On a fixed point theorem of Krasnosel'skii type and application to integral equations

Fixed Point Theory and Applications, Nov 2006

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.

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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)


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Le Thi Phuong Ngoc, Nguyen Thanh Long. On a fixed point theorem of Krasnosel'skii type and application to integral equations, Fixed Point Theory and Applications, 2006, pp. 30847, Volume 2006, Issue 1, DOI: 10.1155/FPTA/2006/30847