Half-linear Dynamic Equations Research Articles

Investigating Oscillations in Higher-Order Half-Linear Dynamic Equations on Time Scales

This study presents novel and generalizable sufficient conditions for determining the oscillatory behavior of solutions to higher-order half-linear neutral delay dynamic equations on time scales. Utilizing the Riccati transformation technique in combination with Taylor monomials, we derive new and comprehensive oscillation criteria that cover a wide range of cases, including super-linear, half-linear, and sublinear equations. These results extend and improve upon existing oscillation criteria found in the literature by introducing more general conditions and providing a broader applicability to different types of dynamic equations. Furthermore, the study highlights the role of symmetry in the underlying equations, demonstrating how symmetry properties can be leveraged to simplify the analysis and provide additional insights into oscillatory behavior. To demonstrate the practical relevance of our findings, we include illustrative examples that show how these new criteria, along with symmetry-based perspectives, can be effectively applied to various time scales.

Oscillation and Asymptotic Behavior of Second Order Half Linear Neutral Dynamic Equations

The oscillation property of the second order half linear dynamic equation was studied, some sufficient conditions were obtained to ensure the oscillation of all solutions of the equation. The results are supported by illustrative examples.

Nonoscillation criteria for damped half-linear dynamic equations with mixed derivatives on a time scale

In this study, we investigate the use of two types of damped half-linear dynamic equations on a single time scale to provide sufficient conditions to guarantee nonoscillation for nontrivial solutions of both ordinary differential equations and difference equations. If the time scale is selected from the sets of all real numbers or all natural numbers, then these equations are half-linear differential and half-linear difference equations. As a feature of the nonoscillation theorems provided, if both the differential and difference equations are linear, the conditions ensuring nonoscillation are similar. However, if both are nonlinear equations with a one-dimensional p-Laplacian, differences exist in the nonoscillatory conditions. An Euler-type dynamic equation with damping is presented as an example to emphasize the similarities and differences in the nonoscillatory conditions described.

On oscillatory behaviour of third-order half-linear dynamic equations on time scales

In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case \(\mathbb{T}=\mathbb{R}\) or \(\mathbb{T}=\mathbb{Z}\).

Diamond alpha Bennett‐Leindler type dynamic inequalities and their applications

In this paper, two kinds of dynamic Bennett‐Leindler type inequalities via the diamond alpha integrals are derived. The first kind consists of eight new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together due to the fact that convex linear combinations of delta and nabla Bennett‐Leindler type inequalities give diamond alpha Bennett‐Leindler type inequalities. The second kind involves four new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Bennett‐Leindler type inequalities. For the second type, choosing or not only yields the same results as the ones obtained for delta and nabla cases but also provides novel results for them. Therefore, both kinds of our results expand some of the known delta and nabla Bennett‐Leindler type inequalities, offer new types of these inequalities, and bind and unify them into one diamond alpha Bennett‐Leindler type inequalities. Moreover, an application of dynamic Bennett‐Leindler type inequalities to the oscillation theory of the second‐order half linear dynamic equation is developed and presented for the first time ever.

Two Weighted Norm Dynamic Inequalities with Applications on Second Order Half-Linear Dynamic Equations

Two Weighted Norm Dynamic Inequalities with Applications on Second Order Half-Linear Dynamic Equations

Hille-Nehari type non-oscillation criteria for half-linear dynamic equations with mixed derivatives on a time scale

This article deals with half-linear dynamic equations that have two types of derivatives, and obtains sufficient conditions for all solutions to be non-oscillatory. The obtained results extend a previous Hille-Nehari type theorems for problems of dynamic equations. To prove our main result, we use a generalized Riccati inequality. As an application, we apply the main result to self-adjoint Euler type linear differential and difference equations with a changing sign coefficient. The equation selected for this application is of Mathieu type. For more information see https://ejde.math.txstate.edu/Volumes/2021/78/abstr.html

New Oscillation Criteria for Second Order Half-Linear Neutral Type Dynamic Equations on Time Scales

In this paper, we will study the oscillatory properties of the second order half-linear dynamic equations with distributed deviating arguments on time scales. We obtain several new sufficient conditions for the oscillation of all solutions of this equation. Our results not only unify the oscillation of second order nonlinear differential and difference equations but also can be applied to different types of time scales with sup T = ∞. Our results improve and extend some known results in the literature. Examples which dwell upon the importance of our results are also included.

Half-linear dynamic equations and investigating weighted Hardy and Copson inequalities

In this paper, we employ some algebraic equations due to Hardy and Littlewood to establish some conditions on weights in dynamic inequalities of Hardy and Copson type. For illustrations, we derive some dynamic inequalities of Wirtinger, Copson and Hardy types and formulate the classical integral and discrete inequalities with sharp constants as particular cases. The results improve some results obtained in the literature.

A connection between weighted Hardy\u2019s inequality and half-linear dynamic equations

In this paper, we give an affirmative answer to the following question: Is the solvability of some nonlinear dynamic equations on a time scale mathbb{T} not only sufficient but in a certain sense also necessary for the validity of some dynamic Hardy-type inequalities with two different weights? In fact, this answer will give a new characterization of the weights in a weighted Hardy-type inequality on time scales. The results contain the results when mathbb{T}=mathbb{R}, mathbb{T}=mathbb{N}, and when mathbb{T}=q^{mathbb{N}_{0}} as special cases. Some applications are given for illustrations.

