EXISTENCE OF POSITIVE SOLUTIONS TO A SECOND-ORDER DIFFERENTIAL EQUATION AT RESONANCE

TitleEXISTENCE OF POSITIVE SOLUTIONS TO A SECOND-ORDER DIFFERENTIAL EQUATION AT RESONANCE
Publication TypeJournal Article
Year of Publication2015
AuthorsKaufmann, ER
Volume19
Issue4
Start Page505
Pagination9
Date Published2015
ISSN1083-2564
AMS34B10, 34B15, 34B18
Abstract

We establish sufficient conditions for the existence of positive solutions to the multipoint boundary value $${ −u ′′ = f(t, u(t)), \ \ \ \ t ∈ (0, 1), \\ \ \ \  u(0) = u(1), \\ \ \ u ′ (0) = u ′ (η). }$$ Since the associated homogeneous boundary value problem is not invertible the problem is said to be at resonance. The main tool employed is a variant of a fixed point index theorem due to Cremins for A-proper semilinear operators defined on cones.

URLhttp://www.acadsol.eu/en/articles/19/4/3.pdf
Refereed DesignationRefereed
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REFERENCES
[1] D. Anderson, R. Avery, J. Henderson, and X. Liu, Existence of positive solutions of a second
order right focal boundary value problem, Comm. Appl. Nonlinear Anal. 18 (2011), no. 3,
41–52.
[2] C. Bai and J. Fang, Existence of positive solutions for three-point boundary value problems
at resonance, J. Math. Anal. Appl. 291 (2004), 538–549.
[3] C. T. Cremins, A fixed-point index and existence theorems for semilinear equations in cones,
Nonlinear Anal. 46 (2001), 789–806.
[4] C. J. Chyan and J. Henderson, A multiplicity result for positive solutions of right focal boundary
value problems, Commun. Appl. Anal. 5 (2001), no. 4, 485–495.
[5] J. Davis and J. Henderson, Triple positive solutions for (k, n − k) conjugate boundary value
problems, Math. Slovaca 51 (2001), no. 3, 313–320.
[6] Z. Du, Solvability of functional differential equations with multi-point boundary value problems
at resonance, Comput. Math. Appl. 55 (2008), no. 11, 2653–2661.
[7] Z. Du, G. Cai, and W. Ge, A class of third order multi-point boundary value problem, Taiwanese
J. Math. 9 (2005), no. 1, 81–94.
[8] Z. Du, X. Lin, and W. Ge, Nonlocal boundary value problem of higher order ordinary differential
equations at resonance, Rocky Mountain J. Math. 36 (2006), no. 5, 1471–1486.
[9] P. W. Eloe, J. Henderson and N. Kosmatov, Countable positive solutions of a conjugate
boundary value problem, Comm. Appl. Nonlinear Anal. 7 (2000), no. 2, 47–55.
[10] R.E. Gaines, J. Santanilla, A coincidence theorem in convex sets with applications to periodic
solutions of ordinary differential equations, Rocky Mountain J. Math. 12 (1982), 669678.
[11] C. P. Gupta, Solvability of a multi-point boundary value problem at resonance, Results Math.
28 (1995), no. 3-4, 270–276.
[12] J. Graef, J. Henderson and B. Yang, Existence and nonexistence of psoitive solutions of an
N-th order nonlocal boundary value problem, Dynamic Systems and Applications. 5 (2008),
186–191.
[13] X. Han, Positive solutions for a three-point boundary value problem at resonance, J. Math.
Anal. Appl. 336 (2007), no. 1, 556-568.
[14] J. Henderson and R. Luca, Positive solutions for singular systems of higher-order multi-point
boundary value problems, Math. Model. Anal. 18 (2013), no. 3, 309–324.
[15] J. Henderson and R. Luca, Positive solutions for systems of second-order integral boundary
value problems, Electron. J. Qual. Theory Differ. Equ. 2013, no 70, 21 pp.
[16] J. Henderson and W. Yin, Positive solutions and nonlinear eigenvalue problems for functional
differential equations, Appl. Math. Lett. 12 (1999), no. 2, 63–68.
[17] G. Infante and M. Zima, Positive solutions of multi-point boundary value problems at resonance,
Nonlinear Anal. 69 (2008), 2458–2465.
[18] W. H. Jiang, Y. P. Guo, and J. Qiu, Solvability of 2n-order m-point boundary value problem
at resonance, Appl. Math. Mech. (English Ed.) 28 (2007), no. 9, 1219–1226.
[19] E. R. Kaufmann, Existence of solutions for a nonlinear second-order equation with periodic
boundary conditions at resonance, Comm. Appl. Anal. 18 (2014), 163–174.
[20] E. R. Kaufmann, A Kolmogorov predator-prey system on a time scale, Dynamic Systems and
Applications 23 (2014), 561–574.
[21] G. L. Karakostas and P. Ch. Tsamatos, On a nonlocal boundary value problem at resonance,
J. Math. Anal. Appl. 259 (2001), no. 1, 209–218.
[22] N. Kosmatov, A multipoint boundary value problem with two critical conditions, Nonlinear
Anal. 65 (2006), 622–633.
[23] N. Kosmatov, Multi-point boundary value problems on an unbounded domain at resonance,
Nonlinear Anal. 68 (2008), 2158–2171.
[24] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, in NSF-CBMS
Regional Conference Series in Mathematics, 40. Amer. Math. Soc., Providence RI 1979.
[25] X. Ni and W. Ge, Multi-point boundary-value problems for the p-Laplacian at resonance,
Electron. J. Differential Equations 2003 (2003), no. 112, 1–7.
[26] D. O’Regan and M. Zima, Leggett-Williams norm-type theorems for coincidences, Arch. Math.
87 (2006), 233–244.
[27] W. V. Petryshyn, Using degree theory for densely defined A-proper maps in the solvability of
semilinear equations with unbounded and noninvertible linear part, Nonlinear Anal. 4 (1980),
no. 2, 259–281.
[28] W. V. Petryshyn, Existence of fixed points of positive k-set-contractive maps as consequences
of suitable boundary conditions, J. London Math. Soc. 38 (1988), no. 2, 503–512.
[29] J. Santanilla, Some coincidence theorems in wedges, cones, and convex sets, J. Math. Anal.
Appl. 105 (1985), 357–371.
[30] F. Wang and F. Zhang, Existence of n positive solutions to second-order multi-point boundary
value problem at resonance, Bull. Korean Math. Soc. 49 (2012), no. 4, 815–827.
[31] J. R. L. Webb and M. Zima, Multiple positive solutions of resonant and non-resonant nonlocal
boundary value problems, Nonlinear Anal. 71 (2009), no. 3-4, 1369–1378.
[32] H. Zhang and J. Sun, Positive solutions of third-order nonlocal boundary value problems at
resonance, Boundary Value Problems 2012 2012:102.