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WOS:000455201800003 The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems.

We shall present new oscillation criteria of second order nonlinear difference equations with a non-positive neutral term of the form \(\Delta(a(t)(\Delta(x(t)-p(t)x(t-k)))^{\gamma})+q(t)x^{\beta}(t+1-m)=0,\) with positive coefficients. Examples are given to illustrate the main results.

In this paper, the authors present some new results on the oscillatory and asymptotic behavior of solutions of the perturbed nonlinear third order functional differential equation $$ \left( b(t)\left( a(t)(x^{\prime }(t))^{\alpha }\right) ^{\prime }\right) ^{\prime }+p(t)f(x(\tau (t)))= h(t, x(t), x(\tau(t)), x'(t)). $$ In addition to other conditions, the authors assume that $uf(u) > 0$ for $u \neq 0$ and $f$ is increasing. Examples to illustrate the results are included.

We study the oscillatory behavior of solutions for integro-differential equations of the form $$x'(t) = e(t) -\int_0^t (t-s)^{\alpha-1}k(t, s)f(s, x(s))\, {\rm ds},\quad t\geq 0,$$ where $0

This paper deals with the asymptotic behavior of nonoscillatory solutions of fractional differential equations of the form CDaαy = e(t) + f(t,x), t ≥ a where 0 < α < 1, a ≥ 0, CDaαy denotes the Caputo fractional derivative of order α of y. The following particular cases are considered: y = (r(t)|x′|δ-1x′)′, (δ ≥ 1), y = x′, y = x. We offer a method that can be applied to investigate more general class of fractional differential equations as well.

The study of oscillation theory for fractional differential equations has been initiated by Grace et al. [5]. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form D a r C ⁢ x ⁢ ( t ) = e ⁢ ( t ) + f ⁢ ( t , x ⁢ ( t ) ) , t > 0 , a > 1 {{}^{C}D_{a}^{r}x(t)=e(t)+f(t,x(t)),t>0,a>1} , where r = α + n - 1 , α ∈ ( 0 , 1 ) {r=\alpha+n-1,\alpha\in(0,1)} , and n ≥ 1 {n\geq 1} is a natural number. We also present the conditions under which all solutions of this equation are asymptotic to t n - 1 {t^{n-1}} as t → ∞ {t\to\infty} .

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

New oscillation criteria for certain third order nonlinear dynamic equations with a nonlinear damping term are established. Examples to illustrate the results are included.

WOS:000529379200014 This paper is concerned with the oscillatory behavior of solutions to a class of even-order nonlinear differential equations with sublinear and superlinear neutral terms. The results are obtained by a comparison with first-order delay differential equations whose oscillatory characters are known as well as by an integral criteria. Examples are provided to illustrate the main results.

New sufficient conditions for oscillation of second-order neutral half-linear delay differential equations are given. Our results essentially improve, complement and simplify a number of related ones in the literature, especially those from a recent paper [R. P. Agarwal, Ch. Zhang, T. Li, Appl. Math. Comput. 274(2016), 178–181.]. An example illustrates the value of the results obtained.

The authors present some new oscillation criteria for the third-order neutral dynamic equation with distributed delays \begin{equation*} \left[ r(t)\left( \left[ x(t)+\int\limits_{a}^{b}p(t,\eta )x\left[ \tau (t,\eta )\right] \Delta \eta \right] ^{\Delta \Delta }\right) ^{\alpha }% \right] ^{\Delta }+\int\limits_{c}^{d}q(t,\xi )f\left( x\left[ \phi (t,\xi )% \right] \right) \Delta \xi =0 \end{equation*} on a time scale $\mathbb{T}$, where $\alpha $ is the quotient of odd positive integers. Using a Riccati type transformation and a comparison technique, they establish some new sufficient conditions to ensure that a solution $x$ of this equation either oscillates or satisfies $\lim_{t\rightarrow\infty}x(t)=0$.

WOS:000485899700001 New sufficient conditions for the oscillation of all solutions to a class of third-order Emden-Fowler differential equations with unbounded neutral coefficients are established. The criteria obtained essentially improve related results in the literature. In particular, as opposed to known results, new criteria can distinguish solutions of third-order differential equations with different behaviors. Examples are also provided to illustrate the results.

We study the oscillation problem for solutions of a class of $n$-th order nonlinear delay differential equations with nonpositive neutral terms. The obtained results improve and correlate many of the known oscillation criteria in the literature for neutral and non-neutral equations.

