Potential and monotone homeomorphisms in Banach spaces

Michał Bełdziński; Marek Galewski

doi:10.1515/anona-2023-0108

Open Access Published by De Gruyter April 1, 2024

Potential and monotone homeomorphisms in Banach spaces

Michał Bełdziński and Marek Galewski

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2023-0108

Abstract

Using the Ekeland variational principle and the mountain pass lemma we prove some properties about potential homeomorphisms between a real Banach space and its dual. In particular, we show that a locally monotone homeomorphism is necessarily strictly monotone if it is potential.

Keywords: Browder-Minty theorem; homeomorphisms in Banach spaces; potential operators; mountain pass lemma; the Ekeland variational principle

MSC 2010: 31C45; 46T20; 49J35

1 Introduction

The celebrated Browder-Minty theorem, which provides sufficient conditions for the existence of solutions for a nonlinear equation involving monotone, continuous, and coercive operator, has numerous applications for various boundary value models governed by diverse differential equations (see, for instance, [11,14]). In addition, the Browder-Minty theorem provides sufficient conditions for a nonlinear operator to be a homeomorphism between a real and reflexive Banach space and its dual. The aim of this study is to investigate further this second assertion, i.e. concentrate on these implications of the Browder-Minty theorem, which provide that the given operator is a homeomorphism. In order to do so, the sole monotonicity methods are insufficient and this is why we resort to variational methods imposing thereby further assumption on potentiality of the operator. Nevertheless, we use classical tools pertaining to the mountain geometry, the theory of monotone operators together with some background on potential operators.

Main features of this study are as follows:

Discussion about multiplicity of critical points in the presence of the mountain geometry under the assumption of the Palais-Smale condition;
Relations between convexity, local convexity, lower boundedness, and strict convexity of the continuously differentiable functional in case its derivative is a homeomorphism;
Provides sufficient conditions for perturbations of homeomorphisms that do not violate the bijectivity.

Conducting the aforementioned research plan, we show that, under additional assumption about potentiality, the Browder-Minty theorem can be partially inverted, i.e. in a certain sense, and that the local behaviour of potential homeomorphism may have impact on general geometry of its potential. Thus, we obtain also some multiplicity of critical points, which be viewed as in the observation made in the study [3], where situations when the mountain pass lemma leads to multiplicity of critical points are discussed. As a byproduct, we also provide conditions under which the derivative of the (coercive) functional is weakly coercive, thereby inverting a result known from the literature.

This article is organized as follows. We start by some general preliminaries, in which we introduce necessary notions from variational calculus and the theory of monotone operators. Then, we move to main results of this article. In Section 3.1 we provide some direct but nontrivial consequences of the Ekeland variational principle to the existence of multiple critical points. Next, in Section 3.2, we consider potential homeomorphisms and investigate the importance of the potentiality assumption also with reference to some perturbations. Finally, in Section 3.3, we show how one can extend the results obtained onto the case of merely Gâteaux differentiable functionals. Some open questions end this article.

2 Preliminaries

For a real Banach space ( X , ‖ ⋅ ‖ ) , we denote by X * the space of all linear and continuous functionals endowed with a norm ‖ u * ‖ * ≔ sup ‖ u ‖ = 1 ∣ ⟨ u * , u ⟩ ∣ . Moreover, for simplicity, we put B r ( u ) ≔ { v ∈ X : ‖ u − v ‖ < r } , B r ≔ B r ( 0 ) , S r ( u ) ≔ { w ∈ X : ‖ u − w ‖ = r } , S r ≔ S r ( 0 ) , and inf A f = inf x ∈ A f ( x ) for a given function f : A ⟶ R .

2.1 Variational tools

We start by recalling the Ekeland variational principle after [6].

Theorem 1

(Ekeland variational principle, [6]) Let ( M , d ) be a complete metric space. Assume that I : M ⟶ R is a continuous functional bounded from below. Then, for every x ∈ M and δ > 0 satisfying

I ( x ) < inf M I + δ ,

there exists a point y ∈ M such that

d ( x , y ) < 1 , I ( y ) ≤ I ( x ) , and I ( z ) > I ( y ) − δ d ( z , y ) , for a l l z ∈ M \ { y } .

Let J : X ⟶ R be locally Lipschitz continuous. We define, following [4], the Clarke directional derivative of J with respect to the direction v by:

J ∘ ( u ; v ) = limsup t → 0 + w → u J ( w + t v ) − J ( w ) t .

Then, we define the Clarke subdifferential of J at u by:

∂ J ( u ) = { u * ∈ X * : ⟨ u * , v ⟩ ≤ J ∘ ( u ; v ) , for all v ∈ X } .

