Abstract
AbstractA new definition of essential fixed points is introduced for a large class of multivalued maps. Two abstract existence theorems are presented for approximable maps on compact ANR-spaces in terms of a nontrivial fixed point index, or a nontrivial Lefschetz number and a zero topological dimension of the fixed point set. The second one is applied to the periodic dissipative Marchaud differential inclusions for obtaining the existence of a discretely essential subharmonic solution. Three simple illustrative examples are supplied.
Highlights
In the present note, we will consider for the first time the notion of essential fixed points to multivalued maps as defined below
We will present two abstract theorems about the existence of essential fixed points to a large class of approximable multivalued maps, on compact ANR-spaces, in terms of a nontrivial fixed point index, or a nontrivial Lefschetz number and a zero topological dimension of the fixed point set. These two theorems can be regarded as a multivalued generalisation of their analogies in our recent paper [ ], where single-valued maps were exclusively examined for the same goal
Let us note that this definition significantly differs from all the other definitions for multivalued maps, because it effectively employs the approximability of given multivalued maps on their graphs by single-valued
Summary
We will consider for the first time the notion of essential fixed points to multivalued maps as defined below. We will present two abstract theorems about the existence of essential fixed points to a large class of approximable multivalued maps, on compact ANR-spaces, in terms of a nontrivial fixed point index, or a nontrivial Lefschetz number and a zero topological dimension of the fixed point set. These two theorems can be regarded as a multivalued generalisation of their analogies in our recent paper [ ] (cf [ ], Section ), where single-valued maps were exclusively examined for the same goal. Andres and Górniewicz Fixed Point Theory and Applications (2016) 2016:78 maps In this way, topological invariants like a fixed point index can be calculated just by means of these single-valued approximations.
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