Existence of bounded solutions of a class of ... - Semantic Scholar
Applied Mathematics and Computation 231 (2014) 478–488
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Existence of bounded solutions of a class of neutral systems of functional differential equations Stevo Stevic´ ⇑ Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
a r t i c l e
i n f o
a b s t r a c t Some results on the existence of bounded solutions together with their first derivatives of a class of neutral systems of functional differential equations with complicated deviations, which extend and unify numerous results in the literature, are proved. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Bounded C 1 solutions System of functional differential equations Iterated deviations Lipschitz derivative
1. Introduction and preliminaries Special cases of the following system of functional differential equations, which is partially solved with respect to the first derivatives of dependent variables,
x0 ðt þ 1Þ ¼ Ax0 ðtÞ þ Uðt; xðtÞ; xðf1 ðt; xðtÞÞÞ; x0 ðf2 ðt; xðtÞÞÞÞ; N 3
N
ð1Þ
N
where t 2 Rþ ¼ ½0; 1Þ; U : Rþ ðR Þ ! R ; f i : Rþ R ! Rþ ; i ¼ 1; 2, have attracted some attention among the experts in the research field (see, for example, [1,14,19,20,22,23,44,47,48]). For some other results on systems/equations not solved with respect to the highest-order derivatives, see, for example, [3–13,16–18,21,24,38,40,46,49]. Based on the idea of iterations of some iterative processes (see, for example, [2,15,25–37,42]) in [38–41,43–47], we proposed the investigation of various types of systems/equations with continuous arguments, whose deviations of an argument depend on an unknown function which depend also of the function and so on, so called, iterated deviations. Motivated by the line of investigations in the papers [4,13,16,17,21,22,38,39,44,46–48], here we investigate the existence of bounded C 1 solutions of the next system of functional differential equations ð1Þ
ðkÞ
ð1Þ
ðlÞ
x0 ðt þ 1Þ ¼ Ax0 ðtÞ þ Uðt; xðv 1 ðtÞÞ; . . . ; xðv 1 ðtÞÞ; x0 ðu1 ðtÞÞ; . . . ; x0 ðu1 ðtÞÞÞ;
ð2Þ
on Rþ , where
v rðjÞ ðtÞ ¼ ujr ðt; xðuj rþ1 ðt; . . . xðujm ðt; xðtÞÞÞ . . .ÞÞÞ; j
upðiÞ ðtÞ ¼ wip ðt; xðwi pþ1 ðt; . . . xðwili ðt; xðtÞÞÞ . . .ÞÞÞ; kþl
j ¼ 1; k; r ¼ 1; mj ; i ¼ 1; l; p ¼ 1; li , U : Rþ ðRN Þ ! RN , ujr ; wip : Rþ RN ! Rþ ; j ¼ 1; k, r ¼ 1; mj ; i ¼ 1; l; p ¼ 1; li ; A is a nonsingular matrix, extending and unifying numerous results in the literature. We use also the following convention ðiÞ v ðjÞ m þ1 ðtÞ ¼ ul þ1 ðtÞ ¼ t; j
i
j ¼ 1; k; i ¼ 1; l:
⇑ Address: Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.151
ð3Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
479
As usual, by CðRþ Þ we denote the space of continuous vector functions on Rþ , while by C 1 ðRþ Þ the space of all continuously differentiable vector functions on Rþ . The subspace of C 1 ðRþ Þ consisting of all bounded vector functions together with their first derivatives on Rþ is denoted by BC 1 ðRþ Þ. The norm on BC 1 ðRþ Þ is
kxkBC 1 ðRþ Þ
( ) 0 0 ¼ max kxk1 ; kx k1 ¼ max supjxðtÞj; supjx ðtÞj ; t2Rþ
t2Rþ
where for y 2 RN ; jyj denotes a norm on RN . The following folklore lemma, which can be found, for example, in [46], will be frequently applied in the proofs of our main results. Lemma 1. Assume that ðan Þn2N and ðbn Þn2N are two sequences of nonnegative numbers, and that sequence ðxn Þn2N satisfies the inequality
xn 6 an þ bn xnþ1 ;
n 2 N:
Then
x1 6
j1 k1 Y k1 X Y aj bi þ xk bi ; j¼1
k 2 N:
i¼1
i¼1
2. Main results First, we give a list of some conditions which will be used in the formulations of the main results in this paper. (a) Vector function Uðt; x1 ; . . . ; xkþl Þ is continuous for t 2 Rþ ; xj 2 RN ; j ¼ 1; k þ l,
Uðt; 0; . . . ; 0Þ 0;
ð4Þ
jUðt; x01 ; . . . ; x0kþl Þ Uðs; x001 ; . . . ; x00kþl Þj 6 c0 ðt; sÞjt sj þ
kþl X
cj ðt; sÞjx0j x00j j;
ð5Þ
j¼1
where cj ðt; sÞ; j ¼ 0; k þ l are continuous and nonnegative functions for t; s 2 Rþ , and x0j ; x00j 2 RN , j ¼ 1; k þ l; (b) ujr ðt; xÞ; j ¼ 1; k; r ¼ 1; mj , and wip ðt; xÞ; i ¼ 1; l; p ¼ 1; li , are continuous and nonnegative functions for t 2 Rþ and x 2 RN , and ð1Þ
ð2Þ
j ¼ 1; k; r ¼ 1; mj ;
ð3Þ
ð4Þ
i ¼ 1; l; p ¼ 1; li ;
jujr ðt; xÞ ujr ðs; yÞj 6 kjr jt sj þ kjr jx yj; jwip ðt; xÞ wip ðs; yÞj 6 kip jt sj þ kip jx yj; N
for every t; s 2 Rþ , and x; y 2 R , and for some positive constants (c) for every j ¼ 0; k þ l, the series
Cj ðt; sÞ ¼
i¼0
i¼0
ð7Þ
ð1Þ kjr ;
Z 1 1 X X jA1 jiþ1 cj ðt þ i; s þ iÞ and Gj ðtÞ ¼ jA1 jiþ1
t
ð6Þ
ð2Þ kjr ;
j ¼ 1; k; r ¼ 1; mj ;
ð3Þ kip ;
ð4Þ kip ;
i ¼ 1; l; p ¼ 1; li ;
1
cj ðs þ i; s þ iÞds;
converge uniformly for t; s 2 Rþ , and for some d 2 ð0; 1Þ, satisfy the condition
(
max
sup
kþl X
kþl X Cj ðt; sÞ; sup Gj ðtÞ
t;s2Rþ j¼0
)
6 d:
ð8Þ
t2Rþ j¼0
Theorem 1. Suppose that conditions (a)–(c) hold. Then for any BC 1 ðRþ Þ solution of system (2), such that
lim jxðt þ 1Þ AxðtÞj ¼ 0;
ð9Þ
t!þ1
and
jx0 ðtÞ x0 ðsÞj 6 Ljt sj
ð10Þ 1
for every t; s 2 Rþ and some L > 0, there is a C vector function a with the Lipschitz first derivative and such that
aðt þ 1Þ ¼ AaðtÞ;
ð11Þ
lim jxðtÞ aðtÞj ¼ 0:
ð12Þ
t!þ1
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
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Proof. Assume x 2 BC 1 ðRþ Þ is a solution of system (2) such that (9) and (10) hold. Set
Z 1 X iþ1 ðA1 Þ
aðtÞ :¼ xðtÞ
1
ðlÞ 0 Uðs þ i; xðv 1ð1Þ ðs þ iÞÞ; . . . ; xðv 1ðkÞ ðs þ iÞÞ; x0 ðuð1Þ 1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds:
t
i¼0
ð13Þ
Using (4) and (5) it is not difficult to see that
Z 1 X 1 ð1Þ ðkÞ ð1Þ ðlÞ 1 iþ1 0 0 ðA Þ U ð s þ i; xð v ð s þ iÞÞ; . . . ; xð v ð s þ iÞÞ; x ðu ð s þ iÞÞ; . . . ; x ðu ð s þ iÞÞÞd s 1 1 1 1 i¼0 t 6
1 X
jA1 jiþ1
Z
1
k X
Z 1 X jA1 jiþ1
j¼1
i¼0
cj ðs þ i; s þ iÞjxðv ðjÞ 1 ðs þ iÞÞjds þ
t
i¼0
6 kxkBC 1 ðRþ Þ
t
1
l X
ckþp ðs þ i; s þ iÞjx0 ðu1ðpÞ ðs þ iÞÞjds
p¼1
kþl X Gj ðtÞ;
ð14Þ
j¼1
which means that the sum in (13) is convergent. Thus, aðtÞ is well-defined on Rþ . Clearly, (13) along with (14) implies
jxðtÞ aðtÞj 6 kxkBC 1 ðRþ Þ
kþl X Gj ðtÞ ! 0;
as t ! þ1;
ð15Þ
j¼1
that is, (12) holds, where the last limit follows from the uniform convergence of series Gj ðtÞ; j ¼ 1; k þ l, as sums of functions in CðRþ Þ tending to zero as t ! þ1. Moreover, due to the uniform convergence of Cj ðt; tÞ and Gj ðtÞ; j ¼ 1; k þ l, relation (13) can be differentiated on Rþ , so that
a0 ðtÞ ¼ x0 ðtÞ þ
1 X iþ1 ð1Þ ðkÞ ð1Þ ðlÞ ðA1 Þ Uðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ:
ð16Þ
i¼0
If we integrate (2) from t to t0 , then let t0 ! þ1, and employ (9), we have that
xðt þ 1Þ ¼ AxðtÞ
Z
1
ðkÞ ð1Þ ðlÞ 0 0 Uðs; xðv ð1Þ 1 ðsÞÞ; . . . ; xðv 1 ðsÞÞ; x ðu1 ðsÞÞ; . . . ; x ðu1 ðsÞÞÞds:
t
ð17Þ
Using (17) we have
aðt þ 1Þ ¼ xðt þ 1Þ ¼ AxðtÞ
Z 1 X iþ1 ðA1 Þ
Z 1 X i ðA1 Þ t
i¼1
¼ A xðtÞ
ðkÞ ð1Þ ðlÞ 0 0 Uðs þ i;xðv ð1Þ 1 ðs þ iÞÞ; .. .; xðv 1 ðs þ iÞÞ;x ðu1 ðs þ iÞÞ;. .. x ðu1 ðs þ iÞÞÞds
ð1Þ ðlÞ 0 0 Uðs; xðv 1ð1Þ ðsÞÞ;. . .; xðv ðkÞ 1 ðsÞÞ;x ðu1 ðsÞÞ; .. .; x ðu1 ðsÞÞÞds
t
tþ1
i¼0 1
Z
1
1
ð1Þ ðlÞ 0 0 Uðs þ i; xðv 1ð1Þ ðs þ iÞÞ;. .. ;xðv ðkÞ 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ;. . .; x ðu1 ðs þ iÞÞÞds
Z 1 X iþ1 ðA1 Þ
1 t
i¼0
! ðkÞ ð1Þ ðlÞ 0 0 Uðs þ i; xðv ð1Þ 1 ðs þ iÞÞ;. .. ;xðv 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ;. .. ;x ðu1 ðs þ iÞÞÞds ¼ AaðtÞ;
which means that aðtÞ satisfies equality (11). Employing (10) in (16), we get
ja0 ðtÞ a0 ðsÞj 6 jx0 ðtÞ x0 ðsÞj þ
1 X ð1Þ ðkÞ ð1Þ ðlÞ jA1 jiþ1 jUðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ i¼0
ð1Þ
ðkÞ
ð1Þ
ðlÞ
Uðs þ i; xðv 1 ðs þ iÞÞ; . . . ; xðv 1 ðs þ iÞÞ; x0 ðu1 ðs þ iÞÞ; . . . ; x0 ðu1 ðs þ iÞÞÞj 6 Ljt sj þ
1 1 k X X X ðjÞ ðjÞ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ jA1 jiþ1 cj ðt þ i; s þ iÞjxðv 1 ðt þ iÞÞ xðv 1 ðs þ iÞÞj i¼0
i¼0
j¼1
1 l X X ðpÞ ðpÞ þ jA1 jiþ1 ckþp ðt þ i; s þ iÞjx0 ðu1 ðt þ iÞÞ x0 ðu1 ðs þ iÞÞj p¼1
i¼0
1 1 k X X X ðjÞ ðjÞ 6 Ljt sj þ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ kx0 k1 jA1 jiþ1 cj ðt þ i; s þ iÞjv 1 ðt þ iÞ v 1 ðs þ iÞj i¼0 1 X
þL
i¼0
jA1 jiþ1
i¼0 l X
ðpÞ ckþp ðt þ i; s þ iÞjuðpÞ 1 ðt þ iÞ u1 ðs þ iÞj:
p¼1
j¼1
ð18Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
481
We have that for each j 2 f1; . . . ; kg and every 1 6 m 6 mj , ðjÞ
ðjÞ
ð1Þ
ð2Þ
ðjÞ
ðjÞ
ðjÞ jv ðjÞ m ðtÞ v m ðsÞj ¼ jujm ðt; xðv mþ1 ðtÞÞÞ ujm ðs; xðv mþ1 ðsÞÞÞj 6 kj m jt sj þ kj m jxðv mþ1 ðtÞÞ xðv mþ1 ðsÞÞj ð1Þ
ð2Þ
ðjÞ
ðjÞ
6 kj m jt sj þ kj m kx0 k1 jv mþ1 ðtÞ v mþ1 ðsÞj: Hence by using Lemma 1 with am ¼ ðjÞ
ðjÞ
jv 1 ðtÞ v 1 ðsÞj 6 jt sj
ð1Þ kj m jt
sj and bm ¼
ð19Þ
ð2Þ kj m kx0 k1 ,
we have that
! jY m m 1 1 X Y ð1Þ ð2Þ ð2Þ ðjÞ ðjÞ jv mþ1 ðtÞ v mþ1 ðsÞj; kx0 kj11 1 kjj1 kji þ kx0 km k 1 ji i¼1
j1 ¼1
ð20Þ
i¼1
holds for each j 2 f1; . . . ; kg and every 1 6 m 6 mj . Choosing m ¼ mj in (20) and by using (3), we get
jv
ðjÞ 1 ðtÞ
v
ðjÞ 1 ðsÞj
6
! mj mj jY 1 1 X Y ð2Þ ð2Þ 0 j1 1 ð1Þ 0 mj kx k1 kjj1 kji þ kx k1 kji jt sj: i¼1
j1 ¼1
ð21Þ
i¼1
Analogously, employing (7) and Lemma 1, it is obtained that ðiÞ
ðiÞ
ju1 ðtÞ u1 ðsÞj 6
! li li j1 X Y Y ð3Þ ð4Þ ð4Þ 0 li jt sj: kx0 kj1 k k þ kx k k 1 1 ij ii1 ii1 j¼1
ð22Þ
i1 ¼1
i1 ¼1
From (18), (21) and (22) it follows that 0
0
ja ðtÞ a ðsÞj 6 ðL þ C0 ðt; sÞÞjt sj þ
! mj mj jY 1 1 X Y ð2Þ ð2Þ 0 j1 ð1Þ 0 mj þ1 Cj ðt; sÞ kx k1 kjj1 kji þ kx k1 kji jt sj
k X
j1 ¼1
j¼1
i¼1
i¼1
! li li j1 l X X Y Y ð3Þ ð4Þ ð4Þ 0 li þ L Ckþi ðt; sÞ kx0 kj1 k k þ kx k k jt sj 6 L1 jt sj; 1 ij 1 ii1 ii1 j¼1
i¼1
i1 ¼1
i1 ¼1
where
L1 ¼ sup
L þ C0 ðt; sÞ þ
t;s2Rþ
k X
mj mj jY 1 1 X Y ð1Þ ð2Þ ð2Þ mj þ1 kx0 kj11 kjj1 kji þ kx0 k1 kji
j¼1
j1 ¼1
Cj ðt; sÞ
i¼1
li li j1 l X X Y Y ð3Þ ð4Þ ð4Þ 0 li þ L Ckþi ðt; sÞ kx0 kj1 k k þ kx k kii1 1 1 ij ii1 j¼1
i¼1
(L1 is finite due to (8)).
i1 ¼1
!!
