Some problems in the fixed point theory

In this paper we present some of my favorite problems, all the time open, in the fixed point theory. These problems are in connection with the following two: • Which properties have the fixed point equations for which an iterative algorithm is convergent ? • Let us have a fixed point theorem, T , and an operator f (single or multivalued) which does not satisfy the conditions in T . In which conditions the operator f has an invariant subset Y such that the restriction of f to Y , f ∣∣ Y , satisfies the conditions of T ?


Introduction
In this paper we present some problems, all the time open problems, in the fixed point theory.These problems are in connection with the following two research directions: (I) Which properties have the fixed point equations for which an iterative algorithm is convergent ?(II) Let us have a fixed point theorem, T , and an operator f (single or multivalued) which does not satisfy the conditions in the theorem T .In which conditions the operator f has an invariant subset Y such that the restriction of f to Y , f Y , satisfies the conditions of T ?

Picard and weakly Picard operators
Let (X, →) be an L-space ( (X, τ )-topological space, τ →; (X, d)-metric space, d →; (X, • )-normed space, • →, ; . ..) and f : X → X be an operator.By definition, f is a weakly Picard operator if the sequence {f n (x)} n∈N converges for all x ∈ X at its limit (which may depend on x) is a fixed point of f .If f is a weakly Picard operator, then we consider the operator f ∞ : X → X, defined by, f ∞ (x) := lim n→∞ f n (x).We remark that the operator f ∞ is a set retraction on the fixed point set of f , F f .If f is a weakly Picard operator and F f = {x * }, then by definition f is called Picard operator.If f is a Picard operator, we have that, and if f is a weakly Picard operator, then, In the case of a metric space and of a contraction we have the following result.
Theorem 2.1 (see [47]).Let (X, d) be a complete metric space and f : X → X be an l-contraction.Then we have: From this result, the following problem rises: Problem 2.2.Let (X, d) be a complete metric space and f : X → X be an operator.Which metric conditions on f imply a similar conclusion as that of Theorem 2.1 ?
Let us consider another result: Theorem 2.3 (see [48]).Let (X, d) be a complete metric space and f : X → X be an operator.We suppose that: (1) There exists l ∈]0, 1[ such that, i.e., f is a graphic contraction. ( ), for all x ∈ X.

Then we have:
(i) f is a weakly Picard operator.
(iv) Let {y n } n∈N be a sequence in X x * , x * ∈ F f .If l < 1 3 and This result suggests the following problem: Problem 2.4 (see [48]).Which metric conditions imposed on an operator f imply a similar conclusion as that in Theorem 2.3 ?
For a better understanding of the above problems, let us consider the following considerations: (a) A weakly Picard operator f : (X, d) → (X, d) satisfies a retraction-displacement condition (see [8]) if there exists an increasing function ψ : R + → R + , ψ(0) = 0 and continuous in 0, such that This condition is useful in studying the data dependence of the fixed point, and of Ulam stability of the fixed point equations (see [44]).
So, conclusions (ii) in Theorems 2.1 and 2.3 are retraction-displacement conditions for the operator f .
(b) Conclusions (iii) in Theorems 2.1 and 2.3 can be formulated as follows: The fixed point problem for the operator f is well posed.
(c) Conclusions (iv) in Theorems 2.1 and 2.3 can be formulated as follows: The operator f has the Ostrowski property.
Problem 2.5.To study similar problems in the case of multivalued operators.

Conjecture on global asymptotic stability
Let (X, →) be an L-space and f : X → X be an operator.A fixed point x * of f is by definition globally asymptotically stable if f is a Picard operator, i.e., f n (x) → x * as n → ∞, for all x ∈ X.
In 1976, J.P. LaSalle presented (see [20]) the following conjecture: Conjecture 1 (LaSalle's Conjecture).Let f : R m → R m be such that: (iii) the spectral radius of the differential of f at x, ρ(df (x)) < 1, for all x ∈ R m .
Then, x * is globally asymptotically stable.
Papers on this conjecture were given by (see [46]): A. Cima -A.Gasull -F.Mañosas (1995,1999,2001,2011,2014) We have the following remark: Let (X, →) be an L-space and f : X → X be an operator.The following statements are equivalent: Starting from this general remark, in [46] the following conjecture is presented.Problem 3.1 (a conjecture).Let X be a real Banach space, Ω ⊂ X be an open, convex subset and f : Ω → Ω be an operator.We suppose that: Then, f is a Picard operator.
In connection with the above conjecture the following problems arise: Problem 3.2.In which conditions we have that: In which conditions we have that: ρ(df (x)) < 1, for all x ∈ Ω ⇒ f is nonexpansive with respect to an equivalent norm on X?
We remember that if (X, • ) is a complex Banach space and f : X → X is a bounded linear operator with the spectrum σ(f ), then (see [17], [5], [14], [4], . . . ) If X is a real Banach space and f : X → X is a bounded linear operator, X C the complexification of X, f C : X C → X C the complexification of f , then by definition, ρ(f ) := ρ(f C ).

