Fixed-Point Theory on a Frechet Topological Vector Space

Abstract

We establish some versions of fixed-point theorem in a Frechet topological vector space 𝐸. The main result is that every map 𝐴=𝐵𝐶 (where 𝐵 is a continuous map and 𝐶 is a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskii's fixed-point theorem for U-contractions and weakly compact mappings, while the second one, by assuming that the family {𝑇(⋅,𝑦)∶𝑦∈𝐶(𝑀) where 𝑀⊂𝐸 and 𝐶∶𝑀→𝐸 a compact operator} is nonlinear 𝜑 equicontractive, we give a fixed-point theorem for the operator of the form 𝐸𝑥∶=𝑇(𝑥,𝐶(𝑥)).

1. Introduction

Fixed-point theorems are very important in mathematical analysis. They are an interesting way to show that something exists without setting it out, which sometimes is very hard, or even impossible to do. Several algebraic and topological settings in the theory and applications of nonlinear operator equations lead naturally to the investigation of fixed-points of a sum of two nonlinear operators, or more generally, fixed-points of mappings on the cartesian product 𝐸×𝐸 into 𝐸, where 𝐸 is some appropriate space.

Fixed-point theorems in topology and nonlinear functional analysis are usually based on certain properties (such as complete continuity, monotonicity, contractiveness, etc.) that the operator, considered as a single entity must satisfy. We recall for instance the Banach fixed-point theorem, which asserts that a strict contraction on a complete metric space into itself has unique fixed-point, and the Schauder principle, which asserts that a continuous mapping 𝐴 on a closed convex set 𝑀 in Hausdorff locally convex topological vector space 𝐸 into 𝑀 such that 𝐴(𝑀) is contained in a compact set, has a fixed-point. In many problems of analysis, one encounters operators which may be split in the form 𝑇=𝐴+𝐵, where 𝐴 is a contraction in some sense, and 𝐵 is completely continuous, and 𝑇 itself has neither of these properties (see [1–3]). Thus neither the Schauder fixed-point theorem nor the Banach fixed-point theorem applies directly in this case, and it becomes desirable to develop fixed-point theorems for such situations. An early theorem of this type was given by Krasnosel’skiĭ [4]: “Let 𝐸 be a Banach space, 𝑀 be a bounded closed convex subset of 𝐸, and 𝐴,𝐵 be operators on 𝑀 into 𝐸 such that 𝐴𝑥+𝐵𝑦∈𝑀 for every pair 𝑥,𝑦∈𝑀. If 𝐴 is a strict contraction and 𝐵 is continuous and compact, then the equation 𝐴𝑥+𝐵𝑥=𝑥 has a solution in 𝑀.” This result has been extended to locally convex spaces in 1971 by Cain and Nashed [5]. There is also another theorem of this type which was given by Amar et al. [6] in 2005 and which extended the Schauder and Krasnoselskii fixed-point theorems in Dunford-Pettis spaces to weakly compact operators. Also in 2010, Amar and Mnif [7] established some new variants of Leray-Schauder type fixed-point theorems for weakly sequentially continuous operators.

In this paper, we give also a generalization of Krasnoselskii fixed-point theorems not in Dunford-Pettis Banach spaces but in Dunford-Pettis Frechet spaces. More precisely, let 𝐸 be a Frechet topological vector space having the property of Dunford-Pettis, 𝑀 a closed bounded convex subset of 𝐸, and 𝐴=𝐵𝐶 (where 𝐵 is a continuous map and 𝐶 is a linear weakly compact operator). If 𝐴 leaves 𝑀 invariant then 𝐴 has a fixed-point in 𝑀 (see Proposition 3.1). In addition, if 𝐵 is a 𝜑-contraction map of 𝑀 into 𝐸, for each 𝑥,𝑦∈𝑀 with 𝐵𝑥+𝐴𝑦∉𝑀, there is a 𝑧∈(𝑥,𝐵𝑥+𝐴𝑦)∩𝑀 such that 𝐵𝑧+𝐴𝑦∈𝑀 and (𝐼−𝐵)⁻¹𝐴(𝑀) is relatively weakly compact, then 𝐴+𝐵 has a fixed-point in 𝑀 (see Proposition 3.3).

