Fixed Points for Some Non-obviously Contractive Operators Defined in a Space of Continuous Functions

Let X be an arbitrary (real or complex) Banach space, endowed with the norm |·|. Consider the space of the continuous functions C ([0, T ] , X) (T > 0), endowed with the usual topology, and let M be a closed subset of it. One proves that each operator A : M → M fulfilling for all x, y ∈ M and for all t ∈ [0, T ] the condition |(Ax) (t) − (Ay) (t)| ≤ β |x (ν (t)) − y (ν (t))| + + k t α t 0 |x (σ (s)) − y (σ (s))| ds,


Introduction
A result due to Krasnoselskii (see, e.g.[1]) ensures the existence of fixed points for an operator which is the sum of two operators, one of them being compact and the other being contraction.A natural question is whether the result continues to hold if the first operator is not compact.In [2] and [3] the case when the compactity is replaced to a Lipschitz condition is considered; the result is proved only in the space of the continuous functions.
More precisely, let X be a (real or complex) Banach space, endowed with the norm |•| .Consider the space C ([0, T ] , X) of the continuous functions from [0, T ] into X (T > 0) , endowed with the usual topology and M a closed subset of C ([0, T ] , X).
Let A : M → M be an operator with the property that there exist α, β ∈ [0, 1), k ≥ 0 such that for every x, y ∈ M , In [2] the authors resume the result contained in [3] and prove that the condition (1.1) ensures the existence in M of a unique fixed point for A; the result is deduced through a subtle technique.Finally, by admitting that (1.1) is fulfilled for every t ∈ IR + , the result is generalized to the space BC (IR + , X) , (where IR + := [0, ∞)), i.e. the space of the bounded and continuous functions from IR + into X.
In the present paper we give an alternative proof of the first result contained in [2], in a more general case, by means of a new approach; more exactly, we use in C ([0, T ] , X) a special norm which is equivalent to the classical norm.Then we extend the result to the space C (IR + , X) .
Define for x ∈ C ([0, T ] , X) , where we denoted It is easily seen that • is a norm on C ([0, T ] , X) and it defines the same topology as the norm • ∞ , where Theorem 2.1 Let M be a closed subset of C ([0, T ] , X) and A : M → M be an operator.If there exist α, β ∈ [0, 1), k ≥ 0 such that for every x, y ∈ M and for every t ∈ [0, T ], where ν, σ Proof.We shall apply the Banach Contraction Principle.To this aim, we show that A is contraction, i.e. there exists δ ∈ [0, 1) such that for any x, y ∈ M, Let t ∈ [0, γ] be arbitrary.Then we have Let t ∈ [γ, T ] be arbitrary.Then we get It follows that By (2.2) and (2.3) we obtain Hence, A is contraction.From the Banach Contraction Principle we conclude that A has exactly one fixed point in M.

The second existence result
As we mentioned in Section 1, in [2] is presented a generalization in the space BC (IR + , X) if (1.1) is fulfilled for every t ∈ IR + .We shall prove that result under slightly more general assumptions.
Consider the space C (IR + , X) and for every n ∈ IN * let γ n ∈ (0, n), λ n > 0. Define the numerable family of seminorms { • n } n∈IN * , where x n := x γn + x λn , for every x ∈ C (IR + , X) , and As it is known, C (IR + , X) endowed with this numerable family of seminorms becomes a Fréchet space, i.e. a metrisable complete linear space.Also, the most natural metric which can be defined is Notice that a sequence {x m } m∈IN ⊂ C (IR + , X) converges to x if and only if Proof.As we have seen within the proof of Theorem 2.1, by choosing conveniently γ n ∈ (0, n) and λ n > 0, there exists δ n ∈ [0, 1) such that for any The proof of Theorem 3.1 is similar to the proof of the Banach Contraction Principle.We build the iterative sequence x m+1 = Ax m , ∀m ∈ IN, where x 0 ∈ M is arbitrary.
Let n ∈ IN * be arbitrary.One has
2004 No. 3, p. 4In addition, a sequence {x m } m∈IN ⊂ C (IR + , X) is fundamental if and only if ∀n ∈ IN * , ∀ε > 0, ∃m 0 ∈ IN, ∀p, q ≥ m 0 , x p − x q n < ε Let M be a closed subset of C (IR + , X) and A : M → M be an operator.If for every n ∈ IN * there exist α n , β n ∈ [0, 1), k n ≥ 0 such that for every x, y ∈ M and for every t ∈ [0, n], : IR + → IR + are continuous functions such that ν (t) ≤ t, σ (t) ≤ t, ∀t ∈ IR + , then A has a unique fixed point in M.