We present the conditions under which every nonoscillator solution x(t) of the forced fractional differential equation $$\begin{aligned} ^{\mathrm{C}}D_{\mathrm{c}}^{\alpha } y ( t ) = e ( t ) +f ( {t, x ( t )} ), c > 1,\alpha \in ( {0,1} ), \quad \mathrm{{and}} \,\, \delta \ge 1, \end{aligned}$$ where $$y(t)= ( {a(t) ( {{x}'(t)} )^{\delta }})^{\prime },c_0 =\frac{y(c)}{\Gamma (1)}= y(c)$$ , is a real constant which satisfies $$\begin{aligned} |x(t)|=O\left( {t^{1/\delta }e^{t}\int _{\mathrm{c}}^{t} {a^{-1/\delta }} (s)\mathrm{d}s} \right) , \quad t \rightarrow \infty \end{aligned}$$ It is shown that the technique can be applied to some related fractional differential equations. Examples are inserted to illustrate the relevance of the obtained results.

We present the conditions under which every positive solution x(t) of the integral equation $$x(t) = a(t) + \int\nolimits_c^t(t - s)^{\alpha-1}[e(s) + k(t, s)f(s, x(s))]\,{\rm d}s,\quad c > 1,\,\alpha > 0$$ Satisfies $$x(t) = O(a(t)) \quad {\rm as}\quad t \to \infty,\quad {\rm i.e.,}\,{\mathop {\rm lim \,sup}\limits_{t \to \infty}} \frac {x(t)}{a(t)} < \infty,$$ From the obtained results, we derive a technique which can be applied to some related integral equations that are equivalent to certain fractional differential equations of Caputo derivative of any order. We also establish new oscillation criteria for such fractional differential equations.

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.

Sufficient conditions are given to ensure that every nonoscillatory solution x(t) of the integro-differential equation x'(t) = a(t) ∫ct (t-s)α-1 [e(s) k(t,s) ƒ (s,x(s))]ds, t≥c, satisfies x(t) = O (tα ∫ct a(s)ds), t→∞, where c≥ 0 and α>0. Additional conditions are also provided to avoid the existence of such oscillatory solutions, resulting in sufficient conditions for oscillation of solutions. The technique can be applied to some related integro-differential equations that are equivalent to certain fractional differential equations with Caputo derivatives. CDcα 1 x(t) = e(t) f(t,x(t)), t≥C. Examples are given to illustrate the relevance of the obtained results.

This paper deals with the asymptotic behavior of positive solutions of certain forced fractional differential equations of the formDcαCyt=et+ft, xt,c>1,α∈0,1, whereyt=atx′t′,c0=y(c)/Γ(1) =yc, andc0is a real constant. From the obtained results, we derive a technique which can be applied to some related fractional differential equations.

We shall present new oscillation criteria of second-order nonlinear differential equations with a nonpositive neutral term of the form: $$\begin{aligned} \left( (a(t)\left( \left( x(t)-p(t)x(\sigma (t) )^{\prime } \right) ^{\gamma } \right) ^{\prime }+q(t)x^{\beta }(\tau (t))=0,\right. \end{aligned}$$ with positive coefficients. The obtained results answer an open problem raised in Li et al. [Adv Differ Equ 35:7, 2015, Remark 4.3 (P2)]. Examples are given to illustrate the main results.

The objective in this paper is to study the oscillatory and asymptotic behavior of the solutions of a linear third-order delay differential equation of the form $$ \bigl( r_{2}(t) \bigl( r_{1}(t)y'(t) \bigr) ' \bigr) '+q(t)y \bigl(\tau(t) \bigr)= 0. $$ We establish new oscillation criteria that can be used to test for oscillations, even when the previously known criteria fail to apply. Examples illustrating the results are also given.

Some new oscillation criteria for third order neutral nonlinear dynamic equations with distributed deviating arguments on time scales are established. The obtained results extend, improve and correlate many known oscillation results for third order dynamic equations

We establish some new criteria for the oscillation of mth order nonlinear difference equations. We study the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions. We also present a sufficient condition for every solution to be asymptotic at 1 to a factorial expression (t) (m 1) .

In this paper, we will consider the higher-order functional dynamic equations of the form on an above-unbounded time scale , where and , . The function is a rd-continuous function such that . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.

In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales.

