In this paper, existence and comparison results are derived for the smallest and greatest solutions of first and second order singular nonlinear initial value problems as well as second order boundary value problems.

Recently, similar problems are studied in ordered Banach spaces, e.g., in 1234, by converting problems into systems of integral equations, integrals in these systems being Bochner-Lebesgue or Henstock-Kurzweil integrals. A novel feature in the present study is that the right-hand sides of the considered differential equations comprise distributions on a compact real interval [a, b]. Every distribution is assumed to have a primitive in the space R [ a , b ] of those functions from [a, b] to ℝ which are left-continuous on (a, b], right-continuous at a, and which have right limits at every point of (a, b). With this presupposition, the considered problems can be transformed into integral equations which include the regulated primitive integral of distributions introduced recently in 5.

The paper is organized as follows. Distributions on [a, b], their primitives, regulated primitive integrals and some of their properties, as well as a fixed point lemma are presented in Section 2. In Section 3, existence and comparison results are derived for the smallest and greatest solutions of first order initial value problems.

A fact that makes the solution space R[a,b] important in applications is that it contains primitives of Dirac delta distributions δ_λ, λ ∈ (a, b). This fact is exploited in Section 4, where results of Section 3 are applied to impulsive differential equations. The continuous primitive integral of distributions introduced in 6 is also used in these applications.

Existence of the smallest and greatest solutions of the second order initial and boundary value problems, and dependence of these solutions on the data are studied in Sections 5 and 6. Applications to impulsive problems are also presented.

Considered differential equations may be singular, distributional and impulsive. Differential equations, initial and boundary conditions and impulses may depend functionally on the unknown function and/or on its derivatives, and may contain discontinuous nonlinearities. Main tools are fixed point theorems in ordered spaces proved in 7 by generalized monotone iteration methods. Concrete problems are solved to illustrate obtained results. Iteration methods and Maple programming are used to determine solutions.

Distributions on a compact real interval [a, b] are (cf. 8) continuous linear functionals on the topological vector space D of functions φ : ℝ → ℝ possessing for every j ∈ ℕ₀ a continuous derivative φ^(j) of order j that vanishes on ℝ\(a, b). The space D is endowed with the topology in which the sequence (φ_k) of D converges to φ ∈ D if and only if φ k ( j ) → φ ( j ) uniformly on (a, b) as k → ∞ and j ∈ ℕ₀. As for the theory of distributions, see, e.g., 910.

In this paper, every distribution g on [a, b] is assumed to have a primitive, i.e., a function G ∈ R [ a , b ] whose distributional derivative G' equals to g, in the function space

R [ a , b ] = { G : lim t → s + G ( s ) e x i s t s , lim s → t - G ( s ) = G ( t ) i f a ≤ s < t ≤ b , a n d G ( a ) : = lim s → a + G ( s ) } .

The value 〈g, φ〉 of g at φ∈D is thus given by

⟨ g , φ ⟩ = ⟨ G ′ , φ ⟩ = - ⟨ G , φ ′ ⟩ = - ∫ a b G ( t ) φ ′ ( t ) d t .

Such a distribution g is called RP integrable. Its regulated primitive integral is defined by

r ∫ s t g : = G ( t ) − G ( s ) ,   a ≤ s ≤ t ≤ b ,   where   G   is a primitive of   g   in   ℛ [ a , b ] .

As noticed in 5, the regulated primitive integral generalizes the wide Denjoy integral, and hence also Riemann, Lebesgue, Denjoy and Henstock-Kurzweil integrals.

Denote by A R [ a , b ] the set of those distributions on [a, b] that are RP integrable on [a, b]. If g ∈ A R [ a , b ] , then the function t ↦ r ∫ a t g is that primitive of g which belongs to the set

P R [ a , b ] = { G ∈ R [ a , b ] : G ( a ) = 0 } .

It can be shown (cf. 5) that a relation ≼, defined by

f ≼ g i n A R [ a , b ] i f a n d o n l y i f r ∫ a t f ≤ r ∫ a t g f o r a l l t ∈ [ a , b ] ,

is a partial ordering on AR[a,b]. In particular,

f = g i n A R [ a , b ] i f a n d o n l y i f r ∫ a t f = r ∫ a t g f o r a l l t ∈ [ a , b ] .

