doi: 10.3934/dcdsb.2018256

Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics

1. 

Department of Mathematics, Ryerson University, Toronto, Ontario, Canada M5B 2K3

2. 

School of Mathematical Sciences and Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China

* Corresponding author: Kunquan Lan

Received  February 2017 Revised  June 2018 Published  October 2018

Fund Project: The first author was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant no. 250187-2013 and 135752-2018, and the Shanghai Key Laboratory of Contemporary Applied Mathematics, and the second author was supported in part by the NNSF of China under grants no. 11322111 and no. 61773125

Lyapunov type inequalities for (linear or nonlinear) Hammerstein integral equations are established and applied to second order differential equations (ODEs) with general separated boundary conditions. These new inequalities provide necessary conditions for the Hammerstein integral equations and these boundary value problems to have nonzero nonnegative solutions. As applications of these inequalities for nonlinear ODEs, we obtain extinction criteria and optimal locations of favorable habitats for populations inhabiting one dimensional heterogeneous environments governed by reaction-diffusion equations with spatially varying growth rates and external forcing.

Citation: Kunquan Lan, Wei Lin. Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018256
References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., bf 18 (1976), 620–709 doi: 10.1137/1018114.

[2]

J. F. BonderJ. P. Pinasco and A. M. Salort, A Lyapunoy type inequality for indefinite weights and eigenvalue homogenization, Proc. Amer. Math. Soc., 144 (2015), 1669-1680. doi: 10.1090/proc/12871.

[3]

G. Borg, On a Lyapunov criterion of stability, Amer. J. Math., 71 (1949), 67-70.

[4]

R. C. Brown and D. B. Hinton, Opial's inequality and oscillation of second-order equations, Proc. Amer. Math. Soc., 125 (1997), 1123-1129. doi: 10.1090/S0002-9939-97-03907-5.

[5]

A. CanadaJ. A. Montero and S. Villeges, Lynapunov inequalities for partial differential equations, J. Funct. Anal., 237 (2006), 176-193. doi: 10.1016/j.jfa.2005.12.011.

[6]

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. doi: 10.1017/S030821050001876X.

[7]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155.

[8]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weigts: Population models in disrupted environments Ⅱ, SIAM J. Appl. Math., 22 (1991), 1043-1064. doi: 10.1137/0522068.

[9]

A. M. Das and A. S. Vatsala, Green function for n-n boundary value problem and an analogue of Hartman's result, J. Math. Anal. Appl., 51 (1975), 670-677. doi: 10.1016/0022-247X(75)90117-1.

[10]

P. L. de Nápoli and J. P. Pinasco, Lyapunov-type inequalities for partial differential equations, J. Funct. Anal., 270 (2016), 1995-2018. doi: 10.1016/j.jfa.2016.01.006.

[11]

P. L. de Nápoli and J. P. Pinasco, A Lyapunov inequality for monotone quasilinear operators, Differential Integral Equations, 18 (2005), 1193-1200.

[12]

W. DingH. FinottiS. LenhartY. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Anal. Real World Applications, 11 (2010), 688-704. doi: 10.1016/j.nonrwa.2009.01.015.

[13]

R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058-1063. doi: 10.1016/j.jmaa.2013.11.025.

[14]

A. M. Fink and D. F. St. Mary, On an inequality of Nehari, Proc. Amer. Math. Soc., 21 (1969), 640-642. doi: 10.1090/S0002-9939-1969-0240388-0.

[15]

C. Ha, Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type, Proc. Amer. Math. Soc., 126 (1998), 3507-3511. doi: 10.1090/S0002-9939-98-05010-2.

[16]

P. Hartman, Ordinary Differential Equations, Boston, 1982.

[17]

M. HintermüllerC. Y. Kao and A. Laurain, Principal eigenvalue minimization for an elliptic problem with indefinite weight and Robin boundary conditions, Appl. Math. Optim., 65 (2012), 111-146. doi: 10.1007/s00245-011-9153-x.

[18]

D. B. Hinton, A criterion for n-n oscillation in differential equations of order 2n, Proc. Amer. Math. Soc., 19 (1968), 511-518. doi: 10.2307/2035825.

[19]

M. Jleli and B. Samet, Lyapunov-type inequality for fractional boundary value problems, Electron. J. Differ. Eqn., (2015), 1-11.

[20]

K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc., 63 (2001), 690-704. doi: 10.1112/S002461070100206X.

[21]

K. Q. Lan and W. Lin, Population models with quasi-constant-yield harvest rates, Math. Biosci. Eng., 14 (2017), 467-490. doi: 10.3934/mbe.2017029.

[22]

K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421. doi: 10.1006/jdeq.1998.3475.

[23]

K. Q. Lan and G. C. Yang, Optimal constants for two point boundary value problems, Discrete Contin. Dyn. Syst. Suppl., (2007), 624-633.

[24]

D. LudwigD. C. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217-258. doi: 10.1007/BF00276310.

[25]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976.

[26]

M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett., 6 (2003), 843-849. doi: 10.1046/j.1461-0248.2003.00493.x.

