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We establish the existence of fast positive semi-wavefront solutions to a delay integro-differential problem cu'(t)=J*u(t)-u(t) + f(u(t-h)), t ∈ R, u(-∞)=0; where the asymmetric kernel J is exponentially bounded, the nonlinearity f ∈ C1([0, + &#8734);R) is monostable, h ≥ 0, and c > 0. [ABSTRACT FROM AUTHOR]

The efficient conditions guaranteeing the existence of a T -periodic solution to the second order differential equation u ″ = h ( t ) g ( u ) are established in the paper. Here, g is a positive and decreasing function which has a strong singularity at the origin, and the weight h ∈ L ( R / T Z ) is a sign-changing function. The obtained results have the form of relation between the multiplicities of the zeroes of the weight function h and the order of the singularity of the nonlinear term. The approach is based on Leray–Schauder degree theory, proving that no T -periodic solution of a certain homotopy appears on the boundary of an unbounded open set during the deformation to an autonomous problem. [ABSTRACT FROM AUTHOR]

Efficient conditions guaranteeing the solvability of multi-point boundary value problems for linear functional-differential equations are established in this paper. The results are proved using the theorems on functional-differential inequalities. [ABSTRACT FROM AUTHOR]

We study the existence of almost periodic solutions for semi-linear abstract parabolic evolution equations with impulse action at state-dependent moments. In particular, we present conditions excluding the beating phenomenon in these systems. The main result is illustrated with an example of impulsive diffusive logistic equation. [ABSTRACT FROM AUTHOR]

The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. It turns out that the existence or not of such solutions depends on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter, we prove the existence of solutions to this boundary value problem. For large values of the same parameter, we prove the nonexistence of solutions. We also provide rigorous bounds for the values of this parameter, which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions applied to an associated two-point boundary value problem. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the existence of a solution to the Dirichlet boundary value problem for the second-order differential equation where the functions satisfy the local Carathéodory conditions and may have singularities both in the time (for and ) and the phase (for and ) variables. Sufficient conditions for the solvability of the above-mentioned problem are established. [ABSTRACT FROM AUTHOR]

We present the formal geometric derivation of a non-equilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of non-equilibrium statistical mechanics. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we continue the study of the periodic problem for the second-order equation , where is a Carathéodory function and are continuous functions on which may have singularities at zero. Both attractive and repulsive singularities are considered. The method relies on a novel technique of construction of lower and upper functions. As an application, we obtain new sufficient conditions for the existence of periodic solutions to the Rayleigh–Plesset equation. [Copyright &y& Elsevier]

Abstract: New criteria for the existence of a maximum or antimaximum principle of a general second order operator with periodic conditions, as well as conditions for nonresonance, are provided and compared with the related literature. [Copyright &y& Elsevier]

We derive some sufficient conditions for the existence of dispersion-managed solitons in a nonlinear Schrödinger equation with periodically varying coefficients. The proof relies on a combination of the variational method with the development of a novel upper and lower function method. [ABSTRACT FROM AUTHOR]

Abstract: Sufficient conditions for the existence of a solution to the problem are established. [Copyright &y& Elsevier]

Biological models inherently contain delay. Mathematical analysis of a delay-induced model is, however, more difficult compare to its non-delayed counterpart. Difficulties multiply if the model contains multiple delays. In this paper, we analyze a realistic HIV-1 infection model in the presence and absence of multiple delays. We consider self-proliferation of CD4+T cells, nonlinear saturated infection rate and recovery of infected cells due to incomplete reverse transcription in a basic HIV-1 in-host model and incorporate multiple delays to account for successful viral entry and subsequent virus reproduction from the infected cell. Both of delayed and non-delayed system becomes disease-free if the basic reproduction number is less than unity. In the absence of delays, the infected equilibrium is shown to be locally asymptotically stable under some parametric space and unstable otherwise. The system may show unstable oscillatory behaviour in the presence of either delay even when the non-delayed system is stable. The second delay further enhances the instability of the endemic equilibrium which is otherwise stable in the presence of a single delay. Numerical results are shown to be in agreement with the analytical results and reflect quite realistic dynamics observed in HIV-1 infected individuals. [ABSTRACT FROM AUTHOR]

In this paper we analyze a delay-induced predator–prey–parasite model with prey harvesting, where the predator–prey interaction is represented by Leslie–Gower type model with type II functional response. Infection is assumed to spread horizontally from one infected prey to another susceptible prey following mass action law. Spreading of disease is not instantaneous but mediated by a time lag to take into account the time required for incubation process. Both the susceptible and infected preys are subjected to linear harvesting. The analysis is accomplished in two phases. First we analyze the delay-induced predator–prey–parasite system in absence of harvesting and proved the local & global dynamics of different (six) equilibrium points. It is proved that the delay has no influence on the stability of different equilibrium points except the interior one. Delay may cause instability in an otherwise stable interior equilibrium point of the system and larger delay may even produce chaos if the infection rate is also high. In the second phase, we explored the dynamics of the delay-induced harvested system. It is shown that harvesting of prey population can suppress the abrupt fluctuations in the population densities and can stabilize the system when it exceeds some threshold value. [ABSTRACT FROM AUTHOR]