Based on our results and other theorems which was given by Sehgal and Singh in 1976 ([8]), we give also an extension of the Krasnoselskii fixed-point theorem: Let 𝐸 be a Frechet topological vector space having the property of Dunford-Pettis (DP), 𝑀⊆𝐸, 𝐶∶𝑀→𝐶(𝑀)⊆𝐸 a compact operator (An operator 𝐶∶𝑀→𝐸 is said to be compact if it is continuous and maps bounded sets into precompact.) and 𝑇 a map defined on the set 𝑀×𝐶(𝑀) and having range in 𝐸. By assuming that the family {𝑇(⋅,𝑦)∶𝑦∈𝐶(𝑀)} is nonlinear 𝜑 equicontractive we prove the existence of a point 𝑥∈𝑀 such that𝑥=𝑇(𝑥,𝐶(𝑥)). (1.1)

Our paper is organized as follows. In Section 2, we give some important definitions and preliminaries which will be used in this paper. Among this preliminaries we cite definition of Dunford-Pettis space, the theorems of Schauder-Tychonoff, Krein-Smulian. The Section 3 is devoted to the generalization of the Krasnoselskii fixed-point theorem in Dunford-Pettis Frechet spaces where our proofs of our two results (Proposition 3.3 and Theorem 3.5) in this section are based on the theorem of Sehgal and Singh and the main result (Proposition 3.1).

2. Preliminaries

In this section, we give the following well-known definitions and results which will be used in this paper.

Definition 2.1. Suppose that 𝐸 and 𝐹 are locally convex spaces. A continuous linear operator 𝐴 from 𝐸 into 𝐹 is said to be weakly compact if 𝐴(𝐵) is relatively weakly compact subset of 𝐹 whenever 𝐵 is a bounded subset of 𝐸.

Theorem 2.2 (Eberlein-Smulian, see [9]). Let 𝐸 be a metrizable locally convex topological vector space, (𝑥_𝑛)_𝑛 a weakly relatively compact sequence in 𝐸. Then from (𝑥_𝑛)_𝑛 may be extracted a weakly convergent subsequence.

Definition 2.3. A subset 𝐶 in a vector space 𝐸 is called balanced if for all 𝑥∈𝐶, 𝜆𝑥∈𝐶 if |𝜆|≤1.

Definition 2.4 (see [9, 10]). A locally convex topological vector space 𝐸 is said to have the Dunford-Pettis (DP) property if any continuous linear map of 𝐸 into a complete locally convex topological vector space 𝐹, which transforms bounded sets into weakly relatively compact sets, transforms each balanced and weakly compact subset of 𝐸 into a relatively compact subset of 𝐹.

Remark 2.5 (see [9]). If 𝐸 is complete, we replace in the precedent definition each balanced and weakly compact subset of 𝐸 by each weakly compact subset of 𝐸.

Theorem 2.6 (see [11]). Let 𝐸 be a locally convex topological vector space and 𝑀 a convex subset of 𝐸. Then 𝑀 is closed if and only if it is weakly closed.

Theorem 2.7 (Krein-Smulian). Let 𝐸 be a metrizable and complete locally convex topological vector space and 𝑀⊂𝐸 weakly compact. Then the closed convex hull of 𝑀 is weakly compact.

Theorem 2.8 (Schauder-Tychonoff [12]). Let 𝑀 be a closed and convex subset of a locally convex topological vector space 𝐸 and 𝐴∶𝑀→𝑀 a continuous mapping such that the range 𝐴(𝑀) is contained in a compact set. Then 𝐴 has a fixed-point.

In the remainder of this section, 𝐸 denotes a Frechet topological vector space having the Dunford-Pettis (DP) property and 𝜗 is a neighborhood basis of the origin consisting of absolutely convex open subsets of 𝐸. Let for each 𝑈∈𝜗,  𝑝_𝑈 the Minkowski's functional of 𝜗.

Let 𝑀 be a nonempty subset of 𝐸. A mapping 𝐴∶𝑀→𝐸 is a 𝑈-contraction (𝑈∈𝜗) if for each 𝜖>0 there is a 𝛿>0 such that if 𝑥,𝑦∈𝑀 and if 𝑥−𝑦∈(𝜖+𝛿)𝑈,then𝐴(𝑥)−𝐴𝑦∈𝜖𝑈. (2.1) If 𝐴∶𝑀→𝐸 is a 𝑈-contraction for each 𝑈∈𝜗, then 𝐴 is a 𝜗-contraction.

