Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line

Definition 2.1

[31] If a function f : [ 0 , T ] → X satisfies the limits

then the function f is called regulated function on [ 0 , T ] . It was proved that the set of discontinuous points of a regulated function is at most countable, regulated functions are bounded, and the space G ( [ 0 , T ] , X ) of regulated functions is a Banach space when endowed with the norm

In order to obtain compactness in G ( [ 0 , T ] , X ) , we need the concept of equiregulated sets.

Definition 2.2

[32] A set A ⊂ G ( [ 0 , T ] , X ) is said to be equi-regulated, 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 − 0 ) ‖ ≤ ε ,

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

Definition 2.3

[33] If f : [ 0 , T ] → X is called Kurzweil-Stieltjes integrable with respect to g : [ 0 , T ] → R if there is a function denoted by ( KS ) ∫ 0 . f ( s ) d g ( s ) : [ 0 , T ] → X such that, for every ε > 0 , there exists a gauge δ ε on [ 0 , T ] with

for every δ ε -fine partition { ( ξ i , [ t i − 1 , t i ) ) : i = 1 , 2 , … , n } of [ 0 , 1 ] . A partition { ( ξ i , [ t i − 1 , t i ) ) : i = 1 , 2 , … , n } is δ ε -fine if for all i = 1 , 2 , … , n , [ t i − 1 , t i ] ⊂ ( ξ i − δ ε ( ξ i ) , ξ i + δ ε ( ξ i ) ) .

Lemma 2.1

[33] Consider the functions f : [ 0 , T ] → X , g is regulated and g : [ 0 , T ] → R such that Kurzweil-Stieltjes integral ∫ 0 T f ( s ) d g ( s ) exists. Then, for every t ∈ [ 0 , T ] , the function h ( t ) = ∫ 0 t f ( s ) d g ( s ) , t ∈ [ 0 , T ] , is regulated and satisfies

where △ + g ( t ) = g ( t + ) − g ( t ) , △ − g ( t ) = g ( t ) − g ( t − ) , g ( t + ) , and g ( t − ) denote the right and left limits of function g at point t.

Proposition 2.1

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

Proposition 2.2

[35] A sequence ( u n ) n ⊂ G ( [ 0 , T ] , X ) is weakly convergent to a function u ∈ G ( [ 0 , T ] , X ) if and only if is (norm) bounded, and for any t ∈ [ 0 , T ] , the sequence ( u n ) n is weakly convergent to u ( t ) in X .

Lemma 2.2

[35] Let ( u n ) n be a sequence of functions from [ 0 , T ] to X. If ( u n ) converges pointwisely to u 0 as n → ∞ and the sequence ( u n ) n is equiregulated, then ( u n ) n converges uniformly to u 0 .

Lemma 2.3

[35] Let X be a Banach space and μ be a finite measure. Then, a bounded, uniformly integrable subset A of L 1 ( μ , X ) such that for each t ∈ [ 0 , T ] , A ( t ) is relatively weakly compact in X, is weakly relatively compact.

Theorem 2.1

[36] Let g : [ 0 , T ] → R be a function of bounded variation. Assume that f , f n : [ 0 , T ] → R , n ∈ N , satisfy the following conditions:

1. The integral ∫ 0 T f n ( s ) d g ( s ) exists for every n ∈ N ;

2. For each τ ∈ [ 0 , T ] , lim n → ∞ f n ( τ ) = f ( τ ) ;

3. There exists a constant K > 0 such that for every division 0 = σ 0 < σ 1 < ⋯ < σ l = T of [ 0 , T ] and every finite sequence m 1 , … , m l ∈ N , we have

   ∑ i = 1 l ∫ σ i − 1 σ i f m i ( s ) d g ( s ) ≤ K .

   Then, the integral ∫ 0 T f ( s ) d g ( s ) exists and

   ∫ 0 T f ( s ) d g ( s ) = lim n → ∞ ∫ 0 T f n ( s ) d g ( s ) .

Theorem 2.2

[37] Let D be a nonempty, compact, and convex subset of a Banach space, and let the correspondence N : D → B ( D ) an upper semi-continuous (u.s.c.) multi-valued mapping with contractible values. Then, the set of fixed points of N is nonempty.

Definition 2.4

[38] A multi-valued mapping F : X → P ( X ) is called weakly u.s.c. if F − 1 ( B ¯ ) is closed for each weakly closed set B ¯ ⊂ X .

Evidently, u.s.c. is stronger than weakly u.s.c., and simple examples show that a weakly u.s.c. function with compact convex values may fail to be u.s.c.

Let us recall the definition of evolution family below, sometimes also called an evolution process or evolution system. For a detailed account and bibliographic references, see, e.g., the survey by Acquistapace [29], Acquistapace and Terreni [30], Engel and Nagel [39], and Goldstein [40].

Lemma 2.4

[29] The family of the linear operator { T ( t , s ) } ( t , s ) ∈ △ satisfies the following properties.

