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Impulsive functional differential inclusions and fuzzy population models
Fuzzy Sets and Systems 138 (2003) 601 – 615 www.elsevier.com/locate/fss
Impulsive functional di#erential inclusions and fuzzy population models Mengshu Guo∗ , Xiaoping Xue, Ronglu Li Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China Received 23 February 2001; received in revised form 5 August 2002; accepted 7 November 2002
Abstract In this paper we shall establish some existence results for the impulsive functional di#erential inclusion and the fuzzy impulsive functional di#erential equation with some conditions, and study the properties of the solution set and the attainable set. Finally, the results will be used to fuzzy population models. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy impulsive functional di#erential inclusion (equation); The solution set; The attainable set; Fuzzy population model
1. Introduction In [5,7,8] the fuzzy di#erential equation (FDE) model was expressed by a family of ordinary di#erential inclusions (DI) in R n as following: x (t) ∈ [G(t; x (t))]
a:e: t ∈ [0; T ];
x (0) ∈ X = [X0 ] ;
∈ [0; 1];
(1.1)
where [G(· ; ·)] : R × R n → PKC (R n ). Diamond [5] proved that under some conditions the solution set (X ; T ) and the attainable set A (X ; T ) are -level sets of 0 (X 0 ; T ) and A 0 (X 0 ; T ), respectively, where ∈ [0; 1]. In this paper under some weaker conditions we shall study the analogous problems for the following fuzzy impulsive functional di#erential equation (FIFDE): x (t) ∈ [G(t; x (t); x (t − (t)))]
a:e: t ∈ R+ ;
∗
Corresponding author. Research supported by HIT.M.D. 2000.24. E-mail address: [email protected] (M. Guo).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 5 2 2 - 5
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Ex (tk ) ∈ [lk (x (tk ))] ; x (t) ∈ [’(t)] ;
k = 0; 1; 2; : : : ;
t ∈ I0 = [t0 ; 0];
(1.2)
where ∈ [0; 1] and t 0 = min t ∈R+ (t − (t)). We shall establish some existence theorems of the solutions to (1.2) and discuss the properties of the solution set. The FIFDE (1.2) is the model for a large class of real objects and processes. In biology, control theory and electronics, etc., many problems can be described by Eq. (1.2) accurately. Example 1 (Some population models): As is well known that the classical population model (P.F. Verhulst model) can be express generally by the following di#erential equation: N (t) N (t) = rN (t) 1 − ; r ¿ 0; P0 where N (t)¿0 is the population biomass (size) at the moment t¿0 and 1−N (t)=P0 is the biological feedback factor. But the social friction factor r=P0 is determined usually by many uncertain factors, e.g., the natural and social resource state, the age structure, the medical treatment and sanitation state. Hence, the more accurate model should be fuzzy. If we consider the delay e#ects of gestation and some other factors, the certain model is (E.M. Wright model) N (t − ) N (t) = rN (t) 1 − ; r; ¿ 0 P0 or more generally, N (t) = N (t)f0 (N (t − )) = f(N (t); N (t − )); N (t) = ’(t);
t ∈ R+ ;
t ∈ I0 = [t0 ; 0]:
And the corresponding fuzzy model is N (t) ∈ [f(t; N (t); N (t − ))] ; N (t) ∈ [’(t)] ;
t ∈ R+ ;
t ∈ I0 ; ∈ [0; 1];
where f is a fuzzy mapping, which is determined by the species growth law and some other uncertain factors. On the other hand, on account of some (usually uncertain) reasons, e.g., natural disaster, the population policy, the population structure, the wars, etc., there may be sharp changes in the amounts of the population at some moment t∗ , i.e., EN (t∗ ) = N (t∗+ ) − N (t∗− ) = C ¿ 0; where N (t∗+ ) = lim t →t∗ + N (t) and N (t∗− ) = limt →t∗ − N (t). Angelova and Dishliev [2] considered some optimization problems from impulsive population dynamics, but the model is certain. There is
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a better model, which is expressed by FIFDE N (t) ∈ [f(t; N (t); N (t − (t)))] ; EN (tk ) ∈ [lk (N (tk ))] ; N (t) ∈ [’(t)] ;
t ∈ R+ ;
k = 0; 1; 2; : : : ; m; ∈ [0; 1];
t ∈ I0 = [t0 ; 0]; t0 = min+ (t − (t)): t ∈R
Example 2 (The model of martial strength between two states): Let [A]; [B] be two states in some locally region, x; y the martial strength of [A] and [B]. The study of the geostrategic shows that the relation of the changes of the martial strength between [A] and [B] may be expressed by x = ky − x + gBA (t); y = lx − y + gAB (t); where gBA (resp. gAB ) is the threat of [B] to [A] (resp. [A] to [B]). But gBA and gAB are generally uncertain and fuzzy. Moreover, there will be sharp changes in the martial strength in a short time when some weapons are equipped. Thus, the impulsive factors and delay should be described in the model. Therefore, the condign model is expressed by FIFDE as x f1 (t; x(t); x(t − ); y(t); y(t − ) (t) ∈ ; y f2 (t; x(t); x(t − ); y(t); y(t − )
Ex Ey
(tk ) ∈
l1k (x(tk ); y(tk )) l2k (x(tk ); y(tk ))
:
Example 3 (Some control systems problems): In [1] some control problems of certain impulsive systems were considered. The corresponding fuzzy problems can be expressed as following: x (t) ∈ [G(t; x (t); x (t − (t)); Ex (tk ) ∈ [lk (x (tk ); u (t))] ; x (t) ∈ [’(t; u (t))] ; y (t) = g(t; u (t));
u (t))] ; t ∈ I; k = 0; 1; 2; : : : ; m;
t ∈ I0 = [t0 ; 0]; ∈ [0; 1];
where g : I × E n → E n ; u ∈ Q(t). Moreover, the di#erential equations and inclusions involving impulse were used to study many complicated dynamical systems including hybrid control systems (see [3,9,12,11]). Many papers discussed the functional di#erential equations (inclusions) and the impulsive differential equations. However, as far as the author knows, few papers considered the corresponding fuzzy problems. It is pointed out in [5,7,8] that the conventional fuzzy di#erential models have some ill properties and should be replaced by the families of the DIs. This paper generalizes the results in [5,7].
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2. Notions and denitions In this paper, let R+ = [0; +∞); R+ = [0; +∞]; I ⊆ R+ be a compact interval, (X; · ) be a Banach space, and (R n ; · ) be the n-dimension Euclid space with usual norm. Denote by PK (X ) the set of all compact subsets of X and by PKC (X ) the set of all compact convex subsets of X . DeJne the Hausdor# metric dH (· ; ·) in PK (X ) by dH (A; B) = max sup dist(a; B); sup dist(b; A) ; a∈ A
b∈ B
and for A ∈ PK (X ) denote |A| = sup{ a : a ∈ A}. Let tk ∈ I; k = 1; 2; : : : : The following spaces will be used: C(I; Rn ) = {u : u is continuous from I to Rn }; AC(I; Rn ) = {u : u is absolutely continuous from I to Rn }; and n
PC(I; R ) = u : u ∈ C(I \{tk }; Rn ); lim u(t) = u(tk ); t →tk −
and lim u(tx) exists; k = 1; 2; : : : ; m : t →tk +
It is easy to show that PC(I; R n ) is a Banach space. For x ∈ PC(I; R n ) and the impulse e#ect tk ∈ I , denote Ex(tk ) = x(tk+ ) − x(tk− ), where x(tk+ ) = limt →tk + x(t) and x(tk− ) = limt →tk − x(t). Some deJnitions and results of fuzzy number spaces can be found in [5,6,10]. Theorem 2.1 (Stacking Theorem; Puri and Ralescu [10]). Let {Y ⊂ R n : 0661} be a family of compact subsets satisfying the following: (1) Y ∈ PKC (Rn ), for all 0661; (2) Y ⊆ Y , for 06 661; (3) Y = ∞ i=1 Yi , for any nondecreasing sequence i → in [0; 1]. Then there is a fuzzy set u ∈ Dn such that [u] = Y . In particular, if the Y are also convex, then u ∈ E n . Conversely, the level sets [u] of any u ∈ Dn satisfy these conditions, while if u ∈ E n the [u] are also convex. The function x(·) ∈ AC(R+ \{tk }; R n ) ∩ PC(I; R n ) is called the solution of DI x (t) ∈ [F(t; x(t))] x(t) = ’(t);
a:e: t ∈ R+ ;
t ∈ I0 ;
(2.1)
if x(·) satisJes the inclusion (2.1) a.e. in R+ , and the solutions of (1.2) may be deJned analogically. Denote the set of all solutions of (2.1) on [0; ] by (’; ) and the attainable set by
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A(’; ) = {x(): x(·) ∈ (’; )}. For any T ∈ [0; ∞) write S(’; T ) = {x(·) ∈ PC([t0 ; T ]; Rn ) : x(t) = ’(t); t ∈ I0 ; x (·) ∈ L∞ ([0; T ]; Rn )}: Let I˜ ⊂ I; M ⊂ R n be open sets and let J be a real compact interval. A mapping F : I˜ × M × J → PKC (R n ) is said to be regularly quasiconcave on J , if (RQ1): for all x ∈ M and almost all t ∈ I˜; F(t; x; ) ⊇ F(t; x; ), where ; ∈ J; 6; (RQ2): k is a nondecreasing sequence in J converging to , denote by k ↑ , then for all x ∈ M and almost all t ∈ I˜, F(t; x; k ) = F(t; x; ) k
and we write F ∈ (RQ). The mapping !(t; u; v) : R+ × R+ × R+ → R+ is called comparison function (or strong function) if ! is measurable in t for all (u; v) ∈ R+ × R+ and continuous nondecreasing in (u; v) for almost all t ∈ I . And let !∗ : R+ × R+ × R+ → R+ be deJned as !(t; u; v) if (t; u; v) ∈ R+ × R+ × R+ ; ∗ ! (t; u; v) = +∞ if u or v = +∞; t ∈ R+ :
3. Basic existence theorems In this section we consider the following impulsive functional di#erential inclusion: x (t) ∈ H (t; x(t); x(t − (t))) Ex(tk ) = lk (x(tk )); x(t) = ’(t);
a:e: t ∈ R+ ;
k = 1; 2; : : : ;
t ∈ I0 = [t0 ; 0]
(3.1)
and fuzzy impulsive di#erential equation: x (t) ∈ [G(t; x (t); x (t − (t)))] Ex (tk ) ∈ [lk (x (tk ))] ; x (t) ∈ [’(t)] = ’ ;
a:e: t ∈ R+ ;
k = 1; 2; : : : ; t ∈ I0 ;
(3.2)
where t 0 = min t ∈R+ (t − (t)). Lemma 3.1 (Tolstonogov [13]). Let J = [t1 ; t2 ] ⊂ R+ and a sequence xn ∈ AC(J; R n ) which satis7es the following conditions (a) the sequence xn (·) converges to some x(·) in C(J; R n ), (b) xn (t) 60(t) a.e. t ∈ J , where 0(·) ∈ L1+ (R1 ).
