It should be remarked that many interesting theorems have been proved in [, ] con- cerning the linear (or **nonlinear**) recurrences. Especially in , the Hyers-Ulam stability of the ﬁrst-order **matrix** diﬀerence **equations** has been proved in [] in a general setting. The substantial diﬀerence of this paper from [] lies in the fact that the stability problems for the ‘backward’ diﬀerence **equations** have been treated in Section of this paper. 2 Hyers-Ulam stability of x i = A x i–1

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Essentially **nonlinear** diﬀerence **equations** in a Euclidean space are considered. Condi- tions for the existence of periodic solutions and solution estimates are derived. Our main tool is a combined usage of the recent estimates for **matrix**-valued functions with the method of majorants.

space of d-dimensional real column vectors with convenient norm ||.||. Let ℝ d × d be the space of all d × d real matrices. By the norm of a **matrix** A Î ℝ d × d , we mean its induced norm ||A|| = sup{||Ax|| |x Î ℝ d , ||x|| = 1}. The zero **matrix** in ℝ d × d is denoted by 0 and the identity **matrix** by I. The vector x and the **matrix** A are non- negative if x i ≥ 0 and A ij ≥ 0,1 ≤ i,j ≤ d, respectively. Sequence (x(n)) n ≥ 0 in ℝ d is

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We obtain some oscillation criteria for solutions of the nonlinear delay diﬀer αj ence equation of the form xn+1 − xn + pn m j=1 xn−k = 0.. 2000 Mathematics Subject Classiﬁcation.[r]

Representations of general solutions to three related classes of **nonlinear** diﬀerence **equations** in terms of specially chosen solutions to linear diﬀerence **equations** with constant coeﬃcients are given. Our results considerably extend some results in the literature and give theoretical explanations for them.

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The aim of our paper is to present some existence results for the problem (1.1). The first result relies on the **nonlinear** alternative of Leray-Schauder type. In the second result, we apply Banach’s contraction principle to prove the uniqueness of the solution of the problem, while the third result is based on Krasnoselskii’s fixed point theorem. The methods used are standard; however, their exposition in the framework of pro- blem (1.1) is new. In Sect. 2, we present some basic material that we need in the sequel and Sect. 3 contains main results of the paper. Some illustrative examples are also discussed.

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It is shown that the decay rates of the positive, monotone decreasing solutions approaching the zero equilibrium of higher-order **nonlinear** diﬀerence **equations** are related to the positive characteristic values of the corresponding linearized equation. If the nonlinearity is suﬃciently smooth, this result yields an asymptotic formula for the positive, monotone decreasing solutions.

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The existence of solution of discrete **equations** subjected to Sturm-Liouville bound- ary conditions was studied by Rodriguez [4], in which the nonlinearity is required to be bounded. For other related results, see Agarwal and O’Regan [5, 6], Bai and Xu [7], Rachunkova and Tisdell [8], and the references therein. However, all of them do not ad- dress the problem under the “asymptotic nonuniform resonance” conditions.

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where { c(n) } is a sequence of nonnegative real numbers and τ is any real number. In Section 3, we investigate the oscillatory property of (1.2) and discuss the case when λ = 1, that is, the linear case. Section 4 is devoted to the study of the oscillatory behavior of (1.3). We also proceed further in this direction and obtain oscillation criteria for second-order **equations** of the form

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Diﬀerence equation or discrete dynamical system is a diverse ﬁeld which impacts almost every branch of pure and applied mathematics. Lately, there has been great interest in investigating the behavior of solutions of a system of **nonlinear** diﬀerence **equations** and discussing the asymptotic stability of their equilibrium points. One of the reasons for this is a necessity for some techniques which can be used in investigating **equations** arising in mathematical models that describe real life situations in population biology, economics, probability theory, genetics, psychology, and so forth, see [3, 5, 8, 9]. Also, similar works in two and three dimensions (limit behaviors) for more general cases, i.e., continuous and discrete cases, have been done by some authors, see [1, 11–13, 16]. There are many pa- pers in which systems of diﬀerence **equations** have been studied, as in the examples given below.

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Rational diﬀerence equation, as a kind of typical **nonlinear** diﬀerence **equations**, is always a subject studied in recent years. Especially, some prototypes for the development of the basic theory of the global behavior of **nonlinear** diﬀerence **equations** of order greater than one come from the results of rational diﬀerence **equations**. For the systematical investiga- tions of this aspect, refer to the monographs 1–3, the papers 4–9, and the references cited therein.

