Fixed points of nonexpansive potential operators in Hilbert spaces

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:

Let be a real Hilbert space and let be a nonexpansive potential operator.

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.

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 [11] 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

for all , while it is nonexpansive 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.

From now on, will be a real Hilbert space.

For each , we put

and

will be a nonexpansive operator, i.e.,

for all . We also assume that there exists a Gâteaux differentiable functional , with , such that , where is the Gâteaux derivative of J. This amounts to say that

for all . It can easily be checked that

for all .

We also put

for all , . For we will simply use the symbol I instead of .

The basic proposition which relates the fixed points of λT () with the functional is as follows.

Proposition 2.1The functionalis strictly convex and coercive for, and convex for. Hence, for each, the fixed points ofλTagree with the global minima of the functional.

Proof Let . For each , we have

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 ([11], pp.247-249). □

Another very useful proposition [7] is as follows.

Proposition 2.2LetYbe a nonempty set, two functions, andλ, μtwo real numbers, with. Letbe a global minimum of the functionand letbe a global minimum of the function.

Then, one has

If eitheroris strict and, then

Let S be a topological space. As usual, given a function and a set , we say that the problem of minimizing (resp. maximizing) f over C is well posed if the following two conditions hold:

– the restriction of f to C has a unique global minimum (resp. maximum), say ;

– every sequence in C such that (resp. ), converges to .

A set of the type is said to be a sub-level set of f.

Given two functionals , for each , we denote by either the set of all global minima of or the empty set according to whether or . We adopt the conventions , . We also set

Note that, by Proposition 2.2, if , one has

In [8], we established the following basic result:

Theorem 2.1Letand let, with. Assume that

and that, for each, the functionalhas weakly compact sub-level sets and admits a unique global minimum inX.

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 [10] that we will apply jointly with Theorem 2.1 in the next section.

Theorem 2.2Assume thatJis sequentially weakly continuous. For each, set

Moreover, letbe an open interval such that, for eachhas no local maxima inand there exists a uniquesatisfying.

Then, the following conclusions hold:

(i) the functionψisand increasing inA;

(ii) for each, one has

Our first result (inspired by [6]) shows the key role which a certain function plays in dealing with the fixed points of T.

Theorem 3.1For each, put

If there issuch that, thenThas a fixed point which lies in.

If, then one has

In any case, one has

Proof Let be such that . So, there exists , such that

By Proposition 2.1, I is weakly lower semicontinuous, and so there exists such that

for all . We claim that . Actually, if , then, by (3.1), taking into account that , we would have

against (3.2). As a consequence, is a local minimum of the functional I, and so it is a fixed point of T, by Proposition 2.1 again.

Now, assume that . Arguing by contradiction, suppose that

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 .

Now, to prove the third assertion, assume that T has a fixed point, say . Then, by Proposition 2.1, we have

for all . Fix . From (3.3), we then obtain

and so

This clearly implies that

Fix . So, λT has a (unique) fixed point. By the previous remark, we clearly infer 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

The next two results come from Theorem 2.1. Clearly, (resp. ) will denote the set of all fixed points of T (resp. −T).

Theorem 3.2Assume that. Set

For each, letbe the unique fixed point of the operatorλT.

Then, the following assertions hold:

() the functionis increasing inand its range is;

() for each, the pointis the unique point of minimal norm oftowards which every minimizing sequence in, for the norm, converges;

() the functionis continuous in.

Proof We apply Theorem 2.1 taking

and

for all . With these choices, we have

and

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

for all . □

Theorem 3.3Assume that. Set

For each, letbe the unique fixed point of the operator.

Then, the following assertions hold:

() the functionis decreasing inand its range is;

() for each, the pointis the unique global maximum oftowards which every maximizing sequence forconverges;

() the functionis continuous in.

If, in addition, the functionalJis sequentially weakly continuous and has no local maxima in, then, withψdefined as in Theorem 2.2, the following further assertions hold:

() the functionψis, increasing and strictly concave in;

() one has

for all;

() one has

for all.

Proof This time, we apply Theorem 2.1 taking

and

for all . With these choices, we have

and

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.

As usual, for a linear operator , we say that

– L is compact if, for each bounded set , the set is compact;

– L is symmetric if

for all .

