First-Order Singular and Discontinuous Differential Equations

Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 507671, 25 pages doi:10.1155/2009/507671 Research Article First-Order Singular and Discontinuous Differential Equations Daniel C. Biles1and Rodrigo L ´ opez Pouso2 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, [email protected] 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. Copyright q2009 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 Let t0,x 0∈Rand L>0 be fixed and let f:t0,t 0L×R→Rbe a given mapping. We are concerned with the existence of solutions of the initial value problem xft, x,t∈I:t0,t 0L,x t0x0.1.1 It is well-known that Peano’s theorem ensures the existence of local continuously differentiable solutions of 1.1in case fis 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 see 1, Theorem 6.2.2for a proof based on the Schauder’s theorem. The reason for believing this is that Peano’s original approach to the problem in 2consisted 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 Goodman 4on 1.1in case fis a Carath´ eodory function. For our purposes in this paper, the importance of Peano’s original ideas is that they can 2 Boundary Value Problems be adapted to prove existence results for 1.1under such weak conditions that standard functional analysis arguments are no longer valid.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 as 5–9,see also the monographs 10,11. On the other hand, singular differential equations have been receiving a lot of attention in the last years, and we can quote 7,12–19. The main objective in this paper is to establish an existence result for 1.1with discontinuous and singular nonlinearities which generalizes in some aspects some of the previously mentioned works. This paper is organized as follows. In Section 2 we 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 for 1.1between given lower and upper solutions and assuming the existence of a L1-bound for fon the sector delimited by the graphs of the lower and upper solutions regular problems, and we give some examples. In Section 4 we 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 case fdoes not have such a strong bound as in Section 3 singular 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 6 and we give some examples in Section 7. Comparison with the literature is provided throughout the paper. 2. Preliminaries In the following definition ACIstands for the set of absolutely continuous functions on I. Definition 2.1. A lower solution of 1.1is a function l∈ACIsuch that lt0≤x0and lt≤ft, lt for almost all a.a.t∈I; an upper solution is defined analogously reversing the inequalities. One says that xis a Carath´ eodorysolution of 1.1if it is both a lower and an upper solution. On the other hand, one says that a solution x∗is the least one if x∗≤xon Ifor any other solution x, 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 fis L1-bounded for all x∈ R,f·,xis measurable, and for a.a. t∈Ift, ·is quasi-semicontinuous, namely, for all x∈R we have lim sup y→x− ft, y≤ft, x≤lim inf y→xft, y.2.1 A similar result was established in 20assuming moreover that fis superpositionally measurable, and the systems case was considered in 5,8. 