Selçuk J. Appl. Math. Selçuk Journal of Vol. 10. No. 2. pp. 15-26, 2009 Applied Mathematics
Stability properties of neutral delay integro-di¤erential equation Ali Fuat Yeniçerio¼glu
Department of Mathematics, The Faculty of Education, Kocaeli University, 41380, Kocaeli, Türkiye
e-mail: fuatyenicerioglu@ ko caeli.edu.tr
Received Date: November 28, 2007 Accepted Date: November 9, 2009
Abstract. Some new stability results are given for a neutral delay integro-di¤erential equation with constant delays. The stability of the trivial solution is described by the use of an appropriate real root of an equation, which is in a sense the corresponding characteristic equation. A basis theorem on the behavior of solutions of neutral delay integro-di¤erential equations is established. As a consequence of this theorem, a stability criterion is obtained.
Key words: Integro-di¤erential equation, Characteristic equation, Stability, Neutral delay.
2000 Mathematics Subject Classi…cation: 39A10, 39A11, 39B72, 39B82, 45J05. 1.Introduction and Preliminaries
This paper deals with the stability of the trivial solution for the …rst order linear neutral delay integro-di¤erential equation with constant delays. An estimate of the solutions is established. The su¢ cient conditions for the stability, the as-ymptotic stability and instability of the trivial solution are given. Our results are derived by the use of a real root (with an appropriate property) of the corresponding characteristic equation. The techniques applied in obtaining our results are originated in a combination of the methods used in [8,9,13,15]. Since the …rst systematic study was carried out by Volterra [14], this type of equa-tions have been investigated in various …elds, such as mathematical biology and control theory (see, e.g., [6,7,10]). This system can be found in a wide variety of scienti…c and engineering …elds such as biology, physics, ecology, medicine and so on (cf. [1,2]). Particularly, it is observed that the delay-integro-di¤erential system plays an important role in modelling of many di¤erent phenomena of circuit analysis and chemical process similation, which have a comprehensive list in [12].
In this paper, we will give a basic theorem on the behavior of solutions of scaler neutral delay integro-di¤erential equations. As a consequence of this the-orem, we will establish a criterion for the stability of the trivial solution (see [5,8,9,13,15] for related stability results). A root of the associated characteristic equation is used in obtaining our results. For the basis theory of integral equa-tions, we choose to refer to the books by Burton [3], Corduneanu [4] and Lak-shmikantham [11]. For the basic theory of neutral delay di¤erential equations, the reader is referred to the books by Hale and Verduyn Lunel [6], Kolmanovski [7] and Lakshmikantham [11].
Note that linear neutral delay di¤erential equations with periodic coe¢ cients have been studied by Philos and Purnaras [13] and the behavior of solutions of linear di¤erential equations with unbounded delay have been obtained by Kordonis and Philos [8].
Let us consider initial value problem for neutral delay integro-di¤erential equa-tion (1) x(t) +P i2I cix(t i) + r R 0 g(s)x(t s)ds 0 = ax(t) + P j2J bjx(t j) + r R 0 k(s)x(t s)ds (2) x(t) = (t) ; r t 0
where I and J are initial segments of natural numbers, a, bj, ci for i 2 I,j 2 J
are real numbers, i for i 2 I are positive real numbers such that i1 6= i2 for
i1; i2 2 I with i1 6= i2, and j for j 2 I are positive real numbers such that j1 6= j2 for j1; j22 J with j16= j2. r is positive number such that
= max
i2I i ; = maxj2J j ; and r = maxf ; g:
k and g are continuous real-valued functions on the interval [0; 1) and (t) is continuous given initial function on the interval [ r; 0].
As usual, a continuous real-valued function x de…ned on the interval [ r; 1) is said to be a solution of the di¤erential equation (1) if the function x(t) + P i2I cix(t i) + r R 0
g(s)x(t s)ds is continuously di¤erentiable for t 0 and x satis…es (1) for all t 0.
It is known (see, for example, [3]) that, for any given initial function , there exists a unique solution x of the neutral delay integro-di¤erential equation (1), where is de…ned in (2), x will be called the solution of the initial value problem (1)-(2).
