Bounded Solutions to Nonhomogeneous Linear Second-Order Difference Equations

Transcription

1 S S symmetry Article Bounded Solutions to Nonhomogeneous Linear Second-Order Difference Equations Stevo Stević, Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 000 Beograd, Serbia; sstevic@ptt.rs Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah 589, Saudi Arabia Received: 9 September 07; Accepted: 08 October 07; Published: 4 October 07 Abstract: By using some solvability methods and the contraction mapping principle are investigated bounded, as well as periodic solutions to some classes of nonhomogeneous linear second-order difference equations on domains N 0, Z \ N and Z. The case when the coefficients of the equation are constant and the zeros of the characteristic polynomial associated to the corresponding homogeneous equation do not belong to the unit circle is described in detail. Keywords: linear second-order difference equation; bounded solution; contraction mapping principle; integer domain MSC: 39A06; 39A; 39A45. Introduction Let Z denote the set of all integers, N k := {n Z : n k}, k Z, and N = N. Investigations of difference equations and systems of difference equations have been conducted for a long time see, for example, [ 37] and the references therein. The solvability of the equations and systems is one of the oldest topics in the area. It is always nice to have formulas for solutions to the equations and systems for themselves, but also since they can frequently help in studying of the long-term behavior of the solutions, as is the case in [7,33,35]. Many classical results, including the ones on solvability of the equations and systems, can be found in the following classical books: [8,4 6,9,0]. Many of solvable difference equations and systems essentially use the solvability of the linear first-order difference equation, that is, of the following difference equation x n+ = q n x n + f n, n N 0, where q n n N0 and f n n N0 are given sequences. We will mention here just a few recent examples; the interested reader can find many other examples in the list of the references of the mentioned papers. In [33] was studied the following difference equation x n = a n x n k b n + c n x x n k, n N 0, which is transformed to an equation of the form in by using the change of variables y n = /x n x x n k+, while [3] studies the following difference equation x n+ = x n x n k x n k+ a + bx n x n k, n N 0, Symmetry 07, 9, 7; doi:0.3390/sym9007

2 Symmetry 07, 9, 7 of 3 which is transformed to an equation of the form in by using the change of variables y n = /x n+ x n k+ for an extension of the equation see [3]. Essentially the same ideas and methods were used in [3,35], while [] studies a special case of Equation in another way. In fact, all the papers use our ideas and methods from a 004 note. Paper [7] presents several related methods and can be considered as a representative one, where a comprehensive list of relevant references on solvability is given. It is also worthy to mention that some nonlinear systems of difference equations were solved by reducing them, by using some suitable changes of variables, to solvable linear ones see, for example, [7,34], as well as the related references therein. The solvability of some product-type equations [7] and systems [8,36,37] has been also shown by using some solvable linear ones, although in a more complex way. In fact, some of the results in papers [8,36,37] use special cases of the following equation z n = b n z a n, n N 0, which is a product-type analog of Equation. All above mentioned examples show the importance of Equation. Here, we will frequently use various things connected to Equation. Some recent applications of this and related solvable equations can be found in [5,6]. Let us also mention that beside showing the solvability of difference equations and systems by finding closed-form formulas for their solutions, in the cases when it is not possible to find them, one can try to find some of their invariants which can be also useful in studying of the long-term behavior of their solutions [,]. Motivated by the recent studies of the solvability, quite recently in [30], we have studied, among other problems, the existence of bounded solutions to the difference equation x n+ q n x n = f n, n N 0, 3 in two different ways. Applying classical method of variation of constants it is easily shown that in the case q n = q C \ {0}, Equation 3 has the general solution in the following form: x n = q c n 0 + q k+ + q n d 0 + k q k+, n N 0, 4 where c 0 and d 0 are arbitrary complex numbers, and q is one of two possible roots of q. By using Formula 4, as well as another method, we have shown in [30], among other results, that the equation in the case q n = q, n N 0, has a unique bounded solution in the case when q >, and used the obtained formula for the bounded solution as a motivation for introducing an operator which along with the contraction mapping principle [4] helps in showing the existence of a unique bounded solution to Equation 3 under some conditions posed on the sequence q n n N0. It is a natural problem to try to use the same ideas and methods in investigating of bounded solutions to some other classes of linear and nonlinear difference equations. One of the aims of the paper is to present some related results to those in [30] for the case of the difference equation x n+ + p n x n+ + q n x n = f n, n N 0. 5 This equation is one of the most important and widely studied difference ones, since it models many real-life quantities and processes, for example, the amplitude of oscillation of the weights on a discretely weighted vibrating string [3] pp For some classical results and methods for studying Equation 5, see, for example, [6,6], as well as the references therein. Note also that by using the differences x n = x n+ x n and x n = x n+ x n+ + x n the equation can be written in the following form x n + p n + x n + p n + q n + x n = f n, n N 0,

3 Symmetry 07, 9, 7 3 of 3 from which it immediately follows that the equation is a discrete variant of a linear second-order differential equation. We first investigate the case when p n = p, q n = q = 0, n N 0, 6 and f n n N0 is a bounded sequence, and after that the case when p n n N0 and q n n N0 are nonconstant sequences. The case when 6 holds can be regarded as a folklore one, but it is difficult to find many of the information provided here in the literature, especially at one place. The case when the zeros of the characteristic polynomial associated to the corresponding homogeneous equation do not belong to the unit circle is described in detail. It should be pointed out that when one of the zeros belongs to the circle then, as usual, very different situations appears. Recall that if in Equation q n =, n N 0, then, the bounded sequence f n n N0 highly influences on the behavior of the solutions to the equation. Namely, a solution to the equation can converge, diverge to infinity, the limit set can be even a whole interval see, for example, [9,4], as well as [] for the case of metric spaces, or it can be even a more complicated set. As in [30], we first use some solvability methods and then the contraction mapping principle in showing the existence of a unique bounded solution to Equation 5 under some conditions posed on the coefficients of the equation. For some other applications of fixed-point theorems in studying difference equations and systems, see, for example, [0,7,8,5] and the related references therein. Note that beside the contraction mapping principle, very frequent situation is application of a variant of the Schauder fixed-point theorem [0,7,8], and since recently the Darbo fixed-point theorem [38] which uses the notion of measure of non-compactness [5]. It should be noted that many of these papers essentially use a similar idea, that is, a combination of a solvability method, which is frequently hidden by some summations, and a fixed point theorem. The existence of periodic solutions in the case when p n n N0 and q n n N0 are constant sequences, while f n n N0 is a periodic sequence is also studied, as well as the relationship between the periodic and non-periodic ones, which is a natural continuation of the investigations in [9]. In our recent paper [9] we have also studied bounded solutions to Equation, but on the set of all integers Z, which has motivated us to conduct a similar investigation for the case of Equation 5. Hence, beside studying bounded solutions on domain N 0, it will be also done on domains Z \ N and Z. One of the reasons, why instead of the domain Z \ N is chosen Z \ N is found in the fact that the initial, that is, end values for the sets N 0 and Z \ N are the same, so that the domains patch each other well. Let us mention that a part of our investigations in [9,30] are motivated by a problem from [9]. Let S Z be an unbounded set. Then the space of bounded sequences f = f n n S on S with the supremum norm f,s = sup f n, 7 n S is Banach s, and is usually denoted by l S. Throughout the paper we will simply use the notations f and l, no matter which set S is used, since at each point it will be clear what the set is. We will also use the standard convention l j=m a j = 0, when m, l Z are such that l < m. It is said that a sequence x n n Nk converges geometrically exponentially to a sequence x n n Nk if there are M > 0 and q [0,, such that x n x n Mq n, for n N k,

