On q-difference asymptotic solutions of a system of nonlinear functional differential equations

Applied Mathematics and Computation 219 (2013) 8295–8301

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

On q-difference asymptotic solutions of a system of nonlinear functional differential equations Stevo Stevic´ Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia

a r t i c l e

i n f o

a b s t r a c t The asymptotic behavior and the existence of a continuously differentiable solution bounded together with its first derivative of a system of nonlinear functional differential equations is studied here. Ó 2013 Elsevier Inc. All rights reserved.

Keywords: System of functional differential equations Iterated deviations q-Difference asymptotic solutions

1. Introduction and preliminaries Nonlinear differential equations, not solved (or partially solved) with respect to the highest-order derivatives, have been studied a lot, see, e.g., [1–11,13–15,19–21,29,31,32,34,35] and the references therein. Particular cases of the next system of functional differential equations

qx0 ðqtÞ ¼ x0 ðtÞ þ Fðt; xðtÞ; xðv 1 ðtÞÞ; x0 ðgðtÞÞÞ;

ð1Þ

where

v j ðtÞ ¼ uj ðt; xðujþ1 ðt; . . . uk ðt; xðtÞÞ   ÞÞÞ;

j ¼ 1; k;

ð2Þ N

N

N

N

N

q is a constant, t 2 Rþ :¼ ½0; 1Þ, and the functions F : Rþ  R  R  R ! R ; uj : Rþ  R ! Rþ ; j ¼ 1; k; g : Rþ ! Rþ , are continuous on their domains, are ones which have attracted some attention (see, e.g. [11,12,16–18,20]). The idea of iterations of some iterative processes which can be found, e.g., in papers [22–28], motivated us to propose studying 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 (see [29–33,?]). Let C 1 ½T; 1Þ; T > 0, denote the space of continuously differentiable functions on the interval ½T; 1Þ. The space BC 1 ½T; 1Þ is defined by

n o BC 1 ½T; 1Þ :¼ x 2 C 1 ½T; 1Þ : kxk1 < 1 ; where the norm kxk1 is defined by

  kxk1 ¼ max kxk1 ; kx0 k1 ¼ max

(

) 0

sup jxðtÞj; sup jx ðtÞj : t2½T;1Þ

t2½T;1Þ

1

In other words, the space BC ½T; 1Þ consists of all C 1 ½T; 1Þ functions which are bounded together with their first derivatives on ½T; 1Þ. The form of BC 1 ½T; 1Þ solutions of system (1) satisfying the next condition

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.02.029

S. Stevic´ / Applied Mathematics and Computation 219 (2013) 8295–8301

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lim ðxðqtÞ  xðtÞÞ ¼ 0

ð3Þ

t!þ1

is studied here. The next lemma from [34] will be used in the proofs of some mains results in this paper. 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 ; i¼1

j¼1

i¼1

for every k 2 N. 2. Main results Now we formulate and prove the main results in this paper. Theorem 1. Assume that 0 < q < 1; T > 0, and that the next conditions hold: (i) the function F : ½T; 1Þ  RN  RN  RN ! RN is continuous,

Fðt; 0; 0; 0Þ  0;

t 2 ½T; 1Þ

ð4Þ

and

jFðt; x; y; zÞ  Fðt; u; v ; wÞj 6 g1 ðtÞjx  uj þ g2 ðtÞjy  v j þ g3 ðtÞjz  wj;

ð5Þ

R1

for x; y; z; u; v ; w 2 R ; t 2 ½T; 1Þ, where the functions gj ðtÞ and t gj ðsÞds; j ¼ 1; 3 are nonnegative, continuous, and bounded on ½T; 1Þ; (ii) the functions uj : ½T; 1Þ  RN ! ½T; 1Þ, j ¼ 1; k, are continuous and N

juj ðt; xÞ  uj ðt; yÞj 6 lj jx  yj;

ð6Þ

for t 2 ½T; 1Þ, and x; y 2 RN , where lj ; j ¼ 1; k, are some non-negative constants; (iii) the function g : ½T; 1Þ ! ½T; 1Þ is continuous; (iv) the series

