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MR2658503 26D15 (65D30)

Huy, Vu Nhat [Vu Nhat Huy] (VN-VNU-NS); Ngo, Qu´oc-Anh (VN-VNU-NS)New inequalities of Simpson-like type involvingn knots and themth derivative. (Englishsummary)Math. Comput. Modelling52 (2010),no. 3-4,522–528.

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MR2672588 26D15 (65D30)

Nhat Huy, Vu [Vu Nhat Huy] (VN-VNU-NS); Ngo, Qu´oc-Anh (VN-VNU-NS)On an Iyengar-type inequality involving quadratures in n knots. (English summary)Appl. Math. Comput.217(2010),no. 1,289–294.

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MR2653143 26E70 (26D15)

Liu, Wenjun [Liu, Wen Jun 3] (PRC-NUIST-MP); Ngo, Qu´oc-Anh (VN-VNU-NS)Some Iyengar-type inequalities on time scales for functions whose second derivatives arebounded. (English summary)Appl. Math. Comput.216(2010),no. 11,3244–3251.

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MR2663354 (Review) 26D15 (26E70 34N05)

Liu, Wenjun [Liu, Wen Jun 3] (PRC-NUIST-MP); Ngo, Qu´oc Anh (VN-VNU-NS);Chen, Wenbing(PRC-NUIST-MP)On new Ostrowski type inequalities for double integrals on time scales. (English summary)Dynam. Systems Appl.19 (2010),no. 1,189–198.

The authors present an Ostrowski-type inequality for∆∆ double integrals on time scales. Theproof of this time scales result relies on a result of N. S. Barnett and S. S. Dragomir [Soochow J.Math.27 (2001), no. 1, 1–10;MR1821346 (2002a:26021)], which was obtained in the context ofthe reals. Further on, the authors specialize their Ostrowski-type inequality to the classical timescalesR, Z andqZ. They also present or mention completely analogous versions of the Ostrowskiinequality for the so-called∆∆, ∆∇,∇∆ and∇∇ cases.

Reviewed byStefan Hilger

References

1. R. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: A survey,Math. Inequal.Appl.4 (2001), pp. 535–557.MR1859660 (2002g:34016)

2. R. Almeida and D. F. M. Torres, Isoperimetric problems on time scales with nabla derivatives,J. Vib. Control, in press (2009). DOI:10.1177/1077546309103268MR2528200 (2010g:49038)

3. F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics,Math.Comput. Modelling43 (2006), pp. 718–726.MR2218315 (2006k:91169)

4. N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applica-tions for cubature formulae,Soochow J. Math.27(2001), pp. 1–10.MR1821346 (2002a:26021)

5. M. Bohner and A. Peterson,Dynamic equations on time scales, Birkhauser, Boston, 2001.MR1843232 (2002c:34002)

6. M. Bohner and A. Peterson,Advances in dynamic equations on time scales, Birkhauser, Boston,2003.MR1962542 (2004d:34003)

7. M. Bohner and G. S. Guseinov, Partial differentiation on time scales,Dynam. Systems Appl.13(2004), pp. 351–379.MR2106411 (2005i:26029)

8. M. Bohner and G. S. Guseinov, Multiple integration on time scales,Dynam. Systems Appl.14(2005), pp. 579–606.MR2179167 (2006g:26024)

9. M. Bohner and G. S. Guseinov, Double integral calculus of variations on time scales,ComputersMath. Applic.54 (2007), pp. 45–57.MR2332777 (2008f:49031)

10. M. Bohner and T. Matthews, The Gruss inequality on time scales,Commun. Math. Anal.3(2007), pp. 1–8.MR2347770 (2008i:26019)

11. M. Bohner and T. Matthews, Ostrowski inequalities on time scales,JIPAM. J. Inequal. PureAppl. Math.9 (2008), Art. 6, 8 pp.MR2391273 (2009b:26025)

12. S. S. Dragomir and S. Wang, Applications of Ostrowski’s inequality to the estimation of error

bounds for some special means and some numerical quadrature rules,Appl. Math. Lett.11(1998), pp. 105–109.MR1490389

13. S. S. Dragomir and S. Wang, A new inequality of Ostrowski’s type inL1 norm and applictionsto some special means and to some numerical quadrature rules,Tamkang J. of Math.28(1997),pp. 239–244.MR1486792 (99a:65034)

14. S. Hilger,Ein Maßkettenkalkul mit Anwendung auf Zentrmsmannigfaltingkeiten, PhD thesis,Univarsi. Wurzburg, 1988.

15. R. Hilscher, A time scales version of a Wirtinger-type inequality and applications,J. Comput.Appl. Math.141(2002), pp. 219–226.MR1908839 (2003d:26019)

16. B. Karpuz and U. M. Ozkan, Generalized Ostrowski inequality on time scales,JIPAM. J.Inequal. Pure Appl. Math.9 (2008), Art. 112, 7 pp.MR2465897 (2009k:26021)

17. V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan,Dynamic systems on measurechains, Kluwer Academic Publishers, 1996.MR1419803 (97j:34002)

18. W. J. Liu, Q. L. Xue and S. F. Wang, Several new perturbed Ostrowski-like type inequalities,JIPAM. J. Inequal. Pure Appl. Math.8 (2007), Art.110, 6 pp.MR2366264 (2008k:26047)

19. W. J. Liu, C. C. Li and Y. M. Hao, Further generalization of some double integral inequal-ities and applications,Acta. Math. Univ. Comenianae77 (2008), pp. 147–154.MR2412405(2009g:26019)

20. W. J. Liu and Q. A. Ngo, An Ostrowski-Gruss type inequality on time scales, arXiv:0804.3231.21. W. J. Liu and Q. A. Ngo, A generalization of Ostrowski inequality on time scales fork points,

Appl. Math. Comput.203(2008), pp. 754–760.MR245899122. W. J. Liu and Q. A. Ngo, An Ostrowski type inequality on time scales for functions whose

second derivatives are bounded, to appear in Inequality Theory and Applications 6, NovaScience Publishers, Inc., New York, 2009.

23. W. J. Liu, Q. A. Ngo and W. B. Chen, A perturbed Ostrowski type inequality on time scales fork points for functions whose second derivatives are bounded,J. Inequal. Appl., Volume 2008,Article ID 597241, 12 pages.