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 that ft, ·is left upper and right lower semicontinuous. Boundary Value Problems 3 On the other hand, the above assumptions imply that the extremal solutions of 1.1 are given as the infimum of all upper solutions and the supremum of all lower solutions, that is, the least solution of 1.1is given by uinftinfut:uupper solution of 1.1,t∈I, 2.2 and the greatest solution is lsuptsup{lt:llower solution of 1.1},t∈I. 2.3 The mappings uinf and lsup turn out to be the extremal solutions even under more general conditions. It is proven in 9that solutions exist even if 2.1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 equation xft, xis the graph of an absolutely continuous function γ:a, b⊂t0,t 0L→ Rsuch that for a.a. t∈a, bone has either γtft, γt,or γt≥ft, γtwhenever γt≥lim inf y→γtft, y,2.4 γt≤ft, γtwhenever γt≤lim sup y→γt− 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 for t00x0and ft, x⎧ ⎨ ⎩ 1,if x<0, −1,if x≥0. 2.6 In this case 2.1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 level x0. Following Binding 21we can say that the equation “jams” at x0. 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>0andδ>0 such that one of the following conditions hold: 1γt≥ft, yεfor a.a. t∈a, band all y∈γt−δ, γtδ, 2γt≤ft, y−εfor a.a. t∈a, band all y∈γt−δ, γtδ. These conditions prevent the differential equation from exhibiting the behavior of the previous example over the line x0 in several ways. First, if γis a solution of xft, xthen any other solution can be continued over γonce they contact each other and independently of the definition of faround the graph of γ. On the other hand, if 1holds then solutions of xft, xcan cross γfrom above to below hence at most once,andif2holds then 4 Boundary Value Problems solutions can cross γfrom below to above, so in both cases the equation does not jam over the graph of γ. 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 over f. Theorem 2.4. Suppose that there exists a null-measure set N⊂Isuch that the following conditions hold: 1condition 2.1holds for all t, x∈I\N×Rexcept, at most, over a countable family of admissible non-quasi-semicontinuity curves; 2there exists an integrable function ggt,t∈I, such that ft, x≤gt∀t, x∈I\N×R.2.7 Then the mapping u∗ inftinfut:uupper solution of 1.1,u≤g1a.e.,t∈I2.8 is absolutely continuous on Iand satisfies u∗ inft0x0and u∗ inftft, u∗ inft for a.a. t∈I\J, where J∪n,m∈NJn,m and for all n, m ∈Nthe set Jn,m :t∈I:u∗ inft−1 n>supft, y:u∗ inft−1 m<y<u ∗ inft 2.9 contains no positive measure set. Analogously, the mapping l∗ suptsuplt:llower solution of 1.1,l≤g1a.e.,t∈I, 2.10 is absolutely continuous on Iand satisfies l∗ supt0x0and l∗ suptft, l∗ supt for a.a. t∈I\K, where K∪n,m∈NKn,m and for all n, m ∈Nthe set Kn,m :t∈I:l∗ supt1 n<infft, y:l∗ supt<y<l ∗ supt1 m 2.11 contains no positive measure set. Note that if the sets Jn,m and Kn,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 in 8and the reader can find its proof in 9. Boundary Value Problems 5 Lemma 2.5. Assume that for a null-measure set N⊂Ithe mapping fsatisfies the following condition. For each q∈Q,f·,qis measurable, and for t, x∈I\N×Rone has minlim sup y→x− ft, y,lim sup y→x ft, y≤ft, x≤maxlim inf y→x−ft, y,lim inf y→xft, y. 2.12 Then the mappings t∈I→ sup{ft, y:x1t<y<x 2t}and t∈I→ inf{ft, y: x1t<y<x 2t}are measurable for each pair x1,x 2∈CIsuch that x1t<x 2tfor all t∈I. Remark 2.6. A revision of the proof of 9, Lemma 2shows that it suffices to impose 2.12for all t, x∈I\N×Rsuch that x1t<x<x 2t. This fact will be taken into account in this paper. As a consequence of Theorem 2.4 and Lemma 2.5 we have a result about existence of extremal Carath´ eodory solutions for 1.1and L1-bounded nonlinearities. Note that the assumptions in Lemma 2.5 include 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 sets Jn,m and Kn,m in Theorem 2.4 are measurable. Therefore, only using the axiom of choice one can find a mapping fin the conditions of Theorem 2.4 which