Topological structure of the solution sets to neutral evolution inclusions driven by measures

Definition 2.1

[32] A multivalued map G : X → P ( X ) is closed (convex) valued if G ( x ) is closed (convex) for all x ∈ X .

Definition 2.2

[32] The multivalued map G : X → P ( W ) is usc at x 0 ∈ X if for every open set V ⊂ W such that G ( x 0 ) ⊂ V , there exists a neighborhood U of x 0 such that for any x ∈ U , G ( x ) ⊂ V .

Definition 2.3

[32] A multivalued map G : X → P ( W ) is said to be usc on X if F − 1 ( V ) = { x ∈ X : F ( x ) ⊂ V } is an open set of X for every open V ⊂ W .

Definition 2.4

[32] A multivalued map G : X → P ( W ) is continuous if it is both usc and lsc.

Definition 2.5

[33] A multivalued map G : X → P ( W ) is quasi-compact if G ( B ) is relatively compact for each compact set B ⊂ W .

Lemma 2.1

[33] Let G : X → P c p , c l ( W ) is a closed and quasi-compact multivalued mapping. Then, G is usc.

Definition 2.6

[29] The e ( C , D ) is called the excess of C over D if for every C , D ∈ P c l , b ( X ) , e ( C , D ) = sup x ∈ C dist ( x , D ) = inf { r > 0 : C ⊂ D + r B } , where B is the close unit ball of X .

Definition 2.7

[29] The h ( C , D ) is said to be Hausdorff distance if for every C , D ∈ P c l , b ( X ) , h ( C , D ) = max { e ( C , D ) , e ( D , C ) } . Denote by ∣ C ∣ = h ( C , { 0 } ) . Therefore, ( P c l , b ( X ) , h ( ⋅ , ⋅ ) ) is a Banach space.

Definition 2.8

[33] A multivalued map G : X → P ( W ) has a fixed point if X ∩ W ≠ ∅ , and exists x ∈ X ∩ W such that x ∈ G ( x ) . The fixed-point set of the multivalued operator G will be denoted by ℱ i x G , i.e., ℱ i x G = { x ∈ X ∩ W : x ∈ G ( x ) } .

Definition 2.9

[32] Let I ⊂ R be a compact interval, μ be a Lebesgue measure on I , and W be a Banach space. A multivalued map G : I → P c p ( W ) is said to be m e a s u r a b l e if for every open subset V ⊂ W the set { t ∈ I : G ( t ) ⊂ V } is measurable.

Definition 2.10

[32] A multivalued map G : I → P c p ( W ) is said to be strongly measurable if there exist a sequence { G n } n = 1 ∞ of step multivalued maps such that h ( G n ( t ) , G ( t ) ) → 0 as n → ∞ for μ – a.e. t ∈ I .

Definition 2.11

[33] A multivalued map G : D ⊂ X → P ( W ) is said to be weakly usc if G − 1 ( B ) is closed in D for every weakly closed set B ⊂ W .

Definition 2.12

[33] A multivalued map G : X → P ( W ) is said to be ε − δ usc if for every x 0 ∈ X and ε > 0 there exists δ > 0 such that G ( x ) ⊂ G ( x 0 ) + B ε ( 0 ) for all x ∈ B δ ( x 0 ) .

Lemma 2.2

[33] Let G : D ⊂ X → P ( W ) be a multi-valued mapping with weakly compact values. Then,

1. G is weakly u s c if G is ε − δ u s c

2. Suppose further that G has convex values and W is reflexive, then, G is weakly usc if and only if for each sequence { ( x n , w n ) } in D × W such that x n → x in X and w n ∈ G ( x n ) , n ≥ 1 , it follows that there exists a subsequence { w n k } of { w n } and w ∈ G ( x ) such that w n k → w weakly in W.

Lemma 2.3

[33] Let D be a non-empty, compact, and convex subset of a Banach space and G : D → P c p ( D ) an u s c multivalued mapping with contractible values. Then, G has at least one fixed-point.

Definition 2.13

[34] Assume that S is a bounded set of X . Then, the Kuratowski measure of noncompactness β ( ⋅ ) defined on S is

where d i a m ( S i ) denotes the diameter of a set S .