4 Symmetry 07, 9, 7 4 of 3 while a sequence x n n Z\Nk converges geometrically exponentially to a sequence x n n Z\Nk if there are M > 0 and q [0,, such that x n x n Mq n, for n k +.. Bounded Solutions to Equation 5 on N 0 First, we prove an auxiliary result in a standard way, for the case when p n n N0 and q n n N0 are constant sequences. The result can be obtained from a formula for the general solution to Equation 5 when the fundamental set of solutions to the corresponding homogeneous equation is known. We will give a proof of it for the completeness, and to avoid frequent troubles with indices which lead to some minor inaccuracies related to the formula as it is the case in [0]. Some consequences of the lemma, which should be folklore, are given. Lemma. Consider the equation x n+ + px n+ + qx n = f n, n N 0, 8 where p C, q C \ {0}, and f n n N0 is a sequence of complex numbers. Then the following statements are true. a If p = 4q, then the general solution to Equation 8 is given by the following formula x n = λ n c 0 λ λ + λ n d 0 + λ λ, 9 for n N 0, where c 0 and d 0 are arbitrary complex numbers, and λ, = p ± p 4q. 0 b If p = 4q, then the general solution to Equation 8 is given by the following formula x n = λ c n 0 k + λ k+ + nλ n d 0 + λ k+, for n N 0, where c 0 and d 0 are arbitrary complex numbers, and λ = p/. Proof. a We solve the equation by the method of variation of constants [8,0]. As it is well-known, the corresponding homogeneous equation, in this case, has the general solution in the following form: x n = cλ n + dλn, n N 0, where λ, are the zeros of the characteristic polynomial P λ = λ + pλ + q, associated to the homogeneous equation, from which 0 follows. Hence, the general solution to 8 is searched for in the following form: where c n n N0 and d n n N0 are two undetermined sequences. x n = c n λ n + d nλ n, n N 0, 3

5 Symmetry 07, 9, 7 5 of 3 The following condition is posed for n N 0, that is, x n+ = c n+ λ n+ + d n+ λ n+ = c n λ n+ + d n λ n+, 4 c n+ c n λ n+ + d n+ d n λ n+ = 0, 5 for n N 0. Employing 3, 4, as well as 4 where n is replaced by n + in 8, and using that p = λ + λ and q = λ λ, we get that is, c n+ λ n+ + d n+ λ n+ λ + λ c n λ n+ + d n λ n+ + λ λ c n λ n + d nλ n = f n, c n+ c n λ n+ + d n+ d n λ n+ = f n, 6 for n N 0. For each fixed n N 0, 5 and 6 jointly can be regarded as a two-dimensional linear system in variables c n+ c n and d n+ d n. By solving the system it is easily obtained c n+ c n = from which it follows that f n λ n+ λ λ and d n+ d n = f n λ n+ λ λ, n N 0, c n = c 0 λ λ and d n = d 0 + λ λ, 7 for n N 0. Using 7 into 3 we get 9. That 9 represents the general solution to 8, follows from the fact that the sequence λ x n := n k λ n k f λ λ k is a particular solution to Equation 8, which is easily verified, while x h n = c 0 λ n + d 0λ n, is the general solution to the corresponding homogeneous equation [8,0]. b The corresponding homogeneous equation, in this case, has the general solution in the following form: x n = cλ n + dnλ n, n N 0, where λ is the double zero of polynomial, that is, λ = p/. So, the general solution to 8 is looked for in the following form: where c n n N0 and d n n N0 are two undetermined sequences. x n = c n λ n + d n nλ n, n N 0, 8

6 Symmetry 07, 9, 7 6 of 3 The following condition is posed for n N 0, that is, x n+ = c n+ λ n+ + d n+ n + λ n+ = c n λ n+ + d n n + λ n+, 9 c n+ c n λ n+ + d n+ d n n + λ n+ = 0, n N 0. 0 Employing 8, 9, as well as 9 where n is replaced by n + in 8, and using that p = λ and q = λ, we get that is, c n+ λ n+ + d n+ n + λ n+ λc n λ n+ + d n n + λ n+ + λ c n λ n + d n nλ n = f n, c n+ c n λ n+ + d n+ d n n + λ n+ = f n, n N 0. For each fixed n N 0, equalities 0 and can be regarded as a two-dimensional linear system in variables c n+ c n and d n+ d n. By solving the system it is easily obtained from which it follows that c n+ c n = n + f n λ n+ and d n+ d n = f n λ n+, n N 0, k + f c n = c 0 k λ k+ and d n = d 0 + λ k+, n N 0. Using into 8 we get. That represents the general solution to 8, follows from the facts that the sequence n k x n := λ k+ n, n N 0, is a particular solution to difference Equation 8, which is easily verified, while the sequence x h n = c 0 λ n + d 0 nλ n, n N 0, is the general solution to the corresponding homogeneous difference equation, as desired. Remark. If q = 0, then Equation 8 is essentially reduced to Equation, when p = 0, or to a very simple equation if p = 0, which is the reason why the condition q = 0 is posed in Lemma. Corollary. Consider Equation 8 where p C, q C \ {0}, x 0 and x are complex numbers, and f n n N0 is a sequence of complex numbers. Then the following statements are true. a If p = 4q, then the solution to Equation 8 with the initial values x 0 and x is given by the following formula x n = λ λ λ n λ x 0 x for n N 0, where λ, are given by 0. λ k+ + λ n x λ x 0 + λ k+, 3

