Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 3, 1-7; http://www.math.u-szeged.hu/ejqtde/

Fixed points for some non-obviously contractive operators defined in a space of continuous

functions

Cezar Avramescu and Cristian Vladimirescu

Abstract 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))| +

+ \frac{k}{t^{α}} ∫_{0}^{t} |x (σ (s)) − y (σ (s))| ds,

(where α, β ∈ [0, 1), k ≥ 0, and ν, σ : [0, T] → [0, T] are continuous functions such that ν (t) ≤ t, σ (t) ≤ t, ∀t ∈ [0, T]) has exactly one fixed point in M. Then the result is extended in C (IR_{+}, X) , where IR_{+} := [0, ∞).

1. 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,

|(Ax) (t) − (Ay) (t)| ≤ β |x (t) − y (t)| +
+\frac{k}{t^{α}} ∫_{0}^{t} |x (s) − y (s)| ds, ∀t ∈ [0, T] . (1.1)

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) .

2. The first existence result

Consider the space C ([0, T] , X), where (X, |·|) is a Banach space, T > 0 and let γ ∈ (0, T) , λ > 0.

Define for x ∈ C ([0, T] , X) ,

∥x∥ := ∥x∥_{γ} + ∥x∥_{λ} ,

where we denoted

∥x∥_{γ} := sup_{t∈[0,γ]} {|x (t)|} , ∥x∥_{λ} := sup_{t∈[γ,T]} {e^{−λ(t−γ)} |x (t)|} .

It is easily seen that ∥·∥ is a norm on C ([0, T] , X) and it defines the same topology as the norm ∥·∥_{∞} , where

∥x∥_{∞} := sup_{t∈[0,T]} {|x (t)|} .

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],

|(Ax) (t) − (Ay) (t)| ≤ β |x (ν (t)) − y (ν (t))| +
+\frac{k}{t^{α}} ∫_{0}^{t} |x (σ (s)) − y (σ (s))| ds,  (2.1)

where ν, σ : [0, T] → [0, T] are continuous functions such that ν (t) ≤ t, σ (t) ≤ t, ∀t ∈ [0, T] , then A has a unique fixed point in M.

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,

∥Ax − Ay∥ ≤ δ ∥x − y∥ .

Let t ∈ [0, γ] be arbitrary. Then we have

|(Ax) (t) − (Ay) (t)| ≤ β |x (ν (t)) − y (ν (t))| +

+\frac{k}{t^{α}} ∫_{0}^{t} |x (σ (s)) − y (σ (s))| ds ≤

≤ β ∥x − y∥_{γ} + t^{1−α}k ∥x − y∥_{γ} ≤
≤ (β + kγ^{1−α}) ∥x − y∥_{γ}

and hence

∥Ax − Ay∥_{γ} ≤ (β + kγ^{1−α}) ∥x − y∥_{γ} .  (2.2)

Let t ∈ [γ, T] be arbitrary. Then we get

|(Ax) (t) − (Ay) (t)| ≤ β |x (ν (t)) − y (ν (t))| +

+\frac{k}{t^{α}} (∫_{0}^{γ} |x (σ (s)) − y (σ (s))| ds+

+ ∫_{γ}^{t} |x (σ (s)) − y (σ (s))| e^{−λ((σ(s))−γ)}e^{λ((σ(s))−γ)}ds)

≤ β |x (ν (t)) − y (ν (t))| + \frac{k}{γ^{α}} (γ ∥x − y∥_{γ} +

+ ∥x − y∥_{λ} ∫_{γ}^{t} e^{λ(σ(s)−γ)}ds)

≤ β |x (ν (t)) − y (ν (t))| + \frac{k}{γ^{α}} (γ ∥x − y∥_{γ} +

+ ∥x − y∥_{λ} ∫_{γ}^{t} e^{λ(s−γ)}ds)

< β |x (ν (t)) − y (ν (t))| + \frac{k}{γ^{α}} (γ ∥x − y∥_{γ} +

+ ∥x − y∥_{λ} \frac{e^{λ(t−γ)}}{λ} ) .

It follows that

|(Ax) (t) − (Ay) (t)| e^{−λ(t−γ)} < β |x (ν (t)) − y (ν (t))| e^{−λ(t−γ)} +

+kγ^{1−α} ∥x − y∥_{γ} + \frac{k}{λ}γ^{−α} ∥x − y∥_{λ}

and therefore

∥Ax − Ay∥_{λ} ≤ β sup_{t∈[γ,T]} {|x (ν (t)) − y (ν (t))| e^{−λ(t−γ)}} +  (2.3)

+kγ^{1−α} ∥x − y∥_{γ} + \frac{k}{λ}γ^{−α} ∥x − y∥_{λ}

≤ β sup_{t∈[γ,T]} {|x (ν (t)) − y (ν (t))| e^{−λ(ν(t)−γ)}} +

+kγ^{1−α} ∥x − y∥_{γ} + \frac{k}{λ}γ^{−α} ∥x − y∥_{λ}

≤ (β + \frac{k}{λ}γ^{−α}) ∥x − y∥_{λ} + kγ^{1−α} ∥x − y∥_{γ} .

