ASYMPTOTIC STABILITY FOR ABSTRACT NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS. (FDE) x(tb)(t) = f(x(tb)(t)) + *(*,(*)), t > 0, x0(<p) = tb.

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volueme 54, January 1976 ASYMPTOTIC STABILITY FOR ABSTRACT NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS G. F. WEBB Abstract. The nonlinear autonomous functional differential equation x(t) = f(x(t)) + g(x,), t > 0, x0 = <f> is investigated by means of the theory of semigroups of nonlinear operators. The properties of the semigroup associated with this equation provide stability information about the solutions. 1. Introduction. The purpose of this paper is to prove some stability properties of the nonlinear autonomous functional differential equation (FDE) x(tb)(t) = f(x(tb)(t)) + *(*,(*)), t > 0, x0(<p) = tb. The notation of (FDE) follows J. Hale [4], that is tb G C = C([-r,0];X) where r > 0 and A' is a Banach space, x(tb)(t): [ r, oo) -* X, and x,(tb) G C is denned for each t > 0 by xt(<?)(0) = x(tb)(t + 9), 0 G [-r,0]. In (FDE) we will require a Lipschitz condition on the nonlinear operator g: C > X and an accretiveness condition on the nonlinear operator /: X > X. The ordinary part of (FDE) corresponding to /will act as a damping term for the equation. Our main result can be summarized as follows: Suppose g has Lipschitz constant B and /+ a I is accretive. If a = B, then (FDE) is stable, and if o < B, then (FDE) is asymptotically stable. As a simple example for our problem one can let X = R,f(x) = jc3 + ax, g(tb) = h(tb( r)), where h has Lipschitz constant B, and then (FDE) is the scalar delay equation x(tb)(t) = -x3(tb)(t) + ax(tb)(t) + h(x(tb)(t - r)). Our approach will be to use the general theory of semigroups of nonlinear operators. By allowing J to be a Banach space our results may be applied to partial functional differential equations as in [7]. For related treatments of our problem one should see [7]-[9]. 2. Definitions. For an arbitrary Banach space Y, a nonlinear operator A from Y to Y is accretive provided (2.1) (/ + XA)x - (I + XA)y\\ > \\x - y\\ for all x, y G D(A), X > 0. Our results rely upon the following general theorem, due to M. Crandall and T. Liggett [1], from nonlinear semigroup theory: Suppose for some y G R, Received by the editors December 26, AMS (MOS) subject classifications (1970). Primary 34K20; Secondary 47H15. Key words and phrases. Autonomous functional differential equation, stability, asymptotic stability, nonlinear accretive operator, nonlinear semigroup of operators. 225 American Mathematical Society 1976

2 226 G. F. WEBB A + yl: Y -* Y is accretive and /?(/ + XA) is onto for all sufficiently small A > 0. Then, _ Apf _ (2.2) lim (/ + t/na) "x = Tit)x exists for x G DiA), t > 0. n»oo Moreover, Tit), t > 0, is a strongly continuous semigroup of nonlinear operators on DiA), that is, (2.3) Ti0)x = x for all x G DiA); (2.4) TX')-* is continuous in / for each fixed x G DiA); (2.5) l\tx + t2) = Titx)Tit2) for tx,t2> 0; (2.6) r(/)je - T(t)y\\ < ei'\\x - y\\ for x, y G (Z), / > 0. Let C0 be the subspace of C given by C0 = {<J> G C: d>(0) = 0}. Define the linear operators (2.7) A0: C0 -» C0 by A0<p = -d>',»(^0) = {* e C0: *' e C0}, (2.8) /1,:C-*C by Ax <f> = -d>', >(/!,) = {4. G C0: <?>' G C}. From the theory of semigroups of linear operators one has that A0 is accretive in C0, #(/ + XA0) = C0 for A > 0, DiA0) is dense in C0, A0 is the infinitesimal generator of a strongly continuous semigroup of linear contractions on C0, and (2.9) lim+(/ + A/f0)_1d> = d> for all tf> G C0. Also, /I, is accretive in C, RQ + XAX) = C for A > 0, but I>(^i) is not dense in C. If A > 0 then (2.10)((/ + \A0)-l4>)(9) = (V)X *~*A*(')*. * G Co> * e 1-^ ]; (2.11) ((/ + A^y-^W) = (j^)j9 e-''x^s)ds, <t>ec,9 G [-r,0]; (2.12) (/ + A/f,rV = (/ + \A0)-*{4>- d,(0)) + (1 - e#xw0), ^C. 3. The nonlinear semigroup for (FDE). In this section we shall construct a nonlinear semigroup which can be associated with (FDE) by exhibiting its generator in the sense of (2.2). In what follows we shall suppose that for some a E R, f; X ^ X such that -f + al is accretive and /?(/ - Xf) = X for 0 < A < l/max{0,a}. From (2.1) this implies that for x, y G Z)(/), A > 0, (3.1) (/ - A/)"1* - (/ - Xf)~]y\\ < (1/(1 - Xa))\\x-y\\,

