Space Of Piecewise Continuous Functions Research Articles

Cauchy problem for impulsive fractional differential equations with almost sectorial operators

We study a class of non-instantaneous impulsive fractional differential equations in a Banach space. We consider the Cauchy problem with Caputo derivative having variable lower limits and almost sectorial operators. We establish the existence of mild solutions in the space of piecewise continuous functions. We employ the properties of semigroup associated with the almost sectorial operator, theory of measure of non-compactness and fixed point theorems.

Topological structure of solution sets for hybrid impulsive fractional differential inclusion in banach algebra

The aim of this paper is to study the topological structure of solution sets of hybrid impulsive fractional differential inclusions on the weighted space of piecewise continuous functions, we apply dhage fixed point theorem for multivalued operator with analysis tools to prove existence result. The paper concludes with an example to illustrate the feasibility of our main result.

Learning sub-patterns in piecewise continuous functions

Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters; however, this implies that the neural network’s activation function must exhibit a degree of continuity which limits the neural network model’s uniform approximation capacity to continuous functions. This paper focuses on the case where the discontinuities arise from distinct sub-patterns, each defined on different parts of the input space. We propose a new discontinuous deep neural network model trainable via a decoupled two-step procedure that avoids passing gradient updates through the network’s only and strategically placed, discontinuous unit. We provide approximation guarantees for our architecture in the space of bounded continuous functions and universal approximation guarantees in the space of piecewise continuous functions which we introduced herein. We present a novel semi-supervised two-step training procedure for our discontinuous deep learning model, tailored to its structure, and we provide theoretical support for its effectiveness. The performance of our model and trained with the propose procedure is evaluated experimentally on both real-world financial datasets and synthetic datasets.

Controllability of Nonlocal Impulsive Functional Differential Equations with Measure of Noncompactness in Banach Spaces

Abstract This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii’s Fixed Point Theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions and mixed-type conditions, respectively.

Controllability of nonlocal impulsive functional differential equations with measure of noncompactness in Banach spaces

This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii's fixed point theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions, and mixed-type conditions, respectively.

Controllability of nonlocal impulsive differential equations with measure of noncompactness

This paper deals with nonlocal conditions of impulsive differential equations with control. Here, we structure the measure of noncompactness and its properties in the space of piecewise continuous functions. Then, we acquire the controllability of impulsive differential equations with nonlocal conditions via compactness condition, Lipschitz conditions by using the Darbo-Sadovskii’s fixed point theorem and the property of noncompactness.

Method for finding a solution to a linear nonstationary interval ОDЕ system

The article analyses a linear nonstationary interval system of ordinary differential equations so that the elements of the matrix of the system are the intervals with the known lower and upper bounds. The system is defined on the known finite time interval. It is required to construct a trajectory, which brings this system from the given initial position to the given final state. The original problem is reduced to finding a solution of the differential inclusion of a special form with the fixed right endpoint. With the help of support functions, this problem is reduced to minimizing a functional in the space of piecewise continuous functions. Under a natural additional assumption, this functional is Gateaux differentiable. For the functional, Gateaux gradient is found, necessary and sufficient conditions for the minimum are obtained. Оn the basis of these conditions, the method of the steepest descent is applied to the original problem. Some examples illustrate the constructed algorithm realization.

Periodic solutions to impulsive stochastic reaction-diffusion neural networks with delays

In this paper, the aims are to study the existence and stability of mild periodic solutions to impulsive stochastic reaction-diffusion neural networks (ISRDNNs) with delays. First, key issues of the Markov property of mild solutions to ISRDNNs with delays are presented in the space of piecewise continuous functions. Next, combining the operator semigroup method with other mathematical techniques, the existence of mild periodic solutions is proposed and some relevant results are generalized. Then, the exponential stability of mild periodic solutions is discussed and some easy-to-test sufficient conditions are obtained by using the Lyapunov method. Finally, numerical simulations are provided to illustrate the effectiveness of our results.

Differential Equations and Inclusions of Fractional Order with Impulse Effects in Banach Spaces

The present paper is concerned with the existence of solutions, in infinite- dimensional Banach spaces, for impulsive differential inclusions and equations of order $$q\in (1,2)$$, with anti-periodic conditions and involving the Caputo derivative whether in the generalized sense (via the Riemann–Liouville fractional derivative,) or in the normal sense and whether the lower limit on each impulsive subinterval $$ (t_{i} ,t_{i+1}],\, i=0,1,\ldots ,m$$ is keep at zero or is set at $$t_{i}$$. Using the technique of fixed point and the properties of the measure of noncompactness, existence results are obtained. Moreover, we derive in reflexive Banach spaces, by using a new version weakly convergent criteria in the space of piecewise continuous functions, an existence result of solutions without assuming any condition on the multivalued function in terms of measure of noncompactness. Some examples will be given to illustrate the obtained results.

Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness

The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators {A(t):t∈[0,b]} generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a C0−semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.

НЕКОТОРЫЕ СВОЙСТВА РЕШЕНИЙ ЗАДАЧИ КОШИ ДЛЯ ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНОГО ВКЛЮЧЕНИЯ С МНОГОЗНАЧНЫМИ ИМПУЛЬСНЫМИ ВОЗДЕЙСТВИЯМИ

Deviation estimates in space of piecewise continuous functions of a set of the generalized decisions from beforehand given function are received. The continuous dependence of the generalized decisions on starting conditions is established.

Fixed point approach for weakly asymptotic stability of fractional differential inclusions involving impulsive effects

We prove the global solvability and weakly asymptotic stability for a semilinear fractional differential inclusion subject to impulsive effects by analyzing behavior of its solutions on the half-line. Our analysis is based on a fixed point principle for condensing multi-valued maps, which is employed for solution operator acting on the space of piecewise continuous functions. The obtained results will be applied to a lattice fractional differential system.

Interpolating Integral Continued Fraction of the Thiele Type

We generalize the well-known results on the interpolation of a function of single variable by the Thiele–Hermite fraction with arbitrary multiplicity of each interpolation node to the case of a functional acting from the space of piecewise continuous functions with finitely many discontinuities of the first kind. We obtain an interpolating integral fraction of the Thiele type on the set of interpolation nodes one of which is continual. We also indicate an efficient approach to the construction of interpolating integral fraction of the Thiele type in the case where all interpolation nodes are continual. This case is important to balance the data used for the construction of the interpolant and its interpolating properties.

Perturbed stable systems on a plane. Part 1

The possibility of extending the concept of rough dynamical systems in the presence of a time-varying perturbation defined up to a functional set in the space of piecewise continuous functions is considered. The constructability of this extension is achieved by the construction of limit cycles giving an estimate of attainability sets for oscillatory systems.

Existence of mild solutions for semilinear differential equations with nonlocal and impulsive conditions

Abstract This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure of noncompactness, and the linear part generates only a strongly continuous evolution system.

A unified approach to nonlocal impulsive differential equations with the measure of noncompactness

This paper is concerned with the existence of mild solutions to impulsive differential equations with nonlocal conditions. We firstly establish a property of the measure of noncompactness in the space of piecewise continuous functions. Then, by applying this property and Darbo-Sadovskii’s fixed point theorem, we get the existence results of impulsive differential equations in a unified way under compactness conditions, Lipschitz conditions and mixed-type conditions, respectively.

Dynamic Systems of Shifts in the Space of Piece-Wise Continuous Functions

In this paper we embark on the study of Dynamic Systems of Shifts in the space of piece-wise continuous functions analogue to the known Bebutov system. We give a formal definition of a topological dynamic system in the space of piece-wise continuous functions and show, by way of an example, stability in the sense of Poisson discontinuous function. We prove that a fixed discontinuous function, f, is discontinuous for all its shifts, whereas the trajectory of discontinuous function is not a compact set.

Existence results for semilinear differential equations with nonlocal and impulsive conditions

This paper is concerned with the existence for impulsive semilinear differential equations with nonlocal conditions. Using the techniques of approximate solutions and fixed point, existence results are obtained, for mild solutions, when the impulsive functions are only continuous and the nonlocal item is Lipschitz in the space of piecewise continuous functions, is not Lipschitz and not compact, is continuous in the space of Bochner integrable functions, respectively.

On the Solvability of Second-Order Impulsive Differential Equations with Antiperiodic Boundary Value Conditions

We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.

Interpolational Integral Continued Fractions

For nonlinear functionals defined on the space of piecewise-continuous functions, we construct an interpolational integral continued fraction on continual piecewise-continuous nodes and establish conditions for the existence and uniqueness of interpolants of this type.