Extremality and Comparison Results for Discontinuous Third Order Functional Initial-Boundary Value Problems

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1 Joural of Mathematical Aalysis a Applicatios 255, oi:.6jmaa , available olie at o Extremality a Compariso Results for Discotiuous Thir Orer Fuctioal Iitial-Bouary Value Problems A. Cabaa Facultae e Matematicas, Depto. e Aalise Matematica, Uiersiae e Satiago e Compostela, 576 Satiago e Compostela, Spai a S. eikkilä Departmet of Mathematical Scieces, Uiersity of Oulu, P.O. Box 3, FIN-94, Uiersity of Oulu, Fila Submitte by William F. Ames Receive August 6, 2 Extremality a compariso results are erive for thir orer fuctioal iitial-bouary value problems. Differetial equatios of problems may ivolve iscotiuous oliearities. 2 Acaemic Press Key Wors: thir orer ifferetial equatios; extremal solutios; epeece o ata; iscotiuity; fuctioal epeece.. INTRODUCTION Recetly, the existece of extremal solutios of the thir orer iitial- bouary value problem IBVP u Ž t. qž u Ž t.. fž t, už t.. for a.e. t I a, b, už a. A, už b. B, u Ž a. C, Ž.. betwee the assume lower a upper solutios a is stuie i X $35. Copyright 2 by Acaemic Press All rights of reprouctio i ay form reserve.

2 96 CABADA AND EIKKILÄ I Sectio 2 of this paper we shall itrouce coitios which esure that the fuctioal iitial-bouary value problem Ž u. Ž t. g t, u, Ž u. Ž t. for a.e. t J t, t, Ž.2. aut Ž. bu Ž t. c, au Ž t. bu Ž t. c, Ž u. Ž t. c 2 has extremal solutios which are icreasig with respect to c a c a ecreasig with respect to g a c 2. The proofs are base o the use of existece, uiqueess, a compariso results erive i, 3 for ifferetial equatios a fixe poit results erive i 3, 5 for mappigs i orere spaces. The fuctio g is allowe to be iscotiuous i all of its variables. Special cases, icluig those where extremal solutios are obtaie by a metho of successive approximatios, are presete i Sectios 3 a 4. The case whe epes fuctioally, a eve iscotiuously, o u is cosiere i Sectio 5. Theoretical a worke examples are give to illustrate the obtaie results. 2. EXISTENCE AND COMPARISON RESULTS I this sectio we erive existece a compariso results for the IBVP ypotheses a Prelimiaries The fuctios a a the costats a j, bj are assume to have the followig properties. : is a icreasig homeomorphism, a : J Ž,. is cotiuous. Ž A. a j, bj, j,, a aa ab ab. Deote Y u C Ž J. Ž u. ACŽ J. 4, Z u CŽ J. u ACŽ J. 4 a equip Y a Z with poitwise orerig. Assume also that CŽ J. is equippe with poitwise orerig a maximum orm. The followig hypotheses are impose o the fuctio g: J CŽ J..

3 TIRD ORDER PROBLEMS 97 Ž g. For all CŽ J. a x the fuctio gž,, x. is measurable, a lim sup gt, Ž, y. gt, Ž, x. lim if gt, Ž, y. y x y x for a.e. t J. Ž g. gt, Ž, x. is ecreasig i for a.e. t J a all x. Ž g2. gt, Ž, x. pt Ž. ŽŽ x.. for all CŽ J. a x a for x a.e. t J, where p L Ž J., : Ž,. is icreasig, a. Ž x. Coitio Ž g. is use i 3, 4, 6. There, existece of solutios for ifferet types of first a seco orer problems has bee obtaie. As far as we kow, this is the first time that existece of solutios is erive for thir orer problems uer such a weak assumptio which, e.g., allows us to avoi a quite abstract assumptio o sup-measurability of gž,,.. We are goig to show that uer the above hypotheses the IBVP Ž.2. has extremal solutios u a u i Y i the sese that if u is ay solutio of Ž.2. i Y, the u u u. Before the proof we itrouce some auxiliary results. Deote t a b yž t. s, Ž s. Ž t. t t a b yž t. s, t J, Ž s. Ž t. t t aa ab ab D s, Ž 2.. Ž s. Ž t. Ž t. kž t, s. t yž t. yž s., t s t, D y Ž t. y Ž s., t s t. D First we covert the IBVP.2 to a pair of a itegral equatio a IVP. if LEMMA 2.. A fuctio u Y is a solutio of the IBVP Ž.2. if a oly cyž t. cyž t. t už t. kž t, s. Ž s. s, t J, Ž 2.2. D t

