Positive solutions of Neumann boundary value problems and applications to logistic type population models
DOI:
https://doi.org/10.12775/TMNA.2021.013Keywords
Neumann boundary value problem, strictly positive solution, $r$-nowhere normal-outward map, one dimensional population modelAbstract
We study the existence of nonzero nonnegative or strictly positive solutions of second order Neumann boundary value problems with nonlinearities which are allowed to take negative values via a recently established fixed point theorem for $r$-nowhere normal-outward maps in Banach spaces. As applications, we obtain results on the existence of strictly positive solutions for some models of population inhabiting one dimensional heterogeneous environments with perfect barriers, where the local rate of change in the population density changes sign.References
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev. 18 (1976), 620–709.
L. Berezansky, E. Braverman and L. Idels, Delay differential logistic equations with harvesting, Math. Comput. Model. 39 (2004), 1243–1259.
R.S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 293–318.
R.S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol. 29 (1991), 315–338.
R.S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weigts: population models in disrupted environments II, SIAM J. Appl. Math. 22 (1991), no. 4, 1043–1064.
R.S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Chichester, 2003.
X. Cheng and G. Dai, Positive solutions of sub-superlinear Sturm–Liouville problems, Appl. Math. Comput. 261 (2015), 351–359.
J. Henderson and N. Kosmatov, Positive solutions of the semipositone Neumann boundary value problem, Math. Model. Anal. 20 (2015), 578–584.
G. Infante and P. Pietramala, Non-trivial solutions of local and non-local Neumann boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 337–369.
M. A. Krasnosel’skiı̆, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964.
K.Q. Lan, Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems, Appl. Math. Comput. 154 (2004), no. 2, 531–542.
K.Q. Lan, Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems, Nonlinear Anal. 71 (2009), 5979–5993.
K.Q. Lan and W. Lin, Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics, Discrete Contin. Dyn. Syst. Ser. B. 24 (2019), no. 4, 1943–1960.
K.Q. Lan and W. Lin, Systems of elliptic boundary value problems and applications to competition models, Appl. Math. Lett. 90 (2019), 86–92.
K.Q. Lan and W. Lin, Positive solutions of elliptic boundary value problems and applications to population dynamics, J. Dynam. Differential Equations 32 (2020), no. 2, 873–894.
K.Q. Lan and J.R.L. Webb, A fixed point index for generalized inward mappings of condensing type, Trans. Amer. Math. Soc. 349 (1997), 2175–2186.
Y. Li, On the existence and nonexistence of positive solutions for nonlinear Sturm–Liouville boundary value problems, J. Math. Anal. Appl. 304 (2005), 74–86.
D. Ludwig, D.C. Aronson and H.F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol. 8 (1979), 217–258.
A.R. Miciano and R. Shivaji, Multiple positive solutions for a class of semi-positone Neumann two point boundary value problems, J. Math. Anal. Appl. 178 (1993), 102–115.
M.G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett. 6 (2003), 843–849.
M.N. Nkashama and J. Santanilla, Existence of multiple solutions for some nonlinear boundary value problems, J. Differential Equations 84 (1990), 148–164.
L. Roques and M.D. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math. 68 (2007), 133–153.
J. Santanilla, Some coincidence theorems in wedges, cones, and convex sets, J. Math. Anal. Appl. 105 (1985), 357–371.
J. Santanilla, Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential eqautions, J. Math. Anal. Appl. 126 (1987), 397–408.
J.P. Sun and W.T. Li, Multiple positive solutions to second-order Neumann boundary value problems, Appl. Math. Comput. 146 (2003), 187–194.
F. Wang, Y.J. Cui and F. Zhang, Existence and nonexistence results for second-order Neumann boundary value problem, Surv. Math. Appl. 4 (2009), 1–14.
J.R.L. Webb and K.Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal. 27 (2006), no. 1, 91–116.
G.C. Yang and K.Q. Lan, A fixed point index theory for nowhere normal-outward compact maps and applications, J. Appl. Anal. Comput. 6 (2016), np. 3, 665–683.
Q.L. Yao, Successively iterative method of nonlinear Neumann boundary value problems, Appl. Math. Comput. 217 (2011), 2301–2306.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Ziyi Cai, Kunquan Lan
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