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Then (1) x ∈ AC(J; R n ).
∞ (2) x (t) ∈ i=1 co ∞ k=i xk (t), a.e. t ∈ J . Let us list some assumptions for problem (3.1) (L): lk ∈ C(R n ; R n ); k = 1; 2; : : : ; and there is a nondecreasing function l˜k ∈ C(R1 ; R1 ) such that lk (u) 6l˜k ( u ) for u ∈ R n ; k = 1; 2; : : : ; and for any bounded interval there are only Jnitely many of the impulse e#ect herein. (A): : R+ → R+ is bounded and continuous, (H1): H : R+ × R n × R n → PKC (R n ), (H2): H (t; · ; ·) is use in (x; y) ∈ R n × R n , for a.e. t ∈ R+ , (H3): H (· ; x; y) is measurable in t for all (x; y) ∈ R n × R n , (H4): there exists a comparison function ! : R+ × R1 × R1 → R1 such that (O1): |H (t; x; y)|6!∗ (t; x ; y ) a.e. t ∈ I , (O2): For some 3 ∈ C(I0 ; R1 ) the scalar impulsive functional di#erential equation u (t) = !(t; u(t)); u(t − (t)) Eu(tk ) = l˜k (u(tk )); u(t) = 3(t);
a:e: t ∈ R+ ;
k = 1; 2; : : : ;
t ∈ I0
(3.3)
has a solution u(t) ∈ I0 ∪ R+ . (V): ’ : I0 → R n is continuous. Theorem 3.2. Suppose that the above assumptions are satis7ed. Then for any ’ ∈ C(I0 ; R n ) with ’(t) 63(t); t ∈ I0 , the impulsive functional di8erential inclusion (3.1) with an initial function ’ has a solution x(t) in I0 ∪ R+ such that x(t) 6u(t); t ∈ R+ , where u is the solution of (3.3) and 3 is the initial function of (3.3). Proof. Let {tk }∞ k=1 be all impulse e#ects of the system (3.1) in [0; ∞) and let t0 = 0; l0 (·) ≡ 0. Set [0; ∞) =
∞
Tk ;
i=0
where Tk = [tk ; tk+1 ]; k = 0; 1; : : : ; for every interval Tk . We shall prove that there is a solution xk (t) on Tk with xk (t) 6u(t), for t ∈ Tk . Assume that the solutions xi (t) were deJned on Ti with xi (t) 6u(t), for t ∈ Ti ; i = 0; 1; : : : ; k − 1. For n ∈ N , we add equably n points in Tk such that t ; j = 0; k t j;k = tk + nj (tk+1 − tk ); j = 1; 2; : : : ; n; j = n + 1: tk+1 ;
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These points will partition interval Tk into n+1 subintervals. Let T j; k = [t j; k ; t j+1; k ]; j = 0; 1; : : : ; n.
n j; the k Thus we have Tk = 0 T . We shall construct a sequence of functions {ym } on Tk , such that ym (t) 6 u(t);
t ∈ Tk :
(3.4)
For the interval T 0; k , let ym (t 0;k ) = xk −1 (tk ) + lk (xk −1 (tk )); By (H2), (H3), we know that H (t; yn (t 0; k ); yn (t 0; k − (t 0; k ))) is measurable on T 0; k , hence by the measurable selection theorem in [4], there exists a measurable function gm(0; k) : T 0; k → R n , such that gm(0;k) ∈ H (t; ym (t 0;k ); yn (t 0;k − (t 0;k )))
for a:e: t ∈ T 0;k :
(3.5)
Since u(·) is nondecreasing on Tk , from an inequality ym (t 0;k ) 6u(tk ) + l˜k (u(tk )) and (3.3) we have gm(0;k) (t) 6!(t; u(t); u(t − (t)))
a:e: t ∈ T 0;k :
Thus gm(0; k) ∈ L1+ (T 0; k ). Let 0;k
ym (t) = ym (t ) +
t t 0;k
gm(0;k) (s) ds;
t ∈ T 0;k :
(3.6)
Then form (3.5), (3.6), we get ym (t) 6u(tk ) + l˜k (u(tk )) +
t tk
!(s; u(s); u(s − (s))) ds = u(t);
t ∈ T 0;k :
Thus in T 0; k we have constructed ym (t) which satisJes (3.4).
i −1 j i −1 j Suppose that ym has been deJned already on j=0 Tk ; 16i¡n and ym (t) 6u(t), for t ∈ j=0 Tk . i; k 0; k i; k Thus, ym (t ) is well deJned. Let us replace ym (t ) by ym (t ) and repeat all above proof observing that gm(i;k) ∈ H (t; ym (t i;k ); yn (t i;k − (t i;k ))) i;k
ym (t) = ym (t ) +
t t i;k
gm(i;k) (s) ds;
for a:e: t ∈ T i;k ; t ∈ T i;k ;
then we can construct ym (t) in T i; k which satisJes (3.4) as above.