Given partially ordered sets X = (X, ≤) and Y = (Y, ≼), we say that a mapping f : X → Y is increasing if f(x) ≼ f(y) whenever x ≤ y in X, and order-bounded if there exist f_± ∈ Y such that f_- f (x) ≼ f₊ for all x ∈ X.

The following fixed point result is a consequence of 11, Theorem A.2.1, or 7, Theorem 1.2.1 and Proposition 1.2.1.

Lemma 2.1. Given a partially ordered set P = (P, ≤), and its order interval [x_-, x₊] = {x ∈ P : x_- ≤ x ≤ x₊}, assume that a mapping G : [x_-, x₊] → [x_-, x₊] is increasing, and that each well-ordered chain of the range G[x_-, x₊] of G has a supremum in P and each inversely well-ordered chain of G[x_-, x₊] has an infimum in P. Then G has the smallest and greatest fixed points, and they are increasing with respect to G.

Remarks 2.1. Under the hypotheses of Lemma 2.1, the smallest fixed point x* of G is by [7, Theorem 1.2.1] the maximum of the chain C of [x_-, x₊] that is well ordered, i.e., every nonempty subset of C has the smallest element, and that satisfies

( I ) x - = min C , a n d i f x - < x , t h e n x ∈ C i f a n d o n l y i f x = sup G [ { y ∈ C : y < x } ] .

The smallest elements of C are Gⁿ(x_-), n ∈ ℕ₀, as long as Gⁿ(x_-) = G(G^n-1(x_-)) is defined and G^n-1(x_-) < Gⁿ(x_-), n ∈ ℕ. If G^n-1(x_-) = Gⁿ (x_-) for some n ∈ ℕ, there is the smallest such an n, and x_* = G^n-1(x_-) is the smallest fixed point of G in [x_-, x₊]. If x ω = sup n ∈ ℕ G n ( x - ) is defined in P and is a strict upper bound of {Gⁿ(x_-)}_n∈ℕ, then x_ω is the next element of C. If x_ω = G(x_ω), then x_* = x_ω, otherwise the next elements of C are of the form Gⁿ(x_ω), n ∈ ℕ, and so on.

The greatest fixed point x* of G is the minimum of the chain D of [x_-, x₊] that is inversely well ordered, i.e., every nonempty subset of D has the greatest element, and that has the following property:

( I I ) x + = max D , a n d i f x < x + , t h e n x ∈ D i f a n d o n l y i f x = inf G [ { y ∈ D : x < y } ] .

The greatest elements of D are n-fold iterates Gⁿ(x₊), as long as they are defined and Gⁿ(x₊) < G^n-1(x₊). If equality holds for some n ∈ ℕ, then x* = G^n-1(x₊) is the greatest fixed point of G in [x_-, x₊].

In this section, existence and comparison results are derived for the smallest and greatest solutions of first order initial value problems. Denote by L l o c 1 ( a , b ] , -∞< a < b < ∞, the space of locally Lebesgue integrable functions from the half-open interval (a, b] to ℝ. Lloc1(a,b] is ordered a.e. pointwise, and its a.e. equal functions are identified.

Given p : [a, b] → ℝ₊, consider the initial value problem (IVP)

( p ⋅ u ) ′ = g ( u ) , lim t → a + ( p ⋅ u ) ( t ) = c ( u ) ,

where c(u) ∈ ℝ, and g ( u ) ∈ A R [ a , b ] . We are looking for solutions of (3.1) from the set

S = { u ∈ L l o c 1 ( a , b ] : lim t → a + ( p ⋅ u ) ( t ) e x i s t s , a n d p ⋅ u ∈ R [ a , b ] } ,

Definition 3.1. We say that a function u ∈ S is a subsolution of the IVP (3.1) if

( p ⋅ u ) ′ ≼ g ( u ) , a n d lim t → a + ( p ⋅ u ) ( t ) ≤ c ( u ) .

If reversed inequalities hold in (3.3), we say that u is a supersolution of (3.1). If equalities hold in (3.3), then u is called a solution of (3.1).

We shall first transform the IVP (3.1) into an integral equation.

