Sturmian comparison theory for half‐linear and nonlinear differential equations via Picone identity

In this paper, Sturmian comparison theory is developed for the pair of second‐order differential equations; first of which is the nonlinear differential equations of the form (1) and the second is the half‐linear differential equations (2) where Φ_α (s ) = |s |^{α  ? 1}s and α ₁ > ? > α _m  > β  > α _{m  + 1} > ? > α _n  > 0. Under the assumption that the solution of  2 has two consecutive zeros, we obtain Sturm–Picone type and Leighton type comparison theorems for  1 by employing the new nonlinear version of Picone formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for  1 . Examples are given to illustrate the relevance of the results. Copyright © 2016 John Wiley & Sons, Ltd.     

Sturm‐Picone comparison theorems for nonlinear impulsive differential equations with discontinuous solutions

The nonlinear versions of Sturm‐Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.     

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

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.     

Picone's formula for linear non-selfadjoint impulsive differential equations

In this paper, we derive a Picone type formula for second-order linear non-selfadjoint impulsive differential equations having fixed moments of impulse actions, and obtain a Wirtinger type inequality, a Leighton type comparison theorem, and a Sturm-Picone comparison theorem for such equations. Moreover, several oscillation criteria are also derived as applications.     

Oscillation result for half‐linear dynamic equations on timescales and its consequences

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 (for differential equations) and (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.     

Positive solutions for asymptotically linear Schrödinger–Kirchhoff‐type equations

In this paper, we focus on the Schrödinger–Kirchhoff‐type equation (SK) where a,b > 0 are constants, may not be radially symmetric, and f(x,u) is asymptotically linear with respect to u at infinity. Under some technical assumptions on V and f, we prove that the problem (SK) has a positive solution. Copyright © 2013 John Wiley & Sons, Ltd.     

Interval oscillation criteria for second order super‐half linear functional differential equations with delay and advanced arguments

Sufficient conditions are established for oscillation of second order super half linear equations containing both delay and advanced arguments of the form where ?_δ (u) = |u |^{δ –1}u; α > 0, β ≥ α, and γ ≥ α are real numbers; k, p, q, e, τ, σ are continuous real‐valued functions; τ (t) ≤ t and σ (t) ≥ t with lim_{t →∞} τ (t) = ∞. The functions p (t), q (t), and e (t) are allowed to change sign, provided that p (t) and q (t) are nonnegative on a sequence of intervals on which e (t) alternates sign. As an illustrative example we show that every solution of is oscillatory provided that either m₁ or m₂ or r₀ is sufficiently large (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applications of variational methods to Sturm–Liouville boundary‐value problem for fourth‐order impulsive differential equations

In this paper, we study Sturm–Liouville boundary‐value problem for fourth‐order impulsive differential equations. Applying variational methods, several new existence results are obtained. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary value problems on half‐line for second‐order nonlinear impulsive differential equations

We obtain sufficient conditions for existence and uniqueness of solutions of boundary value problems on half‐line for a class of second‐order nonlinear impulsive differential equations. Our technique is different than the traditional ones, as it is based on asymptotic integration method involving principal and nonprincipal solutions. Examples are provided to illustrate the relevance of the results.     

New Lyapunov‐type inequalities for a class of even‐order linear differential equations

In this paper, we obtain some new Lyapunov‐type inequalities for a class of even‐order linear differential equations, the results are new and generalize and improve some early results in this field.     

The stability of rarefaction wave for Navier–Stokes equations in the half‐line

This paper studies the stability of the rarefaction wave for Navier–Stokes equations in the half‐line without any smallness condition. When the boundary value is given for velocity u∥_{x = 0} = u_? and the initial data have the state (v₊, u₊) at x→ + ∞, if u_?<u₊, it is excepted that there exists a solution of Navier–Stokes equations in the half‐line, which behaves as a 2‐rarefaction wave as t→ + ∞. Matsumura–Nishihara have proved it for barotropic viscous flow (Quart. Appl. Math. 2000; 58:69–83). Here, we generalize it to the isentropic flow with more general pressure. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability criterion for second order linear impulsive differential equations with periodic coefficients

In this paper we obtain instability and stability criteria for second order linear impulsive differential equations with periodic coefficients. Further, a Lyapunov type inequality is also established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple non semi‐trivial solutions of systems of Kirchhoff‐type equations with discontinuous nonlinearities in RN

In the present paper, we study the problem of multiple non semi‐trivial solutions for the following systems of Kirchhoff‐type equations with discontinuous nonlinearities (1.1) where F∈C¹(R^N×R⁺×R⁺,R),V∈C(R^N,R), and By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi‐trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Periodic solutions of the Navier–Stokes equations in a perturbed half‐space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half‐space and an aperture domain of the dimension n?3. Our proof is based on L^p–L^q estimates of the Stokes semigroup. We apply L^p–L^q estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

We considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein–Gordon differential equation with a nonlinear spectral parameter‐dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S(λ); in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson‐type formula, was obtained. Copyright © 2012 John Wiley & Sons, Ltd.     

Ambarzumyan‐type theorem with polynomially dependent eigenparameter

In this paper, we are concerned with the inverse Sturm–Liouville problem with polynomially dependent eigenparameter in discontinuity and boundary conditions. By using a self‐adjoint operator‐theoretic interpretation for this sort of problem, Ambarzumyan theorem is provided for the mentioned Sturm–Liouville operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Systems of linear ordinary differential equations with bounded coefficients may have very oscillating solutions

An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system.

     

Large‐time behaviour of solutions to non‐linear wave equations: higher‐order asymptotics

The Cauchy problems for the Korteweg–de Vries–Burgers equation and the Benjamin–Bona– Mahony–Burgers equation are studied. Using subtle estimates of solutions to the linearized equations, the higher‐order terms of the asymptotic expansion as of solutions are derived. Copyright © 1999 John Wiley & Sons, Ltd.     

Moment estimates for weak solutions to the Navier–Stokes equations in half‐space

We establish the moment estimates for a class of global weak solutions to the Navier–Stokes equations in the half‐space. Copyright © 2009 John Wiley & Sons, Ltd.