does not satisfy the assumptions in Lemma 2.5 and for which the corresponding problem 1.1lacks the greatest or the leastCarath´ eodory solution. Theorem 2.7 9, Theorem 4.Suppose that there exists a null-measure set N⊂Isuch that the following conditions hold: ifor every q∈Q,f·,qis measurable; iifor every t∈I\Nand all x∈Rone has either 2.1or lim inf y→x−ft, y≥ft, x≥lim sup y→x ft, y,2.13 and 2.1can fail, at most, over a countable family of admissible nonquasisemicontinuity curves; iiithere exists an integrable function ggt,t∈I, such that ft, x≤gt∀t, x∈I\N×R.2.14 Then the mapping uinf defined in 2.2is the least Carath´ eodory solution of 1.1and the mapping lsup defined in 2.3is the greatest one. 6 Boundary Value Problems Remark 2.8. Theorem 4 in 9actually asserts that u∗ inf, as defined in 2.8,istheleast Carath´ eodory solution, but it is easy to prove that in that case u∗ inf uinf, as defined in 2.2. Indeed, let Ube an arbitrary upper solution of 1.1,letgmax{|U|,g}and let v∗ inftinfut:uupper solution of 1.1,u≤g1a.e.,t∈I. 2.15 Theorem 4 in 9implies that also v∗ inf is the least Carath´ eodory solution of 1.1,thusu∗ inf  v∗ inf ≤Uon I. Hence u∗ inf uinf. Analogously we can prove that l∗ sup can be replaced by lsup in the statement of 21, 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 of fbetween a lower and an upper solution. In this section we will prove the following result. Theorem 3.1. Suppose that 1.1has a lower solution αand an upper solution βsuch that αt≤βt for all t∈Iand let E{t, x∈I×R:αt≤x≤βt}. Suppose that there exists a null-measure set N⊂Isuch that the following conditions hold: iα,βfor every q∈Q∩mint∈Iαt,maxt∈Iβt, the mapping f·,qwith domain {t∈I: αt≤q≤βt}is measurable; iiα,βfor every t, x∈E,t/ ∈N, one has either 2.1or 2.13, and 2.1can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained in E; iiiα,βthere exists an integrable function ggt,t∈I, such that ft, x≤gt∀t, x∈E, t / ∈N. 3.1 Then 1.1has extremal solutions in the set α, β:z∈ACI:αt≤zt≤βt∀t∈I.3.2 Moreover the least solution of 1.1in α, βis given by x∗tinfut:uupper solution of 1.1,u∈α, β,t∈I, 3.3 and the greatest solution of 1.1in α, βis given by x∗tsuplt:llower solution of 1.1,l∈α, β,t∈I. 3.4 Boundary Value Problems 7 Proof. Without loss of generality we suppose that αand βexist and satisfy |α|≤g,|β|≤g, α≤ft, α,andβ≥ft, βon I\N. We also may and we doassume that every admissible nqsc curve in condition iiα,β,sayγ:a, b→R, satisfies for all t∈a, b\Neither γt ft, γt or 2.4-2.5. For each t, x∈I×Rwe define Ft, x:⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ft, αt,if x<α t, ft, x,if αt≤x≤βt, ft, βt,if x> β t. 3.5 Claim 1. The modified problem xFt, x,t∈I, xt0x0,3.6 satisfies conditions 1and 2in Theorem 2.4 with freplaced by F. First we note that 2is an immediate consequence of iiiα,βand the definition of F. To show that condition 1in Theorem 2.4 is satisfied with freplaced by F,lett, x∈ I\N×Rbe fixed. The verification of 2.1for Fat t, xis trivial in the following cases: αt<x<βtand fsatisfies 2.1at t, x,x<αt,x>βtand αtxβt.Letus consider the remaining situations: we start with the case xαt<βtand fsatisfies 2.1 at t, x, for which we have Ft, xft, xand lim sup y→x− Ft, yft, αt ft, x≤lim inf y→xft, ylim inf y→xFt, y,3.7 and an analogous argument is valid when αt<βtxand fsatisfies 2.1. The previous argument shows that Fsatisfies 2.1at every t, x∈I\N×Rexcept, at most, over the graphs of the countable family of admissible nonquasisemicontinuity curves in condition iiα,βfor xft, x. Therefore it remains to show that if γ:a, b⊂I→Ris one of those admissible nqsc curves for xft, xthen it is