Duality Fixed Point and Zero Point Theorems and Applications

Abstract

The following main results have been given. (1) Let $E$ be a $p$-uniformly convex Banach space and let $T:E\to E^{*}$ be a $(p-1)\text{-}L$-Lipschitz mapping with condition . Then $T$ has a unique generalized duality fixed point $x^{*}\in E$ and (2) let $E$ be a $p$-uniformly convex Banach space and let $T:E\to E^{*}$ be a $q\text{-}\alpha \text{-}$inverse strongly monotone mapping with conditions $1/p+1/q=1$, . Then $T$ has a unique generalized duality fixed point $x^{*}\in E$. (3) Let $E$ be a $2$-uniformly smooth and uniformly convex Banach space with uniformly convex constant $c$ and uniformly smooth constant $b$ and let $T:E\to E^{*}$ be a $L$-lipschitz mapping with condition . Then $T$ has a unique zero point $x^{*}$. These main results can be used for solving the relative variational inequalities and optimal problems and operator equations.