4 98 CABADA AND EIKKILÄ where Z is a solutio of the IVP t g t, u, t for a.e. t J, t c. 2.3 Ž. 2 Proof. It is easy to see that u Y is a solutio of Ž.2. if a oly if Ž u. Z, a u, are solutios of problems Ž 2.3. a 2.4 Ž u., aut buž t. c, auž t. buž t. c. It follows from 3, Lemma 4.. that for each Z the BVP Ž 2.4. has a uique solutio u Y, give by Ž This proves the assertio. Cosier ext the IVP Ž 2.3. whe u CŽ J. is give. LEMMA 2.2. If the hypotheses Ž., Ž g., Ž g., a Ž g2. hol, the the IVP Ž 2.3. has for each u CŽ J. least a greatest solutios i Z, a they are icreasig with respect to g a c 2, a ecreasig with respect to u. Moreoer, if Z is ay solutio of Ž 2.3., the Ž wž t.. Ž t. Ž wž t.., t J, Ž 2.5. where w is the uique solutio of the IVP w Ž t. pž t. Ž wž t.. a.e. i J, wž t. Ž c 2.. Ž 2.6. Proof. Let u CŽ J. be give. The hypotheses Ž., Ž g., a Ž g2. imply that the hypotheses of 3, Theorem 2..3hol whe Ž t, x. gt, Ž u, x. stas for g. This esures that the IVP Ž 2.3. has least a greatest solutios i Z. Because of the hypothesis Ž g. it follows from 3, Propositio 2.. that these solutios are ecreasig with respect to u, a icreasig with respect to g a c 2. To prove the last assertio, assume that Z is a solutio of the IVP Ž It follows from Ž g2. that t g t, u, t p t t a.e. i J. Ž Ž..

5 TIRD ORDER PROBLEMS 99 Thus t Ž Ž t.. Ž Ž t.. Ž Ž s.. s s t t Ž Ž.. c p s s s 2 for all t J. This implies by 3, Lemma B.7. that ŽŽ t.. wt Ž. o J, where w is the solutio of the IVP Ž I view of this result a the mootoicity of we see that Ž 2.5. hols. Applyig the results of Lemmas 2. a 2.2 we costruct i the followig lemma a orer iterval of CŽ J. which cotais all the possible solutios of Ž.2.. LEMMA 2.3. Assume that the hypotheses, A, g, a g2 hol, a eote cyž t. cyž t. t až t. kž t, s. Ž wž s.. s, t J, D t Ž 2.7. cyž t. cyž t. t bž t. kž t, s. wž s. s, t J, D t t where w is the solutio of the IVP Ž If u is a solutio of the IBVP Ž.2. i Y, the u belogs to the orer iteral a, b u CŽ J. a u b 4. Proof. Let u be a solutio of Ž.2. i Y. The u satisfies Ž 2.2., where is a solutio of Ž 2.3. i Z. Because u CŽ J., it follows from Lemma 2.2 that Ž 2.5. hols. Noticig that the right-ha sie of Ž 2.2. is ecreasig i, we see that u a, b. The followig result is also eee i the proof of our mai theorem. LEMMA 2.4. Gie a orer iteral a, b of CŽ J. a a mappig G: a, b a, b, assume that G is icreasig, a that the sequece Ž G. has a poitwise limit i CŽ J. wheeer Ž. is a mootoe sequece i a, b. The G has the least fixe poit u a the greatest fixe poit u. Moreoer, u miu Gu u4 a u maxu u Gu 4. Ž 2.8. Proof. The give hypotheses a Dii s theorem esure that if Ž. is a mootoe sequece i a, b, the the sequece Ž G. coverges uiformly o J, or equivaletly, i the sup-metric of CŽ J.. Thus the coclusios follow from 5, Theorem.2.2 whe X Y CŽ J..