7 Symmetry 07, 9, 7 7 of 3 b If p = 4q, then the solution to Equation 8 with the initial values x 0 and x is given by the following formula x n = λ x n 0 for n N 0, where λ = p/. k + λ k+ + nλ x λx 0 +, 4 Proof. a Using Formula 9 with n = 0,, and by some calculations, we see that it must be c 0 + d 0 = x 0, λ c 0 + λ d 0 = x. By solving the two-dimensional linear system, we obtain c 0 = λ x 0 x λ λ, d 0 = x λ x 0 λ λ. 5 Using 5 in 9 is obtained 3. b From with n = 0,, and some calculations, we see that it must be from which it follows that Using 6 in is obtained 4. c 0 = x 0, c 0 + d 0 = x λ, c 0 = x 0, d 0 = x λx 0. 6 λ Remark. Corollary, which essentially includes Lemma, can be also obtained by another standard method, the method of decomposition. We would like to point out that the method produces a slightly different formula. Namely, Equation 8 can be written in the following form: By using the change of variables Equation 7 becomes x n+ λ x n+ = λ x n+ λ x n + f n, n N 0. 7 y n = x n+ λ x n, n N 0, y n+ = λ y n + f n, n N 0, 8 which is a special case of Equation, so it is solvable in closed-form, and its solution is from which it follows that y n = λ n y 0 + f j λ j, n N 0, j=0 x n = λ x + λ n x λ x 0 + f j λ n j, n N. 9 j=0

8 Symmetry 07, 9, 7 8 of 3 By solving Equation 9 and after some calculation, in the case λ = λ, we obtain x n = x λ x 0 λ n x λ x 0 λ n n + λ λ j=0 λ n j λ n j f j, n N, 30 λ λ which, on the first site, seems a bit different from the formula in 3. However, since j=0 λ n j λ n j n f j = λ λ j=0 λ n j λ n j f j + f λ λ, λ λ we see that Formulas 3 and 30 are the same. The same situation appears in the case λ = λ. We leave the verification of the fact as an exercise. In fact, a similar situation appears at several points, and at some different contexts, in the paper. We will not mention them, and suggest the reader to have the remark on his mind. The following folklore result is another consequence of Lemma. Corollary. Consider Equation 8, where the zeros λ, of polynomial satisfy the condition M := max{ λ, λ } <, 3 and f n n N0 is a bounded sequence of complex numbers. Then every solution to the equation is bounded. Proof. According to Lemma we know that in the case p = 4q, the general solution to difference Equation 8 is given by Formula 9, while if p = 4q, the general solution is given by. Assume first that p = 4q. Then, by using 9, we have x n λ c n 0 + M n c 0 + d 0 + c 0 + d 0 + λ k+ + λ n d 0 + λ λ f λ λ f λ λ M, λ n k + λ n k λ k+ λ λ for every n N 0, from which the result follows, in this case. Now assume that p = 4q. Since in this case λ = λ = λ = p/ and λ = M, by using, we have x n λ c n 0 + n k λ k+ + n λ n d 0 n λ n c 0 + d 0 + f s λ s s=0 c 0 + d 0 sup nm n + f n N M, for every n N 0, from which along with the boundedness of the sequence nm n n N, the result follows, in this case. If the sequence f n n N0 is T-periodic, that is, f n = f n+t, n N 0, 3

9 Symmetry 07, 9, 7 9 of 3 for some T N for T = is said that f n is eventually constant [3], a natural question is if Equation 8 in this case has periodic solutions, and if so what is the relation between the periodic ones and the other solutions [9]. The following result gives an answer to the question. Theorem. Consider Equation 8, where the zeros λ, of polynomial satisfy condition 3 and f n n N0 is a T-periodic sequence. Then the following statements hold. a There is a unique T-periodic solution to Equation 8. b All the solutions to Equation 8 converge geometrically to the periodic one. Proof. a If x n n N0 is a T-periodic solution to Equation 8, then specially we have x 0 = x T and x = x T+. 33 On the other hand, if 33 holds, then from 8 and 3, we have x T+ = px T+ qx T + f T = px qx 0 + f 0 = x. A simple inductive argument along with a use of 8 shows that x mt+l = x l, for every m N and l {0,,..., T }, that is, such a solution to Equation 8 is T-periodic. Case p = 4q. From this and 3, we see that it is enough to show that the linear system x 0 = λ λ x = λ λ λ T T λ x 0 x λ T+ λ x 0 x T + λ T T x λ x λ T+ x λ x 0 + T, 34 has a unique solution in variables x 0 and x. System 34 can be written in the following form: λ λ λ T λ T + λ λ x 0 + λ T λt x = λ T S λ T S λ λ λ T λt x 0 + λ T+ λ T+ + λ λ x = λ T+ S λ T+ S, 35 where S := T j=0 f j λ j+ and S := T j=0 f j λ j+ After some standard but interesting calculation it is shown that the determinant of system 35 is: λ = λ λ T λ T + λ λ λ T λt λ λ λ T λt λt+ λ T+ + λ λ 36 =λ λ λ T λt = 0,. due to 3 and λ = λ. Also, we have = λ TS λ TS λ T λt λ T+ S λ T+ S λ T+ λ T+ + λ λ =λ λ λ T λt S λ T λt S, 37

10 Symmetry 07, 9, 7 0 of 3 and λ = λ λ T λ T + λ λ λ TS λ TS λ λ λ T λt λt+ S λ T+ S From 36 to 38, it follows that =λ λ λ T+ λ T S λ T+ λ T S. 38 x 0 = λt λt S λ T λt S λ λ λ T λt 39 and x = λt+ λ T S λ T+ λ T S λ λ λ T λt, 40 are the initial values for which is obtained the T-periodic solution to Equation 8 in this case. Case p = 4q. Using 4, we see that 33 becomes the linear system x 0 =λ x T T 0 x =λ T+ x 0 k + λ k+ + Tλ T T x λx T + λ T T k + λ k+ x λx 0 + T. 4 System 4 can be rewritten in the following form: + T λ T x 0 Tλ T x = Tλ T Ŝ λ T Ŝ Tλ T+ x 0 + T + λ T x = T + λ T Ŝ λ T+ Ŝ, 4 where Ŝ := T j=0 f T j j + f j λ j+ and Ŝ := j=0 λ j+. After some calculation it is shown that the determinant of system 4 is: = + T λ T Tλ T Tλ T+ T + λ T = λt = 0, 43 due to 3. Also, we have and = = From 43 45, it follows that Tλ T Ŝ λ T Ŝ T + λ T Ŝ λ T+ Ŝ T + λ T =Tλ T Ŝ + λ T λ T Ŝ, Tλ T + T λ T Tλ T Ŝ λ T Ŝ Tλ T+ T + λ T Ŝ λ T+ Ŝ =λ T + T λ T Ŝ + λ T+ λ T Ŝ x 0 = TλT Ŝ + λ T λ T Ŝ λ T

