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In this short note we indicate some improvement of an article published in this journal by Crespo, Ivorra and Ramos [6]. The techniques used are connected with several smoothing effects associated with linear partial differential operators which give rise to some accretive operators in $L^1(\Omega )$, as well as with some $H^2(\Omega )$ estimates independent on time.

In this short note we indicate some improvement of an article published in this journal by Crespo, Ivorra and Ramos 6. The techniques used are connected with several smoothing effects associated with linear partial differential operators which give rise to some accretive operators in L¹(Ω), as well as with some H²(Ω) estimates independent on time. [ABSTRACT FROM AUTHOR]

We establish existence and multiplicity theorems for a Dirichlet boundary-value problem at resonance. This problem is a nonlinear subcritical perturbation of a linear eigenvalue problem studied by Cuesta, and includes a sign-changing potential. We obtain solutions using the Mountain Pass lemma and the Saddle Point theorem. Our paper extends some recent results of Goncalves, Miyagaki, and Ma.

Our objective in this paper is to develop and utilize a minimax principle for proving the existence of symmetric solutions for the stationary Navier-Stokes equations. Notwithstanding its application to symmetric solutions in this paper, our minimax principle is broad enough to capture other types of solutions provided the equation and the external force are compatible under a family of operations including but not limited to being invariant by compact groups. The subset of functions compatible under this family of operations is not required to be a linear subspace, and being a closed convex set suffices for our purpose. [ABSTRACT FROM AUTHOR]

In this article we study an equation driven by the nonlocal integrodifferential operator -LK in presence of an asymmetric nonlinear term f. Among the main results of the paper we prove the existence of at least a weak solution for this problem, under suitable assumptions on the asymptotic behavior of the nonlinearity f at 1. Moreover, we show the uniqueness of this solution, under additional requirements on f. We also give a non-existence result for the problem under consideration. All these results were obtained using variational techniques and a monotonicity property of the eigenvalues of -LK with respect to suitable weights, that we prove along the present paper. This monotonicity property is of independent interest and represents the nonlocal counterpart of a famous result obtained by de Figueiredo and Gossez [14] in the setting of uniformly elliptic operators. [ABSTRACT FROM AUTHOR]

In this article we prove the existence of solutions to the integrodifferential equation of mixed type x ∆(t) = f t, x(t), Z t 0 k1(t, s)g(s, x(s))∆s, Z a 0 k2(t, s)h(s, x(s))∆s, x(0) = x0, x0, E, t, Ia = [0, a] ∩ T, a > 0, where T denotes a time scale (nonempty closed subset of real numbers R), Ia is a time scale interval. In the first part of this paper functions f, g, h are Carathéodory functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil delta integrals, which generalizes the Henstock-Kurzweil integrals. In the second part f, g, h, x are weaklyweakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis delta integrals. Additionally, functions f, g, h satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. [ABSTRACT FROM AUTHOR]

We consider a perforated domain Ω(ε) of R² with a small hole of size ε and we study the behavior of the solution of a mixed Neumann-Robin problem in Ω(ε) as the size ε of the small hole tends to 0. In addition to the geometric degeneracy of the problem, the nonlinear ε-dependent Robin condition may degenerate into a Neumann condition for ε = 0 and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as ε tends to 0 and to understand how the boundary condition affects the behavior of the solutions when ε is close to 0. The present paper extends to the planar case the results of [36] dealing with the case of dimension n ≥ 3. [ABSTRACT FROM AUTHOR]

In this article, the authors consider a fractional p-Laplacian elliptic Dirichlet problem. Using critical point theory and the variational method, they investigate the existence of at least one, two, and three solutions to the problem. Examples illustrating the results are interspaced in the paper. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of solutions for the nonlocal perturbed Kirchhoff problem − (a + b ∫Ω |∇u|² dx) ∆u = λg(x, u) + f(x, u), in Ω, u = 0, on ∂Ω, where Ω is a bounded smooth domain in RN, N > 4, a, b, λ > 0, and f, g : Ω × R → R are Carathéodory functions, with f subcritical, and g of arbitrary growth. This paper is motivated by a recent results by Faraci and Silva [4] where existence and multiplicity results were obtained when g is subcritical and f is a power-type function with critical exponent. [ABSTRACT FROM AUTHOR]

