Abstract
In this paper, we study Ulam-Hyers-Rassias stability of solutions for nonlocal stochastic Volterra equations. Sufficient conditions for the existence and stability of solutions are derived using the Gronwall lemma. The advantage of our model equation is that it allows for additional measurements leading to better results compared to models with local initial conditions. Examples are solved to illustrate the applications of the results.
Highlights
The study of Volterra differential and integral equations with classical initial conditions have been of interest due to their several applications in various fields of science such as semiconductors, population dynamics, heat conduction, fluid flow, etc., see [1, 2, 7, 9,10,11,12,13,14,15,16,17,18,19,20, 23]
Of interest here is the work of [14], where existence and stability of the solution of stochastic Volterra integral equation is guaranteed if the following conditions are satisfied: given a group G1, a metric group (G2, d), and > 0, there exists a δ > 0 such that the map f : G1 → G2 satisfies ρ(f, f (x)f (y)) > δ ∀ x, y ∈ G1, a homomorphism T : G1 → G2 exists and is stable provided a solution exists for such a problem
We introduce and study the stability of the following nonlocal stochastic Volterra equation dyt = F (t, yt)dt + G(t, yt)dWt y(0) = y0 + g(y), t ∈ [0, T ]
Summary
The study of Volterra differential and integral equations with classical initial conditions have been of interest due to their several applications in various fields of science such as semiconductors, population dynamics, heat conduction, fluid flow, etc., see [1, 2, 7, 9,10,11,12,13,14,15,16,17,18,19,20, 23]. There are not much literature on the theory of stochastic Volterra equations with nonlocal conditions (see [17, 19, 23] and the references therein). [14] extended the work of [8] to a class of nonlinear stochastic integral equation of Volterra type. Of interest here is the work of [14], where existence and stability of the solution of stochastic Volterra integral equation is guaranteed if the following conditions are satisfied: given a group G1, a metric group (G2, d), and > 0, there exists a δ > 0 such that the map f : G1 → G2 satisfies ρ(f (xy), f (x)f (y)) > δ ∀ x, y ∈ G1, a homomorphism T : G1 → G2 exists and is stable provided a solution exists for such a problem.
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