For background, we refer to [4]. We need what follows:

The set ∂ J ( u ) is nonempty, convex, and weakly* compact for all u ∈ X ;
For all u , v ∈ X , we have
J ∘ ( u ; v ) = sup u * ∈ ∂ J ( u ) ⟨ u * , v ⟩ ;
If J is Gâteaux differentiable, then we have ∂ J ( u ) = { J ′ ( u ) } for all u ∈ X .

We say that J satisfies the Palais-Smale condition if every sequence ( u n ) ⊂ X such that

sup n ∈ N ∣ J ( u n ) ∣ < ∞ and inf u * ∈ ∂ J ( u n ) ‖ u * ‖ * → 0

possess a convergent subsequence. The mountain pass lemma, given originally by Ambrosetti and Rabinowitz in [2], has been already extended by many authors (see, for instance, [3,9,15] and [10,13] for surveys). Here, we give a version that is most suitable for our considerations and originally was given by Shi Shuzhong in [16]. In what follows, C ( A , B ) stands for the space of all continuous functions from A to B .

Theorem 2

(Mountain pass lemma, [16]) Assume that J : X ⟶ R is a locally Lipschitz functional satisfying the Palais-Smale condition. If there exist an open neighbourhood Ω of u ∈ X and a point v ∈ X \ Ω such that

max { J ( u ) , J ( v ) } < inf ∂ Ω J ,

then J has a critical point w satisfying

J ( w ) = inf γ ∈ Γ sup 0 ≤ t ≤ 1 J ( γ ( t ) ) ,

where Γ ≔ { γ ∈ C ( [ 0 , 1 ] , X ) : γ ( 0 ) = u , γ ( 1 ) = v } .

Following [12], we give Ky Fan’s Min-Max theorem:

Theorem 3

(Ky Fan’s Min-Max theorem) Assume that Y and V are the Hausdorff topological vector spaces. Take convex sets C ⊂ Y , K ⊂ V and a function φ : C × K ⟶ R such that

φ ( y , ⋅ ) is convex and continuous for all y ∈ C ;
φ ( ⋅ , v ) is concave and continuous for all v ∈ K ;
the set K is compact.

Then,

sup v ∈ K inf y ∈ C φ ( y , v ) = inf y ∈ C sup v ∈ K φ ( y , v ) .

Finally, we recall that a functional J : X ⟶ R is called coercive if lim ‖ u ‖ → ∞ J ( u ) = ∞ .

2.2 Monotone methods

Here, we recall some basics from the theory of monotone operators following [8,17]. An operator A : X ⟶ X * is called monotone if

⟨ A ( u ) − A ( v ) , u − v ⟩ ≥ 0 , for all u , v ∈ X .

If, additionally, ⟨ A ( u ) − A ( v ) , u − v ⟩ = 0 implies u = v , we say that A is strictly monotone. It is well known that the convexity of a functional is equivalent with monotonicity of its derivative (see for instance [17, Proposition 25.10]). An operator A is called coercive if

lim ‖ u ‖ → ∞ ⟨ A ( u ) , u ⟩ ‖ u ‖ = ∞ .

We say that A satisfies the condition (S) if u n → u 0 in X whenever

u n ⇀ u 0 in X and ⟨ A ( u n ) , u n − u 0 ⟩ → 0 .

For a deeper study and other types of similar conditions, we refer to [7,8,17]. Here, let us mention that condition (S) (or even its stronger version, i.e. condition (S)₊) appears in most of the applications concerning non-linear elliptic equations (see, for instance, [14]).

We say that A : X ⟶ X * is radially continuous if for every u , v ∈ X , the function τ ⟼ ⟨ A ( u + τ v ) , v ⟩ is continuous. Now, if A : X ⟶ X * is radially continuous and potential (i.e. A = J ′ holds for some Gâteaux differentiable functional J ), then

J ( u ) − J ( v ) = ∫ 0 1 ⟨ A ( u + τ ( v − u ) ) , v − u ⟩ d τ .

A functional J is called the potential of A . The aforementioned formula shows that J is determined uniquely by A up to an additive constant. The following results are derived from the Browder-Minty theorem (compared with [7, Theorem 7.6] and [8, Theorem 6.4]).

Proposition 1

Let K be a closed, convex, and bounded subset of a real and reflexive Banach space X. Assume that A : K ⟶ X * is radially continuous and strictly monotone. Then, there exists a unique u ∈ K satisfying

(1) ⟨ A ( u ) , v − u ⟩ ≥ 0 , for a l l v ∈ K .

Proposition 2

Let X be a real and reflexive Banach space. Assume that A : X ⟶ X * is continuous, coercive, monotone and that it satisfies condition (S). Then, A is a homeomorphism.