also an admissible nqsc curve for xFt, x. As long as the graph of γremains in the interior of Ewe have nothing to prove because fand Fare the same, so let us assume that γαon a positive measure set P⊂a, b, P∩N∅. Since αand γare absolutely continuous there is a null measure set  Nsuch that αtγtfor all t∈P\ N,thusfort∈P\ Nwe have γt≤ft, γtlim sup y→γt− Ft, y,γ t≤Ft, γt,3.8 so condition 2.5with freplaced by Fis satisfied on P\ N. On the other hand, we have to check whether γt≥Ft, γt for those t∈P\ Nat which we have γt≥lim inf y→γtFt, y.3.9 8 Boundary Value Problems We distinguish two cases: αt<βtand αtβt. In the first case 3.9is equivalent to γt≥lim inf y→γtft, y,3.10 and therefore either γtft, γt or condition 2.4holds, yielding γt≥ft, γt  Ft, γt.Ifαtβtthen we have γtαtβt≥ft, βt  Ft, γt. Analogous arguments show that either γFt, γor 2.4-2.5hold for Fat almost every point where γcoincides with β, so we conclude that γis an admissible nqsc curve for xFt, x. By virtue of Claim 1 and Theorem 2.4 we can ensure that the functions x∗and x∗ defined as x∗tinfut:uupper solution of 3.6,u≤g1a.e.,t∈I, x∗tsuplt:llower solution of 3.6,l≤g1a.e.,t∈I, 3.11 are absolutely continuous on Iand satisfy x∗t0x∗t0x0and x ∗tFt, x∗t for a.a. t∈I\J, where J∪n,m∈NJn,m and for all n, 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Ft, x∗t for a.a. t∈I\K, where K ∪n,m∈NKn,m and for all n, 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 all t∈Iwe have x∗tinfut:uupper solution of 1.1,u∈α, β,u≤g1a.e.,3.14 x∗tsuplt:llower solution of 1.1,l∈α, β,l≤g1a.e..3.15 Let ube an upper solution of 3.6and let us show that ut≥αtfor all t∈I. Reasoning by contradiction, assume that there exist t1,t 2∈Isuch that t1<t 2,ut1αt1and ut<α t∀t∈t1,t 2.3.16 For a.a. t∈t1,t 2we have ut≥Ft, ut ft, αt ≥αt,3.17 Boundary Value Problems 9 which together with ut1αt1imply u≥αon t1,t 2, a contradiction with 3.16. Therefore every upper solution of 3.6is greater than or equal to α, and, on the other hand, βis an upper solution of 3.6with |β|≤ga.e., thus x∗satisfies 3.14. One can prove by means of analogous arguments that x∗satisfies 3.15. Claim 3. x∗is the least solution of 1.1in α, βand x∗is the greatest one. From 3.14and 3.15it suffices to show that x∗and x∗are actually solutions of 3.6. Therefore we only have to prove that Jand Kare null measure sets. Let us show that the set Jis 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 split JA∪B, where A{t∈J:x∗t>αt}and BJ\A{t∈J:x∗t αt}. Let us show that Bis a null measure set. Since αand x∗are absolutely continuous the set Ct∈I:αtdoes not exist ∪t∈I:x ∗tdoes not exist ∪t∈I:αtx∗t,α t/ x ∗t 3.19 is null. If B/ ⊂Cthen there is some t0∈Bsuch that αt0x∗t0and αt0x ∗t0, but then the definitions of Band Fyield αt0>lim sup y→αt0− Ft0,yft0,α t0.3.20 Therefore B\C⊂Nand thus Bis a null measure set. The set Acan be expressed as A∪∞ k1Ak, where for each k∈N Ak⎧ ⎨ ⎩t∈I:x∗t>α t1 k,x  ∗t>lim sup y→x∗t− Ft, y⎫ ⎬ ⎭ ∞  n,m1 Ak∩Jn,m. 3.21 For k,m ∈N,k<m, we have x∗t−1/m > x∗t−1/k, so the definition of Fimplies that Ak∩Jn,m t∈I:x∗t>α t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 of Lemma 2.5 and Remark 2.6. 