Lemma 3.1. Given c(u) ∈ ℝ, u ∈ L l o c 1 ( a , b ] and p : [a, b] → ℝ₊, assume that 1 p ∈ L l o c 1 ( a , b ] , and that g(u)∈AR[a,b]. Then u is a solution of the IVP (3.1) in S if and only if u is a solution of the following integral equation:

u ( t ) = 1 p ( t ) c ( u ) + r ∫ a t g ( u ) , t ∈ ( a , b ] .

Proof: Assume that u is a solution of (3.1) in S. The definition of S and (3.1) ensure by (2.2) that

r ∫ r t g ( u ) =   r ∫ r t ( p   ⋅   u ) ′ = ( p   ⋅   u )   ( t ) − ( p   ⋅   u ) ( r ) ,   a < r ≤ t < b .

Allowing r tend to a+ and applying the initial condition of (3.1) we see that (3.4) is valid. Conversely, let u be a solution of (3.4). According to (3.4) we have

( p ⋅ u ) ( t ) = c ( u ) + r ∫ a t g ( u ) , t ∈ ( a , b ] .

This equation implies that u ∈ S, that the initial condition of (3.1) is valid, and that

( p ⋅ u ) ′ = g ( u ) .

Thus, u is a solution of the IVP (3.1) in S. □

Our first existence and comparison result for the IVP (3.1) reads as follows.

Theorem 3.1. Assume that g : L l o c 1 ( a , b ] → A R [ a , b ] is increasing, that p : [a, b] → ℝ₊, that 1p∈Lloc1(a,b], and that the IVP (3.1) has a subsolution u_- and a supersolution u₊ in S satisfying u_- ≤ u₊. Then (3.1) has the smallest and greatest solutions within the order interval [u_-, u₊] of S. Moreover, these solutions are increasing with respect to g and c.

Proof: Define a mapping G : L l o c 1 ( a , b ] → L l o c 1 ( a , b ] by

G ( u ) ( t ) : = 1 p ( t ) c ( u ) + r ∫ a t g ( u ) , t ∈ ( a , b ] .

Because g is increasing, it follows from (2.3) and (3.6) that G is increasing. Applying (2.3), [5, Theorem 7] and Definition 3.1 we see that if u ∈ L l o c 1 ( a , b ] and u_- ≤ u ≤ u₊, then

( p ⋅ u - ) ( t ) - c ( u ) ≤ ( p ⋅ u - ) ( t ) - lim r → a + ( p ⋅ u - ) ( r ) = lim r → a + r ∫ r t ( p ⋅ u - ) ′ (1) = r ∫ a t ( p ⋅ u - ) ′ ≤ r ∫ a t g ( u - ) ≤ r ∫ a t g ( u ) , a < t ≤ b . (2) (3)

Thus

u - ( t ) ≤ 1 p ( t ) c ( u ) + r ∫ a t g ( u ) = G ( u ) ( t ) , t ∈ ( a , b ] .

Similarly, it can be shown that G(u)(t) ≤ u₊(t) for each t ∈ (a, b]. Thus, G maps the order interval [u_-, u₊] of Lloc1(a,b] into [u_-, u₊]. Let W be a well-ordered or an inversely well-ordered chain in G[u_-, u₊]. It follows from [1, Proposition 9.36] and its dual that sup W and inf W exist in Lloc1(a,b].

The above proof shows that the operator G defined by (3.6) satisfies the hypotheses of Lemma 2.1 when P = L l o c 1 ( a , b ] . Thus G has the smallest fixed point u_* and the greatest fixed point u* in [u_-, u₊]. These fixed points are the smallest and greatest solutions of the integral equation (3.4) in [u_-, u₊]. This result and Lemma 3.1 imply that u_* and u* belong to S, and they are the smallest and greatest solutions of the IVP (3.1) in [u_-, u₊]. Moreover, u_* and u* are by Lemma 2.1 increasing with respect to G. This result implies by (2.3) and (3.6) the last conclusion of Theorem. □

The following result is a consequence of Theorem 3.1.

Proposition 3.1. Assume that mappings g : L l o c 1 ( a , b ] → A R [ a , b ] and c : L l o c 1 ( a , b ] → ℝ are increasing and order-bounded, that p : [a, b] → ℝ₊, and that 1p∈Lloc1(a,b]. Then, the IVP (3.1) has in S the smallest and greatest solutions that are increasing with respect to g and c.

