On a close to symmetric system of difference equations of second order

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Closed form formulas of the solutions to the following system of difference equations: ...... where ......, ......, ......, ......, ......, and initial values ......, ......, ...... are real numbers, are found. The domain of undefinable solutions to the system is described. The long-term behavior of its solutions is studied in detail for the case of constant ......, ......, ...... and ......, .......

Stevi et al. Advances in Dierence Equations (2015) 2015:264 DOI 10.1186/s13662-015-0591-7

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Web End = On a close to symmetric system of difference equations of second order

Stevo Stevi1,2*, Bratislav Irianin3 and Zdenk marda4,5

*Correspondence: [email protected]

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia

2Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi ArabiaFull list of author information is available at the end of the article

Abstract

Closed form formulas of the solutions to the following system of dierence equations:

xn = yn1yn2xn1(an + bnyn1yn2), yn =

xn1xn2

yn1(n + nxn1xn2), n

N0,

where an, bn, n, n, n

N0, and initial values xi, yi, i {1, 2} are real numbers, are

found. The domain of undenable solutions to the system is described. The long-term behavior of its solutions is studied in detail for the case of constant an, bn, n and n, n

N0.

MSC: Primary 39A10; 39A20Keywords: system of dierence equations; closed form solution; long-term behavior; periodic solutions

1 Introduction

Studying concrete nonlinear dierence equations and systems is a topic of a great recent interest (see, e.g., [] and the references therein). Studying systems of dierence equations, especially symmetric and close to symmetric ones, is a topic of considerable interest (see, e.g., [, , , , , , , , , , , , , , ]). Another topic of interest is solvable dierence equations and systems and their applications (see, e.g., [, , , , , , , ]). Renewed interest in the area started after the publication of [] where a formula for a solution of a dierence equation was theoretically explained. The most interesting thing in [] was a change of variables which reduced the equation to a linear one with constant coecients. Related ideas were later used, e.g., in [, , , , , , , ].

Quite recently in [] the following systems of dierence equations were presented:

xn = ynyn xn( ynyn)

,

yn = xnxn yn( xnxn)

()

where xi, yi, i {, } are real numbers, and some formulas for their solutions are given,

some of which are proved by induction.

2015 Stevi et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

, n

N,

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 2 of 17

The next system of dierence equations

xn = ynyn xn(an + bnynyn),

yn = xnxnyn(n + nxnxn), n

N, and initial values xi, yi, i {, }, are real numbers, is a

generalization of the system in (). Our aim is to show that more general system () is solvable by giving a natural method for getting its solutions. The domain of undenable solutions to the system is also described. For the case when an, bn, n, n, n

N, are

N, then the rst equation in () implies that yn = or yn = . If yn = , then xn = or xn = , while if yn = , then xn = or xn = . Repeating this procedure, we get that xi = or yi = for some i {, }. Similarly, if yn = for some n

N, we get xi = or yi = for some i {, }. Hence, for a well-dened solution (xn, yn)n of system (), we have

that

xnyn = , n ()

if and only if xiyi = , i {, }.

Assume now that (xn, yn)n is a solution to system () such that () holds. Then, by multiplying the rst equation in () by xn and the second one by yn, and using the following changes of variables

un = xnxn , vn =

N. ()

From () it follows that

un = annun + ann + bn, ()

vn = nanvn + nbn + n, n

N. ()

()

N,

where an, bn, n, n, n

constant, the long-term behavior of its solutions is investigated in detail.

A solution (xn, yn)n of system () is called periodic, or eventually periodic, with period p if there is n such that

xn+p = xn and yn+p = yn for n n.

For some results in the area, see, e.g., [, , , , , ].

2 Solutions to system (2) in closed form

Assume rst that xi = , yi = , i {, }. Then, by the method of induction and the

equations in (), it follows that for every well-dened solution to system (), xn = and

yn = , for every n

N. On the other hand, if xn = for some n

ynyn , ()

n , system () is transformed in the following one:

un = anvn + bn, vn = nun + n, n

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 3 of 17

This means that (un)nN, (un)nN, (vn)nN, and (vn)nN are solutions to two linear rst-order dierence equations, which are solvable.