By (2.2) and (2.3) we obtain

∥Ax − Ay∥ ≤ (β + 2kγ^{1−α}) ∥x − y∥_{γ} + (β + \frac{k}{λ}γ^{−α}) ∥x − y∥_{λ} .  (2.4)

Since β ∈ [0, 1), for γ ∈ (0, (\frac{1−β}{2k} )^{\frac{1}{1−α}} ) we deduce β + \frac{k}{λ}γ^{1−α} < 1 and for

λ > \frac{k}{1−β} γ^{−α} we deduce γ + \frac{k}{λ}γ^{−α} < 1. Let δ := max \{β + \frac{k}{λ}γ^{1−α}, γ + \frac{k}{λ}γ^{−α}\} . It follows that δ < 1 and, since (2.4),

∥Ax − Ay∥ ≤ δ (∥x − y∥_{γ} + ∥x − y∥_{λ}) = δ ∥x − y∥ .

Hence, A is contraction.

From the Banach Contraction Principle we conclude that A has exactly one fixed point in M.

Remark 2.1 We remark that if ν (t) = t and σ (t) = t, ∀t ∈ [0, T] , then the conditions (1.1) and (2.1) are identical.

3. 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∥_{γ_{n}} + ∥x∥_{λ_{n}} , for every x ∈ C (IR_{+}, X) , and

∥x∥_{γ_{n}} := sup_{t∈[0,γ_{n}]} {|x (t)|} , ∥x∥_{λ_{n}} := sup_{t∈[γ_{n},T]} {e^{−λ(t−γ_{n})} |x (t)|} .

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

d (x, y) := ∑_{n=1}^{∞} \frac{1}{2^{n}} · \frac{∥x − y∥_{n}}{1 + ∥x − y∥_{n}} , ∀x, y ∈ C (IR_{+}, X) .

Notice that a sequence {x_{m}}_{m∈IN} ⊂ C (IR_{+}, X) converges to x if and only if

∀n ∈ IN^{∗}, lim_{m→∞} ∥x_{m} − x∥_{n} = 0.

In 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

or, more easily, if and only if

∀n ∈ IN^{∗}, lim_{p,q→∞} ∥x_{p} − x_{q}∥_{n} = 0.

Theorem 3.1 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],

|(Ax) (t) − (Ay) (t)| ≤ β_{n} |x (ν (t)) − y (ν (t))| +
+\frac{k}{t^{α_{n}}} ∫_{0}^{t} |x (σ (s)) − y (σ (s))| ds,  (3.1)

where ν, σ : IR_{+} → IR_{+} are continuous functions such that ν (t) ≤ t, σ (t) ≤ t, ∀t ∈ IR_{+}, then A has a unique fixed point in M.

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 x, y ∈ M,

∥Ax − Ay∥_{n} ≤ δ_{n} ∥x − y∥_{n}, ∀n ∈ IN^{∗}.  (3.2)

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

∥x_{m+1} − x_{m}∥_{n} = ∥Ax_{m} − Ax_{m−1}∥_{n} ≤ δ_{n} ∥x_{m} − x_{m−1}∥_{n} , ∀m ∈ IN^{∗}

and therefore

∥x_{m+1} − x_{m}∥_{n} ≤ δ_{n}^{m} ∥x_{1} − x_{0}∥_{n} , ∀m ∈ IN.

Similarly,

∥x_{m+p} − x_{m}∥_{n} ≤ (δ_{n}^{m+p} + ... + δ_{n}^{m}) ∥x_{1} − x_{0}∥_{n} <
< \frac{δ_{n}^{m}}{1 − δ_{n}} ∥x_{1} − x_{0}∥_{n} , ∀m ∈ IN, p ∈ IN^{∗}.

So, {x_{m}}_{m∈IN} is fundamental and hence it will be convergent. Let x_{∗} := lim_{m→∞} x_{m} ∈ M. By (3.2) it follows that Ax_{m} → Ax_{∗} or, equivalently, x_{m} → Ax_{∗}. Therefore, x_{∗} = Ax_{∗}.

If A would have another fixed point in M, say x_{∗∗}, it would follow that

∥x_{∗} − x_{∗∗}∥_{n} = ∥Ax_{∗} − Ax_{∗∗}∥_{n} ≤ δ_{n} ∥x_{∗} − x_{∗∗}∥_{n}

and so ∥x_{∗} − x_{∗∗}∥_{n} (1 − δ_{n}) ≤ 0, ∀n ∈ IN^{∗}. But δ_{n} ∈ [0, 1). It follows that x_{∗} = x_{∗∗}.

The proof of Theorem 3.1 is now complete.