!
i¼1
;
i1 ¼1
h
Theorem 2. Suppose that conditions (a)–(c) hold, M :¼ supn2N jAn j < 1, and a is a C 1 vector function satisfying relation (11) and
ja0 ðtÞ a0 ðsÞj 6 L1 jt sj
ð23Þ ð2Þ kj1 ;
ð3Þ ki1 ;
for every t; s 2 Rþ and some L1 > 0. Then for sufficiently small j ¼ 1; k; solution with the Lipschitz first derivative, and satisfying conditions (9) and (12).
ð4Þ ki1 ,
1
i ¼ 1; l, system (2) has a unique BC ðRþ Þ
Proof. Suppose x 2 BC 1 ðRþ Þ is a solution of system (2) satisfying (9) and (12). Let
a1 ðtÞ :¼ xðtÞ
Z 1 X iþ1 ðA1 Þ
1
t
i¼0
ð1Þ ðlÞ 0 0 Uðs þ i; xðv 1ð1Þ ðs þ iÞÞ; . . . ; xðv ðkÞ 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds:
The proof of Theorem 1 shows that xðtÞ is also a solution of the system
xðtÞ ¼ a1 ðtÞ þ
Z 1 X iþ1 ðA1 Þ i¼0
t
1
ðlÞ 0 Uðs þ i; xðv 1ð1Þ ðs þ iÞÞ; . . . ; xðv 1ðkÞ ðs þ iÞÞ; x0 ðuð1Þ 1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds:
Moreover, a1 satisfies relation (11), and
lim ðxðtÞ a1 ðtÞÞ ¼ 0:
t!þ1
Let s 2 ½0; 1Þ. Then, from (12) and (24) it follows that
jaðsÞ a1 ðsÞj ¼ jAn ðaðs þ nÞ a1 ðs þ nÞÞj 6 Mðjaðs þ nÞ xðs þ nÞj þ jxðs þ nÞ a1 ðs þ nÞjÞ ! 0;
ð24Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
482
as n ! þ1, that is, aðsÞ ¼ a1 ðsÞ. From this and (11) we have that aðsÞ ¼ a1 ðsÞ for every s 2 Rþ . Hence
xðtÞ ¼ aðtÞ þ
Z 1 X iþ1 ðA1 Þ
1
t
i¼0
ðkÞ ð1Þ ðlÞ 0 0 Uðs þ i; xðv ð1Þ 1 ðs þ iÞÞ; . . . ; xðv 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds:
ð25Þ
If x 2 BC 1 ðRþ Þ is a solution of system (25), then it is a solution of system (2) satisfying (9) and (12). To show this, first note that (14) implies the uniform convergence of the series
Z 1 X iþ1 ðA1 Þ
1
ðkÞ ð1Þ ðlÞ 0 0 Uðs þ i; xðv ð1Þ 1 ðs þ iÞÞ; . . . ; xðv 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds;
t
i¼0
on Rþ , as well as relation (12). From (4) and (5), we get
X 1 ð1Þ ðkÞ ð1Þ ðlÞ 1 iþ1 ðA Þ Uðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ i¼0 ! 1 k l kþl X 1 iþ1 X X X ðjÞ ðpÞ 0 6 jA j cj ðt þ i; t þ iÞjxðv 1 ðt þ iÞÞj þ ckþp ðt þ i; t þ iÞjx ðu1 ðt þ iÞÞj 6 kxkBC1 ðRþ Þ Cj ðt; tÞ: i¼0
p¼1
j¼1
j¼1
This and the uniform convergence of Cj ðt; tÞ, j ¼ 1; k þ l, implies the uniform convergence of the series 1 X iþ1 ð1Þ ðkÞ ð1Þ ðlÞ ðA1 Þ Uðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ: i¼0
Hence the differentiation of the series in (25) is allowed, and we get 1 X iþ1 ð1Þ ðkÞ ð1Þ ðlÞ ðA1 Þ Uðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ:
x0 ðtÞ ¼ a0 ðtÞ
i¼0
Using this and the relation a0 ðt þ 1Þ ¼ Aa0 ðtÞ, which follows by differentiating (11), we see that xðtÞ is a solution of system (2). Since
Z jxðt þ 1Þ AxðtÞj ¼
1 t
ðkÞ ð1Þ ðlÞ 0 0 Uðs; xðv ð1Þ 1 ðsÞÞ; . . . ; xðv 1 ðsÞÞ; x ðu1 ðsÞÞ; . . . ; x ðu1 ðsÞÞÞds 6 kxkBC 1 ðRþ Þ
kþl X Gj ðtÞ ! 0; j¼1
as t ! þ1, we get that condition (9) holds. Due to just proved equivalence we will consider system of equations (25) instead of (2). Now we define two sequences ðxm ðtÞÞm2N0 and ðx0m ðtÞÞm2N0 , as follows
x0 ðtÞ ¼ aðtÞ;
x00 ðtÞ ¼ a0 ðtÞ;
xm ðtÞ ¼ aðtÞ þ
Z 1 X iþ1 ðA1 Þ
1
t
i¼0
ðkÞ ð1Þ 0 Uðs þ i; xm1 ðv ð1Þ 1 m1 ðs þ iÞÞ; . . . ; xm1 ðv 1 m1 ðs þ iÞÞ; xm1 ðu1 m1 ðs þ iÞÞ; . . . ;
ðlÞ
x0m1 ðu1 m1 ðs þ iÞÞÞds; x0m ðtÞ ¼ a0 ðtÞ
ð26Þ
1 X iþ1 ð1Þ ðkÞ ð1Þ ðA1 Þ Uðt þ i; xm1 ðv 1 m1 ðt þ iÞÞ; . . . ; xm1 ðv 1 m1 ðt þ iÞÞ; x0m1 ðu1 m1 ðt þ iÞÞ; . . . ; i¼0
ðlÞ
x0m1 ðu1 m1 ðt þ iÞÞÞ;
ð27Þ
where
v ðjÞ rm ðtÞ ¼ ujr ðt; xm ðuj rþ1 ðt; . . . xm ðuj m ðt; xm ðtÞÞÞ . . .ÞÞÞ; j
j ¼ 1; k; r ¼ 1; mj ;
and
uðiÞ pm ðtÞ ¼ wip ðt; xm ðwi pþ1 ðt; . . . xm ðwi li ðt; xm ðtÞÞÞ . . .ÞÞÞ;
i ¼ 1; l; p ¼ 1; li :
For every t 2 Rþ we have
jx0 ðtÞj 6 kakBC 1 ðRþ Þ 6
kakBC 1 ðRþ Þ 1d
and jx00 ðtÞj 6 kakBC 1 ðRþ Þ 6
Assume that the following inequalities hold for an m 2 N
kakBC 1 ðRþ Þ 1d
:
ð28Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
jxm1 ðtÞj 6
kakBC 1 ðRþ Þ
and jx0m1 ðtÞj 6
1d
kakBC 1 ðRþ Þ
t 2 Rþ :
;
1d
483
ð29Þ
Then (4), (5), (8), (26) and (29), imply
jxm ðtÞj 6 jaðtÞj þ
Z 1 X jA1 jiþ1
1
t
i¼0
ðkÞ ðlÞ 0 Uðs þ i; xm1 ðv ð1Þ 1 m1 ðs þ iÞÞ; . . . ; xm1 ðv 1 m1 ðs þ iÞÞ; . . . ; xm1 ðu1 m1 ðs þ iÞÞÞjds
1 k Z X X 6 jaðtÞj þ jA1 jiþ1 i¼0
þ
1 X
jA1 j
p¼1
i¼0
1
t
cj ðs þ i; s þ iÞjxm1 ðv ðjÞ 1 m1 ðs þ iÞÞjds
t
j¼1
l Z X iþ1
1
ckþp ðs þ i; s þ iÞjx0m1 ðu1ðpÞm1 ðs þ iÞÞjds
kþl kakBC 1 ðRþ Þ X kakBC 1 ðRþ Þ Gj ðtÞ 6 : 1 d j¼1 1d
6 kakBC 1 ðRþ Þ þ
ð30Þ
From (4), (5), (8), (27) and (29), it follows that
jx0m ðtÞj 6 ja0 ðtÞj þ
1 X
jA1 jiþ1 jUðt þ i; xm1 ðv 1 m1 ðt þ iÞÞ; .. .; xm1 ðv 1 m1 ðt þ iÞÞ; x0m1 ðu1 m1 ðt þ iÞÞ;. .. ;x0m1 ðu1 m1 ðt þ iÞÞÞj ð1Þ
ðkÞ
ð1Þ
ðlÞ
i¼0
6 kakBC 1 ðRþ Þ þ
1 k 1 l X X X X ðjÞ ðpÞ jA1 jiþ1 cj ðt þ i; t þ iÞjxm1 ðv 1 m1 ðt þ iÞÞj þ jA1 jiþ1 ckþp ðt þ i;t þ iÞjx0m1 ðu1 m1 ðt þ iÞÞj i¼0
6 kakBC 1 ðRþ Þ þ
j¼1
1d
p¼1
i¼0
kþl kakBC 1 ðRþ Þ X
Cj ðt; tÞ 6
kakBC 1 ðRþ Þ
j¼1
1d
ð31Þ
:
From (28), (30), (31) and by the induction we have that
jxm ðtÞj 6
kakBC 1 ðRþ Þ
and jx0m ðtÞj 6
1d
kakBC 1 ðRþ Þ
ð32Þ
1d
for every t 2 Rþ and m 2 N0 . By (5), (23), (27) and (32), we have that
jx01 ðtÞ x01 ðsÞj 6 ja0 ðtÞ a0 ðsÞj þ
1 X ð1Þ ðkÞ ð1Þ jA1 jiþ1 jUðt þ i; x0 ðv 1 0 ðt þ iÞÞ; . . . ; x0 ðv 1 0 ðt þ iÞÞ; x00 ðu1 0 ðt þ iÞÞ; . . . ; i¼0
ðlÞ
ð1Þ
ðkÞ
ð1Þ
ðlÞ
x00 ðu1 0 ðt þ iÞÞÞ Uðs þ i; x0 ðv 1 0 ðs þ iÞÞ; . . . ; x0 ðv 1 0 ðs þ iÞÞ; x00 ðu1 0 ðs þ iÞÞ; . . . ; x00 ðu1 0 ðs þ iÞÞÞj 6 ja0 ðtÞ a0 ðsÞj þ
1 1 k X X X ðjÞ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ jA1 jiþ1 cj ðt þ i; s þ iÞjaðv 1 0 ðt þ iÞÞ i¼1
i¼0
j¼1
1 l X X ðjÞ ðpÞ ðpÞ aðv 1 0 ðs þ iÞÞj þ jA1 jiþ1 ckþp ðt þ i; s þ iÞja0 ðu1 0 ðt þ iÞÞ a0 ðu1 0 ðs þ iÞÞj 6 L1 jt sj p¼1
i¼0
1 1 k X X X ðjÞ ðjÞ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ kakBC 1 ðRþ Þ jA1 jiþ1 cj ðt þ i; s þ iÞjv 1 0 ðt þ iÞ v 1 0 ðs þ iÞj þ i¼1
i¼0
j¼1
1 l X X ðpÞ ðpÞ þ L1 jA1 jiþ1 ckþp ðt þ i; s þ iÞju1 0 ðt þ iÞ u1 0 ðs þ iÞj:
ð33Þ
p¼1
i¼0 ðjÞ
ðjÞ
ðpÞ
ðpÞ
ð1Þ
The estimates for jv 1 0 ðt þ iÞ v 1 0 ðs þ iÞj and ju1 0 ðt þ iÞ u1 0 ðs þ iÞj are obtained by Lemma 1 with am ¼ kjm jt sj and ð2Þ ð3Þ ð4Þ bm ¼ kjm kakBC 1 ðRþ Þ , that is, for am ¼ kjm jt sj and bm ¼ kjm kakBC 1 ðRþ Þ . From this and (33), it follows that
jx01 ðtÞ x01 ðsÞj 6 ðL1 þ C0 ðt; sÞÞjt sj þ þ L1
6
l X
li X
i¼1
j¼1
Ckþi ðt; sÞ
L 1 þ C0 þ
Cj
j1 ¼1
j¼1
þ L1
l X i¼1
j¼1
Ckþi
ð3Þ
þ
mj X
j¼1
j1 ¼1
Cj ðt; sÞ
j1 Y i1 ¼1
j
þ
li Y
l
ð4Þ
ð1Þ
1 kakBC k 1 ðR Þ jj1
kii1 þ kakBCi 1 ðR
þÞ
! ð4Þ
kii1
i1 ¼1
mj mj jY 1 1 X Y m þ1 j1 ð1Þ ð2Þ ð2Þ kakBC k kji þ kakBCj 1 ðR Þ kji 1 ðR Þ jj1
k X
li X
kakj1 k BC 1 ðR Þ ij
k X
ð3Þ kj1 k BC 1 ðRþ Þ ij
ka
þ
j1 Y i1 ¼1
þ
i¼1
ð4Þ kii1
li Y
li
þ kakBC 1 ðR
þÞ
i1 ¼1
jY 1 1 i¼1
ð2Þ
m þ1
kji þ kakBCj 1 ðR
þ
! mj Y ð2Þ k jt sj ji Þ i¼1
jt sj !
i¼1
!!
