Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces

Let EE be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let KK be a nonempty closed convex subset of EE, and let T:K⟶ET:K⟶E be a continuous pseudocontraction which satisfies the weakly inward condition. For f:K⟶Kf:K⟶K any contraction map on KK, and every nonempty closed convex and bounded subset of KK having the fixed point property for nonexpansive self-mappings, it is shown that the path x→xt,t∈[0,1)x→xt,t∈[0,1), in KK, defined by xt=tTxt+(1−t)f(xt)xt=tTxt+(1−t)f(xt) is continuous and strongly converges to the fixed point of TT, which is the unique solution of some co-variational inequality. If, in particular, TT is a Lipschitz pseudocontractive self-mapping of KK, it is also shown, under appropriate conditions on the sequences of real numbers {αn},{μn}{αn},{μn}, that the iteration process: z1∈Kz1∈K, zn+1=μn(αnTzn+(1−αn)zn)+(1−μn)f(zn),n∈Nzn+1=μn(αnTzn+(1−αn)zn)+(1−μn)f(zn),n∈N, strongly converges to the fixed point of TT, which is the unique solution of the same co-variational inequality. Our results propose viscosity approximation methods for Lipschitz pseudocontractions.