Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Abstract

Let $E$ be a reflexive Banach space having a weakly sequentially continuous duality mapping $J_{\phi }$ with gauge function $\phi$, $C$ a nonempty closed convex subset of $E$, and $T:C\to 𝒦(E)$ a multivalued nonself-mapping such that $P_T$ is nonexpansive, where $P_T(x)=\{u_x\in Tx:\parallel x-u_x\parallel =d(x,Tx)\}$. Let $f:C\to C$ be a contraction with constant $k$. Suppose that, for each $v\in C$ and $t\in (\mathrm{0,1})$, the contraction defined by $S_{t}x=t{P}_{T}x+(1-t)v$ has a fixed point $x_t\in C$. Let $\{{\alpha }_n\}$, $\{{\beta }_n\},$ and $\{{\gamma }_n\}$ be three sequences in $(\mathrm{0,1})$ satisfying approximate conditions. Then, for arbitrary $x_0\in C$, the sequence $\{x_n\}$ generated by $x_n\in {\alpha }_{n}f(x_{n-1})+{\beta }_n{x}_{n-1}+{\gamma }_n{P}_T(x_n)$ for all $n\ge 1$ converges strongly to a fixed point of $T$.