Gheorghe Moroşanu

Pest regulation by means of impulsive controls

In this paper, we consider an integrated pest management model which is impulsively controlled by means of biological and chemical controls. These controls are assumed to act in a periodic fashion, a nonlinear incidence rate being used to account for the dynamics of the disease caused by the application of the biological control. The Floquet theory for impulsive ordinary differential equations is employed to obtain a condition in terms of an inequality involving the total action of the nonlinear force of infection in a period, under which the susceptible pest-eradication solution is globally asymptotically stable. If the opposite inequality is satisfied, then it is shown that the system under consideration becomes uniformly persistent. A biological interpretation of the persistence condition is also provided.

A proximal point algorithm converging strongly for general errors

In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that Rockafellar (SIAM J Control Optim 14:877–898, 1976) assumed summability for the error sequence to derive weak convergence of PPA in its initial form, and this restrictive condition on errors has been extensively used so far for different versions of PPA. Thus this Note provides a solution to a long standing open problem and in particular offers new possibilities towards the approximation of the minimum points of convex functionals.

Impulsive perturbations of a three-trophic prey-dependent food chain system

The dynamics of an impulsively controlled three-trophic food chain system with general nonlinear functional responses for the intermediate consumer and the top predator are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive controls act in a periodic fashion, the constant impulse (the biological control) and the proportional impulses (the chemical controls) acting with the same period, but not simultaneously. Sufficient conditions for the global stability of resource and intermediate consumer-free periodic solution and of the intermediate consumer-free periodic solution are established, the latter corresponding to the success of the integrated pest management strategy from which our food chain system arises. In this regard, it is seen that, theoretically speaking, the control strategy can be always made to succeed globally if proper pesticides are employed, while as far as the biological control is concerned, its global effectiveness can also be reached provided that the top predator is voracious enough or the (constant) number of top predators released each time is large enough or the release period is small enough. Some situations which lead to chaotic behavior of the system are also investigated by means of numerical simulations.

Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions

We study a boundary value problem of the type in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in (N≥ 3) with smooth boundary and the functions are of the type with , (i = 1, …, N). Combining the mountain pass theorem of Ambrosetti and Rabinowitz and Ekelands variational principle we show that under suitable conditions the problem has two non-trivial weak solutions.

Functional methods in differential equations

INTRODUCTION Function and Distribution Spaces Monotone Operators, Convex Functions, and Subdifferentials Some Elements of Spectral Theory Linear Evolution Equations and Semigroups Nonlinear Evolution Equations ELLIPTIC BOUNDARY VALUE PROBLEMS Non-Degenerate Elliptic Boundary Value Problems Degenerate Elliptic Boundary Value Problems PARABOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC BOUNDARY CONDITIONS Homogeneous Boundary Conditions Nonhomogeneous Boundary Conditions Higher Regularity of Solutions PARABOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC-DIFFERENTIAL BOUNDARY CONDITIONS Homogeneous Boundary Conditions Nonhomogeneous Boundary Conditions Higher Regularity of Solutions HYPERBOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC BOUNDARY CONDITIONS Existence, Uniqueness and Long-Time Behavior of Solutions Higher Regularity of Solutions HYPERBOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC-DIFFERENTIAL BOUNDARY CONDITIONS Existence, Uniqueness and Long-Time Behavior of Solutions Higher Regularity of Solutions THE FOURIER METHOD FOR ABSTRACT DIFFERENTIAL EQUATIONS First Order Linear Equations Semilinear First Order Equations Second Order Linear Equations Semilinear Second Order Equations THE SEMIGROUP APPROACH FOR ABSTRACT DIFFERENTIAL EQUATIONS Semilinear Fist Order Equations Hyperbolic Partial Differential Systems with Nonlinear Boundary Conditions NONLINEAR NON-AUTONOMOUS ABSTRACT DIFFERENTIAL EQUATIONS First Order Differential and Functional Equations Containing Subdifferentials An Application IMPLICIT NONLINEAR ABSTRACT DIFFERENTIAL EQUATIONS Existence of Solution Uniqueness of Solution Continuous Dependence of Solution Existence of Periodic Solutions

Second order difference equations of monotone type

Our aim is to investigate the existence of solutions to some second order difference equations of monotone type. Theorems 1.1 and 1.2 below are the discrete versions of some existence results due to V. Barbu [1] corresponding to the continuous case.

Equations involving a variable exponent Grushin-type operator

In this paper we define a Grushin-type operator with a variable exponent growth and establish existence results for an equation involving such an operator. A suitable function space setting is introduced. Regarding the tools used in proving the existence of solutions for the equation analysed here, they rely on the critical point theory combined with adequate variational techniques.

The coupling of hyperbolic and elliptic boundary value problems with variable coefficients

We consider a one-dimensional coupled problem for elliptic second-order ODEs with natural transmission conditions. In one subinterval, the coefficient e > 0 of the second derivative tends to zero. Then the equation becomes there hyperbolic and the natural transmission conditions are not fulfilled anymore. The solution of the degenerate coupled problem with a flux transmission condition is corrected by an internal boundary layer term taking into account the viscosity e. By using singular perturbation techniques, we show that the remainders in our first-order asymptotic expansion converge to zero uniformly. Our analysis provides an a posteriori correction procedure for the numerical treatment of exterior viscous compressible flow problems with coupled Navier-Stokes/Euler models.

EXISTENCE FOR A NONLINEAR HYPERBOLIC SYSTEM

(1.3) Here L, C and IX are some positive constants. Problems of this type occur in theory of the electrical transmission lines (the equation of telegraphy) and hydraulics (the equation of “water hammer”). Existence, uniqueness and asymptotic behaviour for such problems have been studied by many authors. We mention in this context [l&4] and refer the reader to [5] for further references and significant results. In [6] the first author has proved existence and uniqueness of solutions for general boundary value conditions of local type. The attempt made in [7] to solve problems of the form (l.lt(1.3) failed since the abstract existence scheme developed there does not cover global boundary value conditions of the form (1.3). The main result of this paper, Theorem 1 below, gives existence and uniqueness of a strong solution to system (1.1 t( 1.3) under the following assumptions : (H,) Each of the real-valued functions u + ,4(x, u) and u -+ B(x, U) is continuous and mono- tone increasing on

The study of the evolution of some self‐organized chemical systems

Starting from a self‐organized chemical model with three chemical variables endowed with replication properties, we first get a model that includes the new species X4 with autocatalytic properties. Then we get another model where the additional species X5 appears leading to a model with six chemical variables. We prove that all these models are self‐organized.