• Corpus ID: 14111629

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS BY FIXED POINT THEOREMS

@inproceedings{BurtonASYMPTOTICBO,
  title={ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS BY FIXED POINT THEOREMS},
  author={Theodore A. Burton and Tetsuo Furumochi}
}

• T. Burton, T. Furumochi
• Mathematics

In a series of papers we have studied stability properties of functional differential equations by means of fixed point theory. We enlarge that study now by also considering delay equations which may be unstable when the delay is zero. We continue to focus on challenging examples to illustrate the work, as opposed to attempting to state general theorems. Here, we deal with three different examples. In Part I we obtain asymptotic stability using Schauder’s and Banach’s fixed point theorems; it advances results used by Liapunov techniques in several ways, but particularly by placing no smoothness on the delay. It also advances some of our earlier work with fixed point theory in that it places no smoothness conditions on the nonlinear perturbation. In Part II we prove boundedness and asymptotic stability using Krasnoselskii’s fixed point theorem. Schaefer’s fixed point theorem is used to prove that there is a periodic solution when a periodic forcing function is added to that equation. AMS (MOS) subject classification. 34D20, 34K20, 47H10 1. OVERVIEW Liapunov’s direct method is by far the most general method for dealing with stability, boundedness, and the existence of periodic solution of ordinary and functional differential equations. But over the last hundred years investigators have encountered a number of difficulties which have remained unresolved. Several years Typeset by AMS-TEX ago we began a project [2-5] to see if those problems could be circumvented using fixed point theory. Since all of those problems had originated with specific examples, it seemed that our endeavor should center around difficult examples. Thus, we have offered no general theorems. Instead, we have presented examples which are known to cause severe difficulties using Liapunov’s direct method. Our results here are of two different types. In Part I we look at a typical problem on which there has been much work using Liapunov functionals and we show how fixed point theory can be used to circumvent several difficulties encountered in that method. In Part II we continue studying that problem under conditions which have prevented the use of Liapunov functions; we show that fixed point theory can be used to show boundedness, asymptotic stability, and the existence of periodic solutions. Here is a general overview of the work. The equation x(t) = −a(t)x(t − r1) + b(t)g(x(t − r2(t))) has been studied by means of several Liapunov functionals under different assumptions. In most, but not all, cases it is assumed that (A) r1 = 0 and (B) |g(x)| ≤ |x| for small x. If, in addition, one asks that (C) r2 is a positive constant, then one is led to the condition (D) a(t) ≥ |b(t + r2)| and to the Liapunov functional V (t, xt) = |x(t)| + ∫ t t−r2 |b(s + r2)||g(x(s)|ds with derivative satisfying V (t, xt) ≤ −a(t)|x(t)|+ |b(t)g(x(t− r2))|+ |b(t+ r2)g(x(t))|− |b(t)g(x(t− r2))| ≤ 0. We have V positive definite and V ′ ≤ 0 so the zero solution is stable. With additional conditions asymptotic stability can be obtained. Investigators have successfully removed (C) at some cost. We can ask instead of (C) that there are positive constants α and k with (E) r2(t) continuous and positive, 1 − r ′ 2(t) ≥ α, and (F) a(t) ≥ k ≥ |b(t)|/α which leads to the Liapunov functional V (t, xt) = |x(t)| + k ∫ t t−r2(t) |g(x(s))|ds whose derivative can be made non-positive. Our project seeks to reduce or remove (A)-(F) by means of fixed point theory. In an earlier paper [3] we used contraction mappings by assuming (A) and (B), but eliminating (C) and (E), while reducing (D) to an averaging condition, at the cost of asking (G) that g satisfies a local Lipschitz condition. Part I of this paper uses Schauder’s fixed point theorem to remove (G), resulting in conditions close to (A), (B), and an averaging form of (D) implying asymptotic stability. This is a significantly better result than has been obtained via Liapunov functionals. In Part II we continue with the equation, but we violate every one of the conditions (A) (G) and consider x(t) = −a(t)x(t − r1) + b(t)x (t− r2(t)) where r1 is a positive constant. Notice now that g(x) = x 1/3 and so x does not dominate g. If ri = 0 then it is a Bernoulli equation; with a(t) > 0 and b(t) < 0 a simple Liapunov function will at least yield stability and sometimes asymptotic stability. On the other hand, if a and b are positive constants then the zero solution can be non-unique to the right, and hence, there can be no stability. But, surprisingly, if a(t) > 0 and dominates b(t) sufficiently strongly then the zero solution can be asymptotically stable independent of the sign of b(t). Thus, this equation is very rich in behavior of solutions. When the ri are positive, then it is a challenging problem and provides a good test for the effectiveness of fixed point theory as a tool for proving stability and boundedness. Our first result uses Krasnoselskii’s fixed point theorem to prove boundedness of solutions in the case corresponding to a and b being positive constants mentioned above. Our third result yields asymptotic stability via Krasnoselskii’s fixed point theorem when a dominates b sufficiently strongly. For our second result we suppose that our equation is given a periodic perturbation and that a, b are also periodic. Then, under essentially the same conditions as our first result, we obtain a periodic solution by means of Schaefer’s fixed point theorem. This result is again related to Liapunov theory. It is well known that if a system of differential equations has a globally Lipschitz Liapunov function with negative definite derivative, then the equation can be given a sufficiently small perturbation and that same Liapunov function will yield boundedness. Our second result offers a parallel property. It is to be noted that other authors are also using fixed point theory to prove stability, although in different contexts. For example, Serban [10] has used a Picard operator to prove asymptotic stability of a difference equation. Rus [8] uses Picard operators to prove data dependence of boundary value problems. PART I Asymptotic Stability: Two fixed point theorems 2. PRELIMINARIES In this part, we discuss asymptotic behavior of solutions of functional differential equations by using the following theorem, which can be found in Smart [5; p. 15]. For reference it may be stated as follows. THEOREM 2.1 (Schauder’s first theorem). Let (C, ‖·‖) be a normed space, and let S be a compact convex nonempty subset of C . Then every continuous mapping of S into S has a fixed point. First we introduce a concept of equi-convergence. Let r0 be a fixed nonnegative constant. DEFINITION (Equi-convergence). The set S of real-valued functions on [−r0,∞) is said to be equi-convergent to 0 if there is a function q : R + → R := [0,∞) such that q(t) → 0 as t→ ∞, and that for any φ ∈ S and t ∈ R, |φ(t)| ≤ q(t) holds. Now we prepare an Ascoli-Arzela like lemma. For the proof, see Lemma 3 in Burton and Furumochi [4]. LEMMA. If the set {φk(t)} of real-valued functions on [−r0,∞) is equiconvergent to 0 with respect to a function q : R → R, and is equi-continuous, then the sequence {φk(t)} contains a subsequence that converges uniformly on R + to a continuous function φ(t) with |φ(t)| ≤ q(t) on R. 3. FIRST ORDER HALF-LINEAR EQUATIONS First consider the scalar half-linear equation (3.1) x(t) = −a(t)x(t) − b(t)g(x(t − r(t))), t ∈ R, where a, b, r : R → R := (−∞,∞) and g : R → R are continuous. Let α be any fixed positive number. We assume that there are constants β > 0, γ > 0 and r0 ≥ 0 so that (3.2) |g(x)| ≤ β|x| for |x| ≤ α,