ð4Þ kii1
jt sj 6 ~Ljt sj;
ð34Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
484
where Cj :¼ supt;s2Rþ Cj ðt; sÞ; j ¼ 0; k þ l and
L 1 þ C0 þ ~L ¼ 1
Pk
Pmj
j¼1 Cj
Pl
i¼1 Ckþi
kakBC 1 ðR
j1 ¼1
Pli
j1
ð1Þ Qj1 1 ð2Þ i¼1 kji
kjj1
1d
kakBC 1 ðR
j¼1
þÞ
þÞ
j1
ð3Þ Qj1 ð4Þ i1 ¼1 kii1
kij
1d
kakBC 1 ðR
þ
mj þ1
kakBC 1 ðR
þÞ
li
!
Qm j
ð2Þ
i¼1 kji
1d
þ
þÞ
!
Qli
;
ð4Þ
i1 ¼1 kii1
1d
ð3Þ ð4Þ L is positive. where kii1 ; kii1 ; i ¼ 1; l, i1 ¼ 1; li are chosen such that the denominator of ~ Assume that for some m 2 N and every t; s 2 Rþ
jx0m ðtÞ x0m ðsÞj 6 ~Ljt sj:
ð35Þ
By (5), (27) and (32), we obtain 1 X iþ1 ð1Þ ðkÞ ð1Þ jðA1 Þ Uðt þ i; xm ðv 1 m ðt þ iÞÞ; . . . ; xm ðv 1 m ðt þ iÞÞ; x0m ðu1 m ðt þ iÞÞ; . . . ;
jx0mþ1 ðtÞ x0mþ1 ðsÞj 6 ja0 ðtÞ a0 ðsÞj þ
i¼0 ðlÞ x0m ðu1 m ðt
iþ1
þ iÞÞÞ ðA1 Þ
ðkÞ ð1Þ 0 Uðs þ i; xm ðv ð1Þ 1 m ðs þ iÞÞ; . . . ; xm ðv 1 m ðs þ iÞÞ; xm ðu1 m ðs þ iÞÞ; . . . ;
ðlÞ
x0m ðu1 m ðs þ iÞÞÞj 6 ja0 ðtÞ a0 ðsÞj þ
1 X jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj i¼0
1 k X X ðjÞ ðjÞ jA1 jiþ1 cj ðt þ i; s þ iÞjxm ðv 1m ðt þ iÞÞ xm ðv 1m ðs þ iÞÞj þ i¼0
þ
þ
j¼1
1 X
l X
i¼0
p¼1
jA1 jiþ1
ðpÞ 0 ckþp ðt þ i; s þ iÞjx0m ðuðpÞ 1m ðt þ iÞÞ xm ðu1m ðs þ iÞÞj 6 L1 jt sj
1 1 k X X kakBC 1 ðRþ Þ X ðjÞ ðjÞ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ jA1 jiþ1 cj ðt þ i; s þ iÞjv 1m ðt þ iÞ v 1m ðs þ iÞj 1 d i¼0 i¼0 j¼1
1 l X X ðpÞ ðpÞ þ ~L jA1 jiþ1 ckþp ðt þ i; s þ iÞju1m ðt þ iÞ u1m ðs þ iÞj: i¼0
ð36Þ
p¼1
Since by (32) we have ð1Þ
ð2Þ
ðjÞ
ð1Þ
ð2Þ
kakBC 1 ðRþ Þ
ðjÞ
ðjÞ 0 jv ðjÞ rm ðt þ iÞ v rm ðs þ iÞj 6 kjr jt sj þ kjr kxm k1 jv rþ1 m ðt þ iÞ v rþ1 m ðs þ iÞj
6 kjr jt sj þ kjr
1d
ðjÞ
ðjÞ
jv rþ1 m ðt þ iÞ v rþ1 m ðs þ iÞj
for j ¼ 1; k; r ¼ 1; mj , and ðjÞ
ðjÞ
ð3Þ ð4Þ ðpÞ 0 juðpÞ qm ðt þ iÞ uqm ðs þ iÞj 6 kpq jt sj þ kpq kxm k1 juqþ1 m ðt þ iÞ uqþ1 m ðs þ iÞj ð4Þ 6 kð3Þ pq jt sj þ kpq
kakBC 1 ðRþ Þ 1d
ðjÞ
ðjÞ
juqþ1 m ðt þ iÞ uqþ1 m ðs þ iÞj
ð1Þ
ð2Þ
ð3Þ
for p ¼ 1; l; q ¼ 1; li , applying Lemma 1 with am ¼ kjm jt sj and bm ¼ kjm kakBC 1 ðRþ Þ =ð1 dÞ, and with am ¼ kjm jt sj and ð4Þ bm ¼ kjm kakBC 1 ðRþ Þ =ð1 dÞ, from (36) it easily follows that
jx0mþ1 ðtÞ
x0mþ1 ðsÞj
6 ðL1 þ C0 Þjt sj þ
k X
0
Cj @
j¼1
mj X kakBC 1 ðRþ Þ j1 ¼1
1d
!j 1 ð1Þ kjj1
jY 1 1
ð2Þ kji
þ
i¼1
kakBC 1 ðRþ Þ 1d
!mj þ1
mj Y
1
ð2Þ kji Ajt
sj
i¼1
0 1 !j1 !l l li j1 l i X X Y kakBC 1 ðRþ Þ kakBC 1 ðRþ Þ i Y ð3Þ ð4Þ ð4Þ ~ þ L Ckþi @ kij kii1 þ kii1 Ajt sj 6 ~Ljt sj: 1d 1d i¼1 j¼1 i ¼1 i ¼1 1
So by induction Lipschitz condition (35) holds for every m 2 N0 and t; s 2 Rþ . Let
aj1 ðtÞ ¼ uj1 ðt; aðuj2 ðt; . . . aðujmj ðt; aðtÞÞÞ . . .ÞÞÞ; and
a^ i1 ðtÞ ¼ wi1 ðt; aðwi2 ðt; . . . aðwili ðt; aðtÞÞÞ . . .ÞÞÞ:
1
ð37Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
485
Then by (4), (5), (8) and (26) we get
jx1 ðtÞ x0 ðtÞj 6
1 X
jA1 jiþ1
1
^ 11 ðs þ iÞÞ . . . ; a0 ða ^ l1 ðs þ iÞÞÞjds jUðs þ i; aða11 ðs þ iÞÞ; . . . ; aðak1 ðs þ iÞÞ; a0 ða
t
i¼0
6
Z
1 X
jA1 jiþ1
i¼0
k Z X
cj ðs þ i; s þ iÞjaðaj1 ðs þ iÞÞjds þ
t
j¼1
6 kakBC 1 ðRþ Þ
1
1 l Z X X jA1 jiþ1 p¼1
i¼0
t
1
ckþp ðs þ i; s þ iÞja0 ða^ p1 ðs þ iÞÞjÞds
kþl X Gj ðtÞ 6 kakBC 1 ðRþ Þ d;
ð38Þ
j¼1
while by (4), (5), (8) and (27) we have that
jx01 ðtÞ x00 ðtÞj 6
1 X
^ 11 ðt þ iÞÞ . . . ; a0 ða ^ l1 ðt þ iÞÞÞj jA1 jiþ1 jUðt þ i; aða11 ðt þ iÞÞ; . . . ; aðak1 ðt þ iÞÞ; a0 ða
i¼0
6
1 X
jA1 jiþ1
i¼0
6 kakBC 1 ðRþ Þ
k X
1 l X X ^ p1 ðt þ iÞÞj jA1 jiþ1 ckþp ðt þ i; t þ iÞja0 ða
j¼1
i¼0
cj ðt þ i; t þ iÞjaðaj1 ðt þ iÞÞj þ
p¼1
kþl X
Cj ðt; tÞ 6 kakBC 1 ðRþ Þ d:
ð39Þ
j¼1
Assume that for some m 2 N
jxm ðtÞ xm1 ðtÞj 6 kakBC 1 ðRþ Þ qm
and jx0m ðtÞ x0m1 ðtÞj 6 kakBC 1 ðRþ Þ qm ;
ð40Þ
where
0 q ¼ d @1 þ
max
8 !j j mj <X 1 kakBC 1 ðRþ Þ 1 Y
16j6k;16i6l:
j1 ¼1
1d
li X kakBC 1 ðRþ Þ
ð2Þ kji ; ~L
i¼1
j¼1
1d
!j1
91 j = Y ð4Þ A : kij ; i¼1
From (4), (5), (26), (32), (35) and (40), it follows that
jxmþ1 ðtÞ xm ðtÞj 6
1 X
jA1 jiþ1
i¼0
þ
k Z X j¼1
1 X
jA1 jiþ1
1 X
jA1 jiþ1
i¼0
þ
p¼1
t
k Z X
1
j¼1
1 X
jA1 jiþ1
þ
j¼1
jA1 j
p¼1
1 X
jA1 jiþ1
p¼1
1
ðpÞ 0 ckþp ðs þ i; s þ iÞjx0m ðuðpÞ 1 m ðs þ iÞÞ xm1 ðu1 m1 ðs þ iÞÞjds
ðjÞ cj ðs þ i; s þ iÞjxm ðv ðjÞ 1 m ðs þ iÞÞ xm ðv 1 m1 ðs þ iÞÞjds 1
1
t
l Z X
i¼0
ðjÞ cj ðs þ i; s þ iÞjxm ðv ðjÞ 1 m ðs þ iÞÞ xm1 ðv 1 m1 ðs þ iÞÞjds
t
l Z X iþ1
i¼0
þ
t
k Z X
i¼0 1 X
t
l Z X
i¼0
6
1
1
t
ðjÞ cj ðs þ i; s þ iÞjxm ðv ðjÞ 1 m1 ðs þ iÞÞ xm1 ðv 1 m1 ðs þ iÞÞjds
ðpÞ 0 ckþp ðs þ i; s þ iÞjx0m ðuðpÞ 1 m ðs þ iÞÞ xm ðu1 m1 ðs þ iÞÞjds
ðpÞ 0 ckþp ðs þ i; s þ iÞjx0m ðuðpÞ 1 m1 ðs þ iÞÞ xm1 ðu1 m1 ðs þ iÞÞjds
kþl 1 k Z 1 X X kakBC 1 ðRþ Þ X ðjÞ 6 kakBC 1 ðRþ Þ qm Gj ðtÞ þ jA1 jiþ1 cj ðs þ i; s þ iÞjv ðjÞ 1 m ðs þ iÞ v 1 m1 ðs þ iÞjds 1 d i¼0 j¼1 j¼1 t 1 l Z X X þ ~L jA1 jiþ1 p¼1
i¼0
t
1
ckþp ðs þ i; s þ iÞju1ðpÞm ðs þ iÞ uðpÞ 1 m1 ðs þ iÞjds:
ð41Þ
From (6), (32) and hypothesis (40) we get that for each j 2 f1; . . . ; kg and every 1 6 s 6 mj ðjÞ
ðjÞ
ðjÞ
ð2Þ
ðjÞ
ðjÞ
jv ðjÞ s m ðtÞ v s m1 ðtÞj ¼ juj s ðt; xm ðv sþ1 m ðtÞÞÞ uj s ðt; xm1 ðv sþ1 m1 ðtÞÞÞj 6 kjs jxm ðv sþ1 m ðtÞÞ xm1 ðv sþ1 m1 ðtÞÞj ð2Þ
ðjÞ
ðjÞ
ð2Þ
ðjÞ
ðjÞ
6 kjs jxm ðv sþ1 m ðtÞÞ xm ðv sþ1 m1 ðtÞÞj þ kjs jxm ðv sþ1 m1 ðtÞÞ xm1 ðv sþ1 m1 ðtÞÞj ! kakBC 1 ðRþ Þ ðjÞ ð2Þ ðjÞ m jv sþ1 m ðtÞ v sþ1 m1 ðtÞj : 6 kjs kakBC 1 ðRþ Þ q þ 1d
ð42Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
486
ð2Þ
ð2Þ
From (42) and by using Lemma 1 with as ¼ kjs kakBC 1 ðRþ Þ qm and bs ¼ kjs kakBC 1 ðRþ Þ =ð1 dÞ, we have that for 1 6 s 6 mj
jv
ðjÞ 1 m ðtÞ
v
ðjÞ 1 m1 ðtÞj
6 kakBC 1 ðRþ Þ q
m
s1 kak 1 X BC ðRþ Þ
!j1 1
1d
j1 ¼1
j1 Y
ð2Þ kji
þ jv
ðjÞ s m ðtÞ
v
ðjÞ s m1 ðtÞj
kakBC 1 ðRþ Þ
i¼1
!s1
1d
s1 Y
ð2Þ
ð43Þ
kji :
i¼1
By choosing s ¼ mj in (43), applying (6) and (32) and hypothesis (40) we get that for each j 2 f1; . . . ; kg and every m 2 N mj 1
jv
ðjÞ 1 m ðtÞ
v
ðjÞ 1 m1 ðtÞj
6 kakBC 1 ðRþ Þ q
X kakBC 1 ðRþ Þ
m
1d
j1 ¼1
X kakBC 1 ðRþ Þ
6 kakBC 1 ðRþ Þ q
m
!j1 1
1d
j1 ¼1
ð2Þ kji
þ jv
ðjÞ mj m ðtÞ
v
ðjÞ mj m1 ðtÞj
kakBC 1 ðRþ Þ 1d
j1 Y
ð2Þ
ð2Þ
kji þ kjmj jxm ðtÞ xm1 ðtÞj
!j1 1
1d
j1 Y
!mj 1 m 1 j Y ð2Þ k1i i¼1
!m 1 1 j kakBC 1 ðRþ Þ j mY
i¼1
mj X kakBC 1 ðRþ Þ j1 ¼1
j1 Y i¼1
mj 1
6 kakBC 1 ðRþ Þ qm
!j1 1
1d
ð2Þ
kji
i¼1
ð2Þ
ð44Þ
kji :
i¼1
Similarly to (44) is proved that for each i 2 f1; . . . ; lg and every m 2 N ðiÞ ju1 m ðtÞ
ðiÞ u1 m1 ðtÞj
6 kakBC 1 ðRþ Þ q
m
li X kakBC 1 ðRþ Þ j¼1
!j1
1d
j Y ð4Þ kij :
ð45Þ
i¼1
From (8), (41), (44) and (45), it follows that
0 1 !j j mj k 1 X X kakBC 1 ðRþ Þ 1 Y ð2Þ A @ jxmþ1 ðtÞ xm ðtÞj 6 kakBC 1 ðRþ Þ q Gj ðtÞ 1 þ kji 1d j¼1 j1 ¼1 i¼1 0 1 !j1 li j l X X Y kakBC 1 ðRþ Þ ð4Þ m Gkþi ðtÞ@1 þ ~L kij A 6 kakBC 1 ðRþ Þ qmþ1 : þ kakBC 1 ðRþ Þ q 1 d j¼1 i¼1 i¼1 m
ð46Þ
From (4), (5), (27), (32), (40), (44) and (45), it follows that
jx0mþ1 ðtÞ x0m ðtÞj 6
1 k X X ðjÞ ðjÞ jA1 jiþ1 cj ðt þ i; t þ iÞjxm ðv 1 m ðt þ iÞÞ xm ðv 1 m1 ðt þ iÞÞj i¼0
j¼1
1 k X X ðjÞ ðjÞ þ jA1 jiþ1 cj ðt þ i; t þ iÞjxm ðv 1 m1 ðt þ iÞÞ xm1 ðv 1 m1 ðt þ iÞÞj i¼0
j¼1
1 l X X ðpÞ ðpÞ þ jA1 jiþ1 ckþp ðt þ i; t þ iÞjx0m ðu1 m ðt þ iÞÞ x0m ðu1 m1 ðt þ iÞÞj p¼1
i¼0
1 l X X ðpÞ ðpÞ þ jA1 jiþ1 ckþp ðt þ i; t þ iÞjx0m ðu1 m1 ðt þ iÞÞ x0m1 ðu1 m1 ðt þ iÞÞj p¼1
i¼0
6 kakBC 1 ðRþ Þ q
m
0
k X
Cj ðtÞ@1 þ
j¼1
þ kakBC 1 ðRþ Þ q
m
l X
0
mj X kakBC 1 ðRþ Þ
1d
j1 ¼1
j1 Y
1 ð2Þ kji A
i¼1
li X
kakBC 1 ðRþ Þ
j¼1
1d
Ckþi ðtÞ@1 þ ~L
i¼1
!j 1
!j1
j Y
1 ð4Þ kij A
6 kakBC 1 ðRþ Þ qmþ1 :
ð47Þ
i¼1
From (38), (39), (46) and (47) and the induction we obtain that (40) hold for every m 2 N0 and for all t 2 Rþ . ð2Þ ð4Þ Now note that for sufficiently small kj1 , j ¼ 1; k; ki1 ; i ¼ 1; l,
