Fixed points of nonexpansive potential operators in Hilbert spaces
 Biagio Ricceri^{1}Email author
https://doi.org/10.1186/168718122012123
© Ricceri; licensee Springer 2012
Received: 4 April 2012
Accepted: 10 July 2012
Published: 24 July 2012
Abstract
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 $(X,\u3008\cdot ,\cdot \u3009)$ be a real Hilbert space and let $T:X\to X$ 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 $x\to {\int}_{0}^{1}\u3008T(sx),x\u3009\phantom{\rule{0.2em}{0ex}}ds$ has a unique global maximum towards which each maximizing sequence in S converges.
MSC:47H09, 47H10, 47J30, 47N10, 49K40, 90C31.
Keywords
1 Introduction
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.
for all $x,y\in X$, while it is nonexpansive if and only if ${\parallel L\parallel}_{\mathcal{L}(X)}\le 1$.
The following result subsumes very well the spirit of the ones that we will establish in Section 3:
Theorem 1.1 Let $(X,\u3008\cdot ,\cdot \u3009)$ be a real Hilbert space and let $T:X\to X$ 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 $x\to {\int}_{0}^{1}\u3008T(sx),x\u3009\phantom{\rule{0.2em}{0ex}}ds$ has a unique global maximum towards which each maximizing sequence in S converges.
2 Preliminaries
From now on, $(X,\u3008\cdot ,\cdot \u3009)$ will be a real Hilbert space.
for all $x\in X$.
for all $x\in X$, $\lambda \in \mathbf{R}$. For $\lambda =1$ we will simply use the symbol I instead of ${I}_{1}$.
The basic proposition which relates the fixed points of λT ($\lambda \le 1$) with the functional ${I}_{\lambda}$ is as follows.
Proposition 2.1 The functional ${I}_{\lambda}$ is strictly convex and coercive for $\lambda <1$, and convex for $\lambda =1$. Hence, for each $\lambda \in [1,1]$, the fixed points of λT agree with the global minima of the functional ${I}_{\lambda}$.
From this, it follows that the derivative of the functional ${I}_{\lambda}$ (that is the operator $x\to x\lambda T(x)$) is monotone and that it is uniformly monotone if $\lambda <1$. Now, the conclusion follows from classical results ([11], pp.247249). □
Another very useful proposition [7] is as follows.
Proposition 2.2 Let Y be a nonempty set, $f,g:Y\to \mathbf{R}$ two functions, and λ, μ two real numbers, with $\lambda <\mu $. Let ${\stackrel{\u02c6}{y}}_{\lambda}$ be a global minimum of the function $f+\lambda g$ and let ${\stackrel{\u02c6}{y}}_{\mu}$ be a global minimum of the function $f+\mu g$.

the restriction of f to C has a unique global minimum (resp. maximum), say $\stackrel{\u02c6}{x}$;

every sequence $\{{x}_{n}\}$ in C such that ${lim}_{n\to \mathrm{\infty}}f({x}_{n})={inf}_{C}f$ (resp. ${lim}_{n\to \mathrm{\infty}}f({x}_{n})={sup}_{C}f$), converges to $\stackrel{\u02c6}{x}$.
A set of the type $\{x\in S:f(x)\le r\}$ is said to be a sublevel set of f.
In [8], we established the following basic result:
and that, for each $\lambda \in \phantom{\rule{0.2em}{0ex}}]a,b[$, the functional $\mathrm{\Psi}+\lambda \mathrm{\Phi}$ has weakly compact sublevel sets and admits a unique global minimum in X.
Then, for each $r\in \phantom{\rule{0.2em}{0ex}}]\alpha (\mathrm{\Phi},\mathrm{\Psi},b),\beta (\mathrm{\Phi},\mathrm{\Psi},a)[$, the problem of minimizing Ψ over ${\mathrm{\Phi}}^{1}(r)$ is well posed with respect to the weak topology. More precisely, the unique global minimum of ${\mathrm{\Psi}}_{{\mathrm{\Phi}}^{1}(r)}$, say ${\stackrel{\u02c6}{x}}_{r}$, agrees with the unique global minimum of $\mathrm{\Psi}+\lambda \mathrm{\Phi}$ for some $\lambda \in \phantom{\rule{0.2em}{0ex}}]a,b[$. Moreover, the functions $r\to {\stackrel{\u02c6}{x}}_{r}$ and $r\to \mathrm{\Psi}({\stackrel{\u02c6}{x}}_{r})$ are continuous in $]\alpha (\mathrm{\Phi},\mathrm{\Psi},a),\beta (\mathrm{\Phi},\mathrm{\Psi},b)[$ with 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.
Moreover, let $A\subseteq \phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[$ be an open interval such that, for each $r\in A,J$ has no local maxima in ${B}_{r}$ and there exists a unique ${\stackrel{\u02c6}{x}}_{r}\in {S}_{r}$ satisfying $J({\stackrel{\u02c6}{x}}_{r})=\psi (r)$.