3 Potential homeomorphisms

3.1 On some multiple critical point theorems

In this section, we will use the following assumption

Assumption 1

X is a real Banach space and J : X ⟶ R is a locally Lipschitz functional satisfying the Palais-Smale condition.

The following lemma that will be used in the sequel as a technical tool is somehow related to Theorem 5.7 from [5]. However, since its implications are in fact different and since we cannot prove our next results with the mentioned Theorem 5.7, we decided to include it as well. The main difference here is that we do not require the set C to disconnect U .

Lemma 1

Let Assumption 1 hold. Assume additionally that there exist an open set U and a closed set C with C ⊂ U such that dist ( C , ∂ U ) > 0 and

inf U J = inf C J .

Then, J has a critical point in C.

Proof

Denote r = dist ( C , ∂ U ) and fix ε < r 2 . Then, M = U ¯ equipped with a distance function

d ( x , y ) ≔ 1 ε ‖ x − y ‖

becomes a complete metric space. Since inf C J = inf U J = inf U ¯ J , then for every n ∈ N , there is u n ∈ C such that J ( u n ) < inf M J + ε n . Taking δ = ε n , M = U ¯ , and I = J in the Ekeland variational principle, we obtain v n ∈ U ¯ such that

(2) ‖ u n − v n ‖ ≤ ε , J ( v n ) ≤ J ( u n ) , ∀ w ∈ U ¯ \ { v n } J ( w ) > J ( v n ) − 1 n ‖ w − v n ‖ .

Consequently, v n ∈ U and

J ( w ) − J ( v n ) ‖ w − v n ‖ > − 1 n , for all w ∈ U ¯ \ { v n } .

Taking w = v n + τ h , where h ∈ S 1 and τ > 0 is sufficiently small, we obtain

J ( v n + τ h ) − J ( v n ) τ > − 1 n .

Therefore, again for every h ∈ S 1 , we have J ∘ ( v n ; h ) ≥ − 1 n , and by the properties of Clarke subdifferential, sup u * ∈ ∂ J ( v n ) ⟨ u * , h ⟩ ≥ − 1 n . Consequently,

sup h ∈ B 1 inf u * ∈ ∂ J ( v n ) ⟨ u * , h ⟩ ≤ 1 n ,

which, due to Theorem 3, gives

inf u * ∈ ∂ J ( v n ) ‖ u * ‖ * ≤ 1 n .

Hence, ( v n ) is a Palais-Smale sequence. Therefore, it possesses a convergent subsequence, denoted also by ( v n ) , such that v n → w ε . Moreover,

(3) J ( w ε ) = inf U J , inf y ∈ C ‖ y − w ε ‖ ≤ ε , and 0 ∈ ∂ J ( w ε ) .

Hence, for large n , we obtain a sequence ( w n ) satisfying (3) with ε = 1 n and w ε = w n . Note that ( w n ) is again a Palais-Smale sequence. Therefore, up to a subsequence, w n → w for some w ∈ U ¯ . Then, J ( w ) = inf U J , w ∈ C , and 0 ∈ ∂ J ( w ) .□

The following example shows that the strong separation of C and ∂ U cannot be omitted even in a more regular case.

Example 1

Let J : ℓ 2 ⟶ R be given by the formula:

J ( u ) = − ‖ u ‖ 2 2 .

The functional J satisfies the Palais-Smale condition. Denote by ( e n ) n ∈ N a standard base in ℓ 2 , i.e.

(4) e n ( k ) ≔ 0 , if k ≠ n , 1 , if k = n .

Take C = n n + 1 e n : n ∈ N and let U = B 1 . Then, C = C ¯ ⊂ U and inf C J = inf U J . However, the unique critical point of J is 0. Note that dist ( C , ∂ U ) = 0 .

In the result that we provide below, we show among other things that Lemma 1 can be used to give an alternative, easier proof of some results related to the content given by Pucci and Serin in [15].

Proposition 3

Let Assumption 1 hold.

If 0 is a local minimum of J and if there exists v such that J ( v ) < J ( 0 ) , then there exists a critical point w, with J ( w ) ≥ J ( 0 ) , which is not a local minimum;
If 0 is a strict local minimum of J and if there exists v distinct from 0 such that J ( v ) ≤ J ( 0 ) , then there exists a critical point w, with J ( w ) > J ( 0 ) , which is not a local minimum;
If 0 is a local minimum of J, then either there exists a second critical point, which is not a local minimum, or 0 is a global minimum and the set of global minima is connected;
If J has two local minima, then there is a third critical point;
Let λ be a fixed real number, and suppose that each critical point with a critical value greater than λ is a local minimum. Then, each one is a global minimum and the set of those points is connected.

