axioms Article Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate Lokesh Singh 1,* and Dhirendra Bahuguna 2 1 Department of Mathematics, University of Rijeka, HR-51000 Rijeka, Croatia 2 Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India; [email protected] * Correspondence: [email protected] Abstract: In this article, we construct a C1 stable invariant manifold for the delay differential equation 0 x = Ax(t) + Lxt + f (t, xt) assuming the r-nonuniform exponential dichotomy for the corresponding 1 solution operator. We also assume that the C perturbation, f (t, xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f (t, xt). Keywords: delay differential equation; r-nonuniform exponential dichotomy; stable manifold; arbitrary growth rates MSC: 34D09; 34K19; 34K20; 37D10; 37D25 1. Introduction Citation: Singh, L.; Bahuguna, D. The invariant manifold theory started with the work of Hadamard [1] in 1901 when Smooth Stable Manifold for Delay he constructed a manifold in the solution space of a differential equation with the property Differential Equations with Arbitrary that if the trajectory of a solution starts in the manifold, it will remain in the manifold for Growth Rate. Axioms 2021, 10, 105. all time t > 0. This proved to be of great importance for analyzing complex systems as it https://doi.org/10.3390/axioms10020105 reduces the relevant dimension significantly. Later, Perron [2] and Lyapunov [3] developed another method to construct the invariant manifolds for autonomous differential equations. Academic Editor: Emil Saucan In the 1970s, due to the fundamental works of Hirsch et al. [4,5], Sacker and Sell [6], and Pesin [7,8], this theory became an important instrument for various fields like applied Received: 19 April 2021 mathematics, biology, and engineering. Accepted: 20 May 2021 In the first method for constructing the invariant manifolds, Hadamard [1] used the Published: 25 May 2021 geometrical properties of differential equations. He constructed the manifold over the linearized stable and unstable subspaces. However, Lyapunov [3] and Perron [2] developed Publisher’s Note: MDPI stays neutral an analytical method to construct the invariant manifolds. They obtained the invariant with regard to jurisdictional claims in published maps and institutional affil- manifolds, using the variation of constants formula of the differential equations. In his iations. approach, Perron introduced (and assumed) the notion of (uniform) exponential dichotomy for the solution operators and proved the existence of Lipschitz stable invariant manifolds for the small nonlinear perturbation of autonomous differential equations. The smoothness of these invariant manifolds is proved by Pesin [7]. In 1977, Pesin [8] generalized the notion of uniform hyperbolicity to nonuniform hyperbolicity which allows the rate of expansion Copyright: © 2021 by the authors. and contraction to depend on initial time. Later he proved the stable manifold theorem in Licensee MDPI, Basel, Switzerland. the finite-dimensional settings for nonhyperbolic trajectories. Pugh and Shub [9] proved a This article is an open access article distributed under the terms and similar result for nonhyperbolic trajectories using the method developed by Hadamard. conditions of the Creative Commons Ruelle [10] extended the result by Pesin to the Hilbert space settings in 1982. Attribution (CC BY) license (https:// The exponential dichotomy played an essential role in the development of invariant creativecommons.org/licenses/by/ manifold theory for autonomous differential equations. Barreira and Valls extended the 4.0/). notion of exponential dichotomy for nonautonomous differential equations and called Axioms 2021, 10, 105. https://doi.org/10.3390/axioms10020105 https://www.mdpi.com/journal/axioms Axioms 2021, 10, 105 2 of 14 it nonuniform exponential dichotomy. With the assumption of nonuniform exponential dichotomy, they constructed the Lipschitz invariant manifold [11] and smooth invariant manifold [12] for nonautonomous differential equations. They also obtained the essential conditions for the existence of the nonuniform exponential dichotomy. The book [13] contains all the early works of Barreira and Valls. Barreira and Valls observed that the solution operators corresponding to a class of nonautonomous differential equations show dichotomic behavior and also they have growth or decay rates of ecr(t), for some function r(t). They named this notion as r- nonuniform exponential dichotomy. They showed in article [14] that the class of differential equations for which all the Lyapunov’s exponents are infinite, satisfies r-nonuniform exponential dichotomy for r(t) 6= t. Subsequently they proved the stable manifold theorem for ordinary differential equations assuming r-nonuniform exponential dichotomy in [15]. In the article [16], Pan proved the existence of Lipschitz