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January  2020, 19(1): 221-240. doi: 10.3934/cpaa.2020012

Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

2. 

Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

3. 

Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China

Received  November 2018 Revised  March 2019 Published  July 2019

Fund Project: This work is supported by NSF of China under 11325107 and 11471148

We are concerned with the existence and multiplicity of component-wise positive solutions for nonlinear system of Hammerstein integral equations with the weighted functions and the associated nonlinear eigenvalue problem. Our discussions are based on the product formula of fixed point index on product cones and the fixed point index theory. Moreover, we establish the existence and multiplicity of component-wise positive solutions for the associated nonlinear systems of second-order ordinary differential equations under the mixed boundary value conditions.

Citation: Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012
References:
[1]

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

[2]

X. Cheng, Existence of positive solutions for a class of second-order ordinary differential systems, Nonlinear Anal., 69 (2008), 3042–3049. doi: 10.1016/j.na.2007.08.074. Google Scholar

[3]

X. Cheng and Z. Feng, Existence and multiplicity of positive solutions to systems of nonlinear Hammerstein integral equations, Electron. J. Differential Equations, 2019 (52) (2019), 1–16. doi: 10.1016/j.na.2007.08.074. Google Scholar

[4]

X. Cheng and H. Lü, Multiplicity of positive solutions for a $(p_1, p_2)$-Laplacian system and its applications, Nonlinear Anal. RWA, 13 (2012), 2375–2390. doi: 10.1016/j.nonrwa.2012.02.004. Google Scholar

[5]

X. Cheng and Z. Zhang, Existence of positive solutions to systems of nonlinear integral or differential equations, Topol. Meth. Nonlinear Anal., 34 (2009), 267–277. doi: 10.12775/TMNA.2009.042. Google Scholar

[6]

X. Cheng and C. Zhong, Existence of positive solutions for a second-order ordinary differential system, J. Math. Anal. Appl., 312 (2005), 14–23. doi: 10.1016/j.jmaa.2005.03.016. Google Scholar

[7]

D. Franco, G. Infante and D. O'Regan, Nontrivial solutions in abstract cones for Hammerstein integral systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), 837–850. Google Scholar

[8]

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. Google Scholar

[9]

D. Guo and J. Sun, Nonlinear Integral Equations (in Chinese), Shandong Press of Science and Technology, Jinan, 1987.Google Scholar

[10]

A. Hammerstein, Nichtlineare Intergralgleichungen nebst Anwendungen, Acta Math., 54 (1929), 117–176. doi: 10.1007/BF02547519. Google Scholar

[11]

G. Infante and P. Pietramala, Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations, Nonlinear Anal., 71 (2009), 1301–1310. doi: 10.1016/j.na.2008.11.095. Google Scholar

[12]

M. A. Krasnoselskii, Topological methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford, 1964.Google Scholar

[13]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk, 3 (1948), 3–95. Google Scholar

[14]

K. Q. Lan and W. Lin, Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations, J. London Math. Soc., 83 (2011), 449–469. doi: 10.1112/jlms/jdq090. Google Scholar

[15]

K. Q. Lan and W. Lin, Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli, Nonlinear Anal., 74 (2011), 7184–7197. doi: 10.1016/j.na.2011.07.038. Google Scholar

[16]

K. Q. Lan and W. Lin, Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics, Discrete Contin. Dyn. Syst. Ser. B, doi: 10.3934/dcdsb.2018256 doi: 10.3934/dcdsb.2018256.. Google Scholar

[17]

J. Sun and X. Liu, Computation for topological degree and its applications, J. Math. Anal. Appl., 202 (1996), 785–796. doi: 10.1006/jmaa.1996.0347. Google Scholar

[18]

G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958. Google Scholar

[19]

Z. Yang and Z. Zhang, Positive solutions for a system of nonlinear singular Hammerstein integral equations via nonnegative matrices and applications, Positivity, 16 (2012), 783–800. doi: 10.1007/s11117-011-0146-4. Google Scholar

[20]