Hj ðtÞ ¼

1 X qi gj ðqi tÞ;

Gj ðtÞ ¼

i¼1

Z 1 X qi t

i¼1

1

gj ðqi sÞds;

converge uniformly on ½T; 1Þ and

( ) 3 3 X X max Hj ðtÞ; Gj ðtÞ 6 h < 1; j¼1

for every t 2 ½T; 1Þ:

j¼1

Then for any BC 1 ½T; 1Þ solution xðtÞ of system (1), there is a BC 1 ½T; 1Þ vector function xðtÞ, such that xðqtÞ ¼ xðtÞ and

lim ðxðtÞ  xðtÞÞ ¼ 0

ð7Þ

t!þ1

(i.e. xðtÞ is a q-difference asymptotic solution of system (1)). Proof. Let xðtÞ be a BC 1 ½T; 1Þ solution of system (1) and

xðtÞ ¼ xðtÞ þ

Z 1 X qi i¼1

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds:

t

Since by using (4) and (5) we have that

ð8Þ

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  Z 1 Z 1  X X 1 1     Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds 6 qi g1 ðqi sÞjxðqi sÞj þ g2 ðqi sÞjxðv 1 ðqi sÞÞj þ g3 ðqi sÞjx0 ðgðqi sÞÞj ds  qi   i¼1 t t i¼1 Z 1 3 X 1 3 X X 6 kxk1 qi gj ðqi sÞds ¼ kxk1 Gj ðtÞ; ð9Þ t

j¼1 i¼1

j¼1

we obtain that function xðtÞ is well-defined on ½T; 1Þ and then obviously

xðtÞ ¼ xðtÞ 

Z 1 X qi

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds:

t

i¼1

From (8) and (9) it follows that 3 X Gj ðtÞ:

jxðtÞ  xðtÞj 6 kxk1

ð10Þ

j¼1

The uniform convergence of the series Gj ðtÞ; j ¼ 1; 3, consisting of continuous functions on ½T; þ1Þ, easily implies that Gj ðtÞ ! 0, j ¼ 1; 3, as t ! þ1, from which (7) follows. The uniform convergence of series Hj ðtÞ; Gj ðtÞ; j ¼ 1; 3, along with (4) and (5), and the theorem on differentiating series easily implies that x 2 BC 1 ½T; þ1Þ. Integrating (1) from t to t 1 , then letting in such obtained equality t1 ! þ1 and using condition (3), we get

xðqtÞ ¼ xðtÞ 

Z

1

Fðs; xðsÞ; xðv 1 ðsÞÞ; x0 ðgðsÞÞÞds:

ð11Þ

t

Further, by using the change of variables t ! tq and (11), we have that

xðqtÞ ¼ xðqtÞ þ ¼ xðtÞ 

Z 1 X qiþ1 1

Fðqiþ1 s; xðqiþ1 sÞ; xðv 1 ðqiþ1 sÞÞ; x0 ðgðqiþ1 sÞÞÞds

t

i¼1

Z

1

Fðs; xðsÞ; xðv 1 ðsÞÞ; x0 ðgðsÞÞÞds þ

t

¼ xðtÞ þ

Z 1 X qi

Z 1 X qi i¼0

1

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds

t

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds ¼ xðtÞ;

ð12Þ

t

i¼1

that is, xðqtÞ ¼ xðtÞ, as desired. h Theorem 2. Assume that 0 < q < 1; T > 0, conditions (i)–(iv) in Theorem 1 are satisfied, and xðtÞ is a BC 1 ½T; 1Þ vector function satisfying condition xðqtÞ ¼ xðtÞ. Then for sufficiently small lj , j ¼ 1; k, system (1) has a BC 1 ½T; 1Þ solution satisfying condition (7). Proof. If xðtÞ is a BC 1 ½T; 1Þ solution of system (1) satisfying condition (7), then by the proof of Theorem 1 we see that it satisfies system (11), which can be written in the form

xðtÞ ¼ xðq1 tÞ  q1

Z

1

Fðq1 s; xðq1 sÞ; xðv 1 ðq1 sÞÞ; x0 ðgðq1 sÞÞÞds:

ð13Þ

t

Plugging tqi ; i ¼ 1; N  1 in (13) instead of t, and summing up such obtained equalities, it follows that

xðtÞ ¼ xðqN tÞ 

Z N X qi

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds

t

i¼1

¼ ðxðqN tÞ  xðqN tÞÞ þ xðqN tÞ 

Z N X qi i¼1

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds:

ð14Þ

t

Letting N ! þ1 in (14) and using the assumption q 2 ð0; 1Þ, condition (7) and xðqtÞ ¼ xðtÞ, it follows that xðtÞ satisfies the following system of integral equations

xðtÞ ¼ xðtÞ 

Z 1 X qi i¼1

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds:

ð15Þ

t

On the other hand, using the uniform convergence of series Hj ðtÞ; Gj ðtÞ; j ¼ 1; 3, along with (4) and (5), and the theorem on differentiating series we see that any bounded continuous solution of system (15) is a BC 1 ½T; 1Þ solution of system (1) satisfying condition (7). Hence, it is enough to prove the existence of a BC 1 ½T; 1Þ solution of system (15). We use the method of successive approximations. Let

x0 ðtÞ ¼ xðtÞ;

x00 ðtÞ ¼ x0 ðtÞ;

ð16Þ

S. Stevic´ / Applied Mathematics and Computation 219 (2013) 8295–8301

8298

xm ðtÞ ¼ xðtÞ 

Z 1 X qi t

i¼1

x0m ðtÞ ¼ x0 ðtÞ þ

1

Fðqi s; xm1 ðqi sÞ; xm1 ðv 1;m1 ðqi sÞÞ; x0m1 ðgðqi sÞÞÞds;

1 X qi Fðqi t; xm1 ðqi tÞ; xm1 ðv 1;m1 ðqi tÞÞ; x0m1 ðgðqi tÞÞÞ;

ð17Þ

m 2 N;

ð18Þ

i¼1

where

v j;m ðtÞ ¼ uj ðt; xm ðujþ1 ðt; . . . uk ðt; xm ðtÞÞ . . .ÞÞÞ;

j ¼ 1; k:

1

That xm 2 BC ½T; 1Þ, for every m 2 N0 , is proved similarly as for the bounded continuous solution of system (15) and by using the method of induction. Since x0 ðtÞ ¼ xðtÞ and x00 ðtÞ ¼ x0 ðtÞ, it is obvious that for every t 2 ½T; 1Þ

jx0 ðtÞj 6

kxk1 1h

kxk1 : 1h

and jx00 ðtÞj 6

ð19Þ

Assume that the functions xk ðtÞ and x0k ðtÞ; k ¼ 0; m  1, defined by (17) satisfy the next estimates

jxk ðtÞj 6

kxk1 1h

kxk1 : 1h

and jx0k ðtÞj 6

ð20Þ

Then, by (4), (5), (17) and (20) it follows that

jxm ðtÞj 6 kxk1 þ

Z 1 X qi t

i¼1

6 kxk1 þ 

Z 1 X qi

1 t



jFðqi s; xm1 ðqi sÞ; xm1 ðv 1;m1 ðqi sÞÞ; x0m1 ðgðqi sÞÞÞjds

1

g1 ðqi sÞjxm1 ðqi sÞjds þ

t

i¼1

Z

1

1 X qi i¼1



g2 ðqi sÞjxm1 ðv 1;m1 ðqi sÞÞj þ g3 ðqi sÞjx0m1 ðgðqi sÞÞj ds

6 kxk1 þ

1 kxk1 X qi 1  h i¼1

Z t

1

3 X

gj ðqi sÞds 6 kxk1 þ

j¼1

hkxk1 kxk1 6 1h 1h

ð21Þ

and by (4), (5), (18) and (20)

jx0m ðtÞj 6 kx0 k1 þ

1 X

qi jFðqi t; xm1 ðqi tÞ; xm1 ðv 1;m1 ðqi tÞÞ; x0m1 ðgðqi tÞÞÞj

i¼1

6 kx0 k1 þ

1 X

  qi g1 ðqi tÞjxm1 ðqi tÞj þ g2 ðqi tÞjxm1 ðv 1;m1 ðqi tÞÞj þ g3 ðqi tÞjx0m1 ðgðqi tÞÞj

i¼1 1 3 X kxk1 X hkxk1 kxk1 qi gj ðqi tÞ 6 kx0 k1 þ 6 kx0 k1 þ 6 : 1  h i¼1 1h 1h j¼1

