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April  2019, 24(4): 1943-1960. 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, 2019, 24 (4) : 1943-1960. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.   Google Scholar

[32]

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

[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.   Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.   Google Scholar

[32]

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

[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.   Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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