In this article, we study the global and asymptotic properties of the solutions of the difference equation $$x_{n+1}=Ax_{n}+Bx_{n-k}+(\beta x_{n}+\gamma x_{n-k})/(Cx_{n}+Dx_{n-k}),\quad n=0,1,2,\ldots,$$ where the initial conditions x?k,…,x?1,x0 are arbitrary positive real numbers and the coefficients A,B,C,D,β and γ are positive constants, while k is a positive integer number. Some numerical examples will be given to illustrate our results. 相似文献
In this note we consider the following higher order rational difference equations $$x_{n}=1+\frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}},\quad n=0,1,\ldots,$$ where 1≤k<l<m, and the initial values x?m,x?m+1,…,x?1 are positive numbers. We give some sufficient conditions for the persistence of positive solutions for the above equation, and prove that the positive equilibrium point of this equation is globally asymptotically stable. 相似文献
In this paper, we study the periodicity, the boundedness and the convergence of the following max-type difference equation $$x_n =\max\biggl\{\frac{ 1}{ x_{n-m}} , \frac{A_n }{x_{n-r} }\biggr \},\quad n =0, 1,2,\ldots,$$ where $\{A_{n}\}^{+\infty}_{n=0}$ is a periodic sequence with period k and An??(0,1) for every n??0, m??{1,2} and r??{2,3,??} with m<r, the initial values x?r,??,x?1??(0,+??). The special case when $m = 1, r = 2, \{A_{n}\}^{+\infty}_{ n=0}$ is a periodic sequence with period k and An??(0,1) for every n??0 has been completely investigated by Y.?Chen. Here we extend his results to the general case. 相似文献
In this paper, we study the difference equation $$x_{n+1}=p+\frac{x_{n-1}}{x_n}, \quad n=0,1,\ldots, $$ where initial values x?1,x0∈(0,+∞) and 0<p<1, and obtain the set of all initial values (x?1,x0)∈(0,+∞)×(0,+∞) such that the positive solutions $\{x_{n}\}_{n=-1}^{\infty}$ are bounded. This answers the Open problem 4.8.11 proposed by Kulenovic and Ladas (Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, 2002). 相似文献
In this paper, we investigate the global stability and the periodic nature of solutions of the difference equation $y_{n + 1} = \frac{{\alpha + y_n^p }} {{\beta y_{n - 1}^p }} - \frac{{\gamma + y_{n - 1}^p }} {{\beta y_n^p }},n = 0,1,2,... $ where α, β, γ ∈ (0,∞), α(1 ? p) ? γ > 0, 0 < p < 1, every yn ≠ 0 for n = ?1, 0, 1, 2, … and the initial conditions y?1, y0 are arbitrary positive real numbers. We show that the equilibrium point of the difference equation is a global attractor with a basin that depends on the conditions of the coefficients. 相似文献
Let A be an n-square normal matrix over , and Qm, n be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α,β∈Qm, n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…,m} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i1,…,ik, ik+1,…,im, such that α(ij)=β(ij), j=1,…,k, and {α(ik+1),…,α(im)};∩{β(ik+1),…,β(im)}=ø. Let be the group of n-square unitary matrices. Define the nonnegative number , where |α∩β|=k. Theorem 1 establishes a bound for ?k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that . 相似文献
The main objective of this paper is to study the boundedness character, the periodic character and the global stability of the positive solutions of the following difference equation $x_{n + 1} = \frac{{\alpha x_n + \beta x_{n - 1} + \gamma x_{n - 2} + \delta x_{n - 3} }}{{Ax_n + Bx_{n - 1} + Cx_{n - 2} + Dx_{n - 3} }},n = 0,1,2.....$ where the coefficientsA, B, C, D, α, β, γ, δ, and the initial conditionsx-3,x-2,x-1,x0 are arbitrary positive real numbers. 相似文献
Our aim in this paper is to investigate the global attractivity of the recursive sequence $$x_{n + 1} = \frac{{\alpha - \beta x_{n - k} }}{{\gamma + x_n }},$$ where α, β, γ >0 andk=1,2,… We show that the positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. 相似文献
The aim of this paper is to show that the following difference equation:Xn+1=α+(xn-k/xn-m)^p, n=0,1,2,…, where α 〉 -1, p 〉 O, k,m ∈ N are fixed, 0 ≤ m 〈 k, x-k, x-k+1,…,x-m,…,X-1, x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α+1. It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable. 相似文献
