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This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

This paper deals with the oscillation properties of higher-order nonlinear differential equations with distributed delay $$ \bigl[ b ( \ell ) \bigl( y^{ ( n-1 ) } ( \ell ) \bigr) ^{\gamma} \bigr] ^{\prime}+ \int_{c}^{d}q ( \ell, \xi ) y^{\gamma} \bigl( g ( \ell, \xi ) \bigr) \,d ( \xi ) =0, \quad \ell\geq\ell_{0}, $$ under the condition $$ \int_{\ell_{0}}^{\infty}\frac{1}{b^{\frac{1}{\gamma}} ( \ell ) }\,d\ell< \infty. $$ We obtain new oscillation criteria by employing a refinement of the generalized Riccati transformations and new comparison principles. We provide some examples to illustrate the main results.

In this paper, we present a new sufficient condition for the oscillation of all solutions of linear delay differential equations. The obtained result improves known conditions in the literature. We also give an example to illustrate the applicability and strength of the obtained condition over known ones.

In this paper, a class of fourth-order differential equations with advanced type is studied. Applying the generalized Riccati transformation, integral averaging technique and the theory of comparison, a set of new criteria for oscillation or certain asymptotic behavior of solutions of this equations is given. Our results essentially improve and complement some earlier publications. Some examples are presented to demonstrate the main results.

The aim of this work is to study oscillatory properties of a class of fourth-order delay differential equations. New oscillation criteria are obtained by using generalized Riccati transformations. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results.

In this paper an attempt is made to depict a clear picture of the overall structure of nonoscillatory solutions of the first order half-linear differential system \begin{equation*} x'-p(t)\varphi_{1/\alpha}(y)=0,\qquad y'+q(t)\varphi_{\alpha}(x)=0, \tag{A} \end{equation*} where $\alpha>0$ is a constant, $p(t)$ and $q(t)$ are positive continuous functions on $[0,\infty)$, and $\varphi_{\gamma}(u)=|u|^{\gamma}\textrm{sgn}\;u, \;u\in {\mathbb R},\;\gamma>0$. A systematic analysis of the existence and asymptotic behavior of solutions of (A) is proposed for this purpose. A special mention should be made of the fact that all possible types of nonoscillatory solutions of (A) can be constructed by solving the Riccati type differential equations associated with (A). Worthy of attention is that all the results for (A) can be applied to the second order half-linear differential equation \begin{equation*} (p(t)\varphi_{\alpha}(x'))'+q(t)\varphi_{\alpha}(x)=0, \tag{E} \end{equation*} to build automatically a nonoscillation theory for (E).

Our aim in the present paper is to employ the Riccatti transformation which differs from those reported in some literature and comparison principles with the second-order differential equations, to establish some new conditions for the oscillation of all solutions of fourth-order differential equations. Moreover, we establish some new criterion for oscillation by using an integral averages condition of Philos-type, also Hille and Nehari-type. Some examples are provided to illustrate the main results.

This paper is devoted to obtaining some new sufficient conditions for the oscillation of all solutions of first order nonlinear differential equations with several deviating arguments. Finally, an illustrative example related to our results is given.

In this paper, we study the oscillation of second-order neutral differential equations with delayed arguments. Some new oscillatory criteria are obtained by a Riccati transformation. To illustrate the importance of the results, one example is also given.

In the paper, we study the oscillation of fourth-order delay differential equations, the present authors used a Riccati transformation and the comparison technique for the fourth order delay differential equation, and that was compared with the oscillation of the certain second order differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. Some examples are also presented to test the strength and applicability of the results obtained.

The authors study the boundedness of nonoscillatory solutions of forced fractional differential equations of the form \begin{equation*} ^{C}D_{c}^{\alpha }y(t)=e(t)+f(t,x(t)),\qquad c>1, \quad\alpha \in (0,1), \end{equation*} where $y(t)=\left( a(t)x^{\prime }(t)\right) ^{\prime }$, $c_{0}=\frac{y(c)}{\Gamma (1)}=y(c)$, and $c_{0}$ is a real constant. The technique used in obtaining their results will apply to related fractional differential equations with Caputo derivatives of any order. Examples illustrate the results obtained in this paper.

