Ekeland’s variational principle for interval-valued functions

In this paper, we attempt to propose Ekeland’s variational principle for interval-valued functions (IVFs). To develop the variational principle, we study a concept of sequence of intervals. In the sequel, the idea of gH -semicontinuity for IVFs is explored. A necessary and sufficient condition for an IVF to be gH -continuous in terms of gH -lower and upper semicontinuity is given. Moreover, we prove a characterization for gH -lower semicontinuity by the level sets of the IVF. With the help of this characterization result, we ensure the existence of a minimum for an extended gH -lower semicontinuous, level-bounded and proper IVF. To find an approximate minima of a gH -lower semicontinuous and gH -Gâteaux differentiable IVF, the proposed Ekeland’s variational principle is used.