By using the induction argument, (S t geq 0) holds as long as it exists.
This completes the induction argument.
This completes the base step of the induction argument for the proof of (8).
by the standard induction arguments.
By an induction argument similar to the above we easily obtain begin{aligned}& v_{n+1}geq v_{n}, qquadbiglvert v_{n+1}'(t) bigrvert geqbiglvert v_{n}'(t bigrvert, qquad v_{n+1}"(t)leq v_{n}"(t), & biglvert v_{n+1}"'(t) bigrvert geqbiglvert v_{n}"'(t) bigrvert, quad0leq tleq1, n=0,1,2,ldots.
Then we have lim_{n rightarrow+infty} dbigl(x_{n}, T(x_{n} bigr) = 0, i.e., ({x_{n}}) is an approximate fixed point sequence of T. The first part of the proposition may be found in [7, 32] and is obtained by an induction argument.
By an induction argument, it is easy to show that for all, we have (2.7).
but this follows from (3.11) by an easy induction argument.
Now we complete the actual induction argument.
Since the fixed-point iteration converges in X and the Lipschitz constant of F is proportional to τ, we obtain (11) by a standard induction argument.
