such that Lu = Nu. We shall also consider the positive linear operator J : C( [0, T]) -+ C( [0, 2’1) x R defined by J(u) = (u ) 0).

Now we shall investigate the necessary conditions to assure that hypotheses (HI) - (Ha) hold.

Observe that 0 is not a linear operator. However, it is not difficult to verify that

M4(t> - MyI L MC% - YNW > vt E P, Tl. In consequence, for any z,y E D(L), L(z) - L(y) 5 L(z - y) and hence (Hi) is satisfied.

The existence of the operator G = (L + Al)-’ and its continuous character are given by the following result (see [9]):


LEMMA 3.2. Let X and 6 be real numbers such that

0<6<A. (3.9)

Then, for any 0 E C([O, 2’1) and p E R, the problem

U’(t) + AU(t) + s[@](t) = a(t) : t E [O, T]

U(0) -u(T) = /4 (3.10)

has a unique solution u E C*([O, T]). M oreover, the operator G = (L + XJ)-’ is continuous.

Hence, (17,) is satisfied if the following maximum principle holds: There exists X > 0 such that for any u E C([O,T]),

U’(t) + AU(t) + b@](t) 2 0, t E [O, T] ; u(0) 2 u(T) * 11 >_ 0. (3.11)

But (3.11) is true whenever any of the following condition is satisfied (see [13]):

GTeXh < 1 - (3.12)

Now we shall introduce the concepts of lower and upper solutions for (3.6). A function cr E D(L) is said a lower solution for (3.6) if it satisfies

that is, if I(o) 5 N(o) on X x R. Analogously, an upper solution for (3.6) is a function p E D(L) such that L(p) > N(P)

on X x R. Concerning to the operator IV, we have that N is continuous since f E C([O, T] x R2, R).

Now, let Q and p be respectively a lower and an upper solution of (3.6). Then hypothesis (I&) may be reformulate in the form:

There exists m E R, m > 0, such that:

f(4 u(t), @4(t)) - f(4 v(t), [&l(t)) 2 -m(u(t) - v(t)) - W4(t) - [h](t)) , (3.13)

for all t E [0, T] and u, v E [Q, /3] with v 5 u.

Thus we have the following consequence of Theorem 2.1:

THEOREM 3.4. Let cy and p be respectively a lower and an upper solution of (3.6) with o < p. Assume that there exist m, X,S E R, 0 < m 5 A, such that the conditions (3.9), (3.12) and (3.13) hold.

Then the operator F = G o (N + XJ) : [a, p] --i C( [0, T]) is increasing and F( [cr, /I]) c b, PI.


Finally, to show that F : [a,/31 ---+ [a, /I] is relatively order compact, let {z,},e~ be an increasing or decreasing sequence on [Q, 01.

For 2, E [o, PI, yn = A 2, is the solution of the problem (3.10) with

a(t) = on(t) = f(& G(t), [44(t)) f h&(t) + @?hal(t) . (3.14)

Since f is continuous, 5, E [a,@] and yn E [cr,/3] for every n E N, we have that {Y~}~~N is bounded in C’( [O, 2’1) and h ence Ascoli-Arzela theorem assures that the set {yn : n E N} is relatively compact in C( [0, T]). This fact proves that A is relatively order compact.

As a consequence, we obtain the following corollary of theorem 2.2:

THEOREM 3.5. Assume that the conditions of theorem 3.4 are satisfied. Then there exist monotone sequences {on} and {/3,} such that (Y,-, = o < (Y, 5 ,& 2

PO = p for every n E N, which converge to the minimal and maximal solutions of (3.6) on [(Y, p] respectively.

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POPOV E. P., Theory of Nonlinear Systems of Automatic Regulation and Control. 2nd ed., Nauka, Moscow (1988).

SEDA V., NIETO J.J. & GERA M., Periodic Boundary Value Problems for Nonlinear Higher Order Ordinary Differential Equations. Appl. Math. Comput. 48, 71-82 (1992).

XU H.K. & LIZ E., Boundary Value Problems for Differential Equations with Maxima. Non!. Studies, (To appear).

REFERENCES

monotone iterative techniques in ordered banach spaces

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