11 Symmetry 07, 9, 7 of 3 and x = λt + T λ T Ŝ + λ T+ λ T Ŝ λ T, are the initial values for which is obtained the T-periodic solution to Equation 8 in this case. b If x n n N0 is the T-periodic solution to Equation 8 and x n n N0 is any solution to the equation, then if p = 4q, from 3 we have x n x n = λ λ λ n λ x 0 x λ n λ x 0 x + λ n x λ x λ n x λ x 0 + λ x 0 x 0 + x x λ n + λ x 0 x 0 + x x λ n, λ λ from which along with 3 the statement follows in this case. If p = 4q, then from 4 we have x n x n = λn x 0 k + λ k+ + nλ x λ x 0 + λ x n k + f 0 k λ k+ x 0 x 0 λ n + x x + λ x 0 x 0 n λ n + λ M, nλ x λx 0 + for some M = Mx 0, x, x 0, x, λ, from which the statement follows in this case. The following result solves the problem of existence of a unique bounded solution to Equation 5 for the case p n = p, q n = q, n N 0, when m := min{ λ, λ } >. 46 Theorem. Consider Equation 8, where the zeros λ, of the polynomial satisfy condition 46, and f n n N0 is a bounded sequence of complex numbers. Then, there is a unique bounded solution to the equation. Proof. According to Lemma a we know that in the case p = 4q, the general solution to 8 is given by 9, while if p = 4q, the general solution is given by. Assume first that p = 4q. If in the case there is a bounded solution to Equation 8, then it must be c 0 = λ λ =: S and d 0 = λ λ =: S. 47 Note that sums S and S are finite since due to condition 46 and the boundedness of f n n N0, we have f k j λ λ f < +, j =,. λ λ λ j Indeed, if c 0 = S and d 0 = S, then from 9, we easily get x n max{ λ, λ } n. 48

12 Symmetry 07, 9, 7 of 3 If c 0 = S and d 0 = S, then x n = λ n c 0 λ λ λ n λ λ. we get From this and since λn λ λ f λ λ λ If c 0 = S and d 0 = S, then x n = λ n λ λ + λn x n λ n. 49 d 0 + λ λ. we get From this and since λn λ λ f λ λ λ x n λ n. 50 Hence, in these three cases from it would follow that the solutions would be unbounded, a contradiction. By using 47 in 9, we get x n = λ n k λ n k f λ λ k, n N 0. 5 A direct calculation shows that sequence x n n N0 defined by 5 is a solution to Equation 8. On the other hand, by using the assumptions of the theorem we easily get x n f λ λ m n k = f λ λ m <, n N 0, from which the boundedness of x n n N0, follows. From this and since by 47, c 0, d 0 is uniquely determined it follows that 5 is a unique bounded solution to Equation 8, in this case. Now, assume that p = 4q. If in the case there is a bounded solution to 8, then it must be c 0 = k + f k λ k+ =: S 3 and d 0 = λ k+ =: S 4. 5 Note that sums S 3 and S 4 are also finite since due to condition 46 and the boundedness of f n n N0 we have f k f λ λ < + λ k+

13 Symmetry 07, 9, 7 3 of 3 and k + λ k+ f λ < +. Indeed, if c 0 = S 3 and d 0 = S 4, then from, we easily get If c 0 = S 3 and d 0 = S 4, then x n = λ c n 0 x n n λ n. 53 k + λ k+ nλ n λ k+. From this, since and n = oλ n, we get nλn λ k+ n f λ λ If c 0 = S 3 and d 0 = S 4, then From this, since λn and n = onλ n, we get k + λ k+ x n = λ n f λ x n λ n. 54 k + λ k+ s + n + λ s=0 s + nλ n f d 0 + k λ k+. = f λ n λ λ + λ λ, x n nλ n. 55 Hence, in these three cases from it would follow that the solutions are unbounded, a contradiction. By using 5 in, we get x n = k + n λ k+ n, n N A direct calculation shows that sequence x n n N0 defined by 56 is a solution to Equation 8. On the other hand, by using the assumptions of the theorem we easily get x n f k + n λ n k = f λ <, n N 0, from which the boundedness of x n n N0 follows. From this and since by 5, c 0, d 0 is uniquely determined it follows that 56 is a unique bounded solution to Equation 8, in this case. Theorem 3. Consider Equation 8, where p C, q C \ {0}, the zeros λ, of the polynomial satisfy condition 46 and f n n N0 is a T-periodic sequence. Then, the unique bounded solution to Equation 8 is T-periodic.

14 Symmetry 07, 9, 7 4 of 3 Proof. If x n n N0 is the bounded solution to Equation 8, then by Theorem we see that it is given by 5 when p = 4q and 56 if p = 4q. Hence, if p = 4q, then we have x n+t = = = +T j=n j=n λ n+t k λ n+t k f λ λ k λ n j λ n j f λ λ j+t λ n j λ n j f λ λ j = x n, 57 for n N 0, while if p = 4q, we have x n+t = = = +T j=n j=n k + n T λ k+ n T j + n λ j+ n f j+t j + n λ j+ n f j = x n, 58 for n N 0. From 57 and 58 the result follows. Theorem 4. Consider Equation 8, where p C, q C \ {0}, the zeros λ, of the polynomial satisfy the following condition min{ λ, λ } < < max{ λ, λ }, 59 and f n n N0 is a bounded sequence of complex numbers. Then, the following statements are true. a If λ < < λ, then a solution to Equation 8 is bounded if and only if λ x 0 x = j=0 f j λ j+. 60 b If λ < < λ, then a solution to Equation 8 is bounded if and only if λ x 0 x = j=0 f j λ j+. 6 Proof. a Since λ <, we have that λn λ x 0 x λ x 0 + x λ n + f λ n k λ x 0 + x + f λ <. 6

15 Symmetry 07, 9, 7 5 of 3 From 3, 6, and since λ >, it follows that the boundedness of a solution x n to Equation 8 implies 60. Indeed, since λ >, we have λ k+ λ k+ f λ < +, that is, the last series is absolutely convergent. So, if 60 were not hold, then for the solution would be x n λ n, which would contradict with its boundedness. Now assume that 60 holds. Then from 3 and 60 it follows that the solution in the case must be x n = λ λ Since λ >, we have λn Using 6 and 64 in 63, we have x n λ n λ x 0 x λ k+ λ n λ k+. 63 λ k+ f λ n k = f <. 64 λ λ λ λ x 0 + x + f λ + f, λ from which it follows that the solution to Equation 8 is bounded. b The proof of the statement is similar/dual to the one in a. Hence, it is omitted. Theorem 5. Consider Equation 8, where p C, q C \ {0}, the zeros λ, of the polynomial satisfy condition 59 and f n n N0 is a T-periodic sequence. Then, the following statements are true. a There is a unique T-periodic solution to Equation 8. b All bounded solutions to Equation 8 converge geometrically to the periodic one. Proof. a We may assume that the condition holds λ < < λ, since the other case is essentially the same and is obtained by changing some letters only. By Theorem 4, we see that a solution to Equation 8 is bounded if and only if 60 holds, and that bounded solutions to Equation 8 have the form in 63. If a solution to the equation is T-periodic, then it must be x 0 = x T, that is, x 0 = λ λ λ T T λ x 0 x λ k+ λ T k=t from which, along with 63 and by some calculation is obtained λ k+, 65 x 0 = λ T j=t f j λ λ j+ T f j j=0 + λ λ j+ T T f j j=0 λ j+ λ λ λ T. 66 By using equalities 60 and 66 in 63 and after some calculation it is shown that for such chosen x 0 is obtained a T-periodic solution to Equation 8. Since initial value x 0 is uniquely defined by 66, and consequently by 60 initial value x is also uniquely defined, the T-periodic solution is unique too, as claimed.

