A new continuous dependence result for impulsive retarded functional differential equations

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1 CADERNOS DE MATEMÁTICA 11, May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, São Carlos SP, Brazil federson@icmc.usp.br J. B. Godoy Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, São Carlos SP, Brazil jaquebg@icmc.usp.br We consider a large class of impulsive retarded functional differential equations and we present a new continuous dependence result for this class of equations. May, 2010 ICMC-USP Key Words: Retarded functional differential equations, impulses, existence and uniqueness, continuous dependence. 1. INTRODUCTION We consider a class of impulsive retarded differential equations (we write impulsive RFDEs for short) with Lebesgue integrable righthand sides whose indefinite integrals satisfy Carathéodory- and Lypschitz-type conditions. We also consider that the impulse operators are Lypschitzian functions and that the initial conditions are regulated functions. Then we prove an existence and uniqueness theorem for this class of equations as well as a new continuous dependence type result. In the above setting, the following continuous dependence result for impulsive RFDEs is well-known (see [1], Theorem 4.1). Consider a sequence of impulsive retarded initial value problems (we write IVPs for short) whose righthand sides converge to the righthand side of an impulsive RFDE and whose initial data also converge. Let the sequence of impulse operators be convergent as well. Suppose each element of the sequence of impulsive retarded IVPs admits a unique solution and that the sequence of solutions is uniformly convergent. * Supported by FAPESP grant 2008/ and by CNPq grant / Supported by FAPESP grant 2007/

2 38 M. FEDERSON AND J. B. GODOY Consider also the limiting IVP with limiting righthand side, limiting impulse operators and limiting initial condition. Then the limit of the sequence of solutions is a solution of the limiting IVP, provided certain conditions are fulfilled. In the present paper, we state and prove a certain reciprocal of this result. Roughly speaking, we consider a sequence of IVPs for impulsive RFDEs, in the above setting, with convergent righthand sides, convergent impulse operators and uniformly convergent initial data. We assume the limiting equation is an impulsive RFDE whose initial condition is the uniform limit of the sequence of initial data and whose solution exists and is unique. Then, under certain conditions, for sufficient large indexes, the elements of the sequence of impulsive retarded IVPs admit a unique solution and such a sequence of solutions converges to the solution of the limiting IVP. 2. THE SETTING OF IMPULSIVE RFDES Let X be a Banach space. A function f : [a, b] X is called regulated, if the following limits exist lim f(s) = f(t ) X, t (a, b], and lim f(s) = f(t+) X, s t s t+ t [a, b). In this case, we write f G([a, b], X) and we endow G([a, b], X) with the usual supremum norm f = sup a t b f(t). Then (G([a, b], X), ) is a Banach space. Also, any function in G([a, b], X) is the uniform limit of step functions (see [2]). Define G ([a, b], X) = {u G([a, b], X) : u is left continuous at every t (a, b]}. In G ([a, b], X), we consider the norm induced by G([a, b], X). Given a function y : [ r, + σ] R n, with r > 0 and σ > 0, we consider y t : [ r, 0] R n given by y t (θ) = y (t + θ), θ [ r, 0], t [, + σ]. Then it is clear that for a function y G ([ r, +σ], R n ), we have y t G ([ r, 0], R n ) for all t [, + σ]. We consider the following initial value problem for a retarded functional differential equation (RFDE) with impulses ẏ (t) = f (y t, t), t t k y (t k ) = I k (y (t k )), k = 0, 1,..., m, (1) y t0 = φ, where t k, k = 0, 1,..., m, with < t 1 <... < t k <... < t m + σ, σ > 0, are pre-assigned moments of impulse, for k = 0, 1,..., m, y I k (y) maps R n into itself and y (t k ) := y (t k +) y (t k ) = y (t k +) y (t k ), Sob a supervisão da CPq/ICMC