24. W. J. Liu, Q. A. Ngo and W. B. Chen, Ostrowski type inequalities on time scales for doubleintegrals,Acta Appl. Math., to appear. DOI: 10.1007/s10440-009-9456-ycf. MR2601668

25. N. Martins, D. F. M. Torres, Calculus of variations on time scales with nabla derivatives,Nonlinear Anal., to appear. DOI: doi:10.1016/j.na.2008.11.035cf. MR2671876

26. D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequalities Involving Functions and TheirIntegrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.MR1190927(93m:26036)

27. D. S. Mitrinovic, J. Pecaric and A. M. Fink,Classical and New Inequalities in Analysis, KluwerAcademic, Dordrecht, (1993).MR1220224 (94c:00004)

28. A. M. Ostrowski,Uber die Absolutabweichung einer differentiebaren Funktion von ihremIntegralmitelwert,Comment. Math. Helv.10 (1938), pp. 226–227.

29. U. M. Ozkan and H. Yildirim, Ostrowski type inequality for double integrals on time scales,Acta Appl. Math., to appear. DOI: 10.1007/s10440-008-9407-zcf. MR2601657

30. U. M. Ozkan and H. Yildirim, Gruss type inequalities for double integrals on time scales,Computers Math. Applic.57 (2009), pp. 436–444.MR2488615 (2009k:26029)

31. U. M.Ozkan and H. Yildirim, Steffensen’s integral inequality on time scales,J. Inequal. Appl.,Volume 2007, Article ID 46524, 10 pages.MR2335973 (2008d:26026)

32. U. M.Ozkan and H. Yildirim, Hardy-knopp-type Inequalities on Time Scales,Dynam. SystemsAppl.17 (2008), pp. 477–486.MR2569514 (2010i:26036)

33. U. M. Ozkan, M. Z. Sarikaya and H. Yildirim, Extensions of certain integral inequalities ontime scales,Appl. Math. Lett.21 (2008), pp. 993–1000.MR2450628

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2011

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MR2610536 (Review) 26D15 (65D30)

Huy, Vu Nhat [Vu Nhat Huy] (VN-VNU-NS); Ngo, Qu´oc-Anh (VN-VNU-NS)A new way to think about Ostrowski-like type inequalities. (English summary)Comput. Math. Appl.59 (2010),no. 9,3045–3052.

In this paper, a new method to investigate a class of Ostrowski-like type inequalities involvingn pointsx1, x2, . . . , xn, and themth derivative of a differentiable functionf : I ⊂ R → R with[a, b]⊂ I is presented. Namely, the inequality∣∣∣∣∣ 1

b− a

∫ b

a

f(x) dx− b− a

n

n∑i=1

f(a+xi(b− a))

∣∣∣∣∣≤(2m +5)(b− a)m+1

4(m +1)!(S− s),

whereS = supa≤x≤b f (m)(x) ands = infa≤x≤b f (m)(x) holds. Notice thatm,n ∈ N are arbitrary.So, the estimation in the above expression becomes more accurate asm becomes larger.

Reviewed byJose Luis Dıaz-Barrero

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MR2608576 (2011b:35142)35J66(35J20 35J62 47J30)

Chung, Nguyen Thanh[Nguyen Thanh Chung]; Ngo, Qu´oc-Anh (VN-VNU-NS)Multiple solutions for a class of quasilinear elliptic equations ofp(x)-Laplacian type withnonlinear boundary conditions. (English summary)Proc. Roy. Soc. Edinburgh Sect. A140(2010),no. 2,259–272.

Summary: “Using variational methods we study the non-existence and multiplicity of non-negativesolutions for a class of quasilinear elliptic equations ofp(x)-Laplacian type with nonlinear bound-ary conditions, of the form

−div(|∇u|p(x)−2∇u) + |u|p(x)−2u = 0 in Ω,

|∇u|p(x)−2∂u

∂n= λg(x, u) on∂Ω,

whereΩ is a bounded domain with smooth boundary,n is the outer unit normal to∂Ω andλ isa parameter. Here,g: ∂Ω× [0,∞) → R is a continuous function that may or may not satisfy anAmbrosetti-Rabinowitz-type condition.”

References

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2. A. Ambrosetti and P. H. Rabinowitz. Dual variational methods in critical points theory andapplications.J. Funct. Analysis4 (1973), 349–381.MR0370183 (51 #6412)

3. P. de Napoli and M. C. Mariani. Mountain pass solutions to equations ofp-Laplacian type.Nonlin. Analysis54 (2003), 1205–1219.MR1995926 (2004e:35065)

4. I. Ekeland. On the variational principle.J. Math. Analysis Applic.47 (1974), 324–353.MR0346619 (49 #11344)

5. X. L. Fan. Boundary trace embedding theorems for variable exponent Sobolev spaces.J. Math.Analysis Applic.339(2008), 1395–1412.MR2377096 (2008k:46095)

6. X. L. Fan and X. Han. Existence and multiplicity of solutions forp(x)-Laplacian equations inRN . Nonlin. Analysis59 (2004), 173–188.MR2092084 (2005h:35092)

7. X. L. Fan and Q. H. Zhang. Existence of solutions forp(x)-Laplacian Dirichlet problem.Nonlin. Analysis52 (2003), 1843–1852.MR1954585 (2004f:35060)

8. X. L. Fan and D. Zhao. On the spacesLp(x)(Ω) andW k,p(x)(Ω). J. Math. Analysis Applic.263(2001), 424–446.MR1866056 (2003a:46051)

9. X. L. Fan, J. Shen and D. Zhao. Sobolev embedding theorems for spacesW k,p(x)(Ω). J. Math.Analysis Applic.262(2001), 749–760.MR1859337 (2002g:46050)

10. X. L. Fan, Q. H. Zhang and D. Zhao. Eigenvalues ofp(x)-Laplacian Dirichlet problem.J.Math. Analysis Applic.302(2005), 306–317.MR2107835 (2005m:35213)

11. T. C. Halsey. Electrorheological fluids.Science258(1992), 761–766.12. O. Kovacik and J. Rakosnık. On spacesLp(x) and W k,p(x)(Ω). Czech. Math. J.41 (1991),

592–618.MR1134951 (92m:46047)13. M. Mihailescu. Existence and multiplicity of weak solutions for a class of denegerate non-linear

elliptic equations.Boundary Value Probl.17 (2006), 41295.MR2211397 (2006j:35084)14. M. Mihailescu and V. Radulescu. A multiplicity result for a nonlinear degenerate problem

arising in the theory of electrorheological fluids.Proc. R. Soc. Land.A 462 (2006), 2625–2641.MR2253555 (2007i:35081)