(3.7) (3.8)
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By taking gm (t) = gm(j; k) (t); t ∈ T j; k , we have ym (t) = xk −1 (tk ) + ym (t) 6 u(t);
t tk
gm (s) ds + lk (xk −1 (tk ));
t ∈ Tk :
(3.9)
From (3.7), (3.8), it is easy to see that ym (·) is absolutely continuous on each Tk , hence ym (·) exists on Tk and ym (t) = gm (t), a.e. t ∈ Tk . On the other hand, it is easy to prove that {ym (·)} is equicontinuous on Tk , and ym (t) 6r = supt ∈Tk u(t). Thus, the Arzela–Ascoli theorem implies that {ym (·)} ∈ PK (C(Tk ; R n )). It follows that there exists a subsequence {yms (·)} of the sequence {ym (·)}, which converges to some xk (·) in C(Tk ; R n ). Obviously, xk (t) 6 u(t);
t ∈ Tk :
By Lemma 3.1, xk (·) is absolutely continuous and
x0 (t) ∈
∞
co
∞
gns (t):
(3.10)
s=i
i=1
DeJne the sequence of functions zm (·) by zm (t) = ym (t j; k ) for t ∈ T j; k . From (3.10), we obtain that
xk (t) ∈
∞
co
i=1
∞
H (t; zns (t); zns (t − ((t))):
(3.11)
s=i
Since zns (t) → xk (t)
as s → ∞
for t ∈ Tk ;
hence from (3.11), and (H2) we have xk (t) ∈ H (t; xk (t); xk (t − (t)))
a:e: t ∈ Tk :
By (L), lk is continuous and (3.9) implies that Eym (tk ) = lk (ym (tk )): Thus Exk (tk ) = lk (xk (tk )): So xk (·) is the solution of (3.1) on Tk with xk (t) 6u(t). Similarly, we can prove that (3.1) has the solution xk (·) on every Tk ; k = 0; 1; 2; : : : ; such that Exk (tk ) = lk (xk (tk ));
t ∈ Tk
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and xi (t) 6 u(t);
t ∈ Tk :
Finally, let x(t) = xk (t);
t ∈ Tk ; k = 0; 1; 2; : : : ;
x(t) = ’(t);
t ∈ I0 :
Then x(·) is the solution of (3.1) on [t 0 ; +∞) and x(t) 6 u(t);
t ∈ R+ :
The proof is complete. Theorem 3.3. Suppose that 3(·) ∈ C(I0 ; R1 ); G : R+ × R n × R n → E n satis7es conditions (H2); (H3), ’ ∈ C(I0 ; E n ); lk ∈ C(R n ; E n ) such that |’(t)| 6 3(t);
t ∈ I0 ;
ˆ = |lk (x)| 6 l˜k ( x ); D(lk (x); 8)
for x ∈ Rn ;
where l˜k ∈ C(R1 ; R1 ) is nondecreasing. If G satis7es (H4), taking ˆ |G(t; x; y)| = D(G(t; x; y); 8); then (3.2) has a solution on [t0 ; ∞). Proof. It is a direct consequence of Theorem 3.2 above and Lemma 4.1 of [5]. Remark 3.4. (1) The conditions in Theorem 3.2 is weaker. To see this, let us consider only a simple case of FDE (1.1) without delay and impulsive. Put !(t; u) = M = const, in this time a comparison equation u = !(t; u); u(0) = u0 distinctly has a solution, hence (O2) holds. If, in addition, assumption (O1) is satisJed, then (H4) is weaker than boundedness assumptions which was used in [5]. On the other hand, many functions are unbounded on [0; ∞), or it is not easy to Jnd the bounds. So our results are useful and powerful. (2) In next section we shall point out that the boundedness assumption is also too strong for the study of the properties of solution set. We shall establish similar results to that in [5], under assumptions (H), (L), (A) and (V).
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4. Some properties of the solutions and attainable set In this section we shall study some properties of the solution set of (3.2) and (3.1). Let T ¿0; I = [0; T ]. Theorem 4.1. Suppose that the conditions (H); (L); (A) and (V) are satis7ed on J . Then the solution set and the attainable set of (3.1) is nonempty, compact, but not convex in general. Proof. The proof of compactness is similar to that in [13] and the other results are easy to prove. Theorem 4.2. Assume that the conditions in Theorem 3.3 (resp. Theorem 3.2) hold. Then the solution set and the attainable set are convex, if G(t; x; y) the right side of (3.2) (resp. H (t; · ; ·) the right side of (3.1)) for all t ∈ I and all lk (z); k = 1; 2; : : : are concave in (x; y) ∈ R n × R n and z ∈ R n , respectively. Now consider the family of IFDE: x (t) ∈ H (t; x (t); x (t − (t)); ) Ex (ts ) ∈ Ls (x (ts ); ); x (t) ∈ 9(t; );
a:e: t ∈ I;
s = 1; 2; : : : ;
∈ [0; 1]; t ∈ I0 :
(4.1)
The following theorem generalizes Theorem 3.1 in [5] (see also [7]). Theorem 4.3. Suppose that (H1 ): H : I × R n × R n × [0; 1] → PKC (R n ) satis7es (H2); (H3); (H4) on I × R n × R n , (H2 ): H ∈ (RQ), (L1 ): Lk : R n × [0; 1] → PKC (R n ) is Hausdor8 continuous on R n and there exists nondecreasing L˜ k ∈ (R1 ; R1 ) such that |Lk (u)|6L˜ k ( u ), for u ∈ R n ; k = 1; 2; : : : ; m, (L2 ): Lk ∈ (RQ); k = 1; 2; : : : ; m, (A): ∈ C(I; R+ ) and bounded on I , (V1 ): 9 : I0 × [0; 1] → PKC (R n ) is Hausdor8 continuous, (V2 ): 9 ∈ (RQ). Then the mapping → (9 ; T ) is a regularly quasiconcave mapping from [0; 1] to S(’; T ) and the mapping → A (9 ; T ) is a regularly quasiconcave mapping from [0; 1] to PKC (R n ). Proof. Assume that 0¡t1 ¡t2 ¡; : : : ; ¡tm ¡T are the impulse e#ects. DeJne an operators N = (L1 ; L2 ; : : : ; Lm ; H ; ) : [I × C(I; Rn ) × [0; 1] → (PKC (Rn ))m+1 × [0; 1] and M : I × C(I; Rn ) → (Rn )m+1
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by N (t; x; ) = (L1 (x(t1 )); L2 (x(t2 )); : : : ; Lm (x(tm )); H (t; x(t); x(t − (t)); ); M (t; x) = (Ex(t1 ); Ex(t2 ); : : : ; Ex(tm ); x (t)): Since H; 9; Lk ∈ (RQ); k = 1; 2; : : : ; m, it is easy to see that N satisJes (RQ1). Let B ⊂ R n be a closed unit ball R n and U be an arbitrary neighborhood of the origin in I × C(I; R n ) × [0; 1] × (R n )m+1 . Take suNciently small ;¿0, such that ((t; y; ); N (t; y; ) + ;Bm+1 ) ⊂ Gr N + U: Let x(·) ∈ k k (’k ; T ), then there exists a subset J ⊂ I with <(J ) = 0, and M (t; x) ∈ N (t; x; k ), for all t ∈ I \J and all k ∈ [0; 1]. For any Jxed t ∈ I \J , by upper semicontinuity of lk and H , there is a neighborhood V of the point (x(t1 ); (x(t2 ); : : : ; x(tm ); H (t; x(t); x(t − (t)); ) such that (L1 (a1 ); L2 (a2 ); : : : ; Lm (am ); H (t; y; y ; )) ⊂ (L1 (x(t1 ); L2 (x(t2 ); : : : ; Lm (x(am ); H (t; x(t); x(t − (t)); ) + ;Bm+1 holds for all (a1 ; a2 ; : : : ; am ; y; y ; ) ∈ V . Thus choosing suNciently large k, we have N (t; x; k ) ⊂ N (t; x; ) + ;Bm+1 : It follows that (t; x; k ; M (t; x)) ∈ ((t; x; k ); N (t; x; k )) ⊂ Gr N + U: The convergence theorem implies that M (t; x) ∈ N (t; x(t); );
t ∈ I \J:
By the continuity and compactness of 9 in every t ∈ I0 ; x(t) ∈ 9(t; ); t ∈ I0 . Hence x(·) ∈ (’ ; T ). By closed epigraph theorem (’ ; T ) is usc in , we have k k (’k ; T ) = (’ ; T ). The proof is complete. Theorem 4.4. Assume that the conditions in Theorem 3.3 are satis7ed and for t ∈ I; ∈ [0; 1], write [G(t; x(t); x(t − (t)))] = F(t; x(t); x(t − (t)); ): Then the solution set (’ ; T ) of (3.2) is the level set of a fuzzy set 0 (’0 ; T ), where 0 (’0 ; T ) is the solution set of (3.2) when = 0 de7ned on S(’; T ). Proof. Suppose that for any Jxed T ∈ [0; ∞); tm = max{ti : ti 6T ¡ti+1 ; i + 1; 2; : : :} and
m [0; T ] = k=0 Tk , where Tk = [tk ; tk+1 ); k = 1; 2; : : : ; m − 1; T0 = [0; t1 ); Tm = [tm ; T ]. First we prove that (’ ; T ) is compact in S(’; T ).
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Let {xn } be a sequence of solutions on (’ ; T ), then xn (t) ∈ F(t; xn (t); xn (t − (t)); )
a:e: t ∈ T:
By |F(t; xn (t); xn (t − (t)); )| 6 |G(t; xn (t); xn (t − (t)))| 6 !(t; xn (t) ; xn (t − (t)) ) 6 !(t); u(t); u(t − (t))) = 0(t);
t ∈ Tk ; k = 0; 1; 2; : : : ; m;
(4.2)
we have xn (t) 6 0(t)
for t ∈ Tk :
Hence, by Lemma 3.1, there is a subsequence {xn; 0 (·)} of {xn (·)} uniformly converging to some x0 (·) in T0 , such that
x0 (t) ∈
∞
co
∞
xn;0 ⊂ F(t; x0 (t); x0 (t − (t); );
t ∈ T0 :
n=i
i=1
For {xn; 0 (·)} the inequality (4.2) holds in T1 , again by Theorem 3.2, there is a subsequence {xn;1 (·)} of {xn; 0 (·)} uniformly converging to some x1 (t) in T1 such that x1 (t) ∈
∞
co
∞
xn;1 ⊂ F(t; x1 (t); x1 (t − (t); );
t ∈ T1 :
n=i
i=1
Similarly, we can Jnd a subsequence {xn; m (·)} of {xn; m−1 (·) ⊆ {xn (·)} uniformly converging to some xk (t) in Tk such that xk (t) ∈
∞ i=1
co
∞
xn;m ⊂ F(t; xk (t); xk (t − (t); );
t ∈ Tk ; k = 0; 1; 2; : : : ; m:
n=i
Let xT (t) = xk (t), for t ∈ Tk , then we have xT (t) ∈ F(t; xT (t); xT (t − (t); ): For the sequence {xn; m (·)} we have Exn; m (tk ) ∈ [lk (xn; m (tk ))] . In view of the continuity of lk ; [lk (·)] is also continuous and thus by the left continuity of xn; m in tk the inclusion ExT (tk ) ∈ [lk (xT (tk ))] holds.
