A NEW ℱα-TYPE INTEGRABLE FUNCTIONS AND RETARDED FRACTAL DIFFERENTIAL EQUATIONS

The gauge integral, also known as the Henstock–Kurzweil integral, is a generalization of the Riemann integral. Functions that are not integrable due to singularities in the contexts of Lebesgue or Riemann integration can still be classified as gauge integrable. In this paper, vector-valued gauge integral type [Formula: see text]-integral is introduced. Several properties of the newly introduced integral have been discussed. A measure of noncompactness has been established for fractal sets. In applications, along with some additional conditions and the newly introduced fractal measure of noncompactness, we prove an existence theorem for the fractal retarded differential equation [Formula: see text] whenever [Formula: see text] is a vector-valued [Formula: see text]-integrable and [Formula: see text] is fractal differentiation.