[27]

L. Roques and M. D. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math., 68 (2007), 133-153. doi: 10.1137/060676994.

[28]

L. Roques and F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., 210 (2007), 34-59. doi: 10.1016/j.mbs.2007.05.007.

[29]

P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol., 14 (1999), 401-405. doi: 10.1016/S0169-5347(99)01684-5.

[30]

P. A. StephensW. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190. doi: 10.2307/3547011.

[31]

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), 91-115.

[32]

A. Wintner, On the non-existence of conjugate points, Amer. J. Math., 73 (1951), 368-380. doi: 10.2307/2372182.

[33]

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), 665-683.

[34]

X. J. Yang, On inequalities of Lyapunov type, Appl. Math. Comput., 134 (2003), 293-300. doi: 10.1016/S0096-3003(01)00283-1.

[35]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for a class of even-order linear differential equations, Appl. Math. Comput., 245 (2014), 145-151. doi: 10.1016/j.amc.2014.07.085.

[36]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequalities for a class of higher-order linear differential equations, Appl. lett., 34 (2014), 86-89. doi: 10.1016/j.aml.2013.11.001.

[37]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for a class of linear differential systems, Appl. Math. Comput., 219 (2012), 1805-1812. doi: 10.1016/j.amc.2012.08.019.

[38]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for quasilinear systems, Appl. Math. Comput., 219 (2012), 1670-1673. doi: 10.1016/j.amc.2012.08.007.

[39]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for a class of quasilinear systems, Appl. Math. Model., 53 (2011), 1162-1166. doi: 10.1016/j.mcm.2010.11.083.

[40]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for a class of odd-order differential equations, J. Comput. Appl. Math., 234 (2010), 2962-2968. doi: 10.1016/j.cam.2010.04.008.

[41]

X. J. Yang and K. Lo, New Lyapunov-type inequalities for a class of even-order linear differential equations, Math. Nach., 288 (2015), 1910-1915. doi: 10.1002/mana.201400050.

[42]

X. J. Yang and K. Lo, Lyapunov-type inequalities for a class of higher-order linear differential equations with anti-periodic boundary conditions, Appl. lett., 34 (2014), 33-36. doi: 10.1016/j.aml.2014.03.009.

[43]

X. J. Yang and K. Lo, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884-3890. doi: 10.1016/j.amc.2009.11.032.

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., bf 18 (1976), 620–709 doi: 10.1137/1018114.

[2]

J. F. BonderJ. P. Pinasco and A. M. Salort, A Lyapunoy type inequality for indefinite weights and eigenvalue homogenization, Proc. Amer. Math. Soc., 144 (2015), 1669-1680. doi: 10.1090/proc/12871.

[3]

G. Borg, On a Lyapunov criterion of stability, Amer. J. Math., 71 (1949), 67-70.

[4]

R. C. Brown and D. B. Hinton, Opial's inequality and oscillation of second-order equations, Proc. Amer. Math. Soc., 125 (1997), 1123-1129. doi: 10.1090/S0002-9939-97-03907-5.

[5]

A. CanadaJ. A. Montero and S. Villeges, Lynapunov inequalities for partial differential equations, J. Funct. Anal., 237 (2006), 176-193. doi: 10.1016/j.jfa.2005.12.011.

[6]

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. doi: 10.1017/S030821050001876X.

[7]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155.

[8]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weigts: Population models in disrupted environments Ⅱ, SIAM J. Appl. Math., 22 (1991), 1043-1064. doi: 10.1137/0522068.

[9]

A. M. Das and A. S. Vatsala, Green function for n-n boundary value problem and an analogue of Hartman's result, J. Math. Anal. Appl., 51 (1975), 670-677. doi: 10.1016/0022-247X(75)90117-1.

[10]

P. L. de Nápoli and J. P. Pinasco, Lyapunov-type inequalities for partial differential equations, J. Funct. Anal., 270 (2016), 1995-2018. doi: 10.1016/j.jfa.2016.01.006.

[11]

P. L. de Nápoli and J. P. Pinasco, A Lyapunov inequality for monotone quasilinear operators, Differential Integral Equations, 18 (2005), 1193-1200.

[12]

W. DingH. FinottiS. LenhartY. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Anal. Real World Applications, 11 (2010), 688-704. doi: 10.1016/j.nonrwa.2009.01.015.

[13]

R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058-1063. doi: 10.1016/j.jmaa.2013.11.025.

[14]

A. M. Fink and D. F. St. Mary, On an inequality of Nehari, Proc. Amer. Math. Soc., 21 (1969), 640-642. doi: 10.1090/S0002-9939-1969-0240388-0.

[15]

C. Ha, Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type, Proc. Amer. Math. Soc., 126 (1998), 3507-3511. doi: 10.1090/S0002-9939-98-05010-2.

[16]

P. Hartman, Ordinary Differential Equations, Boston, 1982.

[17]

M. HintermüllerC. Y. Kao and A. Laurain, Principal eigenvalue minimization for an elliptic problem with indefinite weight and Robin boundary conditions, Appl. Math. Optim., 65 (2012), 111-146. doi: 10.1007/s00245-011-9153-x.

