Corollary 3.2 Let ( X, d ) be a complete metric space, ψ ∈ Ψ be a strictly increasing map, x ∗ ∈ X and T : X → CB ( X ) be a multifunction such that H ( T x, T y ) ≤ ψ ( d ( x, y ) ) for all x, y ∈ X with x ∗ ∈ T x ∩ T y.
Corollary 3.3 Let ( X, ⪯, d ) be a complete ordered metric space, ψ ∈ Ψ be a strictly increasing map and T : X → CB ( X ) be a multifunction such that H ( T x, T y ) ≤ ψ ( d ( x, y ) ) for all x, y ∈ X with T x ⪯ T y or T y ⪯ T x.
Corollary 3.5 Let ( X, d ) be a complete metric space, ψ ∈ Ψ be a strictly increasing map, x ∗ ∈ X and T : X → CB ( X ) be a multifunction such that H ( T x, T y ) ≤ ψ ( max { d ( x, y ), d ( x, T x ), d ( y, T y ), d ( x, T y ) + d ( y, T x ) 2 } ). for all x, y ∈ X with x ∗ ∈ T x ∩ T y.
If we let A, D, B, X, Y, Z, K 1, K 2 be as in (QVR α ), f : A × D × B → Z be a vector function, and C : A → 2 Z be a multifunction such that C ( x ) is a closed convex cone with int C ( x ) ≠ ∅, the relation R is defined as follows: R ( x, z, y ) holds iff f ( x, z, y ) ∈ C ( x ).
Let (G Omegatimes Rtimes R^{N}rightarrow{mathcal{P}}_{fc}(R)) be a multifunction such that (i) (forall u,s in Rtimes R^{N} ), (xrightarrow G x,u,s)) is measurable; (ii) (forall xinOmega), ((u,s rightarrow G x,u,s)) is USC; (iii) (forall x,u,s inOmegatimes Rtimes R^{N}), (|G x,u,s)|leq varphi(x)) a.e. with (varphi(x in L_^{q}(Omega)).
Corollary 3.6 Let ( X, ⪯, d ) be a complete ordered metric space, ψ ∈ Ψ be a strictly increasing map and T : X → CB ( X ) be a multifunction such that H ( T x, T y ) ≤ ψ ( max { d ( x, y ), d ( x, T x ), d ( y, T y ), d ( x, T y ) + d ( y, T x ) 2 } ). for all x, y ∈ X with T x ⪯ T y or T y ⪯ T x.
(H Omegatimes Rtimes R^{N}rightarrow {mathcal{P}}_{k}(R)) is a multifunction such that for almost all (xin Omega), ((u,s rightarrow H x,u,s)) is LSC, and (H(F _{4})(i), (iii) holds.
In our next result we prove that the solution set of (5.1) is path-connected under the following assumption: (H(F _{5}):: (H Omegatimes Rtimes R^{N}rightarrow {mathcal{P}}_{k}(R)) is a multifunction such that for almost all (xin Omega), ((u,s rightarrow H x,u,s)) is LSC, and (H(F _{4})(i), (iii) holds.
I, ∀ x ∈ V, p > 2. (H2) F : I × H → P k ( V ∗ ) is a multifunction such that (i) ( t, x ) → F ( t, x ) is graph measurable; (ii) for almost all t ∈ I, x → F ( t, x ) is LSC; (iii) there exist a nonnegative function b 1 ∈ L q ( I ) and a constant C 5 > 0 such that | F ( t, x ) | = sup { ∥ f ∥ V ∗ : f ∈ F ( t, x ) } ≤ b 1 ( t ) + C 5 ∥ x ∥ H k − 1 ∀ x ∈ V a.e.
Next, we consider the convex case, the assumption on F is as follows: (H4) F : I × H → P k c ( V ∗ ) is a multifunction such that (i) ( t, x ) → F ( t, x ) is graph measurable; (ii) for almost all t ∈ I, x → F ( t, x ) has a closed graph; and (H2)(iii) hold. . ( t, x ) → F ( t, x ) is graph measurable; for almost all t ∈ I, x → F ( t, x ) has a closed graph; and (H2)(iii) hold.
We need the following hypothesis: (H5) F : I × H → P w k c ( H ) is a multifunction such that (i) ( t, x ) → F ( t, x ) is graph measurable; (ii) for almost all t ∈ I, x → F ( t, x ) is h-continuous; and (H2)(iii) holds. . ( t, x ) → F ( t, x ) is graph measurable; for almost all t ∈ I, x → F ( t, x ) is h-continuous; and (H2)(iii) holds.
