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ELSEVIH~,
Computers and Mathematics with Applications 51 (2006) 927-936 www.elsevier.com/locate/camwa
Nonhomogeneous Boundary Value Problem for First-Order Impulsive Differential Equations with Delay FENGQIN ZHANG AND MEILI LI Department of Applied Mathematics, Yuncheng University Yuncheng Shanxi 044000, P.R. China zhafq©263, n e t
JURANG YAN Department of Mathematics, Shanxi University Taiyuan Shanxi 030006, P.R. China (Received January 2005; revised and accepted November 2005) Abstract--The authors employ the method of upper and lower solution coupled with the monotone iterative technique to obtain results of existence and uniqueness for a nonhomogeneous boundary value problem of impulsive differential equations with delay. (~) 2006 Elsevier Ltd. All rights reserved. K e y w o r d s - - N o n h o m o g e n e o u s boundary value problem, Monotone iterative technique, Lower solution, Upper solution, Impulsive differential equations with delay.
1. I N T R O D U C T I O N W e are c o n c e r n e d w i t h t h e following n o n h o m o g e n e o u s b o u n d a r y v a l u e p r o b l e m for a first-order i m p u l s i v e differential e q u a t i o n w i t h delay in R,
t E J',
x' (t) = f (t, x (t), z t ) , A x (tk) = Ik (~ (tk)), x (0) = • (0), x
(o) -
~
(T)
=
~ e
k = 1,2,...,m, for a g i v e n 0 E [--T, 0),
(1.1)
n,
w h e r e f : J × R × D --* R, D -- L I ( [ - T , 0 ] , R ) , Ik 6 of x ( t ) at t = tk, i.e., A x ( t k ) = x ( t +) - x(t~-), for all t,~ < T, 5 = m a x ( t k - t k - 1 ; k -- 1 , 2 , . . . , m + 1} h e r e J' = J\{tl,t2,...,tm}; for e v e r y t e J , xt e D is defined
C(R,R), Ax(tk) represents the jump k = 1,2,...,m, 0 < t l < t2 < . - . < to -- O, tm+Z = T; T > O, J =- [0, T], by x t ( s ) = x ( t + s), - ' r < s < O.
This work is supported by the National Sciences Foundation of China (No. 10471040) and the Sciences Foundation of Shanxi (No. 2005Z010) and the Major Subject Foundation of Shanxi (20055024). The authors thank the referee for his valuable suggestion. 0898-1221/06/$ - see front matter (~) 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2005.11.028
Typeset by A~4S-TF_~
F. ZHANG et al.
928
Suppose Jo = [--T, T]. Let P C ( & , R) = {x : Jo --* R, x(t) is continuous for t E J0, t :fi tk and x(t+), x(t~) exist and x(tk) = x(t~-), for k = 1 , 2 , . . . , m } ; P C ' ( J , R ) = {x : J --~ R,x(t) is continuously differentiable for t C J' and x(t+),x(t~) exist and x'(tk) = x'(t-~), for k = 1, 2 , . . . , m}; E = P C ( & , R) n P C ( J , R). Obviously, for any t E J and x E E, we have xt E D and PC(Jo, R) and E are Banach spaces with the norms, Ilallpc(Jo,R) = sup (]x(t)l : t e Jo)
and where
IIx'llPc(J,R) : sup{I x'(t) I: t e J}. By a solution of (1.1), we mean a x E E for which problem (1.1) is satisfied. For x, y C E, we define x < y on J0 if x(t) < y(t), for t E J0. Impulsive differential equations with delay have been extensively studied by m a n y authors; see, e.g., [1-10]. Those results ace applicable in some important cases such as the initial or the periodic case. However, they are not valid, for example, for nonhomogeneous x(0) - x ( T ) : 0The nonhomogeneous boundary value problems have been studied by m a n y authors; see [11,12] and references therein. It is the purpose of the present paper to establish the existence and uniqueness of a solution for (1.1). 2.
COMPARISON
THEOREMS
This section is devoted to comparison theorems, which are needed for the successful employment of the monotone iterative technique. LEMMA 2.1. Let p(t) C E such that
p'(t) f (t, ~, ~) - f (t, u, ~) 1"
_> - M (fi - u) - N
/o
(~t (s) - ~t (s)) ds,
T
whenever v < u < ~ < w, vt
www.sciencedirect.com__
computers
, , = , , , = = ('-~f) ° , . = c T .
