Correction

This article refers to:

Nonsmooth variational inequalities on Hadamard manifolds

Article title: Nonsmooth variational inequalities on Hadamard manifolds

Authors: Qamrul Hasan Ansari, Monirul Islam and Jen-Chih Yao

Journal: APPLICABLE ANALYSIS

DOI: 10.1080/00036811.2018.1495329

When the article was first published online, there were some errors in certain equations. The equations should be set as follows :

1. Errata

Page 3, line +2: “${\gamma }_{xw}\left(t\right)={\gamma }_{xy}\left(st\right)={\exp }_x(st{\exp }_x^{-1}y)$” should be replaced by “${\gamma }_{wx}\left(t\right)={\gamma }_{xy}(-st+s)={\exp }_x(-st+s){\exp }_x^{-1}y$”.

Page 3, line +5: “${\exp }_x(t{\exp }_x^{-1}w)={\exp }_x(st{\exp }_x^{-1}y)$” should be replaced by “${\exp }_w(t{\exp }_w^{-1}x)={\exp }_x(-st+s){\exp }_x^{-1}y$”.

Page 3, line +15: “${\exp }_x^{-1}w=s{\exp }_x^{-1}y$” should be replaced by “${\exp }_w^{-1}x=-s{P}_{w,x}{\exp }_x^{-1}y$”, where P_w,x represents the parallel transport from T_xM to T_wM.

Page 3, line +18: “${\exp }_w^{-1}y=(1-s){\exp }_x^{-1}y$” should be replaced by “${\exp }_w^{-1}y=(1-s)P_{w,x}{\exp }_x^{-1}y$”.

Page 3, line +20: “Since ${\exp }_q^{-1}p=-{\exp }_p^{-1}q$ for all $p,q\in M$, (4) can be written as” should be replaced by “In a similar way, we obtain”.

Page 3, line +22: “${\exp }_y^{-1}w=(1-s){\exp }_y^{-1}x$” should be replaced by “${\exp }_w^{-1}x=s{P}_{w,y}{\exp }_y^{-1}x$”.

Page 7, line +26: “$h({\exp }_{x}t{\exp }_x^{-1}y;{\exp }_y^{-1}x)\le 0\Rightarrow h(x;{\exp }_x^{-1}y)\ge 0$” should be replaced by “$h(w_t;P_{w_t,y}{\exp }_y^{-1}x)\le 0\Rightarrow h(x;{\exp }_x^{-1}y)\ge 0$ where $w_t={\exp }_{x}t{\exp }_x^{-1}y$”.

Page 8, line +10: “By (3), ${\exp }_{y_t}^{-1}\overline{x}=t{\exp }_y^{-1}\overline{x}$” should be replaced by “By (5), ${\exp }_{y_t}^{-1}\overline{x}=t{P}_{y_t,y}{\exp }_y^{-1}\overline{x}$”.

Page 8, line +12: “$h(y_t;{\exp }_y^{-1}\overline{x})\le 0$” should be replaced by “$h(y_t;P_{y_t,y}{\exp }_y^{-1}\overline{x})\le 0$”.

Page 13, line +26: “$h(w;{\exp }_y^{-1}x)<0\text{and}h(w;{\exp }_x^{-1}y)<0$” should be replaced by “$h(w;-P_{w,x}{\exp }_x^{-1}y)<0\text{and}h(w;P_{w,x}{\exp }_x^{-1}y)<0$”.

Page 13, line +29: “$h(w;{\exp }_y^{-1}x)+h(w;{\exp }_x^{-1}y)<0$” should be replaced by “$h(w;-P_{w,x}{\exp }_x^{-1}y)+h(w;P_{w,x}{\exp }_x^{-1}y)<0$”.

Page 13, line +32: “$h(w;{\exp }_y^{-1}x)<-h(w;-{\exp }_y^{-1}x)$” should be replaced by “$h(w;P_{w,x}{\exp }_x^{-1}y)<-h(w;-P_{w,x}{\exp }_x^{-1}y)$”.

Page 14, line -7: “$f_D(w_t;{\exp }_x^{-1}y)\ge f\left(y\right)-f\left(x\right)$” should be replaced by “$f_D(w_t;P_{w_t,x}{\exp }_x^{-1}y)\ge f\left(y\right)-f\left(x\right)$”.

Page 15, line +6: “$f_D(w_t;{\exp }_y^{-1}\overline{x})\ge f\left(\overline{x}\right)-f\left(y\right)$” should be replaced by “$f_D(w_t;P_{w_t,y}{\exp }_y^{-1}\overline{x})\ge f\left(\overline{x}\right)-f\left(y\right)$”.

Page 15, line +8 “By (4), we have ${\exp }_{w_t}^{-1}\overline{x}=(1-t){\exp }_y^{-1}x$” should be replaced by “By (4), we have ${\exp }_{w_t}^{-1}\overline{x}=(1-t)P_{w_t,y}{\exp }_y^{-1}\overline{x}$”.

The authors apologize for this error.