aa r X i v : . [ m a t h . F A ] M a r SOLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE
RADOSŁAW PIETKUNA bstract . We establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue–Bochner spaces with the aid of some recently proven continuation theorem of Leray–Schauder type for theclass of so-called admissible multimaps. Examples illustrating the legitimacy of this approach include theinitial value problem for perturbation of m -accretive mutivalued di ff erential equation, the nonlocal Cauchyproblem for semilinear di ff erential inclusion, abstract integral inclusion of Fredholm and Volterra type andthe two-point boundary value problem for nonlinear evolution inclusion.
1. I ntroduction
This paper aims to formulate quite natural and easily verifiable hypotheses, ensuring solvability of thefollowing inclusion of Hammerstein type(1) u ∈ ( K ◦ N F )( u )in the space L p ( I , E ) of Bochner p -integrable functions. In inclusion (1), N F is the Nemytskiˇı operatorassociated to a multifunction F : I × E ⊸ E , while K is an external set-valued operator of a certain type(defined later).Consideration of such operator inclusion accompany, of course, attempts to grasp the integro-di ff e-rential multivalued problems from a unifying topological point of view. These attempts have been maderepeatedly (see for instance [4, 5, 6, 7, 8, 17, 19, 23]). Our e ff orts follow in the footsteps of authors of [7]by taking into account the situation, where the operator K is not only nonlinear but possibly multivaluedand does not necessarily have a quasi-integral form. The main result (Theorem 6.) regarding the existenceof solutions to inclusion (1) poses an application example of a fixed point approach. Its proof is based onthe principle of Leray–Schauder type [24, Theorem 3.2], which was extended (Theorem 4.) to the case ofadmissible multimaps (in the sense of Górniewicz, [14, Definition 40.1]). Just as in [7] the superposition K ◦ N F may not be a condensing map and our assumptions about K and F are formulated so that theMönch type compactness condition could have been satisfied.In order to apply Eilenberg–Montgomery type fixed point argument directly to the superposition K ◦ N F we need to know that this map is pseudo-acyclic. However, the Nemytskiˇı operator N F : L p ⊸ L q is byno means acyclic. Therefore, the authors of [7] rely on the assumption (SG) that operator H : = K ◦ N F has acyclic values. This is very uncomfortable hypothesis from practical point of view. In general, if K is nonlinear, then the composite map H may not have convex values. Unfortunately, even if K and F has convex values the map H may still have values with “awful” geometry, since the class of acyclicmappings is not closed with respect to the composition law. It turns out that it is enough to take intoaccount a relatively weak assumption regarding convexity or decomposability of fibers of the operator K in addition to the acyclicity of its values, to ensure the fulfillment of condition (SG).The applicability of our abstract existence result is richly illustrated by numerous examples of di ff er-ential and integral inclusions, which may be interpreted as a fixed point problem given by (1). Theseexamples include cases where the operator K is a univalent mild solution operator of the m -accretive Mathematics Subject Classification.
Key words and phrases. acyclic set, admissible map, boundary value problem, evolution inclusion, fixed point, integral inclu-sion, m -accretive, operator inclusion, p -integrable solution. quasi-autonomous problem or the mild solution operator of the semilinear inhomogeneous two-pointboundary value problem. There were also presented examples in which the map K is simply linear. Suchas those, in which it has the form of Volterra or Hammerstein integral operator. And finally, there is alsothe case considered, when the map K constitutes a multivalued strongly upper semicontinuous maximalmonotone operator. 