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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.

In this paper, we study a general class of half-linear difference equations. Applying a version of the discrete Riccati transformation, we prove a non-oscillation criterion for the analyzed equations. In the formulation of the criterion, we do not use auxiliary sequences, but we consider directly the coefficients of the treated equations. Since the obtained criterion is new in many cases, we also formulate new simple corollaries and mention illustrative examples. [ABSTRACT FROM AUTHOR]

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.

Methane draft pressure is a secondary disaster for mine ventilation following coal and gas outbursts, which poses a long-term threat to coal mining and workers' safety. To study the law of methane draft pressure-caused airflow variation, oscillation theory is introduced in this study. The vibration equation is used to deduct the basic equation of airflow oscillation in parallel upward ventilated airways, which regards the entire methane and air as one vibrating object in the loop of the parallel airways. The theoretical critical air velocity to prevent airflow reversion is calculated by the airflow oscillation equation using numerical methods. Furthermore, through dimensional analysis, a formula is given to calculate the critical velocity directly. The theoretical results were compared with experiments and good agreements were obtained, which verifies the rationality of the assumptions in the theoretical deduction process. The conclusion of this study indicates that, in a short time, oscillation theory can be used to predict the critical air velocity to prevent airflow reversion, and be feasible in revealing the essence of methane draft pressure-caused airflow reversion. An airflow oscillation model is built to analyze and predict methane draft pressure-caused airflow reversion based on oscillation theory. A method is carried out to calculate the critical velocity that prevents airflow reversion based on the airflow oscillation model. The conclusions of this paper provide a theoretical basis for further understanding the essence of the airflow reversion caused by methane draft pressure. [ABSTRACT FROM AUTHOR]

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.

It is known that, in the unit disc as well as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc ,A_{k-2}$ of $$ \begin{align*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geqslant 2, \end{align*} $$ determines, under certain growth restrictions, not only the growth but also the oscillation of the equation's nontrivial solutions, and vice versa. A uniform treatment of this principle is given in the disc $D(0,R)$ , $0 , by using several measures for growth that are more flexible than those in the existing literature, and therefore permit more detailed analysis. In particular, the results obtained are not restricted to cases where the solutions are of finite (iterated) order of growth in the classical sense. The new findings are based on an accurate integrated estimate for logarithmic derivatives of meromorphic functions, which preserves generality in terms of three free parameters. [ABSTRACT FROM AUTHOR]

The construction of fundamental solutions in R3 for the equations of harmonic vibrations in the theory of elasticity of anisotropic elastic media is carried out. Solutions are constructed in the form of multipole series. Theorems on the convergence of series in the topology of compact convergence in are proved. The problems on constructing some singular solutions of the theory of vibrations of an anisotropic body are discussed. The fundamental solution of the oscillation equations for an isotropic medium is obtained in a closed form. [ABSTRACT FROM AUTHOR]

In this paper, we analyze oscillatory properties of perturbed half‐linear differential equations (i.e., equations with one‐dimensional p‐Laplacian). The presented research covers the Euler and Riemann–Weber type equations with very general coefficients. We prove an oscillatory result and a nonoscillatory one, which show that the studied equations are conditionally oscillatory (i.e., there exists a certain threshold value that separates oscillatory and nonoscillatory equations). The obtained criteria are easy to use. Since the number of perturbations is arbitrary, we solve the oscillation behavior of the equations in the critical setting when the coefficients give exactly the threshold value. The results are new for linear equations as well. [ABSTRACT FROM AUTHOR]

Kekulé postulated that neighbouring carbon atoms in benzene undergo incessant collision with each other, thereby leading to the interchange of double and single bonds, which amounts to an oscillation between two cyclohexatriene structures in dynamic equilibrium. It has been claimed that Kekulé arrived at a fully symmetric D6h structure of benzene and that the oscillation hypothesis should not be attributed to him. However, Clausius' collision theory, which was known at the time, implies that, when the absolute temperature approaches zero, the collision frequency will tend toward zero too, i.e. collisions will stop, and a static, D3h cyclohexatriene obtains. The classical collision theory did not allow Kekulé to construct the desired D6h structure as the energy minimum. The theory of harmonic oscillators would have allowed it, but that was not attempted at Kekulé's time. [ABSTRACT FROM AUTHOR]

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]

Given a three-coefficient Sturm–Liouville differential expression τ0=r−10[−(d/dx)p0(d/dx)+q0] and its perturbation τq1=τ0+r0−1 q on an interval (a,b)⊂R, we employ the existence of a strictly positive solution u0(λ0,⋅)>0 on (a,b) of τ0u0=λ0u0 to derive a quadratic form inequality for τq1 that naturally generalizes the well-known Hardy inequality and reduces to it in the particular case p0=r0=u0(0,⋅)=1, q0=λ0=0, a∈R,b=∞. [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]

For tridiagonal block Jacobi operators, the standard transfer operator techniques only work if the off-diagonal entries are invertible. Under suitable assumptions on the range and kernel of these off-diagonal operators which assure a homogeneous minimal coupling between the blocks, it is shown how to construct reduced transfer operators that have the usual Krein space unitarity property and also a crucial monotonicity in the energy variable. This allows to extend the results of oscillation theory to such systems. [ABSTRACT FROM AUTHOR]