3 ASYMPTOTIC STABILITY 227 and (3.2) lim (I - Xf)~lx = x for x G D(f) (see [1, Lemma 1.2(H)]). A *0 We shall also suppose that g: C -* X is Lipschitz continuous with Lipschitz constant B. Define A: C -* C by Atb = -tb' with 7>(yl) = {<?> G C: <f>' G C, *(0) G Z>(/),*'"(<)) =/(<K0)) + g(*)}. Proposition 1. A + yl is accretive in C and R(I + XA) = C for 0 < A < 1/y, where y = max{0, a + B}. Proof. We first show I + XA is 1-1 and onto. Let 0 < A < 1/y, xp G C. Define the mapping k: X -> X by (3.3) k(b) = (I- A/)-'(^(0) + Xg(eB'xb + (7 + XA^xp)), b G X. Then A: is a strict contraction on X, since \\k(bx) k(b2)\\ < (XB/(\ - Xa)) \\bi b2\\. Thus, k has a unique fixed point b0, and (3-4) tb(9) = eb/xb0 + ((/ + A^,rV)(0) solves uniquely tb - Xtb' = xp, tb'~(0) = f(tb(0)) + g(tb), that is, (I + XA)tb = xp. Next, we will show that for all xpx, xp2 G C, 0 < A < 1/y, (3.5) (/ + A^r1^, - (7 + \A)-%\\ < (1/(1 - Xy))Ux - fc. Let (7 + A^)< >, = xph (I + XA)tb2 = xp2, and 0 G [-r,0] such that \\tb](0) - ^2(6>) = ^ - <j,2\\. From (3.3),(3.4), and (2.11) we have <j>,-<j>2 = *,(«)-^WH < (e»a/(l - A«))( ^(0) - fc(0) + XdUi ~ <P2W) which implies H*, - tb21 < ((1 - A«+ Xae9lx)/(\ - Aa - XBe9/x))Ux - fc. If 0 < a + B, then (1 - Aa + Aa//X)/(1 - Aa - XBee'x) < 1/(1 - A(a + B)) = 1/(1 - Ay). If a + B < 0, then (1 - Aa + Xae9/x)/(\ - Aa - XBe9'x) < (1 - Aa - XBe9/x)/(\ - Aa- \Be9'x) = 1/(1 - Ay). In either case, (3.5) holds, and this yields the accretiveness of A + y7. Proposition 2. D(A) = {> G C: *(0) G >(/)}. Proof. Let (7 + A^4)* = * as in Proposition 1. Using (3.3) and (3.4) we see

4 228 G. F. WEBB that iwo) - #))ii = ii(/ - A/r'oKo) + \g(4>)) - m\\ (3.6) <(VO-Aa))(/i d,-^ + g^) ) Also, from (2.12), + (7-A/rV(0)-^(0). (3.7) 4, - +\\ < (7 - A^rV - <K0)) - (* - ^0)) + 4»(0) - tfo). From (3.6) and (3.7) we obtain 4, - * < ((1 - A«)/(l - A(«+ (3))) (3.8) ( (7-A/l0r1(^-^0))-(^-^(0)) + (7 - A/r'tMO) - tf0) + (A/(l - Xa)) g(*) ). From the general theory of accretive operators (see [1, Lemma 1.2 (ii)]) we have from Proposition 1 that (3.9) DiA) = (4, G C: Inn (7 + XA)~^ = A Then, (2.9), (3.2), (3.8), and (3.9) imply the conclusion. By virtue of Propositions 1 and 2 we may use formula (2.2) to define a nonlinear semigroup Tit), t > 0, on DiA) with generator A. If 7)(/) = X, then DiA) = C. If / is linear and densely defined in X, then A is exactly the infinitesimal generator of Tit), t > 0 (see [7] or [8]). The question arises as to when the semigroup Tit), t > 0, gives solutions to (FDE). One can use the methods of H. Flaschka and M. Leitman [3] to show that Tit), t > 0, always has the following property: If for each d> G DiA) we define (3.10) x(d>)(r) = 4>(/) for -r < / < 0, = (7X')4>)(0) for t > 0; then Tit)cb = x,(d>). In the case that/is everywhere defined and continuous the methods of [3] can be used to show that the function xi<p)it) in (3.10) satisfies (FDE) for all 4> G DiA). The proof of these facts carries over essentially without change from [3]. In the case that A' is a Hilbert space we can show the following. Proposition 3. If X is a Hilbert space, then for all <j> G DiA) the function jc(d>)(r) in i3.10) satisfies (3.11) x(4>)(/) = /(*(</>)(')) + *(*,(*)) for a.a. t > 0, x0(d») = <#> Proof. We will use the notation and results from [1]. Let d> G DiA). We show first that for all h G A', t > 0,

