1. Introduction
Let H be a real Hilbert space, with an inner product , induced norm , and identity operator I. The study of the existence and approximation of solutions to nonlinear equations is an important topic and an active field of research in nonlinear analysis. However, nonlinear equations, even with strong restrictive conditions imposed, may not have a solution. An important case is the question raised by L. Nirenberg.
Let
. A self-mapping
is said to be expansive (expanding) if
Nirenberg’s question states: “Is any continuous expansive mapping
such that
has nonempty interior, surjective?” [
1]. This question can be formulated as whether for every continuous expansive mapping
T and every
, does the equation
have a solution? In spite of the strong conditions in Nirenberg’s question, one may think that the answer is positive; however, recently, Ives and Preiss [
2] answered this question negatively. Indeed, they provided a counterexample in
, which gives a negative answer to Nirenberg’s problem even in general separable Hilbert spaces. This question had been already asked for more general spaces, such as Banach spaces, where Morel and Steinlein [
3] constructed a counterexample in
. In any case, before this negative answer, many attempts to solve this question ended up giving affirmative answers to Nirenberg’s question under additional conditions. Among them, we point out [
4], where the interior of the range of the expansive mapping is assumed to be unbounded. For more results, see [
5,
6,
7,
8].
From a variational point of view, one can find a correspondence between expansive mappings and nonexpansive operators. We will get back to this correspondence, but before going further. Let us review briefly some classical results on nonexpansive mappings and their variational analysis. A mapping
is nonexpansive if
where
D is a nonempty subset of
H. Nonexpansive mappings are generalizations of contractions (with a Lipschitz constant
); however, their behaviors can be extremely different. One of the basic problems for nonlinear mappings concerns the following:
Every solution to the above problem is called a fixed point of
T, and the set of all fixed points of
T is denoted by
. If
T is nonexpansive, then
is closed and convex. The most important properties of contractions are described by the celebrated Banach contraction principle:
Theorem 1 ([
9]).
Let , and let be a contraction. Then, (i) T has a unique fixed point, say p, and (ii) for each , . This theorem does not hold for nonexpansive mappings without any additional conditions. The following theorem, which extends the first part of Banach’s contraction principle, was independently proved in 1965 by Browder [
10], Kirk [
11] and Göhde [
12]. We state the theorem here in Hilbert space to stay in the framework of our paper; however, the theorem is proved in more general Banach spaces.
Theorem 2. Suppose that is a nonexpansive mapping, where D is a nonempty, closed and convex subset of H. Then, T has a fixed point, and the set of all fixed points of T, which may not be a singleton, is closed and convex.
The second part of Banach’s contraction principle does not hold for nonexpansive mappings either. Indeed, according to Banach’s contraction principle, all orbits of a contraction T converge to the unique fixed point of T, while orbits of a nonexpansive mapping may not converge at all. Baillon, in 1975, proved that the Cesaro means of the Picard iterates of any nonexpansive mapping T always converge weakly to a fixed point of T, provided that .
Theorem 3. Let D be a nonempty, closed, and convex subset of H, and T be a nonexpansive mapping from D into itself. If the set is nonempty, then for each , the Cesaro meansconverge weakly to some . For more details, we refer the reader to [
13] and the beautiful books by Goebel and Kirk [
14], and by Goebel and Reich [
15].
If
D is not convex, then
may be empty, and then Baillon’s proof is not applicable anymore. To avoid the convexity assumption on
D, Djafari Rouhani [
16,
17] introduced the notions of nonexpansive and almost-nonexpansive sequences and curves.
In this survey, after reviewing some backgrounds on nonexpnasive curves and related notions, we take an expansive-type variational approach to problems of the form
where
is a (possibly multivalued) nonlinear operator.
Section 3, briefly, provides some intuition and backgrounds on the celebrated steepest-descent method and its monotone generalizations. In
Section 4, we review some definitions and results on expansive curves. Applying the results in
Section 4,
Section 5 describes the asymptotic behavior of an expansive-type quasi-autonomous system. In
Section 6, we recall discrete versions of the definitions and propositions in
Section 4 and apply them to study the asymptotic behavior of an almost-nonexpansive sequence.
Section 7 studies the periodic behavior of the expansive sequence described in
Section 6.
Section 8 is devoted to the study of continuous- and discrete-time non-monotone expansive-type dynamics. As will be seen later, the system considered in
Section 5 is “strongly ill-posed”. In
Section 9, we introduce new well-posed expansive-type systems, which yield weak and strong convergence to zeros of any maximal monotone operator.
Notation 1. Let u be a curve in H, and .
- (i)
Convergence in weak and strong topologies are, respectively, denoted by → and ⇀.
- (ii)
denotes the closed convex hull of C.
- (iii)
denotes the set of all sequential weak limit points of u.
- (iv)
.
- (v)
The weighted average of u is .
2. Nonexpansive and Almost-Nonexpansive Curves
We recall the following definition from [
17]:
Definition 1. (i) The curve in H is nonexpansive if for all , we have .
(ii) is an almost-nonexpansive curve if for all , we have , where .
The following concept introduced in [
18] will play an important role:
Definition 2. Given a bounded curve in H, the asymptotic center c of is defined as follows: for every , let . Then, ϕ is a continuous and strictly convex function on H, satisfying as . Therefore, ϕ achieves its minimum on H at a unique point c called the asymptotic center of the curve .
To the best of our knowledge, Edelstein [
18] was the first one who applied the technique of an asymptotic center to fixed-point theory. Combining the notion of nonexpansive curves and the concept of an asymptotic center, Djafari Rouhani proved theorems regarding the asymptotic behavior of nonexpansive and almost-nonexpansive curves without assuming the existence of a fixed point.
Theorem 4 ([
17]).
Let be an almost-nonexpansive curve in H. Then, the following are equivalent:- (i)
.
- (ii)
.
- (iii)
converges weakly to .
Moreover, under these conditions, we have:
Browder and Petryshyn [
19] introduced the notion of asymptotically regular mappings. A mapping
is (weakly) asymptotically regular on
D if
They also showed that if
is nonexpansive, then for every
,
is asymptotically regular, and
. Djafari Rouhani extended the notion of asymptotically regular mappings to curves in
H:
Definition 3. (i) The curve in H is asymptotically regular if for all , as .
(ii) is a weakly asymptotically regular curve in H if as .
The following theorem provides sufficient conditions for the weak convergence of asymptotically regular almost-nonexpansive curves:
Theorem 5 ([
17]).
Let be a weakly asymptotically regular almost-nonexpansive curve in H. Then, the following are equivalent:- (i)
.
- (ii)
.
- (iii)
converges weakly to .
3. A Steepest-Descent-like Method
For a smooth function , the gradient operator shows the direction of steepest ascent of a particle traveling along the graph of , hence shows the direction of steepest descent. If we consider the curve as the position of a particle in time t, then the above discussion shows that if the velocity vector equals the value of at , then travels along the steepest-descent direction on the graph of . In this case, if has a minimum point, then it may happen that goes to a minimum point of . This leads to one of the most celebrated methods in optimization:
Let
be convex with a nonempty set of minimizers. Then, every solution trajectory to the following system
converges weakly to a minimizer of
. This method is called the steepest-descent method. A counterexample due to Baillon [
20,
21] shows that, in general, solutions to the above system may not be strongly convergent in
H; see also [
22] (Proposition 3.3). Generalizations of this method to nonsmooth and monotone cases were studied by several authors in the 1970s. If
is nonempty, Baillon and Brézis [
23,
24] proved the weak convergence of the mean of solutions to the following system:
where
A is a maximal monotone operator in
H and
is arbitrary. Bruck [
25] established the weak convergence of solutions to (
2) with an additional condition on
A, which is called demipositivity. Motivated by the approach of nonexpansive curves, Djafari Rouhani studied the convergence analysis of a quasi-autonomous version of (
2) without assuming
to be nonempty.
Theorem 6 ([
17]).
If u is a weak solution (for the notion of weak and strong solutions, see [26]) of the systemon every interval , and satisfies , and if for some , then converges weakly to the asymptotic center of the curve . The following theorems, respectively, study the weak and strong convergence of trajectories of (
3).
Theorem 7 ([
17]).
If u is a weak solution of the system (3) on every interval , and satisfies and for all , as , and if for some , then converges weakly as to the asymptotic center of the curve . Theorem 8 ([
17]).
If u is a weak solution of the system (3) on every interval , and satisfies exists uniformly in , then converges strongly as to the asymptotic center of the curve . 4. Expansive Curves and Autonomous Systems
Now, we are in a position to go back to expansive mappings. In general, contrary to nonexpansive mappings, an expansive mapping may not be continuous. As we have seen, the set of fixed points of a nonexpansive mapping may be empty, but it always remains closed and convex. Djafari Rouhani [
27] provided examples to show that there are expansive self-mappings of the closed unit ball of
H, namely empty, nonconvex, or nonclosed sets of fixed points. The first mean ergodic theorem for expansive mappings was proved by Djafari Rouhani [
27]. A continuous time approach to the orbits of an expansive mapping was considered by Djafari Rouhani, and introduced as the notion of expansive curves.
Definition 4. An expansive curve u in H is a curve satisfying for all .
Expansive curves inherit many properties of orbits of expansive mappings, including the lack of convexity and lack of closedness of the set of their fixed points. In any case, the following two sets, which can be defined for any curve, are closed and convex (or empty) sets.
The following theorem describes the ergodic, weak, and strong convergence of expansive curves in
H:
Theorem 9 ([
27]).
Let u be an expansive curve in H and for .- (i)
If and , then the weak limit q of any weakly convergent subsequence of belongs to .
- (ii)
If in addition to (i), , then u is a bounded curve and converges weakly to the asymptotic center p of . Moreover we have .
- (iii)
If in addition to (ii), u is weakly asymptotically regular, then converges weakly to p as .
- (iv)
If exists, then converges strongly to the asymptotic center p of , and moreover in addition to , we have , where and .
Now, let
A be a monotone operator in
H. If
u is weak solution of
on
for every
, then
u is an expansive curve in
H [
27] (Lemma 5.3); hence, Theorem 9 describes the asymptotic behavior of any weak solution to (
4). Unfortunately, the system (
4) is “strongly ill-posed”. For example, consider the simple linear case of
with Dirishlet boundary conditions, which yields the heat equation with final Cauchy data and is not solvable in general. In
Section 9, we try to fix this problem.
6. Expansive-Type Difference Equations
As we have already explained, the dissipative systems of the form (
3) have a unique weak solution, whereas for solutions to (
4), neither existence nor uniqueness is guaranteed. A similar situation occurs for the backward discretization of (
4):
Hence, we consider the following forward discretization:
which is always well defined.
Similar to the continuous case, by introducing the notion of almost-expansive sequences and studying their asymptotic behavior under some suitable conditions, we describe the asymptotic behavior of the solution to (
6).
Definition 6. A sequence in H is said to be almost-expansive if for all , we havei.e., , such that , , . We note that if
is bounded, then this definition is equivalent to
The sequence of averages of is denoted by and defined by . The following theorem provides a discrete version of Theorem 10.
Theorem 12 ([
27]).
Let be an almost expansive sequence in H.- (i)
If and , then either the weak limit q of any weakly convergent subsequence of belongs to or as .
- (ii)
If in addition to (i), , then is bounded and converges weakly as , to the asymptotic center p of .
- (iii)
Assuming the conditions in (ii), converges weakly as to the asymptotic center p of , if and only if is weakly asymptotically regular.
- (iv)
If , then converges strongly as to the asymptotic center p of . Moreover, we have , where and .
- (v)
If is asymptotically regular, then , where p is the asymptotic center of , and K is defined above.
We still need an additional condition for the sequence
governed by (
6) to be almost expansive.
Proposition 2 ([
29]).
Let be a nondecreasing sequence of positive numbers, such thatIf is a bounded solution to (6), then is almost expansive. Note that the condition (
7) in the above proposition is in particular satisfied if
for some
, and
for some
. For example, the sequence
satisfies the conditions of the above proposition. Now, we are in a position to apply our results on almost-expansive sequences to describe the asymptotic behavior of the sequence
governed by (
6).
Theorem 13 ([
29]).
Assume that is a nondecreasing sequence satisfying the condition (7), and is a bounded solution to (6). Then, the following hold:- (i)
, as , where p is the asymptotic center of .
- (ii)
, as if and only if u is weakly asymptotically regular.
- (iii)
If exists, then , where K is as defined above.
- (iv)
if and only if is asymptotically regular.
In the following theorem, by assuming the zero set of
A to be nonempty, we can obtain stronger results:
Theorem 14 ([
29]).
Let be the sequence generated by (6), where and for some . If is bounded, then there exists some , such that as . Otherwise, as . Note that if the step size
goes to infinity as
, then the existence of a bounded solution to (
6) implies that
. In fact, let
be a bounded solution to (
6) and
. Clearly,
and
. Since
is bounded, there exist some
and a subsequence
, such that
as
. Now, the maximality of
A implies that
.
7. Periodic Solutions in Discrete Time
In this section, we will need the following extended version of expansive mappings
Definition 7. The mapping is said to be α-expansive if If , we say that T is expansive.
Clearly, letting
, the above definition coincides with the definition of an expansive mapping, and if
is
-expansive, then
exists and it is
-Lipschitz continuous. The following theorem provides sufficient conditions for the system (
6) to have a periodic solution.
Theorem 15 ([
29]).
Suppose that A is a single-valued and maximal strongly monotone operator in H. If is a periodic sequence with period N, then there exists an N-periodic solution to (6). The above theorem does not hold for a general maximal monotone operator
A; not even for subdifferentials of proper, convex and lower semicontinuous functions, nor for inverse strongly monotone operators. To see this, let
be the constant function
, and
. Then, (
6) reduces to
, which shows that the sequence
tends to
, as
, for all
. Therefore, it does not have a periodic solution. However, assuming (
6) has a periodic solution, is it possible that (
6) has another solution (by starting from a different initial point) that behaves differently? The following theorem answers this question.
