Convergence theorems of solutions of a generalized variational inequality

The convex feasibility problem (CFP) of finding a point in the nonempty intersection boolean AND(r)(m=1) C-m is considered, where r >= 1 is an integer and each C-m is assumed to be the solution set of a generalized variational inequality. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A(m), B-m : C -> H be relaxed cocoercive mappings for each 1 <= m <= r. It is proved that the sequence {x(n)} generated in the following algorithm: x(1) is an element of C, x(n+1) = alpha(n)u + beta(n)x(n) + gamma(n) Sigma(r)(m=1) delta P-(m,n)(C)(tau(m)B(m)x(n) - lambda(m)A(m)x(n)), n >= 1, where u is an element of C is a fixed point, {alpha(n)}, {beta(n)}, {gamma(n)}, {delta((1,n))}, ... , and {delta({r,n))} are sequences in (0, 1) and {tau(m)}(m=1)(r), {lambda(m)}(m=1)(r) are positive sequences, converges strongly to a solution of CFP provided that the control sequences satisfies certain restrictions.