2. P reliminaries Let ( E , | · | ) be a Banach space, E ∗ its normed dual and σ ( E , E ∗ ) its weak topology. If X is a subsetof a Banach space E , by ( X , w ) we denote the topological space X furnished with the relative weaktopology of E . The symbol ( X , | · | ) stands for the topological space X endowed with the restriction of thenorm-topology of E to X .The normed space of bounded linear endomorphisms of E is denoted by L ( E ). Given T ∈ L ( E ), || T || L is the norm of T . For any ε > A ⊂ E , B E ( A , ε ) ( D E ( A , ε )) stands for an open (closed) ε -neighbourhood of the set A . The (weak) closure and the closed convex envelope of A will be denotedby ( A w ) A and co A , respectively. If x ∈ E we put dist( x , A ) : = inf {| x − y | : y ∈ A } . Besides, for twononempty closed bounded subsets A , B of E the symbol h ( A , B ) stands for the Hausdor ff distance from A to B , i.e. h ( A , B ) : = max { sup { dist( x , B ) : x ∈ A } , sup { dist( y , A ) : y ∈ B }} .We denote by ( C ( I , E ) , || · || ) the Banach space of all continuous maps I → E equipped with themaximum norm. Let 1 p ∞ . By ( L p ([ a , b ] , E ) , || · || p ) we mean the Banach space of all Bochner p -integrable maps f : [ a , b ] → E i.e., f ∈ L p ([ a , b ] , E ) i ff map f is strongly measurable and || f || p = Z ba | f ( t ) | p d t ! p < ∞ if p < ∞ and respectively || f || ∞ = ess sup t ∈ [ a , b ] | f ( t ) | < ∞ provided p = ∞ . Recall that strong measurability is equivalent to the usual measurability in case E isseparable. A subset D ⊂ L p ([ a , b ] , E ) is called decomposable if for every u , w ∈ D and every Lebesguemeasurable A ⊂ [ a , b ] we have u · A + w · A c ∈ D .Given metric space X, a set-valued map F : X ⊸ E assigns to any x ∈ X a nonempty subset F ( x ) ⊂ E . F is (weakly) upper semicontinuous, if the small inverse image F − ( A ) = { x ∈ X : F ( x ) ⊂ A } is open in X whenever A is (weakly) open in E . A map F : X ⊸ E is lower semicontinuous, if the inverse image F − ( A ) is closed in X for any closed A ⊂ E . We say that F : X ⊸ E is upper hemicontinuous if for each p ∈ E ∗ , the function σ ( p , F ( · )) : X → R ∪ { + ∞} is upper semicontinuous (as an extended real function),where σ ( p , F ( x )) = sup y ∈ F ( x ) h p , y i . We have the following characterization ([1, Proposition 2(b)]): a map F : X ⊸ E with convex values is weakly upper semicontinues and has weakly compact values i ff givena sequence ( x n , y n ) in the graph Gr( F ) with x n X −−−−→ n →∞ x , there is a subsequence y k n E −−−− ⇀ n →∞ y ∈ F ( x ) ( ⇀ denotes the weak convergence). A multifunction F : X ⊸ E is compact if its range F ( X ) is relativelycompact in E . It is quasicompact if its restriction to any compact subset A ⊂ X is compact. The set of allfixed points of the map F : E ⊸ E is denoted by Fix( F ).Let H ∗ ( · ) denote the Alexander-Spanier cohomology functor with coe ffi cients in the field of rationalnumbers Q (see [28]). We say that a topological space X is acyclic if the reduced cohomology ˜ H q ( X ) is0 for any q > Definition 1 ([14, Definition 40.1]) . Let F be a Fréchet space and Γ a Hausdor ff topological space. Asurjective continuous map p : Γ → F is called a Vietoris map if p is closed and, for every x ∈ F , the setp − ( x ) is compact acyclic. A set-valued map F : F ⊸ F is admissible, if there is a Hausdor ff topologicalspace Γ and two continuous functions p : Γ → F and q : Γ → F from which p is a Vietoris map such thatF ( x ) = q ( p − ( x )) for every x ∈ F . OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 3
An admissible set-valued map is an upper semicontinuous one with compact connected values. Ifthe aforesaid Vietoris map p : Γ → F is not only proper but open at the same time, then the admissibleset-valued map F : F ⊸ F is also continuous. An upper semicontinuous map F : F ⊸ F is called acyclicif it has compact acyclic values. Evidently, every acyclic map is admissible. Moreover, the compositionof admissible maps is admissible ([14, Theorem 40.6]).A real function γ defined on the family B ( E ) of bounded subsets of E is called a measure of non-compactness (MNC) if γ ( Ω ) = γ (co Ω ) for any bounded subset Ω of E . The following example of MNCis of particular importance: given E ⊂ E and Ω ∈ B ( E ), β E ( Ω ) : = inf ε > x , . . . , x n ∈ E with Ω ⊂ n [ i = B E ( x i , ε ) is the Hausdor ff MNC relative to the subspace E . Recall that this measure is regular i.e., β E ( Ω ) = ff Ω is relatively compact in E ; monotone i.e., if Ω ⊂ Ω then β E ( Ω ) β E ( Ω ) and invariant withrespect to union with compact sets i.e., β E ( A ∪ Ω ) = β E ( Ω ) for any relatively compact A ⊂ E .We recall the reader following results on account of their practical importance. The first is a weakcompactness criterion in L p ( Ω , E ), which originates from [30]. Theorem 1 ([30, Corollary 9.]) . Let ( Ω , Σ , µ ) be a finite measure space with µ being a nonatomic measureon Σ . Let A be a uniformly p-integrable subset of L p ( Ω , E ) with p ∈ [1 , ∞ ) . Assume that for a.a. ω ∈ Ω ,the set { f ( ω ) : f ∈ A } is relatively weakly compact in E. Then A is relatively weakly compact. Remark 1.
