Variation of Parameters for Causal Operator Differential Equations

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 ( ), t x . The initial information is described ( ) ( ) KeywordsNonlinear Operator Differential Equations (NODE), Variation of Parameters, Nonanticipating (Causal), Alekseev Theorem Definitions and Example of Nonanticipative OperatorsAn important feature of ordinary differential equations is that the future behavior of solutions depends only upon the present (initial) values of the solution. There are many physical and social phenomena which have hereditary dependence. That means the future state of the system depends not only upon the present state, but also upon past information (see [1]-[6]).Twins before the time of conception share all of their genetic history and may go to a different path in their future life. We are going to study the phenomenon which can be formulated in principle that the "present" events are independent of the "future". These kinds of events are called nonanticipation or causal events. This system has no after-effect or "no memory". Naylor and Sell 1982 [7] 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. Anticipating (anticausal) Mapping: This is a mapping that the future output Nonanticipating Operator Differential EquationNotations. Let S be the interval of all nonpositive numbers. Let I be the compact According to these two definitions,For any Banach space Y and Z, let When an operator T is nonanticipating, the future values of the input will have no effect on the present state. One can prove that the composition and the Cartesian product of nonanticipating and Lipschitzian operators are Nonanticipating and Lipschitzian. Furthermore, the operator F induced by the function f is a well defined, nonanticipating, and Lipschitzian operator. T y t T z t = for almost all t s < . T T y t T y t T y tThese two equalities will imply that T y T y t T z T z t + = +for almost all t s T T y t T T z t T y t T z t T T y t T T z t T y T y t T z T z t T y t T z t T y t T z tSince both operators are Lipschitzian, the right hand side will berepresents the ess.sup norm in the space measurable1, 2 i = and take the essential supremum norm on the left hand side of the above relation then it will be ( ) ( )for all y and z in the domain D. This proves that the direct sum operator is where r is the growth rate of the species y, and K is called the environment capacity for y. 2) The nonlinear system in this paper includes all evolutionary equations of C 0 semigrop of operators.3) Instead of continuity of the nonlinear functions t y t T y t , wewill replace the more general form of these functions in Banach spaces to be Bochner measurabl...