Solving these equations, we get

un = u

n

[productdisplay]

j=

ajj +

n

[summationdisplay]

i=

(aii + bi)

n

[productdisplay]

s=i+

ass, ()

un = u

n

[productdisplay]

j=

ajj +

n

[summationdisplay]

i=

(aii + bi)

n

[productdisplay]

s=i+

ass, ()

vn = v

n

[productdisplay]

j=

jaj +

n

[summationdisplay]

i=

(ibi + i)

n

[productdisplay]

s=i+

sas, ()

vn = v

n

[productdisplay]

j=

jaj +

n

[summationdisplay]

i=

(ibi + i)

n

[productdisplay]

s=i+

sas. ()

Using () we obtain

xn+i = un+ixn+i =

un+i

un+i x(n)+i, i {, },

and

yn+i = vn+iyn+i =

vn+i

vn+i y(n)+i, i {, },

for n + i , from which it follows that

xm+i = xi

m

[productdisplay]

j=

uj+i

uj+i , ()

ym+i = yi

m

[productdisplay]

j=

vj+i

vj+i ()

N, i {, }.

3 Case of constant coefcients

In this section we consider the case when all the coecients in system () are constant, that is, when

an = a, bn = b, n = , n = , n

N.

for every m

Then () is

xn = ynyn xn(a + bynyn),

yn = xnxnyn( + xnxn), n

N.

()

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 4 of 17

Assume that (xn, yn)n is a solution to system () such that () holds. Then we have

un = avn + b, vn = un + , n

N, ()

and

un = aun + a + b, ()

vn = avn + b + , n

N. ()

From ()-(), we obtain

unl = ul(a)n + (a + b) (a

)n a

= a

+ b + (a)n(ul( a) a b) a ()

N, l {, } when a = , while if a = , we have

unl = ul + (a + b)n, n

N, l {, }, ()

and we also have

vnl = vl(a)n + (b + ) (a

)n a

for n

=

b + + (a)n(vl( a) b ) a , ()

N, l {, } if a = , while if a = , we have

vnl = vl + (b + )n, n

N, l {, }. ()

Now we present formulae for solutions to system ().

Case a = . We have

xm = x

m

[productdisplay]

j=

n

uj

uj = x

m

[productdisplay]

j=

a + b + (a)j(u( a) a b)a + b + (a)j(u( a) a b) , ()

xm+ = x

m

[productdisplay]

j=

uj

uj+ = x

m

[productdisplay]

j=

a + b + (a)j(u( a) a b)a + b + (a)j+(u( a) a b), ()

ym = y

m

[productdisplay]

j=

vj

vj = y

m

[productdisplay]

j=

b + + (a)j(v( a) b )

b + + (a)j(v( a) b ) , ()

ym+ = y

m

[productdisplay]

j=

vj

vj+ = y

m

[productdisplay]

j=

b + + (a)j(v( a) b )

b + + (a)j+(v( a) b ) ()

for every m

N.

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 5 of 17

Case a = . We have

xm = x

m

[productdisplay]

j=

uj

uj = x

m

[productdisplay]

j=

u + (a + b)ju + (a + b)j , ()

xm+ = x

m

[productdisplay]

j=

uj

uj+ = x

m

[productdisplay]

j=

u + (a + b)ju + (a + b)(j + ), ()

ym = y

m

[productdisplay]

j=

vj

vj = y

m

[productdisplay]

j=

v + (b + )jv + (b + )j , ()

ym+ = y

m

[productdisplay]

j=

vj

vj+ = y

m

[productdisplay]

j=

v + (b + )jv + (b + )(j + ) ()

for every m

N.

4 Long-term behavior of solutions to system (14)

Before we formulate and prove the main results regarding the long-term behavior of well-dened solutions to system (), we quote the following well-known asymptotic formula which will be used in the proofs of the main results:

( + x) = x + O[parenleftbig]x[parenrightbig], as x . ()

We also dene the following quantities:

L := u( a

) a bu( a) a b , L :=

u( a) a b a(u( a) a b),

L := v( a

) b v( a) b , L :=

v( a) b a(v( a) b ).

