Nonlinear Stochastic Impulsive Integro-Differential Equations in Locally Convex Space with Fr´echet Topology
Ключевые слова:
Fr´echet space, stochastic differential equation, impulsive system, integrodifferential equation, initial value problem, fixed point, successive approximations, existence and uniquenessАннотация
This paper studies an initial value problem for a nonlinear system of first-order
stochastic impulsive integro-differential equations whose states take values in a Fr´echet space.
The model combines Wiener perturbations, a nonlocal integral interaction, and instantaneous
state jumps at prescribed impulse times. In contrast to the classical Banach-space setting, the
topology of the state space is described by a countable separating family of seminorms, and therefore the solvability analysis must be formulated simultaneously with respect to all seminorms.
We introduce a Fr´echet space of adapted piecewise continuous stochastic processes endowed with
a complete translation-invariant metric generated by a countable family of seminorms. The
original impulsive stochastic differential problem is transformed into an equivalent stochastic
functional-integral equation. Under seminorm-wise Lipschitz conditions on the drift, diffusion,
integral kernel, and impulse operators, together with suitable square-integrability and compactness assumptions, we prove existence of a solution by the Schauder–Tychonoff fixed-point principle. Under a uniform contraction condition with respect to the defining seminorms, uniqueness
is established and the successive approximation method is shown to converge in the Fr´echet topology. The results provide a framework for extending the theory of nonlinear stochastic impulsive
integro-differential equations from Banach spaces to a broad class of complete locally convex
spaces.
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