ASYMPTOTIC PROPERTIES OF SOLUTIONS TO LINEAR NONAUTONOMOUS DELAY DIFFERENTIAL EQUATIONS THROUGH GENERALIZED CHARACTERISTIC EQUATIONS

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1 Electronic Journal of Differential Equations, Vol. 2(2), No. 95, pp. 5. ISSN: URL: or ftp ejde.math.txstate.edu ASYMPTOTIC PROPERTIES OF SOLUTIONS TO LINEAR NONAUTONOMOUS DELAY DIFFERENTIAL EQUATIONS THROUGH GENERALIZED CHARACTERISTIC EQUATIONS CLAUDIO CUEVAS, MIGUEL V. S. FRASSON Abstract. We study some properties concerning the asymptotic behavior of solutions to nonautonomous retarded functional differential equations, depending on the knowledge of certain solutions of the associated generalized characteristic equation.. Introduction We are interested in the study of the asymptotic behavior of solutions to the linear nonautonomous retarded functional differential equation (RFDE) x (t) L(t)x t, t t R, (.) where L(t) is a family of bounded linear functionals on C C([ r, ], C), with r >, depending on the knowledge of certain solutions of the associated generalized characteristic equation (.3), introduced below. For a comprehensive introduction for RFDE see [5]. By the Riesz representation theorem, for each t t, there exists a complex valued function of bounded variation η(t, ) on [, r], normalized so that η(t, ) and η(t, ) is continuous from the right in (, r) such that L(t)ϕ Consider the generalized characteristic equation ( λ(t) d θ η(t, θ) exp d θ η(t, θ)ϕ( θ). (.2) ) λ(s)ds, (.3) The solutions of the generalized characteristic equation (.3) are continuous functions λ( ) defined in [t r, ) which satisfy (.3). 2 Mathematics Subject Classification. 39B99. Key words and phrases. Functional differential equations; generalized characteristic equation; asymptotic behavior. c 2 Texas State University - San Marcos. Submitted May 6, 2. Published July 5, 2. C. Cuevas was partially supported by grant 3365/28- from CNPq/Brazil. M. Frasson was partially supported by grant /28-3 from CNPq/Brazil.

2 2 C. CUEVAS, M. V. S. FRASSON EJDE-2/95 One obtains the generalized characteristic equation (.3) by looking for solutions of (.) the form x(t) exp λ(s)ds. (.4) For autonomous RFDE, the constant solutions of (.3) are the roots of the so called characteristic equation. This work is motivated by Dix, Philos and Purnaras []. These authors studied the asymptotic behavior of solutions of nonautonomous linear function differential equations with discrete delays x (t) a(t)x(t) + b j (t)x(t τ j ), t (.5) j where the coefficients a( ) and b j ( ) are continuous real-valued functions on [, ), τ j > for j, 2,..., k by means of the knowledge of solutions λ(t), defined for t r, of the generalized characteristic equation associated to (.5) λ(t) a(t) + j [ b j (t) exp ] λ(s)ds, t τ j t. (.6) We also find in [] a description of the development of results of the type of Theorem 2.. We would like to mention results of this type are found in [3, 4] too. Dix, Philos and Purnaras extended their results for neutral functional differential equations in [2]. Theorem 2. provides a generalization of [, Thm. 2.3], as it can be applied for instance for RFDE with distributed delay or discrete variable delays, as far as the delays are unifomly bounded. In fact, RFDE (.5) can be written in the form (.) letting L(t)ϕ a(t)ϕ() + b j (t)ϕ(τ j ), ϕ C. j We acknowledge that Theorem 2. is obtained by an adaptation of the proof of [, Thm. 2.3] for the more general case of RFDE (.), together with ideas from [3]. We observe that [, Remarks 2.4, 2.5 and 2.6] can be restated here for RFDE (.) without modification. 2. Results Theorem 2.. Assume that λ(t) is a solution of (.3) such that sup Then for each solution x of (.), we have that the it exists, and Furthermore, λ(s)ds d θ η (t, θ) <. (2.) x(t)e t λ(s)ds (2.2) [x(t)e R ] t λ(s)ds t. (2.3) x (t)e t λ(s)ds λ(t)x(t)e λ(s)ds t if there exists the it in the right hand side of (2.4). (2.4)

3 EJDE-2/95 ASYMPTOTIC PROPERTIES OF SOLUTIONS 3 Proof. Hypothesis (2.) implies that there exists t t such that sup t t λ(s)ds d θ η (t, θ) <. Hence without loss of generality, if necessary translating the initial time to t, we may assume t and µ λ : sup t Let x be a solution of (.), and set λ(s)ds d θ η (t, θ) <. (2.5) y(t) x(t)e λ(s)ds, t r. Differentiating y(t) when t, using that x(t) is a solution of (.), (.3) and the fundamental theorem of calculus, we obtain ( y (t) x (t) x(t)λ(t) )e λ(s)ds ( d θ η(t, θ)x(t θ) x(t) d θ η(t, θ)e λ(s)ds) e λ(s)ds d θ η(t, θ)x(t θ)e R λ(s)ds e λ(s)ds x(t)e λ(s)ds d θ η(t, θ)e λ(s)ds d θ η(t, θ)y(t θ)e λ(s)ds y(t) d θ η(t, θ)[y(t θ) y(t)]e λ(s)ds [ d θ η(t, θ) y (s)ds ]e λ(s)ds, t. d θ η(t, θ)e λ(s)ds (2.6) As a characteristic of RFDE, we have that y (t) is continuous for t, understanding the derivative at t as the derivative from the right. Let M x max t [,r] y (t). (2.7) Let t r arbitrary and suppose that for some A we have Using (2.5) and (2.6), we estimate that [ y (t ) d θ η(t, θ) y (t) A, t r t t. (2.8) A d θ η (t, θ) y (s)ds ]e λ(s)ds y (s)ds e λ(s)ds θ e λ(s)ds d θ η (t, θ) Aµ λ. Since y (t ) Aµ λ < A, the continuity of y (t) implies that y (t) A, t [t r, t + δ].