Lemma 2.4

[34] Let Z, M be bounded sets of X, and λ ∈ R . Then,

1. β ( Z ) = 0 if and only if Z is relatively compact;

2. Z ⊂ M implies β ( Z ) ≤ β ( M ) ;

3. β ( Z ¯ ) = β ( c o Z ) = β ( Z ) , where coZ denotes the convex hull of Z;

4. β ( Z ∪ M ) = max { β ( Z ) , β ( M ) } ;

5. β ( λ Z ) = ∣ λ ∣ β ( Z ) , where λ Z = { λ z : z ∈ Z } ;

6. β ( Z + M ) ≤ β ( Z ) + β ( M ) , where Z + M = { z + m : z ∈ Z , m ∈ M } ;

7. ∣ β ( Z ) − β ( M ) ∣ ≤ 2 h ( Z , M ) , where h ( Z , M ) denotes the Hausdorff distance of Z and M.

Lemma 2.5

[34] Let X be a Banach space, and M ⊂ X be a bounded set. Then, there exists a countable subset M 0 of M such that β ( M ) ≤ 2 β ( M 0 ) .

Lemma 2.6

[34] If N ˜ = { u n } 1 ∞ ⊂ C ( [ 0 , T ] , X ) is a countable set in a Banach space X and there is a function m ∈ L 1 ( [ 0 , T ] , R + ) such that for every n ∈ N ,

then β ( N ˜ ( s ) ) is Lebesgue integrable on [ 0 , T ] and

Throughout, let L 1 ( d g , [ 0 , T ] , X ) stand for the Banach space of all d g – a.e. LS-integrable functions from [ 0 , T ] to X, equipped with the norm

Definition 2.14

[31] Let δ be a gauge. Δ = { ( [ t i − 1 , t i ] , c i ) , i = 1 , 2 , … , n } , where c i ∈ [ t i − 1 , t i ] , i = 1 , 2 , … , n and ⋃ i = 1 n [ t i − 1 , t i ] = [ 0 , T ] , is a partition of [ 0 , 1 ] , then Δ is said to be δ -fine partition if [ t i − 1 , t i ] ⊂ ( c i − δ ( c i ) , c i + δ ( c i ) ) , for any i = 1 , 2 , … , n .

Definition 2.15

[31] A function f : [ 0 , T ] → X is said to be Kurzweil-Stieltjes (KS) integrable w.r.t. g : [ 0 , T ] → R if there exists a vector ( K S ) ∫ 0 T f ( s ) d g ( s ) ∈ X such that, for every ε > 0 , there is a gauge δ ε satisfying

for every δ ε -fine partition.

We point out the following relation:

Lemma 2.7

[35] Let H ˜ 0 ⊆ L 1 ( d g , [ 0 , T ] , X ) . Assume that there exists a positive function θ ∈ L 1 ( d g , [ 0 , T ] , X ) such that ‖ h ˜ ( t ) ‖ ≤ θ ( t ) d g – a.e. for all h ˜ ∈ H ˜ 0 . Then, we have

Definition 2.16

[33] A subset A ⊂ L 1 ( d g , [ 0 , T ] , X ) is called integrably bounded if there exists ς ∈ L 1 ( d g , [ 0 , T ] , X ) such that

for each f ˜ ∈ A .

Definition 2.17

[33] A subset A ⊂ L 1 ( d g , [ 0 , T ] , X ) is called uniformly integrable if for each ε > 0 , there exists δ ( ε ) > 0 such that for each Borel measurable subset I ⊂ [ 0 , T ] with m ( I ) < δ ( ε ) , we have

It is clear that if A ⊂ L 1 ( d g , [ 0 , T ] , X ) is integrably bounded, then A is uniformly integrable.

Lemma 2.8

[36] Let ( Ω , Σ , μ ) be a finite measure space and X be a reflexive Banach space. A subset K of L 1 ( μ , X ) is relatively weakly compact if

1. K is bounded;

2. K is uniformly integrable;

3. for each E ⊂ Σ , the set ∫ E f d μ : f ∈ K is relatively weakly compact.

