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The study of the oscillatory behavior of a general class of neutral Emden-Fowler differential equations is the focus of this work. The main motivations for studying the oscillatory behavior of neutral equations are their many applications as well as the richness of these equations with exciting analytical issues. We obtained novel oscillation conditions in Kamenev-type criteria for the considered equation in the canonical case. We improve the monotonic and asymptotic characteristics of the non-oscillatory solutions to the considered equation and then utilize these characteristics to refine the oscillation conditions. We present, through examples and discussions, what demonstrates the novelty and efficiency of the results compared to previous relevant findings in the literature. In addition, we numerically represent the solutions of some special cases to support the theoretical results.

The study of the oscillatory behavior of a general class of neutral Emden-Fowler differential equations is the focus of this work. The main motivations for studying the oscillatory behavior of neutral equations are their many applications as well as the richness of these equations with exciting analytical issues. We obtained novel oscillation conditions in Kamenev-type criteria for the considered equation in the canonical case. We improve the monotonic and asymptotic characteristics of the non-oscillatory solutions to the considered equation and then utilize these characteristics to refine the oscillation conditions. We present, through examples and discussions, what demonstrates the novelty and efficiency of the results compared to previous relevant findings in the literature. In addition, we numerically represent the solutions of some special cases to support the theoretical results. [ABSTRACT FROM AUTHOR]

We study linear differential equations whose coefficients consist of products of powers of natural logarithm and general continuous functions. We derive conditions that guarantee the non-oscillation of all non-trivial solutions of the treated type of equations. The conditions are formulated as a non-oscillation criterion, which is the counterpart of a previously obtained oscillation theorem. Therefore, from the presented main result, it follows that the analysed equations are conditionally oscillatory. The used method is based on averaging techniques for the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This article is finished by new corollaries and examples.

This article delves into the behavior of solutions to a general class of functional differential equations that contain a neutral delay argument. This category encompasses the half-linear case and the multiple-delay case of neutral equations. The motivation to study this type of equation lies not only in the exciting analytical issues it presents but also in its numerous vital applications in physics and biology. We improved some of the inequalities that play a crucial role in developing the oscillation test. Then, we used an improved technique to derive several criteria that ensure the oscillation of the solutions of the studied equation. Additionally, we established a criterion that did not require imposing monotonic constraints on the delay functions and took into account their effect. We have supported the novelty and effectiveness of the results by analyzing and comparing them with previous results in the literature.

This article delves into the behavior of solutions to a general class of functional differential equations that contain a neutral delay argument. This category encompasses the half-linear case and the multiple-delay case of neutral equations. The motivation to study this type of equation lies not only in the exciting analytical issues it presents but also in its numerous vital applications in physics and biology. We improved some of the inequalities that play a crucial role in developing the oscillation test. Then, we used an improved technique to derive several criteria that ensure the oscillation of the solutions of the studied equation. Additionally, we established a criterion that did not require imposing monotonic constraints on the delay functions and took into account their effect. We have supported the novelty and effectiveness of the results by analyzing and comparing them with previous results in the literature. [ABSTRACT FROM AUTHOR]

We consider semilinear elliptic equations of the form Δu+f(|x|,u)=0 on RN with f(|x|,u)=q(|x|)g(u). These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity f, there exists an oscillating radial solution converging to a zero of the function g. We also study the oscillating and limiting behavior of this solution. [ABSTRACT FROM AUTHOR]

We consider semilinear elliptic equations of the form $ \Delta u + f(|x|, u) = 0 $ on $ {\mathbb{R}}^{N} $ with $ f(|x|, u) = q(|x|)g(u) $. These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity $ f $, there exists an oscillating radial solution converging to a zero of the function $ g $. We also study the oscillating and limiting behavior of this solution.