Next, let > = t ∈I0 [’(t)] . Since xn; m (t) ∈ [’(t)] ; t ∈ I0 , from growth conditions on the initial function ’(t) 63(t); t ∈ I0 , we have xn;m (t) 6 |[’(t)] | 6 3(t) 6 sup 3(t) = r; t ∈ I0
t ∈ I0 ;
which shows that {xn; m } is a bounded set in C(I0 ; >). Moreover, xn; m (t) 63(t) implies that {xn; m } is equicontinuous. By Arzela–Ascoli theorem, {xn; m } is a compact subset in C(I0 ; >), hence there is
M. Guo et al. / Fuzzy Sets and Systems 138 (2003) 601 – 615
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a subsequence {xk } of {xn; m } which converges to some point x’ ∈ C(I0 ; R n ). From the compactness of ’, we have x’ (t) ∈ [’(t)] ; t ∈ I0 . Finally, by taking x(t) = x’ (t), for t ∈ I0 and x(t) = xT (t), for t ∈ T , we have x(t) ∈ (’ ; T ). Therefore, (’ ; T ) is compact in S(’; T ). By Theorem 4.3, to prove that (’ ; T ) satisJes the others two conditions in the deJnition of fuzzy number, it is suNcient to show that F ∈ (RQ). But this is obvious, because all F; lk and ’ are fuzzy number. The proof is complete. Remark 4.5. The attainable set has analogous properties. 5. Some applications for fuzzy population models 5.1. Fuzzy logistic model (isolated population in nonstationary environment) Let us consider the following fuzzy population model: N (t) ∈ rN (t)[1 − (l[w] + c)N (t − 1)] EN (tk ) = lk ;
a:e: t ∈ I = [a; b];
k = 0; 1; 2; : : : ; m; ∈ [0; 1];
N (t) ∈ ct + d + [n(t)] ;
t ∈ I0 = [a − 1; a];
(5.1)
where a¿0; b = a + M + b1 ; M ∈ N; 0¡b1 ¡1, and r; c; d¿0; lk ∈ R1 ; k = 1; 2; : : : ; m. Without loss of generality, assume that a¡t1 ¡t2 ¡ · · · ¡tm ¡b and ti = a + j for all i = 1; 2; : : : ; m; j = 1; 2; : : : ; M . w ∈ E 1 is symmetric triangular fuzzy number with support [−1; 1], i.e., [w] = (1−)[−1; 1]; 0¡l¡c; ∈ [0; 1]. n ∈ C(I; E 1 ) deJned by n (t) = [n(t)] = [2pt + q; (4 − 2 )pt + (2 − )q];
∈ [0; 1]:
Obviously, if we take !(t; u; v) = r|u|(1 + (l + c)|v|), then Eq. (3.3) has a solution on I0 ∪ I . Therefore, (5.1) has a solution in C(I; E 1 ). We can give the solution set (’ ; b) of (5.1) in [a − 1; b] as ct + d + [n(t)] ; t ∈ I0 ; 0 (a + 1); t ∈ T0 ; 1 t ∈ T1 ; (’ ; b) = (a + 2); .. .. . . M (b); t ∈ TM ; where
0 (a + 1) =
u(t) = (ca + d + B (a)) erf0 (t) +
k ∈ S0
lk
: f0 ∈ F0 ; B ∈ n ; t ∈ T1
;
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M. Guo et al. / Fuzzy Sets and Systems 138 (2003) 601 – 615
t
F0 = f0 : f0 (t) =
a
[1 − (lh + c)(cs + d − c + 0 (s − 1))] ds; h ∈
(1 − )[−1; 1]; 0 (t) ∈ n (t) is integrable selection of n (t); t ∈ I0 and
i (a + i + 1) =
u(t) = mi (a + 1) erfi (t) +
Fi =
fi : fi (t) =
lk
; t ∈ Ti ; fi ∈ Fi ; mi ∈ E(n ) ;
k ∈ Si t a+i
[1 − (lh + c)mi (s − 1)] ds; h ∈ (1 − )[−1; 1]; t ∈ Ti :
For a set-value mapping E(n ); mi ∈ E(n ) means that a construct of mi depending on the choice of integrable selections of n . 5.2. Fuzzy Gompertz model (isolated population in stationary environment) The dynamics of simple limitation population with competition in stationary environment is well simulated by Gompertz equation [2] N (t) ∈ N (t)[r − ([W (t)] + G) ln N (t − 1)] EN (tk ) = lk ;
a:e: t ∈ I = [a; b];
k = 0; 1; 2; : : : ; m; ∈ [0; 1];
N (t) = qt + p;
t ∈ I0 = [a − 1; a];
(5.2)
where a¿0; b = a+M +b1 ; M ∈ N; 0¡b1 ¡1, and G; r; p; q¿0; lk ∈ R1 ; k = 1; 2; : : : ; m; W ∈ C(I; E 1 ) is symmetric triangular fuzzy function with support [−1; 1], i.e., [W (t)] = w(t)(1 − )[−1; 1]; w ∈ C (I; R1 ). By taking !(t; u; v) = |u|(r + (|w| + G)|ln v|), then Eq. (3.3) has solution on I0 ∪ I . Therefore, (5.2) has a solution in C(I; E 1 ). Thus, the solution set (’ ; b) of (5.2) in [a − 1; b] is qt + p; t ∈ I0 ; 0 (a + 1); t ∈ T0 ; 1 (’ ; b) = (a + 2); t ∈ T1 ; .. .. . . M (b); t ∈ TM ; where
i (a + i + 1) =
u(t) = mi (a + i) efi (t) +
k ∈ Si
lk
; t ∈ Ti ; fi ∈ Fi
M. Guo et al. / Fuzzy Sets and Systems 138 (2003) 601 – 615
and
Fi =
fi : fi (t) =
t a+i
615
[r − (w(s)h + G)mi−1 (s − 1)] ds; h ∈ (1 − )[−1; 1]; t ∈ Ti :
It is easy to know that (’ ; b) is -level set of 0 (’0 ; b). Acknowledgements The authors would like to express our thanks to Professor P. Diamond for his valuable remarks and suggestions concerning this work. References [1] O. Akinyele, Cone-valued Lyiapunov functions and stability of impulsive control systems, Nonlinear. Anal. 39 (2000) 247–259. [2] J. Angelova, A. Dishliev, Optimization problems for one-impulsive models from population dynamics, Nonlinear Anal. 39 (2000) 483–498. [3] J.P. Aubin, Impulse and Hybrid Control Systems: A Viability Approach, First Preliminary Draft of Lecture Notes of a Mini-course, University of California, Berkeley, 1999. [4] J.P. Aubin, A. Cellina, Di#erential Inclusions, Springer, New York, 1984. [5] P. Diamond, Time-dependent di#erential inclusions, cocycle attractors and fuzzy di#erential equations, IEEE Trans. Fuzzy systems 7 (1999) 734–740. [6] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World ScientiJc, Singapore, 1994. [7] P. Diamond, P. Watson, Regularity of solution sets for di#erential inclusions quasiconcave in a parameter, Appl. Math. Lett. 13 (2000) 31–35. [8] E. Hullermeier, An approach to modeling and simulation of uncertain dynamical systems, Int. J. Uncertainty, Fuzziness, Knowledge-Bases Syst. 5 (1997) 117–137. [9] S.G. Pandit, S.G. Deo, Di#erential Systems Involving Impulses, in: Lecture Notes in Mathematics, Vol. 954, Springer, New York, 1982. [10] M.L. Puri, D.A. Ralescu, Di#erentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552–558. [11] D.E. Stewart, Rigid-body dynamics with friction & impact, SIAM Rev. 42 (2000) 3–39. [12] D.E. Stewart, Reformulations of measure di#erential inclusions and their closed graph properly, J. Di#erential Equations 174 (2001) 109–129. [13] A.A. Tolstonogov, On comprising theorems for di#erential inclusion in locally convex spaces, I and II, Di#erential Equations 17 (1981) 651– 659 and 1016 –1024 (in Russian).