Proof: Because g and c are order-bounded, there exist g ∈ A R [ a , b ] and c_± ∈ ℝ such that g_-≼ g(x) ≼ g₊ and c_- ≤ c(x) ≤ c₊ for all x ∈ L l o c 1 ( a , b ] . Denote

u ( t ) = 1 p ( t ) c + r ∫ a t g , t ∈ ( a , b ] .

Then u_± ∈ S, and

( p ⋅ u - ) ′ = g - ≼ g ( x ) ≼ g + = ( p ⋅ u + ) ′ f o r a l l x ∈ L l o c 1 ( a , b ] ,

and

lim t → a + ( p ⋅ u - ) ( t ) = c - ≤ c ( x ) ≤ c + = lim t → a + ( p ⋅ u + ) f o r a l l x ∈ L l o c 1 ( a , b ] .

Thus u_- is a subsolution and u₊ is a supersolution of (3.1), whence the IVP (3.1) has by Theorem 3.1 the smallest solution u_* and the greatest solution u* in the order interval [u_-, u₊] of S.

If u ∈ S is any solution of (3.1), then

1 p ( t ) c - + r ∫ a t g - ≤ 1 p ( t ) c ( u ) + r ∫ a t g ( u ) ≤ 1 p ( t ) c + + r ∫ a t g + , t ∈ ( a , b ] ,

or equivalently,

u - ( t ) ≤ u ( t ) ≤ u + ( t ) , t ∈ ( a , b ] .

Consequently, u ∈ [u_-, u₊], whence u_* and u* are the smallest and greatest of all the solutions of (3.1) in S. □

In the next proposition, the Henstock-Kurzweil integral K ∫ can be replaced by any of the integrals called Riemann, Lebesgue, Denjoy and wide Denjoy integrals.

Proposition 3.2. Assume that g(x) is RP integrable on [a, b] for every x∈Lloc1(a,b], and that

r ∫ a t g ( x ) = ∑ i = 1 n H i ( t ) K ∫ a t f i ( x ) + H 0 ( t ) ,   t ∈ [ a , b ] ,

Where H 0 ∈ P R [ a , b ] , and for each i = 1,..., n, H_i : [a, b] → [0, ∞) has right limits on [a, b), is left-continuous on (a, b], and f_i : [a, b] → ℝ satisfies the following hypotheses.

(f_i1) f_i(x) is Henstock-Kurzweil integrable on [a, b] for every x∈Lloc1(a,b].

(f_i2) There exist Henstock-Kurzweil integrable functions

f

If c:Lloc1(a,b]→ℝ is increasing and order-bounded, then the IVP (3.1) has in S the smallest and greatest solutions that are increasing with respect to f_i and c.

Proof: The hypotheses imposed above ensure by (2.3) and (3.7) that g is an increasing mapping from Lloc1(a,b] to the order interval [g_-, g₊] of AR[a,b], where

r ∫ a t g − = ∑ i = 1 n H i ( t )   K ∫ a t f ¯ i + H 0 ( t ) ,     r ∫ a t g + = ∑ i = 1 n H i ( t )   K ∫ a t f ¯ i + H 0 ( t ) ,   t ∈ [ a , b ] .

Thus the conclusions follow from Proposition 3.1.

Example 3.1. Assume that

r ∫ a t g ( x ) = H 1 ( t )   K ∫ a t f 1 ( x ) + H 0 ( t ) ,   t ∈ [ 0 , b ] ,

where b ≥ 1, H 0 ∈ R [ 0 , b ] , H₁ is the Heaviside step function, i.e., H 1 ( t ) = 1 , t > 0 , 0 , t ≤ 0 ,

f 1 ( x ) ( t ) = 1 1 0 5 1 0 5 arctan ∫ 1 ∕ 2 1 x ( t ) - H 0 ( t ) p ( t ) d t sin 1 t - 1 t sgn sin 1 t cos 1 t , t ∈ ( 0 , b ] , x ∈ L l o c 1 ( 0 , b ] , [ z ] = max { n ∈ ℤ : n ≤ z } a n d s g n ( z ) = z ∕ | z | , z ≠ 0 , 0 , z = 0 .