Note that if 𝐴 is a 𝜗-contraction, then 𝐴 is continuous. (For a related definition of 𝜗-contraction, see Taylor [13].)

Lemma 2.9 (see [8]). Let 𝐴∶𝑀→𝐸 be a 𝜗-contraction, then 𝐴 is 𝜗-contractive, that is for each 𝑈∈𝜗,  𝑝_𝑈(𝐴(𝑥)−𝐴(𝑦))<𝑝_𝑈(𝑥−𝑦) if 𝑝_𝑈(𝑥−𝑦)≠0 and 0, otherwise.

Theorem 2.10 (Theorem of Sehgal and Singh [8]). Let 𝑀 be a sequentially complete subset of a complete separated locally convex topological vector space 𝐹 and 𝐴∶𝑀→𝐹 be a 𝜗-contraction. If 𝐴 satisfies the condition: (𝑥)((𝑥(𝑧)foreach𝑥∈𝑀with𝐴∉𝑀,thereisa𝑧∈𝑥,𝐴))∩𝑀suchthat𝐴∈𝑀. (2.2) Then 𝐴 has a unique fixed-point in 𝑀.

Definition 2.11. Let 𝑇∶𝑀×𝐸→𝐸 be a map such that 𝑀 be a nonempty subset of 𝐸. The family {𝑇(⋅,𝑦)∶𝑦∈𝐸} is called 𝑈-equicontractive (𝑈∈𝜗) if for each 𝜖>0 there is a 𝛿>0 such that if (𝑥₁,𝑦),(𝑥₂,𝑦) in the domain of 𝑇 and if 𝑥₁−𝑥₂∈(𝜖+𝛿)𝑈,then𝑇𝑥₁,𝑦−𝑇𝑥₂,𝑦∈𝜖𝑈. (2.3)

If {𝑇(⋅,𝑦)∶𝑦} is a 𝑈-equicontractive for each 𝑈∈𝜗, then the family {𝑇(⋅,𝑦)∶𝑦} is a 𝜗-equicontraction. Note that if the family {𝑇(⋅,𝑦)∶𝑦} is a 𝜗-equicontraction, then the operator𝑥→𝑇(𝑥,𝑦) is a 𝜗-contraction for all 𝑦.

Definition 2.12. let 𝜑={𝑝=𝑝_𝑈 for some 𝑈∈𝜗}, ℝ⁺ the nonnegative reals and 𝜓 a family of mapping defined as 𝜓={Φ∶ℝ⁺→ℝ⁺ such that Φ is continuous and Φ(𝑡)<𝑡 if 𝑡>0}. A mapping 𝐴∶𝑀→𝐸 is a nonlinear 𝜑 contraction (see [14]) if for each 𝑝∈𝜑, there is a Φ_𝑝∈𝜓 such that 𝑝(𝐴(𝑥)−𝐴(𝑦))≤Φ_𝑝(𝑝(𝑥−𝑦)) for all 𝑥,𝑦∈𝑀. If this inequality holds with Φ_𝑝(𝑡)=𝛼_𝑝𝑡 such that 0<𝛼_𝑝<1, then 𝐴 is called 𝜑-contraction (see [5]).

Since a nonlinear 𝜑 contraction is a 𝜗-contraction, the following result immediately follows by Theorem 2.10 and provides an extension of a result in [5]:

Theorem 2.13 (see [8]). Let 𝑀 be a sequentially complete subset of a complete separated locally convex topological vector space 𝐹 and 𝐴∶𝑀→𝐹 be a nonlinear 𝜑 contraction. If 𝐴 satisfies (2.2) then 𝐴 has a unique fixed-point in 𝑀.

Definition 2.14. The family {𝑇(⋅,𝑦)∶𝑦∈𝐸} is called nonlinear 𝜑 equicontractive if for each 𝑝∈𝜑, there is a Φ_𝑝∈𝜓 such that if (𝑥₁,𝑦),(𝑥₂,𝑦) in the domain of 𝑇, then 𝑝𝑇𝑥₁,𝑦−𝑇𝑥₂,𝑦≤Φ_𝑝𝑝𝑥₁−𝑥₂. (2.4)

Remark 2.15. Since any nonlinear 𝜑 contraction is a 𝜗-contraction then any nonlinear 𝜑 equicontraction is a 𝜗-equicontraction.