Impulsive functional di#erential inclusions and fuzzy population models Mengshu Guo∗ , Xiaoping Xue, Ronglu Li Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China Received 23 February 2001; received in revised form 5 August 2002; accepted 7 November 2002
Abstract In this paper we shall establish some existence results for the impulsive functional di#erential inclusion and the fuzzy impulsive functional di#erential equation with some conditions, and study the properties of the solution set and the attainable set. Finally, the results will be used to fuzzy population models. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy impulsive functional di#erential inclusion (equation); The solution set; The attainable set; Fuzzy population model
1. Introduction In [5,7,8] the fuzzy di#erential equation (FDE) model was expressed by a family of ordinary di#erential inclusions (DI) in R n as following: x (t) ∈ [G(t; x (t))]
a:e: t ∈ [0; T ];
x (0) ∈ X = [X0 ] ;
∈ [0; 1];
(1.1)
where [G(· ; ·)] : R × R n → PKC (R n ). Diamond [5] proved that under some conditions the solution set (X ; T ) and the attainable set A (X ; T ) are -level sets of 0 (X 0 ; T ) and A 0 (X 0 ; T ), respectively, where ∈ [0; 1]. In this paper under some weaker conditions we shall study the analogous problems for the following fuzzy impulsive functional di#erential equation (FIFDE): x (t) ∈ [G(t; x (t); x (t − (t)))]
a:e: t ∈ R+ ;
∗
Corresponding author. Research supported by HIT.M.D. 2000.24. E-mail address: [email protected] (M. Guo).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 5 2 2 - 5
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Ex (tk ) ∈ [lk (x (tk ))] ; x (t) ∈ [’(t)] ;
k = 0; 1; 2; : : : ;
t ∈ I0 = [t0 ; 0];
(1.2)
where ∈ [0; 1] and t 0 = min t ∈R+ (t − (t)). We shall establish some existence theorems of the solutions to (1.2) and discuss the properties of the solution set. The FIFDE (1.2) is the model for a large class of real objects and processes. In biology, control theory and electronics, etc., many problems can be described by Eq. (1.2) accurately. Example 1 (Some population models): As is well known that the classical population model (P.F. Verhulst model) can be express generally by the following di#erential equation: N (t) N (t) = rN (t) 1 − ; r ¿ 0; P0 where N (t)¿0 is the population biomass (size) at the moment t¿0 and 1−N (t)=P0 is the biological feedback factor. But the social friction factor r=P0 is determined usually by many uncertain factors, e.g., the natural and social resource state, the age structure, the medical treatment and sanitation state. Hence, the more accurate model should be fuzzy. If we consider the delay e#ects of gestation and some other factors, the certain model is (E.M. Wright model) N (t − ) N (t) = rN (t) 1 − ; r; ¿ 0 P0 or more generally, N (t) = N (t)f0 (N (t − )) = f(N (t); N (t − )); N (t) = ’(t);
t ∈ R+ ;
t ∈ I0 = [t0 ; 0]:
And the corresponding fuzzy model is N (t) ∈ [f(t; N (t); N (t − ))] ; N (t) ∈ [’(t)] ;
t ∈ R+ ;
t ∈ I0 ; ∈ [0; 1];
where f is a fuzzy mapping, which is determined by the species growth law and some other uncertain factors. On the other hand, on account of some (usually uncertain) reasons, e.g., natural disaster, the population policy, the population structure, the wars, etc., there may be sharp changes in the amounts of the population at some moment t∗ , i.e., EN (t∗ ) = N (t∗+ ) − N (t∗− ) = C ¿ 0; where N (t∗+ ) = lim t →t∗ + N (t) and N (t∗− ) = limt →t∗ − N (t). Angelova and Dishliev [2] considered some optimization problems from impulsive population dynamics, but the model is certain. There is
M. Guo et al. / Fuzzy Sets and Systems 138 (2003) 601 – 615
603
a better model, which is expressed by FIFDE N (t) ∈ [f(t; N (t); N (t − (t)))] ; EN (tk ) ∈ [lk (N (tk ))] ; N (t) ∈ [’(t)] ;
t ∈ R+ ;
k = 0; 1; 2; : : : ; m; ∈ [0; 1];
t ∈ I0 = [t0 ; 0]; t0 = min+ (t − (t)): t ∈R
Example 2 (The model of martial strength between two states): Let [A]; [B] be two states in some locally region, x; y the martial strength of [A] and [B]. The study of the geostrategic shows that the relation of the changes of the martial strength between [A] and [B] may be expressed by x = ky − x + gBA (t); y = lx − y + gAB (t); where gBA (resp. gAB ) is the threat of [B] to [A] (resp. [A] to [B]). But gBA and gAB are generally uncertain and fuzzy. Moreover, there will be sharp changes in the martial strength in a short time when some weapons are equipped. Thus, the impulsive factors and delay should be described in the model. Therefore, the condign model is expressed by FIFDE as x f1 (t; x(t); x(t − ); y(t); y(t − ) (t) ∈ ; y f2 (t; x(t); x(t − ); y(t); y(t − )
Ex Ey
(tk ) ∈
l1k (x(tk ); y(tk )) l2k (x(tk ); y(tk ))
:
Example 3 (Some control systems problems): In [1] some control problems of certain impulsive systems were considered. The corresponding fuzzy problems can be expressed as following: x (t) ∈ [G(t; x (t); x (t − (t)); Ex (tk ) ∈ [lk (x (tk ); u (t))] ; x (t) ∈ [’(t; u (t))] ; y (t) = g(t; u (t));
u (t))] ; t ∈ I; k = 0; 1; 2; : : : ; m;
t ∈ I0 = [t0 ; 0]; ∈ [0; 1];
where g : I × E n → E n ; u ∈ Q(t). Moreover, the di#erential equations and inclusions involving impulse were used to study many complicated dynamical systems including hybrid control systems (see [3,9,12,11]). Many papers discussed the functional di#erential equations (inclusions) and the impulsive differential equations. However, as far as the author knows, few papers considered the corresponding fuzzy problems. It is pointed out in [5,7,8] that the conventional fuzzy di#erential models have some ill properties and should be replaced by the families of the DIs. This paper generalizes the results in [5,7].
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2. Notions and denitions In this paper, let R+ = [0; +∞); R+ = [0; +∞]; I ⊆ R+ be a compact interval, (X; · ) be a Banach space, and (R n ; · ) be the n-dimension Euclid space with usual norm. Denote by PK (X ) the set of all compact subsets of X and by PKC (X ) the set of all compact convex subsets of X . DeJne the Hausdor# metric dH (· ; ·) in PK (X ) by dH (A; B) = max sup dist(a; B); sup dist(b; A) ; a∈ A
b∈ B
and for A ∈ PK (X ) denote |A| = sup{ a : a ∈ A}. Let tk ∈ I; k = 1; 2; : : : : The following spaces will be used: C(I; Rn ) = {u : u is continuous from I to Rn }; AC(I; Rn ) = {u : u is absolutely continuous from I to Rn }; and n
PC(I; R ) = u : u ∈ C(I \{tk }; Rn ); lim u(t) = u(tk ); t →tk −
and lim u(tx) exists; k = 1; 2; : : : ; m : t →tk +
It is easy to show that PC(I; R n ) is a Banach space. For x ∈ PC(I; R n ) and the impulse e#ect tk ∈ I , denote Ex(tk ) = x(tk+ ) − x(tk− ), where x(tk+ ) = limt →tk + x(t) and x(tk− ) = limt →tk − x(t). Some deJnitions and results of fuzzy number spaces can be found in [5,6,10]. Theorem 2.1 (Stacking Theorem; Puri and Ralescu [10]). Let {Y ⊂ R n : 0661} be a family of compact subsets satisfying the following: (1) Y ∈ PKC (Rn ), for all 0661; (2) Y ⊆ Y , for 06 661; (3) Y = ∞ i=1 Yi , for any nondecreasing sequence i → in [0; 1]. Then there is a fuzzy set u ∈ Dn such that [u] = Y . In particular, if the Y are also convex, then u ∈ E n . Conversely, the level sets [u] of any u ∈ Dn satisfy these conditions, while if u ∈ E n the [u] are also convex. The function x(·) ∈ AC(R+ \{tk }; R n ) ∩ PC(I; R n ) is called the solution of DI x (t) ∈ [F(t; x(t))] x(t) = ’(t);
a:e: t ∈ R+ ;
t ∈ I0 ;
(2.1)
if x(·) satisJes the inclusion (2.1) a.e. in R+ , and the solutions of (1.2) may be deJned analogically. Denote the set of all solutions of (2.1) on [0; ] by (’; ) and the attainable set by
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605
A(’; ) = {x(): x(·) ∈ (’; )}. For any T ∈ [0; ∞) write S(’; T ) = {x(·) ∈ PC([t0 ; T ]; Rn ) : x(t) = ’(t); t ∈ I0 ; x (·) ∈ L∞ ([0; T ]; Rn )}: Let I˜ ⊂ I; M ⊂ R n be open sets and let J be a real compact interval. A mapping F : I˜ × M × J → PKC (R n ) is said to be regularly quasiconcave on J , if (RQ1): for all x ∈ M and almost all t ∈ I˜; F(t; x; ) ⊇ F(t; x; ), where ; ∈ J; 6; (RQ2): k is a nondecreasing sequence in J converging to , denote by k ↑ , then for all x ∈ M and almost all t ∈ I˜, F(t; x; k ) = F(t; x; ) k
and we write F ∈ (RQ). The mapping !(t; u; v) : R+ × R+ × R+ → R+ is called comparison function (or strong function) if ! is measurable in t for all (u; v) ∈ R+ × R+ and continuous nondecreasing in (u; v) for almost all t ∈ I . And let !∗ : R+ × R+ × R+ → R+ be deJned as !(t; u; v) if (t; u; v) ∈ R+ × R+ × R+ ; ∗ ! (t; u; v) = +∞ if u or v = +∞; t ∈ R+ :
3. Basic existence theorems In this section we consider the following impulsive functional di#erential inclusion: x (t) ∈ H (t; x(t); x(t − (t))) Ex(tk ) = lk (x(tk )); x(t) = ’(t);
a:e: t ∈ R+ ;
k = 1; 2; : : : ;
t ∈ I0 = [t0 ; 0]
(3.1)
and fuzzy impulsive di#erential equation: x (t) ∈ [G(t; x (t); x (t − (t)))] Ex (tk ) ∈ [lk (x (tk ))] ; x (t) ∈ [’(t)] = ’ ;
a:e: t ∈ R+ ;
k = 1; 2; : : : ; t ∈ I0 ;
(3.2)
where t 0 = min t ∈R+ (t − (t)). Lemma 3.1 (Tolstonogov [13]). Let J = [t1 ; t2 ] ⊂ R+ and a sequence xn ∈ AC(J; R n ) which satis7es the following conditions (a) the sequence xn (·) converges to some x(·) in C(J; R n ), (b) xn (t) 60(t) a.e. t ∈ J , where 0(·) ∈ L1+ (R1 ).
