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In this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound.

This paper is a contribution to the study of the effective content of LAhyperstructure. In this paper, we introduce the notion of soft interior-hyperideals. Further, we give several basic properties of these notions and provide different important characterizations in terms of soft interior hyperideals.

This paper aims to present some sufficient criteria under which a given function f : X → Y satisfies the error bound property, where X and Y are either topological vector spaces whose topologies are generated by metrics or metrizable subsets of some topological vector spaces. Then, we discuss the Hoffman estimation and obtain some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows to calculate the coefficient of the error bound. The applications of this presentation are illustrated by some examples.

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