Note, that the greatest integer function [·] occurs in the function f₁(x). Prove that the IVP

( p ⋅ u ) ′ = g ( u ) , lim t → 0 + ( p ⋅ u ) ( t ) = 0 ,

where p(t) = t, t ∈ [0, b], has the smallest and greatest solutions, and calculate them.

Solution: Problem (3.9) is of the form (3.1), where c(u) = a = 0 and p(t) ≡ t. The hypotheses (f₁₁) and (f₂₁) are valid when

K ∫ 0 t f - 1 = - 2 t | sin 1 t | + H 0 ( t ) , t ∈ ( 0 , b ] , K ∫ 0 t f ̄ 1 = 2 t | sin 1 t | + H 0 ( t ) , t ∈ ( 0 , b ] .

Thus the IVP (3.9) has by Proposition 3.2 the smallest and greatest solutions. They are the smallest and greatest fixed points of the mapping G defined by

G ( x ) ( t ) : = 1 t r ∫ 0 t g ( x ) , t ∈ ( 0 , b ] , x ∈ L l o c 1 ( 0 , b ] .

G is an increasing mapping from L l o c 1 ( 0 , b ] , to its order interval [u_-, u₊], where

u ( t ) : = 2 | sin 1 t | + H 0 ( t ) t , t ∈ ( 0 , b ] .

Calculating the successive approximations Gⁿ(u_±) we see that G⁷(u_±) = G⁸(u_±). This means by Remark 2.1 that u_* = G⁷(u_-) and u* = G⁷(u₊) are the smallest and greatest fixed points of G in [u_-, u₊]. According to the proof of Proposition 3.1, u_* and u* are also the smallest and greatest solutions, of the initial value problem (3.9) in S. The exact expressions of u_* and u* are:

u * ( t ) = - arctan 6 7 2 2 9 1 0 0 0 0 | sin 1 t | + H 0 ( t ) t , t ∈ ( 0 , b ] , u * ( t ) = arctan 1 6 8 0 7 2 5 0 0 | sin 1 t | + H 0 ( t ) t , t ∈ ( 0 , b ] .

In this section, we assume that Λ is a well-ordered subset of (a, b). Let δ_λ, λ ∈ Λ, denote the translation of Dirac delta distribution for which r ∫ a t δ λ = H ( t − λ ) , t ≥ a, where H is the Heaviside step function. Consider the singular distributional Cauchy problem

( p ⋅ u ) ′ = ∑ λ ∈ Λ I ( λ , u ) δ λ + f ( u ) , lim t → a + ( p ⋅ u ) = c ( u ) ,

where p : [a, b] → ℝ₊ and 1p∈Lloc1(a,b]. The values of f are distributions on [a, b], and the values of I are real numbers.

Definition 4.1. By a solution of (4.1), we mean such a function u ∈ S that satisfies (4.1), for which p · u is continuous on [a, b]\Λ, and has impulses

Δ ( p ⋅ u ) ( λ ) : = ( p ⋅ u ) ( λ + ) - ( p ⋅ u ) ( λ ) = I ( λ , u ) , λ ∈ Λ .

In the study of (4.1), the regulated primitive integral is replaced by the continuous primitive integral presented in 6. A distribution g on [a, b] is called distributionally Denjoy (DD) integrable on [a, b], denote g ∈ A C [ a , b ] , if g has a continuous primitive, i.e., g is a distributional derivative of a function G ∈ C[a, b]. The continuous primitive integral of g is defined by

c ∫ s t g = G ( t ) − G ( s ) ,   a ≤ s ≤ t ≤ b .

A C [ a , b ] is a proper subset of AR[a,b], and for every g∈AC[a,b] its continuous and regulated primitive integrals are equal. As shown in 6, AC[a,b] contains functions that are wide Denjoy integrable, and hence also Riemann, Lebesgue, Denjoy and Henstock-Kurzweil integrable on [a, b]. On the other hand, distributional derivatives of nowhere differentiable Weierstrass function and almost everywhere differentiable Cantor function are distributionally but not wide Denjoy integrable.