16 Boundary Value Problems The above properties of {yn}nimply that the following function is well defined: y∞t⎧ ⎨ ⎩ x0,if tt0, lim ynt,if t∈t0,t 0L. 5.9 Step 3 y∞∈CI∩ACloct0,t 0L.Letε∈0,Lbe fixed. Condition iii∗ α,βimplies that for all n∈Nsuch that tn<t 0εwe have y ntft, ynt≤gtfor a.a.t∈t0ε,t0L,5.10 with g∈L1t0ε,t0L. Hence for s, t ∈t0ε, t0L,s≤t, we have y∞t−y∞slim n→∞t s y n≤t s g, 5.11 and therefore y∞∈ACt0ε,t0L. Since ε∈0,Lwas fixed arbitrarily in the previous arguments, we conclude that y∞∈ACloct0,t 0L. The continuity of y∞at t0follows from the continuity of αand βat t0, the assumption αt0x0βt0, and the relation αt≤y∞t≤βt∀t∈t0,t 0L.5.12 Step 4 y∞is a weak lower solution of 1.1. For ε∈0,Land n∈Nsuch that tn<t 0ε we have 5.10with g∈L1t0ε, t0L, hence lim sup y n∈L1t0ε,t0L,andfors, t ∈ t0ε,t0L,s<t, Fatou’s lemma yields y∞t−y∞slim n→∞t s y n≤t s lim sup yn.5.13 Hence for a.a. t∈t0ε, t0Lwe have y∞t≤lim sup yntlim sup ft, ynt.5.14 Let J1∪n∈NAnwhere An{t∈t0ε,t0L:y∞tynt}and J2t0ε, t0L\J1. For n∈Nand a.a. t∈Anwe have y∞ty ntft, ynt  ft, y∞t,thus y∞tft, y∞t for a.a. t∈J1. Boundary Value Problems 17 On the other hand, for a.a. t∈J2the relation 5.14and the increasingness of {ynt} yield y∞t≤lim sup ft, ynt≤lim sup y→y∞t− ft, y.5.15 Let t0∈J2\Nbe such that 5.15holds. We have two possibilities: either 2.1holds for fat t0,y ∞t0 and then from 5.15we deduce y∞t0≤ft0,y ∞t0,ory∞t0γt0, where γis an admissible curve of non quasisemicontinuity. In the last case we have that either t0belongs to a null-measure set or y∞t0γt0, which, in turn, yields two possibilities: either γt0ft, γt0 and then y∞t0ft, y∞t0,orγt0/ ft, γt0 and then 5.15, with y∞t0γt0and y∞t0γt0, and the definition of admissible curve of non quasisemicontinuity imply that y∞t0≤ft0,y ∞t0. The above arguments prove that y∞t≤ft, y∞t a.e. on t0ε, t0L, and since ε∈0,Lwas fixed arbitrarily, the proof of Step 4 is complete. Conclusion The construction of y∞and Step 1 imply that y∞≥x∗and the definition of x∗and Step 4 imply that x∗≥y∞. Therefore for all n∈Nwe have x∗ynon Inand then x∗is a weak solution of 1.1. Since every weak solution is a weak lower solution, x∗is the greatest weak solution of 1.1in α, βw. The assumption αt0βt0in Theorem 5.3 can be replaced by other types of conditions. The next theorem generalizes the main results in 7,12–14concerning existence of solutions of singular problems of the type of 1.1. Theorem 5.4. Suppose that 1.1with x00has a weak lower solution αand a weak upper solution βsuch that αt≤βtfor all t∈Iand α>0on t0,t 0L. Suppose that there is a null-measure set N⊂Isuch that conditions iα,βand iiα,βin Theorem 3.1 hold for E{t, x∈I×R:αt≤x≤βt},and assume moreover that the following condition holds: iii α,βfor every r∈0,1there exists gr∈L1Isuch that for all t, x∈E,t/ ∈N, and r≤x≤ 1/r one has |ft, x|≤grt. Then the conclusions of Theorem 5.3 hold true. Proof. We start observing that there exists a weak upper solution  βsuch that α≤ β≤βon Iand αt00 βt0.Ifβt00 then it suffices to take  βas β.Ifβt0>0 we proceed as follows in order to construct  β:let{xn}nbe a decreasing sequence in 0,βt0 such that lim xn0 and for every n∈Nlet ynbe the greatest solution between αand βof yft, y,t∈I, yt0xn.5.16 18 Boundary Value Problems claim [ynexists] Let ε∈0,Lbe so small that αt<x n−ε<x nε<βtfor all t∈t0,t 0ε. Condition iii α,βimplies that there exists gε∈L1t0,t 0εsuch that for a.a. t∈t0,t 0εand all x∈xn−ε,xnεwe have |ft, x|≤gεt.Letv±txn± t 0gεsds,t∈t0,t 0εand let δ∈0,εbe such that xn−ε≤v−≤v≤xnεon t0,t 0δ. We can apply Theorem 2.7 to the problem ξft, ξ,t∈t0,t 0δ,ξ t0xn,5.17 and with respect to the lower solution v−and the upper solution v, so there exists ξnthe greatest solution between v−and vof 5.17.Noticethatifxis a solution of 5.17then v−≤x≤v,soξnis also the greatest solution between αand βof 5.17. Now condition iii α,βensures that Theorem 2.7 can be applied to the problem zft, z,t∈t0δ, t0L,z t0δξnt0δ,5.18 with respect to the lower solution αand the upper solution βboth functions restricted to t0δ, t0L. Hence there exists znthe greatest solution of 5.18between αand β. Obviously we