Appl. Math. Inf. Sci. 7, No. 4, 1547-1552 (2013) 1547 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/070439 Some New Classes of Quasi Split Feasibility Problems Muhammad Aslam Noor 1,∗ and Khalida Inayat Noor 2 1,2 Mathematics Department, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan. Received: 28 Nov. 2012, Revised: 5 Jan. 2013, Accepted: 27 Jan. 2013 Published online: 1 Jul. 2013 Abstract: In thispaper, we introduce and consider a new problem of finding u ∈ K(u) such that Au ∈ C, where K : u → K(u) is a closed convex-valued set in the real Hilbert space H 1 , C is closed convex set in the real Hilbert space H 2 respectively and A is linear bounded self-adjoint operator from H 1 and H 2 . This problem is called the quasi split feasibility problem. We show that the quasi feasibility problem is equivalent to the fixed point problem and quasi variational inequality. These s alternative equivalent formulations are used to consider the existence of a solution of the quasi split feasibility problem. Some special cases are also considered. Problems considered in this paper may open further research opportunities in these fields. Keywords: Quasi split feasibility problem, existence of a solution. 1 Introduction The split feasibility problems, introduced and studied by Censor and Elfying [4], have played a fundamental and significant part in the study of several unrelated problems. These problems arise in diverse fields of pure and applied sciences including image reconstruction, medica sciences (medical image), signal processing, image denoising and decomposition, see [1, 2, 3, 4, 5, 6, 13, 26]. It has been shown [1, 2, 3, 4, 5, 6, 13, 21, 26, 28, 29, 30] that the split feasibility problems are equivalent to fixed point problems, variational inequalities and optimization problems. These equivalent alternative formulations of the split feasibility problems have been used to study the existence of a solution as well as to develop various numerical methods. In the formulation of split feasibility problem, the underlying convex sets do not depend on the solution. This fact has motivated us to consider a class of split feasibility problem, which is called quasi split feasibility problem. We would like to emphasized that such type of quasi split feasibility problems have not been investigated up to now. It has been shown that the quasi split feasibility problems are equivalent to the fixed point problems and quasi variational inequalities. These equivalent formulations are used to study the existence of a solution of the quasi split feasibility problem. This result is new and original. Several special cases are also discussed. Our results continue to hold for these cases. Some iterative methods for finding the approximate solutions of the quasi split feasibility problems are suggested. It is expected that the ideas and techniques of this paper may stimulate further research in this area. The interested readers may find new and novel applications of quasi split feasibility problems in image reconstruction, medical imaging and related fields. 2 Preliminaries Let H be real Hilbert space, whose inner product and norm are denoted by 〈.,.〉 and ‖.‖ respectively. Let K be a non- empty, closed and convex set in H. We now recall some basic concepts and results, which are needed. Definition 1. An operator T is said to be strongly monotone, if there exists a constant α > 0 such that 〈Tu − Tv , u − v〉≥ α ‖u − v‖ 2 , ∀u, v ∈ H. Definition 2. An operator T is said to expanding, if and only if ‖Tu − Tv‖≥‖u − v‖, ∀u, v ∈ H. From Definition 2.1 and Definition 2.2, it follows that every strongly monotone operator is expanding, but the converse is not true. ∗ Corresponding author e-mail: noormaslam@hotmail.com c  2013 NSP Natural Sciences Publishing Cor.