Impulsive partial neutral differential equations

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1 Applied Mathematics Letters 19 (26) Impulsive partial neutral differential equations Eduardo Hernández M a,, Hernán R. Henríquez b a Departamento de Matemática, ICMC Universidade de São Paulo, Caixa Postal 668, São Carlos SP, Brazil b Departamento de Matemática, Universidad de Santiago, Casilla 37, Correo-2, Santiago, Chile Received 12 April 24; received in revised form 18 March 25; accepted 15 April 25 Abstract In this work we study the existence and regularity of mild solutions for impulsive first order partial neutral functional differential equations with unbounded delay. 25 Elsevier Ltd. All rights reserved. Keywords: Impulsive differential equations; Neutral differential equations; Semigroups of bounded linear operators 1. Introduction Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the last few decades. The literature related to ordinary neutral functional differential equations is very extensive and we refer the reader to [1,2] andthereferences therein. Partial neutral functional differential equations with delay have been studied in [3 7]. On the other hand, the theory of impulsive differential equations has become an important area of investigation in recent years, stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. Related to this matter we mention [8,9] andthe references in these works. Corresponding author. addresses: lalohm@icmc.sc.usp.br (E. Hernández M), hhenriqu@lauca.usach.cl (H.R. Henríquez) /$ - see front matter 25 Elsevier Ltd. All rights reserved. doi:1.116/j.aml

2 216 E. Hernández M, H.R. Henríquez / Applied Mathematics Letters 19 (26) In this work we study the existence and regularity of mild solutions for a class of abstract neutral functional differential equations (ANFDE) with unbounded delay and impulses described in the form d dt (x(t) + F(t, x t)) = Ax(t) + G(t, x t ), t I =[,σ], (1.1) x = ϕ B, (1.2) x(t i ) = I i (x ti ), i = 1,...,n, (1.3) where A is the infinitesimal generator of an analytic semigroup of linear operators, (T (t)) t,ona Banach space (X, ); thehistory x t : (, ] X, x t (θ) = x(t + θ),belongs to some abstract phase space B defined axiomatically; F, G, I i,areappropriate functions; < t 1 < < t n <σare prefixed points and ξ(t) is the jump ofafunction ξ at t, whichisdefined by ξ(t) = ξ(t + ) ξ(t ). Ordinary impulsive neutral differential systems have been studied recently in several papers; see [1 14]. The existence of solutions for impulsive partial ANFDE is an untreated topic and that is the motivation of this work. In relation with this last remark, we consider it important to observe that some systems similar to (1.1) (1.3) arestudied in [13,14]. However, in these works the authors impose some severe assumptions on the semigroup generated by A which imply that the underlying space X has finite dimension; see [7] fordetails. As a consequence, the systems studied in these works are really ordinary and not partial. In this work, A : D(A) X X is the infinitesimal generator of an analytic semigroup of linear operators (T (t)) t on X, ρ(a), and M is a constant such that T (t) M for every t I. The notation ( A) α,α (, 1), stands for the fractional power of A and X α is the domain of ( A) α endowed with the graph norm. For literature relating to semigroup theory, we suggest [15]. We say that a function u( ) is a normalized piecewise continuous function on [µ, τ] if u is piecewise continuous, and left continuous on (µ, τ ]. Wedenote by PC([µ, τ]; X) the space formed by the normalized piecewise continuous functions from [µ, τ] to X. Thenotation PC stands for the space formed by all functions u PC([,µ]; X) such that u( ) is continuous at t t i, u(ti ) = u(t i ) and u(t i + ) exists for all i = 1,...,n. Inthiswork, (PC, PC ) is the space PC endowed with the norm x PC = sup s I x(s). Itisclear that (PC, PC ) is a Banach space. In this work we will employ an axiomatic definition for the phase space B which is similar to that used in [16]. Specifically, B will be a linear space of functions mapping (, ] to X endowed with a seminorm B and verifying the following axioms: (A) If x : (,µ+ b] X, b >, is such that x µ B and x [µ,µ+b] PC([µ, µ + b] :X), thenfor every t [µ, µ + b] the following conditions hold: (i) x t is in B, (ii) x(t) H x t B, (iii) x t B K (t µ) sup{ x(s) :µ s t}+m(t µ) x µ B, where H > isaconstant; K, M :[, ) [1, ), K is continuous, M is locally bounded and H, K, M are independent of x( ). (B) The space B is complete. Example: The phase space PC r L 2 (g, X). Let g : (, r] R be a positive function verifying the conditions (g-6) and (g-7) of [16]. This means that g( ) is Lebesgue integrable on (, r) and that there exists a non-negative and locally bounded function γ on (, ] such that g(ξ + θ) γ(ξ)g(θ),forallξ andθ (, r) \ N ξ, where N ξ (, r) is a set with Lebesgue measure zero. Let B := PC r L 2 (g; X), r, be the

