First-Order Singular and Discontinuous Differential Equations

Volume 2009, Article ID 507671,25pages doi:10.1155/2009/507671

Research Article

First-Order Singular and Discontinuous Differential Equations

Daniel C. Biles

and Rodrigo L ´opez Pouso

1Department of Mathematics, Belmont University, 1900 Belmont Blvd., Nashville, TN 37212, USA

2Department of Mathematical Analysis, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain

Correspondence should be addressed to Rodrigo L ´opez Pouso,rodrigo.lopez@usc.es Received 10 March 2009; Accepted 4 May 2009

Recommended by Juan J. Nieto

We use subfunctions and superfunctions to derive sufficient conditions for the existence of extremal solutions to initial value problems for ordinary differential equations with discontinuous and singular nonlinearities.

Copyrightq2009 D. C. Biles and R. L ´opez Pouso. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Lett0, x0 ∈RandL >0 be fixed and letf :t0, t0L×R → Rbe a given mapping. We are concerned with the existence of solutions of the initial value problem

xft, x , t∈I: t0, t0L, xt0 x0. 1.1 It is well-known that Peano’s theorem ensures the existence of local continuously differentiable solutions of1.1 in casefis continuous. Despite its fundamental importance, it is probably true that Peano’s proof of his theorem is even more important than the result itself, which nowadays we know can be deduced quickly from standard fixed point theorems see1, Theorem 6.2.2for a proof based on the Schauder’s theorem . The reason for believing this is that Peano’s original approach to the problem in2consisted in obtaining the greatest solution as the pointwise infimum of strict upper solutions. Subsequently this idea was improved by Perron in 3, who also adapted it to study the Laplace equation by means of what we call today Perron’s method. For a more recent and important revisitation of the method we mention the work by Goodman4on1.1 in casefis a Carath´eodory function.

For our purposes in this paper, the importance of Peano’s original ideas is that they can

be adapted to prove existence results for1.1 under such weak conditions that standard functional analysis arguments are no longer valid. We refer to differential equations which depend discontinuously on the unknown and several results obtained in papers as5–9, see also the monographs10,11.

On the other hand, singular differential equations have been receiving a lot of attention in the last years, and we can quote7,12–19. The main objective in this paper is to establish an existence result for1.1 with discontinuous and singular nonlinearities which generalizes in some aspects some of the previously mentioned works.

This paper is organized as follows. InSection 2we introduce the relevant definitions together with some previously published material which will serve as a basis for proving our main results. In Section 3 we prove the existence of the greatest and the smallest Carath´eodory solutions for1.1 between given lower and upper solutions and assuming the existence of aL¹-bound forf on the sector delimited by the graphs of the lower and upper solutionsregular problems , and we give some examples. InSection 4we show that looking for piecewise continuous lower and upper solutions is good in practice, but once we have found them we can immediately construct a pair of continuous lower and upper solutions which provide better information on the location of the solutions. In Section 5 we prove two existence results in casef does not have such a strong bound as inSection 3singular problems , which requires the addition of some assumptions over the lower and upper solutions. Finally, we prove a result for singular quasimonotone systems in Section 6and we give some examples inSection 7. Comparison with the literature is provided throughout the paper.

2. Preliminaries

In the following definition ACI stands for the set of absolutely continuous functions onI.

Definition 2.1. A lower solution of 1.1 is a function l ∈ ACI such that lt0 ≤ x0 and lt ≤ft, lt for almost alla.a. t∈I; an upper solution is defined analogously reversing the inequalities. One says thatxis aCarath´eodory solution of1.1 if it is both a lower and an upper solution. On the other hand, one says that a solutionx_∗is the least one ifx_∗≤xon Ifor any other solutionx, and one defines the greatest solution in a similar way. When both the least and the greatest solutions exist, one calls them the extremal solutions.

It is proven in 8 that 1.1 has extremal solutions if f is L¹-bounded for all x ∈ R, f·, x is measurable, and for a.a.t∈I ft,· is quasi-semicontinuous, namely, for allx∈R we have

lim sup

y→x⁻

f t, y

≤ft, x ≤lim inf

y→x f t, y

. 2.1

A similar result was established in20assuming moreover thatfis superpositionally measurable, and the systems case was considered in5,8. The term “quasi-semicontinuous”

in connection with 2.1 was introduced in 5 for the first time and it appears to be conveniently short and descriptive. We note however that, rigorously speaking, we should say thatft,· is left upper and right lower semicontinuous.

On the other hand, the above assumptions imply that the extremal solutions of1.1 are given as the infimum of all upper solutions and the supremum of all lower solutions, that is, the least solution of1.1 is given by

uinft inf

ut :uupper solution of1.1

, t∈I, 2.2

and the greatest solution is

lsupt sup{lt :llower solution of1.1 }, t∈I. 2.3

The mappings uinf and lsup turn out to be the extremal solutions even under more general conditions. It is proven in9that solutions exist even if2.1 fails on the points of a countable family of curves in the conditions of the following definition.

