Iterative Algorithms for Split Equality Fixed Point Problems and Some Non-Linear Problems in Banach Spaces With Applications

Abstract

Let H1, H2, and H3 be real Hilbert spaces, T and S be non-linear maps defined on H1 and H2, respectively, and with non-empty fixed point sets, F ix(T) and F ix(S), respectively, A and B be linear maps respectively mapping from H1 and H2 to H3. The split equality fixed point problem (SEFPP) considered in this thesis is to find x ∈ F ix(T), y ∈ F ix(S) such that Ax = By. This problem has attracted the attention of numerous researchers due to its vast applications, for instance, in decomposition methods for partial differential equations (PDEs), applications in game theory, and in intensity-modulated radiation therapy, to mention but a few. Few iterative algorithms have been proposed in real Hilbert spaces for approx imating solutions of the SEFPP when they exist. However, the fact that these algorithms are confined in Hilbert spaces is a restriction, since the models of most real-life problems lives in spaces more general than Hilbert spaces. Besides, to guarantee the convergence of these algorithms, it is necessary to impose some compactness-type conditions on some of the involved mappings. In Chapter 3 of this thesis, we proposed the following iterative algorithm that approximates a solution of SEFPP in certain Banach spaces, in particular, in lp spaces 1 < p ≤ 2.  x1 ∈ E1, y1 ∈ E2, zn ∈ JE3 (Axn − Byn) xn+1 = J −1 E1