have ynt⎧ ⎨ ⎩ ξnt,if t∈t0,t 0δ, znt,if t∈t0δ, t0L. 5.19 Analogous arguments to those in the proof of Theorem 5.3 show that  βlim ynis a weak upper solution and it is clear that  βt00αt0. Finally we show that iii∗ α,βholds with βreplaced by  β. We consider a decreasing sequence annsuch that a0t0Land lim ant0.Asαand βare positive on t0,t 0L, we can find ri>0 such that ri≤α≤β≤1/rion ai1,a i. We deduce then from iii α,βthe existence of ψi∈L1ai1,a iso that |ft, x|≤ψitfor a.e. t∈ai1,a iand all x∈αt,βt. The function gdefined by gtψitfor t∈ai1,a iworks. Theorem 5.3 implies that 1.1has extremal weak solutions in α,  βwwhich, moreover, satisfy 5.6and 5.5with βreplaced by  β. Furthermore if xis a weak solution of 1.1in α, βwthen x≤ βon I. Assume, on the contrary, that xs> βsfor some s∈t0,t 0L, then there would exist n∈Nsuch that yns<xsand then ymax{x, yn}would be a solution of 5.16between αand βwhich is strictly greater than ynon some subinterval, a contradiction. Hence 1.1has extremal weak solutions in α, βwwhich, moreover, satisfy 5.6and 5.5. Boundary Value Problems 19 6. Systems Let us consider the following system of n∈Nordinary differential equations: −→ xt−→ ft, −→ xfor a.a.t∈It0,t 0L,−→ xt0−→ x0,6.1 where t0∈R,L>0, −→ xx1,...,x n,−→ x0x0,1,...,x 0,n,and−→ ff1,f 2,...,f n:I×Rn→ Rn. Our goal is to extend Theorem 5.3 to this multidimensional setting, which, as usual, requires the right hand side −→ fto be quasimonotone, as we will define later. We start extending to the vector case the definitions given before for scalar problems. To do so, let ACloct0,t 0L,Rndenote the set of functions −→ yy1,y 2,...,y n:t0,t 0 L→Rnsuch that for each i∈{1,2,...,n}the component yiis absolutely continuous on t0ε,t0Lfor each ε∈0,L.Also,CI,Rnstands for the class of Rn-valued functions which are defined and continuous on I. A weak lower solution of 6.1is a function −→ αα1,α 2,...,α n∈CI,Rn∩ ACloct0,t 0L,Rnsuch that for each i∈{1,2,...,n}we have αi0≤x0,i and for a.a. t∈Iwe have αit≤fit, −→ αt.Weak upper solutions are defined similarly by reversing the relevant inequalities, and weak solutions of 6.1are functions which are both weak lower and weak upper solutions. In the set CI,Rn∩ACloct0,t 0L,Rnwe define a partial ordering as follows: let −→ α,−→ β∈CI,Rn∩ACloct0,t 0L,Rn, we write −→ α≤−→ βif every component of −→ αis less than or equal to the corresponding component of −→ βon the whole of I.If−→ α,−→ β∈CI,Rn∩ ACloct0,t 0L,Rnare such that −→ α≤−→ βthen we define !−→ α,−→ β":−→ η∈CI,Rn∩ACloct0,t 0L,Rn:−→ α≤−→ η≤−→ β.6.2 Extremal least and greatestweak solutions of 6.1in a certain subset of CI,Rn∩ ACloct0,t 0L,Rnare defined in the obvious way considering the previous ordering. Now we are ready to extend Theorem 5.3 to the vector case. We will denote by −→ eithe ith canonical vector. The proof follows the line of that of 8, Theorem 5.1. Theorem 6.1. Suppose that 6.1has weak lower and upper solutions −→ αand −→ βsuch that −→ α≤−→ β, −→ αt0−→ x0−→ βt0, and let E∗{t, −→ x:t∈I, −→ αt≤−→ x≤−→ βt}. Suppose that −→ fis quasimonotone nondecreasing in E∗, that is, for i∈{1,2,...,n}and t, −→ x,t, −→ y∈E∗the relations −→ x≤−→ yand xiyiimply fit, −→ x≤fit, −→ y. Suppose, moreover, that for each −→ ηη1,...,η n∈−→ α,−→ βthe following conditions hold: H1the function −→ f·,−→ η· is measurable; H2for all i∈{1,...,n}and a.a. t∈Ione has either lim sup s→x− fit, −→ ηts−ηit−→ ei≤fit, −→ ηtx−ηit−→ ei ≤lim inf s→xfit, −→ ηts−ηit−→ ei, 6.3 20 Boundary Value Problems or lim inf s→x−fit, −→ ηts−ηit−→ ei≥fit, −→ ηtx−ηit−→ ei ≥lim sup s→x fit, −→ ηts−ηit−→ ei,6.4 and 6.3fails, at most, over a countable family of admissible nqsc curves of the scalar differential equation xfit, −→ ηtx−ηit−→ eicontained in the sector Ei:{t, x: t∈I, αit≤x≤βit}; H3there exists g∈L1 loct0,t 0L such that for i∈{1,2,...,n}and a.a. t∈Ione has |fit, −→ ηt|≤gt. Then 6.1has extremal weak solutions in α, β. Moreover the least weak solution −→ x∗x∗,1,...,x ∗,n is given by x∗,itinfuit:u1,...,u nweak upper solution of 6.1in !