Finally, we give another auxiliary result.

Lemma If a = , a + b = = b + . Then system () has two-periodic solutions.

Proof The equilibrium solution to system () is

un = =

a + b

a = , vn = v =

b +

a = , n

N. ()

From () and () it follows that

xn = a

(a + b)xn = xn, n

N, ()

and

yn = a

N, ()

as desired.

(b + )yn = yn, n

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 6 of 17

The next three results are devoted to the long-term behavior of well-dened solutions to system ().

Theorem Assume that |a| = and (xn, yn)n is a well-dened solution to system ().

Then the following statements are true.(a) If a + b = = b + and |a| < , then (xn, yn) converges to a, not necessarily

prime, two-periodic solution.(b) If u = u = (a + b)/( a), then the sequences (xm)m and (xm+)m are constant.

(c) If v = v = (b + )/( a), then the sequences (ym)m and (ym+)m are constant.

(d) If |a| > and u = (a + b)/( a) = u, then xm and |xm+| , as

m .

(e) If |a| > and u = (a + b)/( a) = u, then xm+ and |xm| , as

m .

(f) If |a| > and v = (a + b)/( a) = v, then ym and |ym+| , as

m .

(g) If |a| > and v = (a + b)/( a) = v, then ym+ and |ym| , as

m .

(h) If |a| > , u = (a + b)/( a) = u and |L| < , then xm , as m .

(i) If |a| > , u = (a + b)/( a) = u and |L| > , then |xm| , as m .

(j) If |a| > , u = (a + b)/( a) = u and L = , then (xm)m is constant.

(k) If |a| > , u = (a + b)/( a) = u and L = , then (xm)m and (xm+)m

are convergent.(l) If |a| > , u = (a + b)/( a) = u and |L| < , then xm+ , as m .

(m) If |a| > , u = (a + b)/( a) = u and |L| > , then |xm+| , as m .

(n) If |a| > , u = (a + b)/( a) = u and L = , then (xm+)m is constant.

(o) If |a| > , u = (a + b)/( a) = u and L = , then (xm+)m and

(xm+)m are convergent.(p) If |a| > , v = (b + )/( a) = v and |L| < , then ym , as m .

(q) If |a| > , v = (b + )/( a) = v and |L| > , then |ym| , as m .

(r) If |a| > , v = (b + )/( a) = v and L = , then (ym)m is constant.

(s) If |a| > , v = (b + )/( a) = v and L = , then (ym)m and (ym+)m

are convergent.(t) If |a| > , v = (b + )/( a) = v and |L| < , then ym+ , as m .

(u) If |a| > , v = (b + )/( a) = v and |L| > , then |ym+| , as m .

(v) If |a| > , v = (b + )/( a) = v and L = , then (ym+)m is constant.

(w) If |a| > , v = (b + )/( a) = v and L = , then (ym+)m and

(ym+)m are convergent.

Proof Let

pm = a

+ b + (a)m(u( a) a b) a + b + (a)m(u( a) a b) ,

a + b + (a)m(u( a) a b) a + b + (a)m+(u( a) a b),

pm =

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 7 of 17

b + + (a)m(v( a) b )

b + + (a)m(v( a) b ) ,

b + + (a)m(v( a) b )

b + + (a)m+(v( a) b )

(a) By using () we have

pm = + (a

)m(u( a) a b)(a + b) + (a)m(u( a) a b)(a + b)

= + (u u)( a)(a + b)(a)m + o[parenleftbig](a)m[parenrightbig], ()

pm =

)(u au a b)

a + b (a

)m(v( a) b )(b + ) + (a)m(v( a) b )(b + )

= + (v v)( a)(b + )(a)m + o[parenleftbig](a)m[parenrightbig], ()

qm =

qm = + (a

+ (a)m(v( a) b )(b + ) + (a)m+(v( a) b )(b + )

= + ( a

for suciently large m.