Definition 2.18

[29] A function f : [ 0 , T ] → X is said to be regulated if there exists the limit f ( t + ) and f ( s − ) for any point t ∈ [ 0 , 1 ) and s ∈ ( 0 , 1 ] .

Definition 2.19

[31] A semigroup { T ( t ) } t ≥ 0 is said to be uniformly continuous on ( 0 , ∞ ) if on this set it is continuous with respect to the uniform operator topology.

It is worthwhile to recall that any compact C 0 -semigroup is uniformly continuous on ( 0 , ∞ ) , but the converse is not necessarily satisfied.

Under these conditions, let us define the fractional power A β , 0 < β ≤ 1 , as a closed linear operator on its domain D ( A β ) . For the analytic semigroup { T ( t ) } t ≥ 0 , the following properties will be used:

1. there is a M ≥ 1 such that M ≔ sup t ≥ 0 ‖ T ( t ) ‖ < ∞ ;

2. for any β ∈ ( 0 , 1 ] , there exists a positive constant C β such that

   ‖ A β T ( t ) ‖ ≤ C β t β , 0 < t < T .

Definition 2.20

[31] A set A ⊂ G ( [ 0 , T ] , X ) is said to be equiregulated if for every ε > 0 and every t 0 ∈ [ 0 , T ] there exists δ > 0 such that

1. for any t 0 − δ < t ′ < t 0 , ‖ x ( t ′ ) − x ( t 0 − ) ‖ < ε ;

2. for any t 0 < t ″ < t 0 + δ , ‖ x ( t ″ ) − x ( t 0 + ) ‖ < ε

Definition 2.21

[31] The operator-valued function T : [ 0 , T ] → L ( X ) is said to be Kurzweil-Stieltjes integrable with respect to h : [ 0 , T ] → X if there exists ∫ 0 T T ( t ) d h ( t ) ∈ X such that for every ε > 0 , there exists a gauge δ ε satisfying

for every δ ε -fine partition of [ 0 , T ] .

In particular, if the gauge δ ε is constant, we obtain the Riemann-Stieltjes integral.

Lemma 2.9

[31] A set K of regulated functions is relatively compact in the space G ( [ 0 , T ] , X ) if and only if it is equiregulated and for every t ∈ [ 0 , T ] , { f ( t ) , f ∈ K } is relatively compact in X.

Lemma 2.10

[31] Let the C 0 -semigroup of uniformly bounded { T ( t ) } t ≥ 0 be uniformly continuous on ( 0 , ∞ ) and f : [ 0 , T ] → X be LS-integrable with respect to g. Then, for every t ∈ [ 0 , T ] , T ( t − ⋅ ) f ( ⋅ ) is KS-integrable with respect to g on [ 0 , t ] and

Definition 2.22

[32] A subset B ⊂ X is called a retract of X if there exists a continuous function r : X → B , such that r ( x ) = x , for every x ∈ B .

Definition 2.23

[32] A subset B ⊂ X is called a neighborhood retract of X if there exists an open subset U ⊂ X such that B ⊂ U and B is retract of U .

Definition 2.24

[32] Let two subsets X , Y ⊂ X , X be embedded in Y if there exists a homeomorphism h : X → Y such that h ( X ) is a closed subset of Y .

Definition 2.25

[32] The space X is an absolute retract ( A R -space) if for any space Y and for any embedding h : X → Y , the set h ( X ) is a retract of Y .

Definition 2.26

[32] The space X is an absolute neighborhood retract ( A N R -space) if for any space Y and for any embedding h : X → Y , the set h ( X ) is a neighborhood retract of Y .

Definition 2.27

[33] A subset B ⊂ X is said to be contractible if there exists a point y 0 ∈ B and a continuous function h : [ 0 , 1 ] × B → B such that h ( 0 , y ) = y 0 and h ( 1 , y ) = y for every y ∈ B .

Definition 2.28

[32] A subset B of a metric space is called an R δ -set if there exists a decreasing sequence B n of compact and contractible sets such that

Any intersection of decreasing sequence of R δ -set is R δ .