In this article, we present new criteria for testing the oscillation of solutions of higher-order neutral delay differential equation. By deriving new monotonic properties of a class of the positive solutions of the studied equation, we establish better criteria for oscillation. Furthermore, we improve these properties by giving them an iterative character, allowing us to apply the criteria more than once. The results obtained in this paper are characterized by the fact that they do not require the existence of unknown functions and do not need the commutation condition to composition of the delay functions, which are necessary conditions for the previous related results. [ABSTRACT FROM AUTHOR]

We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphism on a compact metric space. Given an ergodic probability measure, we study the topological structure of the associated almost sure spectrum. We establish a gap labelling theorem in the spirit of Johnson and Schwartzman. That is, we show that the constant value the integrated density of states takes in a gap of the spectrum must belong to the countable Schwartzman group of the base dynamics. This result is a natural companion to a recent result of Alkorn and Zhang, which established a Johnson-type theorem for the families of Jacobi matrices in question. [ABSTRACT FROM AUTHOR]

This note considers Sturm oscillation theory for regular matrix Sturm-Liouville operators on finite intervals and for matrix Jacobi operators. The number of space oscillations of the eigenvalues of the matrix Prufer phases at a given energy, defined by a suitable lift in the Jacobi case, is shown to be equal to the number of eigenvalues below that energy. This results from a positivity property of the Prufer phases, namely they cannot cross -1 in the negative direction, and is also shown to be closely linked to the positivity of the matrix Prufer phase in the energy variable. The theory is illustrated by numerical calculations for an explicit example.

We use many classical results known for the self-adjoint second-order linear equation and extend them for a three-term even order linear equation with a delay applied to coefficients. We derive several conditions concerning the oscillation and the existence of positive solutions. Our equation for a choice of parameter is diconjugate, and for a different choice can have positive and oscillatory solutions at the same time. However, it is still, in a sense, disconjugate if we use a weaker definition of oscillation.

In this paper, we study the stability and oscillation of fractional differential equations cDαx(t) + ax(t) + ∫0¹ x(s + [t - 1])dR(s) = 0. We discretize the fractional differential equation by variation of constant formula and semigroup property of Mittag-Leffler function, and get the difference equation corresponding to the integer points. From the equivalence analogy of qualitative properties between the difference equations and the original fractional differential equations, the necessary and sufficient conditions of oscillation, stability and exponential stability of the equations are obtained. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are established for the oscillation of all solutions of generalized proportional fractional differential equations of the form \begin{equation*} \left\{ \begin{array}{l} {_{a}D}^{\alpha, \rho}x(t) + \xi_1(t,x(t)) = \mu(t) + \xi_2(t,x(t)),\quad t>a \ge 0,\\[0.3cm] \lim_{t\to a^{+}} {_{a}I}^{j-\alpha, \rho}x(t) = b_j,\quad j=1,2,\ldots,n, \end{array} \right. \end{equation*}where $n = \lceil \alpha \rceil$, ${_{a}D}^{\alpha, \rho}$ is the generalized proportional fractional derivative operator of order $\alpha\in \mathbb{C}$, $Re(\alpha)\ge 0$, $0

Abstract This paper is devoted to the analysis of the oscillatory behavior of Euler type linear and half-linear differential equations. We focus on the so-called conditional oscillation, where there exists a borderline between oscillatory and non-oscillatory equations. The most complicated problem involved in the theory of conditionally oscillatory equations is to decide whether the equations from the given class are oscillatory or non-oscillatory in the threshold case. In this paper, we answer this question via a combination of the Riccati and Prüfer technique. Note that the obtained non-oscillation of the studied equations is important in solving boundary value problems on non-compact intervals and that the obtained results are new even in the linear case.

Abstract Based on the properties of nonlocal fractional calculus generated by conformable derivatives, we establish some sufficient conditions for oscillation of all solutions for fractional differential equations with damping term. Forced oscillation of conformable differential equations in the frame of Riemann, as well as of Caputo type, is established. Examples are provided to demonstrate the effectiveness of the main results.

Abstract Hadamard fractional derivatives are nonlocal fractional derivatives with singular logarithmic kernel with memory, and hence they are suitable to describe complex systems. In this paper, sufficient conditions are established for the oscillation of solutions fractional differential equations in the frame of left Hadamard fractional derivatives of order α∈C,Re(α)≥0 $\alpha\in\mathbb {C}, \operatorname{Re}(\alpha)\geq0$. The results are also obtained for fractional Hadamard derivatives in the Caputo setting. Examples are provided to illustrate the applicability of the main results.

