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We extend classical nabla and delta Hardy–Copson type inequalities from $$\zeta >1$$ to $$0

Oscillation of solutions of even order half-linear differential equations of the form \begin{equation*}\label{eq_Jaros} (\alpha_n,\dots,\alpha_1)x + q(t) |x|^\beta \operatorname{sgn} x = 0, \qquad t \geq a > 0, \tag{*} \end{equation*} where $\alpha_i, \ 1 \leq i \leq n$, and $\beta$ are positive constants, $q$ is continuous functions from $[a,\infty)$ to $(0,\infty)$ and the differential operator $D(\alpha_n,\dots,\alpha_1)$ is defined by $$ D(\alpha_1)x = \frac{d}{dt} \big(|x|^{\alpha_1} \operatorname{sgn} x\big) $$ and $$ D(\alpha_i, \dots, \alpha_1)x = \frac{d}{dt}\big(|D(\alpha_{i-1},\dots,\alpha_1)x|^{\alpha_i} \operatorname{sgn} D(\alpha_{i-1},\dots,\alpha_1)x\big), \qquad i = 2, \dots, n, $$ is proved in the case where $\alpha_1 \cdots \alpha_n = \beta$ through reduction to the problem of oscillation of solutions of some lower order differential equations associated with \eqref{eq_Jaros}.

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.

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.

Using integral average method and properties of conformable fractional derivative, new Kamenev type oscillation criteria are given firstly for conformable fractional differential equations, which improve known results in oscillation theory. Examples are also given to illustrate the effectiveness of the main results.

We study the relative structural stability (relative roughness) of dynamical systems in some subspace of the space of all dynamical systems; moreover, the space of deformations of (dynamical) systems does not coincide with the space of all admissible deformations. We give some examples of relatively rough systems and relatively nonrough systems of different degree of nonroughness arising in rigid body dynamics and oscillation theory.

We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schr\"odinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we demonstrate that the associated linearized operator is nondegenerate when restricted to antiperiodic perturbations, i.e. that its kernel is generated by the translational and gauge symmetries of the governing evolution equation. In the process, we provide a characterization of the antiperiodic ground state eigenfunctions for linear fractional Schr\"odinger operators on $\mathbb{R}$ with real-valued, periodic potentials as well as a Sturm-Liouville type oscillation theory for the higher antiperiodic eigenfunctions., Comment: 46 pages, 2 figures

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.

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, i.e., 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.

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.

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 Prufer angle, we answer this question and we prove that the considered equations are non-oscillatory in the critical borderline case.

The purpose of this paper is to describe the oscillatory properties of second-order Euler-type half-linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non-oscillation of the considered equations, including the so-called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.

Sturm–Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

This paper deals with the oscillation of the fourth-order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third-order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained.

This paper presents a method for expanding horizontal flow variables in data using the free solutions to the shallow-water system as a basis set. This method for equatorial wave expansion of instantaneous flows (EWEIF) uses dynamic constraints in conjunction with projections of data onto parabolic cylinder functions to determine the amplitude of all equatorial waves. EWEIF allows us to decompose an instantaneous wave flow into individual equatorial waves with a presumed equivalent depth without using temporal or spatial filtering a priori. Three sets of EWEIF analyses are presented. The first set is to confirm that EWEIF is capable of recovering the individual waves constructed from theoretical equatorial wave solutions under various scenarios. The other two sets demonstrate the ability of the EWEIF method to derive time series of individual equatorial waves from instantaneous wave fields without knowing a priori exactly which waves exist in the data as well as their spatial and temporal scales using outputs of an equatorial β-channel shallow-water model and ERA-Interim data. The third set of demonstrations shows, for the first time, the continuous evolutions of individual equatorial waves in the stratosphere whose amplitude is synchronized with the background zonal wind as predicted by quasi-biennial oscillation theory.

We investigate oscillation about the positive equilibrium point of a population model with piecewise constant argument. By using linearized oscillation theory for difference equations a necessary and sufficient condition for the oscillation is obtained. Moreover it is showed that every nonoscillatory solution approaches to the equilibrium point as t tends to infinity.

In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form $$ \left \{ \textstyle\begin{array}{l} {}_{q}\nabla_{0}^{\alpha}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x), \quad t>0 ,\\ \lim_{t \to0^{+}}{{}_{q}I_{0}^{j-\alpha}x(t)}=b_{j} \quad(j=1,2,\ldots,m), \end{array}\displaystyle \right . $$ where $m=\lceil\alpha\rceil$ , ${}_{q}\nabla_{0}^{\alpha}$ 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.

