Open Access Research

On singular nonlinear distributional and impulsive initial and boundary value problems

Seppo Heikkilä

Author Affiliations

Department of Mathematical Sciences, University of Oulu, BOX 3000, FIN-90014, Oulu, Finland

Boundary Value Problems 2011, 2011:24  doi:10.1186/1687-2770-2011-24


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/24


Received:29 April 2011
Accepted:16 September 2011
Published:16 September 2011

© 2011 Heikkilä; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Purpose

To derive existence and comparison results for extremal solutions of nonlinear singular distributional initial value problems and boundary value problems.

Main methods

Fixed point results in ordered function spaces and recently introduced concepts of regulated and continuous primitive integrals of distributions. Maple programming is used to determine solutions of examples.

Results

New existence results are derived for the smallest and greatest solutions of considered problems. Novel results are derived for the dependence of solutions on the data. The obtained results are applied to impulsive differential equations. Concrete examples are presented and solved to illustrate the obtained results.

MSC: 26A24, 26A39, 26A48, 34A12, 34A36, 37A37, 39B12, 39B22, 47B38, 47J25, 47H07, 47H10, 58D25

Keywords:
distribution; primitive; integral; regulated; continuous; initial value problem; boundary value problem; singular; distributional

1 Introduction

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 [1-4], 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1">View MathML</a> important in applications is that it contains primitives of Dirac delta distributions δλ, 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.

2 Preliminaries

Distributions on a compact real interval [a, b] are (cf. [8]) continuous linear functionals on the topological vector space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M2">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M2">View MathML</a> is endowed with the topology in which the sequence (φk) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M2">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M3">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M4">View MathML</a> uniformly on (a, b) as k → ∞ and j ∈ ℕ0. As for the theory of distributions, see, e.g., [9,10].

In this paper, every distribution g on [a, b] is assumed to have a primitive, i.e., a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M5">View MathML</a> whose distributional derivative G' equals to g, in the function space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M6">View MathML</a>

(2.1)

The value 〈g, φ〉 of g at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M3">View MathML</a> is thus given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M7">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M8">View MathML</a>

(2.2)

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 onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M9">View MathML</a> the set of those distributions on [a, b] that are RP integrable on [a, b]. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M10">View MathML</a>, then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M11">View MathML</a> is that primitive of g which belongs to the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M12">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M13">View MathML</a>

(2.3)

is a partial ordering on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M9">View MathML</a>. In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M14">View MathML</a>

(2.4)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M15">View MathML</a>

The smallest elements of C are Gn(x-), n ∈ ℕ0, as long as Gn(x-) = G(Gn-1(x-)) is defined and Gn-1(x-) < Gn(x-), n ∈ ℕ. If Gn-1(x-) = Gn (x-) for some n ∈ ℕ, there is the smallest such an n, and x* = Gn-1(x-) is the smallest fixed point of G in [x-, x+]. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M16">View MathML</a> is defined in P and is a strict upper bound of {Gn(x-)}n∈ℕ, then xω is the next element of C. If xω = G(xω), then x* = xω, otherwise the next elements of C are of the form Gn(xω), 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M17">View MathML</a>

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

3 First order initial value problems

In this section, existence and comparison results are derived for the smallest and greatest solutions of first order initial value problems. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a>, -∞< a < b < ∞, the space of locally Lebesgue integrable functions from the half-open interval (a, b] to ℝ. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a> is ordered a.e. pointwise, and its a.e. equal functions are identified.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M19">View MathML</a>

(3.1)

where c(u) ∈ ℝ, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M20">View MathML</a>. We are looking for solutions of (3.1) from the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M21">View MathML</a>

(3.2)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M22">View MathML</a>

(3.3)

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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M23">View MathML</a> and p : [a, b] → ℝ+, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24">View MathML</a>, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M20">View MathML</a>. 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M25">View MathML</a>

(3.4)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M26">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M27">View MathML</a>

(3.5)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M28">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M29">View MathML</a>is increasing, that p : [a, b] → ℝ+, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M30">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M31">View MathML</a>

(3.6)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M32">View MathML</a> and u- u u+, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M33">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M34">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a>.