0
d@1 þ
max
16j6k;16i6l
8 !j j mj <X 1 kakBC 1 ðRþ Þ 1 Y :j
1 ¼1
li X kakBC 1 ðRþ Þ ð2Þ kji ; ~L 1d i¼1 j¼1
1d
!j1
91 j = Y ð4Þ kij A < 1; ; i¼1
that is, q 2 ð0; 1Þ. This with (40) implies the uniform convergence of ðxm ðtÞÞm2N0 and ðx0m ðtÞÞm2N0 on Rþ . Taking the limit in (26), (27), (32) and (35) we get that
xðtÞ :¼ lim xm ðtÞ m!þ1
1
is a C solution of (25) on Rþ such that
jxðtÞj 6
kakBC 1 ðRþ Þ 1d
;
jx0 ðtÞj 6
kakBC 1 ðRþ Þ 1d
ð48Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
487
and
jx0 ðtÞ x0 ðsÞj 6 ~Ljt sj:
ð49Þ 1
Assume (25) has another BC ðRþ Þ solution, say y. Set
v^ jp ðtÞ ¼ ujp ðt; yðuj pþ1 ðt; . . . yðuj m ðt; yðtÞÞÞ . . .ÞÞÞ;
j ¼ 1; k
j
and
^ jp ðtÞ ¼ wjp ðt; yðwj pþ1 ðt; . . . yðwj m ðt; yðtÞÞÞ . . .ÞÞÞ; u j
j ¼ 1; l:
From (4), (5), (6) and (48), we have
jxðtÞ yðtÞj 6
Z 1 X jA1 jiþ1
k X
t
i¼0
1
Z
1
t
cj ðs þ i; s þ iÞjxðv j1 ðs þ iÞÞ xðv^ j1 ðs þ iÞÞjds þ
j¼1
1 X jA1 jiþ1 i¼0
Z 1 X cj ðs þ i; s þ iÞjxðv^ j1 ðs þ iÞÞ yðv^ j1 ðs þ iÞÞjds þ jA1 jiþ1
k X j¼1
i¼0
^ p1 ðs þ iÞÞjÞds þ x0 ð u
Z 1 X jA1 jiþ1
1
t
l X
ckþp ðs þ i; s þ iÞjx0 ðup1 ðs þ iÞÞ
p¼1
l X
ckþp ðs þ i; s þ iÞjx0 ðu^ p1 ðs þ iÞÞ y0 ðu^ p1 ðs þ iÞÞjÞds
t
i¼0
1
p¼1
Z 1X kþl 1 k X kakBC 1 ðRþ Þ X Gj ðtÞ þ jA1 jiþ1 cj ðs þ i; s þ iÞjv j1 ðs þ iÞ v^ j1 ðs þ iÞjds 1 d i¼0 t j¼1 j¼1 Z 1X 1 l X þ ~L jA1 jiþ1 ckþp ðs þ i; s þ iÞjup1 ðs þ iÞ u^ p1 ðs þ iÞjds:
6 kx ykC 1 ðRþ Þ
t
i¼0
ð50Þ
p¼1
Using Lemma 1, similar to (44), we obtain
jv j1 ðtÞ v^ j1 ðtÞj 6 kx ykC1 ðR
mj X kakBC 1 ðRþ Þ
þÞ
j1 ¼1
1d
li X kakBC 1 ðRþ Þ
!j1
1d
j¼1
j1 Y ð2Þ kji
ð51Þ
i¼1
and also
^ i1 ðtÞj 6 kx ykC1 ðR Þ jui1 ðtÞ u þ
!j1 1
j Y
ð4Þ
ð52Þ
kij :
i¼1
Using (51) and (52) into (50) and then applying condition (8) we get
0
0 11 !j j mj k 1 X X kakBC 1 ðRþ Þ 1 Y ð2Þ jxðtÞ yðtÞj 6 kx ykC 1 ðRþ Þ @ Gj ðtÞ@1 þ kji AA 1d j¼1 j1 ¼1 i¼1 0 1 !j1 li j l X X Y kakBC 1 ðRþ Þ ð4Þ A ~ @ Gkþi ðtÞ 1 þ L kij kx ykC 1 ðRþ Þ þ 1d i¼1 i¼1 j¼1 6 qkx ykC1 ðRþ Þ :
ð53Þ
From (4), (5), (6), (8), (49), (51) and (52), it follows that
jx0 ðtÞ y0 ðtÞj 6
1 k 1 k X X X X jA1 jiþ1 cj ðt þ i; t þ iÞjxðv j1 ðt þ iÞÞ xðv^ j1 ðt þ iÞÞj þ jA1 jiþ1 cj ðt þ i; t þ iÞjxðv^ j1 ðt þ iÞÞ i¼0
j¼1
i¼0
j¼1
1 l X X ^ p1 ðt þ iÞÞj yðv^ j1 ðt þ iÞÞj þ jA1 jiþ1 ckþp ðt þ i; t þ iÞjx0 ðup1 ðt þ iÞÞ x0 ðu i¼0
p¼1
1 l X X ^ p1 ðt þ iÞÞ y0 ðu ^ p1 ðt þ iÞÞj jA1 jiþ1 ckþp ðt þ i; t þ iÞjx0 ðu þ p¼1 0 0 11 !j j mj k 1 X X kakBC 1 ðRþ Þ 1 Y ð2Þ kji AA 6 kx ykC 1 ðRþ Þ @ Cj ðtÞ@1 þ 1d j1 ¼1 i¼1 j¼1 0 1 !j1 li j l X X Y kakBC 1 ðRþ Þ ð4Þ A ~ @ 6 qkx ykC1 ðRþ Þ : þ kx ykC1 ðRþ Þ Ckþi ðtÞ 1 þ L kij 1d i¼1 j¼1 i¼1 i¼0
Thus kx ykC 1 ðRþ Þ 6 qkx ykC 1 ðRþ Þ , which with q 2 ð0; 1Þ implies xðtÞ ¼ yðtÞ. h
ð54Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
488
References [1] R. Bellman, K. Cooke, Differential-Difference Equations, Acad. Press, New York, 1963. [2] K. Berenhaut, J. Foley, S. Stevic´, The global attractivity of the rational difference equation yn ¼ 1 þ ðynk =ynm Þ, Proc. Am. Math. Soc. 135 (2007) 1133– 1140. [3] N.I. Blashchak, G.P. Pelyukh, On the existence and uniqueness of periodic solutions of systems of nonlinear differential-functional equations with linear deviation of an argument, Dopov. Nats. Akad. Nauk Ukr. 8 (1997) 10–13. [4] M.A. Cruz, J.K. Hale, Existence, uniqueness, and continuous dependence for hereditary system, Ann. Mat. Pura Appl. 85 (4) (1970) 63–82. [5] B.P. Demidovich, Lectures on the Mathematical Theory of Stability, Nauka, Moscow, 1970. [6] J. Diblik, Asymptotic behaviour of solutions of a differential equation partly solved with respect to its derivative, Sib. Math. J. 23 (5) (1982) 80–91. [7] J. Diblik, Solutions of a singular system of differential equations which is not solved with respect to the derivatives, Demonstratio Math. 17 (4) (1984) 1031–1041 (in Russian). [8] J. Diblik, On conditional stability of the solutions of the systems of differential equations, not solved with respect to the derivatives, Sb.VUT 1–4 (1985) 5–9. [9] J. Diblik, The existence of solution of the singular Cauchy problem for systems of differential equations that are not solved for the derivatives, Publ. Inst. Math. 37 (51) (1985) 73–80. [10] J. Diblik, Functional Differential Equations, University of Zˇilina, 2008. [11] J. Diblik, Z. Svoboda, Z. Šmarda, Retract principle for neutral functional differential equations, Nonlinear Anal. 71 (2009) 1393–1400. [12] R.D. Driver, Existence and continuous dependence of solutions of a neutral functional-differential equation, Arch. Rat. Mech. Anal. 19 (2) (1965) 149– 166. [13] L.J. Grimm, Existence and continuous dependence for a class of nonlinear neutral-differential equation, Proc. Am. Math. Soc. 29 (3) (1971) 467–473. [14] J. Hale, Theory of Functional Differential Equations, Springer, Berlin, 1977. [15] B. Iricˇanin, S. Stevic´, Some systems of nonlinear difference equations of higher order with periodic solutions, Dyn. Contin. Discrete Impuls. Syst. 13a (3– 4) (2006) 499–508. [16] M. Kwapisz, On the existence and uniqueness of solutions of certain integral-functional equation, Ann. Pol. Math. 31 (1) (1975) 23–41. [17] O.P. Oliinychenko, Asymptotic properties of solutions of linear functional differential equations, Nelin. Kolyv. 3 (4) (2000) 489–496. [18] G.P. Pelyukh, On a representation of solutions of nonlinear differential-functional equations in the neighborhood of singular points, Differ. Uravn. 22 (9) (1986) 1628–1630. [19] G.P. Pelyukh, On the structure of the set of solutions for one class of systems of nonlinear differential-functional equations of neutral type, Nonlinear Oscil. 5 (1) (2002) 53–59. [20] G.P. Pelyukh, Asymptotic properties of solutions to systems of nonlinear functional-differential equations, Differ. Uravn. 39 (1) (2003) 46–50. [21] H.P. Pelyukh, O.P. Oliinychenko, Asymptotic properties of global solutions of systems of functional differential equations of neutral type with nonlinear deviations of an argument, Nonlinear Oscil. 5 (4) (2002) 489–503. [22] H.P. Pelyukh, A.V. Vel’hach, On the structure of the set of continuously differentiable solutions of one boundary-value problem for a system of functional differential equations of neutral type, Nonlinear Oscil. 10 (2) (2007) 278–291. [23] A.M. Samoilenko, G.P. Pelyukh, Solutions of systems of nonlinear functional differential equations bounded on the entire real axis and their properties, Ukr. Mat. Zh. 46 (6) (1994) 737–747. [24] Yu.D. Shlapak, On periodic solutions of nonlinear second order differential equations, not solved with respect to the highest derivative, Ukr. Math. J. 26 (6) (1974) 850–854. [25] S. Stevic´, On stability results for a new approximating fixed points iteration process, Demonstratio Math. 34 (4) (2001) 873–880. [26] S. Stevic´, A global convergence results with applications to periodic solutions, Indian J. Pure Appl. Math. 33 (1) (2002) 45–53. Q Q2ðkþ1Þ [27] S. Stevic´, On the recursive sequence xnþ1 ¼ A= ki¼0 xni þ 1= j¼kþ2 xnj , Taiwanese J. Math. 7 (2) (2003) 249–259. [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]
S. Stevic´, On the recursive sequence xnþ1 ¼ an þ ðxn1 =xn Þ II, Dyn. Contin. Discrete Impuls. Syst. 10a (6) (2003) 911–917. S. Stevic´, Stability results for /-strongly pseudocontractive mappings, Yokohama Math. J. 50 (2003) 71–85. S. Stevic´, Approximating fixed points of strongly pseudocontractive mappings by a new iteration method, Appl. Anal. 84 (1) (2005) 89–102. S. Stevic´, Approximating fixed points of nonexpansive mappings by a new iteration method, Bull. Inst. Math. Acad. Sin. 1 (3) (2006) 437–450. S. Stevic´, On the recursive sequence xnþ1 ¼ maxfc; xpn =xpn1 g, Appl. Math. Lett. 21 (8) (2008) 791–796. S. Stevic´, Boundedness character of a class of difference equations, Nonlinear Anal. TMA 70 (2009) 839–848. S. Stevic´, Global stability of a difference equation with maximum, Appl. Math. Comput. 210 (2009) 525–529. S. Stevic´, Periodicity of max difference equations, Util. Math. 83 (2010) 69–71. S. Stevic´, On a nonlinear generalized max-type difference equation, J. Math. Anal. Appl. 376 (2011) 317–328. S. Stevic´, On some solvable systems of difference equations, Appl. Math. Comput. 218 (2012) 5010–5018. S. Stevic´, Bounded solutions of some systems of nonlinear functional differential equations with iterated deviating argument, Appl. Math. Comput. 218 (2012) 10429–10434. S. Stevic´, Existence of bounded solutions of some systems of nonlinear functional differential equations with complicated deviating argument, Appl. Math. Comput. 218 (2012) 9974–9979. S. Stevic´, Globally bounded solutions of a system of nonlinear functional differential equations with iterated deviating argument, Appl. Math. Comput. 219 (2012) 2180–2185. S. Stevic´, Local existence of Lipschitz-continuous solutions of systems of nonlinear functional equations with iterated deviations, Appl. Math. Comput. 218 (2012) 10542–10547. S. Stevic´, On a third-order system of difference equations, Appl. Math. Comput. 218 (2012) 7649–7654. S. Stevic´, On continuous solutions of a class of systems of nonlinear functional difference equations with deviating argument, Appl. Math. Comput. 218 (2012) 10188–10193. S. Stevic´, On solutions of a class of systems of nonlinear functional differential equations of neutral type with complicated deviations of an argument, Appl. Math. Comput. 219 (2012) 3693–3700. S. Stevic´, Unique existence of bounded continuous solutions on the real line of a class of nonlinear functional equations with complicated deviations, Appl. Math. Comput. 218 (2012) 7813–7817. S. Stevic´, Existence of solutions bounded together with their first derivatives of some systems of nonlinear functional differential equations, Appl. Math. Comput. 219 (2013) 6457–6465. S. Stevic´, On bounded continuously differentiable solutions with Lipschitz first derivatives of a system of functional differential equations, Appl. Math. Comput. 219 (2013) 6344–6353. A.V. Vel’hach, Asymptotic properties of solutions of systems of nonlinear functional differential equations of neutral type, Nonlinear Oscil. 12 (1) (2009) 19–26. L.A. Zhivotovskii, On the existence and uniqueness of solutions of differential equations with delay dependent on a solution and its derivative, Differ. Uravn. 5 (5) (1969) 880–889.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Existence of bounded solutions of a class of neutral systems of functional differential equations Stevo Stevic´ ⇑ Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
a r t i c l e
i n f o
a b s t r a c t Some results on the existence of bounded solutions together with their first derivatives of a class of neutral systems of functional differential equations with complicated deviations, which extend and unify numerous results in the literature, are proved. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Bounded C 1 solutions System of functional differential equations Iterated deviations Lipschitz derivative
1. Introduction and preliminaries Special cases of the following system of functional differential equations, which is partially solved with respect to the first derivatives of dependent variables,
x0 ðt þ 1Þ ¼ Ax0 ðtÞ þ Uðt; xðtÞ; xðf1 ðt; xðtÞÞÞ; x0 ðf2 ðt; xðtÞÞÞÞ; N 3
N
ð1Þ
N
where t 2 Rþ ¼ ½0; 1Þ; U : Rþ ðR Þ ! R ; f i : Rþ R ! Rþ ; i ¼ 1; 2, have attracted some attention among the experts in the research field (see, for example, [1,14,19,20,22,23,44,47,48]). For some other results on systems/equations not solved with respect to the highest-order derivatives, see, for example, [3–13,16–18,21,24,38,40,46,49]. Based on the idea of iterations of some iterative processes (see, for example, [2,15,25–37,42]) in [38–41,43–47], we proposed the investigation of various types of systems/equations with continuous arguments, whose deviations of an argument depend on an unknown function which depend also of the function and so on, so called, iterated deviations. Motivated by the line of investigations in the papers [4,13,16,17,21,22,38,39,44,46–48], here we investigate the existence of bounded C 1 solutions of the next system of functional differential equations ð1Þ
ðkÞ
ð1Þ
ðlÞ
x0 ðt þ 1Þ ¼ Ax0 ðtÞ þ Uðt; xðv 1 ðtÞÞ; . . . ; xðv 1 ðtÞÞ; x0 ðu1 ðtÞÞ; . . . ; x0 ðu1 ðtÞÞÞ;
ð2Þ
on Rþ , where
v rðjÞ ðtÞ ¼ ujr ðt; xðuj rþ1 ðt; . . . xðujm ðt; xðtÞÞÞ . . .ÞÞÞ; j
upðiÞ ðtÞ ¼ wip ðt; xðwi pþ1 ðt; . . . xðwili ðt; xðtÞÞÞ . . .ÞÞÞ; kþl
j ¼ 1; k; r ¼ 1; mj ; i ¼ 1; l; p ¼ 1; li , U : Rþ ðRN Þ ! RN , ujr ; wip : Rþ RN ! Rþ ; j ¼ 1; k, r ¼ 1; mj ; i ¼ 1; l; p ¼ 1; li ; A is a nonsingular matrix, extending and unifying numerous results in the literature. We use also the following convention ðiÞ v ðjÞ m þ1 ðtÞ ¼ ul þ1 ðtÞ ¼ t; j
i
j ¼ 1; k; i ¼ 1; l:
⇑ Address: Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.151
ð3Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
479
As usual, by CðRþ Þ we denote the space of continuous vector functions on Rþ , while by C 1 ðRþ Þ the space of all continuously differentiable vector functions on Rþ . The subspace of C 1 ðRþ Þ consisting of all bounded vector functions together with their first derivatives on Rþ is denoted by BC 1 ðRþ Þ. The norm on BC 1 ðRþ Þ is
kxkBC 1 ðRþ Þ
( ) 0 0 ¼ max kxk1 ; kx k1 ¼ max supjxðtÞj; supjx ðtÞj ; t2Rþ
t2Rþ
where for y 2 RN ; jyj denotes a norm on RN . The following folklore lemma, which can be found, for example, in [46], will be frequently applied in the proofs of our main results. Lemma 1. Assume that ðan Þn2N and ðbn Þn2N are two sequences of nonnegative numbers, and that sequence ðxn Þn2N satisfies the inequality
xn 6 an þ bn xnþ1 ;
n 2 N:
Then
x1 6
j1 k1 Y k1 X Y aj bi þ xk bi ; j¼1
k 2 N:
i¼1
i¼1
2. Main results First, we give a list of some conditions which will be used in the formulations of the main results in this paper. (a) Vector function Uðt; x1 ; . . . ; xkþl Þ is continuous for t 2 Rþ ; xj 2 RN ; j ¼ 1; k þ l,
Uðt; 0; . . . ; 0Þ 0;
ð4Þ
jUðt; x01 ; . . . ; x0kþl Þ Uðs; x001 ; . . . ; x00kþl Þj 6 c0 ðt; sÞjt sj þ
kþl X
cj ðt; sÞjx0j x00j j;
ð5Þ
j¼1
where cj ðt; sÞ; j ¼ 0; k þ l are continuous and nonnegative functions for t; s 2 Rþ , and x0j ; x00j 2 RN , j ¼ 1; k þ l; (b) ujr ðt; xÞ; j ¼ 1; k; r ¼ 1; mj , and wip ðt; xÞ; i ¼ 1; l; p ¼ 1; li , are continuous and nonnegative functions for t 2 Rþ and x 2 RN , and ð1Þ
ð2Þ
j ¼ 1; k; r ¼ 1; mj ;
ð3Þ
ð4Þ
i ¼ 1; l; p ¼ 1; li ;
jujr ðt; xÞ ujr ðs; yÞj 6 kjr jt sj þ kjr jx yj; jwip ðt; xÞ wip ðs; yÞj 6 kip jt sj þ kip jx yj; N
for every t; s 2 Rþ , and x; y 2 R , and for some positive constants (c) for every j ¼ 0; k þ l, the series