The genuine formulation of this result assumes the boundedness of the set A . However, thefact that µ is nonatomic means that uniform integrability of A entails its boundedness.The next property is commonly known as the Convergence Theorem for upper hemicontinuous mapswith convex values (the mentioned below version is proved in [25]). Theorem 2.
Let ( Ω , Σ , µ ) be a σ -finite measure space, E a Banach and F : E ⊸ E a closed convexvalued upper hemicontinuous multimap. Assume that functions f n , f : Ω → E and g n , g : Ω → E aresuch that g n ( x ) E −−−−→ n →∞ g ( x ) a.e. on Ω , f n L ( Ω , E ) −−−−−− ⇀ n →∞ f , f n ( x ) ∈ co B ( F ( B ( g n ( x ) , ε n )) , ε n ) a.e. on Ω , where ε n −−−−→ n →∞ + . Then f ( x ) ∈ F ( g ( x )) a.e. on Ω . The third result is a Lefschetz-type fixed point theorem for admissible multimaps.
Theorem 3 ([12, Theorem 7.4]) . Let X be an absolute extensor for the class of compact metrizablespaces and F : X ⊸ X be an admissible map such that F ( X ) is contained in a compact metrizable subsetof X. Then F has a fixed point.
3. F ixed point approach to inclusions of H ammerstein type The subsequent results constitute a generalization of the continuation principle [24, Theorem 3.2] tothe case of admissible multimaps.
Theorem 4.
Let F be a Fréchet space and X a nonempty closed convex subset of F . Assume that U isrelatively open in X and its closure is a retract of X. Assume further that F : U ⊸ X is an admissibleset-valued map and for some x ∈ U the following two conditions are satisfied: (2) Ω ⊂ U , Ω ⊂ co (cid:0) { x } ∪ F ( Ω ) (cid:1) = ⇒ Ω compact RADOSŁAW PIETKUN and (3) x < (1 − λ ) x + λ F ( x ) on U \ U for all λ ∈ (0 , . Then
Fix( F ) is nonempty and compact.Proof. Keeping the notation and notions contained in the proof of [26, Theorem 3.], consider a family { M α } α ∈ A of all fundamental subsets of there defined multimap ˜ F : X ⊸ X , containing x . Recall afterKrasnosel’ski˘ı that the closed convex set M ⊂ X is a fundamental subset of ˜ F if ˜ F ( M ) ⊂ M and for any x ∈ X , it follows from x ∈ co (cid:0) ˜ F ( x ) ∪ M (cid:1) that x ∈ M . Observe that the family { M α } α ∈ A is nonempty(take for example X ). Define M : = T α ∈ A M α . Next, note that M and co (cid:0) ˜ F ( M ) ∪ { x } (cid:1) are fundamental.Whence, M = co (cid:0) ˜ F ( M ) ∪ { x } (cid:1) .Notice that M ∩ U ⊂ M = co (cid:0) ˜ F ( M ) ∪ { x } (cid:1) = co (cid:0) F ( M ∩ U ) ∪ { x } (cid:1) , by the very definition of ˜ F . Thus, M ∩ U is compact, by (2). Eventually, M has compact closure, forthe convex envelope co (cid:0) F ( M ∩ U ) ∪ { x } (cid:1) is compact.As we have seen in the proof of [26, Theorem 3.], ˜ F : M ⊸ M is also an admissible multimap.By virtue of the Dugundji Extension Theorem the domain M is an absolute extensor for the class ofmetrizable spaces. Therefore, the set-valued map ˜ F : M ⊸ M must have at least one fixed point x ∈ M , in view of Theorem 3. Moreover, Fix( ˜ F ) forms a closed subset of the compact domain M andFix( F ) = Fix( ˜ F ). (cid:3) Theorem 5.
