This paper deals with weak solution in weighted Sobolev spaces, of three-point boundary value problems which combine Dirichlet and integral conditions, for linear and quasilinear parabolic equations in a domain with curved lateral boundaries. We, firstly, prove the existence, uniqueness, and continuous dependence of the solution for the linear equation. Next, analogous results are established for the quasilinear problem, using an iterative process based on results obtained for the linear problem.

We consider second-order differential inclusions on a Riemannian manifold with lower semicontinuous right-hand sides. Several existence theorems for solutions of two-point boundary value problem are proved to be interpreted as controllability of special mechanical systems with control on nonlinear configuration spaces. As an application, a statement of controllability under extreme values of controlling force is obtained.

This paper is concerned with existence results for inequality problems of type $F^0\left(u;v\right)+{\Psi }^{\prime }\left(u;v\right)\ge 0$, for all $v\in X$, where $X$ is a Banach space, $F:X\to ℝ$ is locally Lipschitz, and $\Psi :X\to (-\infty +\infty ]$ is proper, convex, and lower semicontinuous. Here $F^0$ stands for the generalized directional derivative of $F$ and ${\Psi }^{\prime }$ denotes the directional derivative of $\Psi$. The applications we consider focus on the variational-hemivariational inequalities involving the $p$-Laplacian operator.

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.