15. J. Musielak.Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034(Springer, 1983).MR0724434 (85m:46028)

16. K. Perera. Multiple positive solutions for a class of quasilinear elliptic boundary-value prob-lems.Electron. J. Diff. Eqns7 (2003), 1–5.MR1958042 (2004a:35078)

17. M. Ruzicka.Electrorheological fluids: modeling and mathematical theory(Springer, 2002).MR1810360 (2002a:76004)

18. M. Struwe.Variational methods, 2nd edn (Springer, 2000).MR1736116 (2000i:49001)19. Q. H. Zhang. Existence of radial solutions forp(x)-Laplacian equation inRN . J. Math. Analysis

Applic.315(2006), 506–516.MR2202596 (2006i:35115)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

c© Copyright American Mathematical Society 2011

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MR2601668 26E70 (26D15)

Liu, Wenjun [Liu, Wen Jun 3] (PRC-NUIST-MP); Ngo, Qu´oc Anh (VN-VNU-NS);Chen, Wenbing(PRC-NUIST-MP)Ostrowski type inequalities on time scales for double integrals. (English summary)Acta Appl. Math.110(2010),no. 1,477–497.

There will be no review of this item.

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2. Almeida, R., Torres, D.F.M.: Isoperimetric problems on time scales with nabla derivatives, J.Vibr. Control (in press). http://arxiv.org/abs/0811.3650MR2528200 (2010g:49038)

3. Atici, F.M., Biles, D.C., Lebedinsky, A.: An application of time scales to economics. Math.Comput. Model.43(7–8), 718–726 (2006)MR2218315 (2006k:91169)

4. Bohner, M., Guseinov, G.S.: Partial differentiation on time scales. Dyn. Syst. Appl.13(3–4),351–379 (2004)MR2106411 (2005i:26029)

5. Bohner, M., Guseinov, G.S.: Multiple integration on time scales. Dyn. Syst. Appl.14(3–4),

579–606 (2005)MR2179167 (2006g:26024)6. Bohner, M., Guseinov, G.S.: Double integral calculus of variations on time scales. Comput.

Math. Appl.54,45–57 (2007)MR2332777 (2008f:49031)7. Bohner, M., Matthews, T.: The Gruss inequality on time scales. Commun. Math. Anal.3(1),

1–8 (2007)MR2347770 (2008i:26019)8. Bohner, M., Matthews, T.: Ostrowski inequalities on time scales. J. Inequal. Pure Appl. Math.

9(1), 8 (2008)MR2391273 (2009b:26025)9. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhauser, Basel (2001)

MR1843232 (2002c:34002)10. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhauser, Basel

(2003)MR1962542 (2004d:34003)11. Dragomir, S.S., Wang, S.: A new inequality of Ostrowski’s type inL1 norm and applications

to some special means and to some numerical quadrature rules. Tamkang J. Math.28,239–244(1997)MR1486792 (99a:65034)

12. Dragomir, S.S., Wang, S.: Applications of Ostrowski’s inequality to the estimation of errorbounds for some special means and some numerical quadrature rules. Appl. Math. Lett.11,105–109 (1998)MR1490389

13. Dragomir, S.S., Cerone, P., Barnett, N.S., Roumeliotis, J.: An inequality of the Ostrowski typefor double integrals and applications for cubature formulae. RGMIA Res. Rep. Collect.2,781–796 (1999)MR1772072 (2001f:26027)

14. Ferreira, R.A.C., Sidi Ammi, M.R., Torres, D.F.M.: Diamond-alpha integral inequalities ontime scales. Int. J. Math. Stat.5(A09), 52–59 (2009)MR2446702 (2009e:26036)

15. Hilger, S.: Ein Maßkettenkalkul mit Anwendung auf Zentrmsmannigfaltingkeiten. Ph.D. thesis,Univarsi. Wurzburg (1988)

16. Hilscher, R.: A time scales version of a Wirtinger-type inequality and applications. J. Comput.Appl. Math.141,219–226 (2002)MR1908839 (2003d:26019)

17. Lakshmikantham, V., Sivasundaram, S., Kaymakcalan, B.: Dynamic Systems on MeasureChains. Kluwer Academic, Dordrecht (1996)MR1419803 (97j:34002)

18. Liu, W.J., Ngo, Q.A.: An Ostrowski-Gruss type inequality on time scales. arXiv:0804.3231MR2554353 (2010h:26028)

19. Liu, W.J., Ngo, Q.A.: A generalization of Ostrowski inequality on time scales fork points.Appl. Math. Comput.203,754–760 (2008)MR2458991

20. Liu, W.J., Ngo, Q.A.: An Ostrowski type inequality on time scales for functions whose sec-ond derivatives are bounded. In: Inequality Theory and Applications, vol 6. Nova SciencePublishers, New York (2009, to appear)cf. MR 2009j:26033

21. Liu, W.J., Xue, Q.L., Wang, S.F.: Several new perturbed Ostrowski-like type inequalities. J.Inequal. Pure Appl. Math.8(4), 6 (2007)MR2366264 (2008k:26047)

22. Liu, W.J., Li, C.C., Hao, Y.M.: Further generalization of some double integral inequalities andapplications. Acta. Math. Univ. Comen.77(1), 147–154 (2008)MR2412405 (2009g:26019)

23. Liu, W.J., Ngo, Q.A., Chen, W.B.: A perturbed Ostrowski type inequality on time scales forkpoints for functions whose second derivatives are bounded. J. Inequal. Appl.2008.Article ID597241, 12 p.MR2465495 (2009j:26030)

24. Martins, N., Torres, D.F.M.: Calculus of variations on time scales with nabla derivatives.Nonlinear Anal. (2008). doi:10.1016/j.na.2008.11.035MR2671876

25. Mitrinovic, D.S., Pecaric, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integralsand Derivatives. Kluwer Academic, Dordrecht (1991)MR1190927 (93m:26036)

26. Mitrinovic, D.S., Pecaric, J., Fink, A.M.: Classical and New Inequalities in Analysis. KluwerAcademic, Dordrecht (1993)MR1220224 (94c:00004)

27. Ostrowski, A.M.:Uber die Absolutabweichung einer differentiebaren Funktion von ihremIntegralmitelwert. Comment. Math. Helv.10,226–227 (1938)MR1509574

28. Ozkan, U.M., Yildirim, H.: Ostrowski type inequality for double integrals on time scales. ActaAppl. Math. (2009). doi:10.1007/s10440–008-9407-zMR2601657

