MULTIPLE POSITIVE SOLUTIONS OF RESONANT AND NON-RESONANT NON-LOCAL FOURTH-ORDER BOUNDARY VALUE PROBLEMS

WEBB, J. R. L. ; ZIMA, M.

Glasgow mathematical journal, Tome 54 (2012) no. 1, pp. 225-240

Résumé

We study the existence of positive solutions for equations of the formwhere 0 < ω < π, subject to various non-local boundary conditions defined in terms of the Riemann–Stieltjes integrals. We prove the existence and multiplicity of positive solutions for these boundary value problems in both resonant and non-resonant cases. We discuss the resonant case by making a shift and considering an equivalent non-resonant problem.

Détail

DOI : 10.1017/S0017089511000590

Mots-clés : Primary 34B10, secondary 34B15, 34B18

Bibliographie

[1] 1.Bai, Z., Li, W. and Ge, W., Existence and multiplicity of solutions for four-point boundary value problems at resonance, Nonlinear Anal. 60 (2005), 1151–1162. Google Scholar | DOI

[2] 2.Cabada, A., Cid, J. A. and Sanchez, L., Positivity and lower and upper solutions for fourth-order boundary value problems, Nonlinear Anal. 67 (2007), 1599–1612. Google Scholar | DOI

[3] 3.Cabada, A. and Enguiça, R. R., Positive solutions of fourth-order problems with clamped beam boundary conditions, Nonlinear Anal. 74 (2011), 3112–3122. Google Scholar | DOI

[4] 4.Cid, J. A., Franco, D. and Minhos, F., Positive fixed points and fourth-order equations, Bull. Lond. Math. Soc. 41 (2009), 72–78. Google Scholar | DOI

[5] 5.Feng, W. and Webb, J. R. L., Solvability of three-point boundary value problems at resonance, Nonlinear Anal. 30 (1997), 3227–3238. Google Scholar | DOI

[6] 6.Gupta, C. P., Existence theorems for a second-order m-point boundary value problem at resonance, Int. J. Math. Math. Sci. 18 (1995), 705–710. Google Scholar | DOI

[7] 7.Han, X., Positive solutions for a three-point boundary value problem at resonance, J. Math. Anal. Appl. 336 (2007), 556–568. Google Scholar | DOI

[8] 8.Infante, G. and Pietramala, P., A cantilever equation with nonlinear boundary conditions, Electron. J. Qual. Theory Differ. Equ. Special Edition I (15) (2009), 14 pp. Google Scholar

[9] 9.Infante, G., Pietramala, P. and Zima, M., Positive solutions for a class of nonlocal impulsive BVPs via fixed point index, Topol. Methods Nonlinear Anal. 36 (2010), 263–284. Google Scholar

[10] 10.Infante, G. and Zima, M., Positive solutions of multi-point boundary value problems at resonance, Nonlinear Anal. 69 (2008), 2458–2465. Google Scholar | DOI

[11] 11.Kosmatov, N., A symmetric solution of a multipoint boundary value problem at resonance, Abstr. Appl. Anal. Art. ID 54121 (2006), 11 pp. Google Scholar

[12] 12.Kosmatov, N., Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear Anal. 68 (2008), 2158–2171. Google Scholar | DOI

[13] 13.Krasnosel'skiĭ, M. A., Positive solutions of operator equations (P. Noordhoff, Groningen, Netherlands, 1964). Google Scholar

[14] 14.Krasnosel'skiĭ, M. A. and Zabreĭko, P. P., Geometrical methods of nonlinear analysis (Springer-Verlag, Berlin, 1984). Google Scholar | DOI

[15] 15.Lan, K. Q. and Webb, J. R. L., Positive solutions of semilinear differential equations with singularities, J. Differ. Equ. 148 (1998), 407–421. Google Scholar | DOI

[16] 16.Liu, B., Solvability of multi-point boundary value problem at resonance, IV, Appl. Math. Comput. 143 (2003), 275–299. Google Scholar

[17] 17.Ma, R., Existence results of a m-point boundary value problem at resonance, J. Math. Anal. Appl. 294 (2004), 147–157. Google Scholar | DOI

[18] 18.Ma, R. and Chen, T., Existence of positive solutions of fourth-order problems with integral boundary conditions, Bound. Value Probl. Art. ID (2011), 17 pp. Google Scholar | DOI

[19] 19.Ma, R. and Yang, Y., Existence result for a singular nonlinear boundary value problem at resonance, Nonlinear Anal. 68 (2008), 671–680. Google Scholar | DOI

[20] 20.Mawhin, J., Equivalence theorems for nonlinear operator equations and coincidence degree theory for mappings in locally convex topological vector spaces, J. Differ. Equ. 12 (1972), 610–636. Google Scholar | DOI

[21] 21.O'Regan, D. and Zima, M., Leggett-Williams norm-type theorems for coincidences, Arch. Math. 87 (2006), 233–244. Google Scholar | DOI

[22] 22.Seneta, E., Non-negative matrices and Markov chains (revised reprint of the second (1981) edn.), Springer Series in Statistics (Springer, New York, 2006). Google Scholar | DOI

[23] 23.Varga, R. S., Matrix iterative analysis (Prentice-Hall, New Jersey, 1963) (second printing). Google Scholar

[24] 24.Webb, J. R. L., Remarks on non-local boundary value problems at resonance, Appl. Math. Comput. 216 (2010), 497–500. Google Scholar

[25] 25.Webb, J. R. L., Solutions of nonlinear equations in cones and positive linear operators, J. Lond. Math. Soc. 82 (2) (2010), 420–436. Google Scholar | DOI

[26] 26.Webb, J. R. L. and Infante, G., Positive solutions of nonlocal boundary value problems: A unified approach, J. Lond. Math. Soc. 74 (2) (2006), 673–693. Google Scholar | DOI

[27] 27.Webb, J. R. L. and Infante, G., Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlinear Differ. Equ. Appl. (NoDEA) 15 (2008), 45–67. Google Scholar | DOI

[28] 28.Webb, J. R. L. and Infante, G., Non-local boundary value problems of arbitrary order, J. Lond. Math. Soc. 79 (2) (2009), 238–258. Google Scholar | DOI

[29] 29.Webb, J. R. L., Infante, G. and Franco, D., Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions, Proc. Roy. Soc. Edinburgh A 138 (2008), 427–446. Google Scholar | DOI

[30] 30.Webb, J. R. L. and Lan, K. Q., Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal. 27 (2006), 91–116. Google Scholar

[31] 31.Webb, J. R. L. and Zima, M., Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems, Nonlinear Anal. 71 (2009), 1369–1378. Google Scholar | DOI