16 Symmetry 07, 9, 7 6 of 3 b If x n n N0 is the T-periodic solution to Equation 8 and x n n N0 is any bounded solution to the equation, then from 60 and 63 we have x n x n = λ λ λn λ λ x 0 + λ n λ λ x 0 + from which the statement follows. f j λ j+ x 0 x 0 λ n, f j λ j+ λ k+ λ n λ n λ k+ Remark 3. Since the sequence f n n N0 in Theorem 5 is T-periodic, then the expression for x 0 in 66 can be written in a somewhat nicer way. Namely, since the series f j j=0 is absolutely convergent, λ j+ we have k+t T T f = = kt+i f j=mt k=m j=kt k=m i=0 λ kt+i+ = i k=m λ kt i=0 λ i+ = λ mt T λ T i=0 f i λ i+ for every m N 0. Using 67 in 66 for m = 0 and m = and after some calculation it follows that, 67 x 0 = λ T λt f j T j=0 λ λ j+ TλT f j T j=0 λ j+ λ λ λ T λt. From this, 60 and some calculation we get x = λ T+ λ T T j=0 f j λ j+ f j λ λ j+ T+ λ T T j=0 λ λ λ T λt. Note that the initial values match with the ones in 39 and 40. Now we are in a position to formulate and prove the main results in this section. The results give some sufficient conditions for the unique existence of bounded solutions to Equation 5, that is, when the sequences p n n N0 and q n n N0, in general, are not constant, and they are in the spirit of the main result in our recent paper [30]. Theorem 6. Assume that p n n N0 and q n n N0 are sequences of complex numbers such that p n + r ˆq := sup + r + q n r r < n N 0 r r r m, 68 for some distinct numbers r and r, such that r m := min{ r, r } >, and f n n N0 is a bounded sequence of complex numbers. Then, Equation 5 has a unique bounded solution. Proof. Write Equation 5 in the following form for n N 0. x n+ r + r x n+ + r r x n = p n + r + r x n+ + r r q n x n + f n, 69

17 Symmetry 07, 9, 7 7 of 3 Let A be the following operator defined on the class of all sequences Au = r n k r n k r r r r q k u k p k + r + r u k n N 0 If u l, then from 70, by using condition 68 and some elementary estimates, it follows that Au = sup sup n N 0 n N 0 r r r r q k u k p k + r + r u k+ + r n k r n k u r r q k + p k + r + r + f r r r k n+ m u r r r m + f r r r m <, which means that operator A maps the Banach space l into itself. On the other hand, for every u, v l we have Au Av = sup r n k r n k r r r r q k u k v k p k + r + r u k+ v k+ n N 0 u v sup ˆq u v, n N 0 r r q k + p k + r + r r r r k n+ m 7 from which along with condition 68 it follows that the operator A : l l is a contraction. By the Banach fixed point theorem [4] it follows that the operator has a unique fixed point, say x = x n n N0 l, that is, Ax = x, which can be written as follows x n = r n k r n k r r r r q k xk p k + r + r xk+ +, 7 for n N 0. A direct calculation shows that this bounded sequence satisfies difference Equation 69, that is, Equation 5 for every n N 0, from which the theorem follows. Theorem 7. Assume that p n n N0 and q n n N0 are sequences of complex numbers such that ˆq := sup n N 0 p n + r + q n r r <, 73 for some number r >, and f n n N0 is a bounded sequence of complex numbers. Then the difference Equation 5 has a unique bounded solution. Proof. Write Equation 5 in the following form for n N 0. x n+ rx n+ + r x n = p n + rx n+ + r q n x n + f n, 74

18 Symmetry 07, 9, 7 8 of 3 Let A be the following operator defined on the class of all sequences Au = If u l, then from 75 it follows that Au = sup sup k + n r k+ n r q k u k p k + ru k n N 0 n N 0 n N 0 k + n r k+ n r q k u k p k + ru k+ + k + n r q k u k + p k + r u k+ + r k+ n r u + f r <, which means that operator A maps the Banach space l into itself. On the other hand, for every u, v l we have Au Av = sup k + n r k+ n r q k u k v k p k + ru k+ v k+ n N 0 n N 0 u v sup ˆq u v. k + n r k+ n r q k + p k + r 76 From 76 and condition 73 it follows that the operator A : l l is a contraction. By the Banach fixed point theorem we get that the operator has a unique fixed point, say x = x n n N0 l, that is, Ax = x or equivalently x n = k + n r k+ n r q k xk p k + rxk+ +, 77 for n N 0. A direct calculation shows that this bounded sequence satisfies difference Equation 74, that is, Equation 5 for every n N 0, from which the theorem follows. 3. Bounded Solutions to Equation 5 on the Domain Z \ N Now we consider Equation 8 on domain Z \ N. Recall that the initial values on the domain are again x 0 and x the values, are, in a way, the end values. To deal with a real second-order difference equation it is natural to assume that q = 0 the case q = 0, p = 0, has been recently studied in [9]. In this case, the equation can be written in the following form or equivalently as follows: x n + p q x n+ + q x n+ = f n, n, 78 q x n + x λ λ + x n = f n, n N, 79 λ λ λ λ where λ and λ are zeros of the polynomial in.

19 Symmetry 07, 9, 7 9 of 3 Equation 79 can be considered by using the change of variables y n = x n which will transform the equation into an equation of the form in 8, but with shifted indices. To avoid some technical problems due to non-symmetricity of domains N 0 and Z \ N, instead of this, we will use the method of decomposition mentioned in Remark see, for example, [5,0]. If we write 79 in the form x n x = x λ λ x n + f n, n N, 80 λ λ λ and multiply the following equation x j x j = x λ λ j x j + f j 8 λ λ λ by λ n j, j =, n, and suming up such obtained equalities, we get for n N. Multiplying the following equality x n = x + λ λ n x 0 x + λ λ λ x i = x i + λ λ i x 0 x + λ λ λ n j= i j= f j λ n j f j λ i j, 8, 83 by λ n i, i =, n, and summing up such obtained equalities, in the case λ = λ, that is, p = 4q, we get x n = x 0 λ n + x 0 x λ λ j=0 λ j λ j + λ λ λ n+ λ n+ =x 0 λ λ x λ n λ n λ λ λ λ λ n+ λ n+ =x 0 λ λ x λ n λ n λ λ λ λ n i= λ n i + λ λ + λ λ = λ n+ x 0 x λ + λ λ n j= f jλ j λ λ λ n+ x 0 x λ + λ λ n j= f jλ j λ λ n j= n j= i j= f j λ i j f j λ n λ j n i λ λ i=j λ j λ j f j λ λ = λ n λ x 0 x + n j= f jλ j λ n λ x 0 x + n j= f jλ j λ λ, 84 for n N. In fact, the formula also holds for n = 0 and n =, which is easily verified by direct calculation and by using the convention for summations mentioned in introduction.

20 Symmetry 07, 9, 7 0 of 3 Now assume that p = 4q, then 84 holds with λ = λ =: λ, from which along with some calculation we get x n = x 0 λ n + x 0 x n λ λ n + λ n+ =λ n+ λx 0 + λx 0 x n + n λ i i= n j= i j= f j λ i j f j n j + λ j, for n N. The last formula can be also written in the following form x n =λ n+ λx 0 n j= + nλ n+ λx 0 x + f j j λ j n j= f j λ j, n N. 85 Note that as in the previous case Formula 85 also holds for n = 0 and n =, which is easily verified by direct calculation and by using the convention for summations mentioned in introduction. As a consequence of the above consideration we have that the following result holds. Lemma. Consider Equation 78 where p C, q C \ {0}, x 0 and x are given complex numbers, and f n n N is a sequence of complex numbers. Then the following statements are true: a If p = 4q, then the solution to Equation 78 with initial/end values x 0 and x is given by x n = λ n λ x 0 x + n j= f jλ j λ n λ x 0 x + n j= f jλ j λ λ, 86 b where λ, are given by 0. If p = 4q, then the solution to Equation 78 with initial/end values x 0 and x is given by where λ = p/. x n = λ n+ λx 0 + λx 0 x n + n j= Remark 4. Note that Formula 86 can be written in the form with f j n j + λ j, 87 x n = ˆx n + x h n, 88 x n h = λ x 0 x λ n λ x 0 x λ n, 89 λ λ x n = n j= f jλ n+j λ n+j λ λ, n N, 90 where x n h is the solution to the homogeneous difference equation corresponding to 79 with initial/end values x 0 and x, while x n is a particular solution to 79, in the case p = 4q.

21 Symmetry 07, 9, 7 of 3 Also, Formula 87 can be written in the form in 88 with x h n =λx 0 + λx 0 x nλ n+, 9 x n = n j= f j n j + λ n+j, n N, 9 where x n h is the solution to the homogeneous difference equation corresponding to 79 with initial/end values x 0 and x, while x n is a particular solution to 79, in the case p = 4q. From 3, 4, 89 and 9 it follows that the solution to Equation 8 with f n = 0, n Z, with initial values x 0 and x is given by x n = λ x 0 x λ n + x λ x 0 λ n λ λ, n Z, when p = 4q, that is, by x n = λx 0 + x λx 0 nλ, n Z, when p = 4q. From 86 and 87 similar to Corollary is proved the following result. Hence, we omit the details. Corollary 3. Consider Equation 78, where the zeros λ, of polynomial satisfy the condition in 46, and f n n N is a bounded sequence of complex numbers. Then every solution to the equation on domain Z \ N is bounded. Theorem 8. Consider Equation 78, where the zeros λ, of polynomial satisfy condition 46 and f n n N is a T-periodic sequence. Then the following statements hold. a There is a unique T-periodic solution to Equation 79 on domain Z \ N. b All the solutions to Equation 79 on domain Z \ N, converge geometrically to the periodic one. Proof. a If x n n is a T-periodic solution to Equation 78, then it must be we have On the other hand, if 93 holds, then from 78 and since x = x T and x 0 = x T. 93 x T+ = p q x T q x T + f T+ q f n = f n T, n N, 94 = p q x 0 q x + f q = x. Using the same argument along with 78 and the method of induction it is proved that x mt l = x l, for every m N and l {, 0,,..., T }, which shows that the solution to Equation 8 is T-periodic.

22 Symmetry 07, 9, 7 of 3 Case p = 4q. This along with 86 shows that it is enough to prove that the linear system x 0 = λ T λ x 0 x + T j= f jλ j λ T λ x 0 x + T j= f jλ j λ λ x = λ T λ x 0 x + T j= f jλ j λ T λ x 0 x + T j= f jλ j, λ λ 95 has a unique solution in variables x 0 and x. Note that system 95 can be written as follows where λ λ T λ λ T + λ λ x 0 + λ T λ T x = λ T S 5 λ T S 6 λ λ T λ λ T x 0 + λ T λ T + λ λ x = λ T S 5 λ T S 6, S 5 := By some calculation is obtained = due to 46 and λ = λ. Also, we have = and T f j λ j T and S 6 := f j λ j. j= j= λ λ T λ λ T + λ λ λ T λ T λ λ T λ λ T λ T λ T + λ λ =λ λ λ T λ T = 0, = From 97 99, it follows that λ T S 5 λ T S 6 λ T λ T λ T S 5 λ T S 6 λ T λ T + λ λ =λ λ λ T λ T S 6 λ T λ T S 5, λ λ T λ λ T + λ λ λ T S 5 λ T S 6 λ λ T λ λ T λ T S 5 λ T S 6 =λ λ λ T λ T S 6 λ T λ T S 5 x 0 = λ T λ T S 6 λ T λ T S 5 λ λ λ T λ T and x = λ T λ T S 6 λ T λ T S 5 λ λ λ T λ T, 0 are the initial values for which is obtained the T-periodic solution to Equation 78 in this case. Case p = 4q. Using 87, we see that 93 becomes the linear system x 0 =λ T+ λx 0 + λx 0 x T + T j= f j T j + λ j x =λ λx T T 0 + λx 0 x T + f j T jλ. j j= 0

23 Symmetry 07, 9, 7 3 of 3 System 0 can be written as follows λ T+ λt + x 0 + Tx = S 7 λtx 0 + T λ T x = S 8, 03 where S 7 := T f j T j + λ j T and S 8 := f j T jλ j. j= j= After some calculation it is shown that the determinant of system 03 is λ = T+ λt + λt T T λ T = λλt = 0, 04 due to 46. Also, we have = S 7 S 8 T T λ T = T λt S 7 TS 8, 05 and = λ T+ λt + S 7 Tλ S 8 = λt+ λt + S 8 TλS From 04 06, it follows that and x 0 = λt + T S 7 + TS 8 λλ T x = TS 7 λ T T S 8 λ T, are the initial values for which is obtained the T-periodic solution to Equation 78 in this case. b If x n n is the T-periodic solution to Equation 78 and x n n is any solution to the equation, then if p = 4q, from 86 we have λ n x n x n = λ x 0 x + n j= f jλ j λ nλ x 0 x + n j= f jλ j λ λ λ n λ x 0 x + n j= f jλ j λ nλ x 0 x + n j= f jλ j λ λ λ x 0 x 0 + x x λ n + λ x 0 x 0 + x x λ n, λ λ for n N 0, from which the statement follows in this case.

