Marek T. Malinowski

On random fuzzy differential equations

We consider a Cauchy problem in a random fuzzy setting. Under the condition of Lipschitzean right-hand side the existence and uniqueness of the solution is proven, also the continuous dependence on the right-hand side and initial condition is shown. Some kind of boundedness of the solution is established.

Interval Cauchy problem with a second type Hukuhara derivative

In this paper we consider interval differential equations. Such the equations can be appropriate in modeling of dynamical systems under presence of uncertainty of parameters. We study an interval initial value problem with a second type Hukuhara derivative. By an example of real-world application we indicate the advantages of the usage of such a kind of interval-valued derivative. A continuous dependence of the solution on initial value and right-hand side of the equation is shown. The existence of approximate local solutions is proven, and then it is used in the derivation of existence of at least one local solution to interval Cauchy problem with second type Hukuhara derivative. The compactness of solutions set is also stated. Finally, the explicit formulae for local solutions to linear interval differential equations are provided.

Interval differential equations with a second type Hukuhara derivative

Abstract In this paper, the convergence theorem and continuous dependence on initial data are proved for first order interval differential equations via comparison principle. Our results generalize some known results under weaker conditions. In this study, we exploit a recently introduced concept of interval-valued derivatives.

Random fuzzy fractional integral equations – theoretical foundations

Abstract This paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order. We consider two different kinds of such equations. Their solutions have different geometrical properties. The equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values. On the other hand, the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values. Firstly, the existence and uniqueness of solutions is investigated. This is showed by using a method of successive approximations. An estimation of error of n th approximation is given. Also a boundedness of the solution is indicated. To show well-posedness of the considered theory, we prove that solutions depend continuously on the data of the equations. Some concrete examples of random fuzzy fractional integral equations are solved explicitly.

Some properties of strong solutions to stochastic fuzzy differential equations

We consider stochastic fuzzy differential equations driven by m-dimensional Brownian motion. Such equations can be useful in modeling of hybrid dynamic systems, where the phenomena are subjected to two kinds of uncertainties: randomness and fuzziness, simultaneously. Under a boundedness condition, which is weaker than linear growth condition, and the Lipschitz condition we obtain existence and uniqueness of solution to stochastic fuzzy differential equations. Solutions, which are fuzzy stochastic processes, and their uniqueness are considered to be in a strong sense. An estimation of error of the Picard approximate solution is established. We give a boundedness type result for the solution defined on finite time interval. Also the stabilities of solution on initial condition and coefficients of the equation are shown. The existence and uniqueness of a solution defined on infinite time interval is proven. Finally, some applications of fuzzy stochastic differential equations are considered. All the results presented in this paper apply to set-valued stochastic differential equations.

Itô type stochastic fuzzy differential equations with delay

Abstract In the paper we give some foundations for the studies of stochastic fuzzy delayed differential equations. We prove the existence and uniqueness of solutions to such the equations. To obtain our result we assume that the coefficients of the equation satisfy the Lipschitz condition together with linear growth condition. We estimate the distance between approximate solution and exact solution. Also the stability of solution with respect to the initial history is shown. An application of stochastic fuzzy delayed differential equations in the modeling of population growth is indicated.

Strong solutions to stochastic fuzzy differential equations of Itô type

Abstract In the paper we consider the fuzzy stochastic integrals and give some of their properties. Then we study the existence of solutions to the stochastic fuzzy differential equations driven by multidimensional Brownian motion. The solutions and their uniqueness are considered to be in a strong sense.

Second type Hukuhara differentiable solutions to the delay set-valued differential equations

Abstract In this paper we consider the delay set-valued differential equations with a recently introduced notion of a second type Hukuhara derivative. Under condition that the right-hand side of the equation is Lipschitzian in the functional variable we obtain the existence and uniqueness of the solution to such the equations. Some properties of the solutions are established. Existence of at least one solution is also proved. Application of the second type Hukuhara differentiability concept to the problems of set-valued differential equations implies that the diameter of the solution values is a function nonincreasing in time.

On set differential equations in Banach spaces - a second type Hukuhara differentiability approach

In this paper we consider set differential equations in Banach spaces and with the second type Hukuhara derivative. We show that a sequence of successive approximations converges uniformly to a unique local solution of a Cauchy problem. Some comparison results are formulated and a continuity of solution with respect to equation data is shown. The existence of a global solution is also established.

Approximation schemes for fuzzy stochastic integral equations

We consider fuzzy stochastic integral equations with stochastic Lebesgue trajectory integrals and fuzzy stochastic Ito trajectory integrals. Some methods of construction of approximate solutions to such the equations are examined. We study the Picard type approximations, the Caratheodory type approximations and the Maruyama type approximations of solutions. In considered framework, the solutions and approximate solutions are mappings with values in the space of fuzzy sets with basis of square integrable random vectors. Under Lipschitz and linear growth conditions each sequence of considered approximate solutions converges to the exact unique solution of the given fuzzy stochastic integral equation. For each type of approximate solutions, we show that the sequence of approximations is uniformly bounded and obtain some bounds for a distance between nth approximation and exact solution.