6 2 CABADA AND EIKKILÄ 2.2. Extremality a Compariso Results for the IBVP.2 Now we are reay to prove our mai existece theorem for the IBVP.2. TEOREM 2.. Assume that the hypotheses Ž., Ž A., Ž g., Ž g., a Ž g2. hol. The the IBVP Ž.2. has extremal solutios u a u i Y; i.e., if u Y is a solutio of Ž.2., the už t. ut u Ž. t o J. Proof. Let a, b give by 2.7. Deote for each u a, b, cyž t. cyž t. t GuŽ t. kž t, s. Ž s. s, t J, Ž 2.9. D t where is the greatest solutio of the IVP Ž 2.3. u i Z. Sice iequality Ž 2.5. hols for, u a, b u, a sice u is ecreasig i u by Lemma 2.2, it follows from Ž 2.9. that Ga, b a, b, a that G is icreasig. Let Ž u. be a mootoe sequece i a, b, a let u be the greatest solutio of the IVP Ž 2.3. with u u. I view of Lemma 2.2 a the sequece Ž. is mootoe a Ž 2.5. hols whe. Sice Ž Gu. u u is a mootoe sequece i a, b, the the limits Ž t. lim Ž t., hž t. lim Gu Ž t., t J, u exist. Applyig the mootoe covergece theorem it the follows from cyž t. cyž t. t Gu Ž t. kž t, s. Ž s. s, t J, as that D t cyž t. cyž t. t hž t. lim GuŽ t. kž t, s. Ž s. s, D t t J. u u I particular, h CŽ J.. The above proof implies that the hypotheses of Lemma 2.4 hol. Thus the operator G has the least fixe poit u. The efiitio Ž 2.9. of G a Lemma 2. imply that u is a solutio of the IBVP Ž.2. i a, b. Assume ow that u Y is a solutio of the IBVP Ž.2.. By Lemma 2., u is of the form Ž 2.2., where is a solutio of the IVP Ž Moreover, u a, b by Lemma 2.3. Deotig by the greatest solutio of Ž 2.3. u, the. It the follows from Ž 2.2. a Ž 2.9. that Gu u. Thus the u

