Stevi´c Advances in Difference Equations (2017) 2017:169 DOI 10.1186/s13662-017-1227-x
RESEARCH
Open Access
Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation Stevo Stevi´c* * Correspondence:
[email protected] Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
Abstract We present some interesting facts connected with the following second-order difference equation: xn+2 – qn xn = fn ,
n ∈ N0 ,
where (qn )n∈N0 and (fn )n∈N0 are given sequences of numbers. We give some sufficient conditions for the existence of a unique bounded solution to the difference equation and present an elegant proof based on a combination of theory of linear difference equations and the Banach fixed point theorem. We also deal with the equation by using theory of solvability of difference equations. A global convergence result of solutions to a linear first-order difference equation is given. Some comments on an abstract version of the linear first-order difference equation are also given. MSC: Primary 39A06; secondary 39A45 Keywords: second-order difference equation; unique bounded solutions; fixed point theorem; linear first-order difference equation
1 Introduction Difference equations and systems of difference equations have been of a great interest in the last several decades. For some classical results in the research area, see, for example, [–], whereas some recent results can be found, for example, in [–] (see also the references therein). Book [] contains a lot of results obtained up to ,. It is a frequent situation that during some investigations in various areas of mathematics and science naturally appeared special cases of the following nonhomogeneous linear second-order difference equation: xn+ – qn xn = fn ,
n ∈ N ,
()
where (qn )n∈N and (fn )n∈N are sequences of real or complex numbers. Recall that equations of type () with fn = , n ∈ N , appeared in many problem books on classical analysis ([, ]), especially in those dealing with sequences and integrals (by using the integra© The Author(s) 2017. 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.
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tion in parts an integral In is frequently presented in terms of a linear function of In– ). However, this case is very simple, so the case fn ≡ , is more interesting. Although equation () is of second order and resembles the following differential equation: x (t) – q(t)x(t) = f (t),
t ≥ ,
()
it is not the one that corresponds quite well to it. The following second-order difference equation: xn – qn xn = fn ,
n ∈ N ,
where xn = xn+ – xn is the forward difference, is one of the discrete cousins of equation (), and is much more interesting for investigation than equation (). Nevertheless, equation () has some nice properties which seem are not so known and will be presented here. As a further motivation, recall that if q is a positive number, then the difference equation n ∈ N ,
xn+ – qxn = ,
()
where x , x are given numbers, is easily solved and it has the following general solution: √ √ xn = cˆ ( q)n + cˆ (– q)n ,
n ∈ N ,
()
where cˆ and cˆ are arbitrary constants ([, , , ]). By using equation () it is immediately seen that the following three statements hold. (a) If q ∈ (, ), then all the solutions to equation () converge to zero. (b) If q = then all the solutions to equation () are bounded. (c) If q > , then all the solutions to equation () are unbounded, except the trivial solution xn = ,
n ∈ N ,
which is obtained for the initial conditions x = x = . In terms of the boundedness these statements claim that all the solutions to equation () are bounded if and only if q ∈ (, ], while for the case q > there is only one bounded solution to the equation. It is a classical problem to see how bounded perturbations of the right-hand side of equation (), as well as of coefficient q influence on the boundedness character of the solutions of such obtained equations. We will present two interesting ways how the problem of the existence of a unique bounded solution can be solved for the case of the difference equation (), where (qn )n∈N and (fn )n∈N are two given bounded sequences. By l∞ we will denote the space consisting of all bounded sequences u = (un )n∈N with the supremum norm u ∞ = sup |un |. n∈N
It is well known that l∞ with norm () is a Banach space.
()
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The paper is partially based on several comments and ideas presented by the author at several international conferences and invited talks during the last several years, which has not been published so far. Some of the results could be known, but we could not find specific references for them. The paper is organized as follows. First, we consider the case when qn = q = for every n ∈ N and (fn )n∈N is a bounded sequence. Then we consider the case when the sequences (qn )n∈N and (fn )n∈N are bounded. By a nice combination of the theory of linear difference equations and the Banach fixed point theorem we present some sufficient conditions for the existence of a unique bounded solution to equation (). In the third section we will present another way for dealing with the unique existence problem by using only the theory of linear difference equations. We finish the paper by giving some comments on the form of general solution to an abstract version of the linear first-order difference equation.