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.

Consideration of partial mappings allows us to expose a noticeable part of the theory of ordinary differential equations at the axiomatic level. This gives us a possibility to cover equations with discontinuous right-hand sides and differential inclusions by standard well-known (but a little revised) theory. Homological properties of the corresponding solution sets allow us to create a new version of the shift method in the theory of boundary value problems as powerful as the application here of the Leray–Schauder theory.

We investigate solutions to nonlinear operator equations which are difficult to investigate with variational methods and obtain some abstract existence results by topology degree methods. These results apply to ordinary differential systems with impulsive effects satisfying nonlinear boundary value conditions, and we obtain some new results.

We discuss the concept of relative oscillation for linear Hamiltonian differential systems. We prove a statement which enables to study this oscillation using the criteria of the classical oscillation theory. We also discuss some other aspects of the problem.

In this paper, we introduce a new modification of the half-linear Prufer 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 .

We review some theorems and mistakes in linearized oscillation results for difference equations with variable coefficients and constant delays, as well as develop linearized oscillation theory when delays are also variable. Main statements are applied to discrete models of population dynamics. In particular, oscillation of generalized Pielou, Ricker and Lasota–Wazewska equations is considered.

In this paper we establish new renormalized oscillation theorems for discrete symplectic eigenvalue problems with Dirichlet boundary conditions. These theorems present the number of finite eigenvalues of the problem in arbitrary interval $(a,b]$ using number of focal points of a transformed conjoined basis associated with Wronskian of two principal solutions of the symplectic system evaluated at the endpoints $a$ and $b.$ We suppose that the symplectic coefficient matrix of the system depends nonlinearly on the spectral parameter and that it satisfies certain natural monotonicity assumptions. In our treatment we admit possible oscillations in the coefficients of the symplectic system by incorporating their nonconstant rank with respect to the spectral parameter., Comment: 28 pages, to be published in Journal of Difference Equations and Applications

The work of Sturm and Liouville on expansion in eigenfunctions is heavily dependent on the study of the zeros of solutions of the equation, nowadays called oscillation theory.

The integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine. In this paper the fuzzy integral equation is considered and the existence and uniqueness theorem, the theorem of continuous dependence on the right-hand side and initial fuzzy set are proved. Also the possibility of using the scheme of full averaging for fuzzy integral equation with a small parameter is considered.

A connection between the oscillation theory and the Weyl--Titchmarsh theory for the second order Sturm--Liouville equation on time scales is established by using the principal solution. In particular, it is shown that the Weyl solution coincides with the principal solution in the limit point case, and consequently the square integrability of the Weyl solution is obtained. Moreover, both limit point and oscillatory criteria are derived in the case of real-valued coefficients, while a~generalization of the invariance of the limit circle case is proven for complex-valued coefficients. Several of these results are new even in the discrete time case. Finally, some illustrative examples are provided.

We consider the following two-dimensional system of non-linear equations: u ′ = g ⁢ ( t ) ⁢ | v | 1 α ⁢ sgn ⁡ v , v ′ = - p ⁢ ( t ) ⁢ | u | α ⁢ sgn ⁡ u , u^{\prime}=g(t)|v|^{\frac{1}{\alpha}}\operatorname{sgn}v,\quad v^{\prime}=-p(t% )|u|^{\alpha}\operatorname{sgn}u, where α > 0 {\alpha>0} , and g : [ 0 , + ∞ [ → [ 0 , + ∞ [ {g\colon{[0,+\infty[}\rightarrow{[0,+\infty[}} and p : [ 0 , + ∞ [ → ℝ {p\colon{[0,+\infty[}\rightarrow\mathbb{R}} are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on [ 0 , + ∞ ] {[0,+\infty]} . We establish new non-oscillation criteria for the considered system, which generalize known results for the corresponding linear system and for second order differential equations. In particular, the presented criteria are in compliance with the results of Hille and Nehari.

The linearization problem of a third-order ordinary differential equation by the tangent transformation is considered in the present paper. This is the first application of tangent (essentially) transformations to the linearization problem of third-order ordinary differential equations. Necessary and sufficient conditions for a third-order ordinary differential equation to be linearizable are obtained.

