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We consider the problem of image recovery by the metric projections in a real Banach space. For a countable family of nonempty closed convex subsets, we generate an iterative sequence converging weakly to a point in the intersection of these subsets. Our convergence theorems extend the results proved by Bregman and Crombez.

Let

Bregman [

Crombez [

Later, Kitahara and Takahashi [

On the other hand, using the hybrid projection method proposed by Haugazeau [

In this paper, we consider this problem by the metric projections, which are one of the most familiar projections to deal with. The advantage of our results is that we use projections onto the given family of subsets only, to generate the iterative scheme. Our convergence theorems extend the results of [

There are a number of results dealing with the image recovery problem from the aspects of engineering using nonlinear functional analysis (see, e.g., [

Throughout this paper, let

Let

For a

Let

Let

Let

Let

Let

For

Firstly, we consider the iteration of Crombez’s type and get the following result.

Let

Let

Using the idea of [

Let

Let

Suppose that the index set

Since a real Hilbert space

Let

Next, we have the following theorem which extends the result of [

Let

The first author was supported by the Grant-in-Aid for Scientific Research no. 22540175 from the Japan Society for the Promotion of Science.