29. Ozkan, U.M., Yildirim, H.: Steffensen’s integral inequality on time scales. J. Inequal. Appl.2007.Article ID 46524, 10 p.MR2335973 (2008d:26026)

30. Ozkan, U.M., Yildirim, H.: Hardy-knopp-type inequalities on time scales. Dyn. Syst. Appl.17,477–486 (2008)MR2569514 (2010i:26036)

31. Ozkan, U.M., Sarikaya, M.Z., Yildirim, H.: Extensions of certain integral inequalities on timescales. Appl. Math. Lett.21,993–1000 (2008)MR2450628

32. Pachpatte, B.G.: On an inequality of Ostrowski type in three independent variables. J. Math.Anal. Appl.249,583–591 (2000)MR1781243 (2001g:26016)

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Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2011

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MR2561013 (2010j:26031)26D15 (26E70)

Liu, Wenjun J. [Liu, Wen Jun 3] (PRC-NUIST-MP);Qu´oc-Anh Ngo [Ngo, Qu´oc Anh] (VN-VNU-NS);Chen, Wenbing B.[Chen, Wenbing] (PRC-NUIST-MP)A new generalization of Ostrowski type inequality on time scales. (English summary)An. Stiint. Univ. “Ovidius” Constanta Ser. Mat.17 (2009),no. 2,101–114.

In this paper the authors extend a generalization of a well-known Ostrowski type integral inequalityon time scales for functions whose derivatives are bounded. This generalization is established byintroducing a parameterλ ∈ [0, 1]. The authors obtain well-known and new results by applyingtheir result to the time scalesT = R, T = Z andT = q(N0), and using differentλ’s.

Reviewed byHassan Ahmed Agwo

c© Copyright American Mathematical Society 2010, 2011

Article

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From References: 1From Reviews: 0

MR2554353 (2010h:26028)26D15 (26E70)

Liu, Wenjun [Liu, Wen Jun 3] (PRC-NUIST-MP); Ngo, Qu´oc-Anh (VN-VNU-NS)An Ostrowski-Gr uss type inequality on time scales. (English summary)Comput. Math. Appl.58 (2009),no. 6,1207–1210.

In this paper the authors prove a new integral inequality of Ostrowski-Gruss type on an arbitrarytime scaleT. This result extends a generalization of a well-known Ostrowski-Gruss type inequalityto time scales. The authors obtain well-known and new results by applying their result in differenttime scales such asT = R, Z andqN∪0.

Reviewed byHassan Ahmed Agwo

c© Copyright American Mathematical Society 2010, 2011

Article

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MR2534004 (Review) 35J62(35J92 47J30 58E05)

Nguyen Thanh Chung; Qu´oc Anh Ngo [Ngo, Qu´oc Anh] (VN-VNU-NS)A multiplicity result for a class of equations ofp-Laplacian type with sign-changingnonlinearities. (English summary)Glasg. Math. J.51 (2009),no. 3,513–524.

The authors consider the boundary value problem−div(a(x,∇u)) = λf(x, u) in Ω,u = 0 on∂Ω.

The vector fielda is assumed to satisfy the growth condition

|a(x, ξ)| ≤ c0(h0(x) +h1(x)|ξ|p−1),

whereh0 ∈ Lp/(p−1), h1 ∈ L1, h1 ≥ 1; previous works cited consider cases whereh0 andh1 arebounded. Further,f can change sign and is not required to satisfy the Ambrosetti–Rabinowitzcondition.

Two results concerning the existence of solutions are proved. First, there is a numberλ suchthat no nontrivial solutions exist if0 < λ < λ. Second, there is a numberλ such that at leasttwo nontrivial solutions exist ifλ ≥ λ. The first is a rather simple consequence of the Poincareinequality and the fact thath1 ≥ 1. In the second, one solution is obtained as a minimizer of asuitable functional, and a second solution is found by applying a version of the mountain passtheorem.

Reviewed byTeemu Lukkari

References

1. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical points theory andapplications,J. Funct. Anal.4 (1973), 349–381.MR0370183 (51 #6412)

2. P. De Napoli and M. C. Mariani, Mountain pass solutions to equations ofp-Laplacian type,Nonlinear Anal.54 (2003), 1205–1219.MR1995926 (2004e:35065)

3. D. M. Duc, Nonlinear singular elliptic equations,J. Lond. Math. Soc.40(2) (1989), 420–440.MR1053612 (91g:35107)

4. D. M. Duc and N. T. Vu, Non-uniformly elliptic equations ofp-Laplacian type,Nonlinear Anal.61 (2005), 1483–1495.MR2135821 (2005k:35111)

5. I. Ekeland, On the variational principle,J. Math. Anal. Appl.47 (1974), 324–353.MR0346619(49 #11344)

6. L. Gasinski and N. S. Papageorgiou,Nonsmooth critical point theory and nonlinear boundaryvalue problems.Series in Mathematical Analysis and Applications, 8. Chapman & Hall/CRC,Boca Raton, FL, 2005.MR2092433 (2006f:58013)

7. M. Mihailescu and V. Radulescu, A multiplicity result for a nonlinear degenerate problemarising in the theory of electrorheological fluids,Proc. R. Soc. Lond. Ser. A462(2006), 2625–2641.MR2253555 (2007i:35081)

8. M. Mihailescu, Existence and multiplicity of weak solutions for a class of denegerate non-linear elliptic equations,Boundary Value Probl.(2006), Art. ID 41295, 17 pp.MR2211397

(2006j:35084)9. Q.-A. Ngo, and H. Q. Toan, Existence of solutions for a resonant problem under Landesman-

Lazer conditions,Electron. J. Diff. Eqns.2008(98) (2008), 1–10.MR2430895 (2009h:35149)10. K. Perera, Multiple positive solutions for a class of quasilinear elliptic boundary-value prob-

lems,Electron. J. Differential Equations7 (2003), 1–5.MR1958042 (2004a:35078)11. M. Struwe,Variational Methods: Applications to nonlinear partial differential equations and

Hamiltonian systems, 4 ed. (Springer-Verlag, Berlin, 2008).MR2431434 (2009g:49002)12. H. Q. Toan and Q.-A. Ngo, Multiplicity of weak solutions for a class of non-uniformly el-

liptic equations ofp-Laplacian type,Nonlinear Anal.70 (2009), 1536–1546.MR2483577(2010b:35127)

13. N. T. Vu, Mountain pass theorem and non-uniformly elliptic equations,Vietnam J. Math33 (4)(2005), 391–408.MR2200236 (2006j:35085)

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MR2542298 (2010g:26025)26D15Liu, Wenjun [Liu, Wen Jun 3] (PRC-NUIST-MP);Qu´oc-Anh Ngo [Ngo, Qu´oc Anh] (VN-VNU-NS); Vu Nhat Huy (VN-VNU-NS)Several interesting integral inequalities. (English summary)J. Math. Inequal.3 (2009),no. 2,201–212.