Theorem 16 ([
29]).
Assume that A is a single-valued and maximal monotone operator in H, and the sequence is periodic with period N. If (6) has an N-periodic solution , then every bounded solution to (6) is also periodic with period N and differs from by an additive constant. In general, the existence of periodic solutions does not imply the boundedness of all solutions to (
6). For this, let
,
, and
. Then, (
6) reduces to
. If we choose
, then
, which is a periodic solution with period
N for all
. However, if we choose
, then
, which clearly goes to
, as
.
8. A Gradient System of Expansive Type
In this section, we consider a particular case of non-monotone operators. This case is motivated by the prominent example of a maximal monotone operator that is the subdifferential of a proper, convex, and lower semicontinuous function. A quasiconvex function is an extension of a convex function, which has found many applications in economics [
30]. Unlike the convex case, quasiconvex functions do not have a convex epigraph, but have convex sublevel sets. This is stated formally in the following definition:
Definition 8. (i) A function is quasiconvex if (ii) A function is strongly quasiconvex if there is such that The notion of a subdifferential has been generalized for nonconvex functions by many authors. Nevertheless, in any circumstance, the subdifferential operator of a quasiconvex function is not monotone. However, in the case where the quasiconvex function
is Gâteaux differentiable, then the following characterization holds:
This characterization will be useful in the rest of this section to make up for the lack of monotonicity.
We consider the expansive system governed by the non-monotone operator
, where
is a differentiable quasiconvex function. Indeed, as in [
31], we consider the following differential equation
where
is a differentiable quasiconvex function, such that
is Lipschitz continuous and
. The Cauchy–Lipschitz theorem implies the existence of a unique solution of the system (
8) with an initial condition, where
is Lipschitz continuous. In order to study the asymptotic behavior of solutions to systems of the form (
8), the authors in [
31] introduced the following set for a function
along a curve
u:
Denoting the set of all global minimizers of
by
, then
. The following proposition shows that if
u is a solution to (
8), then
.
Proposition 3 ([
31]).
Let be a solution to (8). For an arbitrary interval , where , and each , we haveand therefore exists (it may be infinite). Proposition 4 ([
31]).
Let be a solution to (8). If , then- (i)
.
- (ii)
exists and is finite.
- (iii)
.
- (iv)
u is bounded.
The following theorem describes the asymptotic behavior of solutions to (
8).
Theorem 17 ([
31]).
Let be a solution to (8). If , then there exists some , such that as , and if , the convergence is strong. If is unbounded, then as . Note that the above theorem shows that if
, then for any solution to (
8), we have
.
The following two theorems provide sufficient conditions for the strong convergence of solutions to (
8).
Theorem 18 ([
31]).
With either one of the following assumptions, bounded solutions to (8) converge strongly to some point in :- (i)
Sublevel sets of ϕ are compact.
- (ii)
.
Theorem 19 ([
31]).
Assume that is a strongly quasiconvex function and is a bounded solution to (8). Then, is a singleton and converges strongly to the unique minimizer of ϕ. For a differentiable quasiconvex function
whose gradient
is Lipschitz continuous with Lipschitz constant
K, as in
Section 6, we consider the forward finite-difference discrete version of (
8), which yields a well-defined sequence:
where the sequence
belongs to
and
for some
.
In order to study the asymptotic behavior of
, we define the following discrete version of
:
The following proposition is a discrete version of Proposition 3.
Proposition 5 ([
31]).
Let be the sequence generated by (9). For each , and , we haveand consequently exists (it may be infinite). Proposition 6 ([
31]).
Let be a solution to (9), such that . Then, is nonempty if and only if exists, and in this case is bounded. If is convex, we can omit the Lipschtz continuity condition on in Proposition 6.
Proposition 7 ([
31]).
Assume that is a solution to (9), such that . If either one of the following conditions is satisfied, then is nonempty.- (i)
ϕ is convex and the sequence of step sizes is bounded above.
- (ii)
.
In the continuous case, we showed that if , then . However, in the discrete case, it remains an open problem whether without any additional assumption that implies that is nonempty.
The following theorems describe the weak and strong convergence of solutions to (
9).
Theorem 20 ([
31]).
Assume that is the sequence given by (9), and . If , then there is some , such that as , and if , the convergence is strong.If is not bounded, then , as . Theorem 21 ([
31]).
Let be a bounded sequence, which satisfies (9), and let . If either one of the following assumptions holds, then converges strongly to some point in :- (i)
Sublevel sets of ϕ are compact.
- (ii)
.
Example 1. Assume that is defined by and consider (9) with and . Then, it is easy to see that all the assumptions of Theorem 21 are satisfied. In Table 1, we compare 1000 iterations generated by (9) starting from two different initial points, namely and . The numerical results show that for , , and for , slowly goes to infinity.