The following result that provides some bounds for continuous functions of bounded variation may be stated as well.
From the function-theoretic point of view, Hilbert's 13th problem can be exactly characterized as the superposition representability problem for continuous functions of several variables.
We give a necessary and sufficient condition for convergence of this iteration for continuous functions on an arbitrary interval.
Phuengrattana and Suantai [7] considered the convergence of Noor iteration for continuous functions on an arbitrary interval in the real line.
One proves that ([f(H -f(H -fatH {mathcal)]J_{C}}hcal {C}}) is a compact operator for continuous functions, f of compact support.
For continuous functions, a number of particular forms of Theorem 9.4 have appeared for convex subsets of Hausdorff topological vector spaces as follows: (1 Nash [5, Theorem 1] where are subsets of Euclidean spaces, (2 Nikaido and Isoda [60, Theorem 3.2], (3 Fan [10, Theorem 4], (4)Tan et al. [61, Theorem 2.1].
For continuous functions f the value of f(t), for t in the interval, changes only slightly, so it must be very close to f(t).
In these investigations the method of integral inequalities for continuous functions is generalized to the case of piecewise continuous (one-dimensional inequalities) and discontinuous (multidimensional inequalities) functions.
For continuous functions and for convex subsets of Hausdorff topological vector spaces, Theorem 9.6 was due to Ma [12, Theorem 4].
In [1], Gao and Meng established a generalized Gronwall-Bellman type integral inequality, named Mate-Nevai type nonlinear integral inequality for continuous functions, which is one case of inequalities containing integration on infinite intervals for continuous functions.
