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

Published online by Cambridge University Press:  09 December 2011

J. R. L. WEBB  and
M. ZIMA
J. R. L. WEBB
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, UK e-mail: jeffrey.webb@glasgow.ac.uk
M. ZIMA
Affiliation:
Institute of Mathematics, University of Rzeszów, 35-959 Rzeszów, Poland e-mail: mzima@univ.rzeszow.pl
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Abstract

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We study the existence of positive solutions for equations of the form where 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.

Keywords

Primary 34B10secondary 34B1534B18
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 225 - 240
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
3.Cabada, A. and Enguiça, R. R., Positive solutions of fourth-order problems with clamped beam boundary conditions, Nonlinear Anal. 74 (2011), 3112–3122.CrossRefGoogle Scholar
4.Cid, J. A., Franco, D. and Minhos, F., Positive fixed points and fourth-order equations, Bull. Lond. Math. Soc. 41 (2009), 72–78.CrossRefGoogle Scholar
5.Feng, W. and Webb, J. R. L., Solvability of three-point boundary value problems at resonance, Nonlinear Anal. 30 (1997), 3227–3238.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
7.Han, X., Positive solutions for a three-point boundary value problem at resonance, J. Math. Anal. Appl. 336 (2007), 556–568.CrossRefGoogle Scholar
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.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.Infante, G. and Zima, M., Positive solutions of multi-point boundary value problems at resonance, Nonlinear Anal. 69 (2008), 2458–2465.CrossRefGoogle Scholar
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.Kosmatov, N., Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear Anal. 68 (2008), 2158–2171.CrossRefGoogle Scholar
13.Krasnosel'skiĭ, M. A., Positive solutions of operator equations (P. Noordhoff, Groningen, Netherlands, 1964).Google Scholar
14.Krasnosel'skiĭ, M. A. and Zabreĭko, P. P., Geometrical methods of nonlinear analysis (Springer-Verlag, Berlin, 1984).CrossRefGoogle Scholar
15.Lan, K. Q. and Webb, J. R. L., Positive solutions of semilinear differential equations with singularities, J. Differ. Equ. 148 (1998), 407–421.CrossRefGoogle Scholar
16.Liu, B., Solvability of multi-point boundary value problem at resonance, IV, Appl. Math. Comput. 143 (2003), 275–299.Google Scholar
17.Ma, R., Existence results of a m-point boundary value problem at resonance, J. Math. Anal. Appl. 294 (2004), 147–157.CrossRefGoogle Scholar
18.Ma, R. and Chen, T., Existence of positive solutions of fourth-order problems with integral boundary conditions, Bound. Value Probl. Art. ID 297578 (2011), 17 pp.CrossRefGoogle Scholar
19.Ma, R. and Yang, Y., Existence result for a singular nonlinear boundary value problem at resonance, Nonlinear Anal. 68 (2008), 671–680.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
21.O'Regan, D. and Zima, M., Leggett-Williams norm-type theorems for coincidences, Arch. Math. 87 (2006), 233–244.CrossRefGoogle Scholar
22.Seneta, E., Non-negative matrices and Markov chains (revised reprint of the second (1981) edn.), Springer Series in Statistics (Springer, New York, 2006).CrossRefGoogle Scholar
23.Varga, R. S., Matrix iterative analysis (Prentice-Hall, New Jersey, 1963) (second printing).Google Scholar
24.Webb, J. R. L., Remarks on non-local boundary value problems at resonance, Appl. Math. Comput. 216 (2010), 497–500.Google Scholar
25.Webb, J. R. L., Solutions of nonlinear equations in cones and positive linear operators, J. Lond. Math. Soc. 82 (2) (2010), 420–436.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.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.CrossRefGoogle Scholar