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Then (1) x ∈ AC(J; R n ).
∞ (2) x (t) ∈ i=1 co ∞ k=i xk (t), a.e. t ∈ J . Let us list some assumptions for problem (3.1) (L): lk ∈ C(R n ; R n ); k = 1; 2; : : : ; and there is a nondecreasing function l˜k ∈ C(R1 ; R1 ) such that lk (u) 6l˜k ( u ) for u ∈ R n ; k = 1; 2; : : : ; and for any bounded interval there are only Jnitely many of the impulse e#ect herein. (A): : R+ → R+ is bounded and continuous, (H1): H : R+ × R n × R n → PKC (R n ), (H2): H (t; · ; ·) is use in (x; y) ∈ R n × R n , for a.e. t ∈ R+ , (H3): H (· ; x; y) is measurable in t for all (x; y) ∈ R n × R n , (H4): there exists a comparison function ! : R+ × R1 × R1 → R1 such that (O1): |H (t; x; y)|6!∗ (t; x ; y ) a.e. t ∈ I , (O2): For some 3 ∈ C(I0 ; R1 ) the scalar impulsive functional di#erential equation u (t) = !(t; u(t)); u(t − (t)) Eu(tk ) = l˜k (u(tk )); u(t) = 3(t);
a:e: t ∈ R+ ;
k = 1; 2; : : : ;
t ∈ I0
(3.3)
has a solution u(t) ∈ I0 ∪ R+ . (V): ’ : I0 → R n is continuous. Theorem 3.2. Suppose that the above assumptions are satis7ed. Then for any ’ ∈ C(I0 ; R n ) with ’(t) 63(t); t ∈ I0 , the impulsive functional di8erential inclusion (3.1) with an initial function ’ has a solution x(t) in I0 ∪ R+ such that x(t) 6u(t); t ∈ R+ , where u is the solution of (3.3) and 3 is the initial function of (3.3). Proof. Let {tk }∞ k=1 be all impulse e#ects of the system (3.1) in [0; ∞) and let t0 = 0; l0 (·) ≡ 0. Set [0; ∞) =
∞
Tk ;
i=0
where Tk = [tk ; tk+1 ]; k = 0; 1; : : : ; for every interval Tk . We shall prove that there is a solution xk (t) on Tk with xk (t) 6u(t), for t ∈ Tk . Assume that the solutions xi (t) were deJned on Ti with xi (t) 6u(t), for t ∈ Ti ; i = 0; 1; : : : ; k − 1. For n ∈ N , we add equably n points in Tk such that t ; j = 0; k t j;k = tk + nj (tk+1 − tk ); j = 1; 2; : : : ; n; j = n + 1: tk+1 ;
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607
These points will partition interval Tk into n+1 subintervals. Let T j; k = [t j; k ; t j+1; k ]; j = 0; 1; : : : ; n.
n j; the k Thus we have Tk = 0 T . We shall construct a sequence of functions {ym } on Tk , such that ym (t) 6 u(t);
t ∈ Tk :
(3.4)
For the interval T 0; k , let ym (t 0;k ) = xk −1 (tk ) + lk (xk −1 (tk )); By (H2), (H3), we know that H (t; yn (t 0; k ); yn (t 0; k − (t 0; k ))) is measurable on T 0; k , hence by the measurable selection theorem in [4], there exists a measurable function gm(0; k) : T 0; k → R n , such that gm(0;k) ∈ H (t; ym (t 0;k ); yn (t 0;k − (t 0;k )))
for a:e: t ∈ T 0;k :
(3.5)
Since u(·) is nondecreasing on Tk , from an inequality ym (t 0;k ) 6u(tk ) + l˜k (u(tk )) and (3.3) we have gm(0;k) (t) 6!(t; u(t); u(t − (t)))
a:e: t ∈ T 0;k :
Thus gm(0; k) ∈ L1+ (T 0; k ). Let 0;k
ym (t) = ym (t ) +
t t 0;k
gm(0;k) (s) ds;
t ∈ T 0;k :
(3.6)
Then form (3.5), (3.6), we get ym (t) 6u(tk ) + l˜k (u(tk )) +
t tk
!(s; u(s); u(s − (s))) ds = u(t);
t ∈ T 0;k :
Thus in T 0; k we have constructed ym (t) which satisJes (3.4).
i −1 j i −1 j Suppose that ym has been deJned already on j=0 Tk ; 16i¡n and ym (t) 6u(t), for t ∈ j=0 Tk . i; k 0; k i; k Thus, ym (t ) is well deJned. Let us replace ym (t ) by ym (t ) and repeat all above proof observing that gm(i;k) ∈ H (t; ym (t i;k ); yn (t i;k − (t i;k ))) i;k
ym (t) = ym (t ) +
t t i;k
gm(i;k) (s) ds;
for a:e: t ∈ T i;k ; t ∈ T i;k ;
then we can construct ym (t) in T i; k which satisJes (3.4) as above.