The authors present several interesting integral inequalities in the direction that was initiatedby F. Qi [JIPAM. J. Inequal. Pure Appl. Math.1 (2000), no. 2, Article 19, 3 pp.;MR1786406(2001e:26036)]. Some open problems are proposed.

Reviewed byZheng Liu

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MR2536811 (2010e:26023)26D15 (65D30)

Huy, Vu Nhat [Vu Nhat Huy] (VN-VNU-NS); Ngo, Qu´oc-Anh (VN-VNU-NS)New inequalities of Ostrowski-like type involvingn knots and theLp-norm of the m-thderivative. (English summary)Appl. Math. Lett.22 (2009),no. 9,1345–1350.

Let 1≤m,n <∞, 1≤ p≤∞ andp−1 + q−1 = 1, let I ⊂ R be an open interval such that[a, b]⊂I, and letf be anm-times differentiable function such thatf (m) ∈ Lp(a, b). The authors applythe Fundamental Theorem of Calculus, Taylor’s formula and the Holder inequality to establish thefollowing inequality:∣∣∣∣∣

∫ b

a

f(x)dx− b− a

n

n∑i=1

f(a+xi(b− a))

∣∣∣∣∣≤C(m, q)

∥∥∥f (m)∥∥∥

p(b− a)m+ 1

q ,

whereC(m, q) = 1m!((

1mq+1)

1/q + ( 1(m−1)q+1)

1/q), and0 < xi < 1 (i = 1, 2, . . . , n). In addition,some special cases are considered. These results are new, interesting and generalize recent resultsdue to N. Ujevic [Rev. Colombiana Mat.37 (2003), no. 2, 93–105;MR2124725 (2006a:65036);Comput. Math. Appl.48 (2004), no. 10-11, 1531–1540;MR2107109 (2005i:65032)].

Reviewed byMingzhe Gao

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MR2536655 26A24 (39A10)

Ngo, Quoc-Anh (VN-VNU-NS)Some mean value theorems for integrals on time scales. (English summary)Appl. Math. Comput.213(2009),no. 2,322–328.

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MR2513599 (2010g:35105)35J62(35D30 35J20 58E05)

Ngo, Quo′c-Anh (VN-VNU-NS)Existence results for a class of non-uniformly elliptic equations ofp-Laplacian type. (Englishsummary)Anal. Appl. (Singap.)7 (2009),no. 2,185–197.

Summary: “In this paper, we establish the existence of non-trivial weak solutions inW 1,p0 (Ω), 1 <

p <∞, to a class of non-uniformly elliptic equations of the form

−div(a(x,∇u)) = λf(u) +µg(u)

in a bounded domainΩ of RN . Herea satisfies

|a(x, ξ)|5 c0(h0(x) +h1(x)|ξ|p−1)

for all ξ ∈ RN , a.e.x ∈ Ω, h0 ∈ Lp

p−1 (Ω), h1 ∈ L1loc(Ω), h0(x) = 0, h1(x) = 1 for a.e.x in Ω.”

References

1. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory andapplications,J. Funct. Anal.14 (1973) 349–381.MR0370183 (51 #6412)

2. G. Bonanno, Some remarks on a three critical points theorem,Nonlinear Anal.54 (2003)651–665.MR1983441 (2004d:49010)

3. D. G. Costa,An Invitation to Variational Methods in Differential Equations(Birkhauser, 2007).MR2321283 (2008k:58033)

4. G. Dinca, P. Jebelean and J. Mawhin, A result of Ambrosetti–Rabinowitz type forp-Laplacian,in Qualitative Problems for Differential Equations and Control Theory,ed. C. Corduneanu(World Sci. Publ., River Edge, NJ, 1995), pp. 231–242.MR1372755 (96m:35097)

5. G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet prob-lems withp-Laplacian,Protugaliae Math.58 (2001) 340–377.MR1856715 (2002j:35116)

6. D. M. Duc, Nonlinear singular elliptic equations,J. London Math. Soc.40(2) (1989) 420–440.MR1053612 (91g:35107)

7. D. M. Duc and N. T. Vu, Nonuniformly elliptic equations ofp-Laplacian type,Nonlinear Anal.61 (2005) 1483–1495.MR2135821 (2005k:35111)

8. I. Ekeland and N. Ghoussoub, Selected new aspects of the calculus of variations in the large,Bull. Amer. Math. Soc.39(2) (2002) 207–265.MR1886088 (2003b:35048)

9. A. Kristaly, H. Lisei and C. Varga, Multiple solutions forp-Laplacian type equations,NonlinearAnal.61 (2008) 1375–1381.MR2381678 (2009h:35139)

10. M. Mihailescu, Existence and multiplicity of weak solutions for a class of degenerate nonlin-ear elliptic equations,Bound. Value Probl.2006 (2006) Art. ID 41295, 17 pp.MR2211397(2006j:35084)

11. P. de Napoli and M. C. Mariani, Mountain pass solutions to equations ofp-Laplacian type,Nonlinear Anal.54 (2003) 1205–1219.MR1995926 (2004e:35065)

12. Q.-A. Ngo and H. Q. Toan, Existence of solutions for a resonant problem under Landesman-Lazer conditions,Electron. J. Differential Equations2008(2008) No. 98, 10 pp.MR2430895

(2009h:35149)13. Q.-A. Ngo and H. Q. Toan, Some remarks on a class of nonuniformly elliptic equations of

p-Laplacian type,Acta Appl. Math.(2008); doi:10.1007/s10440–008-9291–6.MR249740514. B. Ricceri, On a three critical points theorem,Arch. Math.75 (2000) 220–226.MR1780585

(2001h:49012)15. B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems,Math.