The above proof shows that the operator G defined by (3.6) satisfies the hypotheses of Lemma 2.1 when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M35">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M36">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M37">View MathML</a>are increasing and order-bounded, that p : [a, b] → ℝ+, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M38">View MathML</a> and c± ∈ ℝ such that g-g(x) ≼ g+ and c- c(x) ≤ c+ for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M40">View MathML</a>

Then u± S, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M41">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M42">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M43">View MathML</a>

or equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M44">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M45">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39">View MathML</a>, and that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M46">View MathML</a>

(3.7)

Where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M47">View MathML</a>, and for each i = 1,..., n, Hi : [a, b] → [0, ∞) has right limits on [a, b), is left-continuous on (a, b], and fi : [a, b] → ℝ satisfies the following hypotheses.

(fi1) fi(x) is Henstock-Kurzweil integrable on [a, b] for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39">View MathML</a>.

(fi2) There exist Henstock-Kurzweil integrable functions

    f
i, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M48">View MathML</a>such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M49">View MathML</a>whenever x y in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M37">View MathML</a> is increasing and order-bounded, then the IVP (3.1) has in S the smallest and greatest solutions that are increasing with respect to fi and c.

Proof: The hypotheses imposed above ensure by (2.3) and (3.7) that g is an increasing mapping from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a> to the order interval [g-, g+] of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M9">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M50">View MathML</a>

Thus the conclusions follow from Proposition 3.1.

Example 3.1. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M51">View MathML</a>

(3.8)

where b ≥ 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M52">View MathML</a>, H1 is the Heaviside step function, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M53">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M54">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M55">View MathML</a>

(3.9)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M56">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M57">View MathML</a>

(3.10)

G is an increasing mapping from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M58">View MathML</a>, to its order interval [u-, u+], where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M59">View MathML</a>

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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M60">View MathML</a>

4 Applications to impulsive problems

In this section, we assume that Λ is a well-ordered subset of (a, b). Let δλ, λ ∈ Λ, denote the translation of Dirac delta distribution for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M61">View MathML</a>, t a, where H is the Heaviside step function. Consider the singular distributional Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M62">View MathML</a>

(4.1)

where p : [a, b] → ℝ+ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24">View MathML</a>. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M63">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M64">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M65">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M66">View MathML</a> is a proper subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M9">View MathML</a>, and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M64">View MathML</a> its continuous and regulated primitive integrals are equal. As shown in [6], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M66">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M67">View MathML</a>

(4.2)

is a partial ordering on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M66">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M68">View MathML</a>, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M69">View MathML</a>. Then u is a solution of (4.1) if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M70">View MathML</a>

(4.3)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M71">View MathML</a>

(4.4)

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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M72">View MathML</a>, and the distributional derivative of p · v is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M73">View MathML</a>

(4.5)

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M74">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M75">View MathML</a> is continuous on [a, b], then p · u is continuous on [a, b]\Λ. Because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M76">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M77">View MathML</a>

Moreover <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M78">View MathML</a>, so that u is a solution of the IVP (4.1).

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M79">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M80">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M81">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M82">View MathML</a>

or equivalently, (4.3) holds. □

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M83">View MathML</a>

(4.6)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M84">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M37">View MathML</a>are increasing and order-bounded, if p : [a, b] → ℝ+, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M24">View MathML</a>, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M85">View MathML</a>has the following properties.

(I) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M86">View MathML</a>for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39">View MathML</a>, and x I(λ,x) is increasing for all λ ∈ Λ.

Proof: The given hypotheses imply that (4.6) defines a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M36">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M87">View MathML</a>

(4.7)

combined with the impulsive property:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M88">View MathML</a>

(4.8)

form a special case of the IVP (4.1) when f is the Nemytskij operator associated with the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M89">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M90">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M91">View MathML</a>

(4.9)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M37">View MathML</a>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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39">View MathML</a>.

(q1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M92">View MathML</a>for all t ∈ [a, b] whenever × y in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a>.

(q2) There exist Henstock-Kurzweil integrable functions q± : [a, b] → ℝ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M93">View MathML</a>for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M39">View MathML</a>and t ∈ [a, b].

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M94">View MathML</a>

(4.10)

when q is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M95">View MathML</a>

(4.11)

'[·]' 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], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M96">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M97">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M98">View MathML</a>, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M99">View MathML</a>

(4.12)

Calculating the successive approximations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M100">View MathML</a>

it turns out that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M101">View MathML</a> is strictly increasing, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M102">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M103">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M96">View MathML</a> on x is discontinuous.

