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Various results and techniques, such as Bohl-Perron theorem, a priori solution estimates, M-matrices and the matrix measure, are applied to obtain new explicit exponential stability conditions for the system of vector functional differential equations $$ \dot{x_i}(t)=A_i(t)x_i(h_i(t)) +\sum_{j=1}^n \sum_{k=1}^{m_{ij}} B_{ij}^k(t)x_j(h_{ij}^k(t)) + \sum_{j=1}^n\int\limits_{g_{ij}(t)}^t K_{ij}(t,s)x_j(s)ds,~i=1,\dots,n. $$ Here $x_i$ are unknown vector functions, $A_i, B_{ij}^k, K_{ij}$ are matrix functions, $h_i,h_{ij}^k, g_{ij}$ are delayed arguments. Using these results, we deduce explicit exponential stability tests for second order vector delay differential equations., 14 pages

In this paper, we consider a parabolic second order equation with changing direction of time. Under certain conditions, the existence of regular solutions in a suitable weighted Sobolev space is proven by regularization. The uniqueness of a generalized solution is also established as a consequence.

The middle convolution, introduced by Katz and developed by Dettweiler–Reiter, Oshima and others, defines an operation of reduction of linear ordinary differential equations with polynomial coefficients. In this paper, employing an idea of the exact steepest descent method proposed by Aoki–Kawai–Takei, we study how to determine the complete Stokes geometry and how to obtain the Borel summability of WKB solutions of a higher order linear ordinary differential equation with a large parameter when it is reduced to a second order equation via middle convolution. To show the practical usefulness of the method, we also investigate some concrete examples numerically.

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.

We formulate a well-posedness and approximation theory for a class of generalised saddle point problems. In this way we develop an approach to a class of fourth order elliptic partial differential equations using the idea of splitting into coupled second order equations. Our main motivation is to treat certain fourth order equations on closed surfaces arising in the modelling of biomembranes but the approach may be applied more generally. In particular we are interested in equations with non-smooth right-hand sides and operators which have non-trivial kernels. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.

A set of non-standard discrete models are constructed for the solution of non-homogenous second order equation. The method of non-local approximation and renormalization of the discretization functions have been applied to some examples and the result has shown that the schemes behave qualitatively like the original equation.

When solving acoustic field equations by using numerical approximation technique, absorbing boundary conditions (ABCs) are widely used to truncate the simulation to a finite space. The perfectly matched layer (PML) technique has exhibited excellent absorbing efficiency as an ABC for the acoustic wave equation formulated as a first-order system. However, as the PML was originally designed for the first-order equation system, it cannot be applied to the second-order equation system directly. In this article, we aim to extend the unsplit PML to the second-order equation system. We developed an efficient unsplit implementation of PML for the second-order acoustic wave equation based on an auxiliary-differential-equation (ADE) scheme. The proposed method can benefit to the use of PML in simulations based on second-order equations. Compared with the existing PMLs, it has simpler implementation and requires less extra storage. Numerical results from finite-difference time-domain models are provided to illustrate the validity of the approach.

In this paper new oscillation criteria for the second order neutral difference equation of the form are presented. Gained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. Obtained comparison principles essentially simplify the examination of the studied equations. We cover all possible cases when arguments are delayed, advanced or mixed.

In this paper, we study the existence of global solutions for a class of second order impulsive abstract functional differential equations. The results are obtained by using Leray–Schauder’s Alternative fixed point theorem. An application is provided to illustrate the theory.

The existence of homoclinic orbits is obtained for a class of the second order Hamiltonian systems u ( t ) − L ( t ) u ( t ) + ∇ W ( t , u ( t ) ) = 0 , ∀ t ∈ R , by the local linking lemma, where W ( t , x ) is superquadratic and need not be nonnegative globally.

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t → ∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.

In this paper, under weaker hypothesis, the global existence of oscillatory solutions is established for a forced second order nonlinear delay differential equation. The results of this paper generalize and improve some known results.

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.