Comput. Modelling32 (2000) 1485–1494.MR1800671 (2001j:35220)16. H. Q. Toan and Q.-A. Ngo, Multiplicity of weak solutions for a class of nonuniformly el-

liptic equations ofp-Laplacian type,Nonlinear Anal.70 (2009) 1536–1546.MR2483577(2010b:35127)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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MR2497405 (2011b:35128)35J60(47J30 58E05)

Ngo, Qu´oc Anh (VN-VNU) ; Toan, Hoang Quoc[Hoang Qu´oc Toan] (VN-VNU)Some remarks on a class of nonuniformly elliptic equations ofp-Laplacian type. (Englishsummary)Acta Appl. Math.106(2009),no. 2,229–239.

Summary: “This paper deals with the existence of weak solutions inW 10 (Ω) to a class of elliptic

problems of the form−div(a(x,∇u)) = λ1|u|p−2u + g(u)−h

in a bounded domainΩ of RN . Herea satisfies

|a(x, ξ)| ≤ c0(h0(x) +h1(x)|ξ|p−1)

for all ξ ∈ RN , a.e.x ∈ Ω, h0 ∈ Lp

p−1 (Ω), h1 ∈ L1loc(Ω), h1(x) ≥ 1 for a.e.x in Ω; λ1 is the first

eigenvalue for−∆p on Ω with zero Dirichlet boundary condition andg, h satisfy some suitableconditions.”

References

1. Arcoya, D., Orsina, L.: Landesman–Lazer conditions and quasilinear elliptic equations. Non-linear Anal.28,1623–1632 (1997)MR1430505 (97m:35060)

2. Bouchala, J., Drabek, P.: Strong resonance for some quasilinear elliptic equations. J. Math.Anal. Appl.245,7–19 (2000)MR1756573 (2001c:35031)

3. Costa, D.G.: An Invitation to Variational Methods in Differential Equations. Birkhauser, Basel

(2007)MR2321283 (2008k:58033)4. Duc, D.M., Vu, N.T.: Nonuniformly elliptic equations ofp-Laplacian type. Nonlinear Anal.61,

1483–1495 (2005)MR2135821 (2005k:35111)5. Mihailescu, M.: Existence and multiplicity of weak solutions for a class of degenerate nonlinear

elliptic equations. Bound. Value Probl.41295,1–17 (2006)MR2211397 (2006j:35084)6. De Napoli, P., Mariani, M.C.: Mountain pass solutions to equations ofp-Laplacian type. Non-

linear Anal.54,1205–1219 (2003)MR1995926 (2004e:35065)7. Tang, C.-L.: Solvability for two-point boundary value problems. J. Math. Anal. Appl.216,

368–374 (1997)MR1487269 (98i:34041)8. Tang, C.-L.: Solvability of the forced Duffing equation at resonance. J. Math. Anal. Appl.219,

110–124 (1998)MR1607110 (98j:34079)9. Toan, H.Q., Ngo, Q.-A.: Multiplicity of weak solutions for a class of nonuniformly elliptic equa-

tions ofp-Laplacian type. Nonlinear Anal. (2008). doi:10.1016/j.na.2008.02.033MR2483577(2010b:35127)

10. Vu, N.T.: Mountain pass theorem and nonuniformly elliptic equations. Vietnam J. Math.33(4),391–408 (2005)MR2200236 (2006j:35085)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2011

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From References: 4From Reviews: 0

MR2483577 (2010b:35127)35J60(35J20 47J30)

Toan, Hoang Quoc[Hoang Qu´oc Toan] (VN-VNU-NS); Ngo, Qu´oc-Anh (VN-VNU-NS)Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p-Laplaciantype. (English summary)Nonlinear Anal.70 (2009),no. 4,1536–1546.

Summary: “This paper deals with the multiplicity of weak solutions inW 10 (Ω) to a class of

nonuniformly elliptic equations of the form

−div(a(x,∇u)) = h(x)|u|r−1u + g(x)|u|s−1u

in a bounded domainΩ of RN . Herea satisfies|a(x, ξ)|5 c0(h0(x)+h1(x)|ξ|p−1) for all ξ ∈ RN ,a.e.x ∈ Ω, h0 ∈ L

pp−1 (Ω), h1 ∈ L1

loc(Ω), h1(x) = 1 for a.e.x in Ω, 1 < r < p− 1 < s < (Np−N + p)/(N − p).”

References

1. R.A. Adams, Sobolev Spaces, Academic Press, London, 1975.MR0450957 (56 #9247)2. G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems

with p-Laplacian, Protugaliae Math. 58 (2001) 340–377.MR1856715 (2002j:35116)3. M. Struwe, Variational Methods, Springer, New York, 1996.

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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Citations

From References: 1From Reviews: 0

MR2471390 (2009j:26033)26D15 (39A10)

Ngo, Qu´oc Anh (VN-VNU-NS); Liu, Wenjun [Liu, Wen Jun 3] (PRC-NUIST-MP)A sharp Gruss type inequality on time scales and application to the sharp Ostrowski-Grussinequality. (English summary)Commun. Math. Anal.6 (2009),no. 2,33–41.

In this paper the authors prove sharp Gruss and weighted Gruss type inequalities for general timescales. A sharp Gruss type inequality is used in proving the sharp Ostrowski-Gruss inequality ontime scales. The authors obtain well-known and new results by applying the sharp Gruss typeinequality in different time scalesT = R, Z andqN0.

Reviewed byHassan Ahmed Agwo

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MR2479697 (2010b:35097)35J47(35A01 35J91)

Cardoulis, Laure (F-TOUL-CR); Ngo, Qu´oc Anh (VN-VNU-MIS) ;Toan, Hoang Quoc[Hoang Qu´oc Toan] (VN-VNU-MIS)Existence of non-negative solutions for cooperative elliptic systems involving Schrodingeroperators in the whole space. (English summary)Rostock. Math. Kolloq.No. 63(2008), 63–77.

Summary: “In this paper, we obtain some new results on the existence of non-negative solutionsto systems of the form

(−∆ + qi)ui = µimiui +n∑

j=1,j 6=1

aijuj + fi(x, u1, . . . , un)

in RN , i = 1, . . . , n, where eachqi is a positive potential satisfying

lim|x|→+∞

qi(x) = +∞,

eachmi is a bounded positive weight, eachaij, i 6= j, is a bounded non-negative weight and eachµi is a real parameter. Depending upon some hypotheses onfi, i = 1, . . . , n, we obtain new resultsby using sub- and super-solution methods and the Schauder fixed-point theorem.”

