Nonlinear third-order differential equations with anti-periodic boundary conditions and some optimal control problems (Q1895338)

The paper is divided into two parts. The first part is concerned with the existence and uniqueness of solutions to the anti-periodic boundary value problem \(u''' + f(u')u'' + h(u) = g(x,u,u',u'') + e(x)\), \(0 \leq x \leq 1\), \(u^{(j)} (0) + u^{(j)} (1) = 0\), \(j = 0,1,2\), where \(f\) and \(h\) are real continuous functions such that \(f\) is even, \(h\) is odd, \(g : [0,1] \times \mathbb{R}^3 \to \mathbb{R}\) satisfies Carathéodory type conditions, and \(e \in L^1 (0,1)\). The analysis is based on a Leray-Schauder type argument. An optimal control problem for a third-order linear differential equation in a Hilbert space, that contains a linearized version of \((P)\) as a special case, is discussed in the second part of the article by monotonicity methods.