3 E. Hernández M, H.R. Henríquez / Applied Mathematics Letters 19 (26) space formed of all classes of functions ϕ : (, ] X such that ϕ [ r,] PC([ r, ], X), ϕ( ) is Lebesgue measurable on (, r] and g( ) ϕ( ) p is Lebesgue integrable on (, r]. The seminorm in B is defined by ( r 1/p ϕ B := g(θ) ϕ(θ) dθ) p + sup ϕ(θ). θ [ r,] From the proof of [16, Theorem 1.3.8] it follows that B is a phase space which verifies the axioms (A) and (B) of our work. Moreover, when r =, H = 1, M(t) = γ( t) 1 2 and K (t) = 1 + ( t g(τ )dτ)1 2 for t. Example: The phase space PC g (X). As usual, we say that a function ϕ : (, ] X is normalized piecewise continuous if the restriction of ϕ to any interval [ r, ] is a normalized piecewise continuous function. Let g : (, ] [1, ) be a continuous function which satisfies the conditions (g-1), (g-2) of [16]. We denote by PC g (X) the space formed by the normalized piecewise continuous functions ϕ such that ϕ g is bounded on (, ] and by PC g (X) the subspace of PC g(x) formed by the functions ϕ : (, ] X such that ϕ(θ) g(θ) asθ.itiseasy to see that PC g (X) and PCg (X) ϕ(θ) endowed with the norm ϕ B := sup θ (,] g(θ) are phase spaces in the sense considered in this work. 2. Existence results To study the existence of solutions of (1.1) (1.3), we always assume that the next condition holds. H 1 The functions G, F : I B X and I i : B X, i = 1,...,n, are continuous and satisfy the following conditions: (i) For every x : (,σ] Xsuch that x = ϕ and x I PC,thefunction t G(t, x t ) is strongly measurable and the function t F(t, x t ) belongs to PC. (ii) There are positive constants β (, 1), L F, L G, L i, i = 1,...,n, suchthat F is X β -valued, ( A) β F : I B X is continuous and G(t,ψ 1 ) G(t,ψ 2 ) L G ψ 1 ψ 2 B, ψ i B, t I, ( A) β F(t,ψ 1 ) ( A) β F(t,ψ 2 ) L F ψ 1 ψ 2 B, ψ i B, t I, I i (ψ 1 ) I i (ψ 2 ) L i ψ 1 ψ 2 B, ψ i B. Remark 2.1. Let x : (,σ] X be such that x = ϕ and x I PC,andassume that H 1 holds. From the continuity of s AT(t s) in the uniform operator topology on [, t) and the estimate AT(t s)f(s, x s ) = ( A) 1 β T (t s)( A) β F(s, x s ) C 1 β (t s) 1 β ( A)β F(s, x s ), it follows that the function θ AT(t θ)f(θ, x θ ) is integrable on [, t) for every t >. Proceeding similarly we can assert that the function s T (t s)g(s, x s ) is integrable on [, t] for every t >. Next, we introduce the concepts of mild and strong solutions of (1.1) (1.3).

4 218 E. Hernández M, H.R. Henríquez / Applied Mathematics Letters 19 (26) Definition 2.1. Afunction u : (,σ] X is a mild solution of the impulsive abstract Cauchy problem (1.1) (1.3)ifu = ϕ; u( ) I PC and u(t) = T (t)(ϕ() + F(,ϕ)) F(t, u t ) AT(t s)f(s, u s )ds + T (t s)g(s, u s )ds + T (t t i )[I i (u ti ) + F(t i, u ti )], t I. (2.1) <t i <t Definition 2.2. Afunction u : (,σ] X is a strong solution of (1.1) (1.3)ifu = ϕ; u( ) I PC; the functions u(t) and F(t, u t ) are differentiable a.e. on I with derivatives u (t) and d dt F(t, u t) in L 1 (I ; X);Eq.(1.1) isverified a.e. on I and (1.3)isvalidforeveryi = 1,...,n. Let u( ) be a strong solution and assume that H 1 holds. From the semigroup theory, we get u(t) = T (t)ϕ() which implies that T (t s) d dt F(s, u s)ds + T (t s)g(s, u s )ds, t [, t 1 ), 1 u(t 1 ) = T (t 1)(ϕ() + F(,ϕ)) F(t 1, u t ) AT(t s)f(s, u s )ds T (t s)g(s, u s )ds. By using that u( ) is a solution of (1.1) on(t 1, t 2 ) and that u(t + 1 ) = u(t 1 ) + I 1(u t1 ),weseethat u(t) = T (t t 1 )(u(t + 1 ) + F(t+ 1, u t + 1 )) F(t, u t) + T (t s)g(s, u s )ds t 1 = T (t)(ϕ() + F(,ϕ)) F(t, u t ) + T (t t 1 ) [ I 1 (u t1 ) + F(t 1, u t1 ) ], t (t 1, t 2 ). t 1 AT(t s)f(s, u s )ds AT(t s)f(s, u s )ds + T (t s)g(s, u s )ds Reiterating these procedure, we can conclude that u( ) is also a mild solution of (1.1) (1.3). Remark 2.2. Our definition of a mild solution is different from that introduced in [13,14]. In the following result, themainresult of this work, K σ = sup s I K (s). Theorem 2.1. If Θ = K σ [L F ( ( A) β (1 + 2n M) + C 1 βσ β β ) + M(σ L G + n i=1 L i )] < 1,thenthere exists a unique mild solution of (1.1) (1.3). Proof. On the metric space BPC ={u : (,σ] X; u = ϕ, u I PC} endowed with themetric d(u,v)= u v PC,wedefinetheoperator Γ : BPC BPC by

