
Snezhana HristovaPlovdiv University · Department of Applied Mathematics and Modeling
Snezhana Hristova
PhD, DSci, Professor
impulsive differential equations, fractional differential equations, delay, non-instantaneous impulses, stability
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249
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Introduction
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August 2014 - May 2015
Publications
Publications (249)
In this paper, we study nonlinear differential equations with Caputo fractional derivatives with respect to other functions and impulses. The main characteristic of the impulses is that the time between two consecutive impulsive moments is defined by random variables. These random variables are independent. As the distribution of these random varia...
The Cohen–Grossberg neural network is studied in the case when the dynamics of the neurons is modeled by a Riemann–Liouville fractional derivative with respect to another function and an appropriate initial condition is set up. Some inequalities about both the quadratic function and the absolute values functions and their fractional derivatives wit...
The main aim of the current paper is to be appropriately defined several types of Ulam stability for non-linear fractional differential equation with generalized proportional fractional derivative of Riemann-Liouville type. In the new definitions, the initial values of the solutions of the given equation and the corresponding inequality could not c...
Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initia...
The main goal of the paper is to present and study models of multi-agent systems for which the dynamics of the agents are described by a Caputo fractional derivative of variable order and a kernel that depends on an increasing function. Also, the order of the fractional derivative changes at update times. We study a case for which the exchanged inf...
An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider...
An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding different...
Some inequalities for generalized proportional Riemann–Liouville fractional derivatives (RLGFDs) of convex functions are proven. As a special case, inequalities for the RLGFDs of the most-applicable Lyapunov functions such as the ones defined as a quadratic function or the ones defined by absolute values were obtained. These Lyapunov functions were...
In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for...
The general delay Hopfield neural network is studied. It is considered the case of time-varying delay, continuously distributed delays, time varying coefficients and a special type of a Riemann-Liouville fractional derivative (GPRLFD) with an exponential kernel. The presence of delays and GPRLFD in the model require two special types of initial con...
The general delay Hopfield neural network is studied. We consider the case of time-varying delay, continuously distributed delays, time-varying coefficients, and a special type of a Riemann–Liouville fractional derivative (GRLFD) with an exponential kernel. The kernels of the fractional integral and the fractional derivative in this paper are Sonin...
The main goal of the paper is to use a generalized proportional Riemann–Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved, such as the fractional derivative of a squared function. Additionally, some comparison result...
In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on the application of the monotone-iterative technique combined...
The main goal of the paper is to use a generalized proportional Riemann-Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved such as the fractional derivative of a squared function. Also some comparison results for GPRL...
A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in the special linear case is obtained. This formula is...
In this paper, we study generalized proportional Caputo fractional differential equations via Lyapunov functions. We define conditional boundedness and obtain some sufficient conditions. Several examples are provided to illustrate the application of the proved conditions.KeywordsGeneralized proportional caputo fractional derivativeDifferential equa...
The practical stability of a nonlinear nonautonomous Caputo fractional differential equation is studied using Lyapunov-like functions. The novelty of this paper is based on the new definition of the derivative of a Lyapunov like function along the given fractional differential equation. Comparison results using this definition for scalar fractional...
A nonlinear non-instantaneous impulsive difference equations with maximum of the state variable over a past time interval is investigated. The exponential stability concept is studied and some criteria are derived. These results are also applied for a neural networks with switching topology at certain moments and long time lasting impulses. It is c...
Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of so...
In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as th...
A boundary-value problem for a couple of scalar nonlinear differential equations with a delay and several generalized proportional Caputo fractional derivatives is studied. Ulam-type stability of the given problem is investigated. Sufficient conditions for the existence of the boundary-value problem with an arbitrary parameter are obtained. In the...
A nonlocal boundary value problem for a couple of two scalar nonlinear differential equations with several generalized proportional Caputo fractional derivatives and a delay is studied. The exact solution of the scalar nonlinear differential equation with several generalized proportional Caputo fractional derivatives with different orders is obtain...
It this paper we study a system of nonlinear delay impulsive fractional differential equations with Riemann-Liouville fractional derivatives taken from both, the unknown function at the present time and the unknown function at the variable time delay. We consider the case when the lower limit of the fractional derivative is changed at each impulsiv...