If we look for a solution of (1) of the form x(t) = e tfor t 2 IR, we see that
(3) 0 @1 +X i2I cie i+ r Z 0 g(s)e sds 1 A = a +X j2J bje j + r Z 0 k(s)e sds:
Before closing this section, we will give here the de…nitions of these notions (see, for example, [3]). The trivial solution of (1) is said to be “stable” (at 0) if for every " > 0, there exists a number = (") > 0 such that, for any initial fuction
with
k k max
r t 0j (t)j < ;
the solution x of (1)-(2) satis…es
jx(t)j < " ; for all t 2 [ r; 1):
Otherwise, the trivial solution of (1) is said to be unstable (at 0). Moreover, The trivial solution of (1) is called “asymptotically stable” (at 0) if it is stable in the above sense and in addition there exists a number 0> 0 such that, for
any initial fuction with k k < 0, the solution x of (1)-(2) satis…es
lim
t!1x(t) = 0:
2. Statement of the main results and comments The main results of the paper are Theorems 1, 2 and 3 below.
Theorem 1. Let 0 be a real root of the characteristic equation (3). Assume
that the root 0 has the following property
(4) 0 X i2I (1 + j 0j i) jcije 0 i+ X j2J jbjje 0 j j + r Z 0 (jg(s)j + sjk(s) 0g(s)j) e 0sds < 1:
Then, for any 2 C ([ r; 0]; IR), the solution x of problem (1)-(2) satis…es
(5) e 0tx(t) L 0( ) 0
M 0( ) 0 ; for all t 0;
(6) L 0( ) = (0)+ P i2I ci ( i) 0e 0 i 0 R i e 0s (s)ds ! + P j2J bje 0 j 0 R j e 0s (s)ds + r R 0 g(s) ( s)ds + r R 0 (k(s) 0g(s)) e 0s ( 0 R s e 0u (u)du ) ds; (7) 0= 1 + P i2I (1 0 i) cie 0 i+P j2J bje 0 j j + r R 0 (g(s) + s(k(s) 0g(s))) e 0sds and (8) M 0( ) = max r t 0 e 0t (t) L 0( ) 0 :
Remark. It follows from (4) that 0 > 0.
Proof. Let now x be the solution of (1)-(2). De…ne y(t) = e 0tx(t) ; for t 2 [ r; 1):
Then, for every t 0, we have ( x(t)+P i2I cix(t i) + r R 0 g(s)x(t s)ds 0 ax(t) P j2J bjx (t j) r R 0 k(s)x(t s)ds ) e 0t = y0(t) + 0y(t)+ P i2I cie 0 i(y0(t i) + 0y(t i)) + r R 0 g(s)e 0s (y0(t s) + 0y(t s)) ds ay(t) X j2J bje 0 jy(t j) r Z 0 k(s)e 0sy(t s)ds = 0:
(9) y(t)+P i2I cie 0 iy(t i) + r R 0 g(s)e 0s y(t s)ds 0 = (a 0)y(t) +P j2J bje 0 jy(t j) 0P i2I cie 0 iy(t i) + r R 0 (k(s) 0g(s)) e 0sy(t s)ds:
Moreover, the initial condition (2) can be equivalently written
(10) y(t) = e 0t (t) ; for
t 2 [ r; 0]:
Furthermore, by using the fact that 0 is a root of (3) and taking into account
(6) and (10), we can verify that (9) is equivalent to
y(t)+X i2I cie 0 iy(t i)+ r Z 0 g(s)e 0s y(t s)ds = y(0)+X i2I cie 0 iy( i) + r Z 0 g(s)e 0sy( s)ds + (a 0) t Z 0 y(s)ds +X j2J bje 0 j t Z o y(s j)ds 0 X i2I cie 0 i t Z 0 y(s i)ds + r Z 0 (k(s) 0g(s)) e 0s 8 < : t Z 0 y(u s)du 9 = ;ds = (0)+X i2I ci ( i)+ r Z 0 g(s) ( s)ds+(a 0) t Z 0 y(s)ds+X j2J bje 0 j tZ j j y(s)ds 0 X i2I cie 0 i tZ i i y(s)ds + r Z 0 (k(s) 0g(s)) e 0s 8 < : t s Z s y(u)du 9 = ;ds = (0) +X i2I ci ( i) + r Z 0 g(s) ( s)ds + (a 0) t Z 0 y(s)ds +P j2J bje 0 j ( 0 R j e 0s (s)ds+ tR j 0 y(s)ds ) 0P i2I cie 0 i ( 0 R i e 0s (s)ds + tR i 0 y(s)ds )
+ r Z 0 (k(s) 0g(s)) e 0s 8 < : 0 Z s e 0u (u)du+ t s Z 0 y(u)du 9 = ;ds = (a 0) t Z 0 y(s)ds +X j2J bje 0 j tZ j 0 y(s)ds 0 X i2I cie 0 i tZ i 0 y(s)ds + r Z 0 (k(s) 0g(s)) e 0s 8 < : t s Z 0 y(u)du 9 = ;ds + L 0( ) = 0P i2I cie 0 i P j2J bje 0 j r R 0 (k(s) 0g(s)) e 0sds ! t R 0 y(s)ds +P j2J bje 0 j tR j 0 y(s)ds 0P i2I cie 0 i tR i 0 y(s)ds + r Z 0 (k(s) 0g(s)) e 0s 8 < : t s Z 0 y(u)du 9 = ;ds + L 0( ) = 0 X i2I cie 0 i t Z t i y(s)ds X j2J bje 0 j t Z t j y(s)ds (11) r Z 0 (k(s) 0g(s)) e 0s 8 < : t Z t s y(u)du 9 = ;ds + L 0( ): Next, we de…ne z(t) = y(t) L 0( ) 0 ; for t r:
Then we can see that (11) reduces to the following equivalent equation:
z(t)+X i2I cie 0 iz(t i)+ r Z 0 g(s)e 0sz(t s)ds = 0 X i2I cie 0 i t Z t i z(s)ds (12) X j2J bje 0 j t Z t j z(s)ds r Z 0 (k(s) 0g(s)) e 0s 8 < : t Z t s z(u)du 9 = ;ds:
On the other hand, the initial condition (10) can be equivalently written
(13) z(t) = (t)e 0t L 0( ) 0
; t 2 [ r; 0]: Because of the de…nitions of y ve z, (5) is equivalent to
(14) jz(t)j M 0( ) 0; 8t 0:
The proof will be accomplished by proving (14). Now, in view of (8) and (13), we have
(15) jz(t)j M 0( ) ; for t 2 [ r; 0]:
We will show that M 0( ) is a bound of z on the whole interval [ r; 1), namely
(16) jz(t)j M 0( ) ; for all t 2 [ r; 1):
To this end, let us consider an arbitrary number " > 0. We claim that
(17) jz(t)j < M 0( ) + " ; for every t 2 [ r; 1):
Otherwise, by (15), there exists a t > 0 such that
jz(t)j < M 0( ) + "; for t < t and jz(t )j = M 0( ) + ":
Then from (12), we obtain
M 0( ) + " = jz(t )j X i2I jcije 0 ijz(t i)j + r Z 0 jg(s)je 0sjz(t s)jds +j 0j X i2I jcije 0 i t Z t i jz(s)jds +X j2J jbjje 0 j t Z t j jz(s)jds + r Z 0 jk(s) 0g(s)je 0s 8 < : t Z t s jz(u)jdu 9 = ;ds 8 < : X i2I jcije 0 i+ r Z 0 jg(s)je 0sds + j 0j X i2I jcije 0 i i+ X j2J jbjje 0 j j
+ r Z 0 jk(s) 0g(s)je 0ssds 9 = ;[M 0( ) + "] = 0[M 0( ) + "] < [M 0( ) + "]
which, in view of (4), leads to a contradiction. So, our claim is true. Since (17) holds for every " > 0, it follows that (16) is always satis…ed. By using (16), from (12), we derive for all t 0,
jz(t)j P i2Ijc ije 0 ijz(t i)j + r R 0 jg(s)je 0sjz(t s)jds +j 0jP i2Ijc ije 0 i t R t i jz(s)jds + P j2Jjb jje 0 j t R t j jz(s)jds + r R 0 jk(s) 0g(s)je 0s ( t R t sjz(u)jdu ) ds P i2Ijc ije 0 i+ r R 0 jg(s)je 0sds +j 0jP i2Ijc ije 0 i i+P j2Jjb jje 0 j j+ r R 0 jk(s) 0g(s)je 0ssds ) M 0( ) = 0M 0( )
i.e., (14) holds. This implies the proof of the Theorem 1.
Theorem 2. Let 0be a real root of the characteristic equation (3). Then, for
any 2 C ([ r; 0]; IR), the solution x of problem (1)-(2) satis…es lim
t!1 e
0tx(t) = L 0( ) 0
;
Where L 0( ) and 0 were given in (6) and (7), respectively.
Proof. By the de…nitions of y and z, we need to prove that
(18) lim
t!1z(t) = 0:
In the rest of the proof we will establish (18). By using (12) and taking into account (16) and (14), one can show, by an easy induction, that z satis…es
(19) jz(t)j 0 nM 0( ) for all t nr r; (n = 0; 1; :::):
But, (4) guarantees that 0 < 0 < 1. Thus, from (19) it follows immediately that z tends to zero as t ! 1, i.e. (17) holds.
Let us assume that (20) a+P j2J bj+ r R 0 k(s)ds = 0 and P i2Ijc ij+P j2Jjb jj j+ r R 0(jg(s)j + sjk(s)j) ds < 1:
From the …rst assumption of (20), we see that, 0 = 0 is a (real) root of the
characteristic equation (3). Furthermore, because of the second assumption of (20), it is not di¢ cult to verify that the root of (3) has the property (4). Thus, an application of Theorem with 0= 0 leads to the following results.
