On generalized $\Phi$-strongly monotone mappings and algorithms for the solution of equations of Hammerstein type

Document Type : Research Paper


1 Institute for Systems Science & KZN e-Skills CoLab‎, ‎Durban University of Technology‎, ‎Durban‎, ‎South Africa

2 School of Mathematics‎, ‎Statistics and Computer Science‎, ‎University of KwaZulu-Natal‎, ‎Durban‎, ‎South Africa‎


‎In this paper‎, ‎we consider the class of generalized $\Phi$-strongly monotone mappings and the methods of approximating a solution of equations of Hammerstein type‎. ‎Auxiliary mapping is defined for nonlinear integral equations of Hammerstein type‎. ‎The auxiliary mapping is the composition of bounded generalized $\Phi$-strongly monotone mappings which satisfy the range condition‎. ‎Suitable conditions are imposed to obtain the boundedness and to show that the auxiliary mapping is a generalized $\Phi$-strongly which satisfies the range condition‎. ‎A sequence is constructed and it is shown that it converges strongly to a solution of equations of Hammerstein type‎. ‎The results in this paper improve and extend some recent corresponding results on the approximation of a solution of equations of Hammerstein type‎.


[1] M.O. Aibinu and O.T. Mewomo, Algorithm for Zeros of monotone maps in Banach spaces, Proc. Southern Africa Math. Sci. Assoc. Annual Conf. 21-24 November 2016, University of Pretoria, South Africa, 2017, pp. 35–44.
[2] M. O. Aibinu and O.T. Mewomo, Strong convergence theorems for strongly monotone mappings in Banach spaces, Bol. Soc. Paranaense Mat. 39(1) (2021) 169–187.
[3] M.O. Aibinu, S.C. Thakur and M. Moyo, Algorithm for solutions of nonlinear equations of strongly monotone type and applications to convex minimization and variational inequality problems, Abstr. Appl. Anal. 2020 (2020) Article ID: 6579720.
[4] Ya. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Appl. Math., vol. 178, pp 15–50. Marcel Dekker, New York, 1996.
[5] Y. Alber and I. Ryazantseva, Nonlinear III Posed Problems of Monotone Type, Springer, London, 2006.
[6] H. Brezis and F E. Browder, Some new results about Hammerstein equations, Bull. Amer. Math. Soc. 80 (1974) 567–572.
[7] H. Brezis and F.E. Browder, Existence theorems for nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 81 (1975) 73–78.
[8] H. Brezis and F.E. Browder, Nonlinear integral equations and system of Hammerstein type, Adv. Math. 18 (1975) 115–147.
[9] F.E. Browder and C.P. Gupta, Monotone operators and nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 75 (1969) 1347–1353.
[10] R.Sh. Chepanovich, Nonlinear Hammerstein equations and fixed points, Publ. Inst. Math. (Beograd) (N.S.) 35(49) (1984) 119–123.
[11] C.E. Chidume and N. Djitte, Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators, Abstr. Appl. Anal. 2012 (2012) Article ID 681348.
[12] C.E. Chidume and K.O. Idu, Approximation of zeros of bounded maximal monotone mappings, solutions of Hammerstein integral equations and convex minimization problems, Fixed Point Theory Appl. 2016(97) (2016).
[13] C.E. Chidume, M.O. Nnakwe and A. Adamu, A strong convergence theorem for generalized Φ-strongly monotone maps with applications, Fixed Point Theory Appl. 2019(11) (2019) 19 pages.
[14] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers Group, Dordrecht, 1990.
[15] D. G. De Figueiredo and C. P. Gupta, On the variational method for the existence of solutions of nonlinear equations of Hammerstein type, Proc. Amer. Math. Soc. 40 (1973) 470–476.
[16] N. Djitte, J.T. Mendy and T.M.M. Sow, Computation of zeros of monotone type mappings: on Chidume’s open problem, J. Aust. Math. Soc. 108 (2) (2020) 278–288.
[17] V. Dolezale, Monotone Operators and Applications in Control and Network Theory, Studies in Automation and Control, Elsevier Scientific, New York, USA, 2, (1979).
[18] A. Hammerstein, Nichtlineare integralgleichungen nebst anwendungen, Acta Math. 54(1) (1930) 117–176.
[19] S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in Banach a space, SIAM J. Optim. 13 (2002) 938–945.
[20] K. Kido, Strong convergence of resolvents of monotone operators in Banach spaces, Proc. Am. Math. Soc. 103(3) (1988) 755–7588.
[21] B.T. Kien, The normalized duality mappings and two related characteristic properties of a uniformly convex Banach space, Acta Math. Vietnam. 27(1) (2002) 53–67.
[22] F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces, SIAM J. Optim. 19(2) (2008) 824–835.
[23] S.Y. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2004(1) (2004) 37–47.
[24] D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Editura Academiae, Bucharest, Romania, 1978.
[25] W. Takahashi, Convex Analysis and Approximation Fixed Points, Yokohama Publishers, Yokohama, Japanese, 2000.
[26] W. Takahashi, Fixed Point Theory and its Applications: In Nonlinear Functional Analysis, Yokohama Publishers, 2000.
[27] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16(12) (1991) 1127–1138.
[28] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. II 66(1) (2002) 240–256.
[29] Z.B. Xu and G.F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl. 157 (1991) 189–210.
[30] S. Yekini, Convergence results of forward-backwards algorithms for the sum of monotone operators in Banach spaces, Results Math. 74:138, (2019) 24 pages.
[31] C. Zalinescu, On uniformly convex functions, J. Math. Anal. Appl. 95 (1983) 344–374.
[32] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Part II: Monotone operators, Springer, New York, USA, 1985.
Volume 12, Issue 1
May 2021
Pages 615-632
  • Receive Date: 19 December 2018
  • Revise Date: 17 October 2019
  • Accept Date: 15 January 2020