Nonexpansive operators and graphic contractions
Problem 4.1.Let (X, • ) be a (real or complex) Banach space.Which nonexpansive operators f : X → X are graphic contractions ?
Commentaries: If f is a graphic contraction then inf On the other hand, in the case of nonexpansive operators we have the following Goebel-Karlovitz Lemma (see [12]): Let Ω ⊂ X be a convex, closed and bounded subset.Let D ⊂ Ω be a weakly compact, convex, minimal invariant set for a nonexpansive operator f : Ω → Ω.If for a sequence {x n } n∈N , lim n→∞ x n − f (x n ) = 0, then for any z ∈ D, we have that, . So, the above problem is a hard one.Problem 4.2.Let X be an ordered Banach space.Which increasing, linear and nonexpansive operators f : X → X are graphic contractions ?Problem 4.3.Let X be a Banach space.Which multivalued nonexpansive operators T : X → P (X) are graphic contractions ?

Abstract and concrete Gronwall lemmas
Let (X, →, ≤) be an ordered L-space and f : X → X be an operator.The following results are well known (see [38]: Lemma 5.1 (Abstract Gronwall Lemma for Picard operators).We suppose that: (ii) f is an increasing operator.
Then we have that: Lemma 5.2 (Abstract Gronwall Lemma for weakly Picard operators).We suppose that: (i) f is a weakly Picard operator; (ii) f is an increasing operator Then we have that: The above abstract Gronwall lemmas are very usefully for giving some concrete Gronwall lemmas.On the other hand a large number of concrete Gronwall lemmas are obtained by direct proofs.The following problems are arising: Problem 5.3.In which Gronwall lemmas the upper bounds are fixed points of the corresponding operator ?Problem 5.4.If there are found solutions for the Problem 5.3, which of them are consequences of some abstract Gronwall lemmas ?

Invariant subsets with fixed point property
For a rigorous formulation of a problem (II), from Introduction, we recall a few basic notions and examples of the fixed point structure theory (see [37]).
Let C be a class of structured sets (ordered sets, ordered linear spaces, topological spaces, metric spaces, Hilbert spaces, Banach spaces, ordered Banach spaces, generalized metric spaces, . . .).Let Set * be the class of nonempty sets and if X is a nonempty set, then, P (X) := {Y ⊂ X | Y = ∅}.We also shall use the following notations: ) By a fixed point structure (f.p.s.) on X ⊂ C we understand a triple (X, S(X), M ) with the following properties: Here are some examples of f.p.s.Then, (X, S(X), M ) is a f.p.s.
Example 6.3 (The f.p.s. of contractions).Let C be the class of complete metric spaces.Let Then, (X, S(X), M ) is a f.p.s.Then, (X, S(X), M ) is a f.p.s.Now, our problem (II) takes the following form: Problem 6.5.Let (X, S(X), M ) be a f.p.s. on X ∈ C and f : A → A be an operator with A ⊂ X.In which conditions there exists Y ⊂ A such that We have a similar problem in the case of multivalued operators.

Strict fixed point problems
Let X be a nonempty set and T : X → P (X) be a multivalued operator.Let F T := {x ∈ X | x ∈ T (x)} be the set of fixed point of T and (SF ) T := {x ∈ X | T (x) = {x}} be the strict fixed point set of T .
We have the following result (see [33], p.87): Let (X, d) be a metric space and T : X → P (X) be a multivalued l-contraction.If, (SF ) T = ∅, then, The following problem is arising: Problem 7.1.For which multivalued generalized contractions we have that (SF ) T = ∅ ⇒ F T = (SF ) T = {x * } ?Problem 7.2.Let (X, S(X), M • ) be a multivalued fixed point structure (see [37]) on X ∈ C. Let Y ∈ S(X) and T ∈ M • (Y ).In which conditions we have that F T = (SF ) T ? Commentaries: (1) Let f, g : R → R be such that: Then we have that, F T = (SF ) T .

Example 6 . 4 (
The f.p.s. of Schauder).Let C be the class of Banach spaces.Let S(X) := {Y ∈ P (X) | Y is compact and convex} and M (Y ) := {f : Y → Y | f is continuous}.