Abstract: In this paper, by using topological methods and cone theory, the authorstudies the system of nonlinear
Hammesrstein integral equations

u(x)=∫_G kx,y)f(y,u(y),v(y)) dy,
v(x)=∫_G k(x,y)g(y,u(y),v(y))dy,

The author proves the existence of nontrivial solutions of the above system under appropriate conditions. And the main results are applied
to study the existence of nontrivial solutions of boundary value problems for systems of nonlinear second order ordinary differential equations.

Key words: System of integral equations, Cone, Fixed point, Nontrivial solution.