Table 1 Quandt-Andrews unknown breakpoint test Full sample: 1948Q1-2017Q4, T = 280 observations Trimmed sample (1958Q3-2007Q3): left trim (15%), right trim (15%) Estimated break date: 2004Q1 Number of breaks compared: 197 Ho: no structural break Varying parameters [[alpha].sub.1] [[alpha].sub.1] and [[alpha].sub.2] Test F statistic p value Statistic p value Supremum Wald 18.58 0.0004 18.58 0.0023 Average Wald 6.16 0.0021 7.68 0.0056 Exponential Wald 6.10 0.0000 6.22 0.0027 Table 2 Bai-Perron multiple breakpoint tests Tests of l + 1 versus l globally determined breaks.

The prime objective of this article is to develop a new hybrid scheme on a piecewise-uniform Shishkin mesh for solving the singularly perturbed BVPs of the form (1.1)-(1.2) so that the method is at least second-order uniformly convergent with respect to [epsilon] in the discrete supremum norm.

Now if we define [??] as [mathematical expression not reproducible] for [lambda]([member of] A) [not equal to] v and [mathematical expression not reproducible], then [??] [not member of] S; otherwise, it contradicts that [??] is the supremum of the maximal chain [bar.S].

In the case when L = 0.5, the supremum is reached for all subsets [omega] of the form [omega] = {(r, [theta]) [member of] [0,1] x [0,2[pi]] | [theta] [member of] [[omega].sub.0]} of measure [pi]/2, where [[omega].sub.[theta]] is any measurable subset of [0,2[pi]] such that [omega] and its symmetric image are complementary in [0,2[pi]].

Let b [member of] [Lip.sub.[alpha]], subsequently, if b satisfies [mathematical expression not reproducible], in which the supremum is taken over all x, y [member of] [R.sup.n] and x [not equal to] y.

We note that iterative techniques combined with lower and upper solutions are applied in the literature to approximately solve various problems in ordinary differential equations [19], for second order periodic boundary value problems [10], for differential equations with maxima [3], [14], for difference equations with maxima [7], for impulsive differential equations [9], [12], for impulsive integro-differential equations [15], for impulsive differential equations with supremum [17], for differential equations of mixed type [18], for Riemann-Liouville fractional differential equations [8], [25], [27], for Caputo impulsive fractional differential equations [11] and for non- instantaneous impulsive differential equation [6].