Existence, Uniqueness and C −Differentiability of Solutions in a Non-linear Model of Cancerous Tumor

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 Fr´echet. 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 Fr´echet. In the second result, differentiability is understood in a weaker sense than that of Fr´echet. For the proof we use Hadamard’s theorem of small perturbations of Banach isomorphism of spaces as well as the notion of strict differentiability.