3. Krasnoselskii's Type Theorems

In this section, we will give some new fixed-point results for the sum of two operators where 𝐸 is a Frechet topological vector space having the Dunford-Pettis property. Firstly, we give the following proposition which is a generalization of Theorem 2.1 in [6].

Proposition 3.1. Let 𝐸 be a Frechet topological vector space having the Dunford-Pettis property, 𝑀 a closed, bounded and convex subset of 𝐸 and 𝐵,𝐶 two operators such that:(i)𝐵∶𝐸↦𝐸 a continuous map;(ii)𝐶∶𝐸↦𝐸 a linear weakly compact operator on 𝐸;(iii)𝐵(𝐶(𝑀)) is relatively weakly compact;(iv)𝐴(𝑀)⊂𝑀. Then 𝐴=𝐵𝐶 has a fixed-point in 𝑀.

Proof. We denote by 𝑁=co(𝐴(𝑀)), the closed convex hull of 𝐴(𝑀). Firstly, we show that 𝑁 is a weakly compact subset of 𝐸. Indeed, we have 𝐴(𝑀)⊂𝐵(𝐶(𝑀)). This implies that 𝐴(𝑀) is relatively weakly compact and therefore 𝐴(𝑀) is weakly compact. We have 𝐴(𝑀)⊂𝐴(𝑀)⟹𝑁=co(𝐴(𝑀))⊂co𝐴(𝑀) (3.1) and since 𝐴(𝑀) is weakly compact, then by Krein-Smulian's theorem co(𝐴(𝑀)) is also weakly compact. Since 𝑁 is a closed convex subset of 𝐸, therefore it is weakly closed and this implies that 𝑁 is a weakly closed subset of a weakly compact. Consequently, 𝑁 is weakly compact.
Now, we show that 𝐶(𝑁) is relatively compact. We have 𝑁 is a weakly compact set in 𝐸 and 𝐶 is a weakly compact operator on 𝐸 and since 𝐸 is a Frechet topological vector space having the Dunford-Pettis property, then by Definition 2.4, we have 𝐶(𝑁) is a relatively compact set in 𝐸. Since 𝐵 is a continuous map, then 𝐵𝐶(𝑁) is a relatively compact set in 𝐸.
Moreover, we have𝐴(𝑀)⊂𝑀soco(𝐴(𝑀))⊂co(𝑀). (3.2) Therefore 𝑁=co(𝐴(𝑀))⊂𝑀 (3.3) and this implies that 𝐴(𝑁)⊂𝐴(𝑀)⊂co(𝐴(𝑀))=𝑁, (3.4) where 𝑁 is a closed convex and 𝐴(𝑁)=𝐵𝐶(𝑁) is a relatively compact set. Since 𝐶 is a weakly compact oprator on 𝐸, then by Definition 2.1  𝐶 is continuous and so 𝐴∶𝑁→𝑁 is continuous. Finally, the use of Schauder-Tychonoff's fixed-point theorem shows that 𝐴 has at least one fixed-point in 𝑁⊂𝑀.

Lemma 3.2. Let 𝐸 be a Frechet topological vector space, 𝑀 a sequentially complete subset of 𝐸 and𝐵∶𝑀↦𝐸 a nonlinear 𝜑 contraction. Suppose that for 𝑦∈𝐸 we have: for each 𝑥∈𝑀with 𝐵𝑥+𝑦∉𝑀, there is a 𝑧∈(𝑥,𝐵𝑥+𝑦)∩𝑀such that𝐵𝑧+𝑦∈𝑀. Then, there exists a unique 𝑢(𝑦)∈𝑀 with 𝐵(𝑢(𝑦))+𝑦=𝑢(𝑦), that is (𝐼−𝐵)⁻¹𝑦=𝑢(𝑦)∈𝑀.

Proof. Consequence of Theorem 2.13 (see [8]).

The following proposition is a generalization of Theorem 2.2 in [6].

