On a fixed point theorem of Krasnosel'skii type and application to integral equations
0 Nguyen Thanh Long: Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University , 227 Nguyen Van Cu Street, Dist. 5, Ho Chi Minh, Vietnam E-mail address:
This paper presents a remark on a fixed point theorem of Krasnosel'skii type. This result is applied to prove the existence of asymptotically stable solutions of nonlinear integral equations. t t
1. Introduction
It is well known that the fixed point theorem of Krasnosel’skii states as follows.
Theorem 1.1 (Krasnosel’skii [8] and Zeidler [9]). Let M be a nonempty bounded closed
convex subset of a Banach space (X, · ). Suppose that U : M → X is a contraction and
C : M → X is a completely continuous operator such that
U(x) + C(y) ∈ M,
∀x, y ∈ M.
Then U + C has a fixed point in M.
The theorem of Krasnosel’skii has been extended by many authors, for example, we
refer to [1–4, 6, 7] and references therein.
In this paper, we present a remark on a fixed point theorem of Krasnosel’skii type and
applying to the following nonlinear integral equation:
x(t) = q(t) + f t, x(t) +
V t, s, x(s) ds +
G t, s, x(s) ds, t ∈ R+,
where E is a Banach space with norm | · |, R+ = [0, ∞), q : R+ → E; f : R+ × E → E;
G, V : Δ × E → E are supposed to be continuous and Δ = {(t, s) ∈ R+ × R+, s ≤ t}.
In the case E = Rd and the function V (t, s, x) is linear in the third variable, (1.2) has
been studied by Avramescu and Vladimirescu [2]. The authors have proved the existence
of asymptotically stable solutions to an integral equation as follows:
x(t) = q(t) + f t, x(t) +
V (t, s)x(s)ds +
G t, s, x(s) ds, t ∈ R+,
where q : R+ → Rd; f : R+ × Rd → Rd; V : Δ → Md(R), G : Δ × Rd → Rd are supposed
to be continuous, Δ = {(t, s) ∈ R+ × R+, s ≤ t} and Md(R) is the set of all real quadratic
d × d matrices. This was done by using the following fixed point theorem of Krasnosel’skii
type.
is bounded.
Then the operator C + D admits fixed points.
In [6], Hoa and Schmitt also established some fixed point theorems of Krasnosel’skii
type for operators of the form U + C on a bounded closed convex subset of a locally
convex space, where C is completely continuous and Un satisfies contraction-type conditions.
Furthermore, applications to integral equations in a Banach space were presented.
On the basis of the ideas and techniques in [2, 6], we consider (1.2). The paper consists
of five sections. In Section 2, we prove a fixed point theorem of Krasnosel’skii type. Our
main results will be presented in Sections 3 and 4. Here, the existence solution and the
asymptotically stable solutions to (1.2) are established. We end Section 4 by illustrated
examples for the results obtained when the given conditions hold. Finally, in Section 5, a
general case is given. We show the existence solution of the equation in the form
G t, s, x(s), x χ(s) ds, t ∈ R+,
and in the case π(t) = t, the asymptotically stable solutions to (1.5) are also considered.
The results we obtain here are in part generalizations of those in [2], corresponding to
(1.3).
2. A fixed point theorem of Krasnosel’skii type
Based on the Theorem 1.2 (see [1]) and [6, Theorem 3], we obtain the following theorem.
The proof is similar to that of [6, Theorem 3].
Proof of Theorem 2.1. At first, we note that from the hypothesis (i), the existence and the
continuity of the operator (I − U)−1 follow. And, since a family of seminorms · n is
equivalent with the family | · |n, there exist K1n, K2n > 0 such that
K1n x n ≤ |x|n ≤ K2n x n,
∀n ∈ N∗.
xm − x n = 0 ⇐⇒ mli→m∞
xm − x n = 0,
≤ K1n
≤ K2n
∀x ∈ X, ∀n ∈ N∗.
Hence, lim|x|n→∞(|Cx|n/|x|n) = 0 is equivalent to lim x n→∞( Cx n/ x n) = 0.
Now we will prove that U + C has a fixed point.
For any a ∈ X, define the operator Ua : X → X by Ua(x) = U(x) + a. It is easy to see that
Ua is a k-contraction mapping and so for each a ∈ X, Ua admits a unique fixed point, it
is denoted by φ(a), then
∀y ∈ X.
UCm(x) u0 − u0 n =
UC(x) UCm(−x)1 u0
− U u0 n
UC(x) UCm(−x)1 u0
− U UCm(−x)1 u0
n + U UCm(−x)1 u0
− U u0 n
≤ C(x) n + k UCm(−x)1 u0 − u0 n,
UCm(x) u0 − u0 n ≤ 1 + k + · · · + km−1
where α = 1/1 − k > 1. By the condition (iii) satisfied with respect to
1/4α > 0, there exists M > 0 (we choose M > u0 n) such that
· n as above, for
x n > M =⇒
Choose a positive constant r1n > M + u0 n. Thus, for all x ∈ X, we consider the following
two cases.
x − u0 n + u0 n
Dn = x ∈ X : x n ≤ r2n ,
D =
n∈N∗
For each x ∈ D and for each n ∈ N∗, as above we also consider two cases.
If x − u0 n ≤ r1n, then by (2.7), (2.10),
UCm(x) u0 − u0 n ≤ α C(x) n ≤ αβ < r2n.
We obtain UCm(x)(u0) ∈ D for all x ∈ D.
On the other hand, by UC(x) being a contraction mapping, the sequence UCm(x)(u0)
converges to the unique fixed point φ(C(x)) of UC(x), as m → ∞, it implies that φ(C(x)) ∈
D, for all x ∈ D. Hence, (I − U)−1C(D) ⊂ D.
Applying the Schauder fixed point theorem, the operator (I − U)−1C has a fixed point
in D that is also a fixed point of U + C in D.
3. Existence of solution
Let X = C (...truncated)