Let A be an n × n normal matrix over , and Qm, n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Qm, n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i1,…, ik, ik+1,…, im, such that α(ij) = β(ij), i = 1,…, k, and {α(ik+1),…, α(im) } ∩ {β(ik+1),…, β(im) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let n be the group of n × n unitary matrices. Define the nonnegative number where | α ∩ β| = k. It is proved that Let A be semidefinite hermitian. We conjecture that ρ0(A) ? ρ1(A) ? ··· ? ρm(A). These inequalities have been tested by machine calculations. 相似文献
Let pk(A), k=2,…,n, denote the sum of the permanents of all k×k submatrices of the n×n matrix A. A conjecture of Ðokovi?, which is stronger than the famed van der Waerden permanent conjecture, asserts that the functions pk((1?θ)Jn+;θA), k=2,…, n, are strictly increasing in the interval 0?θ?1 for every doubly stochastic matrix A. Here Jn is the n×n matrix all whose entries are equal . In the present paper it is proved that the conjecture holds true for the circulant matrices A=αIn+ βPn, α, β?0, α+;β=1, and , where In and Pn are respectively the n×n identify matrix and the n×n permutation matrix with 1's in positions (1,2), (2,3),…, (n?1, n), (n, 1). 相似文献
In this paper, we consider the following two-point fractional boundary value problem. We provide sufficient conditions for the existence of multiple positive solutions for the following boundary value problems that the nonlinear terms contain i-order derivative where n?1<α≤n is a real number, n is natural number and n≥2, α?i>1, i∈N and 0≤i≤n?1. ${}^{c}D_{0^{+}}^{\alpha}$ is the standard Caputo derivative. f(t,x0,x1,…,xi) may be singular at t=0. 相似文献
Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx1,x2,…,xn andx*1,x*2,…,x*n?1 denote the roots ofLn(α)(x) andLn(α)′(x) respectively and putx*0=0. In this paper we prove the following theorem: Ify0,y1,…,yn?1 andy1′,…,yn′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ . 相似文献
Every difference equation xn+1 = fn(xn,xn ? 1,…,xn ? k) of order k+1 with each mapping fn being homogeneous of degree 1 on a group G is shown to be equivalent to a system consisting of an equation of order k and a linear equation of order 1. 相似文献
In this paper, we consider the solutions of discrete second-order boundary value problem $$\left\{\begin{array}{l}\Delta ^{2}y(k-1)+a(k)f(k,y(k))=0,\quad k\in \{1,\ldots,T\},\\[2pt]y(0)-\alpha\Delta y(0)=0,\quad y(T+1)=\beta y(n),\end{array}\right.$$ where f is continuous, T≥3 is a fixed positive integer, n∈{2,…,T?1}, constant α,β>0 such that T+1?βn+α(1?β)>0,T+1?βn>0. Under suitable conditions, we accomplish this by using the property of the associate Green’s function and Leggett-Williams fixed point theorem. 相似文献
Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n?0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}n?0⊂[0,1] be such that ∑n?0αn=∞, and ∑n?0αn(kn−1)<∞. Suppose {xn}n?0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n?0, and suppose there exists a strictly increasing continuous function , ?(0)=0 such that 〈Tnx−x∗,j(x−x∗)〉?kn‖x−x∗‖2−?(‖x−x∗‖), ∀x∈K. It is proved that {xn}n?0 converges strongly to x∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n?0 (where {un}n?0 is a bounded sequence in K and {an}n?0, {bn}n?0, {cn}n?0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T. 相似文献
Letx1, …,xn be givenn distinct positive nodal points which generate the polynomial $$\omega _n (x) = \prod\limits_{i = 1}^n {(x - x_i )} .$$ Letx*1, …,x*n?1 be the roots of the derivativeω′n(x) and putx0=0. In this paper, the following theorem is proved: Ify0, …,yn andy′1, …,y′n?1 are arbitrary real numbers, then there exists a unique polynomialP2n?1(x) of degree 2n?1 having the following interpolation properties: $$P_{2n - 1} (x_j ) = y_j (j = 0,...,n),$$ , $$P_{2n - 1}^\prime (x_j^* ) = y_j^\prime (j = 1,...,n - 1).$$ . This result gives the theoretical completion of the original Pál type interpolation process, since it ensures uniqueness without assuming any additional condition. 相似文献
In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z1…,zn), zi ? , i=1, …, n, is uniquely determined from the magnitude of f(x1…,xn): | f(x1…,xn)|, xi ? , i=1,…, n, except for (1) linear shifts: i(α1z1+…+αn2n+β), β, αi?, i=1,…, n; and (2) conjugation: . 相似文献