It can be shown (cf. 6) that relation ≼, defined by

f ≼ g i f a n d o n l y i f c ∫ a t f ≤ c ∫ a t g f o r a l l t ∈ [ a , b ] ,

is a partial ordering on AC[a,b].

Transformation of the Cauchy problem (4.1) into an integral equation is presented in the following lemma.

Lemma 4.1. Assume that u ∈ S, that f ( u ) ∈ A C [ a , b ] , and that ∑ λ ∈ Λ | I ( λ , u ) | < ∞ . Then u is a solution of (4.1) if and only if

u ( t ) = 1 p ( t ) c ( u ) + ∑ λ ∈ Λ I ( λ , u ) H ( t - λ ) + c ∫ a t f ( u ) , t ∈ ( a , b ] .

Proof: Assume first that u ∈ S satisfies (4.3). Because Λ is well-ordered, it follows that if λ ∈ Λ and λ < sup Λ, then H(t - λ) = 1 on (λ, S(λ)], where S(λ) = min{μ ∈ Λ : λ < μ}. This property implies that if the function v : (a, b] → ℝ is defined by

v ( t ) = 1 p ( t ) c ( u ) + ∑ λ ∈ Λ I ( λ , u ) H ( t - λ ) , t ∈ ( a , b ] ,

then the function p · v is constant on every interval (λ, S(λ)], Λ ∋ Λ < sup Λ, on [a, min Λ], and on (sup Λ, b] if sup Λ < b. In particular, p ⋅ v ∈ R [ a , b ] , and the distributional derivative of p · v is

( p ⋅ v ) ′ = ∑ λ ∈ Λ I ( λ , u ) δ λ .

Thus

( p ⋅ u ) ′ = ( p ⋅ v ) ′ + f ( u ) = ∑ λ ∈ Λ I ( λ , u ) δ λ + f ( u ) .

Since t ↦ c ∫ a t f ( u ) is continuous on [a, b], then p · u is continuous on [a, b]\Λ. Because

( p ⋅ v ) ( t ) - ( p ⋅ v ) ( λ ) = I ( λ , u ) H ( t - λ ) = I ( λ , u ) , λ ∈ Λ , t ∈ ( λ , S ( λ ) ] ,

then

Δ ( p ⋅ u ) ( λ ) = ( p ⋅ u ) ( λ + ) - ( p ⋅ u ) ( λ ) = ( p ⋅ v ) ( λ + ) - ( p ⋅ v ) ( λ ) = I ( λ , u ) , λ ∈ Λ .

Moreover lim t → a + ( p ⋅ u ) ( t ) = c ( u ) , so that u is a solution of the IVP (4.1).

Assume next that u ∈ S is a solution of (4.1). Denoting

z ( t ) = u ( t ) - v ( t ) , t ∈ [ a , b ] ,

where v is defined by (4.4), it follows from (4.1) and (4.5) that

( p ⋅ z ) ′ = f ( u ) , lim t → a + ( p ⋅ z ) = 0 .

Because f(u) is DD integrable on [a, b], then

( p ⋅ z ) ( t ) = c ∫ a t f ( u ) , t ∈ [ a , b ] .

Thus

( p ⋅ u ) ( t ) = ( p ⋅ z ) ( t ) + ( p ⋅ v ) ( t ) = c ( u ) + ∑ λ ∈ Λ I ( λ , u ) H ( t - λ ) + c ∫ a t f ( u ) , t ∈ [ a , b ] ,

or equivalently, (4.3) holds. □

Noticing that the IVP (4.1) is a special case of the Cauchy problem (3.1), where

g ( u ) = ∑ λ ∈ Λ I ( λ , u ) δ λ + f ( u ) ,

the results of Section 3 can be applied to study the IVP (4.1). The following result is a consequence of Proposition 3.1.

Proposition 4.1. The distributional IVP (4.1) has the smallest and greatest solutions that are increasing with respect to f and c, if f : L l o c 1 ( a , b ] → A C [ a , b ] and c:Lloc1(a,b]→ℝ are increasing and order-bounded, if p : [a, b] → ℝ₊, if 1p∈Lloc1(a,b], and if I : Λ × L l o c 1 ( a , b ] → ℝ has the following properties.

