Existence And Algorithms Of Solutions For Generalized Strongly Nonlinear Mixed Variational-like Inequalities

In this paper, we introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities in reflexive Banach spaces.﹤N(Tu,Au,Gu),η(v,u)+b(gu,v)-b(gu,u)a(u,v-u)≥0.We construct the following auxiliary problem to prove the existence of solutions for the generalized strongly nonlinear mixed variational-like inequality.﹤N(Tw,Aw,Gw),η(v,w)+b(gu,v)-b(gu,w)+a(w,v-w)≥0.By the KKM theory and fixed point theory, we prove the existence of solutions for the generalized strongly nonlinear mixed variational-like inequality. Applying the auxiliary principle technique, we suggest the following iterative algorithms.﹤h'u_n+1-h'(u_n),v-u_n+1﹥≥-ρ﹤N(Tu_n,Au_n,Gu_n),η(v,u_n+1)﹥-ρb(gu_n,v)+ρb(gu_n,u_n+1)-ρa(u_n+1,v-u_n+1),﹤h'u_n+1-h'(u_n),η(v,u_n+1)﹥≥-ρ﹤N(Tu_n,Au_n,Gu_n),η(v,u_n+1)﹥-ρb(gu_n+1,v)+ρb(gu_n+1,u_n+1)-ρa(u_n+1,v-u_n+1).The algorithms introduce a differentiable function h involving the strongly convexity andη- strongly convexity . Using the properties of h respectively and the KKM theory, we get that the existence and uniqueness of solutions for the algorithms. Under some conditions, applying the monotony-bound principle, we discuss the convergence of iterative sequence generated by the algorithms. Several existence results of solutions involving relaxed cocoercive mapping, strongly monotone mapping, cocoercive mapping and partially relaxed monotone mappings. These results improve and generalize many known results in recent literatures.