Nonlinear Neutral Differential Equations Research Articles

Existence of positive solutions of second-order nonlinear neutral differential equations with positive and negative terms

In this paper, the existence of bounded positive solution of the second-order nonlinear neutral differential equations are studied. Some new sufficient conditions are given. Furthermore, examples given show that the results here are almost sharp.

Kamenev-type oscillation criteria for delay difference equations

Some oscillation criteria are established by Raccati transformation techniques for the following second-order nonlinear neutral difference equation Δ( p n (Δ( x n + c n x n - τ )) γ) + q n x n - σ β = 0,n = 0,1,2 … which extend and include several oscillation criteria in [11], and also correct a theorem and its proof in [10].

Asymptotic behavior of nonoscillatory solutions to [formula omitted]-th order nonlinear neutral differential equations

For a class of n -th order nonlinear neutral differential equations, sufficient conditions for all nonoscillatory solutions to satisfy lim t → + ∞ t 1 − n x ( t ) = a are established. For another class of equations, necessary and sufficient conditions for nonoscillatory solutions to satisfy the above condition are obtained. Our theorems complement and extend recent results reported in the literature.

Existence and iterative approximations of nonoscillatory solutions of higher order nonlinear neutral delay differential equations

This paper studies a few existence results of nonoscillatory solutions for the following higher order nonlinear neutral delay differential equation: dndtnx(t)+cx(t-τ)+(-1)n-i+1didtih(t,x(h1(t)),x(h2(t)),…,x(hk(t)))+(-1)n+1f(t,x(f1(t)),x(f2(t)),…,x(fk(t)))=g(t),t⩾t0, where n is a positive integer, 0⩽i⩽n-1, c∈R⧹{-1}, τ>0, h∈Ci([t0,+∞)×Rk,R), f∈C([t0,+∞)×Rk,R), g∈C([t0,+∞),R), hl∈Ci([t0,+∞),R) and fl∈C([t0,+∞),R) with limt→+∞hl(t)=limt→+∞fl(t)=+∞,l=1,…,k constructs several Mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions. In addition sufficient conditions for the existence of infinitely many nonoscillatory solutions for the above equation are given. Three nontrivial examples are given to illustrate the advantages of our results.

Oscillation criteria for second order quasi-linear neutral delay differential equations

In this paper, some new oscillation criteria for second order neutral nonlinear differential equation [ r ( t ) | ( x ( t ) + p ( t ) x ( ( σ ( t ) ) ) ) ′ | m - 1 ( x ( t ) + p ( t ) x ( ( σ ( t ) ) ) ) ′ ] ′ + q ( t ) f ( x ( τ ( t ) ) ) = 0 are established. New oscillation criteria are different from most known ones in the sense that they are based on a class of new functions Φ ( t , s , l ) defined in the sequel.

Segmented Tau approximation for a parametric nonlinear neutral differential equation

The segmented formulation of the Tau method is used to approximate the solutions of the parametric nonlinear neutral differential equation y ′ ( t ) = ry ( t ) ( a + by ( t - τ ) + cy ′ ( t - τ ) ) , t ⩾ 0 , y ( t ) = Ψ ( t ) , t ⩽ 0 , which represents, for different values of the parameters r, a, b, c and τ, a family of functional differential equations with some of its members arising in areas as different as the number theory, mathematical biology, and population dynamics. For this equation no closed form of analytical solution is available. The numerical results obtained are consistent with the theoretical and practical results reported elsewhere.

New Kamenev-type oscillation criteria for second order neutral nonlinear differential equations

In this paper, some new oscillation criteria for second order neutral nonlinear differential equation [ r ( t ) ( y ( t ) + p ( t ) y ( σ ( t ) ) ) ′ ] ′ + ∑ i = 1 n q i ( t ) f i ( y ( τ i ( t ) ) ) = 0 , t ⩾ t 0 are established. New oscillation criteria are different from most known ones in the sense that they are based on a class of new functions Φ( t, s, l) defined in the sequel. Our results are sharper than some previous results which can be seen by the example at the end of this paper.

Positive periodic solutions in neutral nonlinear differential equations

We use Krasnoselskii’s fixed point theorem to show that the nonlinear neutral differential equation with delay d dt [x(t) ax(t �)] = r(t)x(t) f(t,x(t �)) has a positive periodic solution. An example will be provided as an application to our theorems.

Asymptotic behavior of nonoscillatory solutions of second order nonlinear neutral differential equations

Asymptotic behavior of nonoscillatory solutions of second order nonlinear neutral differential equations

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.

Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses

This paper is concerned with a nonlinear neutral differential equations with impulses of the form (*) [ x ( t ) + C ( t ) x ( t − τ ) ] ′ + P ( t ) f ( x ( t − δ ) ) = 0 , t ⩾ t 0 , t ≠ t k , (**) x ( t k ) = b k x ( t k − ) + ( 1 − b k ) ∫ t k − δ t k P ( s + δ ) f ( x ( s ) ) d s , k = 1 , 2 , … . Sufficient conditions are obtained for every solution of (*) and (**) that tends to a constant as t → ∞ .

