A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces

Abstract

In this paper, we introduce a new viscosity-type iteration process for approximating a common solution of a finite family of split variational inclusion problem and fixed point problem. We prove that the proposed algorithm converges strongly to a common solution of a finite family of split variational inclusion problems and fixed point problem for a finite family of type-one demicontractive mappings between a Hilbert space and a Banach space. Furthermore, we applied our results to study a finite family of split convex minimization problems, and also considered a numerical experiment of our results to further illustrate its applicability. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13:759–775, 2012), Kazmi and Rizvi (Optim. Lett. 8(3):1113–1124, 2014), Moudafi (J. Optim. Theory Appl. 150:275–283, 2011), Shehu and Ogbuisi (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 110(2):503–518, 2016), Takahashi and Yao (Fixed Point Theory Appl. 2015:87, 2015), Chidume and Ezeora (Fixed Point Theory Appl. 2014:111, 2014), and a host of other important results in this direction.

Iterative solution of split equilibrium and fixed point problems in real Hilbert spaces

“…Let E be a smooth Banach space, C a nonempty subset of E, and T : C → C a mapping. Following [3], see also [22] we say that T is of type (P) if…”

A modified contraction method for solving certain class of split monotone variational inclusion problems with application

“…Therefore, SMVIP can be viewed as an important generalization of SVIP. Moreover, the SMVIP is one of the most important problems in monotone operator theory, since its study provides a simple, natural, and unified framework for a general treatment of many important mathematical problems, such as split minimization problems, split common null point problems, evolution systems, split equilibrium problems, complementary problems, monotone inclusions, systems of nonlinear equations, and others (see Ahmad et al 2005;Izuchukwu et al 2019Izuchukwu et al , 2018Jailoka and Suantai 2019;Moudafi 2011;Okeke and Izuchukwu 2019;Suantai et al 2019 and the references therein).…”