From ()-(), by using the condition |a| < and a well-known criterion for the con

vergence of products, the statement easily follows.(b) By using the condition u = u = (a + b)/( a) in () and (), the statement immediately follows.(c) By using the condition v = v = (b + )/( a) in () and (), the statement immediately follows.(d) By using the condition u = (a + b)/( a) = u, we get

pm = a

+ ba + b + (a)m(u( a) a b), ()

a + b + (a)m(u( a) a b)a + b . ()

Letting m in () and () and using the condition |a| > , we have pm and |pm| , from which along with () and () the statement easily follows.

(e) By using the condition u = (a + b)/( a) = u, we get

pm = a

a + ba + b + (a)m+(u( a) a b). ()

qm =

qm =

for m

N.

+ (a)m(u( a) a b)(a + b) + (a)m+(u( a) a b)(a + b)

= + ( a

)m + o[parenleftbig](a)m[parenrightbig], ()

)(v av b )

b + (a

)m + o[parenleftbig](a)m[parenrightbig] ()

pm =

+ b + (a)m(u( a) a b)a + b , ()

pm =

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 8 of 17

Letting m in () and () and using the condition |a| > , we have |pm| and pm , from which along with () and () the statement easily follows.

(f) By using the condition v = (a + b)/( a) = v, we get

qm =

b +

b + + (a)m(v( a) b ), ()

qm =

b + + (a)m(v( a) b )

b + . ()

Letting m in () and () and using the condition |a| > , we have qm and |qm| , from which along with () and () the statement easily follows.

(g) By using the condition v = (a + b)/( a) = v, we get

qm =

b + + (a)m(v( a) b )

b + , ()

qm =

b +

b + + (a)m+(v( a) b ). ()

Letting m in () and () and using the condition |a| > , we have |qm| and qm , from which along with () and () the statement easily follows.

(h), (i) Note that limm pm = L. Hence, from the assumptions |L| < , that is, |L| >

along with (), the statements easily follow.(j) The statement immediately follows by using the condition L = in ().(k) Since L = and by using (), we have that

pm = a

+ b + (a)m(u( a) a b) a + b (a)m(u( a) a b)

+ a

+b (a)m(u(a)ab)

= a

+b (a)m(u(a)ab)

= [parenleftbigg] + (a

+ b) (a)m(u( a) a b) + o

[parenleftbigg]

[parenrightbigg][parenrightbigg]. ()

From (), by using the condition |a| > and a well-known criterion for the convergence

of products, the statement easily follows.(l), (m) Note that limm pm = L. Hence, from the assumptions |L| < , that is, |L| >

along with (), the statements easily follow.(n) The statement immediately follows by using the condition L = in ().(o) Since L = and by using (), we have that

pm =

a + b + (a)m(u( a) a b) a + b (a)m(u( a) a b)

+ a

+b (a)m(u(a)ab)

= a

+b (a)m(u(a)ab)

= [parenleftbigg] + (a

+ b) (a)m(u( a) a b) + o

(a)m

[parenleftbigg]

(a)m

[parenrightbigg][parenrightbigg]. ()

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 9 of 17

From (), by using the condition |a| > and a well-known criterion for the convergence

of products, the statement easily follows.(p), (q) Note that limm qm = L. Hence, from the assumptions |L| < , that is, |L| >

along with (), the statements easily follow.(r) The statement immediately follows by using the condition L = in ().(s) Since L = and by using (), we have that

qm =

b + + (a)m(v( a) b )

b + (a)m(v( a) b )

=

+

b+ (a)m(v(a)b)

b+ (a)m(v(a)b)

= [parenleftbigg] + (

b + ) (a)m(v( a) b ) + o

[parenleftbigg]

[parenrightbigg][parenrightbigg]. ()

From (), by using the condition |a| > and a well-known criterion for the convergence

of products, the statement easily follows.(t), (u) Note that limm qm = L. Hence, from the assumptions |L| < , that is, |L| >

along with (), the statements easily follow.(v) The statement immediately follows by using the condition L = in ().(w) Since L = and by using (), we have that

qm =

b + + (a)m(v( a) b )

b + (a)m(v( a) b )

=

+

(a)m

b+ (a)m(v(a)b)

b+ (a)m(v(a)b)

= [parenleftbigg] + (

b + ) (a)m(v( a) b ) + o

[parenleftbigg]

[parenrightbigg][parenrightbigg]. ()

From (), by using the condition |a| > and a well-known criterion for the convergence

of products, the statement easily follows.