ð22Þ

From (19), (21) and (22), and the method of induction we see that (20) holds for every k 2 N0 . Now we show that the following inequalities hold for t P T and every m 2 N

jxm ðtÞ  xm1 ðtÞj 6 kxk1 ~hm

and jx0m ðtÞ  x0m1 ðtÞj 6 kxk1 ~hm ;

ð23Þ

where

( ~h :¼ max h; sup t2½T;þ1Þ

! s s k  X kxk1 Y G1 ðtÞ þ G2 ðtÞ li þ G3 ðtÞ ; 1  h i¼1 s¼0

sup t2½T;þ1Þ

!) s s k  X kxk1 Y H1 ðtÞ þ H2 ðtÞ li þ H3 ðtÞ ; 1  h i¼1 s¼0

which belongs to the interval ½0; 1Þ for sufficiently small lj ; j ¼ 1; k. By (4), (5), (16), (17) and (20), we have that

jx1 ðtÞ  x0 ðtÞj 6

Z 1 X qi i¼1

6

6 kxk1

jFðqi s; xðqi sÞ; xðv 1;0 ðqi sÞÞ; x0 ðgðqi sÞÞÞjds

t

Z 1 X qi i¼1

1

1



t 1 X i¼1

qi



g1 ðqi sÞjxðqi sÞj þ g2 ðqi sÞjxðv 1;0 ðqi sÞÞj þ g3 ðqi sÞjx0 ðgðqi sÞÞj ds

Z t

1

3 X

3 X Gj ðtÞ 6 kxk1 h;

j¼1

j¼1

gj ðqi sÞds 6 kxk1

ð24Þ

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and

jx01 ðtÞ  x00 ðtÞj 6

1 X

qi jFðqi t; xðqi tÞ; xðv 1;0 ðqi tÞÞ; x0 ðgðqi tÞÞÞj

i¼1

6

1 X

1 3 X X   qi g1 ðqi tÞjxðqi tÞj þ g2 ðqi tÞjxðv 1;0 ðqi tÞÞj þ g3 ðqi tÞjx0 ðgðqi tÞÞj 6 kxk1 qi gj ðqi tÞ

i¼1

i¼1

j¼1

3 X 6 kxk1 Hj ðtÞ 6 kxk1 h j¼1

and, hence, inequalities (23) hold for m ¼ 1. Assume that inequalities (23) hold for an m 2 N. Then, by (4) and (5), we get   Z 1  X 1   jxmþ1 ðtÞ  xm ðtÞj 6  qi ðFðqi s; xm ðqi sÞ; xm ðv 1;m ðqi sÞÞ; x0m ðgðqi sÞÞÞ  Fðqi s; xm1 ðqi sÞ; xm1 ðv 1;m1 ðqi sÞÞ; x0m1 ðgðqi sÞÞÞÞds   i¼1 t Z 1 Z 1 1 1 X X i i i i i i i i 6 q g1 ðq sÞjxm ðq sÞ  xm1 ðq sÞjds þ q g2 ðq sÞjxm ðv 1;m ðq sÞÞ  xm ðv 1;m1 ðq sÞÞjds t

i¼1

Z 1 X þ qi

i¼1 1

t

i¼1

6 kxk1 ~hm

t

g2 ðq sÞjxm ðv 1;m1 ðq sÞÞ  xm1 ðv 1;m1 ðq sÞÞjds þ i

Z 1 X qi

1 t

i¼1

i

3 X

gj ðqi sÞds þ

j¼1

i

1 kxk1 X qi 1  h i¼1

Z

Z 1 X qi i¼1

1

t

t

1

g3 ðqi sÞjx0m ðgðqi sÞÞ  x0m1 ðgðqi sÞÞjds

g2 ðqi sÞjv 1;m ðqi sÞ  v 1;m1 ðqi sÞjds:

ð25Þ

We have

jv j;m ðtÞ  v j;m1 ðtÞj ¼ juj ðt; xm ðv jþ1;m ðtÞÞÞ  uj ðt; xm1 ðv jþ1;m1 ðtÞÞÞj 6 lj jxm ðv jþ1;m ðtÞÞ  xm1 ðv jþ1;m1 ðtÞÞj 6 lj jxm ðv jþ1;m ðtÞÞ  xm ðv jþ1;m1 ðtÞÞj þ lj jxm ðv jþ1;m1 ðtÞÞ  xm1 ðv jþ1;m1 ðtÞÞj   kxk1 6 lj kxk1 ~hm þ jv jþ1;m ðtÞ  v jþ1;m1 ðtÞj : 1h

ð26Þ

hm and bn ¼ ln kxk1 =ð1  hÞ we have that Applying Lemma 1 with an ¼ ln kxk1 ~

s1 Y  k1 Y k1  s k1 X kxk1 kxk1 li þ jv k;m ðtÞ  v k;m1 ðtÞj li 1h 1h s¼1 i¼1 i¼1 s1 Y  k1 Y k1  s k1 X kxk1 kxk1 6 kxk1 ~hm li þ lk jxm ðtÞ  xm1 ðtÞj li 1h 1h s¼1 i¼1 i¼1 s1 Y k  s X kxk1 6 kxk1 ~hm li : 1h s¼1 i¼1

jv 1;m ðtÞ  v 1;m1 ðtÞj 6 kxk1 ~hm

ð27Þ

Using (27) into (25) we obtain

jxmþ1 ðtÞ  xm ðtÞj 6 kxk1 ~hm

3 X j¼1

Gj ðtÞ þ G2 ðtÞ

s s ! k  X kxk1 Y li 6 kxk1 ~hmþ1 : 1  h s¼1 i¼1

ð28Þ

Using (4) and (5), inductive hypothesis and (27) we have that

jx0mþ1 ðtÞ  x0m ðtÞj 6

1 X

qi jFðqi t; xm ðqi tÞ; xm ðv 1;m ðqi tÞÞ; x0m ðgðqi tÞÞÞ

i¼1

 Fðqi t; xm1 ðqi tÞ; xm1 ðv 1;m1 ðqi tÞÞ; x0m1 ðgðqi tÞÞÞj 1 1 X X qi g1 ðqi tÞjxm ðqi tÞ  xm1 ðqi tÞj þ qi g2 ðqi tÞjxm ðv 1;m ðqi tÞÞ  xm ðv 1;m1 ðqi tÞÞj 6 i¼1

i¼1

1 1 X X qi g2 ðqi tÞjxm ðv 1;m1 ðqi tÞÞ  xm1 ðv 1;m1 ðqi tÞÞj þ qi g3 ðqi tÞjx0m ðgðqi tÞÞÞ  x0m1 ðgðqi tÞÞj þ i¼1

i¼1

1 3 1 X X kxk1 X 6 kxk1 ~hm qi gj ðqi tÞ þ qi g2 ðqi tÞjv 1;m ðqi tÞ  v 1;m1 ðqi tÞj 1  h i¼1 i¼1 j¼1 s s ! 3 k  X X kxk1 Y Hj ðtÞ þ H2 ðtÞ li 6 kxk1 ~hmþ1 : 6 kxk1 ~hm 1  h i¼1 s¼1 j¼1

ð29Þ

S. Stevic´ / Applied Mathematics and Computation 219 (2013) 8295–8301

8300

Hence (23) holds for every m 2 N. From this, since ~ h 2 ð0; 1Þ, and by the Weierstrass theorem the sequences xm ðtÞ and x0m ðtÞ; m 2 N0 , converge uniformly on ½T; þ1Þ, to the vector function xðtÞ :¼ limm!þ1 xm ðtÞ 2 BC 1 ½T; 1Þ and its derivative respectively. By letting m ! þ1 in (17) and (18) k ! þ1 in (20) we see that xðtÞ is a solution of system (15) satisfying the conditions

jxðtÞj 6

M 1h

and jx0 ðtÞj 6

M : 1h



Theorem 3. Assume that q > 1; T > 0, conditions (i)–(iii) in Theorem 1 and the following condition are satisfied: (v) the series