Corollary. Let condition (20) holds. Then, for any 2 C ([ r; 0]; IR), the solution x of (1)-(2) satis…es lim t!1x(t) = (0)+P i2I ci ( i)+ P j2J bj 0 R j (s)ds+ r R 0 g(s) ( s)ds+ r R 0 k(s) ( 0 R s (u)du ) ds 1+P i2I ci+P j2J bj j+ r R 0 (g(s)+sk(s))ds :
Theorem 3. Let 0be a real root of the characteristic equation (3). Then, for
any 2 C ([ r; 0]; IR), the solution x of problem (1)-(2) satis…es
(21) jx(t)j 0N 0( )e 0t ; for all t 0; where (22) 0 = 1 + 0 2 0 + 0 and (23) N 0( ) = max r t 0 e 0tj (t)j :
Moreover, the trivial solution of (1) is stable if 0= 0, it is asymptotically stable
if 0< 0 and it is unstable if 0> 0.
Proof. First of all, we observe that, because of (4) and 0 > 0, formula (22) de…nes a real number 0 with 0> 1.
By our theorem, (5) is satis…ed, where L 0( ) and M 0( ) are de…ned by (6)
and (8), respectively. From (5), it follows that
(24) e 0tx(t) jL 0( )j
0
But (8) gives M 0( ) N 0( ) + jL 0( )j 0 : So (24) yields (25) e 0tjx(t)j 1 + 0 0 jL 0( )j + N 0( ) 0; for every t 0:
Furthermore, from (6), we obtain
jL 0( )j j (0)j + P i2Ijc ij j ( i)j + j 0je 0 i 0 R i e 0sj (s)jds ! + ( r R 0jgs)jj ( s)jds + P j2Jjb jje 0 j 0 R j e 0sj (s)jds + r R 0jk(s) 0g(s)je 0s ( 0 R s e 0uj (u)jdu ) ds which gives jL 0( )j 1 + P i2Ijc ij e 0 i+ j 0je 0 i i + r R 0jg(s)je 0sds +P j2Jjb jje 0 j j + r R 0 jk(s) 0g(s)je 0ssds N 0( ) = 1 + 0 N 0( ):
Hence, from (25), we conclude that for t 0,
e 0tjx(t)j 1 + 0 2 0 N 0( ) + 0N 0( ) or e 0tjx(t)j 0N 0( )
and consequently, (21) holds true.
Now, let us assume that 0 0. De…ne jj jj = max
r t 0j (t)j. It follows that
N 0( ) jj jj.
Thus, (21) gives
(26) jx(t)j 0jj jje
0t ; for all t 0:
jx(t)j 0jj jj ; for every t 0:
So, by taking into account the fact that 0 > 1, we have
jx(t)j 0jj jj ; for every t 2 [ r; 1);
which means that the trivial solution of (1) is stable (at 0). Next, if 0< 0, then (26) guarantees that
lim
t!1x(t) = 0;
and so the trivial solution of (1) is asymptotically stable (at 0).
Finally, if 0> 0, then the trivial solution of (1) is unstable (at 0). Otherwise,
there exists a number (1) > 0 such that, for any 2 C ([ r; 0]; IR) with jj jj < , the solution x of problem (1)-(2) satis…es
(27) jx(t)j < 1 for all t r:
De…ne
0(t) = e 0t for t 2 [ r; 0]:
Clearly, 06= 0. Furthermore, by the de…nition of L 0( ), we have
L 0( 0) = 1 + P i2I ci e 0 i 0e 0 i 0 R i e 0se 0sds ! +P j2J bje 0 j 0 R j e 0se 0sds + r R 0 g(s)e 0sds + r R 0 (k(s) 0g(s)) e 0s ( 0 R s e 0ue 0udu ) ds = 1 +P i2I ci(1 0 i) e 0 i+P j2J bje 0 j j+ r R 0 g(s)e 0sds + r R 0 (k(s) 0g(s)) e 0ssds 0 > 0:
Let 2 C ([ r; 0]; IR) be de…ned by
= 1
jj 0jj 0;
where 1is a number with 0 < 1< . Moreover, let x be the solution of (1)-(2).
From Theorem 2 it follows that x satis…es lim t!1 e 0tx(t) = L 0( ) 0 =( 1=jj 0jj) L 0( 0) 0 = 1 jj 0jj > 0:
But, we have jj jj = 1< and hence from (27) and condition 0> 0 it follows that lim t!1 e 0tx(t) = 0: This is a contradiction.
The proof of our Theorem 3 is completed. References
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