Z. Zhang, Existence of nontrivial solutions for superlinear systems of integral equations and applications, Acta Math. Sinica, 15 (1999), 153–162. doi: 10.1007/BF02720490. Google Scholar

[21]

C. Zhong, X. Fan and W. Chen, An Introduction to Nonlinear Functional Analysis (in Chinese), Lanzhou University Press, Lanzhou, 1998.Google Scholar

show all references

References:
[1]

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

[2]

X. Cheng, Existence of positive solutions for a class of second-order ordinary differential systems, Nonlinear Anal., 69 (2008), 3042–3049. doi: 10.1016/j.na.2007.08.074. Google Scholar

[3]

X. Cheng and Z. Feng, Existence and multiplicity of positive solutions to systems of nonlinear Hammerstein integral equations, Electron. J. Differential Equations, 2019 (52) (2019), 1–16. doi: 10.1016/j.na.2007.08.074. Google Scholar

[4]

X. Cheng and H. Lü, Multiplicity of positive solutions for a $(p_1, p_2)$-Laplacian system and its applications, Nonlinear Anal. RWA, 13 (2012), 2375–2390. doi: 10.1016/j.nonrwa.2012.02.004. Google Scholar

[5]

X. Cheng and Z. Zhang, Existence of positive solutions to systems of nonlinear integral or differential equations, Topol. Meth. Nonlinear Anal., 34 (2009), 267–277. doi: 10.12775/TMNA.2009.042. Google Scholar

[6]

X. Cheng and C. Zhong, Existence of positive solutions for a second-order ordinary differential system, J. Math. Anal. Appl., 312 (2005), 14–23. doi: 10.1016/j.jmaa.2005.03.016. Google Scholar

[7]

D. Franco, G. Infante and D. O'Regan, Nontrivial solutions in abstract cones for Hammerstein integral systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), 837–850. Google Scholar

[8]

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. Google Scholar

[9]

D. Guo and J. Sun, Nonlinear Integral Equations (in Chinese), Shandong Press of Science and Technology, Jinan, 1987.Google Scholar

[10]

A. Hammerstein, Nichtlineare Intergralgleichungen nebst Anwendungen, Acta Math., 54 (1929), 117–176. doi: 10.1007/BF02547519. Google Scholar

[11]

G. Infante and P. Pietramala, Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations, Nonlinear Anal., 71 (2009), 1301–1310. doi: 10.1016/j.na.2008.11.095. Google Scholar

[12]

M. A. Krasnoselskii, Topological methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford, 1964.Google Scholar

[13]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk, 3 (1948), 3–95. Google Scholar

[14]

K. Q. Lan and W. Lin, Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations, J. London Math. Soc., 83 (2011), 449–469. doi: 10.1112/jlms/jdq090. Google Scholar

[15]

K. Q. Lan and W. Lin, Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli, Nonlinear Anal., 74 (2011), 7184–7197. doi: 10.1016/j.na.2011.07.038. Google Scholar

[16]

K. Q. Lan and W. Lin, Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics, Discrete Contin. Dyn. Syst. Ser. B, doi: 10.3934/dcdsb.2018256 doi: 10.3934/dcdsb.2018256.. Google Scholar

[17]

J. Sun and X. Liu, Computation for topological degree and its applications, J. Math. Anal. Appl., 202 (1996), 785–796. doi: 10.1006/jmaa.1996.0347. Google Scholar

[18]

G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958. Google Scholar

[19]

Z. Yang and Z. Zhang, Positive solutions for a system of nonlinear singular Hammerstein integral equations via nonnegative matrices and applications, Positivity, 16 (2012), 783–800. doi: 10.1007/s11117-011-0146-4. Google Scholar

[20]

Z. Zhang, Existence of nontrivial solutions for superlinear systems of integral equations and applications, Acta Math. Sinica, 15 (1999), 153–162. doi: 10.1007/BF02720490. Google Scholar

[21]

C. Zhong, X. Fan and W. Chen, An Introduction to Nonlinear Functional Analysis (in Chinese), Lanzhou University Press, Lanzhou, 1998.Google Scholar

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