Cj ðt; sÞ ¼
i¼0
i¼0
ð7Þ
ð1Þ kjr ;
Z 1 1 X X jA1 jiþ1 cj ðt þ i; s þ iÞ and Gj ðtÞ ¼ jA1 jiþ1
t
ð6Þ
ð2Þ kjr ;
j ¼ 1; k; r ¼ 1; mj ;
ð3Þ kip ;
ð4Þ kip ;
i ¼ 1; l; p ¼ 1; li ;
1
cj ðs þ i; s þ iÞds;
converge uniformly for t; s 2 Rþ , and for some d 2 ð0; 1Þ, satisfy the condition
(
max
sup
kþl X
kþl X Cj ðt; sÞ; sup Gj ðtÞ
t;s2Rþ j¼0
)
6 d:
ð8Þ
t2Rþ j¼0
Theorem 1. Suppose that conditions (a)–(c) hold. Then for any BC 1 ðRþ Þ solution of system (2), such that
lim jxðt þ 1Þ AxðtÞj ¼ 0;
ð9Þ
t!þ1
and
jx0 ðtÞ x0 ðsÞj 6 Ljt sj
ð10Þ 1
for every t; s 2 Rþ and some L > 0, there is a C vector function a with the Lipschitz first derivative and such that
aðt þ 1Þ ¼ AaðtÞ;
ð11Þ
lim jxðtÞ aðtÞj ¼ 0:
ð12Þ
t!þ1
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
480
Proof. Assume x 2 BC 1 ðRþ Þ is a solution of system (2) such that (9) and (10) hold. Set
Z 1 X iþ1 ðA1 Þ
aðtÞ :¼ xðtÞ
1
ðlÞ 0 Uðs þ i; xðv 1ð1Þ ðs þ iÞÞ; . . . ; xðv 1ðkÞ ðs þ iÞÞ; x0 ðuð1Þ 1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds:
t
i¼0
ð13Þ
Using (4) and (5) it is not difficult to see that
Z 1 X 1 ð1Þ ðkÞ ð1Þ ðlÞ 1 iþ1 0 0 ðA Þ U ð s þ i; xð v ð s þ iÞÞ; . . . ; xð v ð s þ iÞÞ; x ðu ð s þ iÞÞ; . . . ; x ðu ð s þ iÞÞÞd s 1 1 1 1 i¼0 t 6
1 X
jA1 jiþ1
Z
1
k X
Z 1 X jA1 jiþ1
j¼1
i¼0
cj ðs þ i; s þ iÞjxðv ðjÞ 1 ðs þ iÞÞjds þ
t
i¼0
6 kxkBC 1 ðRþ Þ
t
1
l X
ckþp ðs þ i; s þ iÞjx0 ðu1ðpÞ ðs þ iÞÞjds
p¼1
kþl X Gj ðtÞ;
ð14Þ
j¼1
which means that the sum in (13) is convergent. Thus, aðtÞ is well-defined on Rþ . Clearly, (13) along with (14) implies
jxðtÞ aðtÞj 6 kxkBC 1 ðRþ Þ
kþl X Gj ðtÞ ! 0;
as t ! þ1;
ð15Þ
j¼1
that is, (12) holds, where the last limit follows from the uniform convergence of series Gj ðtÞ; j ¼ 1; k þ l, as sums of functions in CðRþ Þ tending to zero as t ! þ1. Moreover, due to the uniform convergence of Cj ðt; tÞ and Gj ðtÞ; j ¼ 1; k þ l, relation (13) can be differentiated on Rþ , so that
a0 ðtÞ ¼ x0 ðtÞ þ
1 X iþ1 ð1Þ ðkÞ ð1Þ ðlÞ ðA1 Þ Uðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ:
ð16Þ
i¼0
If we integrate (2) from t to t0 , then let t0 ! þ1, and employ (9), we have that
xðt þ 1Þ ¼ AxðtÞ
Z
1
ðkÞ ð1Þ ðlÞ 0 0 Uðs; xðv ð1Þ 1 ðsÞÞ; . . . ; xðv 1 ðsÞÞ; x ðu1 ðsÞÞ; . . . ; x ðu1 ðsÞÞÞds:
t
ð17Þ
Using (17) we have
aðt þ 1Þ ¼ xðt þ 1Þ ¼ AxðtÞ
Z 1 X iþ1 ðA1 Þ
Z 1 X i ðA1 Þ t
i¼1
¼ A xðtÞ
ðkÞ ð1Þ ðlÞ 0 0 Uðs þ i;xðv ð1Þ 1 ðs þ iÞÞ; .. .; xðv 1 ðs þ iÞÞ;x ðu1 ðs þ iÞÞ;. .. x ðu1 ðs þ iÞÞÞds
ð1Þ ðlÞ 0 0 Uðs; xðv 1ð1Þ ðsÞÞ;. . .; xðv ðkÞ 1 ðsÞÞ;x ðu1 ðsÞÞ; .. .; x ðu1 ðsÞÞÞds
t
tþ1
i¼0 1
Z
1
1
ð1Þ ðlÞ 0 0 Uðs þ i; xðv 1ð1Þ ðs þ iÞÞ;. .. ;xðv ðkÞ 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ;. . .; x ðu1 ðs þ iÞÞÞds
Z 1 X iþ1 ðA1 Þ
1 t
i¼0
! ðkÞ ð1Þ ðlÞ 0 0 Uðs þ i; xðv ð1Þ 1 ðs þ iÞÞ;. .. ;xðv 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ;. .. ;x ðu1 ðs þ iÞÞÞds ¼ AaðtÞ;
which means that aðtÞ satisfies equality (11). Employing (10) in (16), we get
ja0 ðtÞ a0 ðsÞj 6 jx0 ðtÞ x0 ðsÞj þ
1 X ð1Þ ðkÞ ð1Þ ðlÞ jA1 jiþ1 jUðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ i¼0
ð1Þ
ðkÞ
ð1Þ
ðlÞ
Uðs þ i; xðv 1 ðs þ iÞÞ; . . . ; xðv 1 ðs þ iÞÞ; x0 ðu1 ðs þ iÞÞ; . . . ; x0 ðu1 ðs þ iÞÞÞj 6 Ljt sj þ
1 1 k X X X ðjÞ ðjÞ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ jA1 jiþ1 cj ðt þ i; s þ iÞjxðv 1 ðt þ iÞÞ xðv 1 ðs þ iÞÞj i¼0
i¼0
j¼1
1 l X X ðpÞ ðpÞ þ jA1 jiþ1 ckþp ðt þ i; s þ iÞjx0 ðu1 ðt þ iÞÞ x0 ðu1 ðs þ iÞÞj p¼1
i¼0
1 1 k X X X ðjÞ ðjÞ 6 Ljt sj þ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ kx0 k1 jA1 jiþ1 cj ðt þ i; s þ iÞjv 1 ðt þ iÞ v 1 ðs þ iÞj i¼0 1 X
þL
i¼0
jA1 jiþ1
i¼0 l X
ðpÞ ckþp ðt þ i; s þ iÞjuðpÞ 1 ðt þ iÞ u1 ðs þ iÞj:
p¼1
j¼1
ð18Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
481
We have that for each j 2 f1; . . . ; kg and every 1 6 m 6 mj , ðjÞ
ðjÞ
ð1Þ
ð2Þ
ðjÞ
ðjÞ
ðjÞ jv ðjÞ m ðtÞ v m ðsÞj ¼ jujm ðt; xðv mþ1 ðtÞÞÞ ujm ðs; xðv mþ1 ðsÞÞÞj 6 kj m jt sj þ kj m jxðv mþ1 ðtÞÞ xðv mþ1 ðsÞÞj ð1Þ
ð2Þ
ðjÞ
ðjÞ
6 kj m jt sj þ kj m kx0 k1 jv mþ1 ðtÞ v mþ1 ðsÞj: Hence by using Lemma 1 with am ¼ ðjÞ
ðjÞ
jv 1 ðtÞ v 1 ðsÞj 6 jt sj
ð1Þ kj m jt
sj and bm ¼
ð19Þ
ð2Þ kj m kx0 k1 ,
we have that
! jY m m 1 1 X Y ð1Þ ð2Þ ð2Þ ðjÞ ðjÞ jv mþ1 ðtÞ v mþ1 ðsÞj; kx0 kj11 1 kjj1 kji þ kx0 km k 1 ji i¼1
j1 ¼1
ð20Þ
i¼1
holds for each j 2 f1; . . . ; kg and every 1 6 m 6 mj . Choosing m ¼ mj in (20) and by using (3), we get
jv
ðjÞ 1 ðtÞ
v
ðjÞ 1 ðsÞj
6
! mj mj jY 1 1 X Y ð2Þ ð2Þ 0 j1 1 ð1Þ 0 mj kx k1 kjj1 kji þ kx k1 kji jt sj: i¼1
j1 ¼1
ð21Þ
i¼1
Analogously, employing (7) and Lemma 1, it is obtained that ðiÞ
ðiÞ
ju1 ðtÞ u1 ðsÞj 6
! li li j1 X Y Y ð3Þ ð4Þ ð4Þ 0 li jt sj: kx0 kj1 k k þ kx k k 1 1 ij ii1 ii1 j¼1
ð22Þ
i1 ¼1
i1 ¼1
From (18), (21) and (22) it follows that 0
0
ja ðtÞ a ðsÞj 6 ðL þ C0 ðt; sÞÞjt sj þ
! mj mj jY 1 1 X Y ð2Þ ð2Þ 0 j1 ð1Þ 0 mj þ1 Cj ðt; sÞ kx k1 kjj1 kji þ kx k1 kji jt sj
k X
j1 ¼1
j¼1
i¼1
i¼1
! li li j1 l X X Y Y ð3Þ ð4Þ ð4Þ 0 li þ L Ckþi ðt; sÞ kx0 kj1 k k þ kx k k jt sj 6 L1 jt sj; 1 ij 1 ii1 ii1 j¼1
i¼1
i1 ¼1
i1 ¼1
where
L1 ¼ sup
L þ C0 ðt; sÞ þ
t;s2Rþ
k X
mj mj jY 1 1 X Y ð1Þ ð2Þ ð2Þ mj þ1 kx0 kj11 kjj1 kji þ kx0 k1 kji
j¼1
j1 ¼1
Cj ðt; sÞ
i¼1
li li j1 l X X Y Y ð3Þ ð4Þ ð4Þ 0 li þ L Ckþi ðt; sÞ kx0 kj1 k k þ kx k kii1 1 1 ij ii1 j¼1
i¼1
(L1 is finite due to (8)).
i1 ¼1
!!
!
i¼1
;
i1 ¼1
h
Theorem 2. Suppose that conditions (a)–(c) hold, M :¼ supn2N jAn j < 1, and a is a C 1 vector function satisfying relation (11) and
ja0 ðtÞ a0 ðsÞj 6 L1 jt sj
ð23Þ ð2Þ kj1 ;
ð3Þ ki1 ;
for every t; s 2 Rþ and some L1 > 0. Then for sufficiently small j ¼ 1; k; solution with the Lipschitz first derivative, and satisfying conditions (9) and (12).
ð4Þ ki1 ,
1
i ¼ 1; l, system (2) has a unique BC ðRþ Þ
Proof. Suppose x 2 BC 1 ðRþ Þ is a solution of system (2) satisfying (9) and (12). Let
a1 ðtÞ :¼ xðtÞ
Z 1 X iþ1 ðA1 Þ
1
t
i¼0
ð1Þ ðlÞ 0 0 Uðs þ i; xðv 1ð1Þ ðs þ iÞÞ; . . . ; xðv ðkÞ 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds:
The proof of Theorem 1 shows that xðtÞ is also a solution of the system
xðtÞ ¼ a1 ðtÞ þ
Z 1 X iþ1 ðA1 Þ i¼0
t
1
ðlÞ 0 Uðs þ i; xðv 1ð1Þ ðs þ iÞÞ; . . . ; xðv 1ðkÞ ðs þ iÞÞ; x0 ðuð1Þ 1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds:
Moreover, a1 satisfies relation (11), and
lim ðxðtÞ a1 ðtÞÞ ¼ 0:
t!þ1
Let s 2 ½0; 1Þ. Then, from (12) and (24) it follows that
jaðsÞ a1 ðsÞj ¼ jAn ðaðs þ nÞ a1 ðs þ nÞÞj 6 Mðjaðs þ nÞ xðs þ nÞj þ jxðs þ nÞ a1 ðs þ nÞjÞ ! 0;
ð24Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
482
as n ! þ1, that is, aðsÞ ¼ a1 ðsÞ. From this and (11) we have that aðsÞ ¼ a1 ðsÞ for every s 2 Rþ . Hence
xðtÞ ¼ aðtÞ þ
Z 1 X iþ1 ðA1 Þ
1
t
i¼0
ðkÞ ð1Þ ðlÞ 0 0 Uðs þ i; xðv ð1Þ 1 ðs þ iÞÞ; . . . ; xðv 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds:
ð25Þ
If x 2 BC 1 ðRþ Þ is a solution of system (25), then it is a solution of system (2) satisfying (9) and (12). To show this, first note that (14) implies the uniform convergence of the series
Z 1 X iþ1 ðA1 Þ
1
ðkÞ ð1Þ ðlÞ 0 0 Uðs þ i; xðv ð1Þ 1 ðs þ iÞÞ; . . . ; xðv 1 ðs þ iÞÞ; x ðu1 ðs þ iÞÞ; . . . ; x ðu1 ðs þ iÞÞÞds;
t
i¼0
on Rþ , as well as relation (12). From (4) and (5), we get
X 1 ð1Þ ðkÞ ð1Þ ðlÞ 1 iþ1 ðA Þ Uðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ i¼0 ! 1 k l kþl X 1 iþ1 X X X ðjÞ ðpÞ 0 6 jA j cj ðt þ i; t þ iÞjxðv 1 ðt þ iÞÞj þ ckþp ðt þ i; t þ iÞjx ðu1 ðt þ iÞÞj 6 kxkBC1 ðRþ Þ Cj ðt; tÞ: i¼0
p¼1
j¼1
j¼1
