In this paper, we show the impact of certain general results by the author on the topic described in the title. Here is a sample:
Then, the following alternative holds: either T has a fixed point, or, for each sphere S centered at 0, the restriction to S of the functional has a unique global maximum towards which each maximizing sequence in S converges.
MSC: 47H09, 47H10, 47J30, 47N10, 49K40, 90C31.
Keywords:nonexpansive operator; potential operator; fixed point; well-posedness
There is no doubt that fixed point theory for nonexpansive mappings is one of the central topics in modern analysis. Actually, since [1,4,5], such a theory has had (and continues to have) a strong development, and several deep (often spectacular) results have been achieved within it in the settings such as abstract harmonic analysis (where the contributions of Professor Lau are fundamental) and the geometry of Banach spaces.
On the other hand, another very important class of operators is that composed of potential operators. That is to say, the operators that can be regarded as the Gâteaux derivative of a suitable functional. Actually, the variational methods to study linear and nonlinear equations are fully based on potential operators.
In the present paper, as the title says, we are interested in fixed point theory for the intersection of the two above classes of operators in the setting of Hilbert spaces. More precisely, we intend to show the impact of certain general results that the author has established in the last years on such a topic.
Referring to  for a thorough introduction to potential operators (with several examples related to them), we recall here a specific situation where one can easily appreciate the relationships between the two classes of operators we are dealing with. Namely, let be a real Hilbert space, a continuous linear operator and . Then, is a potential operator if and only if
The following result subsumes very well the spirit of the ones that we will establish in Section 3:
Theorem 1.1Letbe a real Hilbert space and letbe a nonexpansive potential operator. Then, the following alternative holds: eitherThas a fixed point, or, for each sphereScentered at 0, the restriction toSof the functionalhas a unique global maximum towards which each maximizing sequence inSconverges.
We also put
From this, it follows that the derivative of the functional (that is the operator ) is monotone and that it is uniformly monotone if . Now, the conclusion follows from classical results (, pp.247-249). □
Another very useful proposition  is as follows.
Then, one has
In , we established the following basic result:
Then, for each, the problem of minimizing Ψ overis well posed with respect to the weak topology. More precisely, the unique global minimum of, say, agrees with the unique global minimum offor some. Moreover, the functionsandare continuous inwith respect to the weak topology.
Finally, let us recall the result of M. Schechter and K. Tintarev  that we will apply jointly with Theorem 2.1 in the next section.
Then, the following conclusions hold:
Our first result (inspired by ) shows the key role which a certain function plays in dealing with the fixed points of T.
In any case, one has
Then, we could find a sequence of positive numbers converging to 0 such that for all . But then, for each , there would be a fixed point of T lying in . Hence, would converge to 0 in X and so, by continuity, we would have .
This clearly implies that
and so the conclusion is obtained passing to the limit for λ tending to 1. □
Note the following corollary of Theorem 3.1.
Corollary 3.1IfThas no fixed points, then
Then, the following assertions hold:
Proof We apply Theorem 2.1 taking
By Propositions 2.1 and 2.2, the function g is non-decreasing in and . Now, let be a non-degenerate interval. If g was constant in A, then, by Proposition 2.2 again, the function would be constant in A. Let be its unique value. Then, we would have
for all . This would imply that , and so , against the assumption. Consequently, g is increasing in . Since, for , the functional is weakly lower semicontinuous, coercive and with a unique global minimum (that is ), we are allowed to apply Theorem 2.1. Accordingly, for each , there exists , with , such that
for all , and each sequence in with , weakly converges to . Since X is a Hilbert space, this implies that strongly converges to . Likewise, we get the strong continuity in of the function from its weak continuity that is ensured by Theorem 2.1 too. Now, to get (), (), (), it is enough to observe that
Then, the following assertions hold:
Proof This time, we apply Theorem 2.1 taking
Since and is closed, we have . Of course, for each , the functional is weakly lower semicontinuous, coercive and with a unique global minimum (that is ), and so we can derive (), (), () from Theorem 2.1, reasoning as in the proof of Theorem 3.2. Under the additional assumptions on J, (), () follow directly from Theorem 2.2, taking . Finally, () is a consequence of () and of the fact that . □
Remark 3.1 Of course, Theorem 1.1 is a by-product of Theorem 3.3, as, if T has no fixed points, we have . On the other hand, if, for some , the problem of maximizing J over is not well posed, then T has a fixed point lying in . Indeed, from () it follows that . But, is a closed and convex set. So, it admits a point of minimal norm. By the above inequality, such a point lies in and we are done.
Now, we want to present the form that Theorem 3.3 assumes when T is an affine operator.
– L is symmetric if
Then, the following assertions hold:
If, in addition, Tis compact, then the following further assertions hold:
Before giving the proof of Theorem 3.4, we establish the following
Proof First, observe that the symmetry of L is equivalent to the fact that the functional H is Gâteaux differentiable with derivative given by
for all (, p.235). By the symmetry of L again, it is easy to check that, for each , the inequality
is equivalent to
Now, if (j) holds, then (that is ) and there is such that (3.4) holds for all . So, from (3.5), we have for all and then, by linearity, for all , and this shows (jjj). Vice versa, if (jjj) holds, then (3.5) is satisfied for all and so by (3.4), is a global maximum of H, and the proof is complete. □
Proof of Theorem 3.4 As we observed above, the symmetry of L is equivalent to the fact that L agrees with the derivative of H. So, since , we can apply Theorem 3.3 taking . In such a way, we derive ()-() directly from ()-(). Now, assume that L is also compact. Then, this implies that H is sequentially weakly continuous (, Corollary 41.9). Suppose that H has a local maximum, say . Then, by Proposition 3.1, is a global maximum of H. In particular, this implies that the functional is coercive and hence, by sequential weak lower semicontinuity, it has a global minimum. That is, , by Proposition 2.1. So, by Proposition 2.2, it clearly follows that
Some remarks on Theorem 3.4 are now in order.
Remark 3.4 Note that if L, besides being compact and symmetric, is also positive (i.e., ), then, by classical results, the operator is not surjective, and so there are for which the conclusion of Theorem 3.4 holds with .
In the previous results, the essential assumption is that . In the next (and last) result, to the contrary, we highlight a remarkable uniqueness property occurring when (and so ). Actually, in such a case, 0 is the unique fixed point of λT for each .
We first prove
Proposition 3.2One has
Proof Let . Fix . Let us prove that 0 is the only fixed point of λP. Arguing by contradiction, assume that is a non-zero fixed point of λP. It is not restrictive to assume that (otherwise, we would work with ). Consider now the function defined by
So, in particular, we get
Of course, from this we infer that
and the conclusion clearly follows. □
Proof of Theorem 3.5 First, let us prove that
we also have
From this, it clearly follows that
and so (3.6) follows now from Proposition 3.2.
Now, let us prove that
From this, we infer that
So (3.7) follows from Proposition 3.2, and the proof is complete. □
The author declares that he has no competing interest.
Dedicated to Professor Anthony To-Ming Lau, with esteem and friendship.
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