5 E. Hernández M, H.R. Henríquez / Applied Mathematics Letters 19 (26) ϕ(t), t, t Γ x(t) = T (t)(ϕ() + F(,ϕ)) F(t, x t ) AT(t s)f(s, x s )ds + T (t s)g(s, x s)ds + T (t t i )[I i (x ti ) + F(t i, x ti )], t I. <t i <t From Remark 2.1, weknow that s T (t s)g(s, x s ) and s AT(t s)f(s, x s ) are integrable on [, t) for every t I.Thus, Γ is well defined with values in PC. Letu,v BPC. Usingthat F(t i, u ti ) F(t i,v ti ) 2L F K σ ( A) β (u v) I PC,weget [ ] Γ u(t) Γ v(t) ( A) β C1 β L F L F u t v t B + (t s) 1 β + ML G u s v s B ds + M i u ti v ti B + 2 ( A) t i t(l β L F K σ (u v) I PC ) K σ [ L F ( ( A) β (1 + 2n M) + C 1 βσ β β ) ( + M σ L G + )] n L i (u v) I PC, and hence d(γ u, Γ v) Θd(u,v),whichproves that Γ is a contraction on BPC. Thus, there exists a unique mild solution of (1.1) (1.3). This completes the proof. i=1 In the next result, for x X, X x : (, ] X represents the function defined by X x (θ) = for θ < andx x () = x, and(s(t)) t is the family of linear operators defined by S(t)ψ(θ) = ψ() on [ t, ] and S(t)ψ(θ) = ψ(t + θ) on (, t]. Moreover, for the sake of brevity, we put t =, t n+1 = σ and I =. Theorem 2.2. Assume that the hypotheses of Theorem 2.1 are satisfied and let u( ) be the unique mild solution of (1.1) (1.3). IfXisreflexive, F(I B) D(A) and the next condition holds: (a) u ti + X Ii (u ti ) B, u(t i ) + I i (u ti ) D(A) and t S(t)(u ti + X Ii (u ti )) is Lipschitz on [t i, t i+1 ] for every i =,...n, then u( ) is a strong solution of (1.1) (1.3). Proof. Let u i C([t i, t i+1 ]:X), i =,...,n, besuchthatu i = u on (t i, t i+1 ].Itiseasy to prove that u i ( ) is a mild solution, in the sense introduced in [5], of the abstract Cauchy problem d dt (x(t) + F(t, x t )) = Ax(t) + G(t, x t ), t [t i, t i+1 ], (2.2) x ti = u ti + X Ii (u ti ). (2.3) From [5,Theorem 3.1], we infer that u i ( ) is a strong solution of (2.2)and(2.3). This means that u i and t F(t,(u i ) t ) are differentiable a.e. on [t i, t i+1 ];that(u i ) and d dt F(t, ui t ) belong to L1 ([t i, t i+1 ]; X) and that (2.2) isverified a.e. on [t i, t i+1 ].This is enough for concluding that u( ) is a strong solution of (1.1) (1.3).

