A Boundary Value Problem for a Singularly Perturbed Differential Equation

has a solution «=g(x) for O^x^Xo with g(0)=a and u = h(x) tor xo^x^l with h(l)=b where g(x0)=h(x0). It will be assumed that g'(xo)*h'(xo). The case of (1) with f=l — (y')t and where \a — b\ <1 can be treated explicitly. For small e>0 the solution of (1) tends to the broken line solution of (2) with g(x)=a — x and h = b — 1+x and Xo = (l+a—b)/2. (There is another broken line solution of (2) with g = a+x and h = b+l—x but there is no solution of (1) for e>0 near this broken line solution of (2). As will be seen the criteria below will single out only the first broken line solution.) To formulate the general problem more precisely let yo = g(x0) = h(x0) and let Mi = g'(x0), M2 = ^'(xo). The case ju2>jui will be considered here. (The case pi <pi can be treated in the same way or can be reduced to the first case by replacing y by —y.) Let a, fi, and e0 be positive constants. Let U(x) =g(x) for 0gx^x0 and let U(x)=h(x) for xo^x^l. Let R be the region of (x, y, w, e) space determined by 0^eg«o, and by the union of