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.
We are concerned with the fuzzy stochastic differential equations driven by multidimensional Brownian motion viewed as a tool used to describe the behavior of dynamic systems operating in fuzzy ...environments with stochastic noises. Under the uniform Lipschitz condition, we prove the local uniqueness theorem for the solutions of fuzzy stochastic differential equations. Next we show, assuming the Lipschitz condition is satisfied only locally, that these equations have a unique solution. The fact that the solution is bounded is also proved. We conclude the paper with a number of corresponding results holding for the deterministic fuzzy differential equations and set-valued stochastic differential equations with local Lipschitz condition.
A new sutureless technique used for repositioning and scleral fixation of the capsular bag-intraocular lens (IOL) complex in the surgical treatment of subluxated lenses is described. Iris retractors ...were used not only to induce a tent effect on the capsule but also to permanently fix the capsular bag to the sclera in this method, without the need to prepare scleral or conjunctival flaps. Surgery with the use of a capsular tension ring (CTR) and iris retractors, the ends of which were brought out through the sclera and cauterized, was performed in 7 eyes of 7 patients with moderate or severe subluxation of the crystalline lens. In all cases, simultaneous use of a CTR and iris retractors ensured good centration of the capsular bag-IOL complex. The method was safe and effective in fixing the capsule to the sclera in the case of significant damage to the ligamentous apparatus of the lens.
In the paper, we consider functional set-valued integral equations whose representation contains set-valued integrals occurring symmetrically on both sides of the equation. On the coefficients of the ...equation, we impose certain conditions, more general than the standard Lipschitz condition, which allow the application of the Bihari–LaSalle inequality in the proofs of the obtained theorems. In this way, we obtain a result about the existence and uniqueness of the solution of the equation under consideration and the insensitivity of the solution in the case of minor changes in the parameters of the equation.
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.
In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have ...values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is provided as well. We show the closeness of the unique solutions when the equations differ slightly.
There is experimental evidence of high vibronic activity that accompanies the allowed transition between the ground state and the lowest electronic singlet excited state of oligofurans that contain ...two, three, and four furan rings. The absorption and emission spectra of the three lowest oligofurans measured at liquid nitrogen temperature show distinct fine structures that are reproduced using the projection-based model of vibronic coupling (with Dushinsky rotation included) parameterized utilizing either Density Functional Theory (DFT, with several different exchange-correlation functionals) or ab initio (CC2) quantum chemistry calculations. Using as a reference the experimental data concerning the electronic absorption and fluorescence for the eight lowest oligofurans, we first analyzed the performance of the exchange-correlation functionals for the electronic transition energies and the reorganization energies. Subsequently, we used the best functionals alongside with the CC2 method to explore how the reorganization energies are distributed among the totally symmetric vibrations, identify the normal modes that dominate in the fine structures present in the absorption and emission bands, and trace their evolution with the increasing number of rings in the oligofuran series. Confrontation of the simulated spectra with the experiment allows for the verification of the performance of the selected DFT functionals and the CC2 method.
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.
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.
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.
PDF
