scholarly journals Investigation of the behavior of solutions of differential systems with argument deviation

2018 ◽  
Author(s):  
N. V. Vareh ◽  
O. Y. Volfson ◽  
O. A. Padalka
Keyword(s):  
Differential Equations ◽  
Asymptotic Properties ◽  
Time Interval ◽  
Infinite Time ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Infinite Time Interval

In this paper systems of differential equations with deviation of an argument with nonlinearity of general form in each equation are considered. The asymptotic properties of solutions of systems with a pair and odd number of equations on an infinite time interval are studied

scholarly journals Generalized Memory: Fractional Calculus Approach

2018 ◽  
Vol 2 (4) ◽  
pp. 23 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Power Law ◽  
Memory Function ◽  
Time Interval ◽  
Infinite Time ◽  
Suggested Approach ◽  
Infinite Time Interval ◽  
History Of ◽  
With Memory

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

scholarly journals Investigation of asymptotic behavior of solutions of one class of systems of differential equations with deviation of an argument

2019 ◽  
Author(s):  
N. V. Varekh ◽  
N. L. Kozakova ◽  
A. O. Lavrentieva
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Mathematical Problem ◽  
Asymptotic Properties ◽  
Theoretical Physics ◽  
General Nature ◽  
Asymptotic Behavior Of Solutions ◽  
Time Interval ◽  
Systems Of Differential Equations ◽  
Derivatives Of

In this paper, we study the asymptotic behavior of solutions at an infinite time interval of one class of systems of differential equations with the deviation of an argument, which are a generalization of the Emden-Fowler equation in the sublinear case. Conditions were found under which each solution either oscillates strongly or all its components monotonically end to zero at infinity. Two theorems under different constraints on the deviation of an argument are proved. Equation d(n)y(t)/dtn + δ p(t)f(y(t)) = 0, f(u) = uα, δ = -1 or 1, has been the object of much research. Some cases of this equation are models of processes in theoretical physics (Emden, Fowler, Fermi equations). After that, this physical problem becomes a mathematical problem at an infinite interval. It is found that the asymptotic properties of the solutions depend on the sign δ, type of nonlinearity f(u) (f(u) = uα), (0< α <1 – sublinear case, α = 1 – linear case, α >1 – superlinear), n – even or odd. For this equation, conditions have already been found under which, when δ = 1 and n are even, all solutions oscillates; if n is odd, then each solution either oscillates or monotonically goes to zero indefinitely. If δ = -1, n is even, then each solution oscillates either monotonically to zero or to infinity when t → ∞ together with the derivatives of order (n -1). If δ = -1, n is odd, then each solution oscillates or is monotonically infinite for t → ∞ together with the derivatives of order (n -1). Then, the following results were obtained for differential systems and equations with the general nature of the argument rejection (differential-functional equations). The next stage of the study is to summarize the results for such systems. This article investigates the system of differential equations with the deviation of the argument for the case δ = 1, n = 3. The obtained results are refined and the results obtained earlier are generalized. Two theorems with different assumptions about rejection of the argument by analytical methods are proved. These theorems have different applications. The results of the study are a generalization of the sublinear case for odd n.

scholarly journals Infinite time interval BSDEs and the convergence of g-martingales

2000 ◽  
Vol 69 (2) ◽  
pp. 187-211 ◽  
Author(s):  
Zengjing Chen ◽  
Bo Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Unique Solution ◽  
Backward Stochastic Differential Equations ◽  
Time Interval ◽  
Sufficient Condition ◽  
Infinite Time ◽  
Infinite Time Interval

AbstractIn this paper, we first give a sufficient condition on the coefficients of a class of infinite time interval backward stochastic differential equations (BSDEs) under which the infinite time interval BSDEs have a unique solution for any given square integrable terminal value, and then, using the infinite time interval BSDEs, we study the convergence of g-martingales introduced by Peng via a kind of BSDEs. Finally, we study the applications of g-expectations and g-martingales in both finance and economics.

scholarly journals Stability of Nonlinear Systems of Fractional Order Differential Equations

2010 ◽  
Vol 7 (4) ◽  
pp. 1458-1461
Author(s):  
Baghdad Science Journal
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Fractional Order ◽  
Finite Time ◽  
Time Interval ◽  
Sufficient Condition ◽  
Finite Time Interval ◽  
Infinite Time ◽  
Fractional Order Differential Equations ◽  
Infinite Time Interval

In this paper, a sufficient condition for stability of a system of nonlinear multi-fractional order differential equations on a finite time interval with an illustrative example, has been presented to demonstrate our result. Also, an idea to extend our result on such system on an infinite time interval is suggested.

scholarly journals Lp solutions of infinite time interval backward doubly stochastic differential equations

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1857-1868 ◽  
Author(s):  
Zhaojun Zong ◽  
Feng Hu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Time Interval ◽  
Existence And Uniqueness Theorem ◽  
Infinite Time ◽  
Doubly Stochastic ◽  
Infinite Time Interval ◽  
Lp Solutions

In this paper, we study the existence and uniqueness theorem for Lp (1 < p < 2) solutions to a class of infinite time interval backward doubly stochastic differential equations (BDSDEs). Furthermore, we obtain the comparison theorem for 1-dimensional infinite time interval BDSDEs in Lp.

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