24 Symmetry 07, 9, 7 4 of 3 If p = 4q, then from 87 we have x n x n = λ λ n+ x 0 + λ x 0 x n + n j= λ n+ λx 0 + λx 0 x n + n j= f j n j + λ j f j n j + λ j x 0 x 0 λ n + x x + λ x 0 x 0 n λ n+ n + λ M, λ for n N 0, for some M = M x 0, x, x 0, x, λ, from which the statement follows in this case. Theorem 9. Consider Equation 78, where the zeros λ, of polynomial satisfy condition 3, and f n n N is a bounded sequence of complex numbers. Then, there is a unique bounded solution to the equation on domain Z \ N. Proof. If p = 4q, then from 3 and 86 we see that a solution to Equation 79 is bounded if and only if λ x 0 x = j= f j λ j and λ x 0 x = j= f j λ j, 07 note that both sums are finite due to 3 and the boundedness of f n n N, from which it follows that the solution is x n = λ λ f j λ j λ j. 08 j=n+ By direct calculation is shown that 08 is a solution to Equation 78. Since x n f λ λ M j = j=n+ f λ λ M, for n N, it follows that the solution is bounded. Since by 07, x 0 and x are uniquely defined, the bounded solution is unique. If p = 4q, then from 3 and 87, similar to the corresponding part of the proof of Theorem is obtained that Equation 78 has a unique bounded solution if and only if x 0 = f j j λ j and x = f j jλ j, 09 j= j= note that both sums are also finite due to 3 and the boundedness of f n n N, from which it follows that the solution is x n = λ n+ f j j n λ j. 0 j=n+ By direct calculation is shown that 0 is a solution to Equation 79. Since x n f j n M j n = f M j=n+,

25 Symmetry 07, 9, 7 5 of 3 for n N, it follows that the solution is bounded. Since due to 09, x 0 and x are uniquely defined, the bounded solution is unique. Theorem 0. Consider Equation 78, where p C, q C \ {0}, the zeros λ, of the polynomial are distinct and satisfy condition 3 and f n n N is a T-periodic sequence. Then, the unique bounded solution to Equation 79 is T-periodic. Proof. Let x n n be the unique bounded solution to Equation 79. From Theorem 9 we see that the unique bounded solution to Equation 79 is given by 08 if p = 4q, and it is given by 0 if p = 4q. If p = 4q, then we have x n+t = λ λ f j λ j n T λ j n T j=n+t+ = λ λ + f k T λ k λ k = λ λ f k λ k λ k = x n, + for n, while if p = 4q, we have x n+t =λ n+t+ f j j n T λ j j=n+t+ =λ n+t+ f k T k n λ k+t + for n. From and the result follows. =λ n+ f k k n λ k = x n, + Theorem. Consider Equation 78, where p C, q C \ {0}, the zeros λ, of polynomial satisfy condition 59, and f n n N is a bounded sequence of complex numbers. Then, the following statements are true. a If λ < < λ, then a solution to Equation 78 is bounded if and only if x λ x 0 = f j λ j. 3 j= b If λ < < λ, then a solution to Equation 78 is bounded if and only if x λ x 0 = f j λ j. 4 j= Proof. a Since λ >, we have that λ n λ x 0 x + n j= f j λ j λ x 0 + x λ n n + f λ n+j j= λ x 0 + x + f λ <. 5

26 Symmetry 07, 9, 7 6 of 3 From 86, 5, and since λ <, it follows that the boundedness of a solution x n to Equation 79 implies 3. Now assume that 3 holds. Then from 86 and 3 it follows that the solution in the case must be x n = λ n j=n+ f jλ j λ n λ x 0 x + n j= f jλ j λ λ. 6 Since λ <, we have λ n j=n+ Using 5 and 7 in 6, we have x n f j λ j f λ n+j = f <. 7 λ + λ λ λ x 0 + x + f λ + f, λ from which it follows that the solution to Equation 79 is bounded. b The proof of the statement is similar/dual to the one in a. Hence, it is omitted. Theorem. Consider Equation 78, where p C, q C \ {0}, the zeros λ, of the polynomial satisfy condition 59 and f n n N is a T-periodic sequence. Then, the following statements are true. a There is a unique T-periodic solution to Equation 78 on domain Z \ N. b All bounded solutions to Equation 78 on domain Z \ N, converge geometrically to the periodic one. Proof. a We may assume that the condition holds λ < < λ, since the other case is essentially the same. By Theorem, we see that a solution to Equation 78 is bounded if and only if 3 holds, and that bounded solutions to Equation 78 have the form in 6. If a solution to the equation is T-periodic, then it must be x = x T, that is, x = λ T j=t f jλ j λ T λ x 0 x + T j= f jλ j, λ λ from which, along with 6 and by some calculation is obtained x = λ λ T λ T j= f jλ j + T j= f jλ j T λ j T λ λ λ T. 8 By using equalities 3 and 8 in 6 and after some calculation it is shown that for such chosen x is obtained a T-periodic solution to Equation 78. Since initial value x is uniquely defined by 8, and consequently by 3 initial value x 0 is uniquely defined, the T-periodic solution is unique too, as claimed.

27 Symmetry 07, 9, 7 7 of 3 b If x n n is the T-periodic solution to Equation 78 and x n n is any bounded solution to the equation, then from 6 and some simple estimates, we have x n x n = λ λ λ n λ n from which the statement easily follows. j=n+ j=n+ f j λ j f j λ j λ x 0 x 0 + x x λ λ λ n, λ n λ n λ x 0 x + n j= λ x 0 x + n j= f j λ j f j λ j Remark 5. Since the sequence f n n N in Theorem is T-periodic, then the expression for x in 8 can be written in a somewhat nicer way. Namely, since the series j= f jλ j is absolutely convergent, we have f j λ j k+t = f j λ j T = f i kt λ kt+i j=mt+ i= = k=m k=m j=kt+ λ kt T i= f i λ i k=m = λmt λ T T f i λ i, i= for every m N 0. Using 9 in 8 for m = 0 and after some calculation it follows that x = λ T T j= f jλ j T λ T T j= f jλ j T λ λ λ T λ T. From this, 3 and some calculation we get x 0 = λ λ T T j= f jλ j T λ λ T T j= f jλ j T λ λ λ λ λ T λ T. Note that the initial values match with those in 00 and 0. The following result considers Equation 8 on domain Z. The result is essentially a consequence of some of above mentioned results on domains N 0 and Z \ N, so it is formulated as a corollary, although it is, in fact, an important results concerning the equation. Corollary 4. Consider Equation 8 where p C and q C \ {0}. Let λ, be the zeros of polynomial, f n n Z be a bounded sequence of complex numbers, and one of the conditions 3, 46, 59 holds. Then, the equation has a unique bounded solution on Z. Proof. First, assume that 46 holds, then by Theorem there is a unique bounded solution to Equation 8 on N 0, while by Corollary 3 all the solutions to the equation on Z \ N are bounded, from which the result follows in the case. If 3 holds, then by Corollary we have that all the solutions to the equation on N 0 are bounded, while by Theorem 9 there is a unique bounded solution to the equation on Z \ N, from which the result follows in the case. Now, assume that 59 holds. Then, if λ < < λ by Theorems 4 a and a, it follows that a solution to Equation 8 on Z is bounded if and only if 60 and 3 hold, which is a two-dimensional linear system in x 0 and x with the determinant different from zero, from which it follows that there is a unique pair of initial values x 0, x, such that the solution to Equation 8 is bounded. If λ < < λ, then by Theorems 4 b and b, it follows that a solution to 8 on Z is bounded if and only if 6 and 4 hold, which 9

28 Symmetry 07, 9, 7 8 of 3 is also a two-dimensional linear system in x 0 and x with the determinant different from zero, from which it follows that there is a unique pair of initial values x 0, x, such that the solution to Equation 8 is bounded, in this case, finishing the proof of the theorem. The following two results are the main results in this section and deal with the general Equation 5 on domain Z \ N. They correspond to Theorems 6 and 7, and are proved similarly. We present their proofs for the completeness and benefit of the reader. Theorem 3. Assume that p n n N and q n n N are sequences of complex numbers such that p n /q n + r + r ˆq 3 := sup + r r /q n < n N r r r M, 0 for some nonzero numbers r and r, such that r M := max{ r, r } <, inf q n > δ > 0, n N and f n n N is a bounded sequence of complex numbers. Then, Equation 5 has a unique bounded solution on Z \ N. Proof. Write Equation 5 in the following form x n r + r x + r r x n = p n /q n + r + r x + r r /q n x n + f n /q n, for n N. Let A be the following operator defined on the class of all sequences Au = j=n+ r j p j u r r r r q j + r j q + r u j + f j. 3 j q j n r j If u l, then from, 3, and some simple inequalities, we have Au = sup n N 0 sup n N 0 j=n+ j=n+ r j p j u r r r r q j + r j q + r u j + f j j q j r j r j u r r q j + p j/q j + r + r + f δ M r r u r r r M + f δ r r r M <, which means that operator A maps the Banach space l into itself. On the other hand, for every u, v l we have Au Av = sup r j r j p j u r r r r q j v j + r j q + r u j v j j n N 0 j=n+ u v sup ˆq 3 u v, n N j=n+ r j r r /q j + p j /q j + r + r M r r

29 Symmetry 07, 9, 7 9 of 3 from which along with 0 it follows that A : l l is a contraction. By the Banach fixed point theorem we get that the operator has a unique fixed point, say x = x n n l, that is, Ax = x or equivalently x n = r j r j=n+ r r j x j r r q p j + r j q + r x j + f j, 4 j q j for n. A direct calculation shows that this bounded sequence satisfies difference Equation 5, from which the theorem follows. Theorem 4. Assume that p n n N and q n n N are sequences of complex numbers such that p n /q n + r ˆq := + r /q n sup n N r <, 5 for some number r <, that condition holds, and f n n N is a bounded sequence of complex numbers. Then, Equation 5 has a unique bounded solution on Z \ N. Proof. Write Equation 5 in the following form x n r x + r x p n n = + x q n r + r x q n + f n, 6 n q n for n N. Let A be the following operator defined on the class of all sequences Au = j=n+ j n r j n r If u l, then from 7 it follows that Au = sup n N 0 sup n N 0 j=n+ q j p j u j + u q j r j + f j. 7 q j n N 0 j n r j n r p j u q j + u j q j r j + f j q j u p j + j + q j r u j + f j δ j n r j n r j=n+ r u + f δ r <, which means that operator A maps the Banach space l into itself. On the other hand, for every u, v l we have q j Au Av = sup j n r j n r n N 0 j=n+ u v sup n N 0 ˆq u v. q j j n r j n r j=n+ p j u j v j q j + q j r + p j + q j r u j v j 8 From 8 and condition 5 it follows that the operator A : l l is a contraction.

30 Symmetry 07, 9, 7 30 of 3 By the Banach fixed point theorem we get that the operator has a unique fixed point, say x = x n n l, that is, Ax = x or equivalently x n = j n r j n r j=n+ x j q p j + x j j q j r + f j, 9 q j for n. A direct calculation shows that this bounded sequence satisfies difference Equation 5, from which the theorem follows. 4. Discussion Motivated by some recent investigations of ours here we present how some solvability methods along with the contraction mapping principle can be employed in studying of a nonhomogeneous linear second-order difference equation, with a special attention on the existence of bounded solutions and their relationship with other solutions. Beside the study of the equation on the usual domain N 0, it is also studied on the neglected ones Z \ N and Z. A natural problem is to develop the methods and ideas in the paper so that they can be applied in the study of other related difference equations of second as well as of higher orders on all these domains, which will be one of our further directions in the investigation of difference equations and systems. Author Contributions: The author has contributed solely to the writing of this paper. He read and approved the manuscript. Conflicts of Interest: The author declares that he has no conflicts of interest. References. Andruch-Sobilo, A.; Migda, M. Further properties of the rational recursive sequence x n+ = ax /b + cx n x, Opusc. Math. 006, 6, Ašić, M.D.; Adamović, D.D. Limit points of sequences in metric spaces. Amer. Math. Monthly 970, 77, Atkinson, F.V. Discrete and Continuous Boundary Problems; Academic Press, New York, NY, USA, Banach, S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 9, 3, Berezansky, L.; Braverman, E. On impulsive Beverton-Holt difference equations and their applications. J. Difference Equ. Appl. 004, 0, Berezansky, L.; Migda, M.; Schmeidel, E. Some stability conditions for scalar Volterra difference equations. Opusc. Math. 06, 36, Berg, L.; Stević, S. On some systems of difference equations. Appl. Math. Comput. 0, 8, Brand, L. Differential and Difference Equations; John Wiley & Sons, Inc.: New York, NY, USA, Demidovich, B. Problems in Mathematical Analysis; Mir Publisher: Moscow, Russia, Diblik, J.; Schmeidel, E. On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence. Appl. Math. Comput. 0, 8, Diblik, J.; Schmeidel, E.; Ružičková, R. Asymptotically periodic solutions of Volterra system of difference equations. Comput. Math. Appl. 00, 59, Drozdowicz, A.; Popenda, J. Asymptotic behavior of the solutions of the second order difference equation. Proc. Amer. Math. Soc. 987, 99, Iričanin, B.; Stević, S. Eventually constant solutions of a rational difference equation. Appl. Math. Comput. 009, 5, Jordan, C. Calculus of Finite Differences; Chelsea Publishing Company: New York, NY, USA, Krechmar, V.A. A Problem Book in Algebra; Mir Publishers: Moscow, Russia, Levy, H.; Lessman, F. Finite Difference Equations; Dover Publications: New York, NY, USA, Migda, J. Mezocontinuous operators and solutions of difference equations. Electron. J. Qual. Theory Differ. Equ. 06, 06, 6.

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