1. T ( t , r ) T ( r , s ) = T ( t , s ) , T ( t , t ) = I for 0 ≤ s ≤ t ≤ T ;

2. The map ( t , s ) → T ( t , s ) x is strong continuous for all x ∈ X and 0 ≤ s ≤ t ≤ T ;

3. T ( ⋅ , s ) ∈ C 1 ( ( s , ∞ ) , L ( E ) ) , ∂ T ( t , s ) ∂ t = A ( t ) T ( t , s ) for t > s , and ‖ A k ( t ) T ( t , s ) ‖ ≤ M ( t − s ) − k for 0 < t − s ≤ 1 and k = 0 , 1 ;

4. ∂ T ( t , s ) ∂ t = − T ( t , s ) A ( s ) x for t > s and x ∈ D ( A ( s ) ) with A ( s ) x ∈ D ( A ( s ) ) .

Definition 2.5

[29] An evolution family { T ( t , s ) } ( t , s ) ∈ △ is said to be compact if for all 0 ≤ s ≤ t ≤ T , T ( t , s ) is continuous and maps bounded subsets of X into precompact subsets of X .

Lemma 2.5

[29] Let { T ( t , s ) } ( t , s ) ∈ △ be a compact evolution family on X. Then, for each s ∈ [ 0 , T ] , the function t → T ( t , s ) is continuous by operator norm for t ∈ ( s , T ] .

Lemma 2.6

[41] For each t ∈ [ 0 , T ] and some λ ∈ ρ ( A ( t ) ) , if the resolvent R ( λ , A ( t ) ) is a compact operator, then T ( t , s ) is a compact operator whenever 0 ≤ s ≤ t ≤ T .

Lemma 2.7

[38] Let F : X → P c l , c v ( X ) be a closed and quasi-compact multi-valued mapping. Then, F is u.s.c.

Definition 2.6

[42] A nonempty subset D of a metric space is said to be contractible if there exists a point y 0 ∈ D and a continuous function h : [ 0 , 1 ] × D → D such that h ( 0 , y ) = y 0 and h ( 1 , y ) = y for every y ∈ D .

Definition 2.7

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

Let X be a Banach space and B ( X ) the family of all bounded subsets of X . Then, the function β : B ( E ) → R + defined by

is called the (Hausdorff) measure of noncompactness. It is monotone, nonsingular, real, and regular [43].

Theorem 2.3

[12] Let X be a complete metric space, β denote the Hausdorff measure of noncompactness in X, and let ∅ ≠ A ˜ ⊂ X .

Then, the following statements are equivalent:

1. A ˜ is an R δ -set,

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

3. A ˜ is compact and absolutely neighborhood contractible, i.e., A ˜ is contractible in each neighborhood in Y ∈ A N R , where A ˜ is embedded.

Proposition 2.3

[44] Let S = { X α , π α β , ∑ } be an inverse system. If for every α ∈ ∑ , X α is compact and nonempty, then lim ← S is compact and non-empty.

Proposition 2.4

[22] Let S = { X n , π n p , N } be an inverse system. If for every n ∈ N , X n is an R δ -set, then lim ← S is R δ -set, as well.

Theorem 2.4

[6] Let S = { X α , π α β , ∑ } be an inverse system and φ : lim ← S → lim ← S ′ be a limit map induced by a map { i d , φ α } , where φ α : X α → X α . Then, the fixed point set of φ is a limit of the inverse system generated by the sets Fix ( φ α ) . In particular, if the sets Fix ( φ α ) are compact acyclic (resp. R δ ), then it is compact acyclic (resp. R δ ), as well.

Remark 2.1

[46] Let us recall that by a Frechet space, we mean a locally convex space, which is metrizable and complete. Every Banach space is Frechet space, and every Frechet space is the limit of an inverse system of Banach spaces.

Lemma 3.2

Let conditions (H1) and (H2) be satisfied. Then, Sel F ( u ) is nonempty.

Proof

Let us show that Sel F ( u ) ≠ ∅ for each u ∈ G ( I , X ) . For this purpose, we assume that u ∈ G ( I , X ) and { u n } is the sequence of step functions from I to G ( I , X ) such that

By (H1), we can see that for each n , F ( ⋅ , u n ) has a strongly measurable selection f n ( ⋅ ) . Furthermore, by (3.1), for each n > 0 , t ∈ I , there exists M > 0 , such that ‖ u n ( t ) ‖ ≤ M . It follows from (H2) that f n is integrably bounded in L 1 ( d g , I , X ) . Making use of Lemma 2.3, one can conclude that this family is weakly relatively compact, and so, we can find a subsequence f n (which we do not relabel), which is weakly σ ( L 1 ( d g , I , X ) , L ∞ ( d g , I , X ) ) convergent to some d g -measurable function f , i.e., for every x * ∈ X * , we have