Stabilization with Respect to the Parameter of the Nonlinearity Argument in a Singularly Perturbed Integro-Differential Problem

A nonlinear singularly perturbed second-order ordinary differential equation (ODE) is considered, in which one of the nonlinear arguments is nonlocal and depends on the control parameter. An asymptotic solution to the Neumann problem with an interior transition layer is constructed. The location of the transition point depends on the value of the control parameter. The effect of stabilization of the nonlocal argument with respect to the control parameter is that, as the control parameter changes over a certain range of values, the nonlocal argument changes asymptotically little. The existence of a solution with the asymptotic behavior of this type is proven, and examples of applying the general theory to specific models are considered.