Search

In this paper, the existence of nonoscillatory solutions for a class of higher-order nonlinear differential equations is investigated. Notably, the equations are of mixed neutral type with a forcing term, which distinguished the equations in this paper from the existing ones and made the qualitative analysis of the solution more complicated. By means of the Schauder-Tychonoff fixed point theorem and inequality techniques, some new sufficient conditions for the existence of nonoscillatory solutions were established. The results in this paper improved and generalized some known results in the existing works. Finally, an example was given to illustrate the effectiveness of the proposed method.

This paper deals with the nonoscillatory solutions of the nonlinear differential equation a (t) | x ′ | p (t) - 2 x ′ ′ + b (t) | x | λ - 2 x = 0 involving "singular" p(t)-Laplacian. Sufficient conditions are given for the existence of extremal solutions, which do not exist in the conventional cases. In addition, we prove the coexistence of extremal solutions and weakly increasing solutions. Some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic behavior of the nonoscillatory solutions of a certain forced fractional differential equations with positive and negative terms, involving the Caputo fractional derivative. The results obtained are new and generalize some known results appearing in the literature. Two examples are also provided to illustrate the results. [ABSTRACT FROM AUTHOR]

Abstract 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.

Abstract 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.

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.

Abstract This paper deals with the oscillation properties of higher-order nonlinear differential equations with distributed delay [b(ℓ)(y(n−1)(ℓ))γ]′+∫cdq(ℓ,ξ)yγ(g(ℓ,ξ))d(ξ)=0,ℓ≥ℓ0, $$ \bigl[ b ( \ell ) \bigl( y^{ ( n-1 ) } ( \ell ) \bigr) ^{\gamma} \bigr] ^{\prime}+ \int_{c}^{d}q ( \ell, \xi ) y^{\gamma} \bigl( g ( \ell, \xi ) \bigr) \,d ( \xi ) =0, \quad \ell\geq\ell_{0}, $$ under the condition ∫ℓ0∞1b1γ(ℓ)dℓ

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).

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. [ABSTRACT FROM AUTHOR]

The oscillation of the first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients is investigated, using an iterative procedure. The conditions obtained by this method achieve a marked improvement on all known conditions in the literature. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate oscillatory properties of even-order advanced differential equations. The key idea of our approach is to conduct a comparison with first order equations and use the Riccati transformation technique. Some new oscillation criteria are shown. Two examples are presented in order to clarify the main results. [ABSTRACT FROM AUTHOR]

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 CDα c y(t) + f(t, x(t)) = e(t) + k(t)xŋ(t) + h(t, x(t)), where t ≥ c ≥ 1, α (0, 1), ŋ ≥ 1 is the ratio of positive odd integers, and CDα c y denotes the Caputo fractional derivative of y of order α. The cases y(t) =(a(t)(x'(t))ŋ)' and y(t) = a(t)(x'(t))ŋ 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. [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation of the fourth-order delay differential equation. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and new comparison principles. An example is included to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper deals with the boundedness of nonoscillatory solutions of the forced fractional differential equation with positive and negative terms C D c α y (t) + f (t , x (t)) = e (t) + k (t) x (t) + h (t , x (t)) , where t ≥ c , α ∈ (0 , 1) , c > 1 , and C D c α y denotes the Caputo fractional derivative of y of order α. The cases where y (t) = a (t) x ′ (t) ′ and y (t) = a (t) x ′ (t). are considered. The technique used can be applied to other related fractional differential equations. Examples are inserted to illustrate the relevance of the results obtained. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to prove the existence theorems of the nonoscillatory solution for higher order neutral dynamic equations on time scales. Our results improve and generalize the recent known results by removing some restrictive conditions. [ABSTRACT FROM AUTHOR]

In this paper, we develop sufficient criteria for the existence of a nonoscillatory solution to the fractional neutral functional differential equation of the form: D t α [ x (t) + c x (t - τ) ] ′ + ∑ i = 1 m P i (t) F i (x (t - σ i)) = 0 , t ≥ t 0 , where D t α is Liouville fractional derivatives of order α ≥ 0 on the half-axis, c ∈ R , τ , σ i ∈ R + , P i ∈ C ([ t 0 , ∞) , R) , F i ∈ C (R , R) , i = 1 , 2 , ... , m , m ≥ 1 is an integer. Our results are new and improve many known results on the integer-order functional differential equations. [ABSTRACT FROM AUTHOR]

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

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.

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.

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 $(p_i(n))$, $i=1,\dots,k$, $k\geq 2$.

This article deals with the asymptotic behavior of nonoscillatory solutions of fourth order linear differential equation where the coefficients are perturbations of constants. 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 hypothesis 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 hypotheses, 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.

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

An iterative procedure is used to establish the oscillatory sufficient conditions of first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients. The conditions obtained by this method, achieve a marked improvement on all known conditions in the literature. [ABSTRACT FROM AUTHOR]

In this work, we consider the existence and uniqueness of nonoscillatory solutions to first-order differential equations having both delay and advance terms, known as mixed equations. We use the Banach contraction principle to obtain new suffcient conditions, which are weaker than those known, for the existence and uniqueness of nonoscillatory solutions. [ABSTRACT FROM AUTHOR]

In this paper, by using the classical Fibonacci sequence and the golden ratio, we first give the exact solution of the nonlinear recursive equation xn+1 = xn-1/xn with respect to certain powers of the initial values x-1 and x0: Then we obtain a necessary and sufficient condition on the initial values for which the equation has a non-oscillatory solution. Later we extend our all results to the recursive equations xn+1 = xp n-1=xp n (p > 0) in a similar manner. We also get a characterization for unbounded positive solutions. At the end of the paper we analyze all possible positive solutions and display some graphical illustrations verifying our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to derive sufficient conditions for the oscillation of all solutions of a difference equation with several non-monotone deviating arguments and nonnegative coefficients. Corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the significance of the results are also given. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of nonoscillatory solutions to a class of higher-order nonlinear neutral dynamic equations on time scales by employing Kranoselskii’s fixed point theorem. In addition, two examples are included to illustrate the significance of the results. [ABSTRACT FROM AUTHOR]

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.

