Uniqueness Of Initial Value Problem Research Articles

On classification of singular matrix difference equations of mixed order

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

Langevin Equation Involving Three Fractional Orders

In this paper, the existence and uniqueness of initial value problems for nonlinear Langevin equation involving three fractional orders are discussed. We use a new norm that is convenient for the fractional and singular differential equations. This norm and the contraction mapping principles are the main tools for investigating the existence and uniqueness of the desired issue. The fractional derivatives are described in Caputo sense.

Local Solution of Delay Fractional Differential Equations

By Schauder’s Fixed Point Theorem and some new methods, we study the existence and uniqueness of initial value problems for nonlinear higher fractional equations with delay, and obtain some new results on local solutions.

Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses

In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses is studied. The presence of non-instantaneous impulses require appropriate definitions of impulsive conditions and initial conditions. In the paper several types of initial value problems are considered and their mild solutions are given via integral representations. In the linear case the equivalence of the solution and mild solutions is established. Conditions for existence and uniqueness of initial value problems are presented. Several examples are provided to illustrate the influence of impulsive functions and the interpretation of impulses in the RL fractional case.

On fractional Langevin equation involving two fractional orders

In numerical analysis, it is frequently needed to examine how far a numerical solution is from the exact one. To investigate this issue quantitatively, we need a tool to measure the difference between them and obviously this task is accomplished by the aid of an appropriate norm on a certain space of functions. For example, Sobolev spaces are indispensable part of theoretical analysis of partial differential equations and boundary integral equations, as well as are necessary for the analysis of some numerical methods for the solving of such equations. But most of articles that appear in this field usually use ‖.‖∞ in the space of C[a, b] which is very restrictive. In this paper, we introduce a new norm that is convenient for the fractional and singular differential equations. Using this norm, the existence and uniqueness of initial value problems for nonlinear Langevin equation with two different fractional orders are studied. In fact, the obtained results could be used for the classical cases. Finally, by two examples we show that we cannot always speak about the existence and uniqueness of solutions just by using the previous methods.

Successive approximation of neutral functional stochastic differential equations with variable delays

By using successive approximation, we prove the existence and uniqueness of initial value problem for stochastic differential equations driven by both the cylindrical Brownian motion and by the variable delays in a Hilbert space with non-Lipschitzian coefficients. Moreover, the numerical solutions are shown to converge uniformly to the analytical solutions of the stochastic differential equation.

Successive approximation to solutions of stochastic differential equations with jumps in local non-Lipschitz conditions

In this paper, by using successive approximation, the existence and uniqueness of initial value problem for stochastic differential equations driven by both Wiener process and Poisson process are studied under a local non-Lipschitz conditions. Moreover, the numerical solutions are shown to converge uniformly to the analytical solutions of the stochastic differential equation with jumps.

Initial value problems for arbitrary order fractional differential equations with delay

By fixed point theory the nonlinear alternative of Leray–Schauder type, and the properties of absolutely continuous functions space, we study the existence and uniqueness of initial value problems for nonlinear higher fractional equations with delay, and obtain some new results involving local and global solutions.

Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations

In this paper, by using the fixed point theory, we study the existence and uniqueness of initial value problems for nonlinear fractional differential equations and obtain a new result.

Global Exponential Stability and Global Convergence in Finite Time of Neural Networks with Discontinuous Activations

In this paper, we consider a general class of neural networks, which have arbitrary constant delays in the neuron interconnections, and neuron activations belonging to the set of discontinuous monotone increasing and (possibly) unbounded functions. Based on the topological degree theory and Lyapunov functional method, we provide some new sufficient conditions for the global exponential stability and global convergence in finite time of these delayed neural networks. Under these conditions the uniqueness of initial value problem (IVP) is proved. The exponential convergence rate can be quantitatively estimated on the basis of the parameters defining the neural network. These conditions are easily testable and independent of the delay. In the end some remarks and examples are discussed to compare the present results with the existing ones.

Existence and uniqueness of initial value problems for a class of second-order differential equations

Etude du probleme de la valeur initiale pour les equations differentielles du second ordre de la forme x″(t)=g(x(t), x'(t), x″(t)) qui ne peuvent pas etre resolues pour des derivees d'ordre plus elevees