In this paper, we are concerned with the global structure and stability of positive solutions of periodic boundary value problem − u ″ ( t ) + q ( t ) u ( t ) = λ a ( t ) f ( u ( t ) ) , 0 t 2 π , u ( 0 ) = u ( 2 π ) , u ′ ( 0 ) = u ′ ( 2 π ) , where q ∈ C ( R , [ 0 , ∞ ) ) is of periodic 2 π and q ( t ) ≢ 0 , t ∈ [ 0 , 2 π ] ; a ∈ C ( R , R ) is of periodic 2 π and changes sign. The proof of our main results are based on bifurcation techniques.

In this paper, the existence of at least three positive solutions to impulsive second order differential equations is investigated. Sufficient conditions which guarantee the existence of positive solutions are obtained, by using the Avery–Peterson theorem. An example is added to illustrate the results.

We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the following abstract second order problem u ¨ ( t ) + g ( u ˙ ( t ) ) + M ( u ( t ) ) = 0 , t ∈ R + , where M is the gradient operator of a non-negative functional and g is a non-linear damping operator, under some conditions relating the Łojasiewicz exponent of the functional and the growth of the damping around the origin. The main result is applied to non-linear wave or plate equations, in some cases direct constructive proofs of the Łojasiewicz gradient inequality are given, applicable to some non-analytic functionals in presence of multiple critical points. At the end similar results are obtained when a fast decaying source term is added in the right-hand side.

In this paper, we obtain the global regularity estimates in Orlicz spaces for second-order divergence elliptic and parabolic equations with BMO coefficients in the whole space. In fact, the global result can follow from the local estimates. As a corollary we obtain Lp-type regularity estimates for such equations. Copyright © 2011 John Wiley & Sons, Ltd.

This paper presents a Lyapunov-type inequality for the second order nonlinear equation (r(x)y')' + p(x)f(y(x)) = 0, with r(x), p(x) > 0 and f(y) odd and positive for y > O. It also compares it with similar results. (C) 2010 Elsevier Ltd. All rights reserved., This work has been partially supported by the Spanish M.C.Y.T. grant DPI2010-20891-C02-01.

For the differential equation u ″ = f ( t , u ) in regular as well as in singular cases there are established optimal sufficient conditions of existence for solutions satisfying nonlocal boundary conditions of the type ∫ a b u ( i − 1 ) ( s ) d φ i ( s ) = c i ( i = 1 , 2 ) .

The aim of this paper is to study the oscillation of the second order neutral differential equations (E)(r(t)[x(t)+p(t)x(τ(t))]′)′+q(t)x(σ(t))=0. The obtained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. The obtained comparison principles essentially simplify the examination of the studied equations.

This paper investigates the existence of positive solutions for a second-order differential system by using the fixed point theorem of cone expansion and compression.

In this paper, we study the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. By using the variant fountain theorem, we obtain a new criterion for guaranteeing that second-order Hamiltonian systems has infinitely many homoclinic solutions. Recent results from the literature are generalized and significantly improved. An example is also given in this paper to illustrate our main results.

In this paper we consider a second-order abstract problem involving derivatives of non-integer order. The existence and uniqueness of mild and classical solutions are established in appropriate spaces under weak assumptions.

In this work we study the Gevrey regularity of solutions to a general class of second order quasi-linear equations. Under some kind of sub-ellipticity conditions, we obtain the Gevrey regularity of weak solutions to these equations.

This paper contains two results on the L p L^p regularity problem on Lipschitz domains. For second order elliptic systems and 1 > p > ∞ 1>p>\infty , we prove that the solvability of the L p L^p regularity problem is equivalent to that of the L p ′ L^{p^\prime } Dirichlet problem. For higher order elliptic equations and systems, we show that if p > 2 p>2 , the solvability of the L p L^p regularity problem is equivalent to a weak reverse Hölder condition with exponent p p .

Following Moser’s method we shall give a sufficient condition for removability of singularity for an anisotropic elliptic equation of second order with nonstandard growth.

We establish a new oscillation criterion for the half-linear second order differential equation ( r ( t ) Φ ( x ′ ) ) ′ + c ( t ) Φ ( x ) = 0 , Φ ( x ) ≔ | x | p − 2 x , p > 1 , which improves a result given in [8] .

This paper is concerned with the oscillation of second-order nonlinear neutral dynamic equations of the form r ( t ) ( y ( t ) + p ( t ) y ( τ ( t ) ) ) Δ γ Δ + f ( t , y ( δ ( t ) ) ) = 0 , on a time scale T . By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation results which handle some cases not covered by known criteria.

We prove the H p 1 solvability of second order parabolic equations in divergence form with leading coefficients a i j measurable in ( t , x 1 ) and having small BMO (bounded mean oscillation) semi-norms in the other variables. Additionally we assume a 11 is measurable in x 1 and has small BMO semi-norms in the other variables. The corresponding results for the Cauchy problem are also established. Parabolic equations in Sobolev spaces H q , p 1 with mixed norms are also considered under the same conditions of the coefficients.

We prove the existence of so-called fast solutions of a second-order difference equation related to traveling wave solutions of Fisher–Kolmogorov's equation. Variational approach is used and the fast solutions are obtained as minimizers of an energy functional on a weighted Hilbert space. Numerical experiments are presented.

In this paper, we consider the Neumann boundary value problem { − ( p ( t ) u ′ ( t ) ) ′ + q ( t ) u ( t ) = f ( t , u ( t ) ) , t ∈ ( 0 , 1 ) u ′ ( 0 ) = u ′ ( 1 ) = 0 , where p ∈ C 1 [ 0 , 1 ] , p ( t ) > 0 , q ∈ C [ 0 , 1 ] , q ( t ) ≥ 0 , f ∈ C ( [ 0 , 1 ] × ( − ∞ , + ∞ ) , ( − ∞ , + ∞ ) ) . By using topological degree theory, we investigate the existence of nontrivial solutions. In particular, we prove the existence of positive solutions provided that q ( t ) > 0 .

We study the second order singular boundary value problem u ″ + λ f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = ∫ 0 1 u ( s ) d ξ ( s ) , u ( 1 ) = ∫ 0 1 u ( s ) d η ( s ) . Sufficient conditions are obtained for the existence and uniqueness of positive solutions. The dependence of positive solutions on the parameter λ is also studied. Moreover, application of our theory to a special problem is discussed. To prove our theorem, we utilize some results from the mixed monotone operator theory.

Some existence theorems are obtained for periodic solutions of nonautonomous second-order differential systems with ( q , p ) -Laplacian by using the least action principle and the minimax methods.

By refining the standard integral averaging technique, we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form(r(t)(x(t)+p(t)x([email protected]))^')^'+q(t)f(x(t),x(@s(t)))=0. Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those in the recent paper by Shi and Wang [18] and related contributions to the subject. Examples are provided to illustrate the relevance of new theorems.

In this paper we study the limiting behavior of the value-function for one-dimensional second order variational problems arising in continuum mechanics. The study of this behavior is based on the relation between variational problems on bounded large intervals and a limiting problem on [ 0 , ∞ ) .

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems u − L ( t ) u + W u ( t , u ) = 0 , ∀ t ∈ R , where L ( t ) is unnecessarily positive definite for all t ∈ R , and W ( t , u ) is of subquadratic growth as | u | → ∞ .

This paper is concerned with constructing optimal designs for rational models which are used for modeling problems in Agriculture and other disciplines. Homoscedastic and weighted models are considered. An analytical characterization of these designs is obtained as zeros of a polynomial solution of a second order differential equation.

We establish the existence of periodic solutions for the second order non-autonomous singular coupled systems { x ″ + a 1 ( t ) x = f 1 ( t , y ( t ) ) + e 1 ( t ) , y ″ + a 2 ( t ) y = f 2 ( t , x ( t ) ) + e 2 ( t ) . The proof relies on Schauder’s fixed point theorem. Some recent results in the literature are generalized and improved.

We consider the nonoscillatory half-linear second order differential equation (∗) ( r ( t ) Φ ( x ′ ) ) ′ + c ( t ) Φ ( x ) = 0 , Φ ( x ) = | x | p − 2 x , p > 1 , and we suppose that we know its principal solution x . We establish an asymptotic formula for nonprincipal solutions of (∗) and we use this asymptotic formula to obtain an oscillation criterion for (∗) .

We consider the boundary value problem with nonhomogeneous multi-point boundary condition u ″ + a ( t ) f ( u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = ∑ i = 1 m a i u ( t i ) + λ , u ( 1 ) = ∑ i = 1 m b i u ( t i ) + μ . A sufficient condition is obtained for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters λ and μ is also studied. Our work complements some results in the literature, especially those in our earlier papers [L. Kong, Q. Kong, Second-order boundary value problems with nonhomogeneous boundary conditions I, Math. Nachr. 278 (2005) 173–193; L. Kong, Q. Kong, Second-order boundary value problems with nonhomogeneous boundary conditions II, J. Math. Anal. Appl. 330 (2007) 1393–1411].

The existence of periodic solutions for second-order Duffing equations with p-Laplacian-like operator is studied by applying a new generalization of polar coordinates.

In this Note we analyze a reachability problem for an integro-differential equation by using a harmonic analysis approach. To cite this article: P. Loreti, D. Sforza, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

In this paper, by using the fixed point theorem in cones, the existence of positive solutions for systems of nonlinear second-order semipositone boundary value system { x ″ ( t ) + λ f 1 ( t , x ( t ) , y ( t ) ) + μ g 1 ( t , x ( t ) , y ( t ) ) = 0 , y ″ ( t ) + λ f 2 ( t , x ( t ) , y ( t ) ) + μ g 2 ( t , x ( t ) , y ( t ) ) = 0 , x ( 0 ) = x ( 1 ) = y ( 0 ) = y ( 1 ) = 0 , 0 ≤ t ≤ 1 , which nonlinearity f i and g i ( i = 1 , 2 ) are both semipositone, is obtained under the condition that f i ( i = 1 , 2 ) and g i ( i = 1 , 2 ) are suplinear or sublinear. The results obtained in this paper essentially improve and generalize many well-known results.

Consider the Emden–Fowler sublinear dynamic equation (0.1)xΔΔ(t)+p(t)f(x(σ(t)))=0, where p∈C(T,R), T is a time scale, f(x)=∑i=1maixβi, where ai>0, 00, c>α, and the sublinear q-difference equation xΔΔ(t)+b(−1)nt−cxα(qt)=0,01, b>0, c>1+α.

Let H,V be two real Hilbert spaces such that H ⊂ V with continuous and dense imbedding, and let F ∈ C1(V) be convex. By using differential inequalities, a close-to-optimal ultimate bound of the energy is obtained for solutions in C1(ℝ+,V) ∩ W2,∞loc(ℝ+,V’)u” + cu&rsquo + bu + ∇F(u) = ƒ(t) whenever ƒ ∈ L∞(ℝ,H).

Topological methods are developed for the solvability of vector second-order boundary value problems on noncompact intervals. The solutions are located in given sets and enjoy prescribed properties. The main theorem is supplied by two illustrating examples.

Modeling of many dynamic systems results in second-order delay differential systems. In this paper, we prove that two kinds of delay systems without impulses can be stabilized by proper impulsive control, respectively. Our results have improved and generated some of the known results found in the literature. We also give an example to illustrate the obtained results.

In this paper a necessary and sufficient condition is derived for all positive decreasing solutions of a half-linear second order difference equation to be rapidly varying of index -~. Relations with the standard classification of nonoscillatory solutions and with the notion of recessive solutions are also discussed. The results of this paper are complementary to those of a previous paper by the authors, and lead to a complete characterization of positive decreasing solutions with respect to their regularly or rapidly varying behavior.

It is shown that the empirical pseudo-second order kinetic equation is a very efficient formula to correlate the kinetic data generated by applying theoretical expressions developed from the fundamental SRT (Statistical Rate Theory) approach to the interfacial transport. This is especially true when the most popular linear representation is used in which time/adsorbed amount is plotted vs. time. However, the commonly observed goodness of such linear plots does not necessarily speak for the applicability of the pseudo-second order kinetic equation. A reliable estimation, for instance, of the equilibrium adsorbed amount is possible only when a substantial part of a kinetic isotherms corresponds to the conditions close to equilibrium. Energetic surface heterogeneity increases the goodness of these linear regressions. Then, experimental errors have only little effect on the pseudo-second linear plots.