stable invariant manifold for impulsive nonautonomous differential equations with the assumption of r-nonuniform exponential dichotomy. In this article, we consider a differential equation with the infinite delay given by, 0 x = Ax(t) + Lxt + f (t, xt), xs = f, (1) in a Banach space X. We assume that the solution operator associated with the correspond- ing linear delay differential equation 0 x = Ax(t) + Lxt, xs = f, (2) satisfies r-nonuniform exponential dichotomy and the nonlinear perturbation f (t, xt) is sufficiently small and smooth. With these assumptions, we prove the existence of a C1 stable invariant manifold for the delay differential Equation (1) following the approach of Perron and Lyapunov. We also showed the dependence of invariant manifolds on perturbations. Barreira and Valls used the r-nonuniform exponential dichotomy for the nonau- tonomous differential equations in [17] to construct the Lipschitz stable invariant manifold and in [18] to construct the smooth invariant manifold. Pan considered the impulsive differential equation in [16], to construct the Lipschitz stable invariant manifold assuming r-nonuniform exponential dichotomy. In the article [19], we considered the case of delay differential equations with nonuniform exponential dichotomy and constructed a Lipschitz invariant manifold. However, in this article, we are assuming a more general r-nonuniform exponential dichotomy for differential equation with infinite delay and we are constructing a C1 stable invariant manifold. In the later part of the article, we also show that a small change in perturbation gives rise to a small variation in the manifold. The paper is arranged in the following manner. Our setup and some preliminary results are given in Section2. In the next section, we provide a few examples of differential equations satisfying the r-nonuniform exponential dichotomy. Section3 contains the proof of the existence of the C1 stable invariant manifold, and in Section4, we prove the dependency of the manifold on perturbation. In the end, we present a few more examples satisfying the assumptions of our main theorem. 2. Preliminaries Let (X, k kX) be a Banach space. For any interval J ⊂ R := (−¥, ¥), we denote C(J, X) as a space of X-valued continuous function on J. For a function x : (−¥, a] ! X − − and t ≤ a, we define a function xt : R := (−¥, 0] ! X by xt(q) := x(t + q) for q 2 R . Furthermore, let Cg be a space of continuous functions defined by − gq Cg := y 2 C(R , X) : lim ky(q)kXe = 0 , q!−¥ Axioms 2021, 10, 105 3 of 14 for g > 0. We define a norm on the phase space Cg, gq kykCg := sup ky(q)kXe , y 2 Cg. q2R− Finally, we consider a linear delay differential equation in the Banach space X, 0 x = Ax(t) + Lxt, xs = f, + for (s, f) 2 R × Cg. The linear operator A : X ! X generates a strongly continuous compact semigroup fT(t, s)gt≥0 and L : Cg ! X is a bounded linear operator. Let the evolution operator corresponding to the above differential equation is denoted by V(t, s) and for f 2 Cg, V(t, s) is given by − [V(t, s)f](q) = xt(q, s, f, 0), q 2 R , t ≥ s. (3) One can easily see that the evolution operator defines a strongly continuous semigroup and for every t ≥ s ≥ r ≥ 0, it satisfies the semigroup property given by: V(t, s)V(s, r) = V(t, r), and V(t, t) = I. Here, I is an identity operator on Cg. For a continuous function p : R ! X, we consider the perturbed system of delay differential equation: 0 x = Ax(t) + Lxt + p(t), xs = f. (4) Let the solution of the above delay differential equation is denoted by (xt(s, f, p). Now, we give a representation of xt(s, f, p) depending on the evolution operator fV(t, s)gt≥s. Let us introduce a function Gn given by ( (nq + 1)I , −1 ≤ q ≤ 0, n( ) = X n G q −1 0, q < n , where n is any positive integer and IX is the identity operator on X. It is easy to verify that for y 2 X, −g n n n G y 2 Cg and kG ykCg ≤ maxf1, e gk y kX ≤ k y kX. (5) The next result by Hino and Naito [20] establishes the variation of constants formula for the delay differential Equation (4) in the phase space Cg. + Proposition 1. Let (s, f) 2 R × Cg be given. Then the segment xt(s, f, p) of solution x(·, s, f, p) of non-homogeneous functional differential Equation (4) satisfies the following relation in Cg: Z t n xt(s, f, p) = V(t, s)f + lim V(t, t) G p(t) dt, t ≥ s. (6) n!¥ s Definition 1. [r-nonuniform exponential dichotomy] Let r : [0, ¥) ! [0, ¥) be an increasing function with r(t) ! ¥ as t ! ¥. We say that the linear Equation (2) admits a r-nonuniform exponential dichotomy if for every t ≥ s ≥ 0, there exist projection maps P(t) : Cg ! Cg, constants a < 0 ≤ b, e ≥ 0 and K > 1, such that : (i) P(t)V(t, s) = V(t, s)P(s); (ii) VQ(t, s) := V(t, s) : Q(s)Cg ! Q(t)Cg is invertible, where Q(t) = I − P(t) is the complementary projection ; (iii) a(r(t)−r(s))+er(s) −1 −b(r(t)−r(s))+er(t) kV(t, s)P(s)k ≤ Ke , kVQ(t, s) Q(t)k ≤ Ke .