Abstract In this study, we discuss some theorems related to the oscillatory behavior of nonlinear fractional difference equations equipped with well-known fractional Riemann–Liouville difference operator. Then we give an example for the illustration of the results obtained.

The paper belongs to the qualitative theory of half-linear equations which are located between linear and non-linear equations and, at the same time, between ordinary and partial differential equations. We analyse the oscillation and non-oscillation of second-order half-linear differential equations whose coefficients are given by the products of functions having mean values and power functions. We prove that the studied very general equations are conditionally oscillatory. In addition, we find the critical oscillation constant.

Abstract Recently, Jarad et al. in (Adv. Differ. Equ. 2017:247, 2017) defined a new class of nonlocal generalized fractional derivatives, called conformable fractional derivatives (CFDs), based on conformable derivatives. In this paper, sufficient conditions are established for the oscillation of solutions of generalized fractional differential equations of the form {Dα,ρax(t)+f1(t,x)=r(t)+f2(t,x),t>a,limt→a+aIj−α,ρx(t)=bj(j=1,2,…,m), $$ \textstyle\begin{cases} {}_{a}\mathfrak{D}^{\alpha,\rho}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x),\quad t>a, \\ \lim_{t \to a^{+}}{ {}_{a}\mathfrak{I}^{j-\alpha,\rho}x(t)}=b_{j} \quad (j=1,2,\ldots,m), \end{cases} $$ where m=⌈α⌉ $m=\lceil\alpha\rceil$, Dα,ρa ${}_{a}\mathfrak{D}^{\alpha,\rho}$ is the left-fractional conformable derivative of order α∈C $\alpha\in\mathbb{C}$, Re(α)≥0 $\operatorname{Re}(\alpha)\geq0$ in the Riemann–Liouville setting and Iα,ρa ${}_{a}\mathfrak {I}^{\alpha,\rho}$ is the left-fractional conformable integral operator. The results are also obtained for CFDs in the Caputo setting. Examples are provided to demonstrate the effectiveness of the main result.

Supposing that A (z) is an exponential polynomial of the form A (z) = H 0 (z) + H 1 (z) e ζ 1 z n + ⋯ + H m (z) e ζ m z n , where H j 's are entire and of order < n , it is demonstrated that the function H 0 (z) and the geometric location of the leading coefficients ζ 1 , ... , ζ m play a key role in the oscillation of solutions of the differential equation f ″ + A (z) f = 0. The key tools consist of value distribution properties of exponential polynomials, and elementary properties of the Phragmén-Lindelöf indicator function. In addition to results in the whole complex plane, results on sectorial oscillation are proved. [ABSTRACT FROM AUTHOR]

This note considers Sturm oscillation theory for regular matrix Sturm-Liouville operators on finite intervals and for matrix Jacobi operators. The number of space oscillations of the eigenvalues of the matrix Prüfer phases at a given energy, defined by a suitable lift in the Jacobi case, is shown to be equal to the number of eigenvalues below that energy. This results from a positivity property of the Prüfer phases, namely they cannot cross –1 in the negative direction, and is also shown to be closely linked to the positivity of the matrix Prüfer phase in the energy variable. The theory is illustrated by numerical calculations for an explicit example. [ABSTRACT FROM AUTHOR]

Abstract In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form { ∇ 0 α q x ( t ) + f 1 ( t , x ) = r ( t ) + f 2 ( t , x ) , t > 0 , lim t → 0 + q I 0 j − α x ( t ) = b j ( j = 1 , 2 , … , m ) , $$ \left \{ \textstyle\begin{array}{l} {}_{q}\nabla_{0}^{\alpha}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x), \quad t>0 ,\\ \lim_{t \to0^{+}}{{}_{q}I_{0}^{j-\alpha}x(t)}=b_{j} \quad(j=1,2,\ldots,m), \end{array}\displaystyle \right . $$ where m = ⌈ α ⌉ $m=\lceil\alpha\rceil$ , ∇ 0 α q ${}_{q}\nabla_{0}^{\alpha}$ is the Riemann-Liouville q-differential operator and I 0 m − α q ${}_{q}I_{0}^{m-\alpha}$ is the q-fractional integral. The results are also obtained when the Riemann-Liouville q-differential operator is replaced by Caputo q-fractional difference. Examples are provided to demonstrate the effectiveness of the main result.

