Solutions For Impulsive Differential Equations Research Articles

Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces

<abstract><p>In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.</p></abstract>

(ω,ρ)-BVP Solutions of Impulsive Differential Equations of Fractional Order on Banach Spaces

The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω,ρ)-BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer’s fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results.

Existence of Solutions for Two-Point Integral Boundary Value Problems with Impulses

In this paper, we investigate the existence of at least one solution and at least two nonnegative solutions of impulsive differential equations with the two-point integral boundary conditions. We employ the recent fixed point theorems for the sum of two operators on Banach spaces. The applicability of the results is illustrated through an example.

The existence of one solution for impulsive differential equations via variational methods

We prove the existence of at least one non-trivial weak solution for a nonlinear Dirichlet boundary value problem subject to perturbations of impulsive terms via employing a critical point theorem for differentiable functionals.

Periodic solutions of impulsive differential equations with piecewise alternately advanced and retarded argument of generalized type

We investigate the existence of the periodic solutions of a quasilinear impulsive differential equation with alternately advanced and retarded arguments of generalized type, in short IDEPCAG. By using some fixed point theorems and some new analysis techniques, sufficient conditions are obtained for the existence and uniqueness of periodic solutions of these IDEPCAG systems. A new IDEPCAG’s Gronwall-type lemma is proved. Some examples concerning impulsive biological models such as Lasota–Ważewska model, Nicholson’s blowflies and logistic models are treated.

Factorization of Scalar Piecewise Continuous Almost Periodic Functions

This paper is to characterize piecewise continuous almost periodic functions as the product of Bohr almost periodic functions and sequences. As an application, the result is used to discuss piecewise continuous almost periodic solutions of impulsive differential equations.

New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

On Regulated Solutions of Impulsive Differential Equations with Variable Times

In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined.

Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro‐differential equations

Sufficient conditions for the existence of at least one positive periodic solution are established for a family of scalar periodic differential equations with infinite delay and nonlinear impulses. Our criteria, obtained by applying a fixed‐point argument to an original operator constructed here, allow to treat equations incorporating a rather general nonlinearity and impulses whose signs may vary. Applications to some classes of Volterra integro‐differential equations with unbounded or periodic delay and nonlinear impulses are given, extending and improving results in the literature.

Piecewise-Polynomial Approximations for the Solutions of Impulsive Differential Equations

On the basis of V. Dzyadyk’s approximation method, we consider the problems of construction and theoretical substantiation of high-precision numerical-analytic algorithms for the piecewise polynomial approximation of the solutions of problems with pulsed action.

Existence and Uniqueness of Solution for Abstract Differential Equations with State-Dependent Time Impulses

We study the existence and uniqueness of mild and classical solutions for abstract impulsive differential equations with state-dependent time impulses and an example is presented.

A Study on PC Pseudo almost Automorphic Solution of Impulsive Differential Equation

A Study on PC Pseudo almost Automorphic Solution of Impulsive Differential Equation

Periodic solutions of impulsive differential equations with infinite delay in Banach spaces

Periodic solutions of impulsive differential equations with infinite delay in Banach spaces

Comparison of Quantities of Information in the Human Memory

A mathematical model of changing the amount of information in the abstract human memory is proposed in the presence of the subsequent "external discrete" training (filling the information). Under this model, the amount of information is a solution of impulsive differential equation with fixed moments of impulsive effects and variable structure. Sufficient conditions are proposed related to the moments and magnitudes of the impulsive effects (i.e., to the moments of discrete training and the volume of the received information), where the quantities of information in two different models of learning can be compared.

Some new integral inequalities with mixed nonlinearities for discontinuous functions

In this paper, we establish some new integral inequalities with mixed nonlinearities for discontinuous functions, which provide a handy tool in deriving the explicit bounds for the solutions of impulsive differential equations and differential-integral equations with impulsive conditions.

Existence and multiplicity of solutions for a second-order impulsive differential equation via variational methods

In this paper, we study the existence and multiplicity of solutions for an impulsive differential equation via some critical point theory and the variational method. We extend and improve some recent results and reduce conditions.

Positive solutions of impulsive differential equations

Positive solutions of impulsive differential equations

Bifurcation of Bounded Solutions of Impulsive Differential Equations

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results.

ON EXACT SOLUTIONS FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH NON-INTEGER ORDERS

This paper deals with linear impulsive differential equations with non-integer orders. We provide the explicit representation of solutions of linear impulsive fractional differential equations with constant coefficient by mean of the Mittag-Leffler functions.

The general solution for impulsive differential equations with Hadamard fractional derivative of order q ∈ ( 1 , 2 ) $q \in(1, 2)$

In this paper, we find formulas of general solution for a kind of impulsive differential equations with Hadamard fractional derivative of order $q \in(1, 2)$ by analysis of the limit case (as the impulse tends to zero) and provide an example to illustrate the importance of our results.