Definition 2.2. An admissible non-quasi-semicontinuity nqsc curve for the differential equationxft, x is the graph of an absolutely continuous functionγ:a, b⊂t0, t₀L → Rsuch that for a.a.t∈a, bone has eitherγt ft, γt , or

γt ≥f t, γt

wheneverγt ≥ lim inf

y→γt f t, y

, 2.4

γt ≤f t, γt

wheneverγt ≤lim sup

y→γt ⁻

f t, y

. 2.5

Remark 2.3. The condition 2.1 cannot fail over arbitrary curves. As an example note that 1.1 has no solution fort₀0x₀and

ft, x

⎧⎨

⎩

1, if x <0,

−1, if x≥0. 2.6

In this case2.1 only fails over the line x 0, but solutions coming from above that line collide with solutions coming from below and there is no way of continuing them to the right once they reach the levelx0. Following Binding21we can say that the equation “jams”

atx0.

An easily applicable sufficient condition for an absolutely continuous function γ : a, b ⊂ I → Rto be an admissible nqsc curve is that either it is a solution or there exist ε >0 andδ >0 such that one of the following conditions hold:

1 γt ≥ft, y εfor a.a.t∈a, band ally∈γt −δ, γt δ, 2 γt ≤ft, y −εfor a.a.t∈a, band ally∈γt −δ, γt δ.

These conditions prevent the differential equation from exhibiting the behavior of the previous example over the linex0 in several ways. First, ifγis a solution ofxft, x then any other solution can be continued overγonce they contact each other and independently of the definition off around the graph ofγ. On the other hand, if1 holds then solutions ofx ft, x can crossγ from above to belowhence at most once , and if2 holds then

solutions can crossγ from below to above, so in both cases the equation does not jam over the graph ofγ.

For the convenience of the reader we state the main results in 9. The next result establishes the fact that we can have “weak” solutions in a sense just by assuming very general conditions overf.

Theorem 2.4. Suppose that there exists a null-measure setN⊂Isuch that the following conditions hold:

1 condition2.1 holds for allt, x ∈I\N ×Rexcept, at most, over a countable family of admissible non-quasi-semicontinuity curves;

2 there exists an integrable functionggt ,t∈I, such that

ft, x ≤gt ∀t, x ∈I\N ×R. 2.7

Then the mapping

u^∗_inft inf

ut :uupper solution of1.1 , u ≤g1 a.e.

, t∈I 2.8

is absolutely continuous onIand satisfiesu^∗_inft0 x₀andu^∗_inft ft, u^∗_inft for a.a.t∈I\J, whereJ∪n,m∈NJn,mand for alln, m∈Nthe set

J_n,m:

t∈I:u^∗_inft − 1 n >sup

f

t, y

:u^∗_inft − 1

m< y < u^∗_inft

2.9

contains no positive measure set.

Analogously, the mapping l_sup^∗ t sup

lt :llower solution of1.1 , l ≤g1 a.e.

, t∈I, 2.10

is absolutely continuous onI and satisfiesl^∗_supt0 x₀andl^∗_supt ft, l^∗_supt for a.a.t∈I\K, whereK∪n,m∈NKn,mand for alln, m∈Nthe set

K_n,m:

t∈I:l_sup^∗ t 1 n <inf

f

t, y

:l^∗_supt < y < l^∗_supt 1 m

2.11

contains no positive measure set.

Note that if the sets J_n,m and K_n,m are measurable then u^∗_inf and l^∗_sup immediately become the extremal Carath´eodory solutions of 1.1 . In turn, measurability of those sets can be deduced from some measurability assumptions on f. The next lemma is a slight generalization of some results in8and the reader can find its proof in9.

Lemma 2.5. Assume that for a null-measure set N ⊂ I the mapping f satisfies the following condition.

For eachq∈Q,f·, q is measurable, and fort, x ∈I\N ×Rone has

min

lim sup

y→x⁻ f t, y

,lim sup

y→x

f t, y

≤ft, x ≤max

lim inf

y→x⁻ f t, y

,lim inf

y→x f t, y

. 2.12

Then the mappingst∈I →sup{ft, y :x₁t < y < x₂t }andt∈I → inf{ft, y : x₁t < y < x₂t }are measurable for each pairx₁, x₂ ∈CI such thatx₁t < x₂t for allt∈I.

Remark 2.6. A revision of the proof of9, Lemma 2shows that it suffices to impose2.12 for allt, x ∈I\N ×Rsuch thatx1t < x < x2t . This fact will be taken into account in this paper.

As a consequence ofTheorem 2.4 and Lemma 2.5we have a result about existence of extremal Carath´eodory solutions for1.1 and L¹-bounded nonlinearities. Note that the assumptions inLemma 2.5include a restriction over the type of discontinuities that can occur over the admissible nonqsc curves, but remember that such a restriction only plays the role of implying that the setsJn,mandKn,minTheorem 2.4are measurable. Therefore, only using the axiom of choice one can find a mappingf in the conditions ofTheorem 2.4which does not satisfy the assumptions inLemma 2.5and for which the corresponding problem1.1 lacks the greatestor the least Carath´eodory solution.