Lemma 2.11

[32] Let X be a complete metric space, β denote the Kuratowski measure of noncompactness in X, and let ∅ ≠ B ⊂ X . Then, the following statements are equivalent:

1. B is an R δ -set;

2. B is an intersection of a decreasing sequence { B n } of closed contractible spaces with β ( B n ) → 0 .

Lemma 2.12

[31] Let μ = μ c + μ s be a finite Borel measure on [ 0 , T ] with μ ( { 0 } ) = 0 ; u : [ 0 , T ] → R be regulated and μ -integrable, β : [ 0 , T ] → R + be μ -integrable and α : [ 0 , T ] → R + be nondecreasing and μ -integrable. Suppose that

then

where

Definition 3.1

A regulated function x : I → X is called a mild solution on I of inclusion problem (1.1) if x ( 0 ) = x 0 , and there exists a function f ∈ Sel F ( x ) such that x(t) satisfies the following integral equation

for t ∈ I .

Remark 3.1

Following Lemma 2.10, the last integral in this definition is taken in KS sense and we can write

Since ∫ 0 ⋅ f ( τ ) d g ( τ ) is BV function, by [38, Theorem 3.1], we obtain the regularity and BV-property of ∫ 0 ⋅ T ( t − s ) f ( s ) d g ( s ) . Owing to ‖ T ( t − s ) f ( s ) ‖ ≤ M ‖ f ( s ) ‖ for every s ∈ [ 0 , t ] , the integral exists in Lebesgue-Stieltjes sense as well.

Lemma 3.1

Let ( F 1 ) – ( F 3 ) be satisfied and X be reflexive. Then, Sel F is non-empty.

Proof

We show that Sel F ( x ) ≠ ∅ for each x ∈ G ( I , X ) . Let x ∈ G ( I , X ) and { x n } n ≥ 1 be a sequence of step functions from I to X such that

By ( F 2 ), for each n ≥ 1 , F ( ⋅ , x n ( ⋅ ) ) admits a strongly measurable selection f n ( ⋅ ) . Furthermore, by (3.1), there exists M ˜ > 0 , ∀ n ∈ N , t ∈ I , ‖ x n ( t ) ‖ ≤ M ˜ . By ( F 3 ), we obtain { f n } is integrably bounded in L 1 ( d g , I , X ) . Due to Lemma 2.8, it suffices to prove that (iii) holds. By ( F 3 ), for each t ∈ [ 0 , T ] , we have

Thus, for each t ∈ [ 0 , T ] , there exists a subsequence { f n k } which is σ ( X , X ∗ ) -weakly convergent to some integrable (with respect to g ) function f . i.e., for every x ∗ ∈ X ∗ , we obtain ⟨ x ∗ , f n k ( t ) ⟩ → ⟨ x ∗ , f ( t ) ⟩ , n → ∞ . Making use of Lebesgue controlled convergence theorem, we have

and so, for each t ∈ [ 0 , T ] , { ∫ 0 t f n ( s ) d g ( s ) } n ≥ 1 is relatively weakly compact. Hence, by Lemma 2.8, we may assume that f n → f , n → ∞ weakly in L 1 ( d g , I , X ) . By Mazur’s lemma, we obtain a sequence f ˜ n ∈ c o { f k : k ≥ n } strongly L 1 -convergent: therefore, on a subsequence, d g – a.e. convergent to g . Without loss of generality, we assume that f ˜ n ( t ) → f ( t ) , for a.e. t ∈ I . And then, by ( F 1 ), we have F ( s , ⋅ ) : G w ( I , X ) → P ( G w ( I , X ) ) is usc. To obtain the result, it suffices to prove that f ( t ) ∈ F ( t , x ( t ) ) , for a.e. t ∈ I . Indeed, let N 0 be a null measure set such that F ( s , ⋅ ) : G w ( I , X ) → P ( G w ( I , X ) ) is usc, f n ( t ) ∈ F ( t , x n ( t ) ) and f ˜ n ( t ) → f ( t ) for all t ∈ I \ N 0 and n ∈ N .