5 Second order initial value problems

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M104">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M106">View MathML</a>, p : [a, b] → ℝ+, -∞ < a < b < ∞.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M107">View MathML</a>

(5.2)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108">View MathML</a>and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M109">View MathML</a>for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M110">View MathML</a>. Then u is a solution of the IVP (5.1) in Y if and only if (u, u') = (u, v), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M111">View MathML</a>is a solution of the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M112">View MathML</a>

(5.3)

Proof: Assume that u is a solution of the IVP (5.1) in Y , and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M113">View MathML</a>

(5.4)

The differential equation, the initial conditions of (5.1), the definition (5.2) of Y and the notation (5.4) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M114">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M115">View MathML</a>

Thus, the integral equations of (5.3) hold.

Conversely, let (u, v) be a solution of the system (5.3) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M116">View MathML</a>. The first equation of (5.3) implies that u is a.e. differentiable and v = u', and that the second initial condition of (5.1) is fulfilled. Since v = u', it follows from the second equation of (5.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M117">View MathML</a>

(5.5)

The equation (5.5) implies that p · u' belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1">View MathML</a>, and that the differential equation and first initial condition of (5.1) hold. Thus u is a solution of the IVP (5.1) in Y. □

Assume that Lloc(a, b] is ordered a.e. pointwise, that Y is ordered pointwise, and that the functions p, f, c and d satisfy the following hypotheses:

Our main existence and comparison result for the IVP (5.1) reads as follows.

Theorem 5.1. Assume that p : [a, b] +, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108">View MathML</a>, and that the mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M119">View MathML</a>are increasing and order-bounded. Then, the IVP (5.1) has the smallest and greatest solutions in Y, and they are increasing with respect to f, c and d.

Proof: The hypotheses imposed on f, c and d imply that the following conditions are valid.

(f0) f(u, v) is RP integrable on [a, b] for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M120">View MathML</a>, and there exist such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M121">View MathML</a> that h- f (u1, v1) ≼ f (u2, v2) ≼ h+ for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M122">View MathML</a>, i = 1, 2, u1 u2 and v1 v2.

(c0) c± ∈ ℝ, and c- c(u1, v1) ≤ c(u2, v2) ≤ c+ whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M122">View MathML</a>, i = 1, 2, u1 u2 and v1 v2.

(d0) d± ∈ ℝ, and d- ≤ d(u1, v1) ≤ d(u2, v2) ≤ d+ whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M122">View MathML</a>, i = 1, 2, u1 u2 and v1 v2.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M123">View MathML</a> is ordered componentwise. We shall first show that the vector-functions x+, x- given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M124">View MathML</a>

(5.6)

define functions x± P. Since 1/p is Lebesgue integrable and the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M125">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1">View MathML</a>, then the second components of x± belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a>. This result implies that the first components of x± are defined and continuous, whence they belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a>.

Similarly, by applying also the given hypotheses one can verify that the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M126">View MathML</a>

(5.7)

define an increasing mapping G = (G1, G2) : [x-, x+] → [x-, x+].

Let W be a well-ordered chain in the range of G. The sets W1 = {u : (u, v) ∈ W} and W2 = {v : (u, v) ∈ W} are well-ordered and order-bounded chains in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a>. It then follows from [[1], Proposition 9.36] that the supremums of W1 and W2 exist in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M18">View MathML</a>. Obviously, (sup W1, sup 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 (5.7) satisfies the hypotheses of Lemma 2.1, and therefore G has the smallest fixed point x* = (u*,v*) and the greatest fixed point x* = (u*, v*). It follows from (5.7) that (u*, v*) and (u*, v*) are solutions of the system (5.3). According to Lemma 5.1, u* and u* belong to Y and are solutions of the IVP (5.1).

To prove that u* and u* are the smallest and greatest of all solutions of (5.1) in Y , let u Y be any solution of (5.1). In view of Lemma 5.1, (u, v) = (u, u') is a solution of the system (5.3). Applying the hypotheses (f0), (c0) and (d0) it is easy to show that x = (u, v) ∈ [x-, x+], where x± are defined by (5.6). Thus x = (u, v) is a fixed point of G = (G1, G2) : [x-, x+] → [x-, x+], defined by (5.7). Because x* = (u*, v*) and x* = (u*, v*) are the smallest and greatest fixed points of G, 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 IVP (5.1).