5 ASYMPTOTIC STABILITY 229 (3.12) <(r(0*)(0),a> = <<P(0),h) +/o' ((f(t(s)tb)(0)) + g(t(s)tb))ds, h). As in [3] we have that (Tx(t)tb)(0) = *(0) + /o' (/((A TA(^)(0)) + g(jx Tx(s)tb))ds, and 7X Tx(s)tb converges strongly to T(s)tb as A -» 0. By virtue of the Lebesgue dominated convergence theorem, to establish (3.12) it suffices to show that f((jxtx(s)tb)(0)) is bounded in A and converges weakly to/((7(.y)<j>)(0)) as A» 0. The boundedness follows from the Lipschitz continuity of g, the fact that tb G D(A), and the inequality constm* > \\AJxTx(s)cb\\ > \\(AJxTx(s)tb)(0)\\ = \\f((jxtx(s)tb)(0)) + g(jxtx(s)tb)\\. The weak convergence follows from the fact that / is maximal accretive in X by a well-known argument of accretive operator theory (see [2] or [6]). Then (3.12) implies that x(tb)(t) is weakly differentiable for almost all t > 0. Also, since tb G D(A), T(t)tb is Lipschitz continuous in / (see (1.11) of [1]), and therefore x(<b)(t) is strongly of bounded variation in /. By [5, Theorem 3.8.6, p. 88], x(tb)(t) is strongly differentiable almost everywhere in / and (3.11) holds. Now we consider the stability properties of (FDE). If a = B, then the trajectories of T(t), t > 0, are stable in the sense of (2.6) with y = 0. If a < B, the proposition below yields the asymptotic stability of (FDE) if X is a Hilbert space. Proposition 4. Suppose a < -B and X is a Hilbert space. Then T(t), t > 0, is asymptotically stable in the sense that there exists a unique point xp0 G C such that lim,^^, T(t)xp = xp0for all xp G C. Moreover, xpq is a constant function in C. Proof. Define/: X -> X by j(b) = (I -f)~\b + g(b 1)), where b G X and 1 denotes the constant function identically 1 on [ r, 0]. Since a + B < 0, j is a strict contraction with Lipschitz constant < (1 + B)/(\ a). Let b0 be the unique fixed point of y in X and define *0 = b0 1. Then *0 G D(A) and Axp0 = 0. From formula (2.2) we have that T(t)xp0 = xp0 for all t > 0. To prove the conclusion it suffices to show that (3 13) lim SUP IIT(')^ ~ T(t)M\/U ~ +\\ = 0. Let tb, xp G D(A) and let x(t) and y(t) solve (3.11) for </> and xp, respectively. Then, for almost all t > 0, \(d/dt)\\x{t) -y\t)\\ = <f(x(t)) + g(x,) -f(y(t)) - g(yt),x(t)-y(t)} (3.14) < «IW0-y(t)t + s\\t(t)tb - 7X0*11 IW0 -y(t)\\ <a\\x(t)-y(t)\\2 + BU-xp\\2. By GronwalPs inequality, (3.14) implies that for t > 0,

6 230 G. F. WEBB (3.15) \\xit) - yit)\\ < U ~ +KP/-01 - e2o'(l + [3/-a)f2. Then (3.15) implies that for / > 0, (3.16) \\Tit)4, - 7X0*11 < II* - M\i(3/-a - M^{\ + (3/-a)f2. Using (2.6) with y = 0, we have that (3.16) holds for all d>, * G DiA). But this means that 1\t) is a strict contraction when t > r and this fact, together with (2.5), yields (3.13). In conclusion we remark that all of our propositions carry over easily to the case that / is a multivalued accretive operator, a class of nonlinear operators that is very general and extensively developed (see, e.g., [1] and [2]). Also, Propositions 3 and 4 carry over readily to the case that A' is a uniformly convex Banach space. References 1. M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), MR 44 # M. G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis 3 (1969), MR 39 # H. Flaschka and M. Leitman, Ozz semigroups of non-linear operators and the solution of the functional differential equation xit), = F(xf), J. Math. Anal. Appl. 49 (1975), J. Hale, Functional differential equations, Appl. Math. Series, vol. 3, Springer-Verlag, New York, E. Hille and R. S. Phillips, Functional analysis and semi-groups, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc, Providence, R.I., MR 19, T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), MR 37 # C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), G. Webb, Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl. 46 (1974), , Functional differential equations and nonlinear semigroups in W-spaces, J. Differential Equations (to appear). Department of Mathematics, Vanderbilt University, Nashville, Tennessee Instituto Matematico, Universita di Roma, Roma, Italia r