(I) ∑ λ ∈ Λ | I ( λ , x ) | ≤ M < ∞ for all x∈Lloc1(a,b], and x ↦ I(λ,x) is increasing for all λ ∈ Λ.

Proof: The given hypotheses imply that (4.6) defines a mapping g:Lloc1(a,b]→AR[a,b] that is increasing and order-bounded. Thus, the IVP (3.1) has by Proposition 3.1 the smallest solution u_* and the greatest solution u* in S, and they are increasing with respect to g and c. By Lemma 4.1, u_* and u* are the smallest and greatest solutions of the IVP (4.1), and they are increasing with respect to f, and c, since g is increasing with respect to f. □

The initial value problem

d d t ( p ( t ) u ( t ) ) = q ( t , u ( t ) , u ) a . e . o n [ a , b ] , lim t → a + ( p ( t ) u ( t ) ) = c ( u ) ,

combined with the impulsive property:

Δ ( p ⋅ u ) ( λ ) = ( p ⋅ u ) ( λ + ) - ( p ⋅ u ) ( λ ) = I ( λ , u ) , λ ∈ Λ ,

form a special case of the IVP (4.1) when f is the Nemytskij operator associated with the function q : [ a , b ] × ℝ × L l o c 1 ( a , b ] → ℝ by

f ( x ) : = q ( ⋅ , x ( ⋅ ) , x ) , x ∈ L l o c 1 ( a , b ] .

Considering distributions δ_λ as generalized functions t αδ (t - λ), t ∈ [a, b], we can rewrite the system (4.7), (4.8) as

d d t ( p ( t ) u ( t ) ) = ∑ λ ∈ Λ I ( λ , u ) δ ( t - λ ) + q ( t , u ( t ) , u ) a . e . o n [ a , b ] , lim t → a + ( p ( t ) u ( t ) ) = c ( u ) .

For instance, Proposition 4.1 implies the following result:

Corollary 4.1. The impulsive Cauchy problem (4.9) has the smallest and greatest solutions which are increasing with respect to q and c, if c:Lloc1(a,b]→ℝ is increasing and order- bounded, and if the hypotheses (I) and the following hypotheses are valid.

(q0) q(·, x(·); x) is Henstock-Kurzweil integrable on [a, b] for every x∈Lloc1(a,b].

(q1) K ∫ a t q ( s , x ( s ) , x ) d s ≤ K ∫ a t q ( s , y ( s ) , y ) d s for all t ∈ [a, b] whenever × ≤ y in Lloc1(a,b].

(q2) There exist Henstock-Kurzweil integrable functions q_± : [a, b] → ℝ such that K ∫ a t q - ( s ) d s ≤ K ∫ a t q ( s , x ( s ) , x ) d s ≤ K ∫ a t q + ( s ) d s for all x∈Lloc1(a,b] and t ∈ [a, b].

Example 4.1. Determine the smallest and greatest solutions of the IVP

t u ′ ( t ) + u ( t ) = 1 1 0 4 [ 1 0 4 arctan ( u ( 1 ) ) ] δ t - 1 2 + q ( t , u ) a . e . o n [ 0 , 1 ] , lim t → 0 + ( t u ( t ) ) = 0 ,

when q is defined by

{ q ( t , x ) =   [ 10 4 tanh ( ∫ 2 5 1 ( x ( s ) d s ) ] 10 4 h ( t ) ,   x ∈ L l o c 1 ( 0 , 1 ] ,   t ∈ ( 0 , 1 ] ,   where h ( t ) =   | cos ( 1 t ) | + 1 t sgn ( cos ( 1 t ) ) sin ( 1 t ) ,   t ∈ ( 0 , 1 ] .

'[·]' denotes, as before, the greatest integer function, and 'sgn' the sign function.

Solution: The IVP (4.10) is a special case of (4.6), when a = 0, b = 1, c(u) = 0, p(t) = t, t ∈ [0, 1], I ( 1 2 , x ) = 1 1 0 4 [ 1 0 4 arctan ( x ( 1 ) ) ] , and Λ = { 1 2 } . The validity of the hypotheses of Corollary 4.1 is easy to verify. Thus, the IVP (4.10) has the smallest and greatest solutions. These solutions are the smallest and greatest fixed points of G : L l o c 1 ( 0 , 1 ] → L l o c 1 ( 0 , 1 ] , defined by

G ( x ) ( t ) = 1 t 1 1 0 4 [ 1 0 4 arctan ( x ( 1 ) ) ] H t - 1 2 + K ∫ 0 t q ( t , x ) , x ∈ L l o c 1 ( 0 , 1 ] , t ∈ ( 0 , 1 ] .

