If (T Etomathcal{BC}(X)) is a continuous multivalued mapping of convex type, then (A) holds.
Let C be a nonempty closed convex subset of a real Hilbert space H, let F be a bifunction from C × C to ℝ which satisfies (A1 - A4), and let A F be a multivalued mapping of H into itself defined by A F x = { { z ∈ H : F ( x, y ) ≥ 〈 y − x, z 〉, ∀ y ∈ C }, x ∈ C, ∅, x ∉ C. (3.2).
Let A Θ be a multivalued mapping of H into itself defined by A Θ x = { { z ∈ H : Θ ( x, y ) ≥ 〈 y − x, z 〉 }, x ∈ C, ∅, x ∉ C. Then EP = A Θ − 1 0, and A Θ is a maximal monotone operator with dom ( A Θ ) ⊂ C.
Let C be a nonempty, closed, and convex subset of H, F a bifunction from C×C to ℝ which satisfies (A1 - A4), and A F a multivalued mapping of H into itself defined by A F X = { z ∈ H : F ( x, y ) ≥ ⟨ y - x, z ⟩, ∀ y ∈ C }, x ∈ C, ∅, x ∉ C. (4.3).
Lemma 4.3 Let F be a bifunction from C × C to ℝ which satisfies (A1 - A4), and let A F be a multivalued mapping of H into itself defined by A F x = { { z ∈ H : F ( x, y ) ≥ 〈 y − x, z 〉, ∀ y ∈ C }, x ∈ C, ∅, x ∉ C. (4.4).
Let (A_{f}) be a multivalued mapping of H into itself defined by A_{f}x=left { textstylebegin{array}{l@{quad}l} {zin H: f (x, y geqlangle y-x, zrangle,forall y in C},& xin C, emptyset, &xnotin C. end{array}displaystyle right.
More concretely, 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.
Remark 2.13 The concept of α-admissible multivalued mapping is extension of concept of α ∗ -admissible multivalued mapping due to Asl et al. [10].
Recently, many authors have proposed their fixed point methods for finding a fixed point of both multivalued mapping and a family of multivalued mappings.
Assume that a multivalued mapping D n of G n into G n +1 is surjective and satisfies the condition: the values of any two different elements from G n are disjoint subsets of G n +1.
After all, applying Proposition 2, we obtain the existence of a fixed point of the multivalued mapping φ which represents a solution of problem (13).