(3.7) (3.8)
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By taking gm (t) = gm(j; k) (t); t ∈ T j; k , we have ym (t) = xk −1 (tk ) + ym (t) 6 u(t);
t tk
gm (s) ds + lk (xk −1 (tk ));
t ∈ Tk :
(3.9)
From (3.7), (3.8), it is easy to see that ym (·) is absolutely continuous on each Tk , hence ym (·) exists on Tk and ym (t) = gm (t), a.e. t ∈ Tk . On the other hand, it is easy to prove that {ym (·)} is equicontinuous on Tk , and ym (t) 6r = supt ∈Tk u(t). Thus, the Arzela–Ascoli theorem implies that {ym (·)} ∈ PK (C(Tk ; R n )). It follows that there exists a subsequence {yms (·)} of the sequence {ym (·)}, which converges to some xk (·) in C(Tk ; R n ). Obviously, xk (t) 6 u(t);
t ∈ Tk :
By Lemma 3.1, xk (·) is absolutely continuous and
x0 (t) ∈
∞
co
∞
gns (t):
(3.10)
s=i
i=1
DeJne the sequence of functions zm (·) by zm (t) = ym (t j; k ) for t ∈ T j; k . From (3.10), we obtain that
xk (t) ∈
∞
co
i=1
∞
H (t; zns (t); zns (t − ((t))):
(3.11)
s=i
Since zns (t) → xk (t)
as s → ∞
for t ∈ Tk ;
hence from (3.11), and (H2) we have xk (t) ∈ H (t; xk (t); xk (t − (t)))
a:e: t ∈ Tk :
By (L), lk is continuous and (3.9) implies that Eym (tk ) = lk (ym (tk )): Thus Exk (tk ) = lk (xk (tk )): So xk (·) is the solution of (3.1) on Tk with xk (t) 6u(t). Similarly, we can prove that (3.1) has the solution xk (·) on every Tk ; k = 0; 1; 2; : : : ; such that Exk (tk ) = lk (xk (tk ));
t ∈ Tk
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and xi (t) 6 u(t);
t ∈ Tk :
Finally, let x(t) = xk (t);
t ∈ Tk ; k = 0; 1; 2; : : : ;
x(t) = ’(t);
t ∈ I0 :
Then x(·) is the solution of (3.1) on [t 0 ; +∞) and x(t) 6 u(t);
t ∈ R+ :
The proof is complete. Theorem 3.3. Suppose that 3(·) ∈ C(I0 ; R1 ); G : R+ × R n × R n → E n satis7es conditions (H2); (H3), ’ ∈ C(I0 ; E n ); lk ∈ C(R n ; E n ) such that |’(t)| 6 3(t);
t ∈ I0 ;
ˆ = |lk (x)| 6 l˜k ( x ); D(lk (x); 8)
for x ∈ Rn ;
where l˜k ∈ C(R1 ; R1 ) is nondecreasing. If G satis7es (H4), taking ˆ |G(t; x; y)| = D(G(t; x; y); 8); then (3.2) has a solution on [t0 ; ∞). Proof. It is a direct consequence of Theorem 3.2 above and Lemma 4.1 of [5]. Remark 3.4. (1) The conditions in Theorem 3.2 is weaker. To see this, let us consider only a simple case of FDE (1.1) without delay and impulsive. Put !(t; u) = M = const, in this time a comparison equation u = !(t; u); u(0) = u0 distinctly has a solution, hence (O2) holds. If, in addition, assumption (O1) is satisJed, then (H4) is weaker than boundedness assumptions which was used in [5]. On the other hand, many functions are unbounded on [0; ∞), or it is not easy to Jnd the bounds. So our results are useful and powerful. (2) In next section we shall point out that the boundedness assumption is also too strong for the study of the properties of solution set. We shall establish similar results to that in [5], under assumptions (H), (L), (A) and (V).
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4. Some properties of the solutions and attainable set In this section we shall study some properties of the solution set of (3.2) and (3.1). Let T ¿0; I = [0; T ]. Theorem 4.1. Suppose that the conditions (H); (L); (A) and (V) are satis7ed on J . Then the solution set and the attainable set of (3.1) is nonempty, compact, but not convex in general. Proof. The proof of compactness is similar to that in [13] and the other results are easy to prove. Theorem 4.2. Assume that the conditions in Theorem 3.3 (resp. Theorem 3.2) hold. Then the solution set and the attainable set are convex, if G(t; x; y) the right side of (3.2) (resp. H (t; · ; ·) the right side of (3.1)) for all t ∈ I and all lk (z); k = 1; 2; : : : are concave in (x; y) ∈ R n × R n and z ∈ R n , respectively. Now consider the family of IFDE: x (t) ∈ H (t; x (t); x (t − (t)); ) Ex (ts ) ∈ Ls (x (ts ); ); x (t) ∈ 9(t; );
a:e: t ∈ I;
s = 1; 2; : : : ;
∈ [0; 1]; t ∈ I0 :
(4.1)
The following theorem generalizes Theorem 3.1 in [5] (see also [7]). Theorem 4.3. Suppose that (H1 ): H : I × R n × R n × [0; 1] → PKC (R n ) satis7es (H2); (H3); (H4) on I × R n × R n , (H2 ): H ∈ (RQ), (L1 ): Lk : R n × [0; 1] → PKC (R n ) is Hausdor8 continuous on R n and there exists nondecreasing L˜ k ∈ (R1 ; R1 ) such that |Lk (u)|6L˜ k ( u ), for u ∈ R n ; k = 1; 2; : : : ; m, (L2 ): Lk ∈ (RQ); k = 1; 2; : : : ; m, (A): ∈ C(I; R+ ) and bounded on I , (V1 ): 9 : I0 × [0; 1] → PKC (R n ) is Hausdor8 continuous, (V2 ): 9 ∈ (RQ). Then the mapping → (9 ; T ) is a regularly quasiconcave mapping from [0; 1] to S(’; T ) and the mapping → A (9 ; T ) is a regularly quasiconcave mapping from [0; 1] to PKC (R n ). Proof. Assume that 0¡t1 ¡t2 ¡; : : : ; ¡tm ¡T are the impulse e#ects. DeJne an operators N = (L1 ; L2 ; : : : ; Lm ; H ; ) : [I × C(I; Rn ) × [0; 1] → (PKC (Rn ))m+1 × [0; 1] and M : I × C(I; Rn ) → (Rn )m+1
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by N (t; x; ) = (L1 (x(t1 )); L2 (x(t2 )); : : : ; Lm (x(tm )); H (t; x(t); x(t − (t)); ); M (t; x) = (Ex(t1 ); Ex(t2 ); : : : ; Ex(tm ); x (t)): Since H; 9; Lk ∈ (RQ); k = 1; 2; : : : ; m, it is easy to see that N satisJes (RQ1). Let B ⊂ R n be a closed unit ball R n and U be an arbitrary neighborhood of the origin in I × C(I; R n ) × [0; 1] × (R n )m+1 . Take suNciently small ;¿0, such that ((t; y; ); N (t; y; ) + ;Bm+1 ) ⊂ Gr N + U: Let x(·) ∈ k k (’k ; T ), then there exists a subset J ⊂ I with <(J ) = 0, and M (t; x) ∈ N (t; x; k ), for all t ∈ I \J and all k ∈ [0; 1]. For any Jxed t ∈ I \J , by upper semicontinuity of lk and H , there is a neighborhood V of the point (x(t1 ); (x(t2 ); : : : ; x(tm ); H (t; x(t); x(t − (t)); ) such that (L1 (a1 ); L2 (a2 ); : : : ; Lm (am ); H (t; y; y ; )) ⊂ (L1 (x(t1 ); L2 (x(t2 ); : : : ; Lm (x(am ); H (t; x(t); x(t − (t)); ) + ;Bm+1 holds for all (a1 ; a2 ; : : : ; am ; y; y ; ) ∈ V . Thus choosing suNciently large k, we have N (t; x; k ) ⊂ N (t; x; ) + ;Bm+1 : It follows that (t; x; k ; M (t; x)) ∈ ((t; x; k ); N (t; x; k )) ⊂ Gr N + U: The convergence theorem implies that M (t; x) ∈ N (t; x(t); );
t ∈ I \J:
By the continuity and compactness of 9 in every t ∈ I0 ; x(t) ∈ 9(t; ); t ∈ I0 . Hence x(·) ∈ (’ ; T ). By closed epigraph theorem (’ ; T ) is usc in , we have k k (’k ; T ) = (’ ; T ). The proof is complete. Theorem 4.4. Assume that the conditions in Theorem 3.3 are satis7ed and for t ∈ I; ∈ [0; 1], write [G(t; x(t); x(t − (t)))] = F(t; x(t); x(t − (t)); ): Then the solution set (’ ; T ) of (3.2) is the level set of a fuzzy set 0 (’0 ; T ), where 0 (’0 ; T ) is the solution set of (3.2) when = 0 de7ned on S(’; T ). Proof. Suppose that for any Jxed T ∈ [0; ∞); tm = max{ti : ti 6T ¡ti+1 ; i + 1; 2; : : :} and
m [0; T ] = k=0 Tk , where Tk = [tk ; tk+1 ); k = 1; 2; : : : ; m − 1; T0 = [0; t1 ); Tm = [tm ; T ]. First we prove that (’ ; T ) is compact in S(’; T ).
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Let {xn } be a sequence of solutions on (’ ; T ), then xn (t) ∈ F(t; xn (t); xn (t − (t)); )
a:e: t ∈ T:
By |F(t; xn (t); xn (t − (t)); )| 6 |G(t; xn (t); xn (t − (t)))| 6 !(t; xn (t) ; xn (t − (t)) ) 6 !(t); u(t); u(t − (t))) = 0(t);
t ∈ Tk ; k = 0; 1; 2; : : : ; m;
(4.2)
we have xn (t) 6 0(t)
for t ∈ Tk :
Hence, by Lemma 3.1, there is a subsequence {xn; 0 (·)} of {xn (·)} uniformly converging to some x0 (·) in T0 , such that
x0 (t) ∈
∞
co
∞
xn;0 ⊂ F(t; x0 (t); x0 (t − (t); );
t ∈ T0 :
n=i
i=1
For {xn; 0 (·)} the inequality (4.2) holds in T1 , again by Theorem 3.2, there is a subsequence {xn;1 (·)} of {xn; 0 (·)} uniformly converging to some x1 (t) in T1 such that x1 (t) ∈
∞
co
∞
xn;1 ⊂ F(t; x1 (t); x1 (t − (t); );
t ∈ T1 :
n=i
i=1
Similarly, we can Jnd a subsequence {xn; m (·)} of {xn; m−1 (·) ⊆ {xn (·)} uniformly converging to some xk (t) in Tk such that xk (t) ∈
∞ i=1
co
∞
xn;m ⊂ F(t; xk (t); xk (t − (t); );
t ∈ Tk ; k = 0; 1; 2; : : : ; m:
n=i
Let xT (t) = xk (t), for t ∈ Tk , then we have xT (t) ∈ F(t; xT (t); xT (t − (t); ): For the sequence {xn; m (·)} we have Exn; m (tk ) ∈ [lk (xn; m (tk ))] . In view of the continuity of lk ; [lk (·)] is also continuous and thus by the left continuity of xn; m in tk the inclusion ExT (tk ) ∈ [lk (xT (tk ))] holds.
Next, let > = t ∈I0 [’(t)] . Since xn; m (t) ∈ [’(t)] ; t ∈ I0 , from growth conditions on the initial function ’(t) 63(t); t ∈ I0 , we have xn;m (t) 6 |[’(t)] | 6 3(t) 6 sup 3(t) = r; t ∈ I0
t ∈ I0 ;
which shows that {xn; m } is a bounded set in C(I0 ; >). Moreover, xn; m (t) 63(t) implies that {xn; m } is equicontinuous. By Arzela–Ascoli theorem, {xn; m } is a compact subset in C(I0 ; >), hence there is
M. Guo et al. / Fuzzy Sets and Systems 138 (2003) 601 – 615
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a subsequence {xk } of {xn; m } which converges to some point x’ ∈ C(I0 ; R n ). From the compactness of ’, we have x’ (t) ∈ [’(t)] ; t ∈ I0 . Finally, by taking x(t) = x’ (t), for t ∈ I0 and x(t) = xT (t), for t ∈ T , we have x(t) ∈ (’ ; T ). Therefore, (’ ; T ) is compact in S(’; T ). By Theorem 4.3, to prove that (’ ; T ) satisJes the others two conditions in the deJnition of fuzzy number, it is suNcient to show that F ∈ (RQ). But this is obvious, because all F; lk and ’ are fuzzy number. The proof is complete. Remark 4.5. The attainable set has analogous properties. 5. Some applications for fuzzy population models 5.1. Fuzzy logistic model (isolated population in nonstationary environment) Let us consider the following fuzzy population model: N (t) ∈ rN (t)[1 − (l[w] + c)N (t − 1)] EN (tk ) = lk ;
a:e: t ∈ I = [a; b];
k = 0; 1; 2; : : : ; m; ∈ [0; 1];
N (t) ∈ ct + d + [n(t)] ;
t ∈ I0 = [a − 1; a];
(5.1)
where a¿0; b = a + M + b1 ; M ∈ N; 0¡b1 ¡1, and r; c; d¿0; lk ∈ R1 ; k = 1; 2; : : : ; m. Without loss of generality, assume that a¡t1 ¡t2 ¡ · · · ¡tm ¡b and ti = a + j for all i = 1; 2; : : : ; m; j = 1; 2; : : : ; M . w ∈ E 1 is symmetric triangular fuzzy number with support [−1; 1], i.e., [w] = (1−)[−1; 1]; 0¡l¡c; ∈ [0; 1]. n ∈ C(I; E 1 ) deJned by n (t) = [n(t)] = [2pt + q; (4 − 2 )pt + (2 − )q];
∈ [0; 1]:
Obviously, if we take !(t; u; v) = r|u|(1 + (l + c)|v|), then Eq. (3.3) has a solution on I0 ∪ I . Therefore, (5.1) has a solution in C(I; E 1 ). We can give the solution set (’ ; b) of (5.1) in [a − 1; b] as ct + d + [n(t)] ; t ∈ I0 ; 0 (a + 1); t ∈ T0 ; 1 t ∈ T1 ; (’ ; b) = (a + 2); .. .. . . M (b); t ∈ TM ; where
0 (a + 1) =
u(t) = (ca + d + B (a)) erf0 (t) +
k ∈ S0
lk
: f0 ∈ F0 ; B ∈ n ; t ∈ T1
;
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M. Guo et al. / Fuzzy Sets and Systems 138 (2003) 601 – 615
t
F0 = f0 : f0 (t) =
a
[1 − (lh + c)(cs + d − c + 0 (s − 1))] ds; h ∈
(1 − )[−1; 1]; 0 (t) ∈ n (t) is integrable selection of n (t); t ∈ I0 and
i (a + i + 1) =
u(t) = mi (a + 1) erfi (t) +
Fi =
fi : fi (t) =
lk
; t ∈ Ti ; fi ∈ Fi ; mi ∈ E(n ) ;
k ∈ Si t a+i
[1 − (lh + c)mi (s − 1)] ds; h ∈ (1 − )[−1; 1]; t ∈ Ti :
For a set-value mapping E(n ); mi ∈ E(n ) means that a construct of mi depending on the choice of integrable selections of n . 5.2. Fuzzy Gompertz model (isolated population in stationary environment) The dynamics of simple limitation population with competition in stationary environment is well simulated by Gompertz equation [2] N (t) ∈ N (t)[r − ([W (t)] + G) ln N (t − 1)] EN (tk ) = lk ;
a:e: t ∈ I = [a; b];
k = 0; 1; 2; : : : ; m; ∈ [0; 1];
N (t) = qt + p;
t ∈ I0 = [a − 1; a];
(5.2)
where a¿0; b = a+M +b1 ; M ∈ N; 0¡b1 ¡1, and G; r; p; q¿0; lk ∈ R1 ; k = 1; 2; : : : ; m; W ∈ C(I; E 1 ) is symmetric triangular fuzzy function with support [−1; 1], i.e., [W (t)] = w(t)(1 − )[−1; 1]; w ∈ C (I; R1 ). By taking !(t; u; v) = |u|(r + (|w| + G)|ln v|), then Eq. (3.3) has solution on I0 ∪ I . Therefore, (5.2) has a solution in C(I; E 1 ). Thus, the solution set (’ ; b) of (5.2) in [a − 1; b] is qt + p; t ∈ I0 ; 0 (a + 1); t ∈ T0 ; 1 (’ ; b) = (a + 2); t ∈ T1 ; .. .. . . M (b); t ∈ TM ; where
i (a + i + 1) =
u(t) = mi (a + i) efi (t) +
k ∈ Si
lk
; t ∈ Ti ; fi ∈ Fi
M. Guo et al. / Fuzzy Sets and Systems 138 (2003) 601 – 615
and
Fi =
fi : fi (t) =
t a+i
615
[r − (w(s)h + G)mi−1 (s − 1)] ds; h ∈ (1 − )[−1; 1]; t ∈ Ti :
It is easy to know that (’ ; b) is -level set of 0 (’0 ; b). Acknowledgements The authors would like to express our thanks to Professor P. Diamond for his valuable remarks and suggestions concerning this work. References [1] O. Akinyele, Cone-valued Lyiapunov functions and stability of impulsive control systems, Nonlinear. Anal. 39 (2000) 247–259. [2] J. Angelova, A. Dishliev, Optimization problems for one-impulsive models from population dynamics, Nonlinear Anal. 39 (2000) 483–498. [3] J.P. Aubin, Impulse and Hybrid Control Systems: A Viability Approach, First Preliminary Draft of Lecture Notes of a Mini-course, University of California, Berkeley, 1999. [4] J.P. Aubin, A. Cellina, Di#erential Inclusions, Springer, New York, 1984. [5] P. Diamond, Time-dependent di#erential inclusions, cocycle attractors and fuzzy di#erential equations, IEEE Trans. Fuzzy systems 7 (1999) 734–740. [6] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World ScientiJc, Singapore, 1994. [7] P. Diamond, P. Watson, Regularity of solution sets for di#erential inclusions quasiconcave in a parameter, Appl. Math. Lett. 13 (2000) 31–35. [8] E. Hullermeier, An approach to modeling and simulation of uncertain dynamical systems, Int. J. Uncertainty, Fuzziness, Knowledge-Bases Syst. 5 (1997) 117–137. [9] S.G. Pandit, S.G. Deo, Di#erential Systems Involving Impulses, in: Lecture Notes in Mathematics, Vol. 954, Springer, New York, 1982. [10] M.L. Puri, D.A. Ralescu, Di#erentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552–558. [11] D.E. Stewart, Rigid-body dynamics with friction & impact, SIAM Rev. 42 (2000) 3–39. [12] D.E. Stewart, Reformulations of measure di#erential inclusions and their closed graph properly, J. Di#erential Equations 174 (2001) 109–129. [13] A.A. Tolstonogov, On comprising theorems for di#erential inclusion in locally convex spaces, I and II, Di#erential Equations 17 (1981) 651– 659 and 1016 –1024 (in Russian).