Potential and monotone homeomorphisms in Banach spaces

: 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.


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: (a) Discussion about multiplicity of critical points in the presence of the mountain geometry under the assumption of the Palais-Smale condition; (b) Relations between convexity, local convexity, lower boundedness, and strict convexity of the continuously differentiable functional in case its derivative is a homeomorphism; (c) 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.

Preliminaries
For a real Banach space ( ‖ ‖) ⋅ X , , we denote by X* the space of all linear and continuous functionals endowed with a norm . Moreover, for simplicity, we put for a given function ⟶ f A : .

Variational tools
We start by recalling the Ekeland variational principle after [6].
, be a complete metric space.Assume that ⟶ I M : is a continuous functional bounded from below.Then, for every ∈ x M and > δ 0 satisfying I y I x and I z I y δd z y for all z M y , 1, , , , .
Let ⟶ J X : be locally Lipschitz continuous.We define, following [4], the Clarke directional derivative of J with respect to the direction v by: Then, we define the Clarke subdifferential of J at u by: For background, we refer to [4].We need what follows: (a) The set ( ) ∂J u is nonempty, convex, and weakly* compact for all ∈ u X; (b) For all ∈ u v X , , we have We say that J satisfies the Palais-Smale condition if every sequence 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 : 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 Ω then J has a critical point w satisfying 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 , is concave and continuous for all

Monotone methods
Here, we recall some basics from the theory of monotone operators following [8,17].An operator 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 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 * is radially continuous and potential (i.e.= ′ A J holds for some Gâteaux differentiable functional J ), then Potential and monotone homeomorphisms in Banach spaces  3 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 * is radially continuous and strictly monotone.Then, there exists a unique ∈ u K satisfying 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.

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 : 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 Then, J has a critical point in C.
and fix < ε r 2 .Then, = M U equipped with a distance function .
, for all .
, where ∈ h S 1 and > τ 0 is sufficiently small, we obtain n n Therefore, again for every ∈ h S 1 , we have 1 , and by the properties of Clarke subdifferential, which, due to Theorem 3, gives v n is a Palais-Smale sequence.Therefore, it possesses a convergent subsequence, denoted also by ( ) , and 0 .
Hence, for large n, we obtain a sequence ( ) w n is again a Palais-Smale sequence.Therefore, up to a subsequence, → w w n for some ∈ w U.Then, ( ) and ( ) ∈ ∂J w 0 .□ The following example shows that the strong separation of C and ∂U cannot be omitted even in a more regular case.
Example 1.Let ℓ ⟶ J : 2  be given by the formula: 2 The functional J satisfies the Palais-Smale condition.Denote by ( ) ∈ e n n a standard base in ℓ 2 , i.e.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.(a) 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; (b) 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; (c) 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; (d) If J has two local minima, then there is a third critical point; (e) 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.
(a) Let us denote is open and inf 0 and : 0 and 0 .
V Potential and monotone homeomorphisms in Banach spaces  5 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, ( ) 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 , and ∂ ∩ = ∅ U K .Therefore, by Lemma 1, we obtain ( ) Applying Theorem 2, we obtain the assertion.(b) If 0 is a strict local minimum, then we need to have ( ) > J J inf 0

Sr
for sufficiently small r.Otherwise we can use Lemma 1 to obtain contradiction.Hence, it is enough to use the mountain pass lemma.(c) 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 . Using Theorem 2, we obtain the existence of a critical point, which is not a global minimum, so we have a contradiction.(d) It is an immediate consequence of (a) and (c).(e) 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 ( ) 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.

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 In what follows, we show that local behaviour of J may have impact on the global one.Finally, let us prove (c) ⇒ (a).Assume that (c) holds.Denote Σ is clearly closed.We show that it is also open.Denote for any u the following 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 . Take Since y and v were taken arbitrary, we obtain The functional ψ v is bounded from below on ( ) B w

2
, it satisfies the Palais-Smale condition and v is a unique critical point of ψ v .Moreover, Hence, by Lemma 1, ψ v has a local minimum at v, which is a strict global minimum by Corollary 1.Thus, 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: * 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 is an example of a smooth diffeomorphism, which is strictly monotone on B 1 and coercive on 2 .However, the function f is not monotone on entire 2 , since, for instance, 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. 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 : 2  2 given by ( ) ( ) = − A x y y x , , is a monotone homeomorphism, which is not coercive.

Theorem 4 . 1 .
Let Assumption 2 hold.Then, the following conditions are equivalent: (a) J is strictly convex; (b) There exists an open and convex set U such that J is convex on U; (c) There exists ∈ u X * * such that − J u* has a local minimum; (d) There exists ∈ u X * * such that − J u* is bounded from below.Corollary 2. Let Assumption 2 holds.Then, the following conditions are equivalent: (a) J is strictly convex; (b) J is convex on some open set; (c) J is bounded from below; (d) 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 ( 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.
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,

Theorem 5 .
Let Assumption 2 hold and assume that J is convex.Then,Proof.Since ′ J is bijective, J has a unique critical point ∈ w X. Denote ( ) ( ) ≔ + I uJ u w for all ∈ u X.Note that I is coercive iff J is so.Moreover, a global minimum of I .By Lemma 1 and Corollary 1, we need to have and consequently J are coercive.Moreover, since ⟨ ( ) ⟩ ⟨ ( 2, i.e.