This article deals with the asymptotic behavior of nonoscillatory solutions of fourth order linear differential equation where the coefficients are perturbations of con- stants. We define a change of variable and deduce that the new variable satisfies a third order nonlinear differential equation. We assume three hypotheses. The first hypoth- esis is related to the constant coefficients and set up that the characteristic polynomial associated with the fourth order linear equation has simple and real roots. The other two hypotheses are related to the behavior of the perturbation functions and establish asymptotic integral smallness conditions of the perturbations. Under these general hy- potheses, we obtain four main results. The first two results are related to the application of a fixed point argument to prove that the nonlinear third order equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the nonlinear third order equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise the formulas for the asymptotic behavior of the linear fourth order differential equation. In addition, we present an example to show that the results introduced in this paper can be applied in situations where the assumptions of some classical theorems are not satisfied.

The $k$-dimensional system of neutral type nonlinear difference equations with delays in the following form \begin{equation*} \begin{cases} \Delta \Big(x_i(n)+p_i(n)\,x_i(n-\tau_i)\Big)=a_i(n)\,f_i(x_{i+1}(n-\sigma_i))+g_i(n),\\ \Delta \Big(x_k(n)+p_k(n)\,x_k(n-\tau_k)\Big)=a_k(n)\,f_k(x_1(n-\sigma_k))+g_k(n), \end{cases} \end{equation*} where $i=1,\dots,k-1$, is considered. The aim of this paper is to present sufficient conditions for the existence of nonoscillatory bounded solutions of the above system with various $(p_i(n))$, $i=1,\dots,k$, $k\geq 2$.

In this paper, we consider the following Emden-Fowler type dynamic equations on time scales \begin{equation*} \big(a(t)|x^\Delta(t)|^\alpha \operatorname{sgn} x^\Delta(t)\big)^\Delta+b(t)|x(t)|^\beta \operatorname{sgn}x(t)=0, \end{equation*} when $\alpha

In this work, we study the oscillation of second-order delay differential equations, by employing a refinement of the generalized Riccati substitution. We establish a new oscillation criterion. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. We illustrate the results with some examples.

WOS: 000410453200004, Consider the first-order nonlinear retarded differential equation x'(t) + p(t) f (x(tau(t))) = 0, t >= t(0) where p(t) and tau(t) are function of positive real numbers such that tau(t) = to, and tau(t) = infinity. Under the assumption that the retarded argument is non-monotone, new oscillation results are given. An example illustrating the result is also given.

Oscillation of second order difference equations with bounded phi-Laplacian is studied by comparing it with the discrete Emden–Fowler difference equation. Analogues and discrepancies in behavior of solutions of these equations are also presented and a comparison with the continuous case is given.

By employing Kranoselskii's fixed point theorem, we obtain sufficient conditions for the existence of nonoscillatory solutions of the forced higher-order nonlinear neutral dynamic equation \[ [x(t)+p(t)x(\tau(t))]^{\nabla^m }+\sum^{k}_{i=1}p_{i}(t)f_{i}(x(\tau_{i}(t)))=q(t) \] on a time scale, where $p_{i}(t)$, $f_{i}(t)$ and $q(t)$ may be oscillatory. Then we establish sufficient and necessary conditions for the existence of nonoscillatory solutions to the equation $[x(t)+p(t)x(\tau(t))]^{\nabla^m }+F(t,x(\delta(t)))=q(t)$. Finally, we deal with dynamic equation \[ [x(t)+p(t)x(\tau(t))]^{\nabla^{m-1} \Delta}+\sum^{k}_{i=1}p_{i}(t)f_{i}(x(\tau_{i}(t)))=q(t) \] with mixed $\nabla$ and $\Delta$ derivatives. In particular, some interesting examples are included to illustrate the versatility of our results.

The aim of this paper is to present new summation inequalities and to show that the half-linear difference equation Δ ( a n | Δ x n | α sgn Δ x n ) = b n | x n + 1 | α sgn x n + 1 exhibits not only similarities but also some discrepancies with the linear case, especially as regards the possible coexistence of the so-called extremal solutions. Some of our results are new also for the linear difference equation.

This article is concerned with solutions that behave asymptotically like polynomials for nth order (n > 1) nonlinear ordinary differential equations. For each given integer m with 1 less than or equal to m less than or equal to n - 1, sufficient conditions are presented in order that, for any real polynomial of degree at most m, there exists a solution which is asymptotic at infinity to this polynomial. Conditions are also given, which are sufficient for every solution to be asymptotic at infinity to a real polynomial of degree at most n - 1. The application of the results obtained to the special case of second order nonlinear differential equations leads to improved versions of the ones contained in the recent paper by Lipovan [Glasg. Math. J. 45 (2003) 179] and of other related results existing in the literature. (C) 2004 Elsevier Ltd. All tights reserved. Nonlinear Analysis-Theory Methods & Applications

Some new results are given concerning the behavior of the oscillatory solutions of first or second order delay differential equations. These results establish that all oscillatory solutions x of a first or second order delay differential equation satisfy x (t) = 0 (v(t)) as t --> infinity, where v is a nonoscillatory solution of a corresponding first or second order linear delay differential equation. Some applications of the results obtained are also presented. (C) 2003 Elsevier Inc. All rights reserved. Journal of Mathematical Analysis and Applications

Classification and criteria for the existence of nonoscillatory solutions of nonlinear differential systems are obtained. As a consequence, the existence of positive solutions is discussed.

In this paper, we study the asymptotic behavior of the solutions of a neutral type difference equation of the form% \[ \Delta \bigg[ x(n)+\sum_{j=1}^{w}c_{j}x(\tau _{j}(n))\bigg] +p(n)x(\sigma (n))=0,\quad n\geq 0 \] where $\tau _{j}(n)$, $j=1,\ldots,w$, are general retarded arguments, $\sigma (n) $ is a general deviated argument (retarded or advanced), $c_{j}\in \mathbb{R}$, $j=1,\ldots,w$, $(p(n))_{n\geq 0}$ is a sequence of positive real numbers such that $p(n)\geq p$, $p\in\mathbb{R}_{+}$, and $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)-x(n)$. ¶ We also examine the convergence of the solutions when these are continuous and differentiable with respect to $c_{j}$, $j=1,\ldots,w$.

Bainov, D.D., V. Petrov and V. Proytcheva, Existence and asymptotic behaviour of nonoscillatory solutions of second order neutral differential equations with “maxima”, Journal of Computational and Applied Mathematics. In the paper, sufficient conditions for existence of positive and negative solutions of neutral equations with “maxima” are obtained and their asymptotic behaviour is investigated.

In this paper, we study the asymptotic behavior of neutral advanced difference equations of the form \begin{equation*} \Delta ^{m}\left[ x(n)+cx(n+a)\right] +p(n)x(\sigma (n))=0,\qquad m\in \mathbb{N}, \qquad n\geq 0\text{,} \end{equation*} where $c\in \mathbb{R} $, $ \mathbb{N} \ni a\geq 2$, $(\sigma (n))$ is a sequence of positive integers such that $% (\sigma (n))\geq n+2$\ for all $n\geq 0$, $\left( p(n)\right) _{n\geq 0}$ is a sequence of real numbers, $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)x(n)$, and $\Delta ^{j}$ denotes the $j^{th}$ forward difference operator $\Delta ^{j}\left( x(n)\right) =\Delta \left( \Delta ^{j1}\left( x(n)\right) \right) $ for $j=2,3,...,m$. Examples illustrating the results are also given.

This paper is concerned with a boundary-valne problem on the half-line for nonlinear two-dimensional delay differential systems with positive delays. A theorem is established, which provides sufficient conditions for the existence of positive solutions. The application of this theorem to the special case of second-order nonlinear delay differential equations is given. Also, the application of the theorem to two-dimensional Emden-Fowler-type delay differential systems with constant delays is presented. Moreover, some general examples demonstrating the applicability of the theorem are included. Proceedings of the Edinburgh Mathematical Society

This article is concerned with a boundary value problem on the half-line for nonlinear two-dimensional delay differential systems. By the use of the Schauder-Tikhonov theorem, a result on the existence of solutions is obtained. Also, via the Banach contraction principle, another result concerning the existence and uniqueness of solutions is established. Moreover, these results are applied to the special case of ordinary differential systems and to a certain class of delay differential systems. Furthermore, applications to differential systems of Emden-Fowler type and to linear differential systems are presented, and two specific examples are given. Annales Polonici Mathematici

Second order nonlinear delay differential equations with positive delays are considered, and sufficient conditions are given that guarantee the existence of positive increasing solutions on the half-line with first order derivatives tending to zero at infinity The approach is elementary and is essentially based on an old idea which appeared in the author's paper Arch. Math. (Basel) 36 (1981), 168-178. The application of the result obtained to second order Emden-Fowler type differential equations with constant delays and, especially, to second order linear differential equations with constant delays, is also presented. Moreover, some (general or specific) examples demonstrating the applicability of the main result are given. Glasgow Mathematical Journal

In this paper, effective sufficient conditions for the oscillation of all solutions of impulsive neutral delay differential equations of the form [x(t) − P(t)x(t − τ)]′ + Q(t)|x(t − σ)| λ sgn x(t − σ) = 0, t ≥ t 0 , ((*)) x t + k = b k x(t k ), k = 1, 2, … ((**)) are established. Our results reveal the fact that the oscillatory properties of all solutions of Eqs. (*) and (**) may be caused by the impulsive perturbations (**) though the corresponding neutral delay differential equation without impulses, i.e., Eq. (*), admits a nonoscillatory solution. It is also seen that the oscillatory behavior of all solutions of Eq. (*) can be inherited by Eqs. (*) and (**) under certain impulsive perturbations (**). Some examples are also given to illustrate the applicability of the results obtained.