 (i)
the function ψ is ${C}^{1}$ and increasing in A;
 (ii)for each $r\in A$, one has$T({\stackrel{\u02c6}{x}}_{r})=2{\psi}^{\prime}(r){\stackrel{\u02c6}{x}}_{r}.$
3 Results
Our first result (inspired by [6]) shows the key role which a certain function $\phi :\phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[\phantom{\rule{0.2em}{0ex}}\to [0,+\mathrm{\infty}[$ plays in dealing with the fixed points of T.
If there is $r>0$ such that $\phi (r)<\frac{1}{2}$, then T has a fixed point which lies in ${B}_{r}$.
against (3.2). As a consequence, $\stackrel{\u02c6}{x}$ is a local minimum of the functional I, and so it is a fixed point of T, by Proposition 2.1 again.
Then, we could find a sequence of positive numbers $\{{r}_{n}\}$ converging to 0 such that $\phi ({r}_{n})<\frac{1}{2}$ for all $n\in \mathbf{N}$. But then, for each $n\in \mathbf{N}$, there would be a fixed point ${x}_{n}$ of T lying in ${B}_{{r}_{n}}$. Hence, $\{{x}_{n}\}$ would converge to 0 in X and so, by continuity, we would have $T(0)=0$.
and so the conclusion is obtained passing to the limit for λ tending to 1. □
Note the following corollary of Theorem 3.1.
The next two results come from Theorem 2.1. Clearly, $fix(T)$ (resp. $fix(T)$) will denote the set of all fixed points of T (resp. −T).
For each $\lambda \in \phantom{\rule{0.2em}{0ex}}]1,1[$, let ${\stackrel{\u02c6}{y}}_{\lambda}$ be the unique fixed point of the operator λT.
Then, the following assertions hold:
(${a}_{1}$) the function $\lambda \to g(\lambda ):=J({\stackrel{\u02c6}{y}}_{\lambda})$ is increasing in $]1,1[$ and its range is $]{\eta}_{1},{\theta}_{1}[$;
(${a}_{2}$) for each $r\in \phantom{\rule{0.2em}{0ex}}]{\eta}_{1},{\theta}_{1}[$, the point ${\stackrel{\u02c6}{x}}_{r}:={\stackrel{\u02c6}{y}}_{{g}^{1}(r)}$ is the unique point of minimal norm of ${J}^{1}(r)$ towards which every minimizing sequence in ${J}^{1}(r)$, for the norm, converges;
(${a}_{3}$) the function $r\to {\stackrel{\u02c6}{x}}_{r}$ is continuous in $]{\eta}_{1},{\theta}_{1}[$.
for all $r\in \phantom{\rule{0.2em}{0ex}}]{\eta}_{1},{\theta}_{1}[$. □
For each $\lambda >1$, let ${\stackrel{\u02c6}{u}}_{\lambda}$ be the unique fixed point of the operator $\frac{1}{\lambda}T$.
Then, the following assertions hold:
(${b}_{1}$) the function $\lambda \to h(\lambda ):={\parallel {\stackrel{\u02c6}{u}}_{\lambda}\parallel}^{2}$ is decreasing in $]1,+\mathrm{\infty}[$ and its range is $]0,{\theta}_{2}[$;
(${b}_{2}$) for each $r\in \phantom{\rule{0.2em}{0ex}}]0,{\theta}_{2}[$, the point ${\stackrel{\u02c6}{v}}_{r}:={\stackrel{\u02c6}{u}}_{{h}^{1}(r)}$ is the unique global maximum of ${J}_{{S}_{r}}$ towards which every maximizing sequence for ${J}_{{S}_{r}}$ converges;
(${b}_{3}$) the function $r\to {\stackrel{\u02c6}{v}}_{r}$ is continuous in $]0,{\theta}_{2}[$.
If, in addition, the functional J is sequentially weakly continuous and has no local maxima in ${B}_{{\theta}_{2}}$, then, with ψ defined as in Theorem 2.2, the following further assertions hold:
(${b}_{4}$) the function ψ is ${C}^{1}$, increasing and strictly concave in $]0,{\theta}_{2}[$;
for all $r\in \phantom{\rule{0.2em}{0ex}}]0,{\theta}_{2}[$;
for all $r\in \phantom{\rule{0.2em}{0ex}}]0,{\theta}_{2}[$.