This and the uniform convergence of Cj ðt; tÞ, j ¼ 1; k þ l, implies the uniform convergence of the series 1 X iþ1 ð1Þ ðkÞ ð1Þ ðlÞ ðA1 Þ Uðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ: i¼0
Hence the differentiation of the series in (25) is allowed, and we get 1 X iþ1 ð1Þ ðkÞ ð1Þ ðlÞ ðA1 Þ Uðt þ i; xðv 1 ðt þ iÞÞ; . . . ; xðv 1 ðt þ iÞÞ; x0 ðu1 ðt þ iÞÞ; . . . ; x0 ðu1 ðt þ iÞÞÞ:
x0 ðtÞ ¼ a0 ðtÞ
i¼0
Using this and the relation a0 ðt þ 1Þ ¼ Aa0 ðtÞ, which follows by differentiating (11), we see that xðtÞ is a solution of system (2). Since
Z jxðt þ 1Þ AxðtÞj ¼
1 t
ðkÞ ð1Þ ðlÞ 0 0 Uðs; xðv ð1Þ 1 ðsÞÞ; . . . ; xðv 1 ðsÞÞ; x ðu1 ðsÞÞ; . . . ; x ðu1 ðsÞÞÞds 6 kxkBC 1 ðRþ Þ
kþl X Gj ðtÞ ! 0; j¼1
as t ! þ1, we get that condition (9) holds. Due to just proved equivalence we will consider system of equations (25) instead of (2). Now we define two sequences ðxm ðtÞÞm2N0 and ðx0m ðtÞÞm2N0 , as follows
x0 ðtÞ ¼ aðtÞ;
x00 ðtÞ ¼ a0 ðtÞ;
xm ðtÞ ¼ aðtÞ þ
Z 1 X iþ1 ðA1 Þ
1
t
i¼0
ðkÞ ð1Þ 0 Uðs þ i; xm1 ðv ð1Þ 1 m1 ðs þ iÞÞ; . . . ; xm1 ðv 1 m1 ðs þ iÞÞ; xm1 ðu1 m1 ðs þ iÞÞ; . . . ;
ðlÞ
x0m1 ðu1 m1 ðs þ iÞÞÞds; x0m ðtÞ ¼ a0 ðtÞ
ð26Þ
1 X iþ1 ð1Þ ðkÞ ð1Þ ðA1 Þ Uðt þ i; xm1 ðv 1 m1 ðt þ iÞÞ; . . . ; xm1 ðv 1 m1 ðt þ iÞÞ; x0m1 ðu1 m1 ðt þ iÞÞ; . . . ; i¼0
ðlÞ
x0m1 ðu1 m1 ðt þ iÞÞÞ;
ð27Þ
where
v ðjÞ rm ðtÞ ¼ ujr ðt; xm ðuj rþ1 ðt; . . . xm ðuj m ðt; xm ðtÞÞÞ . . .ÞÞÞ; j
j ¼ 1; k; r ¼ 1; mj ;
and
uðiÞ pm ðtÞ ¼ wip ðt; xm ðwi pþ1 ðt; . . . xm ðwi li ðt; xm ðtÞÞÞ . . .ÞÞÞ;
i ¼ 1; l; p ¼ 1; li :
For every t 2 Rþ we have
jx0 ðtÞj 6 kakBC 1 ðRþ Þ 6
kakBC 1 ðRþ Þ 1d
and jx00 ðtÞj 6 kakBC 1 ðRþ Þ 6
Assume that the following inequalities hold for an m 2 N
kakBC 1 ðRþ Þ 1d
:
ð28Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
jxm1 ðtÞj 6
kakBC 1 ðRþ Þ
and jx0m1 ðtÞj 6
1d
kakBC 1 ðRþ Þ
t 2 Rþ :
;
1d
483
ð29Þ
Then (4), (5), (8), (26) and (29), imply
jxm ðtÞj 6 jaðtÞj þ
Z 1 X jA1 jiþ1
1
t
i¼0
ðkÞ ðlÞ 0 Uðs þ i; xm1 ðv ð1Þ 1 m1 ðs þ iÞÞ; . . . ; xm1 ðv 1 m1 ðs þ iÞÞ; . . . ; xm1 ðu1 m1 ðs þ iÞÞÞjds
1 k Z X X 6 jaðtÞj þ jA1 jiþ1 i¼0
þ
1 X
jA1 j
p¼1
i¼0
1
t
cj ðs þ i; s þ iÞjxm1 ðv ðjÞ 1 m1 ðs þ iÞÞjds
t
j¼1
l Z X iþ1
1
ckþp ðs þ i; s þ iÞjx0m1 ðu1ðpÞm1 ðs þ iÞÞjds
kþl kakBC 1 ðRþ Þ X kakBC 1 ðRþ Þ Gj ðtÞ 6 : 1 d j¼1 1d
6 kakBC 1 ðRþ Þ þ
ð30Þ
From (4), (5), (8), (27) and (29), it follows that
jx0m ðtÞj 6 ja0 ðtÞj þ
1 X
jA1 jiþ1 jUðt þ i; xm1 ðv 1 m1 ðt þ iÞÞ; .. .; xm1 ðv 1 m1 ðt þ iÞÞ; x0m1 ðu1 m1 ðt þ iÞÞ;. .. ;x0m1 ðu1 m1 ðt þ iÞÞÞj ð1Þ
ðkÞ
ð1Þ
ðlÞ
i¼0
6 kakBC 1 ðRþ Þ þ
1 k 1 l X X X X ðjÞ ðpÞ jA1 jiþ1 cj ðt þ i; t þ iÞjxm1 ðv 1 m1 ðt þ iÞÞj þ jA1 jiþ1 ckþp ðt þ i;t þ iÞjx0m1 ðu1 m1 ðt þ iÞÞj i¼0
6 kakBC 1 ðRþ Þ þ
j¼1
1d
p¼1
i¼0
kþl kakBC 1 ðRþ Þ X
Cj ðt; tÞ 6
kakBC 1 ðRþ Þ
j¼1
1d
ð31Þ
:
From (28), (30), (31) and by the induction we have that
jxm ðtÞj 6
kakBC 1 ðRþ Þ
and jx0m ðtÞj 6
1d
kakBC 1 ðRþ Þ
ð32Þ
1d
for every t 2 Rþ and m 2 N0 . By (5), (23), (27) and (32), we have that
jx01 ðtÞ x01 ðsÞj 6 ja0 ðtÞ a0 ðsÞj þ
1 X ð1Þ ðkÞ ð1Þ jA1 jiþ1 jUðt þ i; x0 ðv 1 0 ðt þ iÞÞ; . . . ; x0 ðv 1 0 ðt þ iÞÞ; x00 ðu1 0 ðt þ iÞÞ; . . . ; i¼0
ðlÞ
ð1Þ
ðkÞ
ð1Þ
ðlÞ
x00 ðu1 0 ðt þ iÞÞÞ Uðs þ i; x0 ðv 1 0 ðs þ iÞÞ; . . . ; x0 ðv 1 0 ðs þ iÞÞ; x00 ðu1 0 ðs þ iÞÞ; . . . ; x00 ðu1 0 ðs þ iÞÞÞj 6 ja0 ðtÞ a0 ðsÞj þ
1 1 k X X X ðjÞ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ jA1 jiþ1 cj ðt þ i; s þ iÞjaðv 1 0 ðt þ iÞÞ i¼1
i¼0
j¼1
1 l X X ðjÞ ðpÞ ðpÞ aðv 1 0 ðs þ iÞÞj þ jA1 jiþ1 ckþp ðt þ i; s þ iÞja0 ðu1 0 ðt þ iÞÞ a0 ðu1 0 ðs þ iÞÞj 6 L1 jt sj p¼1
i¼0
1 1 k X X X ðjÞ ðjÞ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ kakBC 1 ðRþ Þ jA1 jiþ1 cj ðt þ i; s þ iÞjv 1 0 ðt þ iÞ v 1 0 ðs þ iÞj þ i¼1
i¼0
j¼1
1 l X X ðpÞ ðpÞ þ L1 jA1 jiþ1 ckþp ðt þ i; s þ iÞju1 0 ðt þ iÞ u1 0 ðs þ iÞj:
ð33Þ
p¼1
i¼0 ðjÞ
ðjÞ
ðpÞ
ðpÞ
ð1Þ
The estimates for jv 1 0 ðt þ iÞ v 1 0 ðs þ iÞj and ju1 0 ðt þ iÞ u1 0 ðs þ iÞj are obtained by Lemma 1 with am ¼ kjm jt sj and ð2Þ ð3Þ ð4Þ bm ¼ kjm kakBC 1 ðRþ Þ , that is, for am ¼ kjm jt sj and bm ¼ kjm kakBC 1 ðRþ Þ . From this and (33), it follows that
jx01 ðtÞ x01 ðsÞj 6 ðL1 þ C0 ðt; sÞÞjt sj þ þ L1
6
l X
li X
i¼1
j¼1
Ckþi ðt; sÞ
L 1 þ C0 þ
Cj
j1 ¼1
j¼1
þ L1
l X i¼1
j¼1
Ckþi
ð3Þ
þ
mj X
j¼1
j1 ¼1
Cj ðt; sÞ
j1 Y i1 ¼1
j
þ
li Y
l
ð4Þ
ð1Þ
1 kakBC k 1 ðR Þ jj1
kii1 þ kakBCi 1 ðR
þÞ
! ð4Þ
kii1
i1 ¼1
mj mj jY 1 1 X Y m þ1 j1 ð1Þ ð2Þ ð2Þ kakBC k kji þ kakBCj 1 ðR Þ kji 1 ðR Þ jj1
k X
li X
kakj1 k BC 1 ðR Þ ij
k X
ð3Þ kj1 k BC 1 ðRþ Þ ij
ka
þ
j1 Y i1 ¼1
þ
i¼1
ð4Þ kii1
li Y
li
þ kakBC 1 ðR
þÞ
i1 ¼1
jY 1 1 i¼1
ð2Þ
m þ1
kji þ kakBCj 1 ðR
þ
! mj Y ð2Þ k jt sj ji Þ i¼1
jt sj !
i¼1
!!
ð4Þ kii1
jt sj 6 ~Ljt sj;
ð34Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
484
where Cj :¼ supt;s2Rþ Cj ðt; sÞ; j ¼ 0; k þ l and
L 1 þ C0 þ ~L ¼ 1
Pk
Pmj
j¼1 Cj
Pl
i¼1 Ckþi
kakBC 1 ðR
j1 ¼1
Pli
j1
ð1Þ Qj1 1 ð2Þ i¼1 kji
kjj1
1d
kakBC 1 ðR
j¼1
þÞ
þÞ
j1
ð3Þ Qj1 ð4Þ i1 ¼1 kii1
kij
1d
kakBC 1 ðR
þ
mj þ1
kakBC 1 ðR
þÞ
li
!
Qm j
ð2Þ
i¼1 kji
1d
þ
þÞ
!
Qli
;
ð4Þ
i1 ¼1 kii1
1d
ð3Þ ð4Þ L is positive. where kii1 ; kii1 ; i ¼ 1; l, i1 ¼ 1; li are chosen such that the denominator of ~ Assume that for some m 2 N and every t; s 2 Rþ
jx0m ðtÞ x0m ðsÞj 6 ~Ljt sj:
ð35Þ
By (5), (27) and (32), we obtain 1 X iþ1 ð1Þ ðkÞ ð1Þ jðA1 Þ Uðt þ i; xm ðv 1 m ðt þ iÞÞ; . . . ; xm ðv 1 m ðt þ iÞÞ; x0m ðu1 m ðt þ iÞÞ; . . . ;
jx0mþ1 ðtÞ x0mþ1 ðsÞj 6 ja0 ðtÞ a0 ðsÞj þ
i¼0 ðlÞ x0m ðu1 m ðt
iþ1
þ iÞÞÞ ðA1 Þ
ðkÞ ð1Þ 0 Uðs þ i; xm ðv ð1Þ 1 m ðs þ iÞÞ; . . . ; xm ðv 1 m ðs þ iÞÞ; xm ðu1 m ðs þ iÞÞ; . . . ;
ðlÞ
x0m ðu1 m ðs þ iÞÞÞj 6 ja0 ðtÞ a0 ðsÞj þ
1 X jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj i¼0
1 k X X ðjÞ ðjÞ jA1 jiþ1 cj ðt þ i; s þ iÞjxm ðv 1m ðt þ iÞÞ xm ðv 1m ðs þ iÞÞj þ i¼0
þ
þ
j¼1
1 X
l X
i¼0
p¼1
jA1 jiþ1
ðpÞ 0 ckþp ðt þ i; s þ iÞjx0m ðuðpÞ 1m ðt þ iÞÞ xm ðu1m ðs þ iÞÞj 6 L1 jt sj
1 1 k X X kakBC 1 ðRþ Þ X ðjÞ ðjÞ jA1 jiþ1 c0 ðt þ i; s þ iÞjt sj þ jA1 jiþ1 cj ðt þ i; s þ iÞjv 1m ðt þ iÞ v 1m ðs þ iÞj 1 d i¼0 i¼0 j¼1
1 l X X ðpÞ ðpÞ þ ~L jA1 jiþ1 ckþp ðt þ i; s þ iÞju1m ðt þ iÞ u1m ðs þ iÞj: i¼0
ð36Þ
p¼1
Since by (32) we have ð1Þ
ð2Þ
ðjÞ
ð1Þ
ð2Þ
kakBC 1 ðRþ Þ
ðjÞ
ðjÞ 0 jv ðjÞ rm ðt þ iÞ v rm ðs þ iÞj 6 kjr jt sj þ kjr kxm k1 jv rþ1 m ðt þ iÞ v rþ1 m ðs þ iÞj
6 kjr jt sj þ kjr
1d
ðjÞ
ðjÞ
jv rþ1 m ðt þ iÞ v rþ1 m ðs þ iÞj
for j ¼ 1; k; r ¼ 1; mj , and ðjÞ
ðjÞ
ð3Þ ð4Þ ðpÞ 0 juðpÞ qm ðt þ iÞ uqm ðs þ iÞj 6 kpq jt sj þ kpq kxm k1 juqþ1 m ðt þ iÞ uqþ1 m ðs þ iÞj ð4Þ 6 kð3Þ pq jt sj þ kpq
kakBC 1 ðRþ Þ 1d
ðjÞ
ðjÞ
juqþ1 m ðt þ iÞ uqþ1 m ðs þ iÞj
ð1Þ
ð2Þ
ð3Þ
for p ¼ 1; l; q ¼ 1; li , applying Lemma 1 with am ¼ kjm jt sj and bm ¼ kjm kakBC 1 ðRþ Þ =ð1 dÞ, and with am ¼ kjm jt sj and ð4Þ bm ¼ kjm kakBC 1 ðRþ Þ =ð1 dÞ, from (36) it easily follows that
jx0mþ1 ðtÞ
x0mþ1 ðsÞj
6 ðL1 þ C0 Þjt sj þ
k X
0
Cj @
j¼1
mj X kakBC 1 ðRþ Þ j1 ¼1
1d
!j 1 ð1Þ kjj1
jY 1 1
ð2Þ kji
þ
i¼1
kakBC 1 ðRþ Þ 1d
!mj þ1
mj Y
1
ð2Þ kji Ajt
sj
i¼1
0 1 !j1 !l l li j1 l i X X Y kakBC 1 ðRþ Þ kakBC 1 ðRþ Þ i Y ð3Þ ð4Þ ð4Þ ~ þ L Ckþi @ kij kii1 þ kii1 Ajt sj 6 ~Ljt sj: 1d 1d i¼1 j¼1 i ¼1 i ¼1 1
So by induction Lipschitz condition (35) holds for every m 2 N0 and t; s 2 Rþ . Let
aj1 ðtÞ ¼ uj1 ðt; aðuj2 ðt; . . . aðujmj ðt; aðtÞÞÞ . . .ÞÞÞ; and
a^ i1 ðtÞ ¼ wi1 ðt; aðwi2 ðt; . . . aðwili ðt; aðtÞÞÞ . . .ÞÞÞ:
1
ð37Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
485
Then by (4), (5), (8) and (26) we get
jx1 ðtÞ x0 ðtÞj 6
1 X
jA1 jiþ1
1
