Mathematical modeling and nonlinear bilateral multivalued stochastic integral equations

Abstract

In this paper, we begin our study by exploring a hypothetical model of stochastic growth of a population, using a single-valued stochastic integral equation that incorporates the control of feeding and harvest. Taking into account the inaccuracies and uncertainties in the measurements, we are led to a nonlinear bilateral multivalued stochastic integral equation that contains multivalued stochastic integrals on both sides of the equation. Due to the possibility of absence of an element opposite to a fixed set, such an equation cannot be reduced to classical unilateral notation with the sign of sum of sets only on one side. The fundamental question arises: Is there a solution to the equation under consideration, and is it the only one? By imposing on the coefficients of the equation the condition of satisfying a certain integral inequality, we prove the existence and uniqueness of solution of the considered equation. The result is preceded by a few lemmas with the sequence of approximate solutions. We also show that solutions have the property of stability. Finally, it has been demonstrated that the results obtained can be applied to establish corresponding theorems for deterministic bilateral multivalued integral equations.