Ravi P. Agarwal, Donal O'Regan, Ram U. Verma

Two-Step Systems For G-H-Relaxed Pseudococoercive Nonlinear Variational Problems Based On Projection Methods

The approximation-solvability of a generalized system of nonlinear variational inequalities (SNVI) involving relaxed pseudococoercive mappings, based on the convergence of a system of projection methods, is presented. The class of relaxed pseudococoercive mappings is more general than classes of strongly monotone and relaxed cocoercive mappings. Let $K_1$ and $K_2$ be nonempty closed convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. The two-step SNVI problem considered here is as follows: find an element $(x^*,y^*)\in H_1\times H_2$ such that $(g(x^*),g(y^*))\in K_1\times K_2$ and
$$\langle S(x^*,y^*),g(x)-g(x^*)\rangle \geq 0\;\;\;\forall\; g(x)\in K_1,$$
$$\langle T(x^*,y^*),h(y)-h(y^*)\rangle \geq 0\;\;\;\forall\; h(y)\in K_2,$$
where $S:H_1\times H_2\to H_1, T: H_1\times H_2\to H_2, g:H_1\to H_1$ and $h:H_2\to H_2$ are nonlinear mappings.

Cocoercive mapping, relaxed cocoercive mapping, relaxed pseudococoercive variational inequality, two-step system of variational inequalities, convergence of projection methods.

MSC 2000: 49J40, 65B05