7 TIRD ORDER PROBLEMS 2 first relatio of Ž 2.8. implies that u u. This shows that u is the least solutio of the IBVP Ž.2.. The existece of the greatest solutio of Ž.2. ca be prove similarly. Next we shall erive results for the epeece of extremal solutios of Ž.2. o g, c, c, a c 2. PROPOSITION 2.. If the hypotheses of Theorem 2. are satisfie, the the extremal solutios of the IBVP Ž.2. are icreasig with respect to c a c, a ecreasig with respect to g a c 2. Proof. Assume that fuctios g, ˆg: J CŽ J. have properties Ž g. a Ž g., that gž t,, x. ˆgŽ t,, x. Ž 2.. for a.e. t J a for all CŽ J. a x, a that c ˆc, c ˆc, c2 ˆc 2. Ž 2.. Replacig c, c, a c i Ž 2.7. by mic, ˆc 4, mic, ˆc 4, a maxc, ˆc i the efiitio of a, a by maxc, c 4, maxc, c 4, a mic, c 4 ˆ ˆ 2 ˆ2 i the efiitio of b, the the relatios Ž 2.9., where u is the greatest solutio of Ž 2.3., a ˆc yž t. ˆc yž t. t ĜuŽ t. kž t, s. ˆ Ž s. s, t J, Ž 2.2. D t u where ˆu is the greatest solutio of the IVP Ž Ž t.. ˆgŽ t, u, Ž t.. for a.e. t J, Ž t. ˆc 2, Ž 2.3. efie icreasig operators G, G: ˆ a, b a, b which satisfy the hypotheses of Lemma 2.4. Moreover, it follows from Ž 2.. Ž 2.2. by Lemma 2.2 that G G ˆ for each a, b. Let u be the least solutio of Ž.2., a let uˆ be the least solutio of the IBVP Ž uˆ. Ž t. ˆg t, u, ˆ Ž uˆ. Ž t. a.e. i J, aut ˆ buˆ Ž t. ˆc, auˆž t. buˆ Ž t. ˆc, Ž ˆ. Ž. ˆ2 u t c. The proof of Theorem 2. implies that u Gu a uˆ Gu. ˆˆ Sice Guˆ Gu, ˆˆ the Guˆ u. ˆ Because u miu Gu u4 by Lemma 2.4, it follows that u u. ˆ This proves the assertio for least solutios. The proof for greatest solutios is similar.

8 22 CABADA AND EIKKILÄ To obtai a existece, uiqueess, a compariso result for the IBVP Ž u. Ž t. g t,, Ž u. Ž t. for a.e. t J t, t, Ž 2.4. aut Ž. bu Ž t. c, au Ž t. bu Ž t. c, Ž u. Ž t. c, 2 we a the followig hypothesis to those of Theorem 2.. Ž gv. For each fixe CŽ J., gt, Ž, y. gt, Ž, x. l Žt, Ž y. Ž x.. for a.e. t J a for all x, y, x y, where l : J, a zt Ž. is the oly absolutely cotiuous a oegative-value fuctio for which z Ž t. l t, zž t. a.e. i J, zž t.. Ž 2.5. a efie a lower solutio of problem 2.3 as a fuctio u Y such that Ž Ž u. Ž t.. gž t,, Ž u. Ž t.. for a.e. t Jt, t, aut bu t c, au t bu Ž. Ž. Ž. Ž t. c, a a upper solutio if the reverse iequalities hol. Ž. 2 u t c, The ext result is a cosequece of Theorem 2., Propositio 2., a, Theorem 3.. PROPOSITION 2.2. Assume that the hypotheses Ž., Ž A., Ž g., Ž g., Ž g2., a Ž gv. hol. If u Y is a lower solutio a w Y is a upper solutio of Ž 2.3. for a fixe CŽ J., the u w. I particular, problem Ž 2.3. has a uique solutio which is icreasig with respect to, c, a c, a ecreasig with respect to g a c CONVERGENCE OF SUCCESSIVE APPROXIMATIONS I this sectio we shall prove that by aig oe-sie cotiuity coitios for the epeece of the fuctio g o its last two argumets to the hypotheses of Theorem 2. we get extremal solutios of the IBVP Ž.2. by the metho of successive approximatios.