Theorem 2.79, Theorem 4 . Suppose that there exists a null-measure setN ⊂ Isuch that the following conditions hold:

i for everyq∈Q,f·, q is measurable;

ii for everyt∈I\Nand allx∈Rone has either2.1 or lim inf

y→x⁻ f t, y

≥ft, x ≥lim sup

y→x f t, y

, 2.13

and2.1 can fail, at most, over a countable family of admissible nonquasisemicontinuity curves;

iii there exists an integrable functionggt ,t∈I, such that

ft, x ≤gt ∀t, x ∈I\N ×R. 2.14

Then the mappinguinfdefined in2.2 is the least Carath´eodory solution of 1.1 and the mapping l_supdefined in2.3 is the greatest one.

Remark 2.8. Theorem 4 in 9 actually asserts that u^∗_inf, as defined in 2.8 , is the least Carath´eodory solution, but it is easy to prove that in that caseu^∗_inf u_inf, as defined in2.2 . Indeed, letUbe an arbitrary upper solution of1.1 , letgmax{|U|, g}and let

v_inf^∗ t inf

ut :uupper solution of1.1 , u ≤g1 a.e.

, t∈I. 2.15

Theorem 4 in9implies that alsov^∗_infis the least Carath´eodory solution of1.1 , thusu^∗_inf v^∗_inf≤UonI. Henceu^∗_infu_inf.

Analogously we can prove that l^∗_sup can be replaced bylsup in the statement of21, Theorem 4.

3. Existence between Lower and Upper Solutions

Condition iii in Theorem 2.7 is rather restrictive and can be relaxed by assuming boundedness offbetween a lower and an upper solution.

In this section we will prove the following result.

Theorem 3.1. Suppose that1.1 has a lower solutionαand an upper solutionβsuch thatαt ≤βt for allt∈Iand letE{t, x ∈I×R:αt ≤x≤βt }.

Suppose that there exists a null-measure setN⊂Isuch that the following conditions hold:

iα,β for every q ∈ Q∩min_t∈Iαt ,max_t∈Iβt , the mappingf·, q with domain{t ∈ I : αt ≤q≤βt }is measurable;

iiα,β for everyt, x ∈E,t /∈N, one has either2.1 or2.13 , and2.1 can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained inE;

iiiα,β there exists an integrable functionggt ,t∈I, such that

ft, x ≤gt ∀t, x ∈E, t /∈N. 3.1

Then1.1 has extremal solutions in the set α, β

:

z∈ACI :αt ≤zt ≤βt ∀t∈I

. 3.2

Moreover the least solution of 1.1 inα, βis given by x_∗t inf

ut :uupper solution of1.1 , u∈ α, β

, t∈I, 3.3

and the greatest solution of 1.1 inα, βis given by x^∗t sup

lt :l lower solution of1.1 , l∈ α, β

, t∈I. 3.4

Proof. Without loss of generality we suppose thatαandβexist and satisfy|α| ≤g,|β| ≤g, α≤ft, α , andβ≥ft, β onI\N. We also mayand we do assume that every admissible nqsc curve in conditioniiα,β , sayγ :a, b → R, satisfies for allt∈a, b\Neitherγt ft, γt or2.4 -2.5 .

For eacht, x ∈I×Rwe define

Ft, x :

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ft, αt , ifx < αt , ft, x , ifαt ≤x≤βt , f

t, βt

, ifx > βt .

3.5

Claim 1. The modified problem

xFt, x , t∈I, xt0 x₀, 3.6

satisfies conditions1 and2 inTheorem 2.4withfreplaced byF. First we note that2 is an immediate consequence ofiiiα,β and the definition ofF.

To show that condition1 inTheorem 2.4is satisfied withfreplaced byF, lett, x ∈ I\N ×Rbe fixed. The verification of2.1 forF att, x is trivial in the following cases:

αt < x < βt andf satisfies2.1 att, x ,x < αt ,x > βt andαt x βt . Let us consider the remaining situations: we start with the casexαt < βt andfsatisfies2.1 att, x , for which we haveFt, x ft, x and

lim sup

y→x⁻ F t, y

ft, αt ft, x ≤lim inf

y→x f t, y

lim inf

y→x F t, y

, 3.7

and an analogous argument is valid whenαt < βt xandfsatisfies2.1 .

The previous argument shows thatFsatisfies2.1 at everyt, x ∈I\N ×Rexcept, at most, over the graphs of the countable family of admissible nonquasisemicontinuity curves in conditioniiα,β forx ft, x . Therefore it remains to show that ifγ :a, b ⊂I → Ris one of those admissible nqsc curves forxft, x then it is also an admissible nqsc curve for xFt, x . As long as the graph ofγ remains in the interior ofEwe have nothing to prove becausefandFare the same, so let us assume thatγαon a positive measure setP⊂a, b, P∩N ∅. Sinceαandγare absolutely continuous there is a null measure setNsuch that αt γt for allt∈P\N, thus for t∈P\Nwe have

γt ≤f t, γt

lim sup

y→γt ⁻

F t, y

, γt ≤F t, γt

, 3.8

so condition2.5 withf replaced byFis satisfied onP\N. On the other hand, we have to check whetherγt ≥Ft, γt for thoset∈P\Nat which we have

γt ≥ lim inf

y→γt F t, y

. 3.9

We distinguish two cases:αt < βt andαt βt . In the first case3.9 is equivalent to γt ≥ lim inf

y→γt f t, y

, 3.10

and therefore eitherγt ft, γt or condition2.4 holds, yielding γt ≥ ft, γt Ft, γt . Ifαt βt then we haveγt αt βt ≥ft, βt Ft, γt .