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MR2465495 (2009j:26030)26D15 (39A10 65D30)

Liu, Wenjun [Liu, Wen Jun 3] (PRC-NUIST-MP);Qu´oc Anh Ngo [Ngo, Qu´oc Anh] (VN-VNU-NS); Chen, Wenbing(PRC-NUIST-MP)A perturbed Ostrowski-type inequality on time scales fork points for functions whose secondderivatives are bounded. (English summary)J. Inequal. Appl.2008,Art. ID 597241, 12pp.

In this paper the authors derive an Ostrowski-type inequality fork points on arbitrary time scales.This result generalizes the corresponding continuous time result from [A. Sofo and S. S. Dragomir,Turkish J. Math.25(2001), no. 3, 379–412;MR1864141 (2002h:41048)]. The discrete time resultis new. The main result is also an extension of this type of result from the number of pointsk =2 in [Q. A. Ngo and W. J. Liu, “An Ostrowski type inequality on time scales for functions whosesecond derivatives are bounded”, inInequality theory and applications. Vol. 6, to appear]. A verysimilar result to the one in this paper can be found in [Appl. Math. Comput.203(2008), no. 2, 754–760;MR2458991], where Liu and Ngo considered the boundedness of the first-order derivativef∆

compared to the boundedness of the second-order derivativef∆∆ in the present paper. The methodof proof is the same as the method for the continuous time case. Examples of special perturbedOstrowski-type inequalities on time scales (as consequences of the main result) are given in thelast section of the paper.

Reviewed byRomanSimon Hilscher

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From References: 3From Reviews: 1

MR2458991 26D15 (39A10)

Liu, Wenjun [Liu, Wen Jun 3] (PRC-NUIST-MP); Ngo, Quoc-Anh (VN-VNU-NS)A generalization of Ostrowski inequality on time scales fork points. (English summary)Appl. Math. Comput.203(2008),no. 2,754–760.

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From References: 0From Reviews: 0

MR2458264 (2009g:26028)26D15 (26A06)

Ngo, Qu´oc-Anh (VN-VNU-NS); Qi, Feng (PRC-HNPU-MIT);Thu, Ninh Van [Ninh Van Thu] (VN-VNU-NS)New generalizations of an integral inequality. (English summary)Real Anal. Exchange33 (2008),no. 2,471–474.

The authors prove the following result: Letf : [a, b] → [0,∞) be a continuous function andg : [a, b]→ [0,∞) be a continuous non-decreasing function such that

(1)∫ b

x

f(t) dt≥∫ b

x

g(t) dt

for all x ∈ [a, b]. Then

(2)∫ b

a

h(f(t)) dt≥∫ b

a

h(g(t)) dt

holds for every convex functionh such thath′ ≥ 0 andh′ is integrable on[0,∞). In the case inwhich f is as above andg : [a, b]→ [0,∞) is a continuous non-increasing function such that thereverse of (1) holds, inequality (2) is valid for every convex functionh such thath′ ≤ 0 andh′ isintegrable on[0,∞).

Reviewed byStamatis Koumandos

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From References: 2From Reviews: 0

MR2430895 (2009h:35149)35J60(35J20 47J30 58E05)

Ngo, Qu´oc Anh (VN-VNU-NS); Toan, Hoang Quoc[Hoang Qu´oc Toan] (VN-VNU-NS)Existence of solutions for a resonant problem under Landesman-Lazer conditions. (Englishsummary)Electron. J. Differential Equations2008,No. 98, 10pp.

Summary: “This article shows the existence of weak solutions inW 10 (Ω) for a class of Dirichlet

problems of the form−div(a(x,∇u)) = λ1|u|p−2u + f(x, u)−h

in a bounded domainΩ⊂ RN . Herea : Ω×RN → RN satisfies

|a(x, ξ)| ≤ c0(h0(x) +h1(x)|ξ|p−1),

for all ξ ∈ RN , a.e.x ∈ Ω, in whichh0 ∈ Lp

p−1 (Ω), h1 ∈ L1loc(Ω) satisfiesh1(x)≥ 1 for a.e.x ∈ Ω,

λ1 is the first eigenvalue for−∆p onΩ with zero Dirichlet boundary condition andf andh satisfysome suitable conditions.”

References

1. R. A. Adams and J. J. F. Fournier;Sobolev spaces, Academic Press, London, 2003.MR2424078(2009e:46025)

2. A. Ambrosetti and Rabinowitz; Dual variational methods in critical point theory and applica-tions,J. Funct. Anal.14 (1973), 349–381.MR0370183 (51 #6412)

3. A. Anane and J. P. Gossez; Strongly nonlinear elliptic problems near resonance: a variationalapproach,Comm. Partial Diff. Eqns.15 (1990), 1141–1159.MR1070239 (91h:35121)

4. D. Arcoya and L. Orsina; Landesman-Lazer conditions and quasilinear elliptic equations,Nonlinear Analysis28 (1997), 1623–1632.MR1430505 (97m:35060)

5. L. Boccardo, P. Drabek and M, Kucera; Landesman-Lazer conditions for strongly nonlinearboundary value problem,Comment. Math. Univ. Carolinae30 (1989), 411–427.MR1031859(90k:35101)

6. D. G. Costa;An invitation to variational methods in differential equations, Birkhauser, 2007.MR2321283 (2008k:58033)

7. G. Dinca, P. Jebelean, J. Mawhin; Variational and topological methods for Dirichlet problemswith p-Laplacian,Protugaliae Math.58 (2001), 340–377.MR1856715 (2002j:35116)

8. D. M. Duc; Nonlinear singular elliptic equations,J. London Math. Soc.(2)40(1989), 420–440.MR1053612 (91g:35107)

9. D. M. Duc and N. T. Vu; non-uniformly elliptic equations ofp-Laplacian type,NonlinearAnalysis61 (2005), 1483–1495.MR2135821 (2005k:35111)

10. E. M. Landesman and A. C. Lazer; Nonlinear perturbations of linear elliptic problems atresonance,J. Math. Mech.19 (1970), 609–623.MR0267269 (42 #2171)

11. Mihai Mihailescu; Existence and multiplicity of weak solutions for a class of degenerate nonlin-ear elliptic equations,Boundary Value ProblemsArticle ID 41295(2006), 1–17.MR2211397(2006j:35084)

