Order Nonlinear Neutral Delay Difference Research Articles

Classifications of Solutions of Second Order Nonlinear Neutral Delay Difference Equations With Positive and Negative Coefficients

Classifications of Solutions of Second Order Nonlinear Neutral Delay Difference Equations With Positive and Negative Coefficients

Positive solutions and iterative approximations of a third order nonlinear neutral delay difference equation

This paper deals with the third order nonlinear neutral delay difference equation with a forced term \t\t\tΔ2(a(n)Δ(x(n)+c(n)x(n−τ)))+f(n,x(n−b1(n)),x(n−b2(n)),…,x(n−bk(n)))=d(n),n≥n0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& \\Delta^{2} \\bigl(a(n)\\Delta \\bigl(x(n)+c(n)x(n-\\tau) \\bigr) \\bigr)+f \\bigl(n,x \\bigl(n-b_{1}(n) \\bigr),x \\bigl(n-b_{2}(n) \\bigr), \\ldots,x \\bigl(n-b_{k}(n) \\bigr) \\bigr) \\\\& \\quad =d(n),\\quad n\\geq n_{0}. \\end{aligned}$$ \\end{document} Using the Banach fixed point theorem, we prove the existence of uncountably many bounded positive solutions for the equation, suggest some Mann iterative schemes and obtain the error estimates between these bounded positive solutions and the sequences generated by the iterative schemes. Five nontrivial examples are also included.

Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients

In this paper some necessary and sufficient conditions are obtained to guarantee the oscillation for bounded and all solutions of second order nonlinear neutral delay difference equations. In Theorem 5 and Theorem 8, We have studied the oscillation criteria as well as the asymptotic behavior, where was established some sufficient conditions to ensure that every solution are either oscillates or |yn |→∞ as n→∞. Examples are given to illustrate the obtained results.

Existence and iterative approximations of nonoscillatory solutions for second order nonlinear neutral delay difference equations

This paper investigates the second order nonlinear neutral delay difference equation $$\begin{aligned}& \Delta \bigl[a_{n}\Delta (x_{n}+bx_{n-\tau}-d_{n} ) \bigr] +\Delta f (n,x_{f_{1}(n)},x_{f_{2}(n)},\ldots,x_{f_{k}(n)} ) \\& \quad{} +g (n,x_{g_{1}(n)},x_{g_{2}(n)},\ldots,x_{g_{k}(n)} )=c_{n},\quad n\ge n_{0}. \end{aligned}$$ By using the Banach fixed point theorem and some new techniques, we establish the existence results of uncountably many bounded nonoscillatory solutions for the above equation, propose a few Mann type iterative approximation schemes with errors and obtain several errors estimates between the iterative approximations and the nonoscillatory solutions. Examples that cannot be solved by known results are given to illustrate our theorems.

Positive Solutions for a Third Order Nonlinear Neutral Delay Difference Equation

The existence, multiplicity, and properties of positive solutions for a third order nonlinear neutral delay difference equation are discussed. Six examples are given to illustrate the results presented in this paper.

On Positive Solutions of a Fourth Order Nonlinear Neutral Delay Difference Equation

The existence results of uncountably many bounded positive solutions for a fourth order nonlinear neutral delay difference equation are proved by means of the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem. A few examples are included.

On Positive Solutions and Mann Iterative Schemes of a Third Order Difference Equation

The existence of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinear neutral delay difference equation are proved. Six examples are given to illustrate the results presented in this paper.

Existence of uncountably many bounded positive solutions for second order nonlinear neutral delay difference equations

This paper deals with the second order nonlinear neutral delay difference equation Δ [ a ( n ) Δ ( x ( n ) + b ( n ) x ( n − τ ) − c ( n ) ) ] + f ( n , x ( f 1 ( n ) ) , x ( f 2 ( n ) ) , … , x ( f k ( n ) ) ) = d ( n ) , n ≥ n 0 . By using Krasnoselskii’s fixed point theorem and some new techniques, we obtain the existence results of uncountably many bounded positive solutions for the above equation. Examples that cannot be solved by known results are given to illustrate the results presented in this paper.

Solvability of a Second Order Nonlinear Neutral Delay Difference Equation

This paper studies the second‐order nonlinear neutral delay difference equation , n ≥ n0. By means of the Krasnoselskii and Schauder fixed point theorem and some new techniques, we get the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions for the equation, respectively. Ten examples are given to illustrate the results presented in this paper.

Existence of uncountably many bounded positive solutions for a third order nonlinear neutral delay difference equation

This paper deals with the solvability of the third order nonlinear neutral delay difference equation Δ2(anΔ(xn+βnxn−τ))+Δ2f(n,xn−b1n,xn−b2n,…,xn−bkn)+Δg(n,xn−c1n,xn−c2n,…,xn−ckn)=h(n,xn−d1n,xn−d2n,…,xn−dkn),n≥n0, relative to the sequence {βn}n∈Nn0⊂R. Using the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem, a few sufficient conditions of the existence of uncountably many bounded positive solutions for the equation are presented. Seven examples are included to demonstrate the advantages and effectiveness of the results presented in this paper.

Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations

In this paper, the Banach fixed-point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference equation Δ a ( n ) Δ x ( n ) + b ( n ) x ( n - τ ) + Δ h ( n , x ( h 1 ( n ) ) , x ( h 2 ( n ) ) , … , x ( h k ( n ) ) ) + f ( n , x ( f 1 ( n ) ) , x ( f 2 ( n ) ) , … , x ( f k ( n ) ) ) = c ( n ) , n ⩾ n 0 , where τ , k ∈ N , n 0 ∈ N 0 , a , b , c : N n 0 → R with a ( n ) > 0 for n ∈ N n 0 , h , f : N n 0 × R k → R and h l , f l : N n 0 → Z with lim n → ∞ h l ( n ) = lim n → ∞ f l ( n ) = + ∞ , l ∈ { 1 , 2 , … , k } . A few Mann type iterative approximation schemes with errors are suggested, and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed. Seven nontrivial examples are given to illustrate the advantages of our results.

Global solvability for a second order nonlinear neutral delay difference equation

This paper studies the global existence of solutions of the second order nonlinear neutral delay difference equation Δ ( a n Δ ( x n + b x n − τ ) ) + f ( n , x n − d 1 n , x n − d 2 n , … , x n − d k n ) = c n , n ≥ n 0 with respect to all b ∈ R . A few results on global existence of uncountably many bounded nonoscillatory solutions are established for the above difference equation. Several nontrivial examples which dwell upon the importance of the results obtained in this paper are also included.

On the oscillation of second order nonlinear neutral delay difference equations

In this paper sufficient conditions are obtained for oscillation of all solutions of a class of nonlinear neutral delay difference equations of the form

Oscillation and nonoscillation of nonlinear neutral delay difference equations

In this paper, the authors establish some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral delay difference equation$$ \Delta^2 (x_n-p_nx_{n-k}) + q_nf(x_{n-\ell}) = 0, ~~n \ge n_0 $$where $ \{p_n\} $ and $ \{q_n\} $ are non-negative sequences with $ 0$

Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero

We find necessary conditions for every solution of the neutral delay difference equationΔ(rnΔ(yn−pnyn−m))+qnG(yn−k)=fnto oscillate or to tend to zero asn→∞, whereΔis the forward difference operatorΔxn=xn+1−xn, andpn, qn, rnare sequences of real numbers withqn≥0, rn>0. Different ranges of{pn}, includingpn=±1, are considered in this paper. We do not assume thatGis Lipschitzian nor nondecreasing withxG(x)>0forx≠0. In this way, the results of this paper improve, generalize, and extend recent results. Also, we provide illustrative examples for our results.

Bounded oscillation for second-order nonlinear neutral difference equations in critical and non-critical states

The sufficient and necessary conditions for the bounded oscillation of the second order nonlinear neutral delay difference equation Δ 2 [ x ( n ) - px ( n - τ ) ] = ∑ i = 1 m q i x ( n - σ i ) + f ( n , x ( n - η 1 ( n ) ) , … , x ( n - η l ( n ) ) ) are obtained in critical and non-critical cases.

Oscillation results for higher order nonlinear neutral delay difference equations

In this work, we shall consider higher order nonlinear neutral delay difference equation of the type Δ m ( y n + p n y n − l ) + q n y n − k α = 0 , where { p n } , { q n } are sequences of nonnegative real numbers, k and l are positive integers and α ∈ ( 0 , 1 ) is a ratio of odd positive integers. We obtain sufficient conditions for the oscillations of all solutions of this equation.

Classification and existence of nonoscillatory solutions of higher order nonlinear neutral difference equations

In this paper, we consider the higher order nonlinear neutral delay difference equation of the form $$\Delta ^r (x_n + px_{n - \tau } ) + f(n,x_{n - \sigma _1 (n)} ,x_{n - \sigma _2 (n)} ,...,x_{n - \sigma _m (n)} ) = 0.$$ We give an integrated classification of nonoscillatory solutions of the above equation according to their asymptotic behaviours. Necessary and sufficient conditions for the existence of nonoscillatory solutions with designated asymptotic properties are also established.

Non-Oscillatory Solutions of Higher Order Nonlinear Neutral Delay Difference Equation

Consider the forced higher order nonlinear neutral delay difference equation L_m( x(n) + cx(n - τ) ) + F ( n, x(σ(n) ) ) = g(n)\quad (n ≥ n_0). We obtain a global result (with respect to c ), which consists of some sufficient conditions for the existence of a non-oscillatory solution of the above equation. Our results improve and extend a number of existing results.

Oscillation of second order nonlinear neutral delay difference equations

In this paper, we obtain sufficient conditions for the oscillation of all solutions of the second order nonlinear delay difference equation Δ[a n( Δ(x n+p nx g(n))) α]+f(n,x σ(n))=0, where Δ x n = x n+1 − x n is the forward difference operator, { p n } is a sequence of nonnegative real numbers and α⩾1 is a quotient of positive odd integers. Our results extend and improve the known results in the literature. An example that dwell up the advantage of our results is included.