Proof

Let us denote
W ≔ ⋃ { V : V is open and inf V J ≥ J ( 0 ) } and K ≔ { u ∈ W ¯ : J ′ ( u ) = 0 and J ( u ) = J ( 0 ) } .
Since J ( v ) < J ( 0 ) for some v ∈ X , the set ∂ W is nonempty. Moreover, the set K is compact by the Palais-Smale condition. Hence, one of the following holds:
- There exists w ∈ ∂ W ∩ K . Then, such a w is a critical point of J and J ( w ) = J ( 0 ) since w ∈ K . Now, suppose that w is an argument of a minimum. Then, inf V J ≥ J ( w ) = J ( 0 ) for some open neighbourhood V of w . Hence, w ∈ V ⊂ W , which means that w is an interior point of W and that contradicts w ∈ ∂ W .
- Sets ∂ W and K are disjoint. By compactness of K , it means that r ≔ dist ( ∂ W , K ) > 0 . Let us put U ≔ K + B r ∕ 2 . Then, dist ( ∂ U , ∂ W ) > 0 , 0 ∈ U ⊂ U ¯ ⊂ W , and ∂ U ∩ K = ∅ . Therefore, by Lemma 1, we obtain inf ∂ U J > J ( 0 ) . Applying Theorem 2, we obtain the assertion.
If 0 is a strict local minimum, then we need to have inf S r J > J ( 0 ) for sufficiently small r . Otherwise we can use Lemma 1 to obtain contradiction. Hence, it is enough to use the mountain pass lemma.
Assume that J has a local minimum at 0 and that all critical points are arguments of a local minimum. Then, the set of critical points needs to be a subset of some level set of J and every critical point needs to be an argument of a global minimum. Otherwise, we will have a contradiction with (1). However, it means that the set M of critical points of J and the set of arguments of a global minima of J coincide. Now, suppose that M is not connected. This means that there exists two disjoint open sets V , W ⊂ X such that M = ( M ∩ V ) ∪ ( M ∩ W ) . Therefore, in particular, M ∩ ∂ V = ∅ . Hence, taking any u ∈ V , C = ∂ V , and U = X in Lemma 1, we obtain inf ∂ V J > J ( u ) . Using Theorem 2, we obtain the existence of a critical point, which is not a global minimum, so we have a contradiction.
It is an immediate consequence of (a) and (c).
Take a critical point u such that J ( u ) > λ . Since each critical point with critical value greater than λ is a local minimum, then, by (a), u needs to be an argument of a global minimum of J . Now, it is enough to use (c) for J ( ⋅ − u ) .□

As a direct consequence of Proposition 3 (c), we obtain the following.

Corollary 1

Let Assumption 1 hold and assume that u is an argument of a local minimum of J. Then, J has a strict global minimum at u or there exists another critical point of J.

3.2 Characterization of homeomorphisms with convex potentials

In this section, we will study some properties of monotone homeomorphisms on Banach spaces. We will pay special attention to the role of potentiality assumption. Again, we will formulate a general assumption, which will be used in this subsection.

Assumption 2

X is a real Banach space and J : X ⟶ R is a functional of class C 1 such that J ′ : X ⟶ X * is a homeomorphism.

In what follows, we show that local behaviour of J may have impact on the global one.

Theorem 4

Let Assumption 2 hold. Then, the following conditions are equivalent:

J is strictly convex;
There exists an open and convex set U such that J is convex on U;
There exists u * ∈ X * such that J − u * has a local minimum;
There exists u * ∈ X * such that J − u * is bounded from below.

Corollary 2

Let Assumption 2 holds. Then, the following conditions are equivalent:

J is strictly convex;
J is convex on some open set;
J is bounded from below;
J has a local minimum.

Proof of Theorem 4

Note that for every u * ∈ X * , the functional J − u * satisfies the Palais-Smale condition. Indeed, since J ′ is a homeomorphism, then ( J ′ − u * ) ′ ( u n ) → 0 gives J ′ ( u n ) → u * , and consequently, u n → ( J ′ ) − 1 ( u * ) . Therefore, every Palais-Smale sequence is convergent.

It is clear that (a) ⇒ (b).

Let us show that (b) ⇒ (c). Assume that (b) holds and take u * = J ′ ( u ) for some u ∈ U . Then, J − u * is convex on U . Since every critical point of a convex functional is an argument of a minimum, (c) holds.

Now, we show the equivalence (c) ⇔ (d). If (c) holds, then J − u * is necessarily bounded from below by Corollary 1, and hence, (d) holds. On the other hand, if (d) is satisfied, then J − u * has a global minimum by Lemma 1. Therefore, (c) is satisfied.

