Results in Mathematics

, Volume 72, Issue 1–2, pp 369–383 | Cite as

Generalized Hammerstein Equations and Applications

  • John GraefEmail author
  • Lingju Kong
  • Feliz Minhós


In this paper the authors study the Hammerstein generalized integral equation
$$\begin{aligned} u(t)=\int _{0}^{1}k(t,s)\text { }g(s)\text { }f(s,u(s),u^{\prime }(s),\dots ,u^{(m)}(s))\,ds, \end{aligned}$$
where \(k:[0,1]^{2}\rightarrow {\mathbb {R}}\) are kernel functions, \(m\ge 1\), \(g:[0,1] \rightarrow [0,\infty )\), and \(f:[0,1]\times {\mathbb {R}}^{m+1} \rightarrow [0,\infty )\) is a \(L^{\infty }-\)Carathéodory function. The existence of solutions of integral equations has been studied in concrete and abstract cases, by different methods and techniques. However, in the existing literature, the nonlinearity depends only on the unknown function. This paper is one of a very few to consider equations having discontinuous nonlinearities that depend on the derivatives of the unknown function and having discontinuous kernels functions that have discontinuities in the partial derivatives with respect to their first variable. Our approach is based on the Krasnosel’skiĭ–Guo compression/expansion theorem on cones and it can be applied to boundary value problems of arbitrary order \(n>m\). The last two sections of the paper contain an application to a third order nonlinear boundary value problem and a concrete example.


Hammerstein integral equation Krasnosel’skiĭ–Guo theorem Boundary value problems Discontinuous kernels Nonlinearities depending on derivatives 

Mathematics Subject Classification

Primary 45G10 Secondary 34B15 47H30 


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  1. 1.
    Agarwal, R.P., O’Regan, D., Wong, P.J.Y.: Constant-sign Solutions of Systems of Integral Equations. Springer, Cham (2013)CrossRefzbMATHGoogle Scholar
  2. 2.
    Amann, H.: Existence theorems for equations of Hammerstein type. Appl. Anal. 1, 385–397 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aziz, W., Leiva, H., Merentes, N.: Solutions of Hammerstein equations in the space \(BV(I_{a}^{b})\). Quaest. Math. 37(3), 359–370 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benmzemai, A., Graef, J.R., Kong, L.: Positive solutions for abstract Hammerstein equations and applications. Commun. Math. Anal. 16, 47–65 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brezis, H., Browder, F.: Existence theorems for nonlinear integral equations of Hammerstein type. Bull. AMS 81, 73–78 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cabada, A., Infante, G., Tojo, F.A.F.: Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications. Topol. Methods Nonlinear Anal. (2016, to appear)Google Scholar
  7. 7.
    Cabada, A., Infante, G., Tojo, F.A.F.: Nontrivial solutions of perturbed Hammerstein integral equations with reflections. Bound. Value Probl. 2013, 86 (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cheng, X., Zhang, Z.: Existence of positive solutions to systems of nonlinear integral or differential equations. Topol. Methods Nonlinear Anal. 34, 267–277 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chidume, C.E., Chidume, C.O., Minjibir, M.: A new method for proving existence theorems for abstract Hammerstein equations. Abstr. Appl. Anal. 2015, Art. ID 627260Google Scholar
  10. 10.
    Chidume, C.E., Shehu, Y.: Iterative approximation of solutions of generalized equations of Hammerstein type. Fixed Point Theory 15, 427–440 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Franco, D., Infante, G., O’Regan, D.: Nontrivial solutions in abstract cones for Hammerstein integral systems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14, 837–850 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988)zbMATHGoogle Scholar
  13. 13.
    Hammerstein, A.: Nichtlineare Integralgleichungen nebst Anwendungen. Acta Math. 54, 117–176 (1930)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Henderson, J., Luca, R.: Positive solutions for systems of second-order integral boundary value problems. Electron. J. Qual. Theory Differ. Equ. 70, 21 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Infante, G., Pietramala, P.: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations. Nonlinear Anal. 71, 1301–1310 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Infante, G., Webb, J.R.L.: Nonzero solutions of Hammerstein integral equations with discontinuous kernels. J. Math. Anal. Appl. 272, 30–42 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)Google Scholar
  18. 18.
    Lan, K.Q.: Multiple positive solutions of Hammerstein integral equations with singularities. Differ. Equ. Dyn. Syst. 8, 175–195 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lan, K.Q., Lin, W.: Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli. Nonlinear Anal. 74, 7184–7197 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lan, K.Q., Lin, W.: Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations. J. Lond. Math. Soc. 83, 449–469 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li-Jun, G., Jian-Ping, S., Ya-Hong, Z.: Existence of positive solutions for nonlinear third-order three-point boundary value problems. Nonlinear Anal. 68, 3151–3158 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Minhós, F., de Sousa, R.: On the solvability of third-order three point systems of differential equations with dependence on the first derivative (2016, to appear)Google Scholar
  23. 23.
    Precup, R.: Componentwise compression-expansion conditions for systems of nonlinear operator equations and applications. In: Mathematical Models in Engineering, Biology and Medicine, pp. 284–293, AIP Conf. Proc., vol. 1124. Amer. Inst. Phys., Melville (2009)Google Scholar
  24. 24.
    Yang, Z.: Positive solutions for a system of nonlinear Hammerstein integral equations and applications. Appl. Math. Comput. 218, 11138–11150 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Yang, Z., O’Regan, D.: Positive solvability of systems of nonlinear Hammerstein integral equations. J. Math. Anal. Appl. 311, 600–614 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yang, Z., Zhang, Z.: Positive solutions for a system of nonlinear singular Hammerstein integral equations via nonnegative matrices and applications. Positivity 16, 783–800 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Tennessee at ChattanoogaChattanoogaUSA
  2. 2.Departamento de Matemática, Escola de Ciências e Tecnologia Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação AvançadaUniversidade de ÉvoraÉvoraPortugal

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