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This paper deals with some new criteria for the oscillation of third order half-linear differential equations. The purpose of the present paper is the linearization of equation 1.1 in the sense that we would deduce oscillation of studied equation from that of the linear form and to provide new oscillation criteria via comparison with first order equations whose oscillatory behavior are known. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature for equation 1.1. The results are illustrated by some examples.

We study the half-linear advanced differential equation $$(r(t)| y'(t)|^{\alpha-1}y'(t))'+q(t)|y|^{\alpha-1}(\sigma(t))y(\sigma(t))= 0, \quad t \geq t_0 >0,$$ where $$\alpha>0, r(t)>0, q(t) >0, \sigma(t)\geq t$$ , and $$R(t):= \int_{t_0}^{t}r^{-1/\alpha}(s) \, {\rm d}s \to \infty$$ as $$t\to \infty$$ . We prove that such an equation is oscillatory if $$ \lambda_*:= \liminf_{t\to \infty}\frac{R(\sigma(t))}{R(t)} \max\{\alpha m^\alpha(1-m)\lambda_*^{-\alpha m}: 0 0.$$ The obtained criteria can be regarded as a natural extension of the well-known Kneser oscillation criterion for half-linear ordinary differential equations. Our oscillation constant is optimal for the correponding half-linear Euler-type delay differential equation.

In the paper, we study some new criteria for the oscillation of the nonlinear second-order delay difference equations of the form $$\varDelta (r\left( t \right) ({\varDelta x\left( t \right) )}^{\alpha })+q\left( t \right) x^{\beta }\left( t-m+1 \right) =0$$ Δ ( r t ( Δ x t ) α ) + q t x β t - m + 1 = 0 , via comparison with a second-order linear difference equation or a first-order linear delay difference equation whose oscillatory behavior is discussed intensively in the literature. The presented results essentially improve existing ones.

A generalized Nicholson blowflies model with harvesting or immigration and random effect is considered. We discuss the existence of positive global solution and provide the estimate on the lower bound of the Lyapunov exponent. Moreover, we show that the nontrivial equilibrium solution is mean square exponential stability and stable in probability. We prove several results using techniques of stochastic calculus. It is evident from the obtained conditions that the noise plays an important role in all qualitative properties of the solution. Numerical simulations are also provided in order to validate the analytical findings. The results are new and compliments the existing ones.

In this paper, we consider three types of fractional difference equations with forcing term. The asymptotic behaviors of positive solutions for the proposed equations are extensively investigated. Our approach mainly relies on the features of discrete fractional calculus and some mathematical inequalities. To conclude our theoretical findings, we give an example demonstrating complete consistency to the main results.

We shall present new oscillation criteria for non-canonical higher order nonlinear neutral dynamic equations of the form $$\begin{aligned} \left( (a(t)\left( y^{\varDelta ^{n-1}}(t)\right) ^{\alpha }\right) ^{\varDelta }+q(t)x^{\beta }(\tau (t))=0, \end{aligned}$$ where $$y(t)=x(t)+p(t)x^{\gamma }(\delta (t)).$$ We also present some oscillation results for certain second order dynamic equations with sublinear as well as super linear neutral terms.

We shall present new oscillation and asymptotic behavior criteria of higher order nonlinear difference equations of the form $$\begin{aligned} \Delta \left( ({a(t)}\left( {\Delta ^{n-1}{x(t)}} \right) ^{\alpha } \right) +{q(t)x}^{\beta }({t}-{m}+1)=0, \end{aligned}$$ with nonnegative real coefficients.

We derive sufficient conditions for the oscillation of solutions second-order delay differential equation containing a sublinear neutral term. Our conditions differ from the earlier ones even in the special cases, linear or nonlinear, and as illustrated with an example, we not only extend but also improve several results in the literature.

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 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.

In this paper, we establish new oscillation criteria for nonlinear second order difference equations with mixed neutral terms. The key idea of our approach is to compare with first order equations whose oscillatory behaviors are already known. The obtained results not only improve and extend existing results reported in the literature but also provide a new platform for the investigation of a wide class of nonlinear second order difference equations. The results are supported by examples to demonstrate the validity of the theoretical findings.

We shall present some new oscillation criteria for third-order nonlinear difference equations with a nonlinear-nonpositive neutral term of the form: $$\begin{aligned} \Delta \left( \left( a(t)\left( \Delta ^{2}\left( x(t)-p(t)x^{\alpha }(t-k) \right) \right) ^{\gamma } \right) +q(t)x^{\beta }(t-m+1)=0,\right. \end{aligned}$$ with positive coefficients via comparison with first-order equations whose oscillatory behavior are known, or via comparison with second-order inequalities with solutions having certain properties. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature even for the case of Eq. (1.1) with $$\hbox {p (t)} = 0$$ . Examples are given to illustrate the main results.

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.

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.

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.

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 $${\mathbb{T}}$$ .

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.

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.

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 criteria for the oscillation of the fourth order functional differential equation (1/a(t)(x″(t))α)″ = q(t)f(x[g(t)]) + p(t)h(x[σ(t)]) are established.

A new nonoscillatory theory is presented for higher order singular nonlinear equations. In particular our nonlinearity may be singular in both the dependent and independent variables.

This paper is concerned with the oscillatory and asymptotic behavior of the solutions of the forced functional differential equations with advanced arguments of the form¶¶\( \frac{d}{dt}\frac{1}{a_{n-1}(t)}\frac{d}{dt}\frac{1}{a_{n-2}(t)}\cdots \frac{d}{dt}\frac{1}{a_1(t)}\frac{d}{dt} x(t)\,+\,\delta f(t,x[g_1(t)],\ldots,x[g_m(t)]) = Q(t) \),(E,\delta)¶where \( \delta =\pm 1 \). Conditions which ensure that every solution of the equation (E,1) is oscillatory or tending to zero as \( t \to \infty \) are given. A classification of all solutions of the equation (E,—1) with respect to their behavior as \( t \to \infty \) and to their oscillatory character is also obtained. Finally, the sufficient conditions so that all solutions of the equation (\( E,\delta \)) are oscillatory or tending to zero as \( t \to \infty \) are presented. The obtained results extend and unify recent ones by Werbowski.

In this paper we discuss the oscillatory behavior of even order retarded differential equations of the form $$\frac{d}{{dt}}\frac{1}{{a_{n - 1} (t)}}\frac{d}{{dt}} \cdots \frac{d}{{dt}}\frac{1}{{a_1 (t)}}\frac{d}{{dt}}x(t) + f\left( {t,x(t - g),\frac{d}{{dt}}x(t - h)} \right) = 0,$$ by comparison to certain differential equations of the same or lower order whose oscillatory behavior is known.

Some new criteria for the oscillation of n th order nonlinear dynamic equations of the form

In this paper we discuss, using the averaging technique and weighted integral functions, the oscillatory behavior of the nonlinear second order differential equation $$(a(t)\dot x(t))^ \bullet + p(t)\dot x(t) + q(t)f(x(t)) = 0,$$ where a, p, q: [t0, ∞) → R, f:R → R are continuous, a(t) > 0 for t ⩾ t0 and xf(x) > 0 for x ≠ 0. The results of this paper extend and improve some of the criteria due to Butler, Coles, Willett and Wong.

Convergence of oscillatory solutions of $$(a(t)\psi (x(t))x \cdot (t)) \cdot + p(t)x \cdot (t) + q(t)f(x(t)) = 0, \left( { \cdot = \frac{d}{{dt}}} \right)$$ is discussed. The results obtained extend some of the results of Graef, Kitamura, Kusano, Onose, Spikes and Singh.