On Strong Convergence for the Split Common Null Point Problem in Two Banach Spaces

Let $E$ be a 2-uniformly convex and uniformly smooth Banach space, $F$ be a smooth, strictly convex and reflexive Banach space and $\eta$ be a real number with $\eta\in (-\infty,1)$. Let $T:E\to E$ be a relatively nonexpansive mapping and let $U\colon F\to F$ be an $\eta$-demimetric and demiclosed mapping with $F(U)\neq\emptyset$, where $F(U)$ is the set of all fixed points of $U$. Let $L\colon E\to F$ be a bounded linear operator such that $L\neq 0$ with $F_0=F(T)\cap L^{-1}F(U)\neq\emptyset$. Then, W.\,Takahashi [{\it A strong convergence theorem for solving the split common fixed point problem in two Banach spaces and applications}, Linear and Nonlinear Analysis 6 (2020) 473--495] considered Halpern-type iteration [see B.\,Halpern, {\it Fixed points of nonexpanding maps}, Bull. Amer. Math. Soc. 73 (1967) 957--961] and proved a strong convergence theorem for the split common fixed point problem for $U$ and $T$. From this, we got several new results for the split convex feasibility problem, the split common null point problem and the split common fixed point problem.
[1mm] In this paper, motivated by the paper of Takahashi cited above, for $p>1$, when $E$ is a p-uniformly convex and smooth Banach space and $F$ is a strictly convex, reflexive and smooth Banach space, we consider the split common null point problem. We propose another Halpern-type iteration [see K.\,Nakajo, {\it Strong convergence for the problem of image recovery by the metric projections in Banach space}, J. Nonlinear Convex Analysis 23 (2022) 357--376] and prove strong convergence theorems, from which we obtain the results for several split problems under weaker conditions than those by Takahashi [loc. cit.].