This article concerns the positive solutions of a boundary-value problem constituted by a linear elliptic partial differential equation, subject to nonlinear mixed boundary conditions containing spatial heterogeneities with arbitrary sign along the boundary. The results obtained in this work provide us the global bifurcation diagram of positive solutions, the pointing behavior of them when the parameters change and the dynamics of the positive solutions of the associated parabolic problem. The main contribution of this paper is to give general results about existence, uniqueness, stability and pointing behavior of positive solutions, for boundary-value problems with nonlinear boundary conditions of mixed type containing spatial heterogeneities. The main technical tools used to develop the mathematical analysis are local and global bifurcation, monotonicity techniques, the Characterization of the Strong Maximum Principle given by Amann and López-Gómez [5] blow up arguments and some of the techniques used in the previous works [19,20,33,34]. The results obtained in this paper are the natural continuation of the previous ones in [11]. [ABSTRACT FROM AUTHOR]

In this article, we discuss the existence of positive solution to a nonlinear p-Laplacian fractional differential equation whose nonlinearity contains a higher-order derivative ... D0+α, D0+β, are the standard Riemann-Liouville fractional derivatives, and f ∈ C((0; 1) x [0+∞)n-1, [0, +∞)). The Green's function of the fractional differential equation mentioned above and its relevant properties are presented, and some novel results on the existence of positive solution are established by using the mixed monotone fixed point theorem and the upper and lower solution method. The interesting of this paper is that the nonlinearity involves the higher-order derivative, and also, two examples are given in this paper to illustrate our main results from the perspective of application. [ABSTRACT FROM AUTHOR]

In this article we study applications of integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive wide range of transmutation operators by this method. Classical integral transforms are involved in the integral transforms composition method (ITCM) as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace and some generalized transforms. The ITCM and transmutations obtaining by it are applied to deriving connection formulas for solutions of singular differential equations and more simple non-singular ones. We consider well-known classes of singular differential equations with Bessel operators, such as classical and generalized Euler-Poisson-Darboux equation and the generalized radiation problem of A. Weinstein. Methods of this paper are applied to more general linear partial differential equations with Bessel operators, such as multivariate Bessel-type equations, GASPT (Generalized Axially Symmetric Potential Theory) equations of Weinstein, Bessel-type generalized wave equations with variable coefficients, ultra B-hyperbolic equations and others. So with many results and examples the main conclusion of this paper is illustrated: the integral transforms composition method (ITCM) of constructing transmutations is very important and effective tool also for obtaining connection formulas and explicit representations of solutions to a wide class of singular differential equations, including ones with Bessel operators. [ABSTRACT FROM AUTHOR]

This paper is an addendum to our earlier paper [8], where a systematic study of quadratic systems of second order ordinary differential equa- tions defined in commutative algebras was presented. Here we concentrate on special solutions and energy considerations of some quadratic systems defined in algebras which need not be commutative, however, we shall throughout assume the algebra to be associative. We here also give a positive answer to an open question, concerning periodic motions of such systems, posed in our earlier paper. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with a class of nonlinear partial differential elliptic equations with Dirichlet boundary data. The key feature of this paper consists in competition effects of two generalized differential operators, which extend the standard operators with variable exponent. This class of problems is motivated by phenomena arising in non-Newtonian uids or image reconstruction, which deal with operators and nonlinearities with variable exponents. We establish an existence property in the framework of small perturbations of the reaction term with indefinite potential. The mathematical analysis developed in this paper is based on the theory of anisotropic function spaces. Our analysis combines variational arguments with energy estimates. [ABSTRACT FROM AUTHOR]

In this article, we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler-Poisson-Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter k, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations. [ABSTRACT FROM AUTHOR]