2 Fixed point approach in dealing with equation (1) In this section we deal with the problem of the existence of a unique bounded solution to the difference equation (). To do this we use fixed point theory. First we deal with the case when qn = q = for every n ∈ N and (fn )n∈N is a bounded sequence. Before we formulate and prove the main result in the section we need an auxiliary result which is incorporated into the following lemma. The lemma is essentially folklore, but we will give a proof of it for completeness, for the benefit of the reader and also as a good motivation and better understanding of some ideas appearing in the proof of the main result. Lemma Consider the difference equation xn+ – qxn = fn ,
n ∈ N ,
()
where q ∈ C \ {}, x and x are given complex numbers, and (fn )n∈N is a given sequence of complex numbers. Then the general solution to equation () is given by the following formula: n– √ n xn = ( q) c + k=
fk √ k+ ( q)
n– (–)k fk √ n + (– q) d + , √ ( q)k+
where c and d are arbitrary numbers, and
n ∈ N ,
()
k=
√
q is one of two possible roots of q.
Proof The difference equation () can be solved. We will demonstrate it by using the method of variation of constants ([, ]). Namely, based on the form of the general solution to the homogeneous equation () given in (), we seek the general solution to equation () in the form √ √ xn = cn ( q)n + dn (– q)n ,
n ∈ N ,
where (cn )n∈N and (dn )n∈N are two (undetermined) sequences.
()
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Further, we pose the following condition: √ √ √ √ xn+ = cn+ ( q)n+ + dn+ (– q)n+ = cn ( q)n+ + dn (– q)n+ ,
()
for n ∈ N , that is, √ √ (cn+ – cn )( q)n+ + (dn+ – dn )(– q)n+ = ,
()
for n ∈ N . Using () with n → n + along with () into equation () is obtained √ √ √ √ cn+ ( q)n+ + dn+ (– q)n+ – q cn ( q)n + dn (– q)n = fn , that is, √ √ (cn+ – cn )( q)n+ + (dn+ – dn )(– q)n+ = fn ,
()
for n ∈ N . For each fixed n ∈ N , equalities () and () can be regarded as a two-dimensional linear system in the variables cn+ – cn and dn+ – dn . By solving the system the following is easily obtained: cn+ – cn =
fn √ n+ ( q)
and dn+ – dn =
(–)n fn , √ ( q)n+
n ∈ N ,
from which it follows that cn = c +
n– k=
fk √ ( q)k+
()
and dn = d +
n– (–)k fk , √ ( q)k+
()
k=
for n ∈ N . Using () and () into () we get equation (). That () represents the general solution to () follows from the facts that the sequence √ xpn := ( q)n n–
k=
fk √ (–)k fk √ k+ + (– q)n √ ( q) ( q)k+ n–
k=
is a particular solution to equation (), which is easily verified, while √ √ xhn = c ( q)n + d (– q)n , is the general solution to the corresponding homogeneous equation ([, ]).
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The following result solves the problem of existence of a unique bounded solution to equation () for the case qn = q, n ∈ N , when |q| > . Theorem Consider the difference equation (), where q is a complex number such that |q| > and f := (fn )n∈N is a given bounded sequence of complex numbers. Then there is a unique bounded solution to the difference equation. Proof By Lemma we know that the general solution to the difference equation () is given by equation (). By using equation () for even and odd indices it follows that xn = q
n
c + d +
n– k=
( + (–)k )fk √ ( q)k+
()
and
√
xn+ = ( q)
n+
n ( – (–)k )fk c – d + , √ ( q)k+
()
k=
