On variational methods for interval-valued functions with some applications

In this paper, we introduce the concept of sequential weak lower semicontinuity of interval-valued functions defined on a Banach space. We focus on the direct methods of the calculus of variations for interval-valued functions and extend some well-known results of F.E. Browder from scalar results to interval-valued cases. Moreover, we obtain some new results on the Palais–Smale condition and the coercivity for interval-valued functions. Finally, we apply the obtained results to interval-valued integral functions, to interval-valued optimal control problems, and to variational problems for interval-valued functions.