In this article, studied the properties of the oscillation of fractional difference equations, and we obtain some results. The results we obtained are an expansion and further development of highly known results. Then we showed them with examples. [ABSTRACT FROM AUTHOR]

In this study, we consider a class of fractional neutral differential equations. We are going to give some new theorems that they complete and improve a number of results in the literature. Then we give an example to illustrating the main results. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the oscillation problem for the nonlinear differential equation (| x ′ | p (t) − 2 x ′) ′ + λ t p (t) | x | p (t) − 2 x = 0 , where λ is a positive parameter and p (t) > 1 is a nondecreasing and smooth function. Some (non)oscillation criteria are given for this equation. The obtained results are the best possible in a certain sense. We use the Riccati technique and the function sequence technique. Moreover, we propose a conjecture concerning the oscillation problem for this equation in the case when p (t) tends to infinity as t → ∞. [ABSTRACT FROM AUTHOR]

By the combination of the modified half-linear Prüfer method and the Riccati technique, we study oscillatory properties of half-linear differential equations. Taking into account the transformation theory of half-linear equations and using some known results, we show that the analysed equations in the Riemann–Weber form with perturbations in both terms are conditionally oscillatory. Within the process, we identify the critical oscillation values of their coefficients and, consequently, we decide when the considered equations are oscillatory and when they are non-oscillatory. As a direct corollary of our main result, we solve the so-called critical case for a certain type of half-linear non-perturbed equations.

Recently, it was shown that the Euler type half-linear differential equations $$ [ r (t) t^{p-1}\Phi(x')]' + \frac{s (t)}{ t \log^p t} \Phi(x) = 0 $$ with periodic coefficients r and s are conditionally oscillatory and the critical oscillation constant was found. Nevertheless, the critical case remains unsolved. The objective of this article is to study the critical case. Thus, we consider the critical value of the coefficients and we prove that any considered equation is non-oscillatory. Moreover, we analyze the situation when the periods of coefficients r and s do not need to coincide.

Recently there has been an increasing interest in studying $p(t)$-Laplacian equations, an example of which is given in the following form (|u'(t)|^{p(t)-2}u'(t))'+c(t)|u(t)|^{q(t)-2}u(t)= 0, \quad t>0. In particular, the first study of sufficient conditions for oscillatory solution of $p(t)$-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations with $p(t)$-Laplacian via Riccati method. The results obtained are new and rare, except for a work of Zhang (2007). We present more detailed results than Zhang (2007).

Applying the modified half-linear Prufer angle, we study oscillation properties of the half-linear differential equation $$ [ r(t) t^{p-1} \Phi(x')]' + \frac{s(t)}{t \log^pt} \Phi(x) = 0, \quad \Phi(x)=|x|^{p-1}\hbox{sgn} x. $$ We show that this equation is conditionally oscillatory in a very general case. Moreover, we identify the critical oscillation constant (the borderline depending on the functions r and s which separates the oscillatory and non-oscillatory equations). Note that the used method is different from the standard method based on the half-linear Prufer angle.

We investigate second-order half-linear Euler-type dynamic equations on time scales with positive periodic coefficients. We show that these equations are conditionally oscillatory, i.e., there exists a sharp borderline (a constant given by the coefficients of the given equation) between oscillation and non-oscillation of these equations. In addition, we explicitly find this so-called critical constant. In the cases that the time scale is $\mathbb{R}$ or $\mathbb{Z}$, our result corresponds to the classical results as well as in the case that the coefficients are replaced by constants and we take into account the linear equations. An example and corollaries are provided as well.

We consider half-linear Euler type differential equations with general periodic coefficients. It is well-known that these equations are conditionally oscillatory, i.e., there exists a border value given by their coefficients which separates oscillatory equations from non-oscillatory ones. In this paper, we study oscillatory properties in the border case. More precisely, we prove that the considered equations are non-oscillatory in this case. Our results cover the situation when the periodic coefficients do not have any common period.