4 4 C. CUEVAS, M. V. S. FRASSON EJDE-2/95 Reasoning as above, we show that y (t) Aµ λ, t [t, t + δ]. Since t y (t) is uniformly continuous on compact intervals, we proceed in this way a finite number of steps and finally conclude that y (t) Aµ λ, t [t, t + r]. (2.9) Taking t nr, n a positive integer, considering (2.7) for n and using (2.9) with A M x (µ λ ) n as induction step, we have proved that We observe that (2.) allows us to conclude that y (t) M x (µ λ ) n, t nr. (2.) y (t) M x (µ λ ) t/r, t. (2.) Letting t, using (2.), we obtain (2.3). We obtain (2.4) by a straight forward application of (2.3), differentiating the quantity in the it and doing simple computations. We proceed to prove (2.2). The cases M x and µ λ are simple, where we have y(t) x() and y(t) y(r) as t, respectively. For < µ λ <, for t T we obtain that T y(t ) y(t) y (s)ds t T M x (µ λ ) s/r ds t M xr µ λ ln µ λ [(µ λ ) T/r (µ λ ) t/r ] as t. By the Cauchy s criterion of convergence, we have that y(t) L x, for some L x. This shows (2.2) and completes the proof. Example 2.2. Consider the linear retarded equation with variable delay x (t) x(t τ(t)) t + c τ(t), t t. (2.2) where c R and τ : [, ) [, r] is a continuous function such that t+c τ(t) > for t t. FDE (2.2) is written in the form (.) letting η(t, ) be given by η(t, θ) for θ < τ(t), η(t, θ) /(t + c τ(t)) for θ τ(t). We have that θ η(t, θ) is increasing and then η η. The generalized characteristic equation associated to (2.2) is given by λ(t) t + c τ(t) exp λ(s)ds (2.3) and we have that a solution of (2.3) is given by t τ(t) λ(t) t + c. (2.4) For (2.2) and λ(t) in (2.4), the left hand side of (2.) reads as sup λ(s)ds τ(t) d θ η (t, θ) sup t + c.

5 EJDE-2/95 ASYMPTOTIC PROPERTIES OF SOLUTIONS 5 and hence the hypothesis (2.) of Theorem 2. is fulfilled and herefore, for all solutions x(t) of (2.2), we have that x(t) [ x(t) ] exists, and t + c. (2.5) t + c Manipulating further the its in (2.5), we are able to state that x(t) O(t) and x (t) o(t) as t. Example 2.3. Consider the linear FDE with distributed delay x (t) x(t θ), t >. (2.6) t θ We write (2.6) in the form (.) by setting η(t, θ) ln t ln(t θ) for t > and θ [, ]. Since θ η(t, θ) is an increasing function, η η. The generalized characteristic equation associated to (2.6) is given by λ(t) t θ exp λ(s)ds dθ (2.7) which has a solution given by λ(t) /t. (2.8) For this λ(t) and for t >, the integral in (2.) reads as θ [ t θ exp ds ] θ dθ s t dθ as t. 2t Hence the hypothesis (2.) of Theorem 2. is fulfilled. Again we obtain that x(t) t exists, [ x(t) ] x (t) and. (2.9) t t References [] Dix, J. G., Philos, C. G., and Purnaras, I. K. An asymptotic property of solutions to linear nonautonomous delay differential equations. Electron. J. Differential Equations (25), no., 9 pp. (electronic). [2] Dix, J. G., Philos, C. G., and Purnaras, I. K. Asymptotic properties of solutions to linear non-autonomous neutral differential equations. J. Math. Anal. Appl. 38, (26), [3] Frasson, M. On the dominance of roots of characteristic equations for neutral functional differential equations. Journal of Mathematical Analysis and Applications 36 (29), [4] Frasson, M. V. S., and Verduyn Lunel, S. M. Large time behaviour of linear functional differential equations. Integral Equations Operator Theory 47, (23), 9 2. [5] Hale, J. K., and Verduyn Lunel, S. M. Introduction to functional-differential equations, vol. 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 993. Claudio Cuevas Departamento de Matemática, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire, S/N, Recife PE, Brazil address: cch@dmat.ufpe.br Miguel V. S. Frasson Departamento de Matemática Aplicada e Estatística, ICMC Universidade de São Paulo, Avenida Trabalhador são-carlense 4, São Carlos SP, Brazil address: frasson@icmc.usp.br