# nonlinear matrix difference equations

## Top PDF nonlinear matrix difference equations: ### Hyers Ulam stability of the first order matrix difference equations

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 ### Periodic solutions of nonlinear vector difference equations

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. ### On the boundedness of the solutions in nonlinear discrete Volterra difference equations

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 ### Oscillation of nonlinear delay difference equations

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 some classes of nonlinear difference equations

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. ### On nonlocal boundary value problems of nonlinear q difference equations

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. ### Positive Decreasing Solutions of Higher Order Nonlinear Difference Equations

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. ### Unbounded Perturbations of Nonlinear Second Order Difference Equations at Resonance

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

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 ### Dynamical behavior of a system of three dimensional nonlinear difference equations

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. ### Dynamical Properties for a Class of Fourth Order Nonlinear Difference Equations

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. ### On the solvability of initial value problems for nonlinear implicit difference equations

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 . ### TWO DIMENSIONAL FRACTIONAL SYSTEM OF NONLINEAR DIFFERENCE EQUATIONS IN THE MODELING COMPETITIVE POPULATIONS

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 . ### Multiple periodic solutions for the second order nonlinear difference equations

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  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. ### Asymptotic decay of nonoscillatory solutions of general nonlinear difference equations

 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.  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. ### Oscillation of solutions of some generalized nonlinear α difference equations

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[.] ### A Cauchy Problem for Some Fractional q Difference Equations with Nonlocal Conditions

Several authors have studied the semi-linear differential equations with nonlocal conditions in Banach space,  . In , 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. ### Periodic solutions of nonlinear finite difference systems with time delays

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.  