Let F be a Fréchet space and X a nonempty closed convex subset of F . Assume that U isrelatively open in X and its closure is a retract of X. Assume further that F : U ⊸ X is either (i) a continuous admissible set-valued mapor (ii) an upper semicontinuous set-valued map with compact convex values.If for some x ∈ U condition (3) together with (4) Ω ⊂ U countable , Ω ⊂ co (cid:0) { x } ∪ F ( Ω ) (cid:1) = ⇒ Ω compactis satisfied, then Fix( F ) is nonempty and compact.Proof. Theorem 5(i) was proven in [26]. Theorem 5(ii) is nothing more than [24, Theorem 3.2]. (cid:3)
Remark 2.
The following properties of F : U ⊸ X imply the Leray–Schauder boundary condition (3)with x ∈ U :(i) if λ ( x − x ) ∈ F ( x ) − x for x ∈ ∂ U , then λ U is convex and F ( ∂ U ) ⊂ U (Rothe’s condition),(iii) | y − x | > | y − x | − | x − x | for each x ∈ ∂ U and y ∈ F ( x ) (Krasnosel’ski˘ı–Altman’s condition),(iv) h y − x , x − x i | x − x | for each x ∈ ∂ U and y ∈ F ( x ) if E is a Hilbert space (Browder’scondition).Assume that p ∈ [1 , ∞ ] and q ∈ [1 , ∞ ). Fix a compact segment I : = [0 , T ] for some end time T > F : I × E ⊸ E be a set-valued map. Throughout the paper we will use the following hypotheses onthe mapping F :(F ) for every ( t , x ) ∈ I × E the set F ( t , x ) is nonempty and convex,(F ) the map F ( · , x ) has a strongly measurable selection for every x ∈ E ,(F ) the graph Gr( F ( t , · )) is sequentially closed in ( E , | · | ) × ( E , w ) for a.a. t ∈ I , OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 5 (F ) F satisfies a sublinear growth condition, i.e. there is b ∈ L q ( I , R ) and c > x ∈ E and for a.a. t ∈ I , || F ( t , x ) || + : = sup {| y | E : y ∈ F ( t , x ) } b ( t ) + c | x | pq , when p ∈ [1 , ∞ ). If p = ∞ , then for every R > b R ∈ L q ( I , R ) such that || F ( t , x ) || + b R ( t ) a.e. on I , for all x ∈ E with | x | R . (F ) for every closed separable linear subspace E of E the map F I × E ( t , · ) ∩ E is quasicompact fora.a. t ∈ I .Recall that the Nemtyskiˇı operator N F : L p ( I , E ) ⊸ L q ( I , E ), corresponding to F , is a multivalued mapdefined by N F ( u ) : = { w ∈ L q ( I , E ) : w ( t ) ∈ F ( t , u ( t )) for a.a. t ∈ I } . Consider also a multivalued external operator K : L q ( I , E ) ⊸ L p ( I , E ). Our hypothesis on the multifunc-tion K is the following:(K ) for every compact C ⊂ E , the map K : ( L q ( I , C ) , w ) ⊸ ( L p ( I , E ) , || · || p ) is acyclic,(K ) the map K : L q ( I , E ) ⊸ L p ( I , E ) is L -Lipschitz with closed values,(K ) for every uniformly q -integrable possessing relatively weakly compact vertical slices a.e. on I subset C ⊂ L q ( I , E ), the map K : ( C , w ) ⊸ ( L p ( I , E ) , || · || p ) is acyclic, Remark 3. (i) For every relatively weakly compact C ⊂ L q ( I , E ), K : ( C , w ) ⊸ ( L p ( I , E ) , || · || p ) is compactvalued upper semicontinuous i ff given a sequence ( x n , y n ) in the graph Gr( K ) with x n C −−−− ⇀ n →∞ x ,there is a subsequence y k n L p ( I , E ) −−−−−→ n →∞ y ∈ K ( x ). (notice that the space ( C , w ) is sequential as a subsetof the angelic space ( L q ( I , E ) , w ))(ii) (K ) ⇒ (K ).Before we will be able to set forth a result concerning the existence of solutions to inclusion (1), wehave to prove a few auxiliary facts. Lemma 1.
