Article Contents
Article Contents

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

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.

Mathematics Subject Classification: 45G15, 37C25, 47H30.

 Citation:

•  [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18 (1976), 620–709. doi: 10.1137/1018114. [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. [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. [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. [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. [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. [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. [8] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. [9] D. Guo and J. Sun, Nonlinear Integral Equations (in Chinese), Shandong Press of Science and Technology, Jinan, 1987. [10] A. Hammerstein, Nichtlineare Intergralgleichungen nebst Anwendungen, Acta Math., 54 (1929), 117–176. doi: 10.1007/BF02547519. [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. [12] M. A. Krasnoselskii, Topological methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford, 1964. [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. [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. [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. [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.. [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. [18] G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958. [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. [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. [21] C. Zhong, X. Fan and W. Chen, An Introduction to Nonlinear Functional Analysis (in Chinese), Lanzhou University Press, Lanzhou, 1998.