Appl. Math. Inf. Sci. 8, No. 5, 2113-2118 (2014) 2113 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080504 Split Algorithms for New Implicit Feasibility Null-Point Problems Abdellatif Moudafi 1 and Muhammad Aslam Noor 2,∗ 1 Universit´ e des Antilles et de Guyane, Ceregmia-DSI 97200 Schoelcher, Martinique, France 2 Mathematics Department, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan Received: 7 Aug. 2013, Revised: 4 Nov. 2013, Accepted: 5 Nov. 2013 Published online: 1 Sep. 2014 Abstract: Inspired by the very recent work by Noor and Noor [9] and given a closed convex set-valued mapping C, we propose a split algorithm for solving the problem of finding an element x ∗ which is a zero of a given maximal monotone operator T such that its image, Ax ∗ , under a linear operator, A, is in a closed convex set C(x ∗ ). Then, we present two strong convergence results and state some examples as applications. The ideas and techniques of this paper may motivate the readers to discover some novel and innovative applications of the implicit split feasibility problems in various branches of pure and applied sciences. Keywords: Fixed-point, monotone operator, implicit feasibility problem. 2010 AMS Subject Classification: Primary, 49J53, 65K10; Secondary, 49M37, 90C25. 1 Introduction and preliminaries Throughout, H is a Hilbert space, 〈·, ·〉 denotes the inner product and ‖·‖ stands for the corresponding norm. The split feasibility problem (SFP) has received much attention due to its applications in image denoising, signal processing and image reconstruction, with particular progress in intensity-modulated therapy. For a complete and exhaustive study on algorithms for solving convex feasibility problem, including comments about their applications and an excellent bibliography see, for example [1] and for split convex feasibility problem see, for instance, the excellent paper [5] and the references therein. Inspired by the idea developed in [9], our interest in this paper is on the study of the convergence of an algorithm for solving a Implicit Feasibility Null-point Problem, i.e., the case where the constrained set, instead of being fixed, is a set-valued mapping. Besides being a more general case, it also has many applications, see for example [2]). Note that by taking A = I , the identity mapping and T = ∂φ , the subdifferential of a strongly convex proper lower semi-continuous function φ , problem (4) reduces to finding a common element in argminφ and the implicit convex set C(x). To be in a position to apply the fixed-point Banach principle and by observing that the fixed-point reformulation of the problem considered in [9] involves the projection operator over convex sets and that the techniques are strongly based on its properties which do not depend on any parameter in contrast to the resolvent and proximal mappings. In this paper, we introduce and consider a new implicit feasibility null-point problems. We suggest and analyze some split algorithms for solving this new feasibility problem. Strong convergence of the proposed algorithm is discussed under some suitable conditions. Some applications of this new problem are given. Comparison of the proposed methods with other techniques is an interesting problem for future research. To begin with, let us recall that the split feasibility problem (SFP) is to find a point x ∈ C such that Ax ∈ Q, (1) where C is a closed convex subset of a Hilbert space H 1 , Q is a closed convex subset of a Hilbert space H 2 , and A : H 1 → H 2 is a bounded linear operator. Assuming that the (SFP) has a solution, it is no hard to see that x ∈ C solves (1) if and only if it solves to fixed-point equation x = P C ( I − γ A ∗ (I − P Q )A ) x, x ∈ C, (2) where P C and P Q are the (orthogonal) projection onto C and Q, respectively, γ > 0 is any positive constant and A ∗ ∗ Corresponding author e-mail: noormaslam@hotmail.com c  2014 NSP Natural Sciences Publishing Cor.