^ 11 ðs þ iÞÞ . . . ; a0 ða ^ l1 ðs þ iÞÞÞjds jUðs þ i; aða11 ðs þ iÞÞ; . . . ; aðak1 ðs þ iÞÞ; a0 ða
t
i¼0
6
Z
1 X
jA1 jiþ1
i¼0
k Z X
cj ðs þ i; s þ iÞjaðaj1 ðs þ iÞÞjds þ
t
j¼1
6 kakBC 1 ðRþ Þ
1
1 l Z X X jA1 jiþ1 p¼1
i¼0
t
1
ckþp ðs þ i; s þ iÞja0 ða^ p1 ðs þ iÞÞjÞds
kþl X Gj ðtÞ 6 kakBC 1 ðRþ Þ d;
ð38Þ
j¼1
while by (4), (5), (8) and (27) we have that
jx01 ðtÞ x00 ðtÞj 6
1 X
^ 11 ðt þ iÞÞ . . . ; a0 ða ^ l1 ðt þ iÞÞÞj jA1 jiþ1 jUðt þ i; aða11 ðt þ iÞÞ; . . . ; aðak1 ðt þ iÞÞ; a0 ða
i¼0
6
1 X
jA1 jiþ1
i¼0
6 kakBC 1 ðRþ Þ
k X
1 l X X ^ p1 ðt þ iÞÞj jA1 jiþ1 ckþp ðt þ i; t þ iÞja0 ða
j¼1
i¼0
cj ðt þ i; t þ iÞjaðaj1 ðt þ iÞÞj þ
p¼1
kþl X
Cj ðt; tÞ 6 kakBC 1 ðRþ Þ d:
ð39Þ
j¼1
Assume that for some m 2 N
jxm ðtÞ xm1 ðtÞj 6 kakBC 1 ðRþ Þ qm
and jx0m ðtÞ x0m1 ðtÞj 6 kakBC 1 ðRþ Þ qm ;
ð40Þ
where
0 q ¼ d @1 þ
max
8 !j j mj <X 1 kakBC 1 ðRþ Þ 1 Y
16j6k;16i6l:
j1 ¼1
1d
li X kakBC 1 ðRþ Þ
ð2Þ kji ; ~L
i¼1
j¼1
1d
!j1
91 j = Y ð4Þ A : kij ; i¼1
From (4), (5), (26), (32), (35) and (40), it follows that
jxmþ1 ðtÞ xm ðtÞj 6
1 X
jA1 jiþ1
i¼0
þ
k Z X j¼1
1 X
jA1 jiþ1
1 X
jA1 jiþ1
i¼0
þ
p¼1
t
k Z X
1
j¼1
1 X
jA1 jiþ1
þ
j¼1
jA1 j
p¼1
1 X
jA1 jiþ1
p¼1
1
ðpÞ 0 ckþp ðs þ i; s þ iÞjx0m ðuðpÞ 1 m ðs þ iÞÞ xm1 ðu1 m1 ðs þ iÞÞjds
ðjÞ cj ðs þ i; s þ iÞjxm ðv ðjÞ 1 m ðs þ iÞÞ xm ðv 1 m1 ðs þ iÞÞjds 1
1
t
l Z X
i¼0
ðjÞ cj ðs þ i; s þ iÞjxm ðv ðjÞ 1 m ðs þ iÞÞ xm1 ðv 1 m1 ðs þ iÞÞjds
t
l Z X iþ1
i¼0
þ
t
k Z X
i¼0 1 X
t
l Z X
i¼0
6
1
1
t
ðjÞ cj ðs þ i; s þ iÞjxm ðv ðjÞ 1 m1 ðs þ iÞÞ xm1 ðv 1 m1 ðs þ iÞÞjds
ðpÞ 0 ckþp ðs þ i; s þ iÞjx0m ðuðpÞ 1 m ðs þ iÞÞ xm ðu1 m1 ðs þ iÞÞjds
ðpÞ 0 ckþp ðs þ i; s þ iÞjx0m ðuðpÞ 1 m1 ðs þ iÞÞ xm1 ðu1 m1 ðs þ iÞÞjds
kþl 1 k Z 1 X X kakBC 1 ðRþ Þ X ðjÞ 6 kakBC 1 ðRþ Þ qm Gj ðtÞ þ jA1 jiþ1 cj ðs þ i; s þ iÞjv ðjÞ 1 m ðs þ iÞ v 1 m1 ðs þ iÞjds 1 d i¼0 j¼1 j¼1 t 1 l Z X X þ ~L jA1 jiþ1 p¼1
i¼0
t
1
ckþp ðs þ i; s þ iÞju1ðpÞm ðs þ iÞ uðpÞ 1 m1 ðs þ iÞjds:
ð41Þ
From (6), (32) and hypothesis (40) we get that for each j 2 f1; . . . ; kg and every 1 6 s 6 mj ðjÞ
ðjÞ
ðjÞ
ð2Þ
ðjÞ
ðjÞ
jv ðjÞ s m ðtÞ v s m1 ðtÞj ¼ juj s ðt; xm ðv sþ1 m ðtÞÞÞ uj s ðt; xm1 ðv sþ1 m1 ðtÞÞÞj 6 kjs jxm ðv sþ1 m ðtÞÞ xm1 ðv sþ1 m1 ðtÞÞj ð2Þ
ðjÞ
ðjÞ
ð2Þ
ðjÞ
ðjÞ
6 kjs jxm ðv sþ1 m ðtÞÞ xm ðv sþ1 m1 ðtÞÞj þ kjs jxm ðv sþ1 m1 ðtÞÞ xm1 ðv sþ1 m1 ðtÞÞj ! kakBC 1 ðRþ Þ ðjÞ ð2Þ ðjÞ m jv sþ1 m ðtÞ v sþ1 m1 ðtÞj : 6 kjs kakBC 1 ðRþ Þ q þ 1d
ð42Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
486
ð2Þ
ð2Þ
From (42) and by using Lemma 1 with as ¼ kjs kakBC 1 ðRþ Þ qm and bs ¼ kjs kakBC 1 ðRþ Þ =ð1 dÞ, we have that for 1 6 s 6 mj
jv
ðjÞ 1 m ðtÞ
v
ðjÞ 1 m1 ðtÞj
6 kakBC 1 ðRþ Þ q
m
s1 kak 1 X BC ðRþ Þ
!j1 1
1d
j1 ¼1
j1 Y
ð2Þ kji
þ jv
ðjÞ s m ðtÞ
v
ðjÞ s m1 ðtÞj
kakBC 1 ðRþ Þ
i¼1
!s1
1d
s1 Y
ð2Þ
ð43Þ
kji :
i¼1
By choosing s ¼ mj in (43), applying (6) and (32) and hypothesis (40) we get that for each j 2 f1; . . . ; kg and every m 2 N mj 1
jv
ðjÞ 1 m ðtÞ
v
ðjÞ 1 m1 ðtÞj
6 kakBC 1 ðRþ Þ q
X kakBC 1 ðRþ Þ
m
1d
j1 ¼1
X kakBC 1 ðRþ Þ
6 kakBC 1 ðRþ Þ q
m
!j1 1
1d
j1 ¼1
ð2Þ kji
þ jv
ðjÞ mj m ðtÞ
v
ðjÞ mj m1 ðtÞj
kakBC 1 ðRþ Þ 1d
j1 Y
ð2Þ
ð2Þ
kji þ kjmj jxm ðtÞ xm1 ðtÞj
!j1 1
1d
j1 Y
!mj 1 m 1 j Y ð2Þ k1i i¼1
!m 1 1 j kakBC 1 ðRþ Þ j mY
i¼1
mj X kakBC 1 ðRþ Þ j1 ¼1
j1 Y i¼1
mj 1
6 kakBC 1 ðRþ Þ qm
!j1 1
1d
ð2Þ
kji
i¼1
ð2Þ
ð44Þ
kji :
i¼1
Similarly to (44) is proved that for each i 2 f1; . . . ; lg and every m 2 N ðiÞ ju1 m ðtÞ
ðiÞ u1 m1 ðtÞj
6 kakBC 1 ðRþ Þ q
m
li X kakBC 1 ðRþ Þ j¼1
!j1
1d
j Y ð4Þ kij :
ð45Þ
i¼1
From (8), (41), (44) and (45), it follows that
0 1 !j j mj k 1 X X kakBC 1 ðRþ Þ 1 Y ð2Þ A @ jxmþ1 ðtÞ xm ðtÞj 6 kakBC 1 ðRþ Þ q Gj ðtÞ 1 þ kji 1d j¼1 j1 ¼1 i¼1 0 1 !j1 li j l X X Y kakBC 1 ðRþ Þ ð4Þ m Gkþi ðtÞ@1 þ ~L kij A 6 kakBC 1 ðRþ Þ qmþ1 : þ kakBC 1 ðRþ Þ q 1 d j¼1 i¼1 i¼1 m
ð46Þ
From (4), (5), (27), (32), (40), (44) and (45), it follows that
jx0mþ1 ðtÞ x0m ðtÞj 6
1 k X X ðjÞ ðjÞ jA1 jiþ1 cj ðt þ i; t þ iÞjxm ðv 1 m ðt þ iÞÞ xm ðv 1 m1 ðt þ iÞÞj i¼0
j¼1
1 k X X ðjÞ ðjÞ þ jA1 jiþ1 cj ðt þ i; t þ iÞjxm ðv 1 m1 ðt þ iÞÞ xm1 ðv 1 m1 ðt þ iÞÞj i¼0
j¼1
1 l X X ðpÞ ðpÞ þ jA1 jiþ1 ckþp ðt þ i; t þ iÞjx0m ðu1 m ðt þ iÞÞ x0m ðu1 m1 ðt þ iÞÞj p¼1
i¼0
1 l X X ðpÞ ðpÞ þ jA1 jiþ1 ckþp ðt þ i; t þ iÞjx0m ðu1 m1 ðt þ iÞÞ x0m1 ðu1 m1 ðt þ iÞÞj p¼1
i¼0
6 kakBC 1 ðRþ Þ q
m
0
k X
Cj ðtÞ@1 þ
j¼1
þ kakBC 1 ðRþ Þ q
m
l X
0
mj X kakBC 1 ðRþ Þ
1d
j1 ¼1
j1 Y
1 ð2Þ kji A
i¼1
li X
kakBC 1 ðRþ Þ
j¼1
1d
Ckþi ðtÞ@1 þ ~L
i¼1
!j 1
!j1
j Y
1 ð4Þ kij A
6 kakBC 1 ðRþ Þ qmþ1 :
ð47Þ
i¼1
From (38), (39), (46) and (47) and the induction we obtain that (40) hold for every m 2 N0 and for all t 2 Rþ . ð2Þ ð4Þ Now note that for sufficiently small kj1 , j ¼ 1; k; ki1 ; i ¼ 1; l,
0
d@1 þ
max
16j6k;16i6l
8 !j j mj <X 1 kakBC 1 ðRþ Þ 1 Y :j
1 ¼1
li X kakBC 1 ðRþ Þ ð2Þ kji ; ~L 1d i¼1 j¼1
1d
!j1
91 j = Y ð4Þ kij A < 1; ; i¼1
that is, q 2 ð0; 1Þ. This with (40) implies the uniform convergence of ðxm ðtÞÞm2N0 and ðx0m ðtÞÞm2N0 on Rþ . Taking the limit in (26), (27), (32) and (35) we get that
xðtÞ :¼ lim xm ðtÞ m!þ1
1
is a C solution of (25) on Rþ such that
jxðtÞj 6
kakBC 1 ðRþ Þ 1d
;
jx0 ðtÞj 6
kakBC 1 ðRþ Þ 1d
ð48Þ
S. Stevic´ / Applied Mathematics and Computation 231 (2014) 478–488
487
and
jx0 ðtÞ x0 ðsÞj 6 ~Ljt sj:
ð49Þ 1
Assume (25) has another BC ðRþ Þ solution, say y. Set
v^ jp ðtÞ ¼ ujp ðt; yðuj pþ1 ðt; . . . yðuj m ðt; yðtÞÞÞ . . .ÞÞÞ;
j ¼ 1; k
j
and
^ jp ðtÞ ¼ wjp ðt; yðwj pþ1 ðt; . . . yðwj m ðt; yðtÞÞÞ . . .ÞÞÞ; u j
j ¼ 1; l:
From (4), (5), (6) and (48), we have
jxðtÞ yðtÞj 6
Z 1 X jA1 jiþ1
k X
t
i¼0
1
Z
1
t
cj ðs þ i; s þ iÞjxðv j1 ðs þ iÞÞ xðv^ j1 ðs þ iÞÞjds þ
j¼1
1 X jA1 jiþ1 i¼0
Z 1 X cj ðs þ i; s þ iÞjxðv^ j1 ðs þ iÞÞ yðv^ j1 ðs þ iÞÞjds þ jA1 jiþ1
k X j¼1
i¼0
^ p1 ðs þ iÞÞjÞds þ x0 ð u
Z 1 X jA1 jiþ1
1
t
l X
ckþp ðs þ i; s þ iÞjx0 ðup1 ðs þ iÞÞ
p¼1
l X
ckþp ðs þ i; s þ iÞjx0 ðu^ p1 ðs þ iÞÞ y0 ðu^ p1 ðs þ iÞÞjÞds
t
i¼0
1
p¼1
Z 1X kþl 1 k X kakBC 1 ðRþ Þ X Gj ðtÞ þ jA1 jiþ1 cj ðs þ i; s þ iÞjv j1 ðs þ iÞ v^ j1 ðs þ iÞjds 1 d i¼0 t j¼1 j¼1 Z 1X 1 l X þ ~L jA1 jiþ1 ckþp ðs þ i; s þ iÞjup1 ðs þ iÞ u^ p1 ðs þ iÞjds:
6 kx ykC 1 ðRþ Þ
t
i¼0
ð50Þ
p¼1
Using Lemma 1, similar to (44), we obtain
jv j1 ðtÞ v^ j1 ðtÞj 6 kx ykC1 ðR
mj X kakBC 1 ðRþ Þ
þÞ
j1 ¼1
1d
li X kakBC 1 ðRþ Þ
!j1
1d
j¼1
j1 Y ð2Þ kji
ð51Þ
i¼1
and also
^ i1 ðtÞj 6 kx ykC1 ðR Þ jui1 ðtÞ u þ
!j1 1
j Y
ð4Þ
ð52Þ
kij :
i¼1
Using (51) and (52) into (50) and then applying condition (8) we get
0
0 11 !j j mj k 1 X X kakBC 1 ðRþ Þ 1 Y ð2Þ jxðtÞ yðtÞj 6 kx ykC 1 ðRþ Þ @ Gj ðtÞ@1 þ kji AA 1d j¼1 j1 ¼1 i¼1 0 1 !j1 li j l X X Y kakBC 1 ðRþ Þ ð4Þ A ~ @ Gkþi ðtÞ 1 þ L kij kx ykC 1 ðRþ Þ þ 1d i¼1 i¼1 j¼1 6 qkx ykC1 ðRþ Þ :
ð53Þ
From (4), (5), (6), (8), (49), (51) and (52), it follows that
jx0 ðtÞ y0 ðtÞj 6
1 k 1 k X X X X jA1 jiþ1 cj ðt þ i; t þ iÞjxðv j1 ðt þ iÞÞ xðv^ j1 ðt þ iÞÞj þ jA1 jiþ1 cj ðt þ i; t þ iÞjxðv^ j1 ðt þ iÞÞ i¼0
j¼1
i¼0
j¼1
1 l X X ^ p1 ðt þ iÞÞj yðv^ j1 ðt þ iÞÞj þ jA1 jiþ1 ckþp ðt þ i; t þ iÞjx0 ðup1 ðt þ iÞÞ x0 ðu i¼0
p¼1
1 l X X ^ p1 ðt þ iÞÞ y0 ðu ^ p1 ðt þ iÞÞj jA1 jiþ1 ckþp ðt þ i; t þ iÞjx0 ðu þ p¼1 0 0 11 !j j mj k 1 X X kakBC 1 ðRþ Þ 1 Y ð2Þ kji AA 6 kx ykC 1 ðRþ Þ @ Cj ðtÞ@1 þ 1d j1 ¼1 i¼1 j¼1 0 1 !j1 li j l X X Y kakBC 1 ðRþ Þ ð4Þ A ~ @ 6 qkx ykC1 ðRþ Þ : þ kx ykC1 ðRþ Þ Ckþi ðtÞ 1 þ L kij 1d i¼1 j¼1 i¼1 i¼0
Thus kx ykC 1 ðRþ Þ 6 qkx ykC 1 ðRþ Þ , which with q 2 ð0; 1Þ implies xðtÞ ¼ yðtÞ. h
ð54Þ
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