The last assertion is an easy consequence of the last conclusion of Lemma 2.1 and the definition (5.7) of G = (G1, G2). □

Consider next the the following special case of (5.1) where the values of f are combined with impulses and a Henstock-Kurzweil integrable function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M127">View MathML</a>

In this case problem (5.1) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M128">View MathML</a>

(5.8)

The next result is a consequence of Theorem 5.1.

Corollary 5.1. Assume that p : [a, b] → ℝ+, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108">View MathML</a>, that functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M106">View MathML</a>are increasing and order-bounded, and that the mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M130">View MathML</a>satisfies the following hypotheses.

(q1) q(·, x) is Henstock-Kurzweil integrable on [a, b] for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M131">View MathML</a>.

(q2) There exist Henstock-Kurzweil integrable functions q± : [a, b] → ℝ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M132">View MathML</a>, whenever × ≤ y in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M116">View MathML</a>.

(I) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M86">View MathML</a>for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M131">View MathML</a>, and × α I(λ, x) is increasing for all λ ∈ Λ.

Then, the impulsive IVP (5.8) has the smallest and greatest solutions that are increasing with respect to q, c and d.

Example 5.1. Determine the smallest and greatest solutions of the following singular impulsive IVP.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M133">View MathML</a>

(5.9)

Solution: System (5.9) is a special case of (5.8) by setting a = 0, b = 3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M97">View MathML</a>, and q, c, d and I are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M135">View MathML</a>

(5.10)

It is easy to verify that the hypotheses of Corollary 5.1 hold. Thus (5.9) has the smallest and greatest solutions. The functions x- and x+ defined by (5.6) can be calculated, and their first components are:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M136">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M137">View MathML</a>

is the Fresnel sine integral. According to Lemma 5.1, the smallest solution of (5.9) is equal to the first component of the smallest fixed point of G = (G1, G2), defined by (5.7), with f, c and d given by (5.10) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M134">View MathML</a>. Calculating the iterations Gnx- it turns out that G4x- = G5x-, whence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M138">View MathML</a> is the smallest solution of (5.9). Similarly, one can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M139">View MathML</a> is the greatest solution of (5.9). The exact expressions of these solutions are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M140">View MathML</a>

6 Second Order Boundary Value Problems

This section is devoted to the study of the second order boundary value problem (BVP)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M141">View MathML</a>

(6.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M142">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M143">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M144">View MathML</a> , and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M109">View MathML</a>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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M145">View MathML</a>

(6.3)

Proof: Assume that u is a solution of the BVP (6.1) in Z, and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M146">View MathML</a>

(6.4)

The differential equation, the boundary conditions of (6.1), the definition (6.2) of Z and the notation (6.4) ensure that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M147">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M148">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M149">View MathML</a>

(6.5)

This equation implies that p · u' belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108">View MathML</a>.

(f1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M142">View MathML</a> is order-bounded, and f (u1, v1) ≼ f (u2, v2) whenever ui, vi 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 ui, vi 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 ui, vi 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M150">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M151">View MathML</a>

(6.6)

We shall first show that the vector-functions x+, x- given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M152">View MathML</a>

(6.7)

belong to P. Since 1/p is Lebesgue integrable and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M153">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M1">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M154">View MathML</a>

(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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M155">View MathML</a>

(6.9)

Corollary 6.1. Assume that p : [a, b] → ℝ+, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M108">View MathML</a>, that functions c, d : L1[a, b]2 → ℝ satisfy the hypotheses (ci) and (di), i = 1, 2, that α : Λ → ℝ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M156">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M157">View MathML</a>such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M158">View MathML</a>, 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.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M159">View MathML</a>

(6.10)

Solution: System (6.10) is a special case of (6.9) when a = 0, b = 3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M134">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M161">View MathML</a>, and g, c, d are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M162">View MathML</a>

(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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M163">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M164">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M165">View MathML</a> is the smallest solution of (6.10). Similarly, one can show that G3x+ = G4x+, whence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M166">View MathML</a> is the greatest solution of (6.10). The exact expressions of these solutions are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M167">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M168">View MathML</a>

Remarks 6.1. The IVP's (3.1) and (5.1) and the BVP (6.1) can be

• singular, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/24/mathml/M169">View MathML</a> 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.

Competing interests

The author declares that they have no competing interests.

Authors' contributions

The work was realized by the author.

Acknowledgements

The author thanks the anonymous referee for a careful review and constructive comments.

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