Abstract:
This paper shows that all positive solutions of a higher-order nonlinear difference equation are bounded, extending some recent results in the literature. 1. Introduction There is a considerable interest in studying nonlinear difference equations nowadays; see, for example, [1–40] and numerous references listed therein. The investigation of the higher-order nonlinear difference equation where and , and , , was suggested by Stevi？ at numerous talks and in papers (see, e.g., [20, 28, 30, 34–38] and the related references therein). In this paper we show that under some conditions on parameters , and all positive solutions of the difference equation where , are bounded. To do this we modify some methods and ideas from Stevi？'s papers [30, 35–37]. Our motivation stems from these four papers. The reader can find results for some particular cases of (1.2), as well as on some closely related equations treated in, for example, [1, 2, 5–11, 18–20, 26, 30, 33–35, 38, 40]. 2. Main Result Here we investigate the boundedness of the positive solutions to (1.2) for the case The following result completely describes the boundedness of positive solutions to (1.2) in this case. The result is an extension of one of the main results in [35]. Theorem 2.1. Assume, and Then every positive solution of (1.2) is bounded if Proof. First note that from (1.2) it directly follows that Using (1.2), it follows that After steps we obtain the following formula Two subcases can be considered now. Case 1 ( ). If , then by (2.2) equality (2.4) implies that for This means that is a bounded sequence.Case 2 ( ). In this case we have From (2.4) and (1.2) we further obtain for each and every where the sequences satisfy the system and the initial values are given by Note that implies that Assume for every . By a direct calculation it follows that , , which, along with (2.8) implies that are strictly increasing sequences. From system (2.8), we have, If it were , , then there was Clearly is a solution of the equation Since and we see that the function attains its maximum at the point Further, by assumption (2.1) we get which along with (2.13) implies that (2.12) does not have solutions on arriving at a contradiction. This implies that there is a fixed index such that From this, inequality (2.2), and identity (2.7) with , it follows that for . From (2.17) the boundedness of the sequence directly follows, as desired. Acknowledgment The research was partially supported by the Serbian Ministry of Science, through The Mathematical Institute of SASA, Belgrade, Project no. 144013.

Abstract:
Let f(z1,…,zk)∈C(Ik,I) be a given function, where I is (bounded or unbounded) subinterval of ℝ, and k∈ℕ. Assume that f(y1,y2,…,yk)≥f(y2,…,yk,y1) if y1≥max{y2, …,yk}, f(y1,y2,…,yk)≤f(y2,…,yk,y1) if y1≤min{y2,…,yk}, and f is non- decreasing in the last variable zk. We then prove that every bounded solution of an autonomous difference equation of order k, namely, xn=f(xn−1,…,xn−k), n=0,1,2,…, with initial values x−k,…,x−1∈I, is convergent, and every unbounded solution tends either to

Abstract:
We prove a useful max-type difference inequality which can be applied in studying of some max-type difference equations and give an application of it in a recent problem from the research area. We also give a representation of solutions of the difference equation . 1. Introduction The investigation of max-type difference equations attracted some attention recently; see, for example, [1–20] and the references therein. In the beginning of the study of these equations the following difference equation was investigated: where ,？？ , are real sequences (mostly constant or periodic), and the initial values are different from zero (see, e.g., monograph [6] or paper [19] and the references therein). The study of the next difference equation where are natural numbers such that , , , and , was proposed by the first author in numerous talks; see, for example, [11, 13]. For some results in this direction see [1, 4, 5, 7, 8, 12, 14–18, 20]. A particular case of the difference equation arises naturally in certain models in automatic control (see [9]). By the change the equation is transformed into the equation which is a special case of (1.2) and which is a natural prototype for the equation. The following result, which extends the main result from the study in [18] was proved by the first author in [17] (see also [16]). Theorem A. Every positive solution to the difference equation where , converges to Here we continue to study (1.5) by considering the cases when some of 's are equal to one. We also give a representation of well-defined solutions of the difference equation where , 2. Main Results In this section we prove the main results of this note. Before this we formulate the following very useful auxiliary result which can be found in [10] and give a definition. Lemma A. Let be a sequence of positive numbers which satisfy the inequality where and are fixed. Then there exists an such that Definition 2.1. For a sequence , , we say that it converges to zero geometrically if there is a and such that for . Now we are in a position to formulate and prove the main results of this note. Proposition 2.2. Assume that is a sequence of nonnegative numbers satisfying the difference inequality where , , , , and if, for some , , then . Then the sequence converges geometrically to zero as . Proof. Let be chosen such that where Then from (2.4) and using the fact that are nonnegative numbers, we have that where . From (2.7), (2.5) and since , we have that Now assume that , for Then from (2.4) we get Inequalities (2.8) and (2.9) along with the method of induction show that Now note

