Sentence examples for mappings under the from inspiring English sources
However, in [15] Abbas and Ilić obtained various common fixed-point and invariant approximation results for such mappings under the assumption of weak compatibility of maps.
In 1941, Hyers [2] considered the case of approximately additive mappings under the assumption that and are Banach spaces.
In this section we give a homotopy result for this class of mappings under the condition (I-C).
Later on, many authors considered different classes of contractive mappings under the P-property (see, e.g., [2 4]).
Our first goal is to study the existence of stationary points for nonexpansive set-valued mappings under the conditions of a normal structure (see [7], Chapter 6).
In the present paper, we do not investigate the fixed point theory of higher-order Lipschitz mappings under the hypotheses in Theorems 1.6, 1.7 and 1.8 above.
We give conditions under which there is a unique fixed point depending differentiably on the parameters; the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of Banach spaces.
It is interesting to note that in all the results obtained so far concerning common fixed points of hybrid mappings the (single-valued and multi-valued) mappings under consideration satisfy either the commutativity condition or one of its generalizations (see, for instance, [6 10]).
Most recently, Bednarczuk [12] defined weak sharp minima of order for vector-valued mappings under an assumption that the order cone is closed, convex, and pointed and used the concept to prove upper Hölderness and Hölder calmness of the solution set-valued mappings for a parametric vector optimization problem.
Moreover the mappings under consideration are α -admissible with respect to η.
In the study of Hyers-Ulam stability problems of monomial functional equations, we have been frequently requested to prove the uniqueness of the monomial mappings under various conditions.
