Abstract
The notions of relaxed submonotone and relaxed monotone mappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.
Highlights
The notions of relaxed submonotone and relaxed monotone mappings in Banach spaces are introduced and many of their properties are investigated
In the recent papers [5, 6], these notions were extended in arbitrary Banach spaces and many of their properties were considered. It was shown in particular in [6] that the subdifferential of Pshenichnyi is almost strictly submonotone almost everywhere in separable Banach spaces. We extend these notions to the so-called relaxed submonotone mappings
In separable Banach spaces, the Clarke subdifferential of every locally Lipschitz real valued functions is almost everywhere relaxed submonotone
Summary
Let (E, · ) be a Banach space with dual E∗ and P(E∗) the set of all nonempty and bounded subsets of E∗. The multifunction F : X → Y , where X and Y are topological spaces, is called USC at x, when for every open set V ⊃ F(x) there exists an open set U x such that F(y) ⊂ V for every y ∈ U. It is called USC when it is USC at every point of D(F) = {x ∈ X : F(x) = ∅}. It is seen that F is RSM (even SRSM) and cannot be represented as a sum of continuous and submonotone mappings
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