Analogous arguments show that eitherγ Ft, γ or2.4 -2.5 hold forF at almost every point whereγcoincides withβ, so we conclude thatγis an admissible nqsc curve for xFt, x .

By virtue of Claim 1 and Theorem 2.4 we can ensure that the functions x_∗ and x^∗ defined as

x_∗t inf

ut :uupper solution of3.6 , u ≤g1 a.e.

, t∈I, x^∗t sup

lt :l lower solution of3.6 , l ≤g1 a.e.

, t∈I,

3.11

are absolutely continuous onIand satisfyx_∗t0 x^∗t0 x₀andx_∗t Ft, x∗t for a.a.

t∈I\J, whereJ∪_n,m∈NJn,mand for alln, m∈Nthe set

Jn,m:

t∈I : x_∗t − 1 n >sup

F

t, y

:x_∗t − 1

m < y < x_∗t

3.12

contains no positive measure set, and x^∗t Ft, x^∗t for a.a. t ∈ I \ K, where K

∪n,m∈NKn,mand for alln, m∈Nthe set

Kn,m:

t∈I:x^∗t 1 n <inf

F

t, y

:x^∗t < y < x^∗t 1 m

3.13

contains no positive measure set.

Claim 2. For allt∈Iwe have x_∗t inf

ut :uupper solution of1.1 , u∈ α, β

, u ≤g1 a.e.

, 3.14 x^∗t sup

lt :llower solution of 1.1 , l∈ α, β

, l ≤g1 a.e.

. 3.15

Letube an upper solution of3.6 and let us show thatut ≥αt for allt∈I. Reasoning by contradiction, assume that there existt1, t2∈Isuch thatt1< t2,ut 1 αt1 and

ut < αt ∀t∈t1, t₂. 3.16

For a.a.t∈t1, t₂we have

ut ≥Ft,ut ft, αt ≥αt , 3.17

which together withut 1 αt1 implyu≥αont1, t2, a contradiction with3.16 . Therefore every upper solution of3.6 is greater than or equal toα, and, on the other hand,βis an upper solution of3.6 with|β| ≤ga.e., thusx_∗satisfies3.14 .

One can prove by means of analogous arguments thatx^∗satisfies3.15 .

Claim 3. x_∗is the least solution of1.1 inα, βandx^∗is the greatest one. From3.14 and 3.15 it suffices to show thatx_∗andx^∗are actually solutions of3.6 . Therefore we only have to prove thatJandKare null measure sets.

Let us show that the setJis a null measure set. First, note that

J

⎧⎨

⎩t∈I:x_∗t > lim sup

y→x∗t ⁻

F t, y⎫

⎬

⎭, 3.18

and we can splitJ A∪B, whereA{t∈J :x_∗t > αt }andBJ\A{t∈J :x_∗t αt }.

Let us show thatBis a null measure set. Sinceαandx_∗are absolutely continuous the set

C

t∈I :αt does not exist

∪

t∈I:x_∗t does not exist

∪

t∈I:αt x_∗t , αt /x_∗t 3.19 is null. IfB /⊂Cthen there is somet0∈Bsuch thatαt0 x_∗t0 andαt0 x_∗t0 , but then the definitions ofBandFyield

αt0 > lim sup

y→αt0 ⁻

F t0, y

ft0, αt0 . 3.20

ThereforeB\C⊂Nand thusBis a null measure set.

The setAcan be expressed asA∪^∞_k1Ak, where for eachk∈N

A_k

⎧⎨

⎩t∈I:x_∗t > αt 1

k, x_∗t > lim sup

y→x∗t ⁻

F t, y⎫

⎬

⎭

^∞

n,m1

A_k∩J_n,m.

3.21

Fork, m∈N,k < m, we havex_∗t −1/m > x_∗t −1/k, so the definition ofFimplies that

Ak∩Jn,m

t∈I:x_∗t > αt 1

k, x_∗t −1 n >sup

f

t, y

:x_∗t − 1

m < y < x_∗t

3.22 which is a measurable set by virtue ofLemma 2.5andRemark 2.6.

SinceJn,mcontains no positive measure subset we can ensure thatAk∩Jn,mis a null measure set for allm ∈ N,m > k, and sinceJ_n,mincreases withnandm, we conclude that A_k ∪^∞_n,m1Ak ∩J_n,m is a null measure set. Finally Ais null because it is the union of countably many null measure sets.

