Geometric inequalities in real Banach spaces with applications

In this paper, new geometric inequalities are established in real Banach spaces. As an application, a new iterative algorithm is proposed  for approximating a solution of a split equality fixed point problem (SEFPP) for a quasi--nonexpansive semigroup. It is proved that the sequence generated by  the algorithm converges {\it strongly} to a solution of the SEFPP in -uniformly convex and uniformly smooth real Banach spaces, . Furthermore, the theorem proved is applied to approximate a solution of a variational inequality problem. All the theorems proved are applicable, in particular, in , and the Sobolev spaces, , for such that