6 22 E. Hernández M, H.R. Henríquez / Applied Mathematics Letters 19 (26) Example Next, we study Example 4.1 of [5] subjected to impulsive conditions. Let X = L 2 ([,π]), B = PC L 2 (g, X), thephase space introduced in Section 1, anda : D(A) X X be the operator Af = f with domain D(A) := { f X : f X, f () = f (π) = }. Itiswellknown that A is the infinitesimal generator of an analytic semigroup (T (t)) t on X. Thespectrum of A is discrete with eigenvalues n 2, n N, andassociated normalized eigenvectors z n (ξ) := ( π 2 )1/2 sin(nξ). Moreover, the set {z n : n N} is an orthonormal basis of X, T (t) f = n=1 e n2t f, z n z n for f X and ( A) α f = n=1 n 2α f, z n z n for f X α.itfollows from these expressions that T (t) e t, ( A) 1 2 T (t) e t 2 t 1 2 for each t > andthat ( A) 1/2 =1. 2 Consider the first order neutral differential equation with unbounded delay and impulses: [ π u(t,ξ)+ b(t s,η,ξ)u(s,η)dηds t + a 1 (t,ξ)+ ] = 2 ξ 2 u(t,ξ)+ a (ξ)u(t,ξ) a(t s)u(s,ξ)ds, t I =[,σ], (3.1) u(t, ) = u(t,π)=, t I, (3.2) u(τ, ξ) = ϕ(τ,ξ), τ, ξ J =[,π], (3.3) u(t i )(ξ) = i p i (t i s)u(s,ξ)ds, ξ J =[,π], (3.4) where < t 1 < < t n <σare prefixed numbers and ϕ B. Tostudy this system we will assume that a : J R, a 1 : I J R are continuous functions and that the following conditions hold: (i) The functions i b(s,η,ξ), i =, 1, are measurable on R J 2, b(s,η,π)= b(s,η,) =, for every ξ i (s,η) R J and N := π π 1 g(τ) (b(τ, η,ζ))2 dηdτ dζ <. (ii) The functions a : R R, p i : R R are continuous, ( pi 2(θ) g(θ) dθ)1 2 < for all i = 1,...,n. By defining the operators G, F, I i by F(t,ψ)(ξ) := π G(t,ψ)(ξ) := a (ξ)ψ(,ξ)+ I i (ψ)(ξ) := b( τ,η,ξ)ψ(τ,η)dηdτ, p i ( s)ψ(s,ξ)ds, a( τ)ψ(τ,ξ)dτ, a 2 ( θ) g(θ) dθ < and L i := we can model (3.1) (3.4) astheabstract system (1.1) (1.3). Moreover, the maps F(t, ), G(t, ), I i are bounded linear operators, F(t, ) N 1 ( 2 ) 1 a( θ), G(t, ) g(θ) dθ 2 + sup ξ I a (ξ) for all t I and I i L i for every i = 1,...,n.

7 E. Hernández M, H.R. Henríquez / Applied Mathematics Letters 19 (26) Proposition 3.1. Assume that the previous conditions are verified. If ( ) g(τ )dτ N (1 + 2n + ( 2e) + σ a( θ) g(θ) dθ + sup ξ J σ a (ξ) + n L i i=1 < 1, where N 1 := π π 1 g(τ) ( ζ b(τ, η,ζ))2 dηdτ dζ,thenthere exists a unique mild solution, u( ), of (3.1) (3.4). Moreover, if ϕ and u( ) verify the conditions in Theorem 2.2, thefunction 2 b(θ,η,ξ) is ξ 2 measurable and N 2 = π π 1 g(s) ( 2 b(s,η,ξ) ) 2 dηdsdξ <, thenu( ) is a strong solution. ξ 2 Proof. A easy estimation using (i) permits us to prove that F is X 1 -valued and that F : I B X 1 is 2 2 continuous. Moreover, ( A) 1 2 F(t, ) is a bounded linear operator and ( A) 1 2 F(t, ) N for each t I.Now,theexistence of a mild solution is a consequence of Theorem 2.1. The regularity assertion follows directly from Theorem 2.2, since under these additional conditions, the function AF : I B X is well defined and continuous. The proof is completed. ) 1 2 Acknowledgements The authors wish to thank the referees for their comments and suggestions. The work of the second author was supported by FONDECYT-CONICYT, Project and Project References [1] S.K. Ntouyas, Y.G. Sficas, P.Ch. Tsamatos, Existence results for initial value problems for neutral functional-differential equations, J. Differential Equations 114 (2) (1994) [2] J.K. Hale, S.M.V. Lunel, Introduction to Functional-Differential Equations, in: Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, [3] M. Adimy, K. Ezzinbi, Strict solutions of nonlinear hyperbolic neutral differential equations, Appl. Math. Lett. 12 (1) (1999) [4] R. Datko, Linear autonomous neutral differential equations in a Banach space, J. Differential Equations 25 (2) (1977) [5] E. Hernández, H.R. Henríquez, Existence results for partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl. 221 (2) (1998) [6] E. Hernández, H.R. Henríquez, Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl. 221 (2) (1998) [7] E. Hernández, Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl. 292 (1) (24) [8] Y.V. Rogovchenko, Impulsive evolution systems: main results and new trends, Dyn. Contin. Discrete Impuls. Syst. 3 (1) (1997) [9] J.H.Liu,Nonlinear impulsive evolution equations, Dyn. Contin. Discrete Impuls. Syst. 6 (1) (1999) [1] M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive neutral functional differential inclusions in Banach spaces, Appl. Math. Lett. 15 (8) (22) [11] M. Benchohra, A. Ouahab, Impulsive neutral functional differential equations with variable times, Nonlinear Anal. 55 (6) (23)

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