} t in [0,1] bigr}, which is called the Nemytskii operator associated with F. Let (F : [0,1] timesmathbb{R} to{mathcal{P}}(mathbb{R})) be a multivalued function with nonempty compact values.
t ∈ [ 0, 1 ] }, which is called the Nemytskii operator associated with F. Definition 4.3 Let F : 0, 1 × ℝ → P ℝ be a multivalued function with nonempty compact values.
Define a multivalued operator F : C ( [ 0, 1 ] × R ) → P ( L 1 ( [ 0, 1 ], R ) ) associated with F as F ( x ) = { v ∈ L 1 ( [ 0, 1 ], R ) : v ( t ) ∈ F ( t, x ( t ) ) for a.e. t ∈ [ 0, 1 ] }, which is called the Nemytskii operator associated with F. Definition 3.4 Let F : [ 0, 1 ] × R → P ( R ) be a multivalued function with nonempty compact values.
Define a multivalued operator F : C ( [ 0, T ] × R ) → P ( L 1 ( [ 0, T ], R ) ) associated with F as F ( x ) = { w ∈ L 1 ( [ 0, T ], R ) : w ( t ) ∈ F ( t, x ( t ) ) for a.e. t ∈ [ 0, 1 ] }, which is called the Nemytskii operator associated with F. Definition 2.10 Let F : [ 0, T ] × R → P ( R ) be a multivalued function with nonempty compact values.
Let K be a non-empty closed subset of a compact separable Banach space E. Let (F Jtimes Erightarrow P_{ck}(E)) be a multivalued function defined by F t,x =frac{e^{-gamma t} Vert x Vert }{lambda (1+ Vert x Vert )}K,x =frac{e^{-gammaain(1,infty)) and λ is a constant such that (sup{ Vert z Vert : zin K}leqlambda).
Define a multivalued operator F : C ( [ 1, e ] × R ) → P ( L 1 ( [ 1, e ], R ) ) associated with F as F ( x ) = { w ∈ L 1 ( [ 1, e ], R ) : w ( t ) ∈ F ( t, x ( t ) ) for a.e. t ∈ [ 1, e ] }, which is called the Nemytskii operator associated with F. Definition 3.6 Let F : [ 1, e ] × R → P ( R ) be a multivalued function with nonempty compact values.