Dipole-type potential arises in the area of nonrelativistic molecular physics. In this paper, we establish the existence and nonexistence of solution to critical Kirchhoff-type equation with dipole-type potential. [ABSTRACT FROM AUTHOR]

Age structure and cell-to-cell transmission are two major infection mechanisms in modeling spread of infectious diseases. We propose an age-structured viral infection model with latency, infection age-structure and cell-to-cell transmission. This paper aims to reveal the basic reproduction number and prove it to be a sharp threshold determining whether the infection dies out or not. Mathematical analysis is presented on relative compactness of the orbit, existence of a global attractor, and uniform persistence of system. We further investigate local and global stability of the infection-free and infection equilibrium. [ABSTRACT FROM AUTHOR]

In this article we introduce a concept of dual problems in metric spaces. Then we state and prove an equivalence result concerning their well-posedness with respect to appropriate Tykhonov triples. We exemplify this result in the study of a history-dependent variational inequality with time-dependent constraints, for which the dual problem is in a form of a history-dependent inclusion. This allows us to deduce a convergence result which provides the continuous dependence of the solution with respect to the data. We end this paper with an example which represents an evidence of our abstract results. [ABSTRACT FROM AUTHOR]

We present a collage-based method for solving inverse problems for elliptic partial differential equations on a perforated domain. The main results of this paper establish a link between the solution of an inverse problem on a perforated domain and the solution of the same model on a domain with no holes. The numerical examples at the end of the paper show the goodness of this approach. [ABSTRACT FROM AUTHOR]

This article concerns the positive solutions of a boundary-value problem constituted by a linear elliptic partial differential equation, subject to nonlinear mixed boundary conditions containing spatial heterogeneities with arbitrary sign along the boundary. The results obtained in this work provide us the global bifurcation diagram of positive solutions, the pointing behavior of them when the parameters change and the dynamics of the positive solutions of the associated parabolic problem. The main contribution of this paper is to give general results about existence, uniqueness, stability and pointing behavior of positive solutions, for boundary-value problems with nonlinear boundary conditions of mixed type containing spatial heterogeneities. The main technical tools used to develop the mathematical analysis are local and global bifurcation, monotonicity techniques, the Characterization of the Strong Maximum Principle given by Amann and Lopez-Gomez [5] blow up arguments and some of the techniques used in the previous works [19,20,33,34]. The results obtained in this paper are the natural continuation of the previous ones in [11].

In this article we study applications of integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive wide range of transmutation operators by this method. Classical integral transforms are involved in the integral transforms composition method (ITCM) as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace and some generalized transforms. The ITCM and transmutations obtaining by it are applied to deriving connection formulas for solutions of singular differential equations and more simple non-singular ones. We consider well-known classes of singular differential equations with Bessel operators, such as classical and generalized Euler-Poisson-Darboux equation and the generalized radiation problem of A.Weinstein. Methods of this paper are applied to more general linear partial differential equations with Bessel operators, such as multivariate Bessel-type equations, GASPT (Generalized Axially Symmetric Potential Theory) equations of Weinstein, Bessel-type generalized wave equations with variable coefficients,ultra B-hyperbolic equations and others. So with many results and examples the main conclusion of this paper is illustrated: the integral transforms composition method (ITCM) of constructing transmutations is very important and effective tool also for obtaining connection formulas and explicit representations of solutions to a wide class of singular differential equations, including ones with Bessel operators.

Porcine pseudorabies is an acute and highly contagious viral disease caused by the pseudorabies virus. It inflicts enormous losses to the pig-breeding industry. In this paper, we propose an age-structured mathematical model. We investigate the dynamics of this model characterized by the basic reproduction number R0=max{R01, R02} by addressing the existence and global stability of equilibria. When R0<1, the disease-free equilibrium is unique and globally asymptotically stable. The boundary equilibrium exists and is globally asymptotically stable under the condition R01<1 and R02>1 or R01>1 and R02<1+ε. If both R01>1 and R0>1+ε, there is a unique disease-endemic equilibrium which is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap between necessary and sufficient conditions for solvability of the Dirichlet problem. [ABSTRACT FROM AUTHOR]

In this article, we study a problem on propagation of coupled porosity, fluid concentration flux, and temperature waves. We use a model formulated in previous papers for porous media saturated by a fluid flow, in the framework of non-equilibrium thermodynamics. We derive three modes of propagation in the one-dimensional and perfect isotropic case, and then we test the validity of the model. The waves propagation velocities are represented in diagrams as functions of the wave number. The derived results have applications in technological sectors such as seismology, medical sciences, geology and nanotechnology. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded smooth domain in ℝN. We study the existence of positive solutions to the Dirichlet problem ... where 10. In particular, we answer to some questions posed in the recent paper [3] where this problem was considered. [ABSTRACT FROM AUTHOR]

In this article, we consider a diffusion system with the Belousov- Zhabotinskii (BZ for short) chemical reaction. The existence and stability of V-shaped traveling fronts for the BZ system in R² had been proved in our previous papers [30, 31]. Here we establish the existence and stability of pyramidal traveling fronts for the BZ system in R³. [ABSTRACT FROM AUTHOR]

In a recent book co-authored by the authors of this article, we studied by semigroup theory methods several classical evolution equations, including the heat and Laplace equations, with real time variable and complex spatial variable, under the hypothesis that the boundary function belongs to the space of analytic functions in the open unit disk and continuous in the closed unit disk, endowed with the uniform norm. Also, in a subsequent paper, the authors have extended the results for the heat and Laplace equations in weighted Bergman spaces on the unit disk. The purpose of this article is to show that the semigroup theory methods work for these two evolution equations of complex spatial variables, under the hypothesis that the boundary function belongs to the weighted Fock space on C, Fpα(C), with 1 ≤ p < +∞, endowed with the Lp-norm. Also, the case of several complex variables is considered. The proofs use the Jensen's inequality, Fubini's theorem for integrals and the Lp-integral modulus of continuity. [ABSTRACT FROM AUTHOR]

This article concerns the existence and uniqueness of a time periodic solution for the non-isentropic quantum hydrodynamic equations with viscosity. By applying the Leray-Schauder theory, subtle energy estimates and a limiting method, we obtain the existence of time periodic solutions under some smallness assumptions on the time periodic external force in R³. The uniqueness can be proved by similar energy estimates. In particular, the quantum effects and the energy equation are taken into account in this paper which play a significant role in the uniform (in the domain R and the positive constant) estimates, especially in the selection of the norm. [ABSTRACT FROM AUTHOR]

A polynomial differential system of degree 2 has no global centers (that is, centers defined in all the plane except the fixed point). In this paper we characterize the global centers of cubic Hamiltonian systems symmetric with respect to the x-axis, and such that the center has purely imaginary eigenvalues. [ABSTRACT FROM AUTHOR]

In previous years the study of the version of Hilbert's 16th problem for piecewise linear differential systems in the plane has increased. There are many papers studying the maximum number of crossing limit cycles when the differential system is defined in two zones separated by a straight line. In particular in [11, 13] it was proved that piecewise linear differential centers separated by a straight line have no crossing limit cycles. However in [14, 15] it was shown that the maximum number of crossing limit cycles of piecewise linear differential centers can change depending of the shape of the discontinuity curve. In this work we study the maximum number of crossing limit cycles of piecewise linear differential centers separated by a conic. [ABSTRACT FROM AUTHOR]

The solution of the biharmonic equation with an homogeneous boundary conditions is decomposed into a regular part and a singular one. The later is written as a coefficient multiplied by the first singular function associated to the bilaplacian operator. In this paper, we consider the dual singular method for finding the value of the leading singular coefficient, and we use the mortar domain decomposition technique with the spectral discretization for its approximation. The numerical analysis leads to optimal error estimates. We present some numerical results which are in perfect coherence with the analysis developed in this paper.

This paper is an addendum to our earlier paper [8], where a systematic study of quadratic systems of second order ordinary differential equations defined in commutative algebras was presented. Here we concentrate on special solutions and energy considerations of some quadratic systems defined in algebras which need not be commutative, however, we shall throughout assume the algebra to be associative. We here also give a positive answer to an open question, concerning periodic motions of such systems, posed in our earlier paper.

In this article, we are concerned with a class of nonlinear partial differential elliptic equations with Dirichlet boundary data. The key feature of this paper consists in competition effects of two generalized differential operators, which extend the standard operators with variable exponent. This class of problems is motivated by phenomena arising in non-Newtonian fluids or image reconstruction, which deal with operators and nonlinearities with variable exponents. We establish an existence property in the framework of small perturbations of the reaction term with indefinite potential. The mathematical analysis developed in this paper is based on the theory of anisotropic function spaces. Our analysis combines variational arguments with energy estimates.

In this article, we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler-Poisson-Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter k, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations.

In this paper, we apply Feng's first-integral method to study trav- eling wave solutions to a two-component generalization of the Ostrovsky sys- tem. We convert the two-component generalization of the Ostrovsky system to an equivalent autonomous system. Then we use the Divisor Theorem of two variables in the complex domain to seek the polynomial first-integral to this autonomous system. Through analyzing the derived first-integral, we obtain traveling wave solutions to the two-component generalization of the Ostrovsky system under certain parametric conditions. [ABSTRACT FROM AUTHOR]

In this article, we study a class of generalized extensible beam equations with a superlinear nonlinearity ∆²u-M(||∇u||²L²)∆u+λV (x)u=f(x, u) in RN,u∈H²(RN),where N≥3, M(t) = atδ+bwith a, δ > 0 and b∈R,λ > 0 is a parameter, V∈C(RN,R)and f∈C(RN×R,R).Unlike most other papers on this problem, we allow the constant bto be nonpositive, which has the physical significance. Under some suitable assumptionson V(x) and f(x, u), when ais small and λis large enough, we prove the existence oftwo nontrivial solutions u(1)a,λ and u(2)a,λ, one of which will blow up as the nonlocal termvanishes. Moreover, u(1)a,λ →u(1)∞and u(2)a,λ →u(2)∞strongly in H²(RN) as λ→ ∞, whereu(1)∞6=u(2)∞∈H²0(Ω) are two nontrivial solutions of Dirichlet BVPs on the bounded domainΩ. It is worth noting that the regularity of weak solutions u(i)∞(i= 1,2) here is explored. Finally, the nonexistence of nontrivial solutions is also obtained for alarge enough. [ABSTRACT FROM AUTHOR]

Let T be a strongly continuous semigroup acting on a complex Banach space X and let A be its infinitesimal generator. It is well-known [29, 33] that the uniform spectral bound s0(A) of the semigroup T is negative provided that all solutions to the Cauchy problems u(t) = Au(t) + eiμtx;t ≥ 0; u(0) = 0; are bounded (uniformly with respect to the parameter μ ∊ R). In particular, if X is a Hilbert space, then this yields all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid. Let X denote the space of all continuous and 1-periodic functions f : B → X whose sequence of Fourier-Bohr coefficients (cm(f))m∊Z belongs to ℓ¹(Z,X). Endowed with the norm ||f||1 := k(cm(f))m∊Zk1 it becomes a non-reexive Banach space [15]. A subspace AT of X (related to the pair (T;X)) is introduced in the third section of this paper. We prove that the semigroup T is uniformly exponentially stable provided that in addition to the above-mentioned boundedness condition, AT = X. An example of a strongly continuous semigroup (which is not uniformly continuous) and fulfills the second assumption above is also provided. Moreover an extension of the above result from semigroups to 1-periodic and strongly continuous evolution families acting in a Banach space is given. We also prove that the evolution semigroup T associated with T on X does not verify the spectral determined growth condition. More precisely, an example of such a semigroup with uniform spectral bound negative and uniformly growth bound non-negative is given. Finally we prove that the assumption AT = X is not needed in the discrete case. [ABSTRACT FROM AUTHOR]

The symmetry breaking phenomenon (SBP) to the Hénon equation was first numerically observed in [4] and then theoretically verified on the unit ball Bn in [8]. Some results on the asymptotic behavior of the ground states to the Hénon equation on Bn are presented in [2, 3, 7]. [8] further discussed SBP to the p-Hénon equation and obtained some results with special value p n on Bn. To inspire theoretical study on more general p, a series of numerical experiments to the p-Hénon equation on a disk and a square are carried out in this paper. Numerical computations are made by the minimax method developed in [9, 10]. Then, SBP, a peak break phenomenon (PBP); i.e., a 1-peak solution, which is symmetric about two axes and two diagonal lines, breaks its peak from 1 to 4, and a 1-peak positive non-ground state solution, which is only symmetric about one axis, on the square are numerically captured and visualized. The peak point and the peak height of the ground states are carefully calculated to study their asymptotic behavior. Several conjectures are made based on the numerical observations to stimulate theoretical analysis. Two of them are proved in this paper. [ABSTRACT FROM AUTHOR]

This article is a sequel to two earlier articles (one of them written jointly with R. Banuelos) on stability results for the Neumann eigenvalues and eigenfunctions of domains in R² with a snowflake type fractal boundary. In particular we want our results to be applicable to the Koch snowflake domain. In the two earlier papers we assumed that a domain Ω ⊆ R² which has a snowflake type boundary is approximated by a family of subdomains and that the Neumann heat kernel of Ω and those of its approximating subdomains satisfy a uniform bound for all sufficiently small t > 0. The purpose of this paper is to extend the results in the two earlier papers to the situations where the approximating domains are not necessarily subdomains of Ω. We then apply our results to the Koch snowflake domain when it is approximated from outside by a decreasing sequence of polygons. [ABSTRACT FROM AUTHOR]

In this note we give a proof of the existence of a solution to the Riemann problem in one-dimensional gasdynamics. Lax's 1957 paper on conservation laws leaves no doubt that such a solution exists, but it seems to us that there may be interest in a brief and explicit proof favorable to numerical computations. Our procedure also allows us to give a simple characterization of those problems in which a given wave is a shock or a rarefaction wave. In the final section we prove a result of Von Neumann's concerning the overtaking of two shocks. This paper was written in 1969 and is being published now at the suggestion of Jerry Goldstein, whose editorial note is included. [ABSTRACT FROM AUTHOR]

In this paper we obtain the existence of quadratic-mean almost periodic solutions to some classes of partial hyperbolic stochastic differential equations. The main result of this paper generalizes in a natural fashion some recent results by authors. As an application, we consider the existence of quadratic-mean almost periodic solutions to the stochastic heat equation with divergence terms. [ABSTRACT FROM AUTHOR]

A method to study asymptotic properties of solutions to systems of differential equations with distributed time-delays and Boolean-type non-linearities (step functions) is offered. Such systems arise in many applications, but this paper deals with specific examples of such systems coming from genetic regulatory networks. A challenge is to analyze stable stationary points which belong to the discontinuity set of the system (thresholds). The paper describes an algorithm of localizing stationary points in the presence of delays as well as stability analysis around such points. The basic technical tool consists in replacing step functions by special smooth functions ("the tempered nonlinearities") and investigating the systems thus obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study a final value problem for the nonlinear parabolic equation ut + Au = h(u(t), t), 0 < t < T u(T) = φ, where A is a non-negative, self-adjoint operator and h is a Lipchitz function. Using the stabilized quasi-reversibility method presented by Miller, we find optimal perturbations, of the operator A, depending on a small parameter ε to setup an approximate nonlocal problem. We show that the approximate problems are well-posed under certain conditions and that their solutions converges if and only if the original problem has a classical solution. We also obtain estimates for the solutions of the approximate problems, and show a convergence result. This paper extends the work by Hetrick and Hughes [11] to nonlinear ill-posed problems. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions so that every solution of (y(t) -- p(t)y(r(t)))(n) + q(t)G(y(g(t))) -- u(t)H(y(h(t))) = f(t) oscillates or tends to zero as t → ∞, where n is an integer n ≥ 2, q > 0, u ≥ 0. Both bounded and unbounded solutions are considered in this paper. The results hold also when u = 0, f(t) Ξ 0, and G(u) Ξ u. This paper extends and generalizes some recent results. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear higher-order three-point eigenvalue problems with first-order derivative on time scales. Under certain growth conditions on the nonlinearity, sufficient conditions for existence and uniqueness of nontrivial solutions, which are easily verifiable, are obtained by using the Leray-Schauder nonlinear alternative. The conditions used in the paper are different from those in [4, 10, 21]. To show applications of our main results, we present some examples. [ABSTRACT FROM AUTHOR]

In this paper, first we prove the existence of global solutions in Sobolev spaces for the initial boundary value problem of the wave equation of φ-Laplacian with a general dissipation of the form (|u'|l-2u')' - Δφu + σ(t)g(u') = 0 in Ω x ℝ+, where Δφ = n ∑ i=1n ∂xi(φ(|∂xi|²)∂xi). Then we prove general stability estimates using multiplier method and general weighted integral inequalities proved by the second author in [18]. Without imposing any growth condition at the origin on g and φ, we show that the energy of the system is bounded above by a quantity, depending on φ, σ and g, which tends to zero (as time approaches infinity). These estimates allows us to consider large class of functions g and φ with general growth at the origin. We give some examples to illustrate how to derive from our general estimates the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize many existing results in the literature, and generate some interesting open problems. [ABSTRACT FROM AUTHOR]

We investigate stability and approximation properties of the lowest nonzero eigenvalue and corresponding eigenfunction of the Neumann Laplacian on domains satisfying a heat kernel bound condition. The results and proofs in this paper will be used and extended in a sequel paper to obtain stability results for domains in R2 with a snowflake type boundary. [ABSTRACT FROM AUTHOR]

When one considers a quadratic differential system, one realizes that it depends on 12 parameters of which one can be fixed by means of a time change. One also can notice that fixing 4 finite real singular points plus 3 infinite real ones (all its possible singular points) implies to fix 11 conditions, that is, 11 equations that the parameters must satisfy. Since these conditions are linear with respect to the parameters, it is obvious to think that the system will be determined, except that the fixed conditions are incompatible with a quadratic differential system having finitely many singular points. In this paper we prove exactly this. That is, if we fix the position of the 7 singular points of a quadratic differential system in a distribution that does not force an infinite number of finite singular points, then the system is completely determined, and consequently its phase portrait is also determined. This determination includes the local behavior of all singular points, even if they are weak focus or centers, the global behavior of separatrices, and even the existence or not of limit cycles. This also implies that limit cycles are sensitive to small perturbations of the coordinates of singular points, even if they are far from the singular points. The result of the paper goes far beyond this, since we state that this result is independent of the fact that the fixed singular points are real or complex, and it does not mind if the infinite singular points are simple or multiple due to the collision of several infinite singular points. Only when some data is lost due to the collision of finite singular points or to the collision of some finite singular points with infinite ones, this adds free parameters to the set of parameters at the same rate than the number of finite singular points are lost. [ABSTRACT FROM AUTHOR]

The Riccati equation has been known since the early 1700s. Numerous papers have been written on the solvability of its special cases. However, to the best of our knowledge, there are no papers that investigate the exact (equation specific) conditions for unbounded growth in finite time of solutions for Riccati initial-value problems. In this paper, we first derive conditions that are necessary and sufficient for the solutions of Riccati problems with constant coefficients to grow unbounded in finite time. Then we use a comparison method to extend these results to Riccati problems with variable coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we consider higher-order Hopfield neural networks (HHNNs) with time-varying delays. Based on the fixed point theorem, Lyapunov functional method, differential inequality techniques, and without assuming the boundedness on the activation functions, we establish sufficient conditions for the existence and local exponential stability of the almost periodic solutions. The results of this paper are new and they complement previously known results. [ABSTRACT FROM AUTHOR]