In this paper we study a general even-order symmetric Sturm–Liouville matrix differential equation, whose leading coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm–Liouville equations with different orders. We identify the so-called normal form of this equation, which allows to transform this equation into a standard (controllable) linear Hamiltonian system. Based on this new transformation we prove that the associated eigenvalue problem with Dirichlet boundary conditions possesses all the traditional spectral properties, such as the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of the eigenfunctions, the oscillation theorem and Rayleigh’s principle, and the Fourier expansion theorem. We also discuss sufficient conditions, which allow to reduce a general even-order symmetric Sturm–Liouville matrix differential equation into the normal form. Throughout the paper we provide several examples, which illustrate our new theory.

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).

The oscillatory properties of half-linear second order Euler type differential equations are studied, where the coefficients of the considered equations can be unbounded. For these equations, we prove an oscillation criterion and a non-oscillation one. We also mention a corollary which shows how our criteria improve the known results. In the corollary, the criteria give an explicit oscillation constant.

This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Numerous examples are given throughout.

By the use of geometric methods for linearizing systems of second-order cubically semi-linear ordinary differential equations and the conditional linearizability of third-order quintically semi-linear ordinary differential equations, we extend to the fourth-order by differentiating the third-order conditionally linearizable equation. This yields criteria for conditional linearizability of a class of fourth-order semi-linear ordinary differential equations, which have not been discussed in the literature previously.

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.

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.

The study of oscillation theory for fractional differential equations has been initiated by Grace et.al. 14. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form c D a α x ( t ) = e ( t ) + f ( t , x ( t ) ) , a 1 , α ? ( 1 , 2 ) .We also present the conditions under which all solutions of this equation are asymptotic to at + b as t ? ∞ for some real numbers a, b. We shall employ a different technique rather than that in 14.

The characterization of the domain of the Friedrichs extension as a restriction of the maximal domain is well known. It depends on principal solutions. Here we establish a characterization as an extension of the minimal domain. Our proof is different and closer in spirit to the Friedrichs construction. It starts with the assumption that the minimal operator is bounded below and does not directly use oscillation theory.

By theorems of Sturm and Lyapunov, the L ∞ and L 1 norms of the coefficient q in a differential equation u ″ + q ( t ) u = 0 limit the oscillation of nontrivial solutions. This paper establishes similar results for the L p norms when 1 p ∞ .

Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems $Ju'=-zHu$ and then use it to discuss semibounded operators from this point of view. Our main new result is a characterization of systems with purely discrete spectrum in terms of the asymptotics of their coefficient functions; we also discuss the exponential types of the transfer matrices.

We suggest a new constructive approach for the solvability analysis and approximate solution of general non-local boundary value problems for non-linear systems of ordinary differential equations with locally Lipschitzian non-linearities. The practical application of the techniques is explained on a numerical example.

We discuss the existence of nontrivial solutions to the boundary value problems for a coupled system of second-order nonlinear difference equations by using the critical point theory. The nontrivial solutions where neither of the components is identically zero are achieved under some sufficient conditions.

Made available in DSpace on 2020-12-12T00:54:25Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-01-01 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) In this paper, we consider one-parameter ( λ>0) families of Liénard differential equations. We are concerned with the study on the asymptotic behavior of periodic solutions for small and large values of λ>0. To prove our main result we use the relaxation oscillation theory and a topological version of the averaging theory. More specifically, the first one is appropriate for studying the periodic solutions for large values of λ and the second one for small values of λ. In particular, our hypotheses allow us to establish a link between these two theories. Universidade Estadual Paulista (UNESP) Faculdade de Engenharia, Ilha Solteira Universidade Estadual de Campinas (UNICAMP) Instituto de Matemática Estatística e Computação Científica Campinas Universidade Estadual Paulista (UNESP) Faculdade de Engenharia, Ilha Solteira FAPESP: 2013/24541-0 FAPESP: 2018/13481-0 FAPESP: 2018/16430-8 FAPESP: 2019/00976-4 FAPESP: 2019/10269-3 CNPq: 306649/2018-7 CNPq: 438975/2018-9 CAPES: 88881.030454/2013-0

We establish in this paper the equivalence between a Volterra integral equation of second kind and a singular ordinary differential equation of third order with two initial conditions and an integral boundary condition, with a real parameter. This equivalence allow us to obtain the solution to some problems for nonclassical heat equation, the continuous dependence of the solution with respect to the parameter and the corresponding explicit solution to the considered problem., 12 pages

In this work a generalised Lienard type system will be considered. We study the problem whether all trajectories of this system intersect the vertical isocline, which is very important in the global asymptotic stability of the origin, oscillation theory, and existence of periodic solutions. Under quite general assumptions we obtain sufficient conditions which are very sharp. We present some new conditions under which the solutions of this system are oscillatory. Some examples are provided to illustrate our results.