Nielsen number and differential equations
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In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations), two main approaches are presented. The first is via Poincar´e’s translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics) are indicated, jointly with some further consequences like the nontrivial Rδ -structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated. 1. Introduction: motivation for differential equations Our main aim here is to show some applications of the Nielsen number to (multivalued) differential equations (whence the title). For this, applicable forms of various Nielsen theories will be formulated, and then applied—via Poincar´e and Hammerstein operators—to associated initial and boundary value problems for differential equations and inclusions. Before, we, however, recall some Sharkovskii-like theorems in terms of differential equations which justify and partly stimulate our investigation. Consider the system of ordinary differential equations x = f (t,x),
f (t,x) ≡ f (t + ω,x),
(1.1)
where f : [0,ω] × Rn → Rn is a Carath´eodory mapping, that is, (i) f (·,x) : [0,ω] → Rn is measurable, for every x ∈ Rn , (ii) f (t, ·) : Rn → Rn is continuous, for a.a. t ∈ [0,ω], (iii) | f (t,x)| ≤ α|x| + β, for all (t,x) ∈ [0,ω] × Rn , where α, β are suitable nonnegative constants. By a solution to (1.1) on J ⊂ R, we understand x ∈ ACloc (J, Rn ) which satisfies (1.1), for a.a. t ∈ J. 1.1. n = 1. For scalar equation (1.1), a version of the Sharkovskii cycle coexistence theorem (see [8, 14, 15, 17]) applies as follows. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:2 (2005) 137–167 DOI: 10.1155/FPTA.2005.137
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Nielsen number and differential equations
Figure 1.1. braid σ.
Theorem 1.1. If (1.1) has an m-periodic solution, then it also admits a k-periodic solution, for every k m, with at most two exceptions, where k m means that k is less than m in the celebrated Sharkovskii ordering of positive integers, namely 3 5 7 · · · 2 · 3 2 · 5 2 · 7 · · · 2 2 · 3 2 2 · 5 22 · 7 · · · 2 m · 3 2 m · 5 2 m · 7 · · · 2 m · · · 22 2 1. In particular, if m = 2k , for all k ∈ N, then infinitely many (subharmonic) periodic solutions of (1.1) coexist. Remark 1.2. Theorem 1.1 holds only in the lack of uniqueness; otherwise, it is empty. On the other hand, f on the right-hand side of (1.1) can be a (multivalued) upperCarath´eodory mapping with nonempty, convex, and compact values. Remark 1.3. Although, for example, a 3ω-periodic solution of (1.1) implies, for every k ∈ N with a possible exception for k = 2 or k = 4,6, the existence of a kω-periodic solution of (1.1), it is very difficult to prove such a solution.
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