In this paper, nonlinear differential equations with a generalized proportional Caputo fractional derivative and finite delay are studied in this paper. The eventual presence of impulses in the equations is considered, and the statement of initial value problems in three cases is defined: namely non-instantaneous impulses, instantaneous impulses an...
In this paper, we study a class of nonlocal functional evolution inclusions in the form of multivalued perturbations of m–dissipative operators. We prove a variant of the important in optimal control relaxation theorem. Illustrative example is provided.
A system of nonlinear neutral delay non-instantaneous impulsive differential equations is studied. It is obtained the integral presentation of the solutions on a finite interval. Also, it is studied the closeness of solutions over a finite time interval. A finite time Lipschitz stability system is defined and sufficient conditions are obtained. Thi...
It this paper we study a system of nonlinear neutral fractional differential equations with Riemann-Liouville fractional derivatives and suppremum deay taken from both, the unknown function at the present time and the unknown function at the time delay and non-instantaneous impulses. We consider the case when the lower limit of the fractional deriv...
The synchronization problem for impulsive fractional-order Cohen–Grossberg neural networks with generalized proportional Caputo fractional derivatives with changeable lower limit at any point of impulse is studied. We consider the cases when the control input is acting continuously as well as when it is acting instantaneously at the impulsive times...
The generalized proportional Caputo fractional derivative is a comparatively new type of derivative that is a generalization of the classical Caputo fractional derivative, and it gives more opportunities to adequately model complex phenomena in physics, chemistry, biology, etc. In this paper, the presence of noninstantaneous impulses in differentia...
This paper deals with multi-agent systems that, due to using the generalized proportional Caputo fractional derivative, possess memories. The information exchange between agents does not occur continuously but only at fixed given update times, and the lower limit of the fractional derivative changes according to the update times. Two types of multi...
Practical stability properties of generalized proportional Caputo fractional differential equations with bounded delay are studied in this paper. Two types of stability, practical stability and exponential practical stability, are defined and considered, and also some sufficient conditions to guarantee stability are presented. The study is based on...
In this manuscript, we examine the existence of solutions to the boundary value problem at resonance of Caputo fractional differential equations of variable order over a finite time interval. A partition of the considered interval is initially given and the variable order of the fractional derivative is a piecewise constant function with a range of...
This paper is dedicated to studying the existence and uniqueness of solutions to a system of fractional differential equations with anti-periodic fractional integral boundary conditions in the frame of the generalized proportional fractional derivatives of Riemann-Liouville type. With the aid of the Schaefer fixed point theorem and the Banach fixed...
A scalar nonlinear integro-differential equation with time-variable and bounded
delays and generalized Caputo proportional fractional derivative is considered. The
main goal of this paper is to study the stability properties of the zero solution. Results
are given concerning stability, exponential stability, asymptotic stability, and
boundedness of...
A model of gene regulatory networks with generalized proportional Caputo fractional
derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functi...
In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some compari...
Caputo fractional differential equations with impulses are a very useful apparatus
for adequate modeling of the dynamics of many real world problems. It requires developments of good and consistent theoretical proofs and the results for various problems. In this note we point out and correct the statement of the boundary value problem with Riemann–...
A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordin...
Difference equations with a special type of delay is studied. This type of delay is characterized by the maximum value of the unknown function on the past time interval with a constant length. It is studied the exponential stability. Some sufficient conditions are theoretically proved and computer simulated on several examples. The influence of the...
Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riem...
A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of the Leray–Schauder nonlinear alte...
Initial and impulsive conditions for initial value problems of systems of nonlinear impulsive Riemann-Liouville fractional differential equations are introduced. The case when the lower limit of the fractional derivative is changed at each time point of the impulses is studied. In the case studied, the solution has a singu-larity at the initial tim...
The present paper is concerned with the existence of solutions of a new class of nonlinear generalized proportional fractional differential inclusions with the right-hand side contains a Carathèodory-type multi-valued nonlinearity on infinite intervals.
The investigation of the proposed inclusion problem relies on the multi-valued form of Leray-Sc...
The main aim of the paper is to present an algorithm to solve approximately initialvalue problems for a scalar non-linear fractional differential equation with generalized proportionalfractional derivative on a finite interval. The main condition is connected with the one sidedLipschitz condition of the right hand side part of the given equation. A...
The object of investigation in this paper is a scalar linear fractional differential equation
with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of sol...
Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and non-instantaneous impulses are studied where initial conditions and impulsive conditions are set up in appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equat...
Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay, depending on both the time and the state variable. The case when the lower limit of the Caputo fractional derivative is fixed at the initial time, and the case when the lower limit of the fractional derivat...
A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value pr...
In this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several...
The synchronization is one of the main problems connected with neural networks. We examine a model of a neural network with time varying delay, and also the case when the connection weights (the influential strength of the j-th neuron to the i-th neuron) are variable in time and unbounded. The rate of change of the dynamic of all neurons is describ...
A system of nonlinear fractional differential equations with the Riemann–Liouville fractional derivative is considered. Lipschitz stability in time for the studied equations is defined and studied. This stability is connected with the singularity of the Riemann–Liouville fractional derivative at the initial point. Two types of derivatives of Lyapun...
In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time po...
Ulam type stability concepts for non-instantaneous impulsive difference equations are introduced and studied. Sufficient conditions for Ulam-Hyers stability, generalized Ulam-Hyers-Rassias stability and Ulam- Hyers-Rassias stability are obtained. Also some conditions for instability are provided. Some of the obtained results are illustrated on exam...
The stability in terms of two different measures for differential equations with supremum and non-instantaneous impulses is studied using Lyapunov like functions and Razumikhin technique. In these differential equation we have impulses, which start abruptly at some points and their action continue on given finite intervals. An example is given to i...
Practical stability for nonlinear differential equations with non-instantaneous impulses and supremum is studied. The impulses start abruptly at some points and their actions continue on given finite intervals. Some sufficient conditions for practical stability and strong practical stability are obtained. An example is given to illustrate some of t...
Linear Riemann-Liouville fractional differential equations with a constant delay and non-instantaneous impulses are studied. Both cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of i...
In this paper, we study a class of nonlocal functional evolution inclusions in the form of multivalued perturbations of m–dissipative operators. We prove some new existence results in case of dissipative type conditions on the right-hand side. Illustrative example is provided.
Lipschitz stability for nonlinear differential equations with non-instantaneous impulses and supremum of the unknown function over a past time interval is studied. The impulses start abruptly at some points and their action continue on given finite intervals. Some sufficient conditions for Lipschitz stability are obtained. Modified Razumikhin metho...
In this paper we study nonlocal semilinear fractional evolution inclusions involving Riemann-Liouville derivative with right-hand side depending on casual operator. The existence of solutions is proved under compactness type conditions on the multivalued term.
Caputo fractional differential equations of order q ∈ (1, 2) with impulses at random moments are set up and investigated in this paper. The main characteristic of the studied equations is that the impulses occur at random moments which are exponentially distributed random variables. It leads to a total change of the behavior of the solution which i...
A system of nonlinear Riemann-Liouville fractional differential equations with a constant delay is studied. The initial condition is set up similarly to the case of ordinary derivative. Sufficient conditions for finite time stability are obtained.
In this paper, the direct Lyapunov method is extended to fractional case of Volterra integro-Caputo fractional differential equations with delays. Two types of homogeneous scalar Voterra integro-Caputo fractional differential equations with delays are studied. Three types of fractional derivatives of Lyapunov functions known in the literature are g...
First, we set up in an appropriate way the initial value problem for nonlinear delay differential equations with a Riemann-Liouville (RL) fractional derivative. We define stability in time and generalize Mittag-Leffler stability for RL fractional differential equations and we study stability properties by an appropriate modification of the Razumikh...
In this paper a model of Hopfield’s graded response neural network is investigated. A network whose neurons are subject to a certain impulsive state displacement at random times is considered. The model is set up and studied. The presence of random moments of impulses in the model leads to a change of the solutions to stochastic processes. Also, we...
Differential equations of second order with impulses at random moments are set up and investigated in this paper. The main characteristic of the studied equations is that the impulses occur at random moments which are exponentially distributed random variables. The presence of random variables in the ordinary differential equation leads to a total...
A nonlinear Riemann–Liouville fractional differential equation with constant delay is studied. Initially, some existence results are proved. Three Ulam type stability concepts are defined and studied. Several sufficient conditions are obtained. Some of the obtained results are illustrated on fractional biological models.
The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, expli...
The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time-varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, expli...
Fractional differential equations with impulses arise in modeling real world phenomena where the state changes instantaneously at some moments. Often, these instantaneous changes occur at random moments. In this situation the theory of Differential equations has to be combined with Probability theory to set up the problem correctly and to study the...
The initial value problem for a special type of scalar nonlinear fractional differential equation with a Riemann–Liouville fractional derivative is studied. The main characteristic of the equation is the presence of the supremum of the unknown function over a previous time interval. This type of equation is difficult to be solved explicitly and we...
The main objective of this paper is to introduce the (α,β)-type ϑ-contraction, (α,β)-type rational ϑ-contraction, and cyclic (α-ϑ) contraction. Based on these definitions we prove fixed point theorems in the complete metric spaces. These results extend and improve some known results in the literature. As an application of the proved fixed point The...
Riemann-Liouville fractional differential equations with a constant delay and impulses are studied in this article. The following two cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point...
This paper studies the leader-following consensus problem in continuous-time multi-agent networks with communications/updates occurring only at random times. The time between two consecutive controller updates is exponentially distributed. Some sufficient conditions are derived to design the control law that ensures the leader-following consensus i...
Abstract A system of linear Riemann–Liouville fractional differential equations with constant delay is studied. The initial condition is set up similar to the case of the ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.
Non-linear scalar Riemann-Liouville fractional differential equation with a constant delay is studied on a finite interval. An initial value problem is set up in appropriate way combining the idea of the initial time interval in ordinary differential equations with delays and the properties of Riemann- Liouville fractional derivatives. The mild sol...
Four Ulam type stability concepts for non-instantaneous impulsive fractional differential equations with state dependent delay are introduced. Two different approaches to the interpretation of solutions are investigated. We study the case of an unchangeable lower bound of the Caputo fractional derivative and the case of a lower bound coinciding wit...
Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the...
The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is...
The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of t...
A discrete Hopfield-type neural network with constant delays, instantaneous switching topologies at certain times and time variable connection weights is studied. Some criteria for exponential stability are derived. The obtained results are illustrated on an example with different activation functions such as tanh, Swish, and the error function. Th...
The main aim of this paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Riemann–Liouville fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the give...
In this paper we consider systems of functional differential equations whose dynamics depends on the maximum value of solution over a prehistory time interval. Such kind of systems are infinite-dimensional and nonlinear. We consider controlled systems with maximum and study their input-to-state stability property. As well we compare stability prope...
In this paper, we study Linear Riemann-Liouville fractional differential equations with a constant delay. The initial condition is set up similarly to the case of ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.
The paper deals with some stability properties of the solutions of impulsive differential equations with supremum. The main characteristic of the impulses in the system is their duration- the impulsive action starts at an arbitrary fixed point and remains active on a finite time interval. Note the impulsive differential systems originate from the r...
We consider the Hopfield’s graded response neural network in the case when the neurons are subject to a certain impulsive state displacement at fixed moments and the duration of this displacement is not negligible small (they are known as non-instantaneous impulses). We examine the case when the present state of any neuron depends on its maximum va...
In this paper some practical stability results for nonlinear differential equations with non-instantaneous impulses and state dependent delays are presented. The impulses start abruptly at some points and their action continue on given finite intervals. The delay depends on both the time and the state variable which is a generalization of time vari...
Recent modeling of real world phenomena give rise to fractional differential equations with non-instantaneous impulses. The main goal of this paper is to provide an existence and uniqueness results for Riemann-Liouville fractional differential equation with non-instantaneous impulses. It is studied the case when the lower bound of the Riemann-Liouv...
Recent modeling of real world phenomena give rise to fractional differential equations with non-instantaneous impulses. The main goal of the paper is to highlight basic points in introducing non-instantaneous impulses in Riemann-Liouville fractional differential equations. The case when the lower limit of the fractional derivative is changed at any...
A leader-following consensus of discrete-time multi-agent systems with nonlinear intrinsic dynamics and impulses is investigated. We propose and prove conditions ensuring a leader-following consensus. Numerical examples are given to illustrate effectiveness of the obtained results. Also, the necessity and sufficiency of the obtained conditions are...