Fixed Point Techniques and Stability for Neutral Nonlinear Differential Equations with Unbounded Delays

Abstract We use the contraction mapping theorem to obtain stability results of the scalar nonlinear neutral differential equation with functional delay 𝑥′(𝑡) = –𝑎𝑥(𝑡) + 𝑏(𝑡)𝑥2(𝑡 – 𝑟(𝑡)) + 𝑐(𝑡)𝑥(𝑡 – 𝑟(𝑡))𝑥′(𝑡 – 𝑟(𝑡)).

Existence and asymptotic behavior of solutions of nonlinear neutral differential equations

The nonlinear neutral differential equation (1) d n d t n [ x ( t ) + h ( t ) x ( τ ( t ) ) ] + f ( t , x ( g ( t ) ) ) = q ( t ) , is considered under the following conditions: n ∈ N ; h ∈ C [ t 0 , ∞ ) ; τ ∈ C [ t 0 , ∞ ) is strictly increasing, lim t → ∞ τ ( t ) = ∞ and τ ( t ) < t for t ≥ t 0 ; g ∈ C [ t 0 , ∞ ) and lim t → ∞ g ( t ) = ∞ ; f ∈ C ( [ t 0 , ∞ ) × R ) ; q ∈ C [ t 0 , ∞ ) . It is shown that if f is small enough in some sense, Eq. (1) has a solution x ( t ) which behaves like the solution of the unperturbed equation d n d t n [ ω ( t ) + h ( t ) ω ( τ ( t ) ) ] = q ( t ) . Several known results in the literature can be obtained from our results.

Differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays

In this paper we consider a class of nonlinear neutral differential equations with state-dependent delays. We study well-posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite-dimensional parameters in a pointwise sense and also in the C- and W 1 , ∞ -norms.

A classification scheme for nonoscillatory solutions of a higher order neutral difference equation

Nonoscillatory solutions of a nonlinear neutral type higher order difference equations are classified by means of their asymptotic behaviors. By means of the Kranoselskii's fixed point theorem, existence criteria are then provided for justification of such classification.

Necessary and sufficient conditions for oscillation of first order nonlinear neutral differential equations

In this paper, we prove that every solution of the first order nonlinear neutral differential equation [ x ( t ) − p x ( t − τ ) ] ′ + q ( t ) ∏ j = 1 m | x ( t − σ j ) | β j sign [ x ( t − σ 1 ) ] = 0 , t ⩾ t 0 , oscillates if and only if ∫ t 0 ∞ q ( s ) exp [ τ −1 ln p ( ∑ j = 1 m β j − 1 ) s ] d s = ∞ , when ( ∑ j = 1 m β j − 1 ) ln p < 0 , and ∫ t 0 ∞ q ( s ) d s = ∞ , when ( ∑ j = 1 m β j − 1 ) ln p > 0 , where p, τ > 0 , β j > 0 , σ j ⩾ 0 , j = 1 , 2 , … , m , q ∈ C ( [ t 0 , ∞ ) , [ 0 , ∞ ) ) .

Oscillation of second-order nonlinear neutral differential equations

In this paper, some sufficient conditions for oscillation and nonoscillation are obtained for the second-order nonlinear neutral differential equation (∗) [ x ( t ) − p ( t ) x ( t − τ ) ] ″ + q ( t ) f ( x ( t − σ ) ) = 0 , t ⩾ 0 , where 0 ⩽ p ( t ) ⩽ 1 , q ( t ) ⩾ 0 , τ , σ > 0 .

Asymptotic trichotomy for positive solutions of a class of fourth-order nonlinear neutral difference equations with quasidifferences

We consider a class of fourth-order nonlinear neutral difference equations with quasidifferences where sequence ( p n ) is uniformly strictly subnormal. The classification of nonoscillatory solutions of this equation are obtained. The conditions under which the eventually positive solutions of considered equation can be classified into three nonempty distinct categories are given. Sufficient conditions are obtained for the difference equation to admit the existence of a minimal solutions. Also necessary and sufficient conditions are obtained for the difference equation to admit the existence of an asymptotically constant and a maximal solutions. Results are illustrated on examples.

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.

Existence for nonoscillatory solutions of higher order nonlinear neutral differential equations

Consider the forced higher-order nonlinear neutral functional differential equation $$\frac{{d^n }}{{dt^n }}\left[ {x\left( t \right) + C\left( t \right)x\left( {t - \tau } \right)} \right] + \sum\limits_{i = 1}^m {Q_i } \left( t \right)f_i \left( {x\left( {t - \sigma _i } \right)} \right) = g\left( t \right),\quad t \geqslant t_0 ,$$ where n,m ≥, 1 are integers, τ, σi ∈ ℝ+ = [0,∞), C,Q i, g ∈ C([t 0,∞), ℝ), fi ∈ C(ℝ, ℝ), (i = 1, 2, ...;, m). Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general Q i(t) (i = 1, 2, ... ,m) and g(t) which means that we allow oscillatory Q i(t) (i = 1, 2, ... ,m) and g(t). Our results improve essentially some known results in the references.