Abstract:
We describe in an elegant and short way the behaviour of positive solutions of the higher-order difference equation , , where and , extending some recent results in the literature. 1. Introduction Studying difference equations has attracted a considerable interest recently, see, for example, [1–39] and the references listed therein. The study of positive solutions of the following higher-order difference equations: and where are natural numbers such that , , and was proposed by Stevi？ in several talks, see, for example, [21, 26]. For some results concerning equations related to (1.1) see, for example, [6, 7, 10, 29, 31, 32, 34, 38], while some results on equations related to (1.2) can be found, for example, in [3, 8, 9, 11–14, 18–20, 22, 25, 29, 32, 33, 35] (see also related references cited therein). Case is of some less interest, since in this case positive solutions of (1.1) and (1.2), by using the change , become solutions of a linear difference equation with constant coefficients. However, some particular results for the case recently appeared in the literature, see [16, 17, 39]. Nevertheless, motivated by the above-mentioned papers, we will describe the behaviour of positive solutions of the higher-order difference equation where and in, let us say, an elegant and short way. Let us introduce the following. Definition 1.1. A solution of (1.3) is said to be eventually periodic with period if there is such that for all If , then we say that the sequence is periodic with period For some results on equations all solutions of which are eventually periodic see, for example, [2, 4, 8, 15, 28, 37] and the references therein. Definition 1.2. One says that a solution of a difference equation converges geometrically to if there exist and such that Now we return to (1.3). First, note that if , then (1.3) becomes from which easily follow the following results:(a)if , then all positive solutions of (1.5) are periodic with period (b)if , then each positive solution of (1.5) converges to zero. Moreover, its subsequences converges decreasingly to zero as (c)if , then each positive solution of (1.5) tends to infinity as . Moreover, its subsequences tend increasingly to infinity as . We may assume that and are relatively prime integers, that is, (the greatest common divisor of numbers and ). Namely, if , then by using the changes , (1.3) is reduced to copies of the following equation: where , and . Further, note that from (1.3), we have that which implies that the sequence satisfies the following simple difference equation: 2. Main Results Here we formulate and prove

Abstract:
Motivated by Iri？anin and Stevi？'s paper (2006) in which for the first time were considered some cyclic systems of difference equations, here we study the global attractivity of some nonlinear k-dimensional cyclic systems of higher-order difference equations. To do this, we use the transformation method from Berenhaut et al. (2007) and Berenhaut and Stevi？ (2007). The main results in this paper also extend our recent results in the work of (Liu and Yang 2010, in press). 1. Introduction Motivated by papers [1, 2], in [3], we proved that the unique positive equilibrium points of the following difference equations: where , and are globally asymptotically stable, respectively. Motivated by paper [4] by Iri？anin and Stevi？, in which for the first time were considered some cyclic systems of difference equations, here we mainly investigate the global attractivity of the -dimensional system of higher-order difference equations where , and , as well as the following counterpart difference equation system: where Furthermore, we also present some similar results regarding the -dimensional cyclic difference equation system: where and as well as the following counterpart difference equation system where and For some recent papers on systems of difference equations, see, for example, [4–10] and the related references therein. Some related scalar equations have been studied mainly by the semicycle structure analysis which was unnecessarily complicated and only useful for lower-order equations, as it was shown by Berg and Stevi？ in [11] (see also papers [12–22]). Thus in this paper we will investigate systems (1.2)–(1.5) by the transformation method from [1, 2] which makes the proofs more concise and elegant, and moreover, it is also effective for higher-order ones. 2. Preliminary Lemmas Before proving the main results in Section 3, in this section we will first present some useful lemmas which are extensions of those ones in [1]. Lemma 2.1. Define a mapping by where Then,(1) is nondecreasing in if and strictly decreasing in if (2) is nondecreasing in if and strictly decreasing in if Proof. The results follow directly from the following fact: For simplicity, systems (1.2) and (1.3) can be respectively rewritten in the following forms: Lemma 2.2. Denote a transformation Then for the mapping defined in Lemma 2.1, we have Proof. It is easy to see that defined in Lemma 2.1 is invariant, if both arguments are replaced by the reciprocal ones, but it turns over into if only one argument is replaced by its reciprocal value. Note that from which the result directly follows by

Abstract:
Based on the principles derived from Campbell’s theorem, this paper carries out an analysis of the possibilities of Campbell’s mean square value signal processing system. The mean square value mode is especially suitable for measurements performed in a mixed radiation field, because the quantities of electrical charge involved in the interactions of the two types of radiation are substantially different. The measuring detector element may be an adequate ionization chamber and/or semiconductor components for mixed n-γ fields. An examination of the discrimination of gamma in relation to the neutron component in the signal of the detector output was carried out, calculated according to the theoretical model of radiation interaction with the detector. The advantage of the mean square value method was confirmed and it was concluded that the order of n-γ discrimination in mean square value signal processing is greater than the one rendered by the classical measuring method.

Abstract:
The following difference equation =？？？1, ∈？0, where ,∈？, <, gcd(,)=1, and the initial values ？,…,？2,？1 are real numbers, has been investigated so far only for some particular values of and . To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long-term behavior of the solutions of the equation for all values of parameters and , where the initial values satisfy the following condition min？,…,？2,？1.