9 TIRD ORDER PROBLEMS 23 If a sequece Ž. of CŽ J. coverges uiformly to u, eote if Ž. is icreasig a if is ecreasig. The similar otatios are use also for coverget sequeces of. PROPOSITION 3.. Assume that the hypotheses Ž., Ž A., Ž g., Ž g., a Ž g2. hol, a let a, b CŽ J. be efie by Ž The the relatios u a, cyž t. cyž t. t už t. kž t, s. u Ž s. s, t J,, D t Ž 3.. where u is the greatest solutio of the IVP Ž t. g t, u, Ž t. for a.e. t J, Ž t. c 2, Ž 3.2. efie a icreasig sequece Ž u. i a, b, which coerges uiformly o J to the least solutio of the IBVP Ž.2. if Ž g3. gt, Ž, x. gt, Ž, x. for a.e. t J as i CŽ J. a x xi. Proof. The sequece Ž u., give by Ž 3.., is equal to the iteratio sequece ŽGa., where G is efie as i the proof of Theorem 2.. I particular, the sequece Ž u. is cotaie i a, b. Sice G is icreas- ig, the ŽGa. Ž u. is a icreasig sequece i a, b. It the follows from the proof of Theorem 2. that the sequece Ž u. Ž Gu. coverges poitwise o J to a fuctio u CŽ J.. I view of Dii s theorem the covergece is uiform. Because Ž u. is icreasig, it follows from Lemma 2.2 that Ž. u is a ecreasig a boue sequece i Z, a hece coverges poitwise to a fuctio L Ž J.. These results a the hypothesis Ž g3. imply that Sice u equatio Ž. lim g t, u, u t g t, u, t for a.e. t J. is a solutio of the IVP Ž 2.3. with u u, it satisfies the itegral t Ž u. Ž 2. Ž u. t t c g s, u, s s, t J,. Whe i this equatio, we get, by oticig the covergece results erive above a applyig the omiate covergece theorem, Ž t. Ž c2. g s, u, Ž s. s, t J. t This implies that is a solutio of the IVP Ž t

10 24 CABADA AND EIKKILÄ Deote by ˆ the greatest solutio of Ž Sice ˆ is a lower solutio of Ž 2.3. with u u for each, the Ž. t Ž. t, t J,, by ˆ 3, Theorem This implies whe that ˆ. The reverse iequality hols because is a solutio of Ž 2.3. a ˆ is its greatest solutio. Thus ; ˆ i.e., is the greatest solutio of Ž Whe i Ž 3.., we get cyž t. cyž t. t už t. kž t, s. Ž s. s, t J. D t This result a the efiitio of G imply that u Gu. The above proof shows that u Ga i CŽ J., equippe with the topology of uiform covergece, a that u Gu. Thus u is by 3, Propositio..3 the least fixe poit of G. This implies by the proof of Theorem 2. that u is the least solutio of the IBVP Ž.2.. Remarks 3.. The result of Propositio 3. hols also whe the hypothesis Ž g3. is replace by the followig weaker oe: Ž g4. lim gt, Ž u, Ž t.. gt, Ž u, Ž t.. u for a.e. t J, where the sequeces Ž u., Ž. u are as i Propositio 3. a u, are their limits. Replacig a by b, a the greatest solutios of Ž 3.2. by least solutios, we obtai mootoe a coverget sequeces Ž u., Ž. u. If the hypotheses of Theorem 2. are vali, a if Ž g4. hols, the greatest solutio of the IBVP Ž.2. is obtaie as the uiform limit of the sequece Ž u.. The problem 4. SPECIAL CASES AND EXAMPLES Ž u. Ž t. q Ž u. Ž t. g t, u, Ž u. Ž t. for a.e. t J t, t, Ž 4.. aut Ž. bu Ž t. c, au Ž t. bu Ž t. c, Ž u. Ž t. c, 2 ca be reuce to the IBVP.2 if q:, satisfies the followig hypothesis. z q q a q belog to Lloc, a qž z..

