Simultaneous diagonal p-adic equations

It is known that a system of r additive equations of degree k with greater than 2rk variables has a non-trivial p-adic solution for all p > k(2r+2). I n this paper we consider the same system with more than crk variables, c > 2, and show the existence of a non-trivial solution for all p > r(2)k(2+(2/ (c-2))) if r not equal 1 and p > k(2+ (2/(c-1))) if r = 1.