ON WEAK HORIZONTAL QUASI-CONTINUITY AND JOINT QUASI-CONTINUITY OF MULTIVALUED MAPPINGS
Authors: Nesterenko V. , Fotij O. 
Published: December 2024

The joint upper (lower) quasi-continuity of multivalued mappings from two variables is investigated. Some results on joint quasi-continuity of functions of two variables are transferred to the case of multivalued mappings. For this purpose, the concept of upper (lower) weak horizontal quasi-continuity is first introduced. With the help of this concept, sufficient conditions are established under which the multivalued mapping from two variables is joint quasi-continuous. In particular, it is established that if $X$ is a Baire space, a space $Y$ has a countable pseudobase, $Z$ a regular space, and the multivalued mapping $F:X\times Y \to Z$ is upper and lower weakly horizontally quasi-continuous and lower quasi-continuouswith respect to the second variable for the values of the first variable from some residual set in $X$, then $F$ is a joint lower quasi-continuous mapping. A similar result was established for the joint upper quasi-continuity: if $X$ is a Baire space, a space $Y$ has a countable pseudobase, $Z$ a normal space, and $F:X \times Y\to Z$ is a closed-valued mapping that is upper and lower weakly horizontally quasi-continuous and upper quasi-continuous with respect to of the second variable at the values of the first variable from some residual set in $X$, then $F$ is an upper quasi-continuous mapping .The joint upper (lower) quasi-continuity of multivalued mappings from two variables is investigated. Some results on joint quasi-continuity of functions of two variables are transferred to the case of multivalued mappings. For this purpose, the concept of upper (lower) weak horizontal quasi-continuity is first introduced. With the help of this concept, sufficient conditions are established under which the multivalued mapping from two variables is joint quasi-continuous. In particular, it is established that if $X$ is a Baire space, a space $Y$ has a countable pseudobase, $Z$ a regular space, and the multivalued mapping $F:X\times Y \to Z$ is upper and lower weakly horizontally quasi-continuous and lower quasi-continuouswith respect to the second variable for the values of the first variable from some residual set in $X$, then $F$ is a joint lower quasi-continuous mapping. A similar result was established for the joint upper quasi-continuity: if $X$ is a Baire space, a space $Y$ has a countable pseudobase, $Z$ a normal space, and $F:X \times Y\to Z$ is a closed-valued mapping that is upper and lower weakly horizontally quasi-continuous and upper quasi-continuous with respect to of the second variable at the values of the first variable from some residual set in $X$, then $F$ is an upper quasi-continuous mapping . Necessary and sufficient conditions are also obtained that the multivalued mapping from two variables is joint upper (lower) quasi-continuous. In particular, it is established that if $X$ is a Baire space,  $Y$ a second countable space, $Z$ a metric separable space, then the compact-valued multivalued mapping $F: X\times Y \to Z$ is joint upper and lower quasi-continuous if and only if $F$ is upper and lower weakly horizontally quasicontinuous and $F^x$ is upper and lower quasicontinuous for of all $x$ from some residual set in $X$.