  This paper is the first in a series of paper where we describe the
differential operators on general nonlinear metric measure spaces, namely, the
Finsler spaces. We try to propose a general method for gradient estimates of
the positive solutions to nonlinear parabolic (or elliptic) equations on both
compact and noncompact forward complete Finsler manifolds. The key of this
approach is to find a valid comparison theorem, and particularly, to find
related suitable curvature conditions. Actually, we define some non-Riemannian
tensors and a generalization of the weighted Ricci curvature, called the mixed
weighted Ricci curvature. With the assistance of these concepts, we prove a new
kind of Laplacian comparison theorem. As an illustration of such method, we
apply it to Finslerian Schr\"odinger equation. The Schr\"odinger equation is a
typical class of parabolic equations, and the gradient estimation of their
positive solution is representative. In this paper, based on the new Laplacian
comparison theorem, we give global and local Li-Yau type gradient estimates for
positive solutions to the Finslerian Schr\"odinger equation with respect to two
different mixed weighted Ricci curvature conditions.
