Sublinear Term Research Articles

On the Oscillatory Behavior of Solutions of Second-Order Damped Differential Equations with Several Sub-Linear Neutral Terms

The oscillation and asymptotic behavior of solutions of a general class of damped second-order differential equations with several sub-linear neutral terms is considered. New sufficient conditions are established to fulfill a part of the gap in the oscillation theory for the case of sub-linear neutral equations. Our main results improve and generalize some of those recently published in the literature. Several examples are given to support our results.

Second-Order Neutral Differential Equations with a Sublinear Neutral Term: Examining the Oscillatory Behavior

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations under study. By employing the Riccati technique in various ways, we derive results that expand the scope of previous research and enhance the cognitive understanding of this mathematical domain. Additionally, we provide three illustrative examples to demonstrate the validity and applicability of our findings.

Second-Order Noncanonical Delay Differential Equations with Sublinear and Superlinear Terms: New Oscillation Criteria via Canonical Transform and Arithmetic–Geometric Inequality

Second-Order Noncanonical Delay Differential Equations with Sublinear and Superlinear Terms: New Oscillation Criteria via Canonical Transform and Arithmetic–Geometric Inequality

Oscillatory behavior of second‐order nonlinear delay dynamic equations with multiple sublinear neutral terms utilizing canonical transformation

This paper aims to present innovative criteria for characterizing the oscillatory behavior of second‐order nonlinear delay dynamics expressed in noncanonical form with sublinear neutral terms. We achieve this by converting the noncanonical equation into an equivalent canonical form. The practical examples presented throughout this paper illustrate the significance of our findings and contribute to enhancing, generalizing, and augmenting the existing literature on this subject.

Oscillatory Features of Fourth-Order Emden–Fowler Differential Equations with Sublinear Neutral Terms

This article examines the oscillatory characteristics of a fourth-order Emden–Fowler differential equation, specifically when it includes a sublinear neutral term. Our methodology centers on establishing multiple theorems that introduce innovative conditions to guarantee that there are no positive solutions to the examined equation. Due to the symmetry between non-oscillatory solutions, we obtain oscillation conditions by excluding only positive solutions. We employ the Riccati technique in various ways to achieve this objective. The criteria presented in this study complement and generalize many findings published in the literature. We support the efficiency of our findings by applying them to an example.

Discrete Schrödinger equations and systems with mixed and concave-convex nonlinearities

In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models.

New oscillation criteria of nonlinear second order delay dynamic equations with sublinear neutral term

New oscillation criteria of nonlinear second order delay dynamic equations with sublinear neutral term

New results on the oscillation of second-order damped neutral differential equations with several sub-linear neutral terms

New results on the oscillation of second-order damped neutral differential equations with several sub-linear neutral terms

Oscillation Criteria for Even-Order Nonlinear Dynamic Equations with Sublinear and Superlinear Neutral Terms on Time Scales

Oscillation Criteria for Even-Order Nonlinear Dynamic Equations with Sublinear and Superlinear Neutral Terms on Time Scales

Oscillation Theorems for Nonlinear Second-Order Delay Differential Equations with Some Sublinear Neutral Terms via Canonical Transform

Abstract New sufficient conditions for oscillation of a second-order nonlinear differential equation with some sublinear neutral terms are established via canonical transform and integral averaging method. Examples are provided to illustrate the significance and novelty of the presented results.

Oscillatory behaviour of a class of second order Emden-Fowler differential equations with a sublinear neutral term

This article is dedicated to the oscillatory properties of a second order Emden-Fowler delay differential equation with a sublinear neutral term. The presented criteria extend and improve several well-known results reported recently in the literature. Four illustrating examples are included to verify the validity of the results derived.

Multiplicity of solutions for the stationary quantum Zakharov system with combined nonlinearities

In this article, we prove the existence and multiplicity of nontrivial solutions for the stationary quantum Zakharov system with a sign-changing potential and a general nonlinearity involving a combination of a superlinear f ( x , u ) source term and a sublinear term ( K ( x ) | u | q − 2 u , K ∈ L 2 2 − q ( R 3 ) , where 1 < q < 2 ) .

Multiple solutions for a boundary singular semilinear equation with sublinear term involving Hardy potential and Hardy–Sobolev exponent

Multiple solutions for a boundary singular semilinear equation with sublinear term involving Hardy potential and Hardy–Sobolev exponent

Oscillation criteria for second-order delay dynamic equations with a sub-linear neutral term on time scales

This paper addresses the oscillatory behavior of the solutions of second-order dynamic equations with a sublinear neutral term. Using Riccati transformation and comparison principles, we obtain new oscillation criteria. The obtained results essentially improve, complement, and simplify some of the previous ones in the literature. Some examples have been provided herein to illustrate our main results.

OSCILLATION OF SECOND-ORDER HALF-LINEAR NEUTRAL NONCANONICAL DYNAMIC EQUATIONS

In this paper, we shall establish some new criteria for the oscillation of certain second-order noncanonical dynamic equations with a sublinear neutral term. This task is accomplished by reducing the involved nonlinear dynamic equation to a second-order linear dynamic inequality. We also establish some new oscillation theorems involving certain integral conditions. Three examples, illustrating our results, are presented. Our results generalize results for corresponding differential and difference equations.

New oscillation criteria for second-order delay dynamic equations with a sub-linear negative neutral term on time scales

New oscillation criteria for second-order delay dynamic equations with a sub-linear negative neutral term on time scales

Oscillation for second order nonlinear differential equations with a sub-linear neutral term

We study the oscillatory behavior of solution to the second order nonlinear differential equations with a sub-linear neutral term $$ \big(a(t)[(x(t)+p(t)x^{\alpha}(\tau(t)))']^{\gamma}\big)'+q(t)x^{\beta}(\sigma(t))=0, \quad t\geq t_0&gt;0. $$ A new criterion is established that improves related results reported in the literature. Moreover, some examples are provided to illustrate the main results.

Ramsey numbers of several Kt,s and a large Km,n

For graphs G and H, denote by rk+1(G;H) the minimum N such that any edge-coloring of KN by k+1 colors contains either a monochromatic G in the first k colors or a monochromatic H in the last color. As usual, we write r2(G;H) as r(G,H). We show that if integers s≥t≥m≥1, then rk+1(Kt,s;Km,n)≤n+(1+o(1))(s−t+1)1/tkmn1−1/t as n→∞. The upper bound is shown to be sharp up to the asymptotical sub-linear term for rk+1(K2,s;K1,n), rk+1(K3,3;K1,n), r(K2,s,K2,n) and r(K3,3,Km,n) for m≤3.

Distributed Online Convex Optimization With an Aggregative Variable

This article investigates distributed online convex optimization in the presence of an aggregative variable without any global/central coordinators over a multiagent network. In this problem, each individual agent is only able to access partial information of time-varying global loss functions, thus requiring local information exchanges between neighboring agents. Motivated by many applications in reality, the considered local loss functions depend not only on their own decision variables, but also on an aggregative variable, such as the average of all decision variables. To handle this problem, an online distributed gradient tracking algorithm (O-DGT) is proposed with exact gradient information and it is shown that the dynamic regret is upper bounded by three terms: 1) a sublinear term; 2) a path variation term; and 3) a gradient variation term. Meanwhile, the O-DGT algorithm is also analyzed with stochastic/noisy gradients, showing that the expected dynamic regret has the same upper bound as the exact gradient case. To our best knowledge, this article is the first to study online convex optimization in the presence of an aggregative variable, which enjoys new characteristics in comparison with the conventional scenario without the aggregative variable. Finally, a numerical experiment is provided to corroborate the obtained theoretical results.

Oscillation of second-order non-canonical non-linear dynamic equations with a sub-linear neutral term

Oscillation of second-order non-canonical non-linear dynamic equations with a sub-linear neutral term