The aim of this paper is to mention a generalization of the adapted Riccati equation and, using this method, to prove a non-oscillatory result concerning half-linear differential equations with coefficients having mean values. Note that this result is new even for linear equations. [ABSTRACT FROM AUTHOR]

The oscillation and nonoscillation theories for nonlinear systems have recently received a lot of attention. We consider a two-dimensional time-scale system and find the oscillation criteria for solutions of the system by using some improper integrals and inequalities. We also give a few examples in order to highlight our main results. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form where $m=\lceil\alpha\rceil$ , ${}_{q}\nabla_{0}^{\alpha}$ is the Riemann-Liouville q-differential operator and ${}_{q}I_{0}^{m-\alpha}$ is the q-fractional integral. The results are also obtained when the Riemann-Liouville q-differential operator is replaced by Caputo q-fractional difference. Examples are provided to demonstrate the effectiveness of the main result. [ABSTRACT FROM AUTHOR]

We find very general classes of linear and half-linear difference equations which are conditionally oscillatory. We identify the critical oscillation constant whose value implies the oscillation or non-oscillation of studied equations. Our results are divided into oscillatory and non-oscillatory theorems which determine the critical oscillation constant for coefficients given by sequences having mean values. In addition, our approach enables to analyse also the oscillatory properties of equations whose coefficients are not given by sequences with mean values. We point out that the obtained results are new even for linear equations with periodic coefficients. Such consequences are formulated at the end of this paper. [ABSTRACT FROM AUTHOR]

Normality arguments are applied to study the oscillation of solutions of f'' +Af = 0, where the coefficient A is analytic in the unit disc D and supz∈D(1−|z|²)²|A(z)| < ∞. It is shown that such a differential equation may admit a non-normal solution having prescribed uniformly separated zeros. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new modification of the half-linear Prüfer angle. Applying this modification, we investigate the conditional oscillation of the half-linear second order differential equation ( ∗ ) t α − 1 r ( t ) Φ ( x ′ ) ′ + t α − 1 − p s ( t ) Φ ( x ) = 0 , Φ ( x ) = | x | p − 1 sgn x , where p > 1 , α ≠ p , and r , s are continuous functions such that r ( t ) > 0 for large t . We present conditions on the functions r , s which guarantee that Eq. ( ∗ ) behaves like the Euler type equation [ t α − 1 Φ ( x ′ ) ] ′ + λ t α − 1 − p Φ ( x ) = 0 , which is conditionally oscillatory with the oscillation constant λ 0 = | p − α | p ∕ p p . [ABSTRACT FROM AUTHOR]

The propellant form of hydrogen peroxide is a non-toxic alternative to hydrazine. However, catalysts used to induce hydrogen peroxide decomposition are often plagued by low melting point, high pressure drop and poor activity. 3D-printing may offer solutions to many of these issues, through the realisation of complex geometries such as triply-periodic minimal surfaces (TPMS). In this work, methods to print and coat such ceramic catalyst supports were developed. A flow reactor was used to quickly screen the catalysts and assess the influence of the chosen active phase, precursor, and support sintering temperature on their intrinsic activity. Platinum prepared from H 2 PtCl 6 emerged as the most active catalyst, with a support sintering temperature of 1200 °C yielding the highest BET surface area. The latter was strongly correlated to catalytic activity. On a per weight basis, the printed Pt catalyst outperformed conventional silver screens and also matched the performance of Pt on a high surface area commercial support, due to its egg-shell distribution. These findings have laid the groundwork for future testing in a thruster configuration, where the proposed advantages of the 3D-printed structures will be evaluated. [Display omitted] • Porous ceramic catalyst supports can be generated via FDM and partial sintering. • Wet impregnation of the support results in an egg-shell type catalyst distribution. • Platinum highly effective at decomposing hydrogen peroxide. • Activity of the 3D-printed catalysts exceeds that of traditional silver screens. [ABSTRACT FROM AUTHOR]

In this paper, we consider the third-order neutral functional differential equation with distributed deviating arguments. We give sufficient conditions for the oscillatory behavior of this functional differential equation.