Finally, let us prove (c) ⇒ (a). Assume that (c) holds. Denote

Σ ≔ { u ∈ X : J ( v ) ≥ J ( u ) + ⟨ J ′ ( u ) , v − u ⟩ , for all v ∈ X } .

Σ is clearly closed. We show that it is also open. Denote for any u the following ψ u ≔ J − J ′ ( u ) . Let w ∈ Σ be fixed. By Assumption 2, ψ w satisfies the Palais-Smale condition. Moreover, by what we have already shown, w is the unique critical point of ψ w . Hence,

inf B 2 ( w ) \ B 1 ( w ) ψ w > ψ w ( w ) .

Indeed, if it is not the case, we can apply Lemma 1 to obtain the existence of another critical point of ψ w , which contradicts the bijectivity of J ′ . By continuity of J ′ , we can find an open neighbourhood W of w such that

‖ J ′ ( v ) − J ′ ( w ) ‖ < ε 4 , for all v ∈ W ,

where ε ≔ inf B 2 ( w ) \ B 1 ( w ) ψ w − ψ w ( w ) . Take y ∈ B 2 ( w ) \ B 1 ( w ) and v ∈ W ∩ B 1 ( w ) . Then,

ψ v ( y ) − ψ v ( w ) = ψ w ( y ) − ψ w ( w ) + ψ v ( y ) − ψ w ( y ) + ψ w ( w ) − ψ v ( w ) ≥ inf B 2 ( w ) \ B 1 ( w ) ψ w − ψ w ( w ) + ⟨ J ′ ( v ) − J ′ ( w ) , y − w ⟩ ≥ ε − ‖ J ′ ( v ) − J ′ ( w ) ‖ ‖ y − w ‖ ≥ ε − ε 4 ⋅ 2 = ε 2 .

Since y and v were taken arbitrary, we obtain

inf B 2 ( w ) \ B 1 ( w ) ψ v > ψ w ( w ) .

The functional ψ v is bounded from below on B 2 ( w ) , it satisfies the Palais-Smale condition and v is a unique critical point of ψ v . Moreover,

inf B 2 ( w ) ψ v = inf B 1 ( w ) ¯ ψ v .

Hence, by Lemma 1, ψ v has a local minimum at v , which is a strict global minimum by Corollary 1. Thus,

ψ v ( u ) = J ( u ) − ⟨ J ′ ( v ) , u ⟩ ≥ J ( v ) − ⟨ J ′ ( v ) , v ⟩ = ψ v ( v ) , for all u ∈ X ,

and hence, v ∈ Σ . Since v and w were taken arbitrary, Σ is open. Finally, note that Σ is nonempty by (c). Hence, Σ is a connected component of X . Since X is connected, Σ is an entire space. Consequently, (c) ⇒ (a).□

Note that from Corollary 2, we obtain the following statement: If J ′ is a homeomorphism, which is monotone on some open set, then it is strictly monotone on the entire space. One may ask whether this statement can be extended onto general operator setting, i.e. ask if the following statement is true: if f : X ⟶ X * is a homeomorphism, which is monotone on some open set, then it is strictly monotone on the entire space. A negative answer is provided by:

Example 2

Denote by M θ a rotation in R 2 , i.e.

M θ ≔ cos θ − sin θ sin θ cos θ .

Moreover, let φ be a smooth function satisfying φ ≡ 0 on ( − ∞ , 1 ] ∪ [ 4 , ∞ ) and φ ≡ π on [ 2 , 3 ] . Then, f ( x ) = M φ ( ∣ x ∣ 2 ) x is an example of a smooth diffeomorphism, which is strictly monotone on B 1 and coercive on R 2 . However, the function f is not monotone on entire R 2 , since, for instance,

⟨ f ( 3 , 0 ) − f ( 2 , 0 ) , ( 3 , 0 ) − ( 2 , 0 ) ⟩ = 2 6 − 5 < 0 .

In the following two results, we will assume the convexity of J . In fact, any equivalent condition listed in Theorem 4 can be considered. The theorem which we provide now serves as a partial inverse of Proposition 2.

Theorem 5

Let Assumption 2 hold and assume that J is convex. Then,

(5) lim ‖ u ‖ → ∞ ⟨ J ′ ( u ) , u ⟩ = ∞ .

Proof

Since J ′ is bijective, J has a unique critical point w ∈ X . Denote I ( u ) ≔ J ( u + w ) for all u ∈ X . Note that I is coercive iff J is so. Moreover,

I ( u ) − I ( 0 ) = ∫ 0 1 ⟨ I ′ ( τ u ) , u ⟩ d τ ,

and 0 is a global minimum of I . By Lemma 1 and Corollary 1, we need to have

inf S 1 I > I ( 0 ) .