12. P. De Napoli and M. C. Mariani; Mountain pass solutions to equations ofp-Laplacian type,

Nonlinear Analysis54 (2003), 1205–1219.MR1995926 (2004e:35065)13. Q.-A. Ngo and H. Q. Toan, Some remarks on a class of non-uniformly elliptic equations of

p-Laplacian type,submitted.14. P. H. Rabinowitz;Minimax methods in critical point theory with applications to differential

equations, A.M.S., 1986.MR0845785 (87j:58024)15. H.Q. Toan and Q.-A.Ngo; Multiplicity of weak solutions for a class of non-uniformly ellip-

tic equations ofp-Laplacian type,Nonlinear Analysis (2008), doi:10.1016/j.na.2008.02.033.MR2483577 (2010b:35127)

16. N.T. Vu; Mountain pass theorem and non-uniformly elliptic equations,Vietnam J. of Math.33:4 (2005), 391–408.MR2200236 (2006j:35085)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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Article

Citations

From References: 0From Reviews: 0

MR2320614 (2008e:26023)26D15Quo′c Anh Ngo [Ngo, Qu´oc Anh] (VN-VNU-MIS) ; Pham Huy Tung (5-MELB-MS)Notes on an open problem of F. Qi and Y. Chen and J. Kimball. (English summary)JIPAM. J. Inequal. Pure Appl. Math.8 (2007),no. 2,Article 41, 4pp. (electronic).

In this paper the authors give an answer to an open problem proposed by F. Qi [JIPAM. J. Inequal.Pure Appl. Math.1 (2000), no. 2, Article 19, 3 pp. (electronic);MR1786406 (2001e:26036)]and Y. Chen and J. F. Kimball [JIPAM. J. Inequal. Pure Appl. Math.7 (2006), no. 1, Article 4,4 pp. (electronic);MR2217167 (2007b:26032)]. In Theorem 2.2 the authors prove the following:Let n be a positive integer. Supposef(x) has a continuous derivative of then-th order on theinterval[a, b] such thatf (i)(a) = 0, where0≤ i≤ n− 1, andf (n)(x)≥ n!

(n+1)(n−1) . Then∫ b

a

fn+2 (x) dx≥(∫ b

a

f (x) dx

)n+1

.

In this theorem the authors use a technique that was introduced by Qi [op. cit.].Reviewed byBozidar Tepes

c© Copyright American Mathematical Society 2008, 2011

Article

Citations

From References: 0From Reviews: 10

MR2268574 (2007g:26035)26D15Ngo, Quo′c Anh (VN-VNU-MIS) ; Thang, Du Duc[Du Duc Thang] (VN-VNU-MIS) ;Dat, Tran Tat [Tran Tat Dat] (VN-VNU-MIS) ;Tuan, Dang Anh [Dang Anh Tuan] (VN-VNU-MIS)Notes on an integral inequality. (English summary)JIPAM. J. Inequal. Pure Appl. Math.7 (2006),no. 4,Article 120, 5pp. (electronic).

From the introduction: “Letf(x) be a continuous function on[0, 1] satisfying

(1.1)∫ 1

x

f(t)dt≥ 1−x2

2, ∀x ∈ [0, 1].

First, we consider the integral inequality (1.2) below. Lemma 1.1. If (1.1) holds then we have

(1.2)∫ 1

0[f(x)]2dx≥

∫ 1

0xf(x)dx.

“The aim of this paper is to generalize (1.2) in order to obtain some new integral inequalities.In the first part of this paper, we will prove Lemma 1.1 and present some preliminary results. Ourmain results are Theorem 2.1 and Theorem 2.2, which will be proved in Section 2; and Theorem3.2 and Theorem 3.3, which will be proved in Section 3. Finally, an open question is proposed.”

c© Copyright American Mathematical Society 2007, 2011

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Citations

From References: 0From Reviews: 0

MR2181273 (2006e:35077)35J55(35J50 35J60)

Ngo, Quo′c Anh (VN-VNU-MIS)An application of the Lyapunov-Schmidt method to semilinear elliptic problems. (Englishsummary)Electron. J. Differential Equations2005,No. 129, 11pp. (electronic).

Summary: “In this paper we consider the existence of nonzero solutions for the undecouplingelliptic system

−∆u = λu + δv + f(u, v),−∆v = θu + γv + g(u, v),

on a bounded domain ofRn, with zero Dirichlet boundary conditions. We use the Lyapunov-Schmidt method and the fixed-point principle.”

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elliptic boundary value problems at resonance,Indiana Univ. Math. J.25 (1976), 933–944.MR0427825 (55 #855)

2. A. Ambrosetti and G. Mancini, Existence and multiplicity results for nonlinear elliptic problemswith linear part at resonance. The case of the simple eigenvalue,J. Diff. Equations28 (1978),220–245.MR0492839 (58 #11901)

3. A. Anane,Etude des valeurs propers et la resonance pour le operateurP -Laplacian, Ph. D.Thesis, Univ. Bruxelles, 1988.

4. K. J. Brown, Spatially inhomogeneous steady-state solutions for systems of equations describ-ing interacting populations,J. Math. Anal. Appl.95(1983), 251–264.MR0710432 (85a:35032)

5. H. Berestycki and D. De Figueiredo, Double resonance in semilinear elliptic problems,Comm.Partial Diff. Equations6 (1981), 91–120.MR0597753 (82f:35078)

6. P. Bartolo and V. Benci and D. Fortunato, Abstract critical point theorems and applications tosome nonlinear problems with strong resonance,Nonlinear Analysis T.M.A.7 (1983), no. 9,981–1012.MR0713209 (85c:58028)

7. A. Capozzi and D. Lupo and S. Solimini, On the existence of a nontrivial solution to nonlinearproblem at resonance,Nonlinear Analysis T.M.A.13 (1989), no. 2, 151–163.MR0979038(90c:35086)

8. L. Cesari and R. Kannan, Qualitative study of a class of nonlinear boundary value problems atresonance,J. Diff. Equations56 (1985), 63–81.MR0772121 (86g:47083)

9. R. Chiappinelli and J. Mawhin and R. Nugari, Bifurcation from infinity and multiple solutionsfor some Dirichlet problems with unbounded nonlinearities,Nonlinear Analysis T.M.A, inpress.

10. S. Chow and J. Hale,Methods of Bifurcation Theory, Springer-Verlag, 1982.MR0660633(84e:58019)

11. D. Costa and Magalh aes, A variational approach to subquadratic perturbations of ellipticsystems,J. Diff. Equations111(1994), no. 1, 103–122.MR1280617 (95f:35082)

12. D. De Figueiredo and R. Chiappinelli, Bifurcation from infinity and multiple solutions for anelliptic system,Differential and Integral Equations6 (1993), no. 4, 757–771.MR1222299(94e:35025)

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