We study the oscillation of Euler type half-linear difference equations with asymptotically periodic and bounded coefficients. Applying the adapted Riccati technique, we prove a non-oscillation criterion for the treated equations. Since the obtained criterion is new in several cases, we formulate many consequences explicitly. In particular, the criterion is new for intensively studied linear equations (e.g., with periodic coefficients). [ABSTRACT FROM AUTHOR]

Oscillation of solutions of f (k) + a k - 2 f (k - 2) + ⋯ + a 1 f ′ + a 0 f = 0 is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors, and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained. [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 work, we examine the oscillation of a class fractional differential equations in the frame of generalized nonlocal fractional derivatives with function dependent kernel type. We present sufficient conditions to prove the oscillation criteria in both of the Riemann-Liouville (RL) and Caputo types. Taking particular cases of the nondecreasing function appearing in the kernel of the treated fractional derivative recovers the oscillation of several proven results investigated previously in literature. Two examples, where the kernel function is quadratic and cubic polynomial, have been given to support the validity of the proven results for the RL and Caputo cases, respectively. [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.

The goal of this paper is to study the oscillation of the following second‐order nonlinear functional dynamic equation [r(t)(yΔ(t))α]Δ+Υ(t)yβ(δ(t))=0,t∈𝕋0=[t0,∞)∩𝕋.The main aim is to establish the necessary and sufficient conditions for the oscillation of the equation. The results are obtained when α > β and α < β. At the end, examples are provided to illustrate the analytical findings. We point out that the results are new even for the case 𝕋=ℝ and 𝕋=ℤ. [ABSTRACT FROM AUTHOR]

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.

We deal with Euler‐type half‐linear second‐order differential equations, and our intention is to derive conditions in order their non‐trivial solutions are non‐oscillatory. This paper connects to the article P. Hasil, J. Šišoláková, M. Veselý: Averaging technique and oscillation criterion for linear and half‐linear equations, Appl. Math. Lett. 92 (2019), 62–69, where the corresponding oscillatory counterpart is studied and an oscillation criterion is established. The used effective technique for this investigation is the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This paper is completed by two corollaries and an example concerning the linear case when we obtain new results as well. [ABSTRACT FROM AUTHOR]

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]

In this paper we investigate the energy integral which is obtained in the problem of the motion of a material point in a central field within thе frame of the general theory of relativity. Applied are the method of the small parameter in a combination with the balance method. Derived is a compact formula, describing the trajectory of the motion. This formula gives a correct quantitative description of the basic relativistic effects. We prove the shortening of the major axis of the orbit in comparison with the case where we do not take into account relativistic effects. This result can be useful for analysing the structure of planet systems around massive stars. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation of the second-order Emden–Fowler neutral differential equations with advanced and delay arguments (r (t) (x (t) + p (t) x (τ (t))) ′) ′ + q (t) x γ (σ (t)) = 0 , t ⩾ t 0 , where τ (t) ⩽ t , σ (t) ⩾ t . Sufficient conditions for oscillation of the equation are obtained by using the inequality principle, comparison principle and Riccati transform. Two examples are given to illustrate the results. [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]

In this paper, we study the oscillation of the impulsive Riemann–Liouville fractional differential equation [ r (t) D t k + α x (t) ] ′ + q (t) f d + ∫ t k + t (t - s) - α x (s) d s = 0 , t ∈ (t k , t k + 1 ] , k = 0 , 1 , 2 ... , 1 d D t k + α x (t k +) - D t k - 1 + α x (t k -) d + ∫ t k - 1 + t k - (t k - - s) - α x (s) d s = - b k , k = 1 , 2 , ... Philos-type oscillation criteria of the equation are obtained. We are interested in finding adequate impulsive controls to make the fractional system with Riemann–Liouville derivatives oscillate. An example of the change from non-oscillation to oscillation under the impulsive conditions is found. [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.

We study oscillatory properties of half-linear difference equations with asymptotically periodic coefficients, i.e., coefficients which can be expressed as the sums of periodic sequences and sequences vanishing at infinity. Using a special variation of the discrete Riccati technique, we prove that the non-oscillation of the studied equations can be determined directly from their coefficients. Thus, the studied equations can be widely used as testing equations. Our main result is new even for linear equations with periodic coefficients. This fact is illustrated by simple corollaries and examples at the end of this paper. [ABSTRACT FROM AUTHOR]

We consider the Sturm–Liouville differential equations with a power of the independent variable and sums of periodic functions as coefficients (including the case when the periodic coefficients do not have any common period). Using known results, one can show that the studied equations are conditionally oscillatory, i.e., there exists a threshold value which can be expressed by the coefficients and which separates oscillatory equations from non-oscillatory ones. It is very complicated to specify the behaviour of the treated equations in the borderline case. In this paper, applying the method of the modified Prüfer angle, we answer this question and we prove that the considered equations are non-oscillatory in the critical borderline case. [ABSTRACT FROM AUTHOR]

We study oscillatory properties of half‐linear dynamic equations on timescales. Via the combination of the Riccati technique and an averaging method, we find the domain of oscillation for many equations. The presented main result is not the conversion of a known result from the theory of differential or difference equations, ie, we obtain new results for the timescales T=R (for differential equations) and T=Z (for difference equations). Half‐linear equations generalize linear equations (in fact, they coincide with certain one‐dimensional PDEs with p‐Laplacian), but the main result is new also for linear differential and difference equations. The corresponding corollaries and examples are given as well. [ABSTRACT FROM AUTHOR]

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).