Nonlinear third-order differential equations with anti-periodic boundary conditions and some optimal control problems
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.
third-order linear differential equation
scientific article; zbMATH DE number 786130