Put a ≔ inf S 1 I − I ( 0 ) . Since I ′ is monotone, we have ⟨ I ′ ( α u ) , u ⟩ ≥ ⟨ I ′ ( β u ) , u ⟩ whenever α ≥ β . Therefore, for every u ∈ X with ‖ u ‖ ≥ 1 , one has

I ( u ) − I ( 0 ) = ∫ 0 1 ⟨ I ′ ( τ u ) , u ⟩ d τ = ‖ u ‖ ∫ 0 1 I ′ ( τ u ) , u ‖ u ‖ d τ ≥ ‖ u ‖ ∫ 0 1 I ′ τ u ‖ u ‖ , u ‖ u ‖ d τ = ‖ u ‖ I u ‖ u ‖ − I ( 0 ) ≥ a ‖ u ‖ .

Hence, I and consequently J are coercive. Moreover, since ⟨ J ′ ( u ) , u ⟩ ≥ ⟨ J ′ ( τ u ) , u ⟩ for every τ ∈ [ 0 , 1 ] by monotonicity, we have

⟨ J ′ ( u ) , u ⟩ = ∫ 0 1 ⟨ J ′ ( u ) , u ⟩ d τ ≥ ∫ 0 1 ⟨ J ′ ( τ u ) , u ⟩ d τ = J ( u ) − J ( 0 ) .

Hence, the assertion holds.□

Let us mention that the coercivity of J is provided by the Palais-Smale condition and by the boundedness from below, see [10, Proposition 15.7]. However, to obtain the coercivity of J ′ , convexity of J is crucial. This is in contrast to the relation between the coercivity of a potential, bounded, and demicontinuous operator and its potential. We must mention that not every monotone homeomorphism satisfies (5). The following example shows the importance of the potentiality assumption.

Example 3

An operator A : R 2 ⟶ R 2 given by A ( x , y ) = ( − y , x ) is a monotone homeomorphism, which is not coercive.

It is easy to check that monotonicity (or even strict monotonicity) of an operator cannot be violated by a monotone perturbation. However, it is no longer the case if we consider a bijectivity property.

Example 4

Take X = R 2 , A ( x , y ) = ( e x − y , x ) , and B ( x , y ) = ( y , − x ) . Then, both A and B are homeomorphisms. However, they are not potential. Moreover, ( A + B ) ( x , y ) = ( e x , 0 ) is clearly not a bijection.

The following result shows that monotone perturbations of a monotone homeomorphism are again a bijection if we assume the potentiality of a homeomorphism.

Theorem 6

Let X be a real and reflexive Banach space. Take two monotone and radially continuous operators A , B : X ⟶ X * . If A is a potential homeomorphism, then A + B is bijective.

Proof

Fix u * ∈ X * and define T ( u ) ≔ A ( u ) + B ( u ) − u * . Then, A − u * + B ( 0 ) is also a potential homeomorphism. By Theorem 5, we have

lim ‖ u ‖ → ∞ ( ⟨ A ( u ) , u ⟩ − ⟨ u * , u ⟩ + ⟨ B ( 0 ) , u ⟩ ) = ∞ .

Hence, for every u ∈ X , one has

⟨ A ( u ) , u ⟩ + ⟨ B ( u ) , u ⟩ − ⟨ u * , u ⟩ = ⟨ A ( u ) , u ⟩ + ⟨ B ( u ) − B ( 0 ) , u ⟩ + ⟨ B ( 0 ) , u ⟩ − ⟨ u * , u ⟩ ≥ ⟨ A ( u ) , u ⟩ + ⟨ B ( 0 ) , u ⟩ − ⟨ u * , u ⟩ → ∞ as ‖ u ‖ → ∞ .

Therefore, there exists R > 0 such that ⟨ T ( u ) , u ⟩ > 0 for all u ∈ ∂ B R . Proposition 1 applied to T and B R ¯ yields existence of u ∈ B R ¯ satisfying

(6) ⟨ T ( u ) , v − u ⟩ ≥ 0 , for all v ∈ B R ¯ .

Taking v = 0 , we see that u ∈ B R . Therefore, B r ( u ) ⊂ B R for sufficiently small r > 0 . Hence, taking v = u + w in (6) with w ∈ B r , we obtain ⟨ T ( u ) , w ⟩ ≥ 0 for all w ∈ B r . Consequently, T ( u ) = 0 , which means A ( u ) + B ( u ) = u * . Finally, since A is strictly monotone by Theorem 4, T is also strictly monotone. Therefore, u is determined uniquely, and since u * was taken arbitrary, the assertion follows.□

3.3 Final remarks

Partially, results from Sections 3.1 and 3.2 can be shifted into the setting of Gâteaux differentiable functionals with strong-to-weak* continuous derivatives. Functionals of this type have been already studied in many articles, for instance, in [9], where the authors provide a suitable version of the mountain pass lemma. Let us mention that in contrast to the Fréchet differentiability, Gâteaux differentiability does not provide the continuity of a functional. Therefore, we provide a lemma whose proof follows readily from the known results about differentiability in abstract spaces:

Lemma 2

Let X be a real Banach space. If J : X ⟶ R is Gâteaux differentiable and J ′ is strong-to-weak* continuous, then J is continuous.

