Abstract: In this paper we define a fixed point index theory for locally Lip., completely continuous and weakdy inward mappings defined on closed convex sets in general Banach spaces where no other artificial conditions are imposed. This makes us to deal with these kinds of mappings more easily. As obvious applications, some results in[3],[5],[7],[9],[10] are deepened and extended.

Key words: Weakly inward mappings, Fixed point index theory, Differential equations in Banach spaces