11 TIRD ORDER PROBLEMS 25 This is a cosequece of the ext two lemmas, the first oe beig a obvious cosequece of the properties assume for q i Ž q., a the proof of the seco oe beig similar to the proof of, Lemma 5.2. LEMMA 4.. If q hols, the the fuctio :, efie by x z Ž x., x, Ž 4.2. qž z. is a icreasig homeomorphism, a both a are locally Lipschitz cotiuous. LEMMA 4.2. If Ž q. hols, the u Y is a solutio of the ifferetial equatio of Ž 4.. if a oly if u is a solutio of the ifferetial equatio of Ž.2., where : is efie by Ž As a cosequece of Lemmas 4. a 4.2 a results erive i Sectio 2 we obtai the followig propositio. PROPOSITION 4.. Assume that : J Ž,. is cotiuous, that q: has property Ž q., a that g: J CŽ J. satisfies the hypotheses Ž g., Ž g., a Ž g2.. The the IBVP Ž 4.. with a, b, a, b, aa ab ab has for all choice of c, c, c2 extremal solutios u a u i Y. Moreoer, they are icreasig with respect to c a c, a ecreasig with respect to g a c 2. EXAMPLE 4.. Defie a fuctio q: Ž,. by m m Ž 2 k z k z. qž z. Ý Ý 2 Ž km. m k ž / 2 si, m m k z k z where x eotes the greatest iteger x. It is easy to see that q is iscotiuous at for all, k, m, 2,.... Moreover, qž z. k m 4 6 for each z, so that q has property Ž q.. The fuctio :, efie by x Ž x. qž s. si s s, x, is a icreasig homeomorphism.

12 26 CABADA AND EIKKILÄ Assumig that the fuctio Ž x. ½ 2 x, x, x, x 2, 2 is extee perioically to, the the fuctio : J, efie by Ž t. Ý Ž 4 t. 4, t J, is cotiuous but owhere ifferetiable. I particular, a satisfy coitio Ž.. The fuctio where ž / m m hž s. Ý Ý 2 sgž s / 2, m ½, t, sgž t., t,, t, is icreasig a iscotiuous at each ratioal poit s. Deotig Ž 4. g t,, x h t max t t J x, t J, CŽ J., x, it is easy to see that g satisfies the hypotheses Ž g., Ž g., a Ž g2.. Thus the extremality a compariso results erive above hol for problems Ž.2. a Ž 4.. whe the fuctios,, q, a g are efie as above. EXAMPLE 4.2. Cosier the IBVP u Ž t. 2t u Ž t. Ž u Ž t. t. u Ž t. 2t už t. Ž 4.3. a.e. i J,, už t. už. u Ž., už. u Ž., u Ž.,

13 TIRD ORDER PROBLEMS 27 where eotes the eavisie fuctio, x Ž x. ½, x, a z eotes the greatest iteger z. Problem Ž 4.3. is of the form Ž.2., where Ž t. x 2t gž t,, x. Ž x t., x 2t Ž t. Ž x. x, Ž t.. The hypotheses of Theorem 2. are satisfie, whece problem Ž 4.3. has extremal solutios. We shall show that they ca be obtaie by the metho of successive approximatios Ž 3... Notice first that, sice gt, Ž, x. 3 for all Ž t,, x. J CŽ J., the hypothesis Ž g2. hols with pt 3 a Ž x.. Because is the ietity fuctio, the fuctio wt 3t is the solutio of Ž Thus the first equatio of Ž 2.7. ca be writte i the form t 2 t t až t. Ž s.ž 3s. s Ž 2 s.ž 3s. s, 3 3 t 2 7 Ž. Ž. 2 3 which yiels at t t. Choosig u a, we have u t 35. Thus is the greatest solutio of the IVP 24 u Ž t. 2t 2 Ž t. Ž Ž t. t. Ž t. 2t 3 a.e. i J, Ž.. Ž 4.4. To etermie this, oe ca try the metho of successive approximatios, Ž. t wt Ž. 3t, Ž t. 2t 2 Ž t. Ž Ž t. t. Ž t. 2t 3 a.e. i J, Ž..

