Let H be an ordered real Hilbert space with an ordering given by a closed cone P and suppose that the gradient of a given functional has the expression . Obviously, the critical points of the functional Φ are the fixed points of the operator A, and vice versa.
We will show that, under additional assumptions on the operator A, Φ satisfies the Palais-Smale compactness condition on a closed convex set , which ensures the existence of a critical point of Φ, see [15, 16].
Every sequence , satisfying the conditions has a subsequence which converges strongly in M.
The next lemma, the mountain pass lemma [17, 18] on a closed convex subset of a Hilbert space H, is crucial in the proof of our first result.
Lemma 2.1 Assume that H is a Hilbert space, M is a closed convex subset of H, Φ is a functional defined on H, can be expressed in the form , and . Assume also that Φ satisfies the PS condition on M, Ω is an open subset of M, and there are two points , such that . Then
is a critical value of Φ and there is at least one critical point in M corresponding to this value, where .
Theorem 2.1 Let X, H be two ordered real Hilbert spaces. Suppose that satisfies the following hypotheses.
Φ satisfies the PS condition on H and its gradient admits the decomposition such that , where is quasi-additive on lattice, is a positive bounded linear operator satisfying:
for , ;
that is, φ is the normalized first eigenfunction.
There exist such that , where denotes the norm of .
and a positive number
Then A has a nontrivial fixed point.
Remark 2.1 From Theorem 3.1 below, we point out that conditions (i) and (iv) of Theorem 2.1 appear naturally in the applications for nonlinear differential equations and integral equations.
Proof of Theorem 2.1 It follows from condition (v) that
Thus, we have
By virtue of (2.1) and (2.2), we have
where and . Since , Lemma 1.1 yields that is a positive linear operator. Let and , then and . This shows . For , by we infer . On account of (2.3) and (2.4), we arrive at
It is easy to see that is a closed convex subset of H.
By the definition of gradient operator and , we have
Noticing , we claim that there is a mountain surrounding θ and , and with . Indeed, from (v), we have such that
Replacing x by in the above inequality, we have . Thus,
Similarly, we have
Thus by (i), for , we have
Let . Then by (iv) we get
Since , Lemma 1.1 yields that exists and
It follows from that . So we know that
Combining with (2.5), we have
Therefore , such that .
Next, we take , where is the normalized first eigenfunction of B, and is to be determined. Set
From (2.3), we have
So we have as . Therefore there exists satisfying , set , is as required. Hence Theorem 2.1 holds by Lemma 2.1. □