On the convergence of a modified superquadratic method for generalized equations

Year 2018, Volume: 1 Issue: 3, 202 - 214, 30.09.2018

Mohammed Harunor Rashid Md. Zulfiker Ali

https://doi.org/10.32323/ujma.415225

Abstract

Let $X$ and $Y$ be Banach spaces. Let $\Omega$ be an open subset of $X$. Suppose that $f:X\to{Y}$ is Fr\'{e}chet differentiable in $\Omega$ and $\mathcal F:X\rightrightarrows2^Y$ is a set-valued mapping with closed graph. In the present paper, a modified superquadratic method (MSQM) is introduced for solving the generalized equations $0\in{f(x)+\mathcal F(x)}$, and studied its convergence analysis under the assumption that the second Fr\'{e}chet derivative of $f$ is H\"{o}lder continuous. Indeed, we show that the sequence, generated by MSQM, converges super-quadratically in both semi-locally and locally to the solution of the above generalized equation whenever the second Fr\'{e}chet derivative of $f$ satisfies a H\"{o}lder-type condition.

Keywords

Generalized equations, Lipschitz--like mappings, Semi-local convergence, Set-valued mapping

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