We consider a three-dimensional nonlinear system of first order dynamic equations on time scales and show the existence and asymptotic behavior of nonoscillatory solutions by using the most well-known fixed point theorems. Examples are also provided, which validates our theoretical claims. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillatory behavior of first-order differential equations with several non-monotone delay arguments and non-negative coefficients. A sufficient condition, involving lim sup , which guarantees the oscillation of all solutions is established. Also, an example illustrating the significance of the result is given. [ABSTRACT FROM AUTHOR]

In this paper we consider the existence of nonoscillatory solutions for system of higher-order neutral differential equations with distributed coefficients and delays. We use the B a n a c h contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider the existence of nonoscillatory solutions for a system of higher-order neutral differential equations with distributed coefficients and delays. We use the $Banach$ contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The oscillatory behavior of the solutions to a differential equation with several non-monotone delay arguments and non-negative coefficients is studied. A new sufficient oscillation condition, involving lim sup , is obtained. An example illustrating the significance of the result is also given. [ABSTRACT FROM AUTHOR]

The system of neutral type difference equations with delays...is considered. The aim of this paper is to present sufficient conditions for the existence of nonoscillatory bounded positive solutions of the considered system with various (p(n)). [ABSTRACT FROM AUTHOR]

In this paper the nonlinear second-order neutral difference equation of the following form: $\Delta ( a_{n}\Delta(x_{n}-p_{n}x_{n-1}) ) + q_{n}f(x_{n-\tau})=0$ is considered. By suitable substitution the above equation is transformed into a new one, which is a third-order non-neutral difference equation. Using results obtained for the new equation, the asymptotic properties of the neutral difference equation are studied. Some classification of nonoscillatory solutions is presented, as well as an estimation of the solutions. Finally, we present necessary and sufficient conditions for the existence of solutions to both considered equations being asymptotically equivalent to the given sequences. [ABSTRACT FROM AUTHOR]

Consider the first-order retarded difference equation Δ x ( n ) + p ( n ) x τ ( n ) = 0 , n ∈ N 0 where ( p ( n ) ) n ⩾ 0 is a sequence of nonnegative real numbers, and ( τ ( n ) ) n ⩾ 0 is a sequence of integers such that τ ( n ) ⩽ n - 1 , n ⩾ 0 , and lim n → ∞ τ ( n ) = ∞ . Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving lim inf , is established. An example illustrates the case when the result of the paper implies oscillation while previously known results fail. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the oscillation of all solutions to the retarded difference equationand the (dual) advanced difference equationwhere $${(p_{i}(n)), 1\leq i\leq m}$$ are sequences of nonnegative real numbers, $${(\tau _{i}(n)), 1 \leq i\leq m}$$ are sequences of integers such that $${(\sigma _{i}(n)), 1\leq i \leq m}$$ are sequences of integers such that $${\Delta }$$ denotes the forward difference operator $${\Delta x(n) = x(n + 1) - x(n)}$$ and $${\nabla}$$ denotes the backward difference operator $${\nabla x(n) = x(n) - x(n - 1)}$$ . Examples illustrating the results are also given. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of neutral advanced difference equations of the form Δm[x(n) + cx(n + a)]+p(n)x(σ(n)) = 0, m ∈ ℕ, n ≥ 0, where c ∈ ℝ, ℕ ∋ a ≥ 2, (σ(n)) is a sequence of positive integers such that (σ(n)) ≥ n + 2 for all n ≥ 0, (p(n))n≥0 is a sequence of real numbers, Δ denotes the forward difference operator Δx(n) = x(n + 1) - x(n), and Δj denotes the jth forward difference operator Δj(x(n)) = Δ(Δj-1 (x(n))) for j = 2,3,...,m. Examples illustrating the results are also given. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain an oscillation criterion of solutions and the existence of nonoscillatory solutions to a class of fuzzy delay differential equation (FDDE) of second order. Some examples are given to illustrate the main theorems. [Copyright &y& Elsevier]

Abstract: For a class of -th order nonlinear neutral differential equations, sufficient conditions for all nonoscillatory solutions to satisfy 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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 on a time scale. To dwell upon the importance of our results, three interesting examples are also included. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation where is an integer, , , , (), are studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general and () which means that we allow oscillatory and (). In particular, our results improve essentially and extend some known results in the recent references. [Copyright &y& Elsevier]

Abstract: A complete characterization for nonoscillatory solutions of a coupled nonlinear difference system is presented. Nonoscillatory solutions are classified into several classes according to their limit behavior, and necessary and sufficient conditions for the existence in each of these classes are established. The multiplicity of solutions is also examined and explicit asymptotic formulas are derived. Possible generalizations and comparisons of our results with existing ones are discussed as well. [Copyright &y& Elsevier]