State-dependent Impulses Research Articles

A pest control model with state-dependent impulses

In this paper, a pest control model with state-dependent impulses is firstly established, which relies on releasing of natural enemies, together with spraying pesticides. By using the successor function of differential equation geometry rules, the existence of order one periodic solution is discussed. According to the Analogue of Poincaré's Criterion, the orbitally asymptotic stability of the order one periodic solution is obtained. Furthermore, we investigated the global attractor of the system. From a biological point of view, our results indicate that: (1) the pest population can be controlled below some threshold; (2) compared to single measure, it is more efficient to take two measures for reducing the level of the pests.

Periodic Solutions Generated by Impulses for State-Dependent Impulsive Differential Equation

We study the existence of periodic solutions for a class of state-dependent impulsive differential systems via geometrical analysis methods. Our results show that these periodic solutions are generated by impulses. Moreover, numerical simulations are used to examine the existence of the periodic solutions.

Fixed point problem associated with state-dependent impulsive boundary value problems

The paper investigates a fixed point problem in the space which is connected to boundary value problems with state-dependent impulses of the form , a.e. , , . Here, the impulse instants are determined as solutions of the equations , . We assume that , , the vector function f satisfies the Carathéodory conditions on , the impulse functions , , are continuous on , and the barrier functions , , are continuous on . The operator ℓ is an arbitrary linear and bounded operator on the space of left-continuous regulated on vector valued functions and is represented by the Kurzweil-Stieltjes integral. Provided the data functions f and are bounded, transversality conditions which guarantee that this fixed point problem is solvable are presented. As a result it is possible to realize the construction of a solution of the above impulsive problem. MSC: 34B37, 34B10, 34B15.

Existence principle for higher-order nonlinear differential equations with state-dependent impulses via fixed point theorem

The paper provides an existence principle for a general boundary value problem of the formn=0 aj(t)u (j) (t )= h(t, u(t), ... , u (n-1) (t)), a.e. t ∈ (a, b) ⊂ R, � k(u, u � , ... , u (n-1) )= ck, k =1 , ... , n, with the state-dependent impulses u (j) (t+) - u (j) (t-) = Jij(u(t-), u � (t-), ... , u (n-1) (t-)), where the impulse points t are determined as solutions of the equations t = γi(u(t-), u � (t-), ... , u (n-2) (t-)), i =1 , ... , p, j =0 , ... , n -1 . Here,n, p ∈ N, c1, ... , cn ∈ R,

Second order BVPs with state dependent impulses via lower and upper functions

AbstractThe paper deals with the following second order Dirichlet boundary value problem with p ∈ ℕ state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ i+) − z′(τ i−) = I i(τ i, z(τ i)), τ i = γ i(z(τ i)), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.

First-order nonlinear differential equations with state-dependent impulses

The paper deals with the state-dependent impulsive problem z ′ ( t ) = f ( t , z ( t ) ) for a.e. t ∈ [ a , b ] , z ( τ + ) − z ( τ ) = J ( τ , z ( τ ) ) , γ ( z ( τ ) ) = τ , ℓ ( z ) = c 0 , where [a,b]⊂R, c 0 ∈R, f fulfils the Carathéodory conditions on [a,b]×R, the impulse function is continuous on [a,b]×R, the barrier function γ has a continuous first derivative on some subset of ℝ and ℓ is a linear bounded functional which is defined on the Banach space of left-continuous regulated functions on [a,b] equipped with the sup-norm. The functional ℓ is represented by means of the Kurzweil-Stieltjes integral and covers all linear boundary conditions for solutions of first-order differential equations subject to state-dependent impulse conditions. Here, sufficient and effective conditions guaranteeing the solvability of the above problem are presented for the first time.MSC:34B37, 34B15.

A new approach to BVPs with state-dependent impulses

The paper deals with the second-order Dirichlet boundary value problem with one state-dependent impulse Proofs of the main results contain a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space . Sufficient conditions for the existence of solutions to the problem are given here. The presented approach can be extended to more impulses and to other boundary conditions. MSC:34B37, 34B15.

Impulsive system of ODEs with general linear boundary conditions

The paper provides an operator representation for a problem which consists of a system of ordinary differential equations of the first order with impulses at fixed times and with general linear boundary conditions z (t) = A(t)z(t) + f(t, z(t)) for a.e. t ∈ [a, b] ⊂ R, z(ti+)− z(ti) = Ji(z(ti)), i = 1, . . . , p, l(z) = c0, c0 ∈ R n . Here p, n ∈ N, a < t1 < . . . < tp < b, A ∈ L ([a, b];R), f ∈ Car([a, b] × R;R), Ji ∈ C(R ;R), i = 1, . . . , p, and l is a linear bounded operator on the space of left-continuous regulated functions on interval [a, b]. The operator l is expressed by means of the KurzweilStieltjes integral and covers all linear boundary conditions for solutions of the above system subject to impulse conditions. The representation, which is based on the Green matrix to a corresponding linear homogeneous problem, leads to an existence principle for the original problem. A special case of the n-th order scalar differential equation is discussed. This approach can be also used for analogical problems with state-dependent impulses. Mathematics Subject Classification 2010: 34B37, 34B15, 34B27

ANALYSIS OF A DELAYED PREDATOR-PREY SYSTEM WITH IMPULSIVE STATE FEEDBACK

A delayed dynamical system model incorporating modified Leslie–Gower and Holling-type II schemes with state-dependent impulses is investigated for the case of small delay. The boundedness of solutions is analyzed by applying the comparison argument. Existence and stability of the semi-trivial periodic solution and the positive period-1 solution are discussed by using the variational principle, constructing Poincaré map and applying Floquet multiplier theory. Flip bifurcation is also investigated and some numerical simulation is presented in detail.

Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments

The state-dependent delay differential equation x ˙ ( t ) + ∑ i = 1 m p i ( t ) x ( t − ( H i x ) ( t ) ) = f ( t ) , t ∈ [ 0 , ∞ ) , x ( ξ ) = φ ( ξ ) , ξ < 0 , with state-dependent impulses is under consideration. Sufficient conditions for positivity of solutions to the Cauchy and periodic problems as well as conditions for positivity of solutions to the problem with a condition on the right end of the interval [ 0 , ω ] are given. Sufficient conditions for nonoscillation of solutions to the homogeneous equation ( f = 0 , φ = 0 ) on the half-line are formulated.

Periodic Liénard-type delay equations with state-dependent impulses

Conditions are obtained for Liénard-type equations with delay and state-dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation (and consequently with Lebesgue integrable second derivative). The results are applied to Josephson's equation and the nonconservative forced pendulum equation.

Stability of nontrivial solution of delay differential equations with state-dependent impulses

This paper considers the stability problem of nontrivial solutions of delay differential equations with state-dependent impulses. By using the concepts of reduced system and quasi-stability, the stability problem of a nontrivial solution of the impulsive delay differential equation with state-dependent impulses is converted to that of the trivial solution of the impulsive delay differential equation with fixed impulses. Several stability criteria are then established utilizing the known results for systems with fixed impulses.

Existence of periodic solution of order one of planar impulsive autonomous system

Based on the situation that all the recent research about state-dependent impulsive differential equations is focused on systems which have explicit solutions, this paper try to consider those systems with state-dependent impulsion which have not explicit solutions. We get an existence theorem of periodic solution of order one for a general planar autonomous impulsive system, and, by applying it to a special state-dependent impulsive differential equations we get the concrete conditions of existence of one-order periodic solution of that special system.

Periodic Duffing delay equations with state dependent impulses

Conditions are obtained for the Duffing equation with delay and state dependent impulses to admit a periodic solution.

Existence and continuability of solutions for differential equations with delays and state-dependent impulses

Existence and continuability of solutions for differential equations with delays and state-dependent impulses

State-dependent impulsive delay-differential equations

The paper deals with existence and uniqueness of solutions to delay-differential equations with infinitely many state-dependent impulses. A simple transformation allows us to show that the problem can be treated as a delay-differential equation without impulses. A fixed-point approach is then applied with an appropriate norm.

Existence, uniqueness and approximation of the solution of an ode with (infinitely many) state-dependent impulses via fixed mesh galerkin formulation

A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (infinitely many) state-dependent impulses on the right-hand side. This approach gives a natural numerical scheme to approximate the solution. The convergence of the approximation is proved and its asymptotic order obtained.