Lower bounds for the eigenvalues of first-order nonlinear Hamiltonian systems on time scales

We study first-order nonlinear planar Hamiltonian boundary value problems on time scales. Estimates on lower bounds for the eigenvalues of the problems are established by way of the Lyapunov inequality method. Our results are interpreted to nonlinear differential and difference planar Hamiltonian boundary value problems. As a special case, an estimate on lower bounds for eigenvalues of half-linear dynamic equations is obtained which generalizes and improves the existing ones to nonlinear Hamiltonian systems. Based on the main results, we establish existence and uniqueness of solutions of a related linear boundary value problem.

Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments

We study oscillatory behavior of solutions to a class of second-order half-linear dynamic equations with deviating arguments under the assumptions that allow applications to dynamic equations with delayed and advanced arguments. Several improved Fite–Hille–Wintner-type criteria are obtained that do not need some restrictive assumptions required in related results. Illustrative examples and conclusions are presented to show that these criteria are sharp for differential equations and provide sharper estimates for oscillation of corresponding q-difference equations.

Opial and Lyapunov inequalities on time scales and their applications to dynamic equations

We prove some weighted inequalities for delta derivatives acting on products and compositions of functions on time scales and apply them to obtain generalized dynamic Opial-type inequalities. We also employ these inequalities to establish some new dynamic Lyapunov-type inequalities, which are essential in studying disfocality, disconjugacy, lower bounds of eigenvalues, and distance between generalized zeros for half-linear dynamic equations. In particular, we solve an open problem posed by Saker in [Math. Comput. Modelling 58 (2013), 1777-1790]. Moreover, the results presented in this paper generalize, improve, extend, and unify most of known results not only in the discrete and continuous analysis but also on time scales.

Oscillation criteria for third-order functional half-linear dynamic equations

In this paper, we study the third-order functional dynamic equation {r2(t)ϕα2([r1(t)ϕα1(xΔ(t))]Δ)}Δ+q(t)ϕα(x(g(t)))=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl\\{ r_{2}(t)\\phi_{\\alpha_{2}} \\bigl( \\bigl[ r_{1}(t) \\phi _{\\alpha _{1}} \\bigl( x^{\\Delta}(t) \\bigr) \\bigr] ^{\\Delta} \\bigr) \\bigr\\} ^{\\Delta}+q(t)\\phi_{\\alpha} \\bigl( x\\bigl(g(t)\\bigr) \\bigr) =0, $$\\end{document} on an upper-unbounded time scale mathbb{T}. We will extend the so-called Hille and Nehari type criteria to third-order dynamic equations on time scales. This work extends and improves some known results in the literature on third-order nonlinear dynamic equations and the results are established for a time scale mathbb{T} without assuming certain restrictive conditions on mathbb{T}. Some examples are given to illustrate the main results.

Oscillation criteria of second order half linear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation $${\left( {r\left( t \right)g\left( {{x^\Delta }\left( t \right)} \right)} \right)^\Delta } + p\left( t \right)f\left( {x\left( {\tau \left( t \right)} \right)} \right) = 0,$$ on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=ℝ, difference equations when T=ℕ but can be applied on different types of time scales such as when T=qℕ for q > 1 and also improve most previous results. Finally, we give some examples to illustrate our main results.

Generalized Philos-type oscillation criteria for second order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation w(s) and showing that [H(σ(t),s)w(s)]Δs≤0, we establish new Philos-type oscillation criteria for second order nonlinear dynamic equations on time scales. The obtained results essentially generalize and improve the well-known oscillation results for half-linear dynamic equations such as Philos-type and Kamenev-type oscillation criteria. We illustrate the versatility of our results by means of examples.

Oscillation criteria for second order nonlinear delay dynamic equations on time scales

By employing the generalized Riccati transformation, we establish some new oscillation criteria for second order delay dynamic equations on time scales. The obtained results essentially improve the well-know oscillation results for half-linear dynamic equations such as Kamenev-type and Philos-type oscillation criteria. Some interesting examples are given to illustrate the versatility of our results.

A new Picone’s dynamic inequality on time scales with applications

In this paper, we will derive a new dynamic Picone-type inequality for half-linear dynamic equations and dynamic inequalities of second order on an arbitrary time scale T. As a consequence, we will apply this new Picone inequality to get a new Wirtinger-type inequality on time scales with two different weighted functions. The results contain the Wirtinger inequalities formulated by Beesack, Lee and Jaroš for the continuous case. For the discrete case our results are also new.

On the Oscillation for Second-Order Half-Linear Neutral Delay Dynamic Equations on Time Scales

We discuss oscillation criteria for second-order half-linear neutral delay dynamic equations on time scales by using the generalized Riccati transformation and the inequality technique. Under certain conditions, we establish four new oscillation criteria. Our results in this paper are new even for the cases of𝕋=ℝand𝕋=ℤ.

Oscillation Theorems for Second-Order Half-Linear Neutral Delay Dynamic Equations with Damping on Time Scales

We establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.