This may not always be true, but when it is verified, it allows the scientist to do mathematical physics by making successive approximations
. If the phenomenon does not stand the test of verification, then the scientist must search for something analogous to reach the elementary phenomenon, for example, an observed motion can be decomposed into simple ones, such as sound into its harmonics.
Nevertheless, I see no alternative to a process of successive approximations
. We must tolerate some exceptions early on, with the expectation that they will eventually be embraced as our analyses evolve.
The displacement fluctuations u'([??]), with respect to which the equation is being solved, are both on the left and on the right side of the expression, so the successive approximations
are used for the solution.
Accordingly, the successive approximations
of [y.sup.k.sub.i] (t), k [greater than or equal to] 0, for the PIM iterative relation will be obtained readily in the auxiliary parameter h.
In this paper we prove the existence as well as approximation of the solutions of a certain generalized quadratic integral equation with maxima via an algorithm based on successive approximations
dveloped in Dhage iteration method under weak partial Lipschitz and compactness type conditions.
We obtain that the convergence of the sequences of successive approximations
is of order n where n is 2k + 1 or 2k.
In all cases except for L =10 and [beta] = 0.1, where the algorithm was stopped when the distance between successive approximations
reached the threshold, the algorithm was stopped when meeting the 3[sigma] stopping criterion.
Explicit Expressions of Various Estimates for Each Step of Successive Approximations
. To the best of our knowledge, there has been no study where explicit expressions of successive estimates have been derived [6-13].
Tricomi  was the first to introduce the successive approximations
method for nonlinear integral equations.
In order to prove the existence of mild solution for system (1), let us consider the sequence of successive approximations
defined as follows:
As stated before, we can select [u.sub.0] = w(v, 0) = Asec hv; using the iteration (11) and the mathematica software, we obtain the following successive approximations
And from (29), one can obtain the successive approximations
of p(t) as follows: