In this paper, we introduce a new projection in a Banach space and show an example of the projections. Then, we study the Mosco convergence of a sequence of nonempty sets concerning the projections in a Banach space.

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.

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