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2. Two-Point Boundary Value Problems
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With this notation at hand, we notice that the formula for the exact

solution given by (2.9) defines a mapping from C (0, 1) into C (0, 1) .
The following result is a summary of the discussion so far.

Theorem 2.1 For every f ∈ C (0, 1) there is a unique solution u ∈

C (0, 1) of the boundary value problem (2.1). Furthermore, the solution
2
0
u admits the representation (2.9) above.
Having established this result, further smoothness of the solution can be

derived by using the differential equation. More precisely, if f ∈ C

(0, 1) and
for m ≥ 1, then u ∈ C
m+2
u (m+2) = −f (m) , 2 0 m (0, 1) ,
Hence, the solution is always smoother than the “data,” and for f ∈ C ,
∞
∞
we have u ∈ C .
Example 2.3 Consider the problem (2.1) with f(x)=1/x. Note that

f ∈ C (0, 1) , but f/∈ C [0, 1] since f(0) does not exist. It is easy to
verify directly that the solution u is given by
u(x)= −x ln (x),
and

u (x)= −1 − ln (x).

Hence, u ∈ C (0, 1) . However, note that u and u are not continuous at

2
0
zero.
2.1.3 A Maximum Principle
The solution of (2.1) has several interesting properties. First we shall con-
sider what is often referred to as a monotonicity property. It states that
nonnegative data, represented by the right-hand side f, is mapped into a
nonnegative solution. Secondly, we will derive a maximum principle for the
solution of the two-point boundary value problem. This principle states
how large the solution of the problem, measured by its absolute value, can
be for a given right-hand side f.
The following monotonicity property is derived using the representation
of the solution given by (2.9).

Proposition 2.1 Assume that f ∈ C (0, 1) is a nonnegative function.
Then the corresponding solution u of (2.1) is also nonnegative.
Proof: Since G(x, y) ≥ 0 for all x, y ∈ [0, 1], this follows directly from
(2.9).