Proposition 3.3. Let 𝐸 be a Frechet topological vector space having the Dunford-Pettis property, 𝑀 a closed, bounded and convex subset of 𝐸 and 𝐴,𝐵 two operators such that:(i)𝐴∶𝐸↦𝐸 a linear weakly compact operator on 𝐸;(ii)𝐵∶𝑀↦𝐸be a 𝜑-contraction;(iii)For each 𝑥,𝑦∈𝑀 with 𝐵𝑥+𝐴𝑦∉𝑀, there is a 𝑧∈(𝑥,𝐵𝑥+𝐴𝑦)∩𝑀 such that 𝐵𝑧+𝐴𝑦∈𝑀;(iv)(𝐼−𝐵)⁻¹𝐴(𝑀) is relatively weakly compact.Then there exists 𝑦 in 𝑀 such that 𝐴𝑦+𝐵𝑦=𝑦

Proof. Firstly, we have 𝐵 is a 𝜑-contraction then 𝐵 is a continuous function and for any 𝑥,𝑦∈𝑀 we have 𝑝_𝑈(𝐼−𝐵)𝑥−(𝐼−𝐵)𝑦≥𝑝_𝑈𝑥−𝑦−𝑝_𝑈𝐵𝑥−𝐵𝑦≥1−𝛼_𝑝𝑝_𝑈𝑥−𝑦 (3.5) with 𝛼_𝑝∈(0,1) which gives the continuity of (𝐼−𝐵)⁻¹.
Now, by Lemma 3.2 equation 𝑧=𝐵𝑧+𝐴𝑦 has a unique solution 𝑧∈𝑀 for all 𝑦∈𝑀. It follows, that𝑧=(𝐼−𝐵)⁻¹𝐴𝑦∈𝑀, (3.6) so (𝐼−𝐵)⁻¹𝐴(𝑀)⊂𝑀. (3.7) For conclusion, we have (𝐼−𝐵)⁻¹ is a continuous mapping, 𝐴 a linear weakly compact operator on 𝐸 and (𝐼−𝐵)⁻¹𝐴(𝑀) is relatively weakly compact on 𝐸 where (𝐼−𝐵)⁻¹𝐴(𝑀)⊂𝑀. So, by Proposition 3.1, we prove that (𝐼−𝐵)⁻¹𝐴 has a fixed-point in 𝑀 and this implies that, there exists 𝑦∈𝑀 such that 𝐴𝑦+𝐵𝑦=𝑦.

We will now take 𝐶∶𝑀→𝐶(𝑀)⊆𝐸 a compact operator and 𝑇 a map defined on the set 𝑀×𝐶(𝑀) and having range in 𝐸. We are interested to the existence of a point 𝑥∈𝑀⊂𝐸 such that 𝑥=𝑇(𝑥,𝐶(𝑥)). (H)

Proposition 3.4. Let 𝐸 be a Frechet topological vector space, 𝑀 a bounded sequentially complete subset of 𝐸 and 𝑇∶𝑀×𝐸⟶𝐸 (3.8) a map such that the family {𝑇(⋅,𝑦)∶𝑦∈𝐸} is nonlinear 𝜑 equicontractive, for all 𝑥∈𝑀, 𝑦→𝑇(𝑥,𝑦) is continuous and which satisfies the condition: for each (𝑥,𝑦)∈𝑀×𝐸 with 𝑇(𝑥,𝑦)∉𝑀, there is a 𝑧∈𝑥,𝑇𝑥,𝑦∩𝑀suchthat𝑇𝑧,𝑦∈𝑀. (3.9)
Then there exists a continuous map 𝐹_𝑇∶𝐸→𝑀 such that𝑇𝐹_𝑇𝑦,𝑦=𝐹_𝑇𝑦. (3.10)

Proof. We start from an arbitrary point 𝑦∈𝐸. Since the family {𝑇(⋅,𝑦)∶𝑦∈𝐸} is a nonlinear 𝜑 equicontractive then the operator 𝑥⟶𝑇𝑥,𝑦∶𝑀⟶𝐸isanonlinear𝜑contraction (3.11) which satisfy for each 𝑥∈𝑀 with 𝑇(𝑥,𝑦)∉𝑀, there is a 𝑧∈𝑥,𝑇𝑥,𝑦∩𝑀suchthat𝑇𝑧,𝑦∈𝑀. (3.12) Then by Theorem 2.13, there is a unique point 𝑥=𝐹_𝑇(𝑦)∈𝑀 that satisfies the operator equation: 𝑇𝐹_𝑇𝑦,𝑦=𝐹_𝑇𝑦. (3.13) We will show that the mapping 𝑦↦𝐹_𝑇(𝑦)∶𝐸→𝑀 is continuous. To do this we let (𝑦_𝑛) be a sequence in 𝐸, with lim𝑦_𝑛=𝑦₀∈𝐸. We suppose that 𝐹_𝑇(𝑦_𝑛) does not converge to 𝐹_𝑇(𝑦₀). Then there exist 𝑝∈𝜑, an 𝜖>0 and 𝜚(𝑛) such that 𝑝𝐹_𝑇𝑦_𝜚(𝑛),𝐹_𝑇𝑦₀>𝜖∀𝑛∈ℕ. (3.14) Since {𝑝(𝐹_𝑇(𝑦_𝜚(𝑛)),𝐹_𝑇(𝑦₀))>𝜖, 𝑛∈ℕ} is a bounded real subsequence, it has a subsequence {𝑝(𝐹_𝑇(𝑦_{𝜚(𝜚1(𝑛))}),𝐹_𝑇(𝑦₀)),𝑛∈ℕ}→𝑟≥0. However, we have 𝑝𝐹_𝑇𝑦_{𝜚(𝜚1(𝑛))}−𝐹_𝑇𝑦₀=𝑝𝑇𝐹_𝑇𝑦_{𝜚(𝜚1(𝑛))},𝑦_{𝜚(𝜚1(𝑛))}−𝑇𝐹_𝑇𝑦₀,𝑦₀≤𝑝𝑇𝐹_𝑇𝑦_{𝜚(𝜚1(𝑛))},𝑦_{𝜚(𝜚1(𝑛))}−𝑇𝐹_𝑇𝑦₀,𝑦_𝜚((𝑛))+𝑝𝑇𝐹_𝑇𝑦₀,𝑦_{𝜚(𝜚1(𝑛))}−𝑇𝐹_𝑇𝑦₀,𝑦₀≤Φ_𝑝𝑝𝐹_𝑇𝑦_{𝜚(𝜚1(𝑛))}−𝐹_𝑇𝑦₀+𝑝𝑇𝐹_𝑇𝑦₀,𝑦_{𝜚(𝜚1(𝑛))}−𝑇𝐹_𝑇𝑦₀,𝑦₀ (3.15) which implies that 𝑟=0. This contradicts (3.14) and consequently 𝐹_𝑇 is continuous.

In what follows, we give also another result of Krasnoselskii type.

Theorem 3.5. Let 𝑀 be a closed, bounded and convex subset of a Frechet topological vector space having the Dunford-Pettis property 𝐸 and 𝐶∶𝑀→𝐸 a linear weakly compact operator such that the image of 𝐶(𝑀) by any continuous mapping is contained in a weakly compact subset of 𝐸. Let 𝑇∶𝑀×𝐶(𝑀)⟶𝐸 (3.16) be a map such that the family {𝑇(⋅,𝑦)∶𝑦∈𝐶(𝑀)} is nonlinear 𝜑 equicontractive, for all 𝑥∈𝑀, 𝑦↦𝑇(𝑥,𝑦) is continuous on 𝐶(𝑀) and which satisfies that for each (𝑥,𝑦)∈𝑀×𝐶(𝑀) with 𝑇(𝑥,𝑦)∉𝑀, there is a 𝑧∈𝑥,𝑇𝑥,𝑦∩𝑀suchthat𝑇𝑧,𝑦∈𝑀. (3.17) Then (H) admits a solution in 𝑀.

Proof. We start from an arbitrary point 𝑦∈𝐶(𝑀). By Proposition 3.4 we prove that there exists a unique point 𝑥=𝐹_𝑇(𝑦)∈𝑀 that satisfies the operator equation 𝑇𝐹_𝑇𝑦,𝑦=𝐹_𝑇𝑦, (3.18) where the mapping 𝑦↦𝐹_𝑇(𝑦)∶𝐶(𝑀)→𝑀 is continuous. Then the operator 𝐹_𝑇𝐶 maps the set 𝑀 into itself. We have by hypothesis that 𝐹_𝑇(𝐶(𝑀)) is contained in a weakly compact subset of 𝐸. Therefore, by Proposition 3.1, we prove the existence of a point 𝑥∈𝑀 such that 𝐹_𝑇(𝐶(𝑥))=𝑥. This means that 𝑇(𝑥,𝐶(𝑥))=𝑇𝐹_𝑇(𝐶(𝑥)),𝐶(𝑥)=𝐹_𝑇(𝐶(𝑥))=𝑥. (3.19)