Let

M := u(u b a

)

u(u b a) , M :=

(a)m

v(v b)

v(v b) .

Theorem Assume that a = and (xn, yn)n is a well-dened solution to system (). Then the following statements are true.

(a) If |M| < , then xm and |xm+| , as m .

(b) If |M| > , then xm+ and |xm| , as m .

(c) If M = , then (xn)n is four-periodic.(d) If M = , then (xn)n is eight-periodic.(e) If |M| < , then ym and |ym+| , as m .

(f) If |M| > , then ym+ and |ym| , as m .

(g) If M = , then (yn)n is four-periodic.(h) If M = , then (yn)n is eight-periodic.

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 10 of 17

Proof First, note that since a = , from ()-() we have

xm = xMm, xm+ = xMm+, xm+ = xMm , xm+ =

x

Mm+ , ()

ym = yMm, ym+ = yMm+, ym+ = y

Mm , ym+ =

y

Mm+ , ()

for m

N. From () and () all the statements easily follow.

Let

N := u

u , N :=

v

v .

Theorem Assume that a = and (xn, yn)n is a well-dened solution to system (). Then the following statements hold true.

(a) If a + b = and |N| < , then xm and |xm+| , as m ;

(b) If a + b = and |N| > , then |xm| and xm+ , as m ;

(c) If a + b = and N = , then (xm)m and (xm+)m are constant;(d) If a + b = and N = , then (xm+i)m, i = , , are constant.(e) If a + b = and (u u)/(a + b) > , then |xm| , as m ;

(f) If a + b = and (u u)/(a + b) < , then xm , as m ;

(g) If a + b = and u = u, then (xm)m is constant;

(h) If a + b = and (u u)/(a + b) > , then |xm+| , as m ;

(i) If a + b = and (u u)/(a + b) < , then xm+ , as m ;

(j) If a + b = and u u = a + b, then (xm+)m is constant;

(k) If b + = and |N| < , then ym and |ym+| , as m ;

(l) If b + = and |N| > , then |ym| and ym+ , as m ;

(m) If b + = and N = , then (ym)m and (ym+)m are constant;(n) If b + = and N = , then (ym+i)m, i = , , are constant.(o) If b + = and (v v)/(b + ) > , then |ym| , as m ;

(p) If b + = and (v v)/(b + ) < , then ym , as m ;

(q) If b + = and v = v, then (ym)m is constant.

(r) If b + = and (v v)/(b + ) < , then ym+ , as m ;

(s) If b + = and (v v)/(b + ) > , then |ym+| , as m ;

(t) If b + = and v v = b + , then (ym+)m is constant.

Proof Let

rm = u + (a

+ b)mu + (a + b)m , rm =

u + (a + b)m u + (a + b)(m + ),

sm = v + (

b + )mv + (b + )m ,m =

v + (b + )mv + (b + )(m + ), m

N.

(a)-(d) Since in this case we have

xm = x[parenleftbigg]u u

m+

, xm+ = x[parenleftbigg] u u

m+

, m

N,

these statements easily follow.

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 11 of 17

(e), (f) By using () we have

rm = u + (a

+ b)m u + (a + b)m =

[parenleftbigg] + u

(a + b)m

[parenrightbigg][parenleftbigg] + u

(a + b)m

= [parenleftbigg] + u

(a + b)m + O

[parenleftbigg] m

[parenrightbigg][parenrightbigg][parenleftbigg] u

(a + b)m + O

[parenleftbigg] m

[parenrightbigg][parenrightbigg]

