On the fixed point theorem of

We formulate a version of the fixed point theorem of Krasnoselskii and Sobolev in locally convex spaces. We apply this result to establish the existence of solutions of an integral equation defined in an abstract space. Mathematics Subject Classification Numbers 2010: Primary 47H10; Secondary 45N05


Fixed Points of Monotone Operators
In this work we are concerned with the existence of fixed points for a class of monotone operators defined on locally convex spaces.For generalities about locally convex spaces we refer the reader to [1].
Let X be a Hausdorff locally convex space.Let K ⊆ X be a closed convex cone such that K ∩ (−K) = {0}.The cone K defines the partial ordering in X given by x ≤ y if y − x ∈ K.An operator Γ : X → X is called monotone if x, y ∈ X, x ≤ y implies that Γx ≤ Γy.The operator Γ is said to be monotonically limit compact on a bounded set M ⊆ X if for each sequence (x n ) n in M such that we have that the sequence (Γ n x n ) n is convergent.We refer the reader to [2] for a discussion about the properties of monotonically limit compact operators.In particular, the following result [2,Theorem 38.2] has been established by Krasnoselskii and Sobolev.
Theorem 1.1.Let X be a Banach space.Let Γ : X → X be a monotone operator which is monotonically limit compact on a closed bounded set M ⊆ X such that ΓM ⊆ M. If there is an x 0 ∈ M with Γx 0 ≤ x 0 , then Γ has a fixed point in M.
Henceforth we will assume that (X, τ ) is a Hausdorff locally convex space which satisfies the following property: (P) For every closed and bounded set A ⊆ X the induced topology is metrizable and complete.
It is clear that every Fréchet space satisfies the property (P).In addition, if X is a reflexive separable Banach space, then (X, σ(X, X * )) also satisfies the property (P).
The main result of this note is the following extension of Theorem 1.1.
Theorem 1.2.Let X be a real Hausdorff locally convex space which satisfies the property (P).Let Γ : X → X be a monotone operator which is monotonically limit compact on a closed bounded set M ⊆ X such that ΓM ⊆ M. If there is an Proof.We only need to modify slightly the construction carried out in the proof of [2,Theorem 38.2].Let N ⊆ X be a closed bounded absolutely convex set such that M − M ⊆ N. Let ρ be a metric on N which induces the relative topology τ in N. We can assume that ρ is bounded.We set For x ∈ M 0 and j ∈ N we define It is easy to see that α j (x) makes sense and that the sequence (α j (x)) j is nonincreasing.Hence we can define α(x) = lim j→∞ α j (x).It follows from the definition that α(Γx) ≤ α(x).
We assert that inf{α(u) : u ∈ M 0 , u ≤ x} = 0 for any x ∈ M 0 .In fact, if we assume that this assertion is false, proceeding inductively we can construct a sequence such that ρ(Γ 2j w j − Γ 2j v j , 0) > β for some β > 0. This implies the sequence does not converge, which contradicts our hypothesis that Γ is a monotonically limit compact operator.
Proceeding similarly we can construct a sequence (x n ) n in M 0 such that Using again that Γ is a monotonically limit compact operator, we can affirm that there exists z ∈ M such that Γ n x n → z as n → ∞.Since Γ n x n ≥ Γ n+1 x n ≥ Γz we have that z ≥ Γz.Moreover, for each n ∈ N there exists m n ∈ N such that ρ(Γ n+mn x n+mn − Γ n+mn z, 0) ≤ 1 n .In view of that z − Γz ≤ Γ n+mn x n+mn − Γ n+mn z it follows that z ≤ Γz.Consequently, z = Γz.

Applications
The fixed point Theorem 1.2 can be applied to study the existence of solutions of many ordinary or integral equations, with or without delay, defined by a noncontinuous function, on an unbounded interval of R.
In this section we will show the usefulness of the Theorem 1.2 to establish sufficient conditions for the existence of solutions of an abstract integral equation on R.
To specify the problem under consideration, we will assume that (X, • ) is a Banach space and that K ⊆ X is a closed convex cone such that K ∩ (−K) = {0}.We will assume that K is a fully regular cone, which means that every norm bounded nondecreasing sequence in X is convergent.We denote by ≤ the partial ordering induced by K in X such as was established in the Section 1.Let C(R, X) be the space consisting of the continuous functions x : R → X endowed with the compact-open topology.It is well known that C(R, X) is a Fréchet space.Moreover, Hence we can consider C(R, X) as an ordered locally convex space.We consider the following integral equation EJQTDE, 2011 No. 5, p. 3 where x, h : R → X and k are continuous functions.Moreover, the values h(s) ≤ 0 for all s ∈ R, and the function f : R × X → X verifies the following conditions: (f1) The function f is Borel-measurable.
(f2) For every separable set A ⊆ X the set f (R × A) is separable.
(f3) There exists a locally integrable function g : R → [0, ∞) such that f (s, x) ≤ g(s) for all s ∈ R, x ∈ X, and For example, a function f which is the pointwise limit of a sequence of continuous functions satisfies conditions (f1) and (f2).
Remark 2.1.If f satisfies conditions (f1)-(f3) and x : R → X is a continuous function, it follows from (f1), (f2) and [3,Proposition 2.2.6] that the function s → f (s, x(s)) is strongly measurable on (−∞, t] for every t ∈ R. Combining this property with (f3) and the Lebesgue dominated convergence theorem [3,Théorème 2.4.7]we infer that the function s → f (s, x(s)) is integrable on (−∞, t] for every t ∈ R. Proof.We define the map Γ : It follows from the preceding remark that Γ is well defined.Moreover, if x(•) ≤ y(•), it follows from (f4) that f (s, x(s)) ≤ f (s, y(s)), which means that k(t, s)[f (s, y(s))− f (s, x(s))] ∈ K. Let a < t be fixed.Since K is a closed convex cone, applying the mean value theorem for the Bochner integral we obtain that

Theorem 2 . 3 .
Under the above assumptions, there exists a continuous solution of the equation (2.1).