Since $0\notin fix(T)$ and $fix(T)$ is closed, we have ${\theta}_{2}>0$. Of course, for each $\lambda >1$, the functional $x\to \frac{\lambda}{2}{\parallel x\parallel}^{2}J(x)$ is weakly lower semicontinuous, coercive and with a unique global minimum (that is ${\stackrel{\u02c6}{u}}_{\lambda}$), and so we can derive (${b}_{1}$), (${b}_{2}$), (${b}_{3}$) from Theorem 2.1, reasoning as in the proof of Theorem 3.2. Under the additional assumptions on J, (${b}_{4}$), (${b}_{5}$) follow directly from Theorem 2.2, taking $A=\phantom{\rule{0.2em}{0ex}}]0,{\theta}_{2}[$. Finally, (${b}_{6}$) is a consequence of (${b}_{5}$) and of the fact that $T({\stackrel{\u02c6}{v}}_{r})={h}^{1}(r){\stackrel{\u02c6}{v}}_{r}$. □
Remark 3.1 Of course, Theorem 1.1 is a byproduct of Theorem 3.3, as, if T has no fixed points, we have ${\theta}_{2}=+\mathrm{\infty}$. On the other hand, if, for some $r>0$, the problem of maximizing J over ${S}_{r}$ is not well posed, then T has a fixed point lying in ${\overline{B}}_{r}$. Indeed, from (${b}_{2}$) it follows that $r\ge {\theta}_{2}$. But, $fix(T)$ is a closed and convex set. So, it admits a point of minimal norm. By the above inequality, such a point lies in ${\overline{B}}_{r}$ and we are done.
Now, we want to present the form that Theorem 3.3 assumes when T is an affine operator.

L is compact if, for each bounded set $C\subset X$, the set $\overline{L(C)}$ is compact;

L is symmetric if$\u3008L(x),u\u3009=\u3008L(u),x\u3009$
for all $x,u\in X$.
Theorem 3.4 Let $L:X\to X$ be a symmetric continuous linear operator, with norm 1, and let $z\in X\setminus \{0\}$.
Then, the following assertions hold:
(${c}_{1}$) the function $\lambda \to k(\lambda ):={\parallel {w}_{\lambda}\parallel}^{2}$ is decreasing in $]1,+\mathrm{\infty}[$ and its range is $]0,\theta [$;
(${c}_{2}$) for each $r\in \phantom{\rule{0.2em}{0ex}}]0,\theta [$, the point ${\stackrel{\u02c6}{\omega}}_{r}:={\stackrel{\u02c6}{w}}_{{k}^{1}(r)}$ is the unique global maximum of ${H}_{{S}_{r}}$ towards which every maximizing sequence for ${H}_{{S}_{r}}$ converges;
(${c}_{3}$) the function $r\to {\stackrel{\u02c6}{\omega}}_{r}$ is continuous in $]0,\theta [$.
If, in addition, T is compact, then the following further assertions hold:
(${c}_{4}$) the function δ is ${C}^{1}$, increasing and strictly concave in $]0,\theta [$;
for all $r\in \phantom{\rule{0.2em}{0ex}}]0,\theta [$;
for all $r\in \phantom{\rule{0.2em}{0ex}}]0,\theta [$.
Before giving the proof of Theorem 3.4, we establish the following
Proposition 3.1 Let $L:X\to X$ be a symmetric continuous linear operator and let H be defined as in Theorem 3.4.
 (j)
$\tilde{x}$ is a local maximum of H.
(jj) $\tilde{x}$ is a global maximum of H.
(jjj) $L(\tilde{x})=z$ and ${sup}_{x\in X}\u3008L(x),x\u3009\le 0$.
Now, if (j) holds, then ${H}^{\prime}(\tilde{x})=0$ (that is $L(\tilde{x})=z$) and there is $r>0$ such that (3.4) holds for all $x\in {B}_{r}$. So, from (3.5), we have $\u3008L(x),x\u3009\le 0$ for all $x\in {B}_{r}$ and then, by linearity, for all $x\in X$, and this shows (jjj). Vice versa, if (jjj) holds, then (3.5) is satisfied for all $x\in X$ and so by (3.4), $\tilde{x}$ is a global maximum of H, and the proof is complete. □
In other words, H has no local maxima in ${B}_{\theta}$. At this point, (${c}_{4}$)(${c}_{6}$) come directly from (${b}_{4}$)(${b}_{6}$). □
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 $x\to \u3008L(x),x\u3009$ is sequentially weakly continuous. So, Theorem 3.4 actually holds under such a weaker condition.
Remark 3.3 A natural question is: if assertions (${c}_{1}$)(${c}_{6}$) hold, must the operator L be symmetric and the functional $x\to \u3008L(x),x\u3009$ sequentially weakly continuous?
Remark 3.4 Note that if L, besides being compact and symmetric, is also positive (i.e., ${inf}_{x\in X}\u3008L(x),x\u3009\ge 0$), then, by classical results, the operator $x\to L(x)x$ is not surjective, and so there are $z\in X$ for which the conclusion of Theorem 3.4 holds with $\theta =+\mathrm{\infty}$.
In the previous results, the essential assumption is that $T(0)\ne 0$. In the next (and last) result, to the contrary, we highlight a remarkable uniqueness property occurring when ${sup}_{X}J=0$ (and so $T(0)=0$). Actually, in such a case, 0 is the unique fixed point of λT for each $\lambda \in \phantom{\rule{0.2em}{0ex}}]0,3[$.
We have:
We first prove
and the conclusion clearly follows. □
and so (3.6) follows now from Proposition 3.2.
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.
Declarations
Acknowledgement
Dedicated to Professor Anthony ToMing Lau, with esteem and friendship.
Authors’ Affiliations
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