Some results about conformable derivatives in banach spaces and an application to the partial differential equations

In the paper we discuss conformable derivative behavior in arbitrary Banach spaces and clear connection between two derivatives of different order. As a consequence obtain important result that an abstract function has at point (which does not coincide with lower terminal derivative) if only it first order same point. application obtained results prove existence weak solution mixed (initial/boundary) problem for parabolic partial differential equation on time is equivalent to appropriate considered integer derivative.