## Method of Quasi-Upper and Lower Solutions in Abstract Cones

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1981-05##### Author

Lakshmikantham, V.

Vatsala, A. S.

Leela, S.

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Let E be a real Banach space with ||•|| and let E* denote the dual of E. Let K C E be a cone, that is, a closed convex subset such
that ^K C K for every ^ ≥ 0 and K ^ {-K} = {O}, By means of K a partial order ≤ is defined as v ≤ u if u - v E K. We let K* =[φ E E*:
φ(u) ≥ 0 for all u E K].
A cone K is said to be normal if there exists a real number N > 0 such that 0 ≤ v ≤ u implies ||v|| ≤ N||u|| where N is independent
of u,v. We shall always assume in this paper that K is a normal cone.
Let a denote the Kuratowski's measure of noncompactness, the properties of which may be found in [2,4].
For any [see pdf for notation] such that [see pdf for notation] on I where I = [0,T], we define the conical segment [see pdf for
notation] and the set [see pdf for notation]
Let us consider the IVP (1.1)[see pdf for notation] where [see pdf for notation]. Suppose that [see pdf for notation] and (1.2)
[see pdf for notation] on I. Then [see pdf for notation] are called lower and upper solutions of (1.1) defined in a natural way.
A function f is said to be quasimonotone relative to K if v ≤ u and φ(v-u) = 0, φ E K* implies φ(f(t,v)) ≤ φ(f(t,u)). If E = Rn
and [see pdf for notation], the standard cone, the inequalities induced by K are componentwise and the quasimonotonicity of f is
reduced to v ≤ u and [see pdf for notation] implies [see pdf for notation].
In this special case one can prove the following result.
Theorem A. [see pdf for notation]. Suppose that [see pdf for notation] satisfy (1.2) with [see pdf for notation] on I and that f
is quasimonotone. Then there exists a solution u(t) of (1.1) on I such that [see pdf for notation] on I provided [see pdf for notation].
If f is not known to be quasimonotone, we need to strengthen lower and upper solutions as follows: for each i,1≤i≤n, (1.3)
[see pdf for notation]. We then have the following classical result of Müller.
Theorem B. [see pdf for notation]. Suppose that [see pdf for notation] satisfy (1.3) Then the conclusion of Theorem A holds. See
for the details of proofs [1,7].
We observe that the proofs of Theorems A and B depend crucially on the modification of f, that f where f(t,u) = f(t,p(t,u)) and [see pdf for notation]
for each i. Clearly this modification makes sense only when [see pdf for notation].
If K is an arbitrary cone, the inequalities (1.2) need no change. On the other hand, the inequalities (1.3) can be formulated in terms
of functionals from K*, namely, for φEK* (1.4) [see pdf for notation]. This version of condition (1.3) allows us to consider cones K
other than the standard cone. The question is whether Theorems A and B hold even when K is an arbitrary cone. Theorem B may not be valid
even in Rn as was shown by Volkmann [8]. Consider the example in R3. Let [see pdf for notation]. Take [see pdf for notation] and
[see pdf for notation], otherwise. But the solution through u0 = (1,0,1) is u(t) = (1,t,1) which does not remain in the sector [see pdf for notation].
Note also that f is Lipschitzian. Thus it is evident that f being Lipschitzian is not sufficient to prove Theorem B in the set up
corresponding to (1.4).
Recently results corresponding to Theorems A and B are proved in [3.5]. In this paper, we consider a very general situation where f admits
a mixed quasimonotone property and the results obtained include and unify several special cases.