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Recently [1, 3], a notion of index 1 linear implicit diﬀerence **equations** (LIDEs) has been introduced and the solvability of initial-value problems (IVPs), as well as multipoint boundary-value problems (MBVPs) for index 1 LIDEs, has been studied. In this paper, we propose a natural definition of index for LIDEs so that it can be extended to a class of **nonlinear** IDEs. The paper is organized as follows. Section 2 is concerned with index 1 LIDEs and their reduction to ordinary diﬀerence **equations**. In Section 3, we study the index concept and the solvability of IVPs for **nonlinear** IDEs. The result of this paper can be considered as a discrete version of the corresponding result of [4].

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In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to **nonlinear** **difference** **equations** and system of **nonlinear** **difference** **equations** .

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Diﬀerence **equations** are widely found in mathematics itself and in its applications to combinatorial analysis, quantum physics, chemical reactions, and so on. Many authors were interested in diﬀerence **equations** and obtained many signiﬁcant conclusions; see, for instance, the papers [1–3, 5–20]. Various methods have been used to deal with the existence of solutions to discrete problems, we refer to the ﬁxed point theorems in cones in [12] and the variational method in [2, 3, 5–11, 13, 14, 18–20]. In 2003, in [10, 11] Yu and Guo made a new variational structure to handle discrete **equations** and obtained good conclusions on the solvability condition of a periodic solution. This new variational struc- ture represents an important advance as it allows us to prove multiplicity results as well.

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[2] R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Diﬀerence **Equations**, Mathematics and its Applications, vol. 404, Kluwer, Dordrecht, 1997. MR 98i:39001. Zbl 878.39001. [3] S. S. Cheng, H. J. Li, and W. T. Patula, Bounded and zero convergent solutions of second-order diﬀerence **equations**, J. Math. Anal. Appl. 141 (1989), no. 2, 463–483. MR 90g:39001. Zbl 698.39002.

Manuel et al Advances in Difference Equations 2014, 2014 109 http //www advancesindifferenceequations com/content/2014/1/109 R ES EARCH Open Access Oscillation of solutions of some generalized nonline[.]

Several authors have studied the semi-linear differential **equations** with nonlocal conditions in Banach space, [15] [16]. In [17], Dong et al. studied the existence and uniqueness of the solutions to the nonlocal problem for the fractional differential equation in Banach space. Motivated by these studied, we explore the Cauchy problem for **nonlinear** fractional q-**difference** **equations** according to the following hypotheses.

In this paper a coupled system of **nonlinear** finite **difference** **equations** corresponding to a class of periodic- parabolic systems with time delays and with **nonlinear** boundary conditions in a bounded domain is investigated. Using the method of upper-lower solutions two monotone sequences for the finite **difference** system are con- structed. Existence of maximal and minimal periodic solutions of coupled system of finite **difference** **equations** with **nonlinear** boundary conditions is also discussed. The proof of existence theorem is based on the method of upper-lower solutions and its associated monotone iterations. It is shown that the sequence of iterations converges monotonically to unique solution of the **nonlinear** finite **difference** system with time delays under consideration.

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In this paper, two theorems on the distribution of oscillation zeros for second-order non- linear neutral delay diﬀerence **equations** are obtained by means of inequality techniques, speciﬁc function sequences and non-increasing solutions for corresponding ﬁrst-order diﬀerence inequality. Comparing with the corresponding diﬀerential equation, it is more complex to deal with the lower bound of summation. Function a(t) 1 t–1 s=1 (1 + a(s)) is in- variant after derivation in diﬀerence equation, which is equivalent to e x in diﬀerential equation. That is the diﬃculty we address and the innovation of this paper. We study a second-order equation under the canonical form, and it is also of great signiﬁcance for the study of non-canonical forms. Moreover, this paper can be extended to the dynamic equation on time scale.

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In the last few years, there has been a constantly increasing interest in developing the theory and numerical approaches for HPD (Hermitian positive definite) solutions to different classes of **nonlinear** **matrix** **equations** (see [8-21]). In this study, we consider the following problem: Find (X 1 , X 2 , ..., X m ) Î (P(n)) m solution to the following system

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