A projection method for pseudomonotone equilibrium problems (Q2910619)

The authors introduce a new iteration algorithm for solving equilibrium problems, given as:NEWLINENEWLINEFind \(x^* \in C\) such that \(f(x^*,y) \geq 0\) for every \(\mathcal{Y} \in C\) \(EP (f,C)\),NEWLINENEWLINEFind \(x^* \in C\) such that \(\langle F(x^*), y-x^* \rangle\) for every \(\mathcal{Y} \in C\) \(VI (F,C)\),NEWLINENEWLINEfor pseudomonotone and Lipschitz-type continuous bifunctions on a real Hilbert space \(\mathcal{H}\). Here \(C\) is a nonempty closed convex subset of \(\mathcal{H}\), \(f\) is a bifunction from \(C \times C\) to \(\mathcal{R}\) such that \(f(x,x)=0\) for every \(x \in C\), and for the problem \(VI(F,C),\) \(f\) is defined by \(f(x,y) = \langle F(x), y-x \rangle\), where \(F:C \rightarrow \mathcal{R}^n\). The presented algorithm can be considered as an improvement of extragradient-type iteration algorithms in recent papers of the authors, via the fixed point techniques. For each \(x^0 \in C\) it is proved that the sequences converge strongly to the projection of \(x^0\) on the solution set \(Sol (f,C)\) and also it is obtained a strong convergence theorem for all the sequences generated by this process.NEWLINENEWLINEMain result: A strong convergence of the sequences \(\{x^n\}\), \(\{y^n\}\), \(\{z^n\}\) and \(\{t^n\}\) defined by the algorithm (choose \(x^0 \in C\), positive sequences \(\{\lambda_n\}\) and \(\{\alpha_n\}\) satisfying the conditions: \(\{ \lambda_n \subset [a,b] \subset (0, \min \{ \frac{1}{2C_1}, \frac{1}{2C_2}\}), \{\alpha_n\} \subset [0,C]\) for some \(c \in (0,1)\)) based on the extragradient-type method and fixed point techniques for solving problem \(EP (f,C)\) in a real Hilbert space \(\mathcal{H}\), is shown. As \(C\) is a convex subset of \(\mathcal{H}\) and \(g:C \rightarrow \mathcal{R}\) is convex and subdifferentiable on \(C\), \(x^*\) is then a solution of the convex problem, \( \min \{g(x):x+C\}\), if and only if \(0 \in \partial g (x^*) + N_C (x)\), (\(\partial g (\cdot)\) denotes the subdifferentiable of \(g\) and \(N_C(x^*)\) is the normal cone of \(C\) at \(x^*\)).