Remark 1

We provide a short proof while being aware that some other proofs can be known but we have not found any in the known literature. The aforementioned lemma also says why we cannot use fully the content of Proposition 2, i.e., examine inversions of radially continuous maps.

Proof of Lemma 2

Let u n → u 0 . Then, the set { τ u n : n ∈ N 0 , 0 ≤ τ ≤ 1 } is compact and since J ′ is strong-to-weak* continuous, there exists R > 0 such that

∣ ⟨ J ′ ( τ u n ) , u n ⟩ ∣ ≤ R

for all n ∈ N 0 . Moreover, for every τ ∈ [ 0 , 1 ] , we have

⟨ J ′ ( τ u n ) , u n ⟩ → ⟨ J ′ ( τ u 0 ) , u 0 ⟩ .

Consequently, the Lebesgue dominated convergence theorem yields

□ J ( u n ) = ∫ 0 1 ⟨ J ′ ( τ u n ) , u n ⟩ d τ → ∫ 0 1 ⟨ J ′ ( τ u 0 ) , u 0 ⟩ d τ = J ( u 0 ) .

Following [1], we say that a Gâteaux differentiable functional J : X ⟶ R satisfies the weak Palais-Smale condition if every sequence ( u n ) ⊂ X satisfying

sup n ∈ N ∣ J ( u n ) ∣ < ∞ and J ′ ( u n ) → 0

has a convergent subsequence. The following version of the mountain pass lemma was proved in [9]. The authors require continuity of a functional. However, this assumption can be omitted by Lemma 2.

Theorem 7

Assume that J : X ⟶ R is Gâteaux differentiable satisfying the weak Palais-Smale condition and that J ′ : X ⟶ X * is strong-to-weak* continuous. If there exist an open set Ω , u ∈ Ω , and v ∈ X \ Ω such that

max { J ( u ) , J ( v ) } < inf ∂ Ω J ,

then J has a critical point w satisfying

J ( w ) = inf γ ∈ Γ sup 0 ≤ t ≤ 1 J ( γ ( t ) ) ,

where Γ ≔ { γ ∈ C ( [ 0 , 1 ] , X ) : γ ( 0 ) = u , γ ( 1 ) = v } .

Then, a counterpart of Lemma 1 reads.

Proposition 4

Assume that J : X ⟶ R is Gâteaux differentiable with strong-to-weak* continuous derivative and that it satisfies the weak Palais-Smale condition. If there exists a strong-open set U, its strong-closed subset C satisfying dist ( ∂ U , C ) > 0 and if

inf U J = inf C J ,

then J has a critical point in C.

To prove Proposition 4, it suffices to follow the lines of the proof of Lemma 1 replacing ∂ J by J ′ (with some necessary changes of notations). Unfortunately, for weak-to-weak* homeomorphisms, the results from Section 3.2 cannot be shifted.

It remains to mention that Theorem 4 covers the case of functionals bounded from below by some linear functionals. It seems to be interesting what happens if we omit this assumption. Let us consider the simplest case when F : R 2 ⟶ R is C 2 functional such that F ′ is a diffeomorphism of R 2 onto R 2 . Then, F ″ ( x ) is a linear bijection at every point. Since F ″ ( x ) is symmetric by potentiality, then for every point x ∈ R 2 , there exists a real orthogonal matrix S x and real numbers λ x 1 , λ x 2 ≠ 0 satisfying

(7) F ″ ( x ) = S x T λ x 1 0 0 λ x 2 S x

If λ x 1 and λ x 2 have the same sign for at least one x ∈ R 2 (and consequently, F − F ′ ( x ) has a local minimum), then using Theorem 4, we obtain that F is convex or concave. Then, we obtain the following result.

Proposition 5

Assume that F : R 2 ⟶ R is of class C 2 and that F ′ is a diffeomorphism. Then, one of the following holds:

F is convex;
F is concave;
for every x ∈ R 2 , there exist a real orthogonal matrix S x and real numbers λ x 1 < 0 < λ x 2 satisfying (7).

We do not know if S x can be taken uniformly for each x ∈ R 2 . If this were the case, we would arrive at the following:

Hypothesis 1

Assume that F : R 2 ⟶ R is of class C 2 and that F ′ is a diffeomorphism. Then, there exist closed linear subspaces U , V ⊂ X satisfying U ⊕ V = R 2 such that U ∋ w ⟼ F ( w + v ) is convex and V ∋ w ⟼ F ( u + w ) is concave for all u ∈ U and v ∈ V .

