This section is devoted to the study of the second order boundary value problem (BVP)
(6.1)
where , c, d : L1[a, b]2 → ℝ, and p : [a, b] → ℝ+, -∞ < a < b < ∞. Now we are looking for the smallest and greatest solutions of (6.1) from the set
(6.2)
The BVP (6.1) can be transformed into a system of integral equations as follows.
Lemma 6.1. Assume that p : [a, b] → ℝ+, that , and that for all u, v ∈ L1[a, b]. Then u is a solution of the IVP (6.1) in Z if and only if (u, u') = (u, v), where (u, v) ∈ L1[a, b]2 is a solution of the system
(6.3)
Proof: Assume that u is a solution of the BVP (6.1) in Z, and denote
(6.4)
The differential equation, the boundary conditions of (6.1), the definition (6.2) of Z and the notation (6.4) ensure that
and
Thus the integral equations of (6.3) hold.
Conversely, let (u, v) be a solution of the system (6.3) in L1[a, b]2. The first equation of (6.3) implies that u is a.e. differentiable and v = u', and that the second boundary condition of (6.1) holds. Since v = u', it follows from the second equation of (6.3) that
(6.5)
This equation implies that p · u' belongs to , and that the differential equation and first boundary condition of (6.1) are satisfied. Thus u, is a solution of the BVP (6.1) in Z. □
Assume that L1[a, b] is ordered a.e. pointwise, that Z is ordered pointwise. We shall impose the following hypotheses for the functions p, f, c, and d.
(p1) p : [a, b] → ℝ+, and .
(f1) is order-bounded, and f (u1, v1) ≼ f (u2, v2) whenever u
i
, v
i
∈ L1[a, b], i = 1, 2, u1 ≤ u2, and v1 ≥ v2.
(c1) c : L1[a, b]2 → ℝ is order-bounded, and c(u2, v2) ≤ c(u1, v1) whenever u
i
, v
i
∈ L1[a, b], i = 1, 2, u1 ≤ u2, and v1 ≥ v2.
(d1) d : L1[a, b]2 → ℝ is order-bounded, and d(u1, v1) ≤ d(u2, v2) whenever u
i
, v
i
∈ L1[a, b], i = 1, 2, u1 ≤ u2 and v1 ≥ v2.
The next theorem is our main existence and comparison result for the BVP (6.1).
Theorem 6.1. Assume that the hypotheses (p1), (f1), (c1), and (d1) hold. Then, the BVP (6.1) has the smallest and greatest solutions in Z, and they are increasing with respect to f and d and decreasing with respect to c.
Proof: Because f, c and d are order-bounded, then the following conditions are valid.
(f0) There exist such that h- ≼ f (u, v) ≼ h+ for all u, v ∈ L1[a, b].
(c0) There exist c± ∈ ℝ such that c
-
≤ c(u, v) ≤ c+ whenever u, v ∈ L1[a, b].
(d0) There exist d
±
∈ ℝ such that d
-
≤ d(u, v) ≤ d+ whenever u, v ∈ L1[a, b].
Assume that P = L1[a, b]2 is ordered by
(6.6)
We shall first show that the vector-functions x+, x
-
given by
(6.7)
belong to P. Since 1/p is Lebesgue integrable and the function belongs to , then the second component of x+ is Lebesgue integrable on [a, b]. Similarly one can show that the second component of x
-
belongs to L1[a, b]. These results ensure that the first components of x
±
are defined and continuous in t, and hence are in L1[a, b].
Similarly, by applying the given hypotheses one can verify that the relations
(6.8)
define an increasing mapping G = (G1, G2) : [x
-
, x+] → [x
-
, x+].
Let W be a well-ordered chain in the range of G. The set W1 = {u : (u, v) ∈ W} is well ordered, W2 = {v : (u, v) ∈ W } is inversely well-ordered, and both W1 and W2 are order-bounded in L1[a, b]. It then follows from [1, Lemma 9.32] that the supremum of W1 and the infimum of W2 exist in L1[a, b]. Obviously, (sup W1, inf W2) is the supremum of W in (P, ≤). Similarly, one can show that each inversely well-ordered chain of the range of G has the infimum in (P, ≤).