Let p ∈ [1 , ∞ ] . If the multimap F : I × E ⊸ E satisfies conditions (F ) – (F ) , then theNemytskiˇı operator N F : L p ( I , E ) ⊸ L q ( I , E ) is a weakly upper semicontinuous multivalued map withnonempty convex weakly compact values.Proof. For any u ∈ L p ( I , E ) one can always define a sequence ( u n ) n > of simple functions, which con-verges to u almost everywhere and for which | u n ( t ) | | u ( t ) | for every t ∈ I (cf. the proof of [10, TheoremIII.2.22]). Consequently, vertical slices { u n ( t ) } ∞ n = are relatively compact in E for a.a. t ∈ I .Accordingly to the assumption (F ) we can indicate a strongly measurable map w n : I → E such that w n ( t ) ∈ F ( t , u n ( t )) for a.a. t ∈ I . Thanks to condition (F ) we know that the sequence ( w n ) ∞ n = is q -integrably bounded. Let E be a closed separable linear subspace of E such that { u n ( t ) } ∞ n = ∪ { w n ( t ) } ∞ n = ⊂ E a.e. on I . By (F ), the vertical slices { w n ( t ) } ∞ n = are relatively compact a.e. on I . In view of Theorem1 the sequence ( w n ) ∞ n = is relatively weakly compact in L q ( I , E ). Hence we may assume, passing to asubsequence if necessary, that w n L q ( I , E ) −−−−− ⇀ n →∞ w .Observe that for a.a. t ∈ I , the multimap F ( t , · ) is compact valued upper semicontinuous. Considersequences ( x n ) ∞ n = and ( y n ) ∞ n = satisfying x n E −−−−→ n →∞ x and y n ∈ F ( t , x n ). Put E : = span (cid:16) { x n } ∞ n = ∪ { y n } ∞ n = (cid:17) .Then { y n } ∞ n = ⊂ F (cid:16) t , { x n } ∞ n = (cid:17) ∩ E . The latter is relatively compact, in view of (F ). Thus, there exists( y k n ) ∞ n = such that y k n E −−−−→ n →∞ y . Assumption (F ) implies y ∈ F ( t , x ). RADOSŁAW PIETKUN
Applying Theorem 2 one gets w ( t ) ∈ F ( t , u ( t )) for a.a. t ∈ I . In this way we have shown that theNemytskiˇı operator N F has nonempty values.Applying similar reasoning one may prove that given a sequence ( u n , w n ) ∞ n = in the graph Gr( N F ) with u n L p ( I , E ) −−−−−→ n →∞ u , there is a subsequence w k n L q ( I , E ) −−−−− ⇀ n →∞ w ∈ N F ( u ). Indeed, since the family {| u n ( · ) | p } ∞ n = isuniformly integrable, the sequence ( w n ) ∞ n = must be uniformly q -integrable as well ( q -integrably boundedin case of p = ∞ ). Therefore, we may apply weak compactness criterion (Theorem 1) to extract asubsequence ( w k n ) ∞ n = with w k n L q ( I , E ) −−−−− ⇀ n →∞ w . Convergence theorem (Theorem 2) entails w ∈ N F ( u ). (cid:3) Corollary 1.
Let E be a reflexive Banach space space. If the set-valued map F : I × E ⊸ E fulfillsconditions (F ) - (F ) , then the thesis of Lemma 1 holds.Proof. By (F ) the map F ( t , · ) is locally bounded a.e. on I . Consider a sequence ( x n , y n ) n > in the graphGr( F ( t , · )) with x n → x in the norm of E . Since E is reflexive, there must be a subsequence y k n ⇀ y .Bearing in mind (F ), i.e. that Gr( F ( t , · )) is strongly-weakly closed, we obtain y ∈ F ( t , x ). Therefore, F ( t , · ) is weakly upper semicontinuous and possesses weakly compact values for a.a. t ∈ I .Retaining the notation of the previous proof it may be observed that { w n ( t ) } ∞ n = forms a subset of aweakly compact set F (cid:16) t , { u n ( t ) } ∞ n = (cid:17) for a.a. t ∈ I . Now, in manner fully analogous to the mentionedproof, we can use Theorem 1 and Theorem 2 to justify the thesis. (cid:3) Lemma 2.