Analogous arguments show thatK is a null measure set, thus the proof ofClaim 3is complete.

Claim 4. x_∗satisfies3.3 andx^∗satisfies3.4 . LetU∈α, βbe an upper solution of1.1 , let

gmax{|U|, g}, and for allt∈Ilet

y_∗t inf

ut :u upper solution of3.6 , u ≤g1 a.e.

. 3.23

Repeating the previous arguments we can prove that also y_∗ is the least Carath´eodory solution of1.1 inα, β, thusx_∗y_∗≤UonI. Hencex_∗satisfies3.3 .

Analogous arguments show thatx^∗satisfies3.4 .

Remark 3.2. Problem3.6 may not satisfy conditioni inTheorem 2.7as the compositions f·, α· andf·, β· need not be measurable. That is why we usedTheorem 2.4, instead of Theorem 2.7, to establishTheorem 3.1.

Next we show that even singular problems may fall inside the scope ofTheorem 3.1if we have adequate pairs of lower and upper solutions.

Example 3.3. Let us denote byzthe integer part of a real numberz. We are going to show that the problem

x 1

t|x|

x sgnx

2 , for a.a. t∈0,1, x0 0 3.24

has positive solutions. Note that the limit of the right hand side ast, x tends to the origin does not exist, so the equation is singular at the initial condition.

In order to applyTheorem 3.1we consider1.1 witht₀0x₀,L1, and

ft, x

⎧⎪

⎨

⎪⎩ 1

tx

x1

2, ifx >0, 1

2, ifx≤0.

3.25

It is elementary matter to check thatαt 0 andβt t,t∈I, are lower and upper solutions for the problem. Condition2.1 only fails over the graphs of the functions

γ_nt 1

n−t, t∈

0,1 n

, n∈N, 3.26

which are a countable family of admissible nqsc curves at which condition2.13 holds.

Finally note that

ft, x ≤ 3

2 ∀t, x ∈I×R,0≤x≤t, 3.27

so conditioniiiα,β is satisfied.

Theorem 3.1ensures that our problem has extremal solutions betweenαandβwhich, obviously, are different from zero almost everywhere. Therefore3.24 has positive solutions.

The result ofTheorem 3.1may fail if we assume that conditioniiα,β is satisfied only in the interior of the setE. This is shown in the following example.

Example 3.4. Let us consider problem1.1 witht0 x0 0,L 1 andf : 0,1×R → R defined as

ft, x

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

1, if x <0, 1

2, if x0,

−1, if x >0.

3.28

It is easy to check thatαt 0 andβt tfor allt ∈0,1are lower and upper solutions for this problem and that all the assumptions ofTheorem 3.1are satisfied in the interior ofE.

However this problem has no solution at all.

In order to complete the previous information we can say that conditioniiα,β in the interior of E is enough if we modify the definitions of lower and upper solutions in the following sense.

Theorem 3.5. Suppose thatαandβare absolutely continuous functions onI such thatαt < βt for allt∈t0, t₀L,αt0 ≤x₀ ≤βt0 ,

αt ≤ lim inf

y→αt f t, y

for a.a. t∈I, βt ≥lim sup

y→βt ⁻

f t, y

for a.a. t∈I, 3.29

and letE{t, x ∈I×R:αt ≤x≤βt }.

Suppose that there exists a null-measure setN⊂Isuch that conditionsiα,β andiiiα,β hold and, moreover,

ii^◦_α,β for everyt, x ∈E^◦,t /∈N, one has either2.1 or2.13 , and2.1 can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained inE.^◦

Then the conclusions ofTheorem 3.1hold true.

Proof (Outline)

It follows the same steps as the proof ofTheorem 3.1but replacingFby

Ft, x

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

0, iftt₀,

lim inf

y→αt f t, y

, ift > t₀, x≤αt , ft, x , ift > t0, αt < x < βt , lim sup

y→βt ⁻

f t, y

, ift > t₀, x≥βt .

3.30

Note that condition2.1 withfreplaced byFis immediately satisfied over the graphs ofα andβthanks to the definition ofF.

Remarks

i The functionαinExample 3.4does not satisfy the conditions inTheorem 3.5.

ii Whenft,· satisfies2.1 everywhere or almost allt∈Ithen every couple of lower and upper solutions satisfies the conditions in Theorem 3.5, so this result is not really new in that casewhich includes the Carath´eodory and continuous cases .

4. Discontinuous Lower and Upper Solutions

Another modification of the concepts of lower and upper solutions concerns the possibility of allowing jumps in their graphs. Since the task of finding a pair of lower and upper solutions is by no means easy in general, and bearing in mind that constant lower and upper solutions are the first reasonable attempt, looking for lower and upper solutions “piece by piece” might make it easier to find them in practical situations. Let us consider the following definition.

Definition 4.1. One says thatα :I → Ris a piecewise continuous lower solution of1.1 if there existt₀ < t₁<· · ·< t_nt₀Lsuch that

a for alli∈ {1,2, . . . , n}, one hasα∈ACti−1, t_i and for a.a.t∈I

αt ≤ft, αt , 4.1

b lim_t→t₀αt αt0 ≤x0, for alli∈ {1,2, . . . , n}

t→limt⁻_i αt αti > lim

t→t_i αt , 4.2

and lim_t→t⁻_nαt αtn .

