Modified viscosity implicit rules for proximal split feasibility and fixed point problems
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Modified viscosity implicit rules for proximal split feasibility and fixed point problems R. Pant1 · C. C. Okeke1
· C. Izuchukwu2
Received: 10 February 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract The purpose of this paper is to present a modified implicit rules for finding a common element of the set of solutions of proximal split feasibility problem and the set of fixed point problems for ϑ-strictly pseudo-contractive mappings in Hilbert spaces. First, we prove strong convergence results for finding a point which minimizes a convex function such that its image under a bounded linear operator minimizes another convex function which is also a solution to fixed point of ϑ-strictly pseudo-contractive mapping. Our second algorithm generates a strong convergent sequence to approximate common solution of non-convex minimization feasibility problem and fixed point problem. In all our results in this work, our iterative scheme is proposed by a way of selecting the step size such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of an operator norm is not an easy task. Finally, we gave numerical example to study the efficiency and implementation of our schemes. Keywords Generalized implicit rule · Proximal split feasibility problems · Pseudo-contractive mapping · Fixed point · Hilbert space Mathematics Subject Classification 47H09 · 47H10 · 49J20 · 49J40
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C. C. Okeke [email protected]; [email protected] R. Pant [email protected] C. Izuchukwu [email protected]; [email protected]
1
Mathematics and Applied Mathematics Department, University of Johannesburg, Auckland Park Kingsway Campus, PO Box 524, 2006 Johannesburg, South Africa
2
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
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R. Pant et al.
1 Introduction Let H1 and H2 be two real Hilbert spaces. Suppose that f : H1 → R ∪ {+∞}, g : H2 → R ∪ {+∞} are two proper, convex and lower semicontinuous functions and A1 : H1 → H2 is a bounded linear operator. In this paper, we consider the following problem: find a solution x¯ ∈ H1 such that min { f (x) + gλ (Ax)},
(1.1)
x∈H1
1 u − y2 } stands for the Moreau-Yosida approxiwhere gλ (y) = minu∈H2 {g(u) + 2λ mate of the function g of parameter λ. Based on an idea introduced in Lopez et al. [14], Moudafi and Thakur [18] proved weak convergence results for solving (1.1) in the case when arg min f ∩ A−1 (arg min) = ∅, or in other words: in finding a minimizer x¯ of f such that A x¯ minimizes g, namely
x¯ ∈ arg min f such that A x¯ ∈ arg min g,
(1.2)
f , g being two proper, lower semicontinuous convex functions, arg min f := {x¯ ∈ ¯ ≤ f (x), ∀x ∈ H1 } and arg min g := { y¯ ∈ H2 : g( y¯ ) ≤ g(y), ∀y ∈ H1 : f (x) H2 }. We shall denote the solution set of (1.2) by . Concerning problem (1.2), moudafi and Thakur [18] introduced a new way of selecting the step size: Set θ (x) := ∇h(x)2 + ∇l(x)2 with h(x) = 21 (I
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