# zbMATH — the first resource for mathematics

Existence results for superlinear semipositone BVP’s. (English) Zbl 0857.34032
The authors consider the existence of a positive solution to the Sturm-Liouville boundary value problem $$(p(t)\cdot u')'+\lambda\cdot f(t,u)=0$$, $$r<t<R$$, $$a\cdot u(r)-b\cdot u'(r)=0$$, $$c\cdot u(R)+d\cdot u'(R)=0$$, for $$\lambda>0$$ small under the condition $\lim_{u\to\infty} {{f(t,u)}\over u}=\infty$ uniformly on a compact subinterval of $$(r,R)$$. The main results of the paper are related to the cases: $$f$$ is a positive possibly singular function, and $$f$$ is a regular optionally positive function. The proofs of the results are based on fixed point theorems in a cone.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory
Full Text:
##### References:
 [1] V. Anuradha and R. Shivaji, A quadrature method for classes of multi-parameter two point boundary value problems, Applicable Anal. (to appear). · Zbl 0838.34020 [2] John V. Baxley, Some singular nonlinear boundary value problems, SIAM J. Math. Anal. 22 (1991), no. 2, 463 – 479. · Zbl 0719.34038 · doi:10.1137/0522030 · doi.org [3] Alfonso Castro and R. Shivaji, Nonnegative solutions for a class of nonpositone problems, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 3-4, 291 – 302. · Zbl 0659.34018 · doi:10.1017/S0308210500014670 · doi.org [4] A. M. Fink, Juan A. Gatica, Gastón E. Hernández, and Paul Waltman, Approximation of solutions of singular second order boundary value problems, SIAM J. Math. Anal. 22 (1991), no. 2, 440 – 462. · Zbl 0722.34015 · doi:10.1137/0522029 · doi.org [5] Xabier Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 70 (1987), no. 1, 69 – 92. · Zbl 0651.35033 · doi:10.1016/0022-0396(87)90169-0 · doi.org [6] J. A. Gatica, Vladimir Oliker, and Paul Waltman, Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations 79 (1989), no. 1, 62 – 78. · Zbl 0685.34017 · doi:10.1016/0022-0396(89)90113-7 · doi.org [7] G. B. Gustafson and K. Schmitt, Method of nonlinear analysis in the theory of differential equations, Lecture notes, University of Utah, 1975. [8] M. A. Krasnosel$$^{\prime}$$skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. [9] Steven D. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal. 3 (1979), no. 6, 897 – 904. · Zbl 0421.34021 · doi:10.1016/0362-546X(79)90057-9 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.