~ j ðtÞ ¼ H

1 X qi gj ðqi tÞ;

~ j ðtÞ ¼ G

i¼0

Z 1 X qi i¼0

1

gj ðqi sÞds; j ¼ 1; 3;

t

converge uniformly on ½T; 1Þ and

( ) 3 3 X X ~ ~ Hj ðtÞ; max Gj ðtÞ 6 h < 1; j¼1

for every t 2 ½T; 1Þ:

j¼1

Then for any BC 1 ½T; 1Þ solution xðtÞ of system (1), there is a BC 1 ½T; 1Þ vector function xðtÞ, such that xðqtÞ ¼ xðtÞ and condition (7) holds. Proof. Let xðtÞ be a BC 1 ½T; 1Þ solution of system (1) and

xðtÞ ¼ xðtÞ 

Z 1 X qi

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds:

t

i¼0

Similar to the proof of Theorem 1 it follows that vector function xðtÞ is well-defined and belongs to BC 1 ½T; 1Þ, and that the solution xðtÞ satisfies relations (7) and (11). By using (11), we obtain

xðqtÞ ¼ xðqtÞ  ¼ xðtÞ 

Z 1 X qiþ1 1

Fðqiþ1 s; xðqiþ1 sÞ; xðv 1 ðqiþ1 sÞÞ; x0 ðgðqiþ1 sÞÞÞds

t

i¼0

Z

1

Fðs; xðsÞ; xðv 1 ðsÞÞ; x0 ðgðsÞÞÞds 

t

¼ xðtÞ 

Z 1 X qi i¼1

Z 1 X qi

1

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds

t

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds ¼ xðtÞ;

t

i¼0

that is, xðqtÞ ¼ xðtÞ, finishing the proof of the theorem. h Theorem 4. Assume that q > 1; T > 0, conditions (i)–(iii) and (v) are satisfied. Then, for any BC 1 ½T; 1Þ vector function xðtÞ satisfying condition xðqtÞ ¼ xðtÞ, there is a BC 1 ½T; 1Þ solution xðtÞ of system (1) satisfying relation (7). Proof. Let xðtÞ be a BC 1 ½T; 1Þ solution of system (1) satisfying condition (7). Then from the proof of Theorem 1 we know that (11) hold. Plugging tqi , i ¼ 1; N  1 in (11) instead of t, and summing up such obtained equalities, it follows that

xðtÞ ¼ xðqN tÞ þ

Z N1 X qi

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds

t

i¼0

¼ ðxðqN tÞ  xðqN tÞÞ þ xðqN tÞ þ

Z N 1 X qi i¼0

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds:

ð30Þ

t

Letting N ! þ1 in (30) and using the assumption q > 1, conditions (7) and xðqtÞ ¼ xðtÞ, it follows that xðtÞ satisfies the following system of integral equations

xðtÞ ¼ xðtÞ þ

Z 1 X qi i¼0

1

Fðqi s; xðqi sÞ; xðv 1 ðqi sÞÞ; x0 ðgðqi sÞÞÞds:

ð31Þ

t

Using conditions (i)-(iii) and (v) it is not difficult to prove that any BC 1 ½T; 1Þ solution of system (31), where x 2 BC 1 ½T; 1Þ satisfies the condition xðqtÞ ¼ xðtÞ, is a solution of system (1) satisfying condition (7). Hence it is enough to prove the existence of a BC 1 ½T; 1Þ solution of system (31). This can be done by using the method of successive approximations where the approximations are defined by

S. Stevic´ / Applied Mathematics and Computation 219 (2013) 8295–8301

x0 ðtÞ ¼ xðtÞ;

x00 ðtÞ ¼ x0 ðtÞ;

xm ðtÞ ¼ xðtÞ þ

Z 1 X qi i¼0

x0m ðtÞ ¼ x0 ðtÞ 

t

1

8301

Fðqi s; xm1 ðqi sÞ; xm1 ðv 1;m1 ðqi sÞÞ; x0m1 ðgðqi sÞÞÞds;

1 X qi Fðqi t; xm1 ðqi tÞ; xm1 ðv 1;m1 ðqi tÞÞ; x0m1 ðgðqi tÞÞÞ; i¼0

m 2 N. The rest of the proof is similar to corresponding part of the proof of Theorem 2, so it is omitted. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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