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In this article the pseudo-amenability and pseudo-contractibility of restricted semigroup algebra $l_r^1(S)$ and semigroup algebra, l1(Sr) on restricted semigroup, Sr are investigated for different classes of inverse semi-groups such as Brandt semigroup, and Clifford semigroup. We particularly show the equivalence between pseudo-amenability and character amenability of restricted semigroup algebra on a Clifford semigroup and semigroup algebra on a restricted semigroup. Moreover, we show that when S = M0(G, I)is a Brandt semigroup, pseudo-amenability of l1(Sr) is equivalent to its pseudo-contractibility.

In this paper, we study the problem of finding the solution of monotone variational inclusion problem (MVIP) with constraint of common fixed point problem (CFPP) of strict pseudocontractions. We propose a new viscosity method, which combines the inertial technique with self-adaptive step size strategy for approximating the solution of the problem in the framework of Hilbert spaces. Unlike several of the existing results in the literature, our proposed method does not require the co-coerciveness and Lipschitz continuity assumptions of the associated single-valued operator. Also, our method does not involve any linesearch technique which could be time-consuming, rather we employ a self-adaptive step size technique that generates a nonmonotonic sequence of step sizes. Moreover, we prove strong convergence result for our algorithm under some mild conditions and apply our result to study other optimization problems. We present several numerical experiments to demonstrate the computational advantage of our proposed method over the existing methods in the literature. Our result complements several of the existing results in the current literature in this direction.

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