14 28 CABADA AND EIKKILÄ Ž. 3 Whe we obtai t 6 t. It turs out that 2, so that 3 Ž. t t. Thus Ž 3.. u 6, with, takes the form / ž / t 2 t 3 t 3 už t. Ž s. ž s s Ž 2 s. s s t t t Ž t Sice u t, the is also a greatest solutio of 4.4 ; i.e.,. But the u u, whece the least solutio of Ž 4.3. is 75 Ž. 44 u u u už t. už t. t t Ž t Aalogous but more teious calculatios show that the greatest solutio of the IBVP 4.3 is t t Ž t., t, u Ž t t 2 t, t Remark 4.. The fuctio :, efie by p2 Ž x. x x, x, Ž 4.5. is a icreasig homeomorphism for each p. Because is ot locally Lipschitz-cotiuous if p, 2, a is ot locally Lipschitz-cotiuous if p 2, it follows from Lemma 4. that the fuctio efie by Ž 4.5. is of the form Ž 4.2., where q has property Ž q., oly whe p 2. Thus the ifferetial equatio of Ž.2. is more geeral tha that of Ž TE CASE WEN DEPENDS FUNCTIONALLY ON TE SOLUTION If fuctio g is icreasig with respect to its last variable, we ca cosier problem Ž.2. i the case whe epes fuctioally o u. More precisely, we stuy the existece of extremal solutios of the prob-

15 TIRD ORDER PROBLEMS 29 lem i the set u, Ž u. Ž t. g t, u, Ž u. Ž t. for a.e. t J t, t, Ž 5.. aut Ž. bu Ž t. c, au Ž t. bu Ž t. c, u, Ž u. Ž t. c, 2 Ž. 4 Y u C J u, u AC J, by assumig coitio Ž A. for costats a, a, b, b a the followig properties for fuctios : J, : CŽ J., a g: J CŽ J. : is cotiuous, Ž u,. is a icreasig homeomorphism o for each u CŽ J., Ž, x. is icreasig for each x, a for each m there exists a M such that x Ž u, x. m4 M, M for each u CŽ J.. Ž ga. For all CŽ J. a x the fuctio gž,, x. is measurable. Ž gb. gt, Ž, x. is ecreasig i a icreasig i x for a.e. t J. Ž gc. gt, Ž, x. pt Ž. for all CŽ J. a x a for a.e. t J, where p L Ž J.. Deotig Ž. 4 Z C J u, AC J, the followig result ca be prove as Lemma 2.. if LEMMA 5.. A fuctio u Y is a solutio of the IBVP Ž 5.. if a oly cyž t. cyž t. t už t. kž t, s. Ž s. s, t J, Ž 5.2. D t where Z is a solutio of the IVP u, t g t, u, t for a.e. t J, u, t c. Ž. 2 Ž 5.3.

16 2 CABADA AND EIKKILÄ Cosier ext the IVP Ž 5.3. whe u CŽ J. is give a efie a lower solutio for such a problem if Z a Ž u, Ž t.. gž t, u, Ž t.. for a.e. t J t, t, u, t c. Ž. 2 If the reverse iequalities hol we say that is a upper solutio. LEMMA 5.2. If the hypotheses Ž., Ž ga., Ž gb., a Ž gc. hol, the the IVP Ž 5.3. has for each u CŽ J. least a greatest solutios i Z, a they are icreasig with respect to g a c 2, a ecreasig with respect to u. If Z is ay solutio of Ž 5.3., the M Ž t. M, Ž 5.4. where M is a costat i the hypothesis correspoig to m c 2 t pt Ž. t. Proof. Let u CŽ J. be give. The hypotheses Ž., Ž ga., a Ž gc. imply that the hypotheses of 3, Theorem 2..3 hol whe x Ž u, x. stas for a Ž t, x. gt, Ž u, x. stas for g. This esures that the IVP Ž 5.3. has least a greatest solutios i Z. Deotig x Ž u, x. u,it follows from 3, Lemma 2.. that is a solutio, a upper solutio, or a lower solutio of problem Ž 5.3. if a oly if u is a solutio, a upper solutio, or a lower solutio of the IVP t g t, u, u t for a.e. t J, t c Sice u is icreasig i u by, the u is ecreasig i u. This a the hypothesis Ž gb. imply Žcf. the proof of 3, Propositio 2... that the extremal solutios of Ž 5.5., a hece those of Ž 5.3., are ecreasig with respect to u, a icreasig with respect to g a c 2. To prove the last assertio, assume that Z is a solutio of the IVP Ž It follows from Ž gc. that Thus u, t g t, u, t p t a.e. i J. 2 u, Ž t. c pž s. s, t J. t t This iequality a coitio imply that 5.4 hols.

