Mathematical Models, Methods and Applications by Abul Hasan Siddiqi Pammy Manchanda & Rashmi Bhardwaj

Author:Abul Hasan Siddiqi, Pammy Manchanda & Rashmi Bhardwaj
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

where λ > 0 and ρ > 0 are two constants.

Proof

The conclusion can be obtained directly from the definitions of and .□

Based on Lemma 4.1, we now define an iterative algorithm for approximating a solution of problem (4.1).

Algorithm 4.1

Let X 1 , X 2 , A 1 , A 2 , B 1 , B 2 , H 1 , H 2 , M, N, F, and G are same as Lemma 4.1. For any given initial (a 0 , b 0 ) ∊ X 1 × X 2 , we define the following iterative scheme:

for n = 0, 1, 2,….., where λ > 0 and ρ > 0 are two constants.

Now, we show the existence of solution of problem (4.1) and analyze the convergence of iterative Algorithm 4.1.

Theorem 4.1

Let X 1 and X 2 be two real Hilbert spaces. Let A 1 , B 1 :X 1 → X 1 , A 2 , B 2 :X 2 → X 2 be the single-valued mappings. Let H 1 :X 1 × X 1 → X 1 be a single-valued mapping such that H 1 (A 1 , B 1 ) is cocoercive with respect to A 1 with constant μ 1 > 0 and relaxed cocoercive with respect to B 1 with constant γ 1 > 0, A 1 is α 1 -expansive and B 1 is β 1 -Lipschitz continuous, α 1 > β 1 and μ 1 > γ 1 . Let H 2 :X 2 × X 2 → X 2 be also a single-valued mapping such that H 2 (A 2 , B 2 ) is cocoercive with respect to A 2 with constant μ 2 > 0 and relaxed cocoercive with respect to B 2 with constant γ 2 > 0, A 2 is α 2 -expansive and B 2 is β 2 -Lipschitz continuous, α 2 > β 2 and μ 2 > γ 2 . Let is set-valued, H 1 (∙,∙)-cocoercive mapping and is set-valued, H 2 (∙,∙)-cocoercive mapping. Assume that H 1 (A 1 , B 1 ) is r 1 -Lipschitz continuous with respect to A 1 and r 2 -Lipschitz continuous with respect to B 1 , F:X 1 × X 2 → X 1 is τ 1 -Lipschitz continuous with respect to the first argument and τ 2 -Lipschitz continuous with respect to the second argument, H 2 (A 2 , B 2 ) is r 3 -Lipschitz continuous with respect to A 2 and r 4 -Lipschitz continuous with respect to B 2 , G:X 1 × X 2 → X 2 is τ 1 ’-Lipschitz continuous with respect to first argument and τ 2 ’-Lipschitz continuous with respect to second argument. F(x, ·) is m 1 -strongly monotone with respect to H 1 (A 1 , B 1 ) and G( ·, y) is m 2 -strongly monotone with respect to H 2 (A 2 , B 2 ). If the following conditions are satisfied:

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