for every n ∈ N . Now note that from the assumptions of the theorem we have ∞ ∞ ( + (–)k )f f ∞ f ∞ k ≤ √ k+ = √ √ √ ( q)k+ ( |q|) |q|( |q| – ) k= k= and ∞ ∞ ( – (–)k )f f ∞ f ∞ k √ k+ = √ √ √ k+ ≤ ( q) ( |q|) |q|( |q| – ) k= k= so that these two series are absolutely convergent. Using this fact, (), () and the assumption |q| > we see that for a bounded solution (xn )n∈N to equation () we must have c + d = –
∞ ∞ fj ( + (–)k )fk √ k+ = – √ j+ ( q) ( q) j=
()
k=
and c – d =
∞ ∞ fj+ ((–)k – )fk √ k+ = – √ j+ ( q) ( q) j=
()
k=
(at the moment we do not know if a bounded solution to () exists, but if it does, then its subsequences (xn ) and (xn+ ) will be bounded, which will imply that () and () must hold). From () and () it follows that ∞
c = –
fk √ ( q)k+ k=
()
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and ∞
d =
(–)k+ fk . √ ( q)k+
()
k=
If we use () and () in (), we get ∞
∞
fk √ √ (–)k+ fk xn = –( q)n √ k+ + (– q)n √ ( q) ( q)k+ k=n
√ = ( q)n
∞ k=n
k=n
n+k+
((–) – )fk , √ k+ ( q)
()
for n ∈ N . A direct calculation shows that sequence (xn )n∈N defined by () is a solution to equation (). On the other hand, by using the assumptions of the theorem we easily get |xn | ≤ √
f ∞ < ∞, √ |q|( |q| – )
n ∈ N ,
from which the boundedness of (xn )n∈N , follows. From this and since (c , d ) is a unique solution to the linear system ()-(), (xn )n∈N defined by () is a unique bounded solution to equation (). Now we are in a position to formulate and prove the main result in the section in an elegant way. Theorem Consider the difference equation (), where < a ≤ qn ≤ b,
n ∈ N ,
()
or –b ≤ qn ≤ –a < –,
n ∈ N ,
()
for some positive numbers a and b, and (fn )n∈N is a bounded sequence of complex numbers. Then equation () has a unique bounded solution. Proof We will prove the theorem under condition (). The proof when () holds is similar/dual so is omitted. Let q be a positive number such that q ∈ max a, (b + )/ , b .
()
Write equation () in the following form: xn+ – qxn = (qn – q)xn + fn , for n ∈ N .
()
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Let A be the following operator defined on the class of all sequences:
∞
((–)n+k+ – )((qk – q)uk + fk ) A(u) = ( q) √ ( q)k+ √
n
k=n
.
()
n∈N
If u ∈ l∞ , then from () it follows that ∞ √ n+k+
((–) – )((q – q)u + f ) k k k
A(u) = sup ( q)n √ ∞ k+ ( q) n∈N k=n
∞ (qk + q)|uk | + |fk | ≤ sup √ | q|k+–n n∈N k=n
(b + q) u ∞ + f ∞ < ∞, ≤ √ √ q( q – ) which means that the operator A maps the Banach space l∞ into itself. On the other hand, for every u, v ∈ l∞ we have ∞ √ n+k+
((–) – )(q – q)(u – v ) k k k
A(u) – A(v) = sup ( q)n √ k+ ∞ ( q) n∈N k=n ∞ √ (qn+j – q)(un+j – vn+j ) n = sup ( q) √ ( q)n+j+ n∈N j= ≤ sup
∞ |qn+j – q||un+j – vn+j |
n∈N j=
≤
qj+
max{q – a, b – q} u – v ∞ . q–
()
Since a > we have q – a < q – . On the other hand, from () we see that < b – q < q – . From this and (), we have
A(u) – A(v) ≤ qˆ u – v ∞ , ∞
()
for every u, v ∈ l∞ , and for qˆ :=
max{q – a, b – q} ∈ (, ), q–
that is, A : l∞ → l∞ is a contraction. By the Banach fixed point theorem we see that the operator has a unique fixed point, say x∗ = (x∗n )n∈N ∈ l∞ , that is, A(x∗ ) = x∗ or equivalently ∞
√ ((–)n+k+ – )((qk – q)x∗k + fk ) , x∗n = ( q)n √ ( q)k+
()
k=n
for n ∈ N . A direct calculation shows that this bounded sequence satisfies the difference equa tion () for every n ∈ N , from which the theorem follows.
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Remark Beside the choice of operator (), which is naturally imposed, a crucial point in the proof of Theorem is the choice of constant q in () to get the contractivity of the operator. It is also expected that a modification of the above arguments can be applied to some other related difference equations.
3 Equation (1) versus the linear first-order difference equation Another approach in dealing with equation () is to note that it is obtained from a linear first-order difference equation by scaling the indices. The linear first-order difference equation is of the form xn+ = qn xn + fn ,
n ∈ N ,
()
where (qn )n∈N and (fn )n∈N are arbitrary real (or complex) sequences and x ∈ R (or x ∈ C). The main feature of the equation is that it is solvable. How equation () is solved can be found, for example, in [, , ]. On periodic solutions to equation () see []. If qn = for every n ∈ N , then general solution to equation () can be obtained, for exam ple, as follows. Dividing both sides of () by nj= qj , we obtain xn fn xn+ n = n– + n , j= qj j= qj j= qj
n ∈ N .
()
Summing the equalities in () from to n – , we easily get the following formula for a general solution to equation ():
xn =
n–
qj x +
j=
n– i=
i
fi
j= qj
.
()
It is important to note that many nonlinear difference equations and systems can be reduced to some special cases of equation (), which means that they are solvable too. Some interest in the area has been renewed recently. For some recent classes of solvable difference equations see, for example, [, , , , , , ], while some related systems of difference equations can be found in [, , , ]. For some recent results on solvable product-type systems of difference equations, see [, –] and the references therein. For some classical equations and systems which can be reduced to () or solved by some other methods; see, for example, [, , , , ]. How useful and powerful equation () is shows the following small but nice result, which is an old result by the author which has never been published so far but has been presented at several talks. Theorem Consider equation (). Assume that < qn ≤ p < , for n ∈ N , (fn )n∈N is a sequence of real numbers, and that there is a finite lim
n→∞
fn . – qn
Then every real solution (xn )n∈N to the equation is convergent.
()
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Proof Let the limit in () be equal to l. Since qn = for every n ∈ N we see that every solution to equation () can be written in the form as in (). Since ≤ n– < n , n p j= qj j= qj
n ∈ N ,
it follows that the sequence Qn = n– j=
qj
n ∈ N,
,
increasingly tends to +∞ as n → ∞. Using the Stoltz theorem ([, ]) and condition () we have x + lim xn = lim
n→∞
n→∞
n–
fi i= i q j= j
n–
fn– n–
= lim
j= qj
j= qj
–qn– n→∞ n– q j= j
= l,
which implies that limn→∞ xn = l, for every solution to equation (), from which the theorem follows. Remark Special cases of Theorem frequently appear at problem books or students’ competitions (see, for example, []), but in all the cases that we have met so far the existence of the limit limn→∞ fn is assumed. However, if () holds, then the limit need not exist. For example, if we assume that sequences (qn )n∈N and (fn )n∈N are defined as follows: qn = q ,
qn+ = q ,
fn = f ,
fn+ = f ,
n ∈ N ,
where the following conditions are satisfied: f f = , – q – q f = = f = f and q , q ∈ (, ), then it is clear that lim
n→∞
f fn = , – qn – q
but there is no limn→∞ fn . If we assume that inf |qn | =: q > ,
()
n∈N
then n– qj → +∞, j=
as n → ∞
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(we can assume here also lim infn→∞ |qn | > ). Hence, in this case a solution to equation () will be bounded, only if
x = –
∞
i
fi
j= qj
i=
,
()
and using () in (), we see that it will be given by the following formula:
xn = –
n–
qj
∞
j=
i
fi
j= qj
i=n
n ∈ N .
,
()
In order that () holds the convergence of the series in () is necessary, which will be so, for example, if (fn )n∈N is a bounded sequence. Now note that if (xn )n∈N is a solution to equation (), then its subsequences (xn )n∈N and (xn+ )n∈N are solutions to the equations n ∈ N ,
yn+ = qn yn + fn ,
()
and n ∈ N ,
yn+ = qn+ yn + fn+ ,
()
respectively, which are nothing but two difference equations of the form in (), and consequently they are solvable. By applying the arguments from () to (), it is easy to see that the following result holds. Theorem Consider equation (), where condition () holds and (fn )n∈N is a bounded sequence. Then the equation has a unique bounded solution given by the formulas
xn = –
n– j=
xn+ = –
n– j=
qj
∞
i
i=n
qj+
∞ i=n
fi
j= qj
i
,
fi+
j= qj+
,
for every n ∈ N . Remark Theorem also shows that equation () has a unique bounded solution if () holds and (fn )n∈N is a bounded sequence, and gives it explicitly, although in not so nice way. Moreover, the conditions in Theorem are somewhat weaker so that its result is somewhat stronger than the one in Theorem , which again shows the importance and usefulness of equation (). Nevertheless, both approaches in dealing with equation () are interesting, each of them in its own way.