A piecewise continuous upper solution of1.1 is defined reversing the relevant inequalities.

The existence of a pair of well-ordered piecewise continuous lower and upper solutions implies the existence of a better pair of continuous lower and upper solutions. We establish this more precisely in our next proposition. Note that the proof is constructive.

Proposition 4.2. Assume that all the conditions inTheorem 3.1hold with piecewise continuous lower and upper solutionsαandβ.

Then the following statements hold:

i there exist a lower solutionαand an upper solutionβsuch that

α≤α≤β≤β onI; 4.3

ii ifuis an upper solution of 1.1 withα ≤u ≤βthenα ≤ u, and iflis a lower solution withα≤l≤βthenl≤β.

In particular, the conclusions of Theorem 3.1 remain valid and, moreover, every solution of 1.1 betweenαandβlies betweenαandβ.

Proof. We will only prove the assertions concerningαbecause the proofs forβare analogous.

To constructαwe simply have to join the pointstk, αtk ,k ∈ {1, . . . , n−1}with the graph ofα_|tk,tk1 by means of an absolutely continuous curve with derivative less than or equal to−g a.e.,g being the function given iniiiα,β . It can be easily proven that thisαis a lower solution of1.1 that lies betweenαandβ.

Moreover, ifuis an upper solution of1.1 betweenαandβthen we have

ut ≥ft, ut ≥ −gt a.e.ont0, t0 L, 4.4

so it cannot go belowα.

Piecewise continuous lower and upper solutions in the sense ofDefinition 4.1were already used in 15, 22. It is possible to generalize further the concept of lower and upper solutions, as a piecewise continuous lower solution is a particular case of a bounded variation function that has a nonincreasing singular part. Bounded variation lower and upper solutions with monotone singular parts were used in23, 24, but it is not clear whether Theorem 3.1is valid with this general type of lower and upper solutions. Anyway, piecewise continuous lower and upper solutions are enough in practical situations, and since these can be transformed into continuous ones which provide better information we will only consider from now on continuous lower and upper solutions as defined inDefinition 2.1.

5. Singular Differential Equations

It is the goal of the present section to establish a theorem on existence of solutions for1.1 between a pair of well-ordered lower and upper solutions and in lack of a localL¹ bound.

Solutions will be weak, in the sense of the following definition. By AC_loct0, t₀L we denote the set of functionsξsuch thatξ_|t0ε,t0L∈ACt0ε, t0L for allε∈0, L , and in a similar way we defineL¹_loct0, t₀L .

Definition 5.1. We say thatα ∈ CI ∩ACloct0, t0L is a weak lower solution of1.1 if αt0 ≤x₀andαt ≤ ft, αt for a.a.t ∈I. A weak upper solution is defined analogously reversing inequalities. A weak solution of 1.1 is a function which is both a weak lower solution and a weak upper solution.

We will also refer to extremal weak solutions with obvious meaning.

Note that lower/upper solutions, as defined in Definition 2.1, are weak lower/upper solutions but the converse is false in general. For instance the singular linear problem

x x

t −cos1/t

t , t∈0,1, x0 0, 5.1

has exactly the following weak solutions:

xat tsin 1

t

at, t∈0,1, xa0 0 a∈R , 5.2

and none of them is absolutely continuous on0,1. Another example, which uses lower and upper solutions, can be found in15, Remark 2.4.

However weaklower/upper solutions are of Carath´eodory type provided they have bounded variation. We establish this fact in the next proposition.

Proposition 5.2. Leta, b∈Rbe such thata < band leth:a, b → Rbe continuous ona, band locally absolutely continuous ona, b.

A necessary and sufficient condition forhto be absolutely continuous ona, bis thathbe of bounded variation ona, b.

Proof. The necessary part is trivial. To estalish the sufficiency of our condition we use Banach- Zarecki’s theorem, see18, Theorem 18.25. LetN⊂a, bbe a null measure set, we have to prove thathN is also a null measure set. To do this letn₀ ∈ Nbe such thata1/n₀ < b.

Sincehis absolutely continuous ona1/n₀, bthe sethN∩a1/n, b is a null measure set for eachn≥n0. ThereforehN is also a null measure set because

hN ⊂ {ha } ∪ _∞

nn0

h

N∩

a1 n, b

. 5.3

Next we present our main result on existence of weak solutions for1.1 in absence of integrable bounds.

Theorem 5.3. Suppose that1.1 has a weak lower solutionαand a weak upper solutionβsuch that αt ≤βt for allt∈Iandαt0 x₀βt0 .