= + u u(a + b)m + O

[parenleftbigg] m

[parenrightbigg] ()

for suciently large m.From (), by using the fact that for every k

N

j , as m , ()

and a known criterion for convergence of products, the statements easily follow.(g) Using the condition u = u in (), the statement immediately follows.(h), (i) By using () we have

rm =

u + (a + b)mu + (a + b)(m + ) =

m

[summationdisplay]

j=k

[parenleftbigg] + u

(a + b)m

[parenrightbigg][parenleftbigg] + u + a

+ b (a + b)m

= [parenleftbigg] + u

(a + b)m

[parenrightbigg][parenleftbigg] u + a

+ b(a + b)m + O

[parenleftbigg] m

[parenrightbigg][parenrightbigg]

= + u u a

b(a + b)m + O

[parenleftbigg] m

[parenrightbigg] ()

for suciently large m.

From (), (), () and a known criterion for convergence of products, the statements easily follow.(j) Using the condition u = u + a + b in (), the statement immediately follows.(k)-(n) Since in this case we have

ym = y[parenleftbigg]v v

m+

, ym+ = y[parenleftbigg] v v

m+

, m

N,

these statements easily follow.(o), (p) By using () we have

sm = v + (

b + )m v + (b + )m =

[parenleftbigg] + v

(b + )m

[parenrightbigg][parenleftbigg] + v

(b + )m

= [parenleftbigg] + v

(b + )m

[parenrightbigg][parenleftbigg] v

(b + )m + O

[parenleftbigg] m

[parenrightbigg][parenrightbigg]

= + v v

(b + )m + O

[parenleftbigg] m

[parenrightbigg] ()

for suciently large m.

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 12 of 17

From (), (), () and a known criterion for convergence of products, the statements easily follow.(q) Using the condition v = v in (), the statement immediately follows.(r), (s) By using () we have

m =

v + (b + )mv + (b + )(m + ) =

[parenleftbigg] + v

(b + )m

[parenrightbigg][parenleftbigg] + v +

b + (b + )m

= [parenleftbigg] + v

(b + )m

[parenrightbigg][parenleftbigg] v +

b + (b + )m + O

[parenleftbigg] m

[parenrightbigg][parenrightbigg]

= + v v

b (b + )m + O

[parenleftbigg] m

[parenrightbigg] ()

for suciently large m.

From (), (), () and a known criterion for convergence of products, the statements easily follow.(t) Using the condition v = v + b + in (), the statement immediately follows.

5 Domain of undenable solutions to system (2)

In Section we proved that solutions to system (), for which xj = or yj = for some j {, }, are not dened. The set of all such initial values is characterized here.

Denition Consider the system of dierence equations

xn = f (xn, . . . , xns, yn, . . . , yns, n),

yn = g(xn, . . . , xns, yn, . . . , yns, n), n

N,

()

R, i = , s. The string of vectors

(xs, ys), . . . , (x, y), (x, y), . . . , (xn, yn),

where n , is called an undened solution of system () if

xj = f (xj, . . . , xjs, yj, . . . , yjs, j)

and

yj = g(xj, . . . , xjs, yj, . . . , yjs, j)

for j < n + , and xn+ or yn+ is not a dened number, that is, the quantity

f (xn, . . . , xns+, yn, . . . , yns+, n + )

or

g(xn, . . . , xns+, yn, . . . , yns+, n + )

is not dened.

where s

N, and xi, yi

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 13 of 17

The set of all initial values (xs, ys), . . . , (x, y) which generate undened solutions to system () is called domain of undenable solutions of the system.

The next result characterizes the domain of undenable solutions to system () when anbnnn = , n

N.

Theorem Assume that anbnnn = , n

N. Then the domain of undenable solutions

to system () is the following set:

U = [uniondisplay]

m

N

[braceleftbigg](x, x, y, y)

R :

xx = g f gm f m gm f m+()

or

xx = g f gm f m gm()

or

yy = f g f m gm f m()

or

yy = f g gm f m gm+()

[bracerightbigg]

[braceleftbig](x, x, y, y)

R :

x = or x = or y = or y = [bracerightbig], ()

where

fn(t) = ant + bn, gn(t) = nt + n, n

N.

Proof We have already proved that the set

(x, x, y, y)

R : x = or x = or y = or y = [bracerightbig]

belongs to the domain of undenable solutions to system ().

If xj = = yj, j = , (i.e., xn = = yn for every n ), then such a solution (xn, yn)n

is not dened if and only if

an + bnynyn = or n + nxnxn = ()

for some n

N, which is equivalent to

vn = bn/an or un = n/n ()

for some n

N.

Note that

f n() = bn/an and gn() = n/n, n

N. ()

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 14 of 17

We have

vm = (gm fm f g f)(v), () vm = (gm fm g f g)(u), ()

um = (fm gm g f g)(u), () um = (fm gm f g f)(v) ()

for m

From () and () we have that

bmam = vm = (gm fm f g f)(v)

for some m

N if and only if

xx = g f gm f m gm f m+(). ()

From () and () we have that

N if and only if

xx = g f gm f m gm(). ()

From () and () we have that

N if and only if

yy = f g gm f m gm+(). ()

From ()-() we see that the rst union in () also belongs to the domain of undenable solutions, nishing the proof of the theorem.

N.

N if and only if

yy = f g f m gm f m(). ()

From () and () we have that

bm+am+ = vm = (gm fm g f g)(u)

for some m

mm = um = (fm gm g f g)(u)

for some m

m+m+ = um = (fm gm f g f)(v)

for some m

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 15 of 17

Remark Quantities

g f gm f m gm f m+(), () g f gm f m gm(), ()

f g f m gm f m(), () f g gm f m gm+() ()

can be calculated for every m

N.

Indeed, note that

g f gm f m gm f m+() =

[parenleftBigg] m

j=

[productdisplay] gj f j+[parenrightbig]

[parenrightBigg](t)[vextendsingle][vextendsingle][vextendsingle]t=

, ()

g f gm f m gm() =

m

j=

[productdisplay] gj f j+[parenrightbig]

[parenrightBigg](t)[vextendsingle][vextendsingle][vextendsingle]t=gm()

, ()

f g f m gm f m() =

m

j=

[productdisplay] f j gj+[parenrightbig]

[parenrightBigg](t)[vextendsingle][vextendsingle][vextendsingle]t=f

m(), ()

f g gm f m gm+() =

[parenleftBigg] m

j=

[productdisplay] f j gj+[parenrightbig]

[parenrightBigg](t)[vextendsingle][vextendsingle][vextendsingle]t=

, ()

and also that

gj f j+[parenrightbig](t) =t

jaj+

bj+ jaj+

jj , j

N, ()

and

f j gj+[parenrightbig](t) =t

bj

ajj+

j+

ajj+

aj , j

N. ()

On the other hand, if

hj(t) = cjt + dj, j

N,

it is easy to see that

(h h hn)(t) =

[parenleftBigg] n

[productdisplay]

j=

cj[parenrightBigg]t +

n

[summationdisplay]

i=

dj

i

[productdisplay]

j=

cj, n

N. ()

From ()-() explicit formulas for the quantities in ()-() are easily obtained.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Stevi et al. Advances in Dierence Equations (2015) 2015:264 Page 16 of 17

Author details

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia. 2Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 3Faculty of Electrical Engineering, Belgrade University, Bulevar Kralja Aleksandra 73, Beograd, 11000, Serbia. 4CEITEC - Central European Institute of Technology, Brno University of Technology, Technick 3058/10, Brno, CZ-616 00, Czech Republic. 5FEEC - Faculty of Electrical Engineering and Communication, Department of Mathematics, Brno University of Technology, Technick 3058/10, Brno, CZ-616 00, Czech Republic.

Acknowledgements

The work of the rst and the second authors was supported by the Serbian Ministry of Education and Science, project III 41025. The work of the rst author was also supported by the Serbian Ministry of Education and Science, project III 44006. The work of the second author was also supported by the Serbian Ministry of Education and Science, project OI 171007. The work of the third author was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by project CZ.1.05/1.1.00/02.0068 nanced from the European Regional Development Fund. The third author was also supported by the project FEKT-S-14-2200 of Brno University of Technology.

Received: 11 June 2015 Accepted: 3 August 2015

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