−→ α,−→ β" 6.5 and the greatest weak solution −→ x∗x∗ 1,...,x ∗ nis given by x∗ itsuplit:l1,...,l nweak lower solution of 6.1in !−→ α,−→ β".6.6 Proof. Let −→ LL1,...,L n∈−→ α,−→ βbe a weak lower solution of 6.1,andletg∈L1 loct0,t 0 L be as in H3and such that |Li|≤ga.e. on Ifor all i∈{1,...,n}.Nowlet−→ ξ∗ξ∗ 1,...,ξ∗ n be defined for i∈{1,...,n}as ξ∗ itsuplit:l1,...,l nweak lower solution in !−→ α,−→ β",l i≤ga.e.on I.6.7 In particular, −→ ξ∗≥−→ L. Further, every possible solution of 6.1in −→ α,−→ βis less than or equal to −→ ξ∗by 6.7and H3, independently of −→ L. Claim 1 −→ ξ∗∈CI,Rn∩ACloct0,t 0L,Rn.Ifl1,...,l nis a weak lower solution in −→ α,−→ β with |l i|≤ga.e. on Ithen for s, t ∈t0,t 0L,s<t, we have |lit−lis|≤t s grdr, 6.8 which implies ξ∗ it−ξ∗ is≤t s grdr, 6.9 and, therefore, ξ∗ i∈ACloct0,t 0L. Further ξ∗ iis continuous at t0because αit≤ξ∗ it≤βit for all t∈I,αiand βiare continuous at t0and αit0βit0. Boundary Value Problems 21 Claim 2. −→ ξ∗is the greatest weak solution of 6.1in −→ α,−→ β.For each weak lower solution −→ l∈−→ α,−→ βsuch that |l i|≤ga.e., the quasimonotonicity of −→ fyields l it≤fi#t, −→ ξ∗tlit−ξ∗ it−→ ei$for a.a.t∈I. 6.10 Hence liis a weak lower solution between αiand βiof the scalar problem xfi#t, −→ ξ∗tx−ξ∗ it−→ ei$for a.a.t∈I, xt0x0,i,6.11 and then Theorem 5.3 implies that li≤y∗ i, where y∗ iis the greatest weak solution of 6.11in αi,β i. Then −→ ξ∗≤−→ y∗y∗ 1,...,y∗ n. On the other hand, we have y∗ itfi#t, −→ ξ∗ty∗ it−ξ∗ it−→ ei$≤fi#t, −→ y∗$for a.a.t∈I, 6.12 hence −→ y∗is a weak lower solution of 6.1in −→ α,−→ βwith |y i|≤ga.e. on I,thus−→ ξ∗≥−→ y∗. Therefore −→ ξ∗is a weak solution of 6.1,and,by6.7and H3, it is the greatest one in −→ α,−→ β. In particular, the greatest weak solution of 6.1in −→ α,−→ βexists and it is greater than or equal to −→ L. Claim 3. The greatest weak solution of 6.1in −→ α,−→ β,−→ x∗,satisfies6.6. The weak lower solution −→ L∈−→ α,−→ βwas fixed arbitrarily, so −→ x∗is greater than or equal to any weak lower solution in −→ α,−→ β. On the other hand, −→ x∗is a weak lower solution. Analogously, the least weak solution of 6.1in −→ α,−→ βis given by 6.5. 7. Examples Example 7.1. Let us show that the following singular and non-quasisemicontinous problem has a unique positive Carath´ eodory solution: xfx1 x,t∈I0,1 2,x 00.7.1 Here square brackets mean integer part, and by positive solution we mean a solution which is positive on 0,1/2. First note that 7.1has at most one positive weak solution because the right hand side in the differential equation is nonincreasing with respect to the unknown on 0,∞,thusat no point can solutions bifurcate. For all x>0 we have 1/x≤1/x and therefore βt√2t,t∈I, is an upper solution of 7.1as it solves the majorant problem x1 x,x 00.7.2 22 Boundary Value Problems On the other hand it is easy to check that for 0 <x≤1 we have 1/x≥1/2xand then αt√t,t∈I, is a lower solution. The function fis continuous between the graphs of αand βexcept over the lines γnt1/n,t∈n−2/2,n −2, which are admissible nqsc curves for all n∈N,n≥2note that γ1is not an admissible nqsc curve but it does not lie between αand β. Finally, for r∈0,1we have fx≤n−1 for max{r, αt}≤x≤βt,7.3 where n∈Nis such that 1/n < r. Therefore Theorem 5.4 implies