Ereú, J.; Pérez, L.; Pineda, E.; Rodríguez, L. A study of solutions of some nonlinear integral equations in the space of functions of bounded second variation in the sense of Shiba. (English) Zbl 1495.45002 Mediterr. J. Math. 19, No. 4, Paper No. 151, 23 p. (2022). The authors study the existence and uniqueness of solutions for nonlinear Hammerstein equations of the form \[ x(t)=g(t)+\lambda\int_{I}K(t,s)f(x(s))ds,\quad \lambda\in I,\ t\in I=[0,b],\ b>0, \tag{1} \] Volterra-Hammerstein equations of the form \[ x(t)=g(t)+\int_{0}^{t}K(t,s)f(x(s))ds,\quad t\in[0,b],\tag{2} \] and Volterra equations of the form \[ x(t)=g(t)+\int^{t}_{a}K(t,s)f(x(s))ds,\quad t\in[a,b],\ a<b, \tag{3} \] in the space of functions of bounded second variation in the sense of Shiba, viz, \(\bigwedge^{2}_{p}BV([a,b])\). For the proofs of the theorems for Equations (1) and (2), the authors use the Banach contraction principle, while for equations of type (3) the Leray-Schauder alternative theorem is used. A section is devoted to applications, where a nonlinear Hammerstein-Volterra integral equation is solved by means of numerical methods. Reviewer: Narahari Parhi (Bhubaneswar) MSC: 45G10 Other nonlinear integral equations 45D05 Volterra integral equations 47N20 Applications of operator theory to differential and integral equations 26A45 Functions of bounded variation, generalizations 65R20 Numerical methods for integral equations Keywords:integral equation; bounded second variation in the sense of Shiba; Volterra-Hammerstein equation; existence and uniqueness PDFBibTeX XMLCite \textit{J. Ereú} et al., Mediterr. J. Math. 19, No. 4, Paper No. 151, 23 p. (2022; Zbl 1495.45002) Full Text: DOI References: [1] Lardy, LJ, A variation of Nystrom’s method for Hammerstein equations, J. Integral Equ., 3, 143-60 (1981) · Zbl 0471.65096 [2] Tricomi, F., Integral Equations (1985), New York: Dover Publications, New York [3] Brunner, H., Implicitly linear collocation methods for nonlinear Volterra equations, Appl. Numer. Math., 9, 235-247 (1992) · Zbl 0761.65103 · doi:10.1016/0168-9274(92)90018-9 [4] Kumar, K.; Sloan, IH, A new collocation-type method for Hammerstein integral equations, Math. Comp., 48, 178, 585-593 (1987) · Zbl 0616.65142 · doi:10.1090/S0025-5718-1987-0878692-4 [5] Ereú, J.; Giménez, J.; Pérez, L., On solutions of nonlinear integral equations in the space of functions of Shiba-bounded variation, Appl. Math. Inf, 14, 393-404 (2020) · doi:10.18576/amis/140305 [6] Giménez, J.; Merentes, N.; Pineda, E.; Rodriguez, L., Uniformly bounded superposition operators in the space of second bounded variation functions in the sense of Shiba, Pubicaciones en Ciencias y tecnología, 10, 49-58 (2016) [7] Pérez, L.; Pineda, E.; Rodriguez, L., Some results on the space of bounded second variation functions in the sense of Shiba, Adv. Pure Math., 10, 245-258 (2020) · doi:10.4236/apm.2020.105015 [8] K. Maleknejad, P. Torabi, Appication of fixed point method for solving nonlinear Volterra-Hammestein integral equation, U.P.B. Sci. Bull., Series Avol.74, (2012) 45-56 · Zbl 1249.65287 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.