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# Fixed point theorems and convergence theorems for monotone ($\alpha,\beta$)-nonexpansive mappings in ordered Banach spaces

### Abstract

creative_2017_26_2_163_180_abstract

In this paper, we introduce the notion of a monotone $(\alpha,\beta)$-nonexpansive mapping in an ordered Banach space $E$ with the partial order $\leq$ and prove
some existence theorems of fixed points of a monotone $(\alpha,\beta)$-nonexpansive mapping in a uniformly convex ordered Banach space. Also, we prove
some weak and strong convergence theorems of Ishikawa type iteration under the control condition
$$\limsup_{n\to\infty}s_n(1-s_n) > 0\quad and \quad \liminf_{n\to\infty}s_n(1-s_n) > 0.$$
Finally, we give an numerical example to illustrate the main result in this paper.