In this paper, an optimal q-homotopy analysis method (Oq-HAM) is proposed. We present some examples to show the reliability and efficiency of the method. It is compared with the one-step optimal homotopy analysis method. The results reveal that the Oq-HAM has more accuracy to determine the convergence-control parameter than the one-step optimal HAM.

We establish some new criteria for the oscillation of fourth order nonlinear dynamic equations on a time scale. We investigate the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions. Some examples are given here to illustrate our main results.

WOS:000425348300018 The authors establish some new criteria for the oscillation of second order damped nonlinear neutral differential equations with distributed deviating arguments. Two examples are also provided to illustrate the results.

This article has been retracted: please see Elsevier Policy on Article Withdrawal (http://www.elsevier.com/locate/withdrawalpolicy).This article has been retracted at the request of the Editor-in-Chief and Authors.The article was found to contain errors which will be corrected in a revised version to be submitted to Applied Mathematics Letters.

We present some new criteria for the oscillation of even order dynamic equation $$\left(a(t)({x^\Delta}^{n-1}(t))^\alpha\right)^\Delta +q(t)(x^\sigma(t))^\lambda = 0$$ on an unbounded above time scale $${\mathbb{T}}$$ , where α and λ are the ratios of positive odd integers, a and q is a real valued positive rd-continuous functions defined on $${\mathbb{T}}$$ .

This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation $(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$ on a time scale $\mathbb{T}$ where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on $\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}$ and $\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty $. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.

In this paper we initiate the oscillation theory for fractional differential equations. Oscillation criteria are obtained for a class of nonlinear fractional differential equations of the form $$D_a^q x + f_1 (t,x) = v(t) + f_2 (t,x),\mathop {\lim }\limits_{t \to a} J_a^{1 - q} x(t) = b_1 $$ , where Daq denotes the Riemann-Liouville differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo’s differential operator.

Some new criteria for the oscillation of solutions of the second-order half-linear dynamic equation ( a ( xΔ )α)Δ (t)+q(t)xα (t) = 0 are established when ∫ ∞ a−1/α (s)Δs = ∞ . Mathematics subject classification (2010): 34C10, 39A10, 34N05, 34C15, 34K11, 39A21.

We present some new necessary and sufficient conditions for the oscillation of second order nonlinear dynamic equation $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)=0$$ on an arbitrary time scale \(\mathbb{T}\) , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on \(\mathbb{T}\) . Comparison results with the inequality $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)\leqslant 0\quad (\geqslant 0)$$ are established and application to neutral equations of the form $$\bigl(a(t)\bigl(\bigl[x(t)+p(t)x[\tau (t)]\bigr]^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }+q(t)x^{\beta }\bigl[g(t)\bigr]=0$$ are investigated.

We establish some new criteria for the oscillation of second-order nonlinear dynamic equations on a time scale. We study the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions.

We obtain some oscillation criteria for solutions to the second-order half-linear dynamic equation when or . These criteria unify and extend known criteria for corresponding half-linear differential and difference equations. Some of our results are new even in the continuous and the discrete cases.

Some new criteria for the oscillation of nonlinear dynamic equations of the form $$\bigl(a(t)(x^{\Delta}(t))^{\alpha}\bigr)^{\Delta}+f(t,x^{\sigma}(t))=0$$ on a time scale \(\mathbb{T}\) are established.

Some new criteria for the oscillation of third order nonlinear difference equations $$\begin{array}{l}\Delta^{2}\bigl(\frac{1}{a(k)}(\Delta x(k))^{\alpha}\bigr)+q(k)f(x[g(k)])=0\quad\mbox{and}\\[6pt]\Delta^{2}\bigl(\frac{1}{a(k)}(\Delta x(k))^{\alpha}\bigr)=q(k)f(x[g(k)])+p(k)h(x[\sigma(k)])\end{array}$$ are established.

The authors investigate the oscillatory behavior of all solutions of the fourth order functional differential equations $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})+q(t)f(x[g(t)])=0$ and $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})=q(t)f(x[g(t)])+p(t)h(x[\sigma(t)])$ in the case where ∫ ∞ a −1/α (s)ds

Some new oscillation theorems for higher-order nonlinear functional differential equations of the form >0, are established.

WOS:000383088500001 This paper discusses oscillatory and asymptotic properties of solutions of a class of second-order nonlinear damped neutral differential equations. Some new sufficient conditions for any solution of the equation to be oscillatory or to converge to zero are given. The results obtained extend and improve some of the related results reported in the literature. The results are illustrated with examples.