This paper is concerned with sequences that satisfy a class of fourth order linear recurrence equations. Basic properties of such sequences are derived. In addition, we discuss the oscillatory and nonoscillatory behavior of such sequences.

Classification schemes for nonoscillatory solutions of a class of nonlinear two-dimensional nonlinear difference systems are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also provided.

In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation [ r ( t ) ( x ( t ) + P ( t ) x ( t − τ ) ) ′ ] ′ + ∑ i = 1 m Q i ( t ) f i ( x ( t − σ i ) ) = 0 , t ⩾ t 0 , where m ⩾ 1 is an integer, τ > 0 , σ i ⩾ 0 , r , P , Q i ∈ C ( [ t 0 , ∞ ) , R ) , f i ∈ C ( R , R ) ( i = 1 , 2 , … , m ), are studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general P ( t ) and Q i ( t ) ( i = 1 , 2 , … , m ) which means that we allow oscillatory P ( t ) and Q i ( t ) ( i = 1 , 2 , … , m ). In particular, our results improve essentially and extend some known results in the recent references.

In this paper, we give an analogue of the Arzela–Ascoli theorem on time scales. Then, we establish the existence of nonoscillatory solutions to the neutral dynamic equation [ x ( t ) + p ( t ) x ( g ( t ) ) ] Δ + f ( t , x ( h ( t ) ) ) = 0 on a time scale. To dwell upon the importance of our results, three interesting examples are also included.

In this paper, we consider the following forced higher-order nonlinear neutral difference equation Δ m (x n +c n x n−k )+ ∑ s=1 u p (s) n f s (x n−r s )=q n ,n⩾n 0 , where m,u⩾1, k⩾0, and rs⩾0 are integers, {cn}, {p(s)n} (s=1,2,…,u) and {qn} are sequence of real numbers and f s ∈C( R , R ) (s=1,2,…,u). By using Krasnoselskii's fixed point theorem and some new techniques, we obtain sufficient conditions for the existence of nonoscillatory solutions for general {p(s)n} (s=1,2,…,u) and {qn} which means that we allow oscillatory {p(s)n} (s=1,2,…,u) and {qn}.

This paper deals with nonoscillatory behaviour of solutions of third-order nonlinear functional differential equations of the form y ‴ + p ( t ) y ′ + q ( t ) F ( y ( g ( t ))) = 0. It has been shown that under certain conditions on coefficient functions, the nonoscillatory solutions of this equation tends to either zero or ∓∞ as t → ∞.

The asymptotic behavior of nonoscillatory solutions of the half-linear differential equation is studied. In particular, two Wronskian-type functions, which have some interesting properties, similar to the one of the Wronskian in the linear case, are given. Using these properties and suitable integral inequalities, the existence of the so-called intermediate solutions is examined and an open problem is solved.

In this paper, we consider the following higher-order neutral delay difference equations with positive and negative coefficients: Δ m ( x n + cx n − k ) + p n x n − r − q n x n − l = 0, n ≥ n 0 , where c ϵ R , m ⩾ 1, k ⩾ 1, r , l ⩾ 0 are integers, and { p n } ∞ n = n 0 and { q n } n = n 0 ∞ are sequences of nonnegative real numbers. We obtain the global results (with respect to c ) which are some sufficient conditions for the existences of nonoscillatory solutions.

This paper establishes an apparently overlooked relationship between the pair of fourth order linear equationsyiv−p(x)y=0andyiv+p(x)y=0, wherepis a positive, continuous function defined on[0,∞). It is shown that if all solutions of the first equation are nonoscillatory, then all solutions of the second equation must be nonoscillatory as well. An oscillation criterion for these equations is also given.

This work considers differential equations of the form(py″)′​′+qy″+ry=0wherep,qandrare positive continuous functions defined on[0,∞). The main concentration is on the oscillatory and asymptotic behavior of the solutions. Such an investigation is important because the above equation often arises in the study of mechanical vibrations.

In this paper we consider the existence of nonoscillatory solutions of higher-order neutral differential equations with distributed coefficients and delays. We use the $Banach$ contraction principle to obtain new sufficient condition for the existence of nonoscillatory solutions.