&
mathematics with applications
ELSEVIH~,
Computers and Mathematics with Applications 51 (2006) 927-936 www.elsevier.com/locate/camwa
Nonhomogeneous Boundary Value Problem for First-Order Impulsive Differential Equations with Delay FENGQIN ZHANG AND MEILI LI Department of Applied Mathematics, Yuncheng University Yuncheng Shanxi 044000, P.R. China zhafq©263, n e t
JURANG YAN Department of Mathematics, Shanxi University Taiyuan Shanxi 030006, P.R. China (Received January 2005; revised and accepted November 2005) Abstract--The authors employ the method of upper and lower solution coupled with the monotone iterative technique to obtain results of existence and uniqueness for a nonhomogeneous boundary value problem of impulsive differential equations with delay. (~) 2006 Elsevier Ltd. All rights reserved. K e y w o r d s - - N o n h o m o g e n e o u s boundary value problem, Monotone iterative technique, Lower solution, Upper solution, Impulsive differential equations with delay.
1. I N T R O D U C T I O N W e are c o n c e r n e d w i t h t h e following n o n h o m o g e n e o u s b o u n d a r y v a l u e p r o b l e m for a first-order i m p u l s i v e differential e q u a t i o n w i t h delay in R,
t E J',
x' (t) = f (t, x (t), z t ) , A x (tk) = Ik (~ (tk)), x (0) = • (0), x
(o) -
~
(T)
=
~ e
k = 1,2,...,m, for a g i v e n 0 E [--T, 0),
(1.1)
n,
w h e r e f : J × R × D --* R, D -- L I ( [ - T , 0 ] , R ) , Ik 6 of x ( t ) at t = tk, i.e., A x ( t k ) = x ( t +) - x(t~-), for all t,~ < T, 5 = m a x ( t k - t k - 1 ; k -- 1 , 2 , . . . , m + 1} h e r e J' = J\{tl,t2,...,tm}; for e v e r y t e J , xt e D is defined
C(R,R), Ax(tk) represents the jump k = 1,2,...,m, 0 < t l < t2 < . - . < to -- O, tm+Z = T; T > O, J =- [0, T], by x t ( s ) = x ( t + s), - ' r < s < O.
This work is supported by the National Sciences Foundation of China (No. 10471040) and the Sciences Foundation of Shanxi (No. 2005Z010) and the Major Subject Foundation of Shanxi (20055024). The authors thank the referee for his valuable suggestion. 0898-1221/06/$ - see front matter (~) 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2005.11.028
Typeset by A~4S-TF_~
F. ZHANG et al.
928
Suppose Jo = [--T, T]. Let P C ( & , R) = {x : Jo --* R, x(t) is continuous for t E J0, t :fi tk and x(t+), x(t~) exist and x(tk) = x(t~-), for k = 1 , 2 , . . . , m } ; P C ' ( J , R ) = {x : J --~ R,x(t) is continuously differentiable for t C J' and x(t+),x(t~) exist and x'(tk) = x'(t-~), for k = 1, 2 , . . . , m}; E = P C ( & , R) n P C ( J , R). Obviously, for any t E J and x E E, we have xt E D and PC(Jo, R) and E are Banach spaces with the norms, Ilallpc(Jo,R) = sup (]x(t)l : t e Jo)
and where
IIx'llPc(J,R) : sup{I x'(t) I: t e J}. By a solution of (1.1), we mean a x E E for which problem (1.1) is satisfied. For x, y C E, we define x < y on J0 if x(t) < y(t), for t E J0. Impulsive differential equations with delay have been extensively studied by m a n y authors; see, e.g., [1-10]. Those results ace applicable in some important cases such as the initial or the periodic case. However, they are not valid, for example, for nonhomogeneous x(0) - x ( T ) : 0The nonhomogeneous boundary value problems have been studied by m a n y authors; see [11,12] and references therein. It is the purpose of the present paper to establish the existence and uniqueness of a solution for (1.1). 2.
COMPARISON
THEOREMS
This section is devoted to comparison theorems, which are needed for the successful employment of the monotone iterative technique. LEMMA 2.1. Let p(t) C E such that
p'(t) f (t, ~, ~) - f (t, u, ~) 1"
_> - M (fi - u) - N
/o
(~t (s) - ~t (s)) ds,
T
whenever v < u < ~ < w, vt