Suppose that there is a null-measure setN ⊂ I such that conditions iα,β and iiα,β in Theorem 3.1hold forE{t, x ∈I×R:αt ≤x≤βt },and assume moreover that the following condition holds:

iii^∗_α,β there existsg∈L¹_loct0, t0L such that for allt, x ∈E,t /∈N, one has|ft, x | ≤gt . Then1.1 has extremal weak solutions in the set

α, β_w:

z∈CI ∩AC_loct0, t0L :αt ≤zt ≤βt ∀t∈I

. 5.4

Moreover the least weak solution of 1.1 inα, βis given by x_∗t inf

ut :uweak upper solution of1.1 , u∈ α, β

w

, t∈I, 5.5

and the greatest weak solution of 1.1 inα, β_wis given by x^∗t sup

lt :l weak lower solution of1.1 , l∈ α, β

w

, t∈I. 5.6

Proof. We will only prove that5.6 defines the greatest weak solution of1.1 inα, β_w, as the arguments to show that5.5 is the least one are analogous.

First note thatαis a weak lower solution betweenαandβ, sox^∗is well defined.

Let {tn}_n be a decreasing sequence in t0, t₀ L such that limt_n t₀.Theorem 2.7 ensures that for everyn∈Nthe problem

yf t, y

, t∈tn, t0L :In, ytn x^∗tn , 5.7

has extremal Carath´eodory solutions betweenα_|Inandβ_|In. Letyndenote the greatest solution of5.7 betweenα_|In andβ_|In. By virtue ofTheorem 2.7we also know thaty_nis the greatest lower solution of5.7 betweenα_|In andβ_|In.

Next we prove in several steps thatx^∗ynonInfor eachn∈N.

Step 1 yn ≥ x^∗ onI_n for each n ∈ N . The restriction toI_n of each weak lower solution betweenαandβis a lower solution of5.7 betweenα_|In andβ_|In, thusy_nis, on the interval In, greater than or equal to any weak lower solution of1.1 betweenαandβ. The definition ofx^∗implies then thaty_n≥x^∗onI_n.

Step 2 yn1 ≥ y_n on I_n for all n ∈ N . First, since y_n1 ≥ x^∗ on I_n1 we have y_n1tn ≥ x^∗tn yntn . Reasoning by contradiction, assume that there existss∈tn, t0L such that y_n1s < y_ns . Then there is somer∈tn, s such thaty_n1r y_nr andy_n1< y_nonr, s , but then the mapping

yt

⎧⎨

⎩

y_n1t , if t∈t_n1, r,

y_nt , if t∈r, t0L, 5.8

would be a solution of5.7 withnreplaced byn1 betweenα_|In1andβ_|In1which is greater thany_n1onr, s , a contradiction.

The above properties of{yn}_nimply that the following function is well defined:

y_∞t

⎧⎨

⎩

x₀, if tt₀,

limy_nt , if t∈t0, t₀L. 5.9

Step 3y_∞ ∈CI ∩ACloct0, t0L . Letε∈0, L be fixed. Conditioniii^∗_α,β implies that for alln∈Nsuch thatt_n< t₀εwe have

y_nt f

t, ynt ≤gt for a.a. t∈t0ε, t0L, 5.10

withg∈L¹t0ε, t0L . Hence fors, t∈t0ε, t0L,s≤t, we have

y_∞t −y_∞s lim

n→ ∞

_t

s

y_n ≤

_t

s

g, 5.11

and thereforey_∞ ∈ACt0ε, t0L . Sinceε∈0, L was fixed arbitrarily in the previous arguments, we conclude thaty_∞∈AC_loct0, t₀L .

The continuity ofy_∞att₀follows from the continuity ofαandβatt₀, the assumption αt0 x0βt0 , and the relation

αt ≤y_∞t ≤βt ∀t∈t0, t₀L. 5.12

Step 4y_∞is a weak lower solution of1.1 . Forε ∈ 0, L andn ∈ Nsuch thatt_n < t₀ε we have5.10 withg ∈ L¹t0ε, t₀L , hence lim supy_n ∈L¹t0ε, t₀L , and fors, t ∈ t0ε, t0L ,s < t, Fatou’s lemma yields

y_∞t −y_∞s lim

n→ ∞

_t

s

y_n≤ _t

s

lim supy_n. 5.13

Hence for a.a.t∈t0ε, t0Lwe have

y_∞t ≤lim supy_nt lim supf

t, y_nt

. 5.14

LetJ₁∪n∈NA_nwhereA_n{t∈t0ε, t0L:y_∞t y_nt }andJ₂ t0ε, t0L\J1. For n ∈ Nand a.a. t ∈ An we have y_∞t y_nt ft, ynt ft, y∞t , thus y_∞t ft, y_∞t for a.a.t∈J₁.

On the other hand, for a.a.t∈J2the relation5.14 and the increasingness of{ynt } yield

y_∞t ≤lim supf

t, ynt

≤ lim sup

y→y∞t ⁻

f t, y

. 5.15

Lett₀ ∈J₂\Nbe such that5.15 holds. We have two possibilities: either2.1 holds forfatt0, y_∞t0 and then from5.15 we deduce y_∞t0 ≤ft0, y_∞t0 , ory_∞t0 γt0 , whereγis an admissible curve of non quasisemicontinuity. In the last case we have that either t0 belongs to a null-measure set or y_∞t0 γt0 , which, in turn, yields two possibilities:

eitherγt0 ft, γt0 and then y_∞t0 ft, y_∞t0 , orγt0 /ft, γt0 and then5.15 , with y_∞t0 γt0 and y_∞t0 γt0 , and the definition of admissible curve of non quasisemicontinuity imply that y_∞t0 ≤ft0, y_∞t0 .

The above arguments prove that y_∞t ≤ ft, y_∞t a.e. ont0ε, t₀L, and since ε∈0, L was fixed arbitrarily, the proof ofStep 4is complete.

Conclusion

The construction ofy_∞ andStep 1 imply thaty_∞ ≥ x^∗ and the definition ofx^∗ and Step 4 imply thatx^∗ ≥ y_∞. Therefore for alln ∈ Nwe havex^∗ y_n onI_n and thenx^∗ is a weak solution of1.1 . Since every weak solution is a weak lower solution,x^∗is the greatest weak solution of1.1 inα, β_w.

The assumption αt0 βt0 in Theorem 5.3 can be replaced by other types of conditions. The next theorem generalizes the main results in7,12–14concerning existence of solutions of singular problems of the type of1.1 .

Theorem 5.4. Suppose that1.1 withx00 has a weak lower solutionαand a weak upper solution βsuch thatαt ≤βt for allt∈Iandα >0 ont0, t₀L.

Suppose that there is a null-measure setN ⊂ I such that conditions iα,β and iiα,β in Theorem 3.1hold forE{t, x ∈I×R:αt ≤x≤βt },and assume moreover that the following condition holds:

iii_α,β for everyr ∈0,1 there existsg_r ∈L¹I such that for allt, x ∈E,t /∈N, andr ≤x≤ 1/rone has|ft, x | ≤grt .

Then the conclusions ofTheorem 5.3hold true.

Proof. We start observing that there exists a weak upper solutionβsuch thatα ≤ β ≤ βon I and αt0 0 βt 0 . Ifβt0 0 then it suffices to takeβasβ. Ifβt0 > 0 we proceed as follows in order to constructβ: let {xn}_n be a decreasing sequence in0, βt0 such that limx_n0 and for everyn∈Nlety_nbe the greatest solution betweenαandβof

yf t, y

, t∈I, yt0 x_n. 5.16

claim [ynexists]

Letε ∈0, L be so small thatαt < x_n−ε < x_nε < βt for allt∈ t0, t₀ε. Condition iii_α,β implies that there exists g_ε ∈ L¹t0, t₀ ε such that for a.a. t ∈ t0, t₀ εand all x∈xn−ε, x_nεwe have|ft, x | ≤g_εt . Letv_±t x_n± ^t₀g_εs ds,t∈t0, t₀εand let δ ∈0, εbe such thatx_n−ε≤v₋ ≤v ≤x_nεont0, t₀δ. We can applyTheorem 2.7to the problem

ξft, ξ , t∈t0, t₀δ, ξt0 x_n, 5.17

and with respect to the lower solutionv₋ and the upper solution v, so there exists ξ_n the greatest solution betweenv₋ and v of 5.17 . Notice that if x is a solution of 5.17 then v₋≤x≤v, soξ_nis also the greatest solution betweenαandβof5.17 .

Now conditioniii_α,β ensures thatTheorem 2.7can be applied to the problem

zft, z , t∈t0δ, t₀L, zt0δ ξ_nt0δ , 5.18

with respect to the lower solutionαand the upper solutionβboth functions restricted to t0δ, t0L . Hence there existsznthe greatest solution of5.18 betweenαandβ.

Obviously we have

ynt

⎧⎨

⎩

ξ_nt , if t∈t0, t₀δ,

z_nt , if t∈t0δ, t₀L. 5.19

Analogous arguments to those in the proof ofTheorem 5.3show thatβ limynis a weak upper solution and it is clear thatβt 0 0αt0 .

Finally we show thatiii^∗_α,β holds with β replaced by β. We consider a decreasing sequencean nsuch thata0 t0Land liman t0. Asαandβare positive ont0, t0L, we can findr_i > 0 such thatr_i ≤ α≤ β ≤ 1/r_iona_i1, a_i. We deduce then fromiii_α,β the existence ofψ_i∈L¹a_i1, a_i so that|ft, x | ≤ψ_it for a.e.t∈a_i1, a_iand allx∈αt , βt . The functiongdefined bygt ψit fort∈ai1, aiworks.

Theorem 5.3implies that1.1 has extremal weak solutions inα,β _wwhich, moreover, satisfy5.6 and 5.5 with βreplaced byβ. Furthermore if xis a weak solution of1.1 in α, β_wthen x ≤ βonI. Assume, on the contrary, that xs > βs for somes ∈ t0, t0 L , then there would existn ∈ Nsuch thatyns < xs and theny max{x, yn}would be a solution of5.16 betweenαandβwhich is strictly greater thany_n on some subinterval, a contradiction. Hence1.1 has extremal weak solutions inα, β_wwhich, moreover, satisfy 5.6 and5.5 .