The above proof shows that the operator G = (G1, G2) defined by (6.8) satisfies the hypotheses of Lemma 2.1, whence G has the smallest fixed point x* = (u*, v*) and a greatest fixed point x* = (u*, v*). It follows from (6.8) that (u*, v*) and (u*, v*) are solutions of the system (6.3). According to Lemma 6.1, u* and u* belong to Z and are solutions of the BVP (6.1).
To prove that u* and u* are the smallest and greatest of all solutions of (6.1) in Z, let u ∈ Z be any solution of (6.1). In view of Lemma 6.1, (u, v) = (u, u') is a solution of the system (6.3). Applying the properties (f0), (c0), and (d0) it is easy to show that x = (u, v) ∈ [x
-
, x+], where x
±
are defined by (6.7). Thus, x = (u, v) is a fixed point of G = (G1, G2) : [x-, x+] → [x
-
, x+], defined by (6.8). Because x* = (u*, v*) and x* = (u*, v*) are the smallest and greatest fixed points of G, respectively, then (u*, v*) ≤ (u, v) ≤ (u*, v*). In particular, u* ≤ u ≤ u*, whence u* and u* are the smallest and greatest of all solutions of the BVP (6.1).
The last assertion is an easy consequence of the last conclusion of Lemma 2.1, and the definition (6.8) of G = (G1, G2). □
Consider next a special case of (6.1) where the values of f combined with impulses and Henstock-Kurzweil integrable functions:
(6.9)
Corollary 6.1. Assume that p : [a, b] → ℝ+, , that functions c, d : L1[a, b]2 → ℝ satisfy the hypotheses (c
i
) and (d
i
), i = 1, 2, that α : Λ → ℝ, , and that g satisfies the following hypotheses.
(g1) g(u, v) is Henstock-Kurzweil integrable on [a, b] for all u, v ∈ L1[a, b].
(g2) There exist Henstock-Kurzweil integrable functions such that , whenever u1 ≤ u2 and v1 ≥ v2 in L1[a, b].
Then, the impulsive BVP (6.9) has the smallest and greatest solutions that are increasing with respect to g, d and decreasing with respect to c.
Example 6.1. Determine the smallest and greatest solutions of the following singular impulsive BVP.
(6.10)
Solution: System (6.10) is a special case of (6.9) when a = 0, b = 3, , , and g, c, d are given by
(6.11)
It is easy to verify that the hypotheses of Corollary 6.1 are valid. Thus (6.10) has the smallest and greatest solutions. The functions x- and x+ defined by (6.7) can be calculated, and their first components are:
and
where FresnelS is the Fresnel sine integral.
According to Lemma 6.1 the smallest solution of (6.10) is equal to the first component of the smallest fixed point of G = (G1, G2), defined by (6.3). Calculating the first iterations Gnx- it turns out that G6x- = G7x- . Thus is the smallest solution of (6.10). Similarly, one can show that G3x+ = G4x+, whence is the greatest solution of (6.10). The exact expressions of these solutions are
and
Remarks 6.1. The IVP's (3.1) and (5.1) and the BVP (6.1) can be
-
singular, since is allowed;
-
nonlocal, because the functions g, c, d, and f may depend functionally on u and/or u';
-
discontinuous, since the dependencies of g, c, d and f on u and/or u' can be discontinuous;
-
distributional, since the values of g and f can be distributions;
-
impulsive, since the values of g and f can contain impulses.
A theory for first order nonlinear distributional Cauchy problems is presented in [12]. Linear distributional differential equations are studied in [13, 8]. Singular ordinary differential equations are studied, e.g., in [11, 14, 15]. Initial value problems in ordered Banach spaces are studied, e.g., in [1–4, 7]. As for the study of impulsive differential equations, see, e.g. [1, 16, 17]. The case of well-ordered set of impulses is studied first time in [18].
The solutions of examples have been calculated by using simple Maple programming.