[18]

D. B. Hinton, A criterion for n-n oscillation in differential equations of order 2n, Proc. Amer. Math. Soc., 19 (1968), 511-518. doi: 10.2307/2035825.

[19]

M. Jleli and B. Samet, Lyapunov-type inequality for fractional boundary value problems, Electron. J. Differ. Eqn., (2015), 1-11.

[20]

K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc., 63 (2001), 690-704. doi: 10.1112/S002461070100206X.

[21]

K. Q. Lan and W. Lin, Population models with quasi-constant-yield harvest rates, Math. Biosci. Eng., 14 (2017), 467-490. doi: 10.3934/mbe.2017029.

[22]

K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421. doi: 10.1006/jdeq.1998.3475.

[23]

K. Q. Lan and G. C. Yang, Optimal constants for two point boundary value problems, Discrete Contin. Dyn. Syst. Suppl., (2007), 624-633.

[24]

D. LudwigD. C. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217-258. doi: 10.1007/BF00276310.

[25]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976.

[26]

M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett., 6 (2003), 843-849. doi: 10.1046/j.1461-0248.2003.00493.x.

[27]

L. Roques and M. D. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math., 68 (2007), 133-153. doi: 10.1137/060676994.

[28]

L. Roques and F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., 210 (2007), 34-59. doi: 10.1016/j.mbs.2007.05.007.

[29]

P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol., 14 (1999), 401-405. doi: 10.1016/S0169-5347(99)01684-5.

[30]

P. A. StephensW. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190. doi: 10.2307/3547011.

[31]

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), 91-115.

[32]

A. Wintner, On the non-existence of conjugate points, Amer. J. Math., 73 (1951), 368-380. doi: 10.2307/2372182.

[33]

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), 665-683.

[34]

X. J. Yang, On inequalities of Lyapunov type, Appl. Math. Comput., 134 (2003), 293-300. doi: 10.1016/S0096-3003(01)00283-1.

[35]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for a class of even-order linear differential equations, Appl. Math. Comput., 245 (2014), 145-151. doi: 10.1016/j.amc.2014.07.085.

[36]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequalities for a class of higher-order linear differential equations, Appl. lett., 34 (2014), 86-89. doi: 10.1016/j.aml.2013.11.001.

[37]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for a class of linear differential systems, Appl. Math. Comput., 219 (2012), 1805-1812. doi: 10.1016/j.amc.2012.08.019.

[38]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for quasilinear systems, Appl. Math. Comput., 219 (2012), 1670-1673. doi: 10.1016/j.amc.2012.08.007.

[39]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for a class of quasilinear systems, Appl. Math. Model., 53 (2011), 1162-1166. doi: 10.1016/j.mcm.2010.11.083.

[40]

X. J. YangY. Kim and K. Lo, Lyapunov-type inequality for a class of odd-order differential equations, J. Comput. Appl. Math., 234 (2010), 2962-2968. doi: 10.1016/j.cam.2010.04.008.

[41]

X. J. Yang and K. Lo, New Lyapunov-type inequalities for a class of even-order linear differential equations, Math. Nach., 288 (2015), 1910-1915. doi: 10.1002/mana.201400050.

[42]

X. J. Yang and K. Lo, Lyapunov-type inequalities for a class of higher-order linear differential equations with anti-periodic boundary conditions, Appl. lett., 34 (2014), 33-36. doi: 10.1016/j.aml.2014.03.009.

[43]

X. J. Yang and K. Lo, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884-3890. doi: 10.1016/j.amc.2009.11.032.

Figure 1.  (a) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{1}$, where $k = 3$, $a = 0$, $\beta = 4$, $\delta = 1$, and $T = 0.8$. (b) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{2}$, where $k = 3$, $a = 0$, $\beta = 2.5$, $\delta = 2$, and $T = 0.8$. (c) The monotonically decreasing curve of $\eta_{1}$, as defined in (4.12), with respect to the variable $a\in[0, 1-T] = [0, 0.2]$ for $(\beta, \delta, T)$ in different area. Here, the curve, plotted by squares, corresponds to the parameters in (a) and the curve, plotted by dots, to the parameters in (b)
Figure 2.  (a) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{3}$, where $k = 3$, $a = 1-T$, $\beta = 2.5$, $\delta = 2$, $T = 0.3$, and $a_{1}(T) = 0.1$. (b) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{4}$, where $k = 3$, $a = 1-T$, $\beta = \delta = 3$, $T = 0.3$, and $a_{1}(T) = \frac{1-T}{2} = 0.35$. (c) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{5}$, where $k = 3$, $a = 1-T$, $\beta = 3$, $\delta = 3.5$, $T = 0.3$, and $a_{1}(T) = 0.6$. (d) The unimodal curve of $\eta_{1}$, as defined in (4.12), with respect to the variable $a\in[0, 1-T] = [0, 0.7]$ for $(\beta, \delta, T)$ in different area. Here, the curve, plotted by the dash line, corresponds to the parameters in (a), the curve, plotted by the solid line, to the parameters in (b), and the curve, plotted by the dotted line, to the parameters in (c)
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