Bulletin of the Iranian Mathematical Society https://doi.org/10.1007/s41980-020-00448-9 ORIGINAL PAPER Modified Hybrid Projection Methods with SP Iterations for Quasi-Nonexpansive Multivalued Mappings in Hilbert Spaces Watcharaporn Chaolamjiak 1 · Damrongsak Yambangwai 1 · Hasanen A. Hammad 2 Received: 25 September 2019 / Revised: 18 June 2020 / Accepted: 1 August 2020 © Iranian Mathematical Society 2020 Abstract In this paper, we present a modified SP iteration with the inertial technical term for three quasi-nonexpansive multivalued mappings in a Hilbert space. We then obtain weak convergence theorem under suitable conditions. The strong convergence theorems are given using CQ and shrinking projection methods with our modified iteration. Finally, we test some numerical experiments to illustrate that our inertial forward–backward method with the inertial technique term has a more effective convergence than that of the standard forward–backward method and Halpern algorithm. Keywords Weak and strong convergence · Common fixed point · Quasi-nonexpansive multivalued mappings · SP iteration · Inertial technical term Mathematics Subject Classification 54H25 · 47H10 1 Introduction Let H be a real Hilbert space with inner product 〈., .〉 and norm ‖.‖, respectively. Let C be a nonempty closed and convex subset of H . A subset C ⊂ H is said to be proximinal if, for each x ∈ H , there exists y ∈ C , such that: Communicated by Fatemeh Panjeh Ali Beik. B Hasanen A. Hammad h_elmagd89@yahoo.com Watcharaporn Chaolamjiak watcharaporn.ch@up.ac.th Damrongsak Yambangwai damrongsak.ya@up.ac.th 1 School of Science, University of Phayao, Phayao 56000, Thailand 2 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt 123