Korean J. Math. Vol. 25 No. 1 (2017) pp.37-43
DOI: https://doi.org/10.11568/kjm.2017.25.1.37

Positive solutions of superlinear and sublinear boundary value problems

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Juan A. Gatica
Yun-Ho Kim


We study the existence of positive solutions of second order nonlinear separated boundary value problems of superlinear as well as sublinear type without imposing monotonicity restrictions on the problem. The type of problem investigated cannot be analyzed using the linearization about the trivial solution because either it does not exist (the sublinear case) or is trivial (the superlinear case). The results follow from a known fixed point theorem by noticing that the concavity of the solutions provides an important condition for the applicability of the fixed point result.

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[1] J. A. Gatica, V. Oliker and P. Waltman, Iterative procedures for nonlinear second order boundary value problems, Ann. Mat. Pura Appl. 157 (1990), 1–25. Google Scholar

[2] J. A. Gatica and H. L. Smith, Fixed point techniques in a cone with applications, J. Math. Anal. Appl. 61 (1977) (1), 58–71. Google Scholar

[3] G. B. Gustafson and K. Schmitt, Methods of Nonlinear Analysis in the Theory of Differential Equations, Lecture Notes, Department of Mathematics, University of Utah, 1975. Google Scholar

[4] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441–467. Google Scholar

[5] C. D. Luning and W. L. Perry, Convergence of Berriman’s Iterative Method for Some Emden-Fowler Equations, J. Math. Phys. 22 (1981) (8), 1591–1595. Google Scholar

[6] C. D. Luning and W. L. Perry, An iterative technique for solution of the Thomas- Fermi equation utilizing a nonlinear eigenvalue problem, Quart. Appl. Math. 35 (1977/78) (2), 257–268. Google Scholar

[7] J. S. W. Wong, On the generalized Emden-Fowler equation, SIAM Rev. 17 (1975), 339–360. Google Scholar