Existence results for superlinear semipositone BVP’s
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- by V. Anuradha, D. D. Hai and R. Shivaji PDF
- Proc. Amer. Math. Soc. 124 (1996), 757-763 Request permission
Abstract:
We consider the existence of positive solutions to the BVP \begin{gather*} (p(t)u’)’ + \lambda f(t,u)=0,\qquad r<t<R,\ au(r)-bp(r)u’(r)=0,\ cu(R) +dp(R)u’(R)=0, \end{gather*} where $\lambda >0$. Our results extend some of the existing literature on superlinear semipositone problems and singular BVPs. Our proofs are quite simple and are based on fixed point theorems in a cone.References
- V. Anuradha and R. Shivaji, A quadrature method for classes of multi-parameter two point boundary value problems, Applicable Anal. (to appear).
- John V. Baxley, Some singular nonlinear boundary value problems, SIAM J. Math. Anal. 22 (1991), no. 2, 463–479. MR 1084968, DOI 10.1137/0522030
- 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. MR 943804, DOI 10.1017/S0308210500014670
- 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. MR 1084967, DOI 10.1137/0522029
- Xabier Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 70 (1987), no. 1, 69–92. MR 904816, DOI 10.1016/0022-0396(87)90169-0
- 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. MR 997609, DOI 10.1016/0022-0396(89)90113-7
- G. B. Gustafson and K. Schmitt, Method of nonlinear analysis in the theory of differential equations, Lecture notes, University of Utah, 1975.
- M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
- Steven D. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal. 3 (1979), no. 6, 897–904. MR 548961, DOI 10.1016/0362-546X(79)90057-9
Additional Information
- V. Anuradha
- Affiliation: Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, Arkansas 72212
- D. D. Hai
- Email: dang@math.msstate.edu
- R. Shivaji
- Affiliation: Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762
- MR Author ID: 160980
- Email: shivaji@math.msstate.edu
- Received by editor(s): June 10, 1994
- Additional Notes: The third author was partially supported by NSF Grants DMS-9215027. This author also thanks the CDSNS at Georgia Institute of Technology, Atlanta, GA, for providing a Visiting Research Scientist position (Fall 1993) during which time this work was completed
- Communicated by: Hal Smith
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 757-763
- MSC (1991): Primary 34B15
- DOI: https://doi.org/10.1090/S0002-9939-96-03256-X
- MathSciNet review: 1317029