On the convergence of a modified superquadratic method for generalized equations

Authors : Mohammed Harunor RASHİD, Md Zulfiker ALİ

Pages : 202-214

Doi:10.32323/ujma.415225

View : 12 | Download : 6

Publication Date : 2018-09-30

Article Type : Research Paper

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 insert ignore into journalissuearticles values(MSQM); is introduced for solving the generalized equations $0\in{finsert ignore into journalissuearticles values(x);+\mathcal Finsert ignore into journalissuearticles values(x);}$, 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