3 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES 39 that is, y is left continuous at t = t k and the lateral limit y(t k +) exists, for k = 0, 1, 2,..., m. We also consider φ G ([ r, 0], R n ). It is known that the impulsive system (1) is equivalent to the integral equation y(t) = y( ) + f(y s, s)ds + y t0 = φ, t k < t I k (y(t k )) whenever the integral exists in some sense. We will consider Lebesgue integration in (2). Let P C 1 G ([ r, + σ], R n ) be an open set (in the topology of locally uniform convergence in G ([ r, + σ], R n ) with the following property: if y is an element of P C 1 and t [ r, + σ], then ȳ given by (2) ȳ (t) = { y (t), t0 r t t, y ( t), t < t, (3) is also an element of P C 1. In particular, any open ball in G ([ r, + σ], R n ) has this property. We assume that f : (ψ, t) G ([ r, 0], R n ) [, + σ] R n and that the mapping t f (y t, t) is Lebesgue integrable. We also assume the following conditions: (A) There is a Lebesgue integrable function M : [, +σ] R such that for all x P C 1 and all u 1, u 2 [, + σ], u2 u 1 u2 f (x s, s) ds M (s) ds; (B) There is a Lebesgue integrable function L : [, + σ] R such that for all x, y P C 1 and all u 1, u 2 [, + σ], u2 u 1 u2 [f (x s, s) f (y s, s)] ds L (s) x s y s ds. For the impulse operators I k : R n R n, k = 1, 2,..., we assume the following conditions: (A ) there is a constant K 1 > 0 such that for all k = 0, 1, 2,..., and all x R n, u 1 I k (x) K 1 ; (B ) there is a constant K 2 > 0 such that for all k = 0, 1, 2,..., and all x, y R n, u 1 I k (x) I k (y) K 2 x y.

4 40 M. FEDERSON AND J. B. GODOY Note that the Carathéodory- and Lipschitz-type conditions (A) and (B) are required for the indefinite integral of f only and not for the function f itself. Thus the standard requirement that f(ψ, t) is continuous in ψ does not need to be fulfilled. Also, the mapping t f (y t, t) does not need to be piecewise continuous. Let us recall the concept of a solution of problem (1). Definition 2.1. A function y P C 1 for which (y t, t) G ([ r, 0], R n ) [, + σ] for t < + σ and the conditions (i) ẏ (t) = f (y t, t), almost everywhere (in Lebesgue s sense), whenever t t k. (ii) y (t k +) = y (t k ) + I k (y (t k )), k = 0, 1, 2,..., m. (iii) y t0 = φ are satisfied is called a solution of (1) in [, + σ] (or sometimes also in [ r, + σ]) with initial condition (φ, ). In [1], it was proved that under the conditions (A), (B), (A ) and (B ), system (1) can be identified in a one-to-one correspondence with a system of generalized ordinary differential equations taking values in a Banach space. Local existence and uniqueness of solutions are guaranteed by Theorems 2.15, 3.4 and 3.5 from [1]. In what follows, we give a direct proof of an existence and uniqueness theorem for the impulsive RFDE (1) without employing the theory of generalized ODEs. Nevertheless, our proof is inspired in the proof of [1], Theorem Theorem 2.1. Consider problem (1) and suppose conditions (A), (B), (A ) and (B ) are fulfilled. Then there is a > 0 such that on the interval [, + ] there exists a unique solution y : [ r, + ] R n of problem (1) for which y t0 = φ. Proof. For t [, + σ], define h 1 (t) = [M(s) + L(s)]ds and h 2 (t) = max(k 1, K 2 ) m H tk (t), k=0 where H tk denotes the left continuous Heaviside function concentrated at t k, that is, { 0, for t tk H tk (t) = 1, for t k > t. Let h = h 1 +h 2. Then h is nondecreasing and left continuous. Besides, for s [, +σ], we have f(y t, t)dt + I j (y(t j )) = s m f(y t, t)dt + I j (y(t j ))H tj (s) t j<s j=0 Sob a supervisão da CPq/ICMC

