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An example of solving a boundary-value problem for a homogeneous Monge-Ampère equation is given, which produces a Bellman function for an extremal problem on the space BMO. The paper contains a step-by-step instruction for calculation of this function. Cases of rather complicated foliations are considered. This illustrates the technique elaborated in a paper by Ivanishvili, Stolyarov, Vasyunin, and Zatitskiy. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

The present paper is devoted to optimal control problems whose behavior is described by quasilinear first-order differential equations on the plane with nonlocal Bitsadze-Samarski boundary conditions. A theorem on the existence and uniqueness of a generalized solution in the space $$ {C}_{\mu}\left(\overline{G}\right) $$ is proved for quasilinear differential equations; necessary optimality conditions are obtained in terms of the maximum principle; the Bitsadze-Samarski boundary-value problem is examined for a first-order linear differential equation; the existence of a solution in the space $$ {C}_{\mu}^p\left(\overline{G}\right) $$ is proved, and an a priori estimate is derived. A necessary and sufficient optimality condition is proved for a linear optimal control problem. [ABSTRACT FROM AUTHOR]

This paper investigates some nonlinear third-order ordinary differential equations and inclusions with anti-periodic type integral boundary conditions and multi-strip boundary conditions. To establish the existence results for the given problems, we apply standard tools of fixed point theory for single-valued and multi-valued problems. The results are illustrated with the aid of examples. [ABSTRACT FROM AUTHOR]

The combined compact difference (CCD) scheme has better spectral resolution than many other existing compact or noncompact high-order schemes, and is widely used to solve many differential equations. However, due to its implicit nature, very little theoretical results on the CCD method are known. In this paper, we provide a rigorous theoretical proof for the unique solvability of the CCD scheme for solving the convection-diffusion equation with variable convection coefficients subject to periodic boundary conditions. [ABSTRACT FROM AUTHOR]