How do you think the ticks should be "bended"? I think you mean with this that they should move to the "inner part" of the axis, but this is not (easily) possible I think.
Here I present two solutions without "bending" the ticks. For more details have a look at the comments in the code.
% used PGFPlots v1.14
\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
% define some macros that hold the min and max values for the axes ...
\pgfmathsetmacro{\xmin}{-10}
\pgfmathsetmacro{\xmax}{10}
\pgfmathsetmacro{\ymin}{-1}
\pgfmathsetmacro{\ymax}{1}
\begin{axis}[
% and apply the values here
xmin=\xmin,
xmax=\xmax,
ymin=\ymin,
ymax=\ymax,
grid=both,
axis line style={rounded corners}, % = rounded corners=4pt
% -----
% activate the `set layers' feature to be sure that the extra
% ticks are drawn below the axis lines
set layers,
% then add extra ticks in the four corners
% (unfortunately `\pgfkeysvalueof{/pgfplots/xmin}' and friends
% can't be used in the axis options, so you have to provide the
% values manually or you have do define commands like I did,
% so you only have one place if you want to change the values)
extra x ticks={\xmin,\xmax},
extra y ticks={\ymin,\ymax},
% and finally set the extra tick style to draw thick white lines
% ("thick" because the grid lines are drawn thick by default)
% (of course this means that you will see the white lines,
% if you change the background color!)
extra tick style={
grid=none,
tick style={
draw=white,
thick,
},
},
% -----
% % alternatively just enlarge the limits a bit, so the ticks
% % are outside of the "rounded corners part"
% % (the default value of `rounded corners' is 4pt)
% enlargelimits={abs=4pt},
% -----
]
\addplot [
samples=100,
color=blue,
very thick,
smooth,
domain=-10:10,
] {cos(x^(3))};
\end{axis}
\end{tikzpicture}
\end{document}