Abstract
The operator T from a domain D into the space of measurable functions is called a nonanticipating (causal) operator if the past information is independent from the future outputs. We will study the solution x(t) of a nonlinear operator differential equation where its changes depends on the causal operator T, and semigroup of operator A(t), and all initial parameters (t0, x0) . The initial information is described x(t)=φ(t) for almost all t ≤t0 and φ(t0) =φ0. We will study the nonlinear variation of parameters (NVP) for this type of nonanticipating operator differential equations and develop Alekseev type of NVP.
Highlights
The operator T from a domain D into the space of measurable functions is called a nonanticipating operator if the past information is independent from the future outputs
We will study the solution x(t) of a nonlinear operator differential equation where its changes depends on the causal operator T, and semigroup of operator A(t), and all initial parameters (t0, x0 )
We will study the nonlinear variation of parameters (NVP) for this type of nonanticipating operator differential equations and develop Alekseev type of NVP
Summary
An important feature of ordinary differential equations is that the future behavior of solutions depends only upon the present (initial) values of the solution. Notice that when a mapping is not nonanticipating it will be an anticipating mapping, meaning that the past and the present depend on the future. Ahangar is independent of the past input, meaning that the mapping T : Z → Z is said to be (anticausal) or anticipating if for fixed s in I = [0,a] , (Tx)(t ) = (Ty)(t ) for t > s , whenever x (t ) = y (t ) for t > s. For fixed real number s, the fact that x1 (t ) = x2 (t ) for t > s implies y1 (t ) = y2 (t ) for t > s and means that the future input {y (t ) : t > s} will affect the past
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