17 TIRD ORDER PROBLEMS 2 Applyig the results of Lemmas 5. a 5.2 we prove that all the possible solutios of 5. belog to a orer iterval of C J. LEMMA 5.3. Assume that the hypotheses, A, ga, gb, a gc hol, a eote cyž t. cyž t. t až t. M kž t, s. s, t J, D t Ž 5.6. cyž t. cyž t. t bž t. M kž t, s. s, t J, D t where M is a costat i the hypothesis correspoig to m c 2 t pt Ž.. If u is a solutio of the IBVP Ž 5.. t. i Y, the u belogs to the orer iteral a, b u CŽ J. a u b 4. Proof. Let u be a solutio of Ž 5.. i Y. The u satisfies Ž 5.2., where is a solutio of Ž 5.3. i Z. Because u CŽ J., it follows from Lemma 5.2 that Ž 5.4. hols. Noticig that the right-ha sie of Ž 5.2. is ecreasig i, we see that u a, b. The followig result ca be prove as Theorem 2. a Propositio 2. whe Lemmas 2., 2.2, a 2.3 are replace by Lemmas 5., 5.2, a 5.3. TEOREM 5.. Assume that the hypotheses Ž., Ž A., Ž ga., Ž gb., a Ž gc. hol. The the IBVP Ž 5.. has extremal solutios u a u i Y; i.e., if u Y is a solutio of Ž 5.., the už. t ut Ž. u Ž. t o J. Moreoer, u a u are icreasig with respect to c a c a ecreasig with respect to g a c 2. Remark 5.. As i, by a simple chage of variables oe ca obtai aalogous results for problems Ž Ž u. Ž t.. gž t, u, Ž u. Ž t.. for a.e. t J t, t, aut bu t c, au t bu Ž. Ž. Ž. Ž t. c, Ž. 2 u t c,

18 22 CABADA AND EIKKILÄ a Ž u, Ž u. Ž t.. gž t, u, Ž u. Ž t.. for a.e. t J t, t, aut bu t c, au t bu Ž. Ž. Ž. Ž t. c, Ž u, Ž u. Ž t.. c. 2 EXAMPLE 5.. Let the fuctios q, h, g, a be as i Example 4. a ž / t t Ž, x. h Ž s. s qž s. si s s, x, It is easy to see that g satisfies the hypotheses Ž ga., Ž gb., a Ž gc. a coitio Ž.. Thus the extremality a compariso results erive above hol for problem Ž 5... x REFERENCES. A. Cabaa a S. eikkila, Uiqueess, compariso a existece results for thir orer fuctioal iitial-bouary value problems, Comput. Math. Appl., to appear. 2. A. Cabaa a S. Lois, Existece of solutio for iscotiuous thir orer bouary value problems, J. Comput. Appl. Math. Ž 999., S. Carl a S. eikkila, Noliear Differetial Equatios i Orere Spaces, Chapma & all, LooNew York a CRC, Boca Rato, FL, E. R. assa a W. Rzymowski, Extremal solutios of a iscotiuous ifferetial equatio, Noliear Aal. 37, No. 8 Ž 999., S. eikkila a V. Lakshmikatham, Mootoe Iterative Techiques for Discotiuous Noliear Differetial Equatios, Dekker, New York, R. L. Pouso, Upper a lower solutios for first-orer iscotiuous oriary ifferetial equatios, J. Math. Aal. Appl. 244 Ž 2.,