In this paper, we prove the existence and uniqueness of the weak solution of a system of nonlinear equations involved in the mathematical modeling of cancer tumor growth with a non homogeneous divergence condition. We also present  a new concept of generalized differentiation of non linear operators : C-differentiability. Through this notion, we also prove the uniqueness and the C-differentiability of the solution when the system is perturbed by a certain number of parameters. Two results have been established. In the first one, differentiability is according to Fréchet. The proof is given uses the theorem of reciprocal functions in Banach spaces. First of all, we give the proof of strict differentiability of a direct mapping, according to Fréchet. In the second result, differentiability is understood in a weaker sense than that of Fréchet. For the proof we use Hadamard's theorem of small perturbations of Banach isomorphism of spaces as well as the notion of strict differentiability.

The main goal of this paper is to establish the first order necessary optimality conditions for a tumor growth model that evolves due to cancer cell proliferation. The phenomenon is modeled by a system of three-dimensional partial differential equations. We prove the existence and uniqueness of optimal control and necessary conditions of optimality are established by using the variational formulation.