Calculating the successive approximations

y n + 1 = G ( y n ) , y 0 = x - a n d z n + 1 = G ( z n ) , z 0 = x + , w h e r e x ( t ) = 2 t H t - 1 2 + 1 2 K ∫ 0 t h ( s ) d s = 2 t H t - 1 2 1 t cos ( 1 t ) , t ∈ ( 0 , 1 ] ,

it turns out that ( y n ) n = 0 1 7 is strictly increasing, that ( z n ) n = 0 1 6 is strictly decreasing, that y₁₇ = G(y₁₇), and that z₁₆ = G(z₁₆). Thus u_* = y₁₇ and u* = z₁₆ are by Remark 2.1 the smallest and greatest solutions of (4.1) with c(u) = 0. The exact formulas of u_* and u* are

u * ( t ) = - 4 4 3 9 H ( t - 1 2 ) 5 0 0 0 t - 6 3 1 3 1 0 0 0 0 cos ( 1 t ) , t ∈ ( 0 , 1 ] , u * ( t ) = 2 2 1 9 H ( t - 1 2 ) 2 5 0 0 t + 6 3 1 1 1 0 0 0 0 cos ( 1 t ) , t ∈ ( 0 , 1 ] .

Remarks 4.1. The function (t, x) α q(t, x), defined in (4.11), has the following properties.

• It is Henstock-Kurzweil integrable, but it is not Lebesgue integrable with respect to the independent variable t if x ≠ 0, because h is not Lebesgue integrable on [0,1].

• Its dependence on the variables t and x is discontinuous, since the signum function sgn, the greatest integer function [·], and the function h are discontinuous.

• Its dependence on the unknown function x is nonlocal, since the integral of function x appears in the argument of the tanh-function.

• Its dependence on x is not monotone, since h attains positive and negative values in an infinite number of disjoint sets of positive measure. For instance, y*(t) > y*(t) for all t ∈ (0, 1], but the difference function t α q(t, y*) -q(t, y_*) is neither nonnegative-valued nor Lebesgue integrable on [0, 1].

Notice also that in Example 4.1 dependence of the function I(12,x)=1104[104 arctan(x(1))] on x is discontinuous.

We shall study the second order initial value problem in this section

( p ⋅ u ′ ) ′ = f ( u , u ′ ) , lim t → a + ( p ⋅ u ′ ) ( t ) = c ( u , u ′ ) , lim t → a + u ( t ) = d ( u , u ′ ) ,

where f : L l o c 1 ( a , b ] 2 → A R [ a , b ] , c ,   d : L l o c 1 ( a , b ] 2 → ℝ , p : [a, b] → ℝ₊, -∞ < a < b < ∞.

We are looking for the smallest and greatest solutions of (5.1) from the set

Y = { u : ( a , b ] → ℝ : u ′ ∈ L l o c 1 ( a , b ] , lim t → a + u ( t ) a n d lim t → a + ( p ⋅ u ′ ) ( t ) e x i s t , a n d p ⋅ u ′ ∈ R [ a , b ] } .

The IVP (5.1) can be converted to a system of integral equations which does not contain derivatives.

Lemma 5.1. Assume that p : [a, b] →ℝ₊, that 1 p ∈ L 1 [ a , b ] and that f ( u , v ) ∈ A R [ a , b ] for all u , v ∈ L l o c 1 ( a , b ] . Then u is a solution of the IVP (5.1) in Y if and only if (u, u') = (u, v), where ( u , v ) ∈ L l o c 1 ( a , b ] 2 is a solution of the system

u ( t ) = d ( u , v ) + ∫ a t v ( s ) d s , t ∈ ( a , b ] , v ( t ) = 1 p ( t ) c ( u , v ) + r ∫ a t f ( u , v ) , t ∈ ( a , b ] .