the existence of a weak solution of 7.1between αand β. Moreover, this weak solution between αand βis increasing, so Proposition 5.2 ensures that it is, in fact, a Carath´ eodory solution on I. It is possible to extend the solution on the right of t1/2tosomet∗where the solution will assume the value 1. The solution cannot be extended further on the right of t∗,as7.1 with x00 replaced by x01 has no solution on the right of 0. We owe to the anonymous referee the following remarks. Problem 7.1is autonomous, so it falls inside the scope of the results in 21, which ensure that if we find α>0 such that α 0 ds 1/s<∞,7.4 then 7.1has a positive absolutely continuous solution defined implicitly by xt 0 ds 1/st∀t∈0,α 0 ds 1/s.7.5 Since 1 0 ds 1/sπ2−6 6≈0.644934,7.6 we deduce that the solution xtis defined at least on 0,t ∗, where t∗π2−6/6and xt∗1. Example 7.2. Let ε:0,1→Rbe measurable and 0 <εt≤1 for a.a. t∈0,1. We will prove that for each k∈R,k≥1, the problem xft, x1 xk−1 tkεt,t∈I0,1,x 007.7 has a unique positive Carath´ eodory solution. Note that the equation is not separable and fassumes positive and negative values on every neighborhood of the initial condition. Moreover the equation is singular at the initial condition with respect to both of its variables. Boundary Value Problems 23 Once again the right hand side in the differential equation is nonincreasing with respect to the unknown xon 0,∞, thus we have at most one positive weak solution. Lower and upper solutions are given by, respectively, αtt/3andβttfor t∈I. For each t∈0,1the function ft, ·is continuous between the graphs of αand β except over the lines γntn−1/k,t∈n−1/k,T, where Tmin{3n−1/k,1},n∈N.Letus show that fis positive between αand β,thusγnwill be an admissible nqsc curve for each n∈N. For t∈0,1and n1−1/k <x≤n−1/k,n∈N, we have ft, xn−1 tkεt,7.8 and if, moreover, we restrict our attention to those t>0 such that αt≤x≤βtthen we have n1−1/k <t≤3n−1/k which implies n 3≤1 tk≤n, 7.9 and thus for t∈0,1,n1−1/k <x≤n−1/k,andαt≤x≤βt, we have εt≤ft, x≤n−n 31≤2 3n2.7.10 This shows that fis positive between αand βand, moreover, we can say that for r∈0,1 it suffices to take n∈Nsuch that n1−1/k <rto have |ft, x|≤2/3n2 for all t, x between the graphs of αand βand r≤x≤1/r. Therefore Theorem 5.4 implies the existence of a weak solution of 7.7between αand β. Moreover, since fis positive between αand βthe solution is increasing and, therefore, it is a Carath´ eodory solution. The previous two examples fit the conditions of Theorems 5.3 and 5.4. Next we show an example where Theorem 5.3 can be used but it is not clear whether or not we can also apply Theorem 5.4. Example 7.3. Let a>0 be fixed and consider the problem xft, x1 xat,t∈I0,1,x 00.7.11 Lower and upper solutions are given by α≡0andβt√2t,t∈I. Since fis nonnegative between αand βthe lines γnt−at1/n,n∈N, are admissible nqsc curves for the differential equation. Finally it is easy to check that 0 ≤ft, x≤n1ifa−1n1−1≤t≤1 and x≥0, thus one can construct g∈L1 loc0,1 such that |ft, x|≤gtfor a.a. t∈Iand 0≤x≤√2t. Theorem 5.3 ensures that 7.11has extremal weak solutions between αand β. Moreover 7.11has a unique solution between αand βas fis nonincreasing with respect to the unknown. Further, the unique solution is monotone and therefore it is a Carath´ eodory solution. 24 Boundary Value Problems Acknowledgments The research of Rodrigo L´ opez Pouso is partially supported by Ministerio de Educaci´ on y Ciencia, Spain, Project MTM2007-61724, and by Xunta de Galicia, Spain, Project PGIDIT06PXIB207023PR. References 1D. R. 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