Assume (K ) and (K ) are satisfied. Let M be a countable subset of L q ( I , E ) such that M ( t ) is relatively compact in E for a.a. t ∈ I. Then the image K ( M ) is relatively compact in L p ( I , E ) and Kis upper semicontinuous from M furnished with the relative weak topology of L q ( I , E ) to L p ( I , E ) with itsnorm topology.Proof. Let M : = { w n } ∞ n = . In view of Pettis measurability theorem there exists a closed separable subspace E of E with w n ( t ) ∈ E a.e. on I for every n >
1. For each k > k -dimensional linear subspace E k ⊂ E such that E k ⊂ E k + and E = S k > E k . Let ρ > Θ ⊂ I such that | w n ( t ) | ρ for every t ∈ I \ Θ and n >
1. Consequently, ∀ n > ∀ k > ∀ t ∈ I \ Θ dist( w n ( t ) , E k ) = dist( w n ( t ) , D E (0 , ρ ) ∩ E k ) . Take ε >
0. Using the formula for the Hausdor ff MNC in a separable Banach space [21, Proposition 2]one sees that β E (cid:16) { w n ( t ) } ∞ n = (cid:17) = lim k →∞ sup n > dist( w n ( t ) , E k ). Thus, in view of Egorov’s theorem, ∃ Θ ⊂ I \ Θ ∃ k ∈ Ž ∀ k > k ∀ t ∈ I \ ( Θ ∪ Θ ) sup n > dist( w n ( t ) , D E (0 , ρ ) ∩ E k ) < β E (cid:16) { w n ( t ) } ∞ n = (cid:17) + ε . Referring once more to Egorov’s theorem we can indicate a measurable Θ ⊂ I and a simple function˜ w n : I → E such that | w n ( t ) − ˜ w n ( t ) | < ε w n ( t ) , D E (0 , ρ ) ∩ E k ) < ε for all t ∈ I \ ( Θ ∪ Θ ∪ Θ ), k > k and n >
1. The latter property comes down eventually to thefollowing:(5) ∀ ε > ∃ k ∈ Ž ∀ k > k ∀ n > ∃ w n , k ∈ L q ( I , D E (0 , ρ ) ∩ E k ) such that || w n − w n , k || q < ε. Fix ε > k > k . The range { w n , k } ∞ n = is relatively weakly compact in L q ( I , E ) in view of Theorem1. Condition (K ) implies the relative compactness of K (cid:16) { w n , k } ∞ n = (cid:17) in L p ( I , E ). Thus, making use of (5) OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 7 and (K ) we arrive at (cid:12)(cid:12)(cid:12)(cid:12) β L p (cid:16) K (cid:16) { w n } ∞ n = (cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) β L p (cid:16) K (cid:16) { w n } ∞ n = (cid:17)(cid:17) − β L p (cid:16) K (cid:16) { w n , k } ∞ n = (cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) h ∞ [ n = K ( w n ) , ∞ [ n = K ( w n , k ) sup n > h ( K ( w n ) , K ( w n , k )) sup n > L || w n − w n , k || q ε L . Since ε was arbitrary, the image K (cid:16) { w n } ∞ n = (cid:17) must be relatively compact.Assume that ( w n , v n ) ∈ Gr( K ) with w n M −−−− ⇀ n →∞ w . As we have shown above the set K ( { w n } ∞ n = ) isrelatively compact. Thus, there exists a subsequence (again denoted by) ( v n ) ∞ n = such that v n L p ( I , E ) −−−−−→ n →∞ v .Our aim is to show that v ∈ K ( w ). Take ε >
0. As previously, we can indicate a sequence ( w ε n ) ∞ n = and a compact subset C ε ⊂ E such that { w ε n } ∞ n = ⊂ L q ( I , C ε ) and || w n − w ε n || q ε L . In view of theweak compactness criterion (Theorem 1), we may assume that w ε n L q ( I , E ) −−−−− ⇀ n →∞ w ε , passing once again to asubsequence if necessary. Clearly, || w − w ε || q ε L due to the weak lower semicontinuity of the norm.Choose y ε n ∈ K ( w ