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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 initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied. [ABSTRACT FROM AUTHOR]

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 information between agents occurs only at initially given update times. Two types of linear variable-order Caputo fractional models are studied. We consider both multi-agent systems without a leader and multi-agent systems with a leader. In the case of multi-agent systems without a leader, two types of models are studied. The main difference between the models is the fractional derivative describing the dynamics of agents. In the first one, a Caputo fractional derivative with respect to another function and with a continuous variable order is applied. In the second one, the applied fractional derivative changes its constant order at each update time. Mittag–Leffler stability via impulsive control is defined, and sufficient conditions are obtained. In the case of the presence of a leader in the multi-agent system, the dynamic of the agents is described by a Caputo fractional derivative with respect to an increasing function and with a constant order that changes at each update time. The leader-following consensus via impulsive control is defined, and sufficient conditions are derived. The theoretical results are illustrated with examples. We show with an example the leader's influence on the consensus. [ABSTRACT FROM AUTHOR]

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 differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ -tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ -tempered Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

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 combined with a modification of the Razumikhin method to study the stability properties of the Cohen–Grossberg model of neural networks with both time-variable and continuously distributed delays, time-varying coefficients, and RLGFDs. The initial-value problem was set and studied. Upper bounds by exponential functions of the solutions were obtained on intervals excluding the initial time. The asymptotic behavior of the solutions of the model was studied. Some of the obtained theoretical results were applied to a particular example. [ABSTRACT FROM AUTHOR]

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 fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained. [ABSTRACT FROM AUTHOR]

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 Sonine kernels and satisfy the first and the second fundamental theorems in calculus. The presence of delays and GRLFD in the model require a special type of initial condition. The applied GRLFD also requires a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. An inequality for Lyapunov type of convex functions with the applied GRLFD is proved. It is combined with the Razumikhin method to study stability properties of the equilibrium of the model. As a partial case we apply quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included. [ABSTRACT FROM AUTHOR]

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 results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time, the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given, illustrating the meaning of the equilibrium as well as its stability properties. [ABSTRACT FROM AUTHOR]

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 with the method of lower and upper solutions. In connection with this, initially, the linear fractional differential equation with a boundary condition is studied, and its explicit solution is obtained. An appropriate integral fractional operator for the nonlinear problem is constructed and it is used to define the mild solutions, upper mild solutions and lower mild solutions of the given problem. Based on this integral operator we suggest a scheme for obtaining two monotone sequences of successive approximations. Both sequences consist of lower mild solutions and lower upper solutions of the studied problem, respectively. The monotonic uniform convergence of both sequences to mild solutions is proved. The algorithm is computerized and applied to a particular example to illustrate the theoretical investigations. [ABSTRACT FROM AUTHOR]

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 a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results. [ABSTRACT FROM AUTHOR]

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 solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations. [ABSTRACT FROM AUTHOR]

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 the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditions for approaching zero are obtained. Lyapunov functions defined as a sum of squares with their generalized proportional Caputo fractional derivatives are applied and a comparison result for a scalar linear generalized proportional Caputo fractional differential equation with several constant delays is presented. Lyapunov functions and the comparison principle are then combined to establish our main results. [ABSTRACT FROM AUTHOR]

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 study of Ulam-type stability, this parameter was chosen to depend on the solution of the corresponding fractional differential inequality. We provide sufficient conditions for Ulam–Hyers stability, Ulam–Hyers–Rassias stability and generalized Ulam–Hyers–Rassias stability for the given problem on a finite interval. As a partial case, sufficient conditions for Ulam-type stability for a couple of multi-term delay, Caputo fractional differential equations are obtained. An example is illustrating the results. [ABSTRACT FROM AUTHOR]

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 obtained. A mild solution of the boundary value problem for the multi-term nonlinear couple of the given fractional equations is defined. The connection between the mild solution and the solution of the studied problem is discussed. As a partial case, several results for the nonlocal boundary value problem for the linear and non-linear multi-term Caputo fractional differential equations are provided. The results generalize several known results in the literature. [ABSTRACT FROM AUTHOR]

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 and no impulses. The relations between these three cases are discussed. Additionally, some stability properties are investigated. We apply the Mittag–Leffler function which plays a vital role and which gives well-known bounds on the norm of the solutions. The symmetry of this function about a line and the bounds is a property that plays an important role in stability. Several sufficient conditions are presented via appropriate new comparison results and the modified Razumikhin method. The results generalize several known results in the literature. [ABSTRACT FROM AUTHOR]

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. We defined the global Mittag–Leffler synchronization as a generalization of exponential synchronization. We obtained some sufficient conditions for Mittag–Leffler synchronization. Our results are illustrated with examples. [ABSTRACT FROM AUTHOR]

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 differential equations with generalized proportional Caputo fractional derivatives is discussed. Generalized proportional Caputo fractional derivatives with fixed lower limits at the initial time as well as generalized proportional Caputo fractional derivatives with changeable lower limits at each impulsive time are considered. The statements of the problems in both cases are set up and the integral representation of the solution of the defined problem in each case is presented. Ulam-type stability is also investigated and some examples are given illustrating these concepts. [ABSTRACT FROM AUTHOR]

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 the application of Lyapunov like functions and their generalized proportional Caputo fractional derivatives among solutions of the studied system where appropriate Razumikhin like conditions are applied (appropriately modified in connection with the fractional derivative considered). The theory is illustrated with several nonlinear examples. [ABSTRACT FROM AUTHOR]

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 solutions. The investigations are based on an application of a quadratic Lyapunov function, its generalized Caputo proportional derivative, and a modification of the Razumikhin approach. Some auxiliary properties of the generalized Caputo proportional derivative are proved. Five illustrative examples are included. [ABSTRACT FROM AUTHOR]

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 sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results. [ABSTRACT FROM AUTHOR]

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 with the point of jump for the solution. In both cases we obtain sufficient conditions for Ulam type stability. An example is also provided to illustrate both approaches. [ABSTRACT FROM AUTHOR]

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 Lyapunov functions among the studied fractional equations are applied to obtain sufficient conditions for the defined stability property. Some examples illustrate the results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Caputo fractional delay differential equations with non-instantaneous impulses are studied. Initially a brief overview of the basic two approaches in the interpretation of solutions is given. A generalization of Mittag-Leffler stability with respect to non-instantaneous impulses is given and sufficient conditions are obtained. Lyapunov functions and the Razumikhin technique will be applied and appropriate derivatives among the studied fractional equations is defined and applied. Examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

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 variable delay. Some sufficient conditions for practical stability and strong practical stability are obtained by the help with the appropriate modification of Razumikhin method and an appropriate definition of the derivative of the Lyapunov function. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

We use Lyapunov functions to study stability of the first-order Volterra integro-differential equation with Caputo fractional derivative ... For the Lyapunov functions, we consider three types of fractional derivatives. By means of these derivatives, we obtain new sufficient conditions for stability and uniformly stability of solutions We consider both constant and time variable bounded delays, and illustrated our results with an example. [ABSTRACT FROM AUTHOR]

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 several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues. [ABSTRACT FROM AUTHOR]

Lipschitz stability and Mittag-Leffler stability with initial time difference for nonlinear nonautonomous Caputo fractional differential equation are defined and studied using Lyapunov like functions. Some sufficient conditions are obtained. The fractional order extension of comparison principles via scalar fractional differential equations with a parameter is employed. The relation between both types of stability is discussed theoretically and it is illustrated with examples. [ABSTRACT FROM AUTHOR]

Recent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of the solutions. Both approaches are compared and their advantages and disadvantages are illustrated with examples. Also some existence results are derived. [ABSTRACT FROM AUTHOR]

Nonlinear differential equations with non-instantaneous impulses are studied. The impulses start abruptly at some points and their actions continue on given finite intervals. We pursue the study of Lipschitz stability using Lyapunov functions. Some sufficient conditions for Lipschitz stability, uniform Lipschitz stability, and uniform global Lipschitz stability are obtained. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]