Remark 3.1 If the relation (1.1) holds for all t ∈ IR_{+}, then the relation (3.1) holds.

In particular, the condition (3.1) is fulfilled if for every x, y ∈ M and t ∈ [0, n],

|(Ax) (t) − (Ay) (t)| ≤ β (t) |x (ν (t)) − y (ν (t))| +

+\frac{k (t)}{t^{α(t)}} ∫_{0}^{t} |x (σ (s)) − y (σ (s))| ds,

where α : IR_{+} → [0, 1), β : IR_{+} → [0, 1), and k : IR_{+} → IR_{+}, are continuous functions.

Indeed, in this case we can set

β_{n} := sup_{t∈[0,n]} {β (t)} , k_{n} := sup_{t∈[0,n]} {k (t)} , α_{n} := inf_{t∈[0,n]} {α (t)}, ∀n ∈ IN^{∗}.

Remark 3.2 Within the proof of Theorem 3.1 we have get the fixed point of A as limit of the iterative sequence. It is interesting to remark that the fixed point of A can be obtained as limit of other sequences.

We present in the sequel an example.

Consider the space C ([0, n] , X) and let

M_{n} := {x |_{[0,n]}, x ∈ M}

i.e. M_{n} is the set of the restrictions of x ∈ M to [0, n] , ∀n ∈ IN^{∗}.

Let n ∈ IN^{∗} be arbitrary. One has obviously AM_{n} ⊂ M_{n}. By applying Theorem 2.1, A has a unique fixed point x_{n} ∈ M_{n}. We extend x_{n} to IR_{+} by continuity: for example, one could set

x̃_{n} (t) := \{ \begin{matrix}x_{n} (t) , if t ∈ [0, n] \\ x_{n} (n) , if t ≥ n \end{matrix}

and hence x̃_{n} ∈ C (IR_{+}, X) .

By the uniqueness property of the fixed point we have

x̃_{n} (t) = x̃_{m} (t) , ∀m ≤ n, ∀t ∈ [0, m] ,  (3.3)

which allows us to conclude that \left\{{ \atop x_{n}}\right\}_{n∈IN^{∗}} converges in C (IR_{+}, X) to the function x^{∗} : IR_{+} → X defined by

x^{∗} (t) = x̃_{n} (t) , ∀t ∈ [0, n] .  (3.4)

Notice that x^{∗} is well defined due to (3.3) .

Let t ∈ IR_{+} be arbitrary. Then there exists n_{0} ∈ IN^{∗} such that t ∈ [0, n_{0}] . But

x^{∗} (t) = x̃_{n_{0}} (t) = (Ax̃_{n_{0}}) (t) = (Ax^{∗}) (t) ,

and so x^{∗} (t) = (Ax^{∗}) (t) . Since t was arbitrary in IR_{+}, it follows x^{∗} = Ax^{∗}.

4. Applications

A particular case when the previous existence results can be applied is the following.

Consider an integral equation of the type

x (t) = F (t, x (ν (t))) + \frac{1}{t^{α(t)}} ∫_{0}^{t} K (t, s, x (σ (s))) ds,  (4.1)

where α ∈ [0, 1) and F : J × IR^{N} → IR^{N}, K : Δ → IR^{N}, α : J → [0, 1) are continuous functions. Here,

J = [0, T] or J = IR_{+}, Δ = {(t, s, x) | t, s ∈ J, 0 ≤ s ≤ t, x ∈ IR^{N}}

and ν, σ : J → J are continuous functions such that ν (t) ≤ t, σ (t) ≤ t, ∀t ∈ J. Consider the continuous functions β : J → [0, 1), γ : J → IR_{+}. If

|F (t, x) − F (t, y)| ≤ β (t) |x − y| , ∀x, y ∈ IR^{N}, t ∈ J,
|K (t, s, x) − K (t, s, y)| ≤ k (t) |x − y| , ∀ (t, s, x) , (t, s, y) ∈ Δ,

then the equation (4.1) has exactly one solution.

Indeed, it is easily checked the hypotheses of Theorem 2.1 and Theorem 3.1.

References

[1] M.A. Krasnoselskii, Some problems of nonlinear analysis, Amer. Math. Soc. Translations, 10(2) 345-409 (1958).

[2] E. De Pascale, L. De Pascale, Fixed points for some non-obviously contractive operators, Proc. Amer. Math. Soc., 130(11), 3249-3254 (2002).

[3] B. Lou, Fixed points for operators in a space of continuous functions and applications, Proc. Amer. Math. Soc., 127, 2259-2264 (1999).

Authors’ addresses:

Cezar AVRAMESCU and Cristian Vladimirescu Department of Mathematics, University of Craiova Al.I. Cuza Street, No. 13, Craiova RO-200585, Romania Tel. & Fax: (+40) 251 412 673 E-mail: cezaravramescu@hotmail.com, zarce@central.ucv.ro vladimirescucris@hotmail.com, cvladi@central.ucv.ro