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

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

**Abstract:**

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.

**Keywords:**

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

**MSC 2000:** 49J40, 65B05