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3.1 An abstract form of equation (29) Equation () is not only ‘solvable’ for the case of real or complex initial values and coefficients. Namely, let S be a set equipped with two binary operations and ⊕ (two maps which send elements of the Cartesian product S × S to S), or in terminology of abstract algebra, let (S, ) and (S, ⊕) be two groupoids/magmas, such that operation is leftdistributive over ⊕, that is, x (y ⊕ z) = (x y) ⊕ (x z),
()
for every x, y, z ∈ S. Then the ‘difference’ equation xn = (an– xn– ) ⊕ bn– ,
n ∈ N,
()
where x ∈ S, an , bn ∈ S, n ∈ N , is solvable. Indeed, for n = , we get x = (a x ) ⊕ b . From equality () with n = along with (), is obtained x = a (a x ) ⊕ b ⊕ b = a (a x ) ⊕ (a b ) ⊕ b . An inductive argument shows that xn = an– an– · · · (a x ) · · · ⊕ an– an– · · · (a b ) · · · ⊕ · · · ⊕ (an– bn– ) ⊕ bn– ,
()
for every n ∈ N, which means that () is a formula for general solution to equation () under the posed conditions. If operation has precedence over ⊕, that is, if x y ⊕ z = (x y) ⊕ z, for every x, y, z ∈ S, then equation () can be written in the following form: xn = an– an– · · · (a x ) · · · ⊕ an– an– · · · (a b ) · · · ⊕ · · · ⊕ an– bn– ⊕ bn– ,
n ∈ N.
If further (S, ) is a semigroup; a groupoid where operation is associative, that is, (x y) z = x (y z), for every x, y, z ∈ S, then from the well known fact that in this case for every set d , d , . . . , dk of elements from S the product k j=
dj := d d · · · dk
()
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is unambiguous, that is, the same element is obtained regardless of how parentheses are inserted in product (), equation () can be written in the following form: xn = an– an– · · · a x ⊕ an– an– · · · a b ⊕ · · · ⊕ an– bn– ⊕ bn– n– n– n––j = an––j x ⊕ an––i bj , n ∈ N, j=
j=
i=
where if d , d , . . . , dk ∈ S, then k
dj := d ⊕ d ⊕ · · · ⊕ dk .
()
j=
Note that in () and () the order of elements is important, since we do not assume commutativity of these operations on S. If further the operation is commutative, that is, x y = y x, for every x, y ∈ S, then equation () can be written in the following form: n– n– n– xn = aj x ⊕ a i bj , j=
j=
n ∈ N,
i=j+
from which if S = R or S = C and and ⊕ are multiplication and addition in R or C, respectively, is obtained equation ().
Competing interests The author declares that he has no competing interests. Author’s contributions The author has contributed solely to the writing of this paper. He read and approved the manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 13 March 2017 Accepted: 1 June 2017 References 1. Brand, L: A sequence defined by a difference equation. Am. Math. Mon. 62(7), 489-492 (1955) 2. Brand, L: Differential and Difference Equations. Wiley, New York (1966) 3. Jordan, C: Calculus of Finite Differences. Chelsea, New York (1956) 4. Karakostas, G: Convergence of a difference equation via the full limiting sequences method. Differ. Equ. Dyn. Syst. 1(4), 289-294 (1993) 5. Krechmar, VA: A Problem Book in Algebra. Mir, Moscow (1974) 6. Levy, H, Lessman, F: Finite Difference Equations. Dover, New York (1992) 7. Mitrinovi´c, DS, Adamovi´c, DD: Sequences and Series. Nauˇcna Knjiga, Beograd (1980) (in Serbian) 8. Mitrinovi´c, DS, Keˇcki´c, JD: Methods for Calculating Finite Sums. Nauˇcna Knjiga, Beograd (1984) (in Serbian) 9. Agarwal, RP, Popenda, J: Periodic solutions of first order linear difference equations. Math. Comput. Model. 22(1), 11-19 (1995) 10. Berg, L, Stevi´c, S: On some systems of difference equations. Appl. Math. Comput. 218, 1713-1718 (2011) 11. Papaschinopoulos, G, Stefanidou, G: Asymptotic behavior of the solutions of a class of rational difference equations. Int. J. Difference Equ. 5(2), 233-249 (2010)
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