Theorem 3.4Letbe a symmetric continuous linear operator, with norm 1, and let.

For each, letbe the unique fixed point of the operator. Moreover, set

and, for each,

where

Then, the following assertions hold:

() the functionis decreasing inand its range is;

() for each, the pointis the unique global maximum oftowards which every maximizing sequence forconverges;

() the functionis continuous in.

If, in addition, Tis compact, then the following further assertions hold:

() the functionδis, increasing and strictly concave in;

() one has

for all;

() one has

for all.

Before giving the proof of Theorem 3.4, we establish the following

Proposition 3.1Letbe a symmetric continuous linear operator and letHbe defined as in Theorem 3.4.

Then, for, the following are equivalent:

(j) is a local maximum ofH.

(jj) is a global maximum ofH.

(jjj) and.

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 ([11], 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 ([11], 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

In other words, H has no local maxima in . At this point, ()-() come directly from ()-(). □

Some remarks on Theorem 3.4 are now in order.

Remark 3.2 Note that the compactness of L serves only to guarantee that the functional is sequentially weakly continuous. So, Theorem 3.4 actually holds under such a weaker condition.

Remark 3.3 A natural question is: if assertions ()-() hold, must the operator L be symmetric and the functional sequentially weakly continuous?

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 .

More precisely, for each real Hilbert space , we denote by the class of all nonexpansive potential operators such that

Set

We have:

Theorem 3.5For any real Hilbert space, with, one has

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

Clearly, . Let . If , we have

If , we have

So, in particular, we get

which contradicts the fact . From what we have just proven, it clearly follows that

Now, fix any . Continue to consider the function χ defined above (for a fixed ). Clearly, the function belongs to and

Of course, from this we infer that

and the conclusion clearly follows. □

Proof of Theorem 3.5 First, let us prove that

To this end, fix any and any such that

for some . Fix also , with , and consider the operator defined by

for all . Clearly, . Finally, set

Of course, . Since

we also have

and so

From this, it clearly follows that

and so (3.6) follows now from Proposition 3.2.

Now, let us prove that

To this end, fix , and such that

Then, consider the function defined by

for all . Clearly, the primitive of Q vanishing at 0 is non-positive in R. Moreover, for each , we have

This shows that Q is nonexpansive, and so . Now, from (3.8), we get

that is

From this, we infer that

So (3.7) follows from Proposition 3.2, and the proof is complete. □

For specific consequences of Theorem 3.5 concerning nonlinear elliptic equations, we refer to the very interesting papers [2] and [3] where a problem asked in [9] was solved.

The author declares that he has no competing interest.

Dedicated to Professor Anthony To-Ming Lau, with esteem and friendship.

1. Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA. 54, 1041–1044 (1965). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

2. Fan, XL: A remark on Ricceri’s conjecture for a class of nonlinear eigenvalue problems. J. Math. Anal. Appl.. 349, 436–442 (2009). Publisher Full Text

3. Fan, XL: On Ricceri’s conjecture for a class of nonlinear eigenvalue problems. Appl. Math. Lett.. 22, 1386–1389 (2009). Publisher Full Text

4. Göhde, D: Zum Prinzip der kontraktiven Abbildung. Math. Nachr.. 30, 251–258 (1965). Publisher Full Text

5. Kirk, WA: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon.. 72, 1004–1006 (1965). Publisher Full Text

6. Ricceri, B: A general variational principle and some of its applications. J. Comput. Appl. Math.. 113, 401–410 (2000). Publisher Full Text

7. Ricceri, B: Uniqueness properties of functionals with Lipschitzian derivative. Port. Math.. 63, 393–400 (2006)

8. Ricceri, B: Well-posedness of constrained minimization problems via saddle-points. J. Glob. Optim.. 40, 389–397 (2008). Publisher Full Text

9. Ricceri, B: A remark on a class of nonlinear eigenvalue problems. Nonlinear Anal.. 69, 2964–2967 (2008). Publisher Full Text

10. Schechter, M, Tintarev, K: Spherical maxima in Hilbert space and semilinear elliptic eigenvalue problems. Differ. Integral Equ.. 3, 889–899 (1990)

11. Zeidler, E: Nonlinear Functional Analysis and Its Applications, Springer, Berlin (1985)