If V = { 0 } (or U = { 0 } ), the case of convexity (or concavity) is covered. In general, it is possible that the following result is true.

Hypothesis 2

Assume that J is a functional of class C 1 and that J ′ : X ⟶ X * is a homeomorphism. Then, there exist closed linear subspaces U , V ⊂ X satisfying U ⊕ V = X such that U ∋ w ⟼ J ( w + v ) is convex and V ∋ w ⟼ J ( u + w ) is concave for all u ∈ U and v ∈ V .

Funding information: There are no funders to report for this submission.
Conflict of interest: This work does not have any conflicts of interest.

References

[1] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, United States, 1984. Search in Google Scholar

[2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), no. 4, 349–381. 10.1016/0022-1236(73)90051-7Search in Google Scholar

[3] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. Theory Meth. Appl. Ser A Theory Meth. 75 (2012), no. 5, 2992–3007. 10.1016/j.na.2011.12.003Search in Google Scholar

[4] F. H. Clarke, Optimization and nonsmooth analysis, Can. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley, New York, NY, 1983. Search in Google Scholar

[5] D. G. De Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, vol. 81, Springer, Berlin, 1989. Search in Google Scholar

[6] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), no. 2, 324–353. 10.1016/0022-247X(74)90025-0Search in Google Scholar

[7] J. Francu, Monotone operators. A survey directed to applications to differential equations, Appl. Math. 35 (1990), no.4, 257–301. 10.21136/AM.1990.104411Search in Google Scholar

[8] M. Galewski, Basic monotonicity methods with some applications, Compact Textbooks in Mathematics, Birkhäuser, Springer Nature, 2021. 10.1007/978-3-030-75308-5Search in Google Scholar

[9] N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Annales de laInstitut Henri Poincaré C, Analyse non linéaire 6 (1989), no. 5, 321–330. 10.1016/s0294-1449(16)30313-4Search in Google Scholar

[10] Y. Jabri, The Mountain Pass Lemma: Variants, Generalizations and Some Applications, vol. 95, Cambridge University Press, Cambridge, 2003. 10.1017/CBO9780511546655Search in Google Scholar

[11] D. Motreanu and V. D. Rădulescu, Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems, Springer, US, 2003. 10.1007/978-1-4757-6921-0Search in Google Scholar

[12] L. Nirenberg, Variational and topological methods in nonlinear problems, Bulletin Am. Math. Soc. 4 (1981), no. 3, 267–302. 10.1090/S0273-0979-1981-14888-6Search in Google Scholar

[13] P. Pucci and V. D. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital. (9) 3 (2010), no. 3, 543–582. Search in Google Scholar

[14] N. S. Papageorgiou, V. D. Rădulescu, and D. D. Repovš, Nonlinear Analysis - Theory and Methods, Springer International Publishing, Switzerland, 2019. 10.1007/978-3-030-03430-6Search in Google Scholar

[15] P. Pucci and J. Serrin, Extensions of the Mountain Pass Lemma, J. Funct. Anal. 59 (1984), no. 2, 185–210. 10.1016/0022-1236(84)90072-7Search in Google Scholar

[16] S. Shuzhong, Ekeland’s variational principle and the mountain pass lemma, Acta Math. Sinica 1 (1985), no. 4, 348–355. 10.1007/BF02564843Search in Google Scholar

[17] E. Zeidler, Nonlinear Functional Analysis and Its Applications: II/B: Nonlinear Monotone Operators, Springer Science & Business Media, United States, 2013. Search in Google Scholar

Received: 2022-07-08

Revised: 2023-05-16

Accepted: 2023-09-12

Published Online: 2024-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Potential and monotone homeomorphisms in Banach spaces

Abstract

1 Introduction

2 Preliminaries

2.1 Variational tools

Theorem 1

Theorem 2

Theorem 3

2.2 Monotone methods

Proposition 1

Proposition 2

3 Potential homeomorphisms

3.1 On some multiple critical point theorems

Assumption 1

Lemma 1

Proof

Example 1

Proposition 3

Proof

Corollary 1

3.2 Characterization of homeomorphisms with convex potentials

Assumption 2

Theorem 4

Corollary 2

Proof of Theorem 4

Example 2

Theorem 5

Proof

Example 3

Example 4

Theorem 6

Proof

3.3 Final remarks

Lemma 2

Remark 1

Proof of Lemma 2

Theorem 7

Proposition 4

Proposition 5

Hypothesis 1

Hypothesis 2

References

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