5 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES 41 M(t)dt + K 1 m H tj (s) h(s) h( ), j=0 by conditions (A) and (A ). We will consider two cases: when is a point of continuity of h : [, + σ] R n and otherwise. At first, let be a point of continuity of h. Assume that > 0 is such that [, + ] [, + σ) and h( + ) h( ) < 1 2. Let Q be the set of functions z : [, + ] R n such that z BV ([, + ], R n ) and z(t) y( ) h(t) h( ) for t [, + ]. It is easy to show that the set Q BV ([, + ], R n ) is closed. For s [, + ] and z Q, define T z(s) = y( ) + f(z t, t)dt + <t j s I j (z(t j )). Then, T z(s) y( ) = f(z t, t)dt + t j<s I j (z(t j )) h(s) h(), s [, + ]. Also, the fact that T z belongs to BV ([, + ], R n ) is not difficult to prove. Thus, it follows that T maps Q into itself. Take s 1 < s 2 + and z 1, z 2 Q. Using conditions (B) and (B ), we obtain T z 2 (s 2 ) T z 1 (s 2 ) [T z 2 (s 1 ) T z 1 (s 1 )] = 2 = [f(z 2, t) f(z 1, t)]dt + [I j (z 2 (t j )) I j (z 1 (t j ))] s 1 s 1 t j<s 2 2 m L(t) z 2 (t) z 1 (t) dt + K 2 z 2 (t j ) z 1 (t j ) [ H tj (s 2 ) H tj (s 1 ) ] s 1 j=0 2 m [ sup z 2 (s) z 1 (s) L(t)dt + K 2 Htj (s 2 ) H tj (s 1 ) ] s [,+ ] s 1 z 2 z 1 BV ([t0,+ ])[h(s 2 ) h(s 1 )]. Recall that z BV ([t0,+ ]) = z( ) +vart t0+ 0 (z) defines a norm in BV ([, + ], R n ), where vart t0+ 0 (z) denotes the variation of z on the interval [, + ]. Therefore T z 2 T z 1 BV ([t0,+ ]) z 2 z 1 BV ([t0,+ ])[h( + ) h( )] j=0 < 1 2 z 2 z 1 BV ([t0,+ ])

6 42 M. FEDERSON AND J. B. GODOY and hence T is a contraction. Then, by the Banach fixed-point theorem, the result follows. Now, we consider the case where is not a point of continuity of h (or of h 2 ). Define { h2 (t), t = t h 2 (t) = 0 h 2 (t) h 2 ( +) + h 2 ( ) = h 2 (t) h 2 ( +), t + σ, and h = h 1 + h 2. Then h is continuous at t =, left continuous and nondecreasing. Define an impulse operator I 0 from R n to R n such that, for y G ([, + σ], R n ), I 0 (y(t)) = { I0 (y(t)), t = I 0 (y(t)) I 0 (y( +)) + I 0 (y( )), t + σ and consider the impulsive RFDE ż (t) = f (z t, t), t t k z (t k ) = I k (z (t k )), k = 1,..., m, z ( ) = I 0 (z ( )), z t0 = φ, (4) where φ(θ) = { φ(θ), r θ < 0 y( +), θ = 0. Then, for s [, + σ], we have f(z t, t)dt + I 0 (z( )) + <t j<s I j (z(t j )) h(s) h(). As in the previous case, let > 0 be such that [, + ] [, + σ) and h( + ) h( ) < 1 2. Then, define Q as the set of functions z : [, + ] R n such that z BV ([, + ], R n ) and z(t) y( +) h(t) h( ) for t [, + ] and consider the operator T defined on Q and given by T z(s) = y( +) + f(z t, t)dt + I 0 (z( )) + <t j s I j (z(t j )). Following the procedure of the previous case, it can be shown that equation (4) admits a unique solution on [, + ]. Then, defining y t0 = φ and y(t) = z(t), for t >, we obtain a unique solution of (1), for which y t0 = φ in [, + ]. For the next theorem, we consider the following sequence of initial value problems: ẏ (t) = f p (y t, t), t t k y (t k ) = I p k (y (t k)), k = 0, 1,..., m, y t0 = φ p, (5) Sob a supervisão da CPq/ICMC

7 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES 43 where < t 1 <... < t k <... < t m + σ, and for each p = 1, 2,..., x I p k (x) maps R n into itself and y (t k ) := y (t k +) y (t k ) = y (t k +) y (t k ), k = 0, 1, 2,..., m. We will show that, under conditions (A), (B), (A ) and (B ) for f p and I p k, k = 0, 1, 2,..., m, the sequence {y p } p 1 of solutions of (5) is equibounded and of uniformly bounded variation on some closed subinterval of [, + σ]. Theorem 2.2. Assume that for p = 0, 1,..., φ p G ([ r, 0], R n ) and moreover f p : G ([ r, 0], R n ) [, + σ] R n and I p k : Rn R n satisfy conditions (A), (B), (A ) and (B ) for the same functions M, L and the same constants K 1, K 2. Then there is a > 0 such that y p : [ r, + ] R n is a solution of (5), for each p, and the sequence {y p } p 1 is equibounded and of uniformly bounded variation on [, + ]. Proof. From the proof of Theorem 2.1, it is clear that a unique > 0 can be obtained such that a solution y p : [ r, + ] R n of (5) exists and is unique, independently of p. Thus, for p = 0, 1,..., we have y p (s) = y p ( ) + f p ((y p ) t, t)dt + I p j (y p(t j )), < t j s s [, + ]. Consider a partition = s 0 < s 1 <... < s n = + of [, + ]. Then y p (s i ) y p (s i 1 ) = i s i 1 f p ((y p ) t, t)dt + Therefore, conditions (A) and (A ) imply n y p (s i ) y p (s i 1 ) i=1 i=1 0+ n i f p ((y p ) t, t)dt s i 1 + M(s)ds + K 1 l, s i 1< t j s i I j (y p (t j )), i = 1, 2,..., n. < t j + where l is the number of impulse moments in the interval [, + ]. Then I j (y p (t j )) 0+ vart t0+ 0 (y p ) M(s)ds + K 1 l, for all p N. (6) where vart t0+ 0 (y p ) denotes the variation of y p [, + ], and hence {y p } p 1 is of uniformly bounded variation on [, + ]. Now, we are going to show that {y p } p 1 is equibounded. Again, since y p (t) = y p ( ) + f((y p ) s, s)ds + <t j t I j (y p (t j ))

8 44 M. FEDERSON AND J. B. GODOY for each t [, + ] and each p = 1, 2, 3,..., we have y p (t) y p ( ) + f p ((y p ) s, s)ds + y p ( ) + M(s)ds + K 1 l y p ( ) + 0+σ M(s)ds + K 1 l < t j + I j (y p (t j )) Therefore {y p } p 1 is equibounded on [, + ] and the result follows. 3. CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES In general, one cannot expect that an impulsive RFDE depends on the initial data. We mention [3] for an elucidative discussion on the continuous dependence of solutions of an impulsive RFDE whose impulse operators also involve delays. The next theorem is a continuous dependence result which, together with Theorem 2.2, are important to prove the theorem following it. A proof of it can be found in [1], Theorem 4.1. Theorem 3.1. Assume that for p = 0, 1,..., φ p G ([ r, 0], R n ) and moreover f p : G ([ r, 0], R n ) [, + σ] R n and I p k : Rn R n, k = 0, 1, 2,..., satisfy conditions (A), (B), (A ) and (B ) for the same functions M, L and the same constants K 1, K 2. Let the relations ϑ lim sup [f p ϑ [,+σ] p (y s, s) f 0 (y s, s)]ds = 0 (7) for every y P C 1 and lim p Ip k (x) = I0 k(x) (8) for every x R n, k = 0, 1,..., m be satisfied. Assume that y p : [ r, + σ] R n, p = 1, 2,..., is a solution on [ r, + σ] of the problem such that ẏ (t) = f p (y t, t), t t k y (t k ) = I p k (y (t k)), k = 1,..., m y t0 = φ p, lim y p(s) = y(s) uniformly on [ r, + σ]. (10) p (9) Sob a supervisão da CPq/ICMC

9 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES 45 Then y : [ r, + σ] R n is a solution on [ r, + σ] of the problem ẏ (t) = f 0 (y t, t), t t k y (t k ) = Ik 0 (y (t k)), k = 1,..., m y t0 = φ 0. (11) The assumptions (7) and (8) in Theorem 2.2 ensure that if the sequence {y p } p 1, y p : [ r, + ] R n, p = 1, 2,..., of solutions of (5) converges uniformly to a function y : [ r, + ] R n, then this limit is a solution of (11). The next result says that adding an uniqueness condition to the limiting equation, then for sufficient large p N, y p : [ r, + ] R n is a solution of (5), provided the sequence of initial data {φ p } p 1 converges uniformly on [ r, 0]. Theorem 3.2. Assume that f p (φ, t) : G ([ r, 0], R n ) [, + σ] R n, p = 0, 1, 2,..., satisfies conditions (A) and (B) for the same functions M and L. Let I k p : R n R n k = 1, 2,..., p = 0, 1, 2,..., be impulse operators which satisfy conditions (A ) and (B ) for the same constants K 1 and K 2. Assume that for every y P C 1, and lim p [f p (y s, s) f 0 (y s, s)]ds = 0, t [, + σ] (12) lim p Ip k (x) = I0 k(x) (13) for every x R n and k = 1,..., m. Let y : [ r, + σ] R n be a solution of ẏ (t) = f 0 (y t, t), t t k y (t k ) = Ik 0 (y (t k)), k = 1,..., m y t0 = φ 0, (14) on [ r, + σ]. Assume that if there exists ρ > 0 such that sup θ [ r,0] u(θ) φ 0 (θ) < ρ, then u G ([ r, 0], R n ). Assume further that φ p φ 0 uniformly on [ r, 0] as p. Then, for sufficiently large p N, there exists a solution y p of on [ r, + σ] and ẏ (t) = f p (y t, t), t t k y (t k ) = I p k (y (t k)), k = 1,..., m y t0 = φ p, (15) lim y p(s) = y(s), s [ r, + σ] (16) p

10 46 M. FEDERSON AND J. B. GODOY Proof. The present proof is inspired in the proof of [4], Theorem 8.6, for generalized ODEs. We strongly use the fact that the functions f p, p = 0, 1, 2,..., take values in a finite dimensional space so that we can apply Helly s choice principle. Because φ p φ 0 uniformly on [ r, 0] as p, there exists ρ > 0 and k N such that sup φ p (θ) φ 0 (θ) < ρ, for p > k. (17) θ [ r,0] Thus, by the assumption, φ p G ([ r, 0], R n ), for p > k. Then this implies by the existence theorem (Theorem 2.1) that for p > k, there exists a > 0 such that y p is a solution of (15) on [ r, + ]. We assert that lim y p(s) = y(s), for s [ r, + σ]. (18) p By Theorem 3.1, if the sequence {y p } p 1 admits a convergent subsequence, then since φ p φ 0, it will follow by the uniqueness of solutions that there is a > 0 such that lim p y p (s) = y(s), for s [ r, + ], where y t0 = φ 0. We will use Helly s choice principle to prove that in fact {y p } p 1 admits a convergent subsequence. By Theorem 2.2 the sequence {y p }, p > k, of functions on [, + ] is equibounded and of uniformly bounded variation. Thus, by the Helly s choice principle, the sequence {y p }, p > k, contains a pointwise convergent subsequence and hence y(t) is the only accumulation point of the sequence y k (t) for every t [, + ]. Therefore the theorem holds on [, + ], > 0. It also holds on [ r, ], since φ p φ 0. Now, let us assume that the convergence result does not hold on the whole interval [ r, +σ]. Thus there exist a, 0 < < σ, such that for every < and for p > k, there is a solution y p of (15) on [ r, + ], with (y p ) t0 = φ 0, and lim p y p (t) = y(t) for t [ r, + ], but this does not hold on [ r, + ], whenever >. By the proof of Theorem 2.1, y k (s 2 ) y k (s 1 ) < h(s 2 ) h(s 1 ) for every s 2, s 1 [ r, + ) and every p > k. Hence the limits y p (( + ) ) = lim ε 0 y p ( + + ε), p > k, exist and y p (( + ) ) = y( + ), for p > k, since y is left continuous. Defining y p ( + ) = y p (( + ) ), for p > k, then lim p y p ( + ) = y( + ). Therefore the theorem holds on [ r, + ] as well. Then, using + < + σ as the starting point, it can be proved analogously that the theorem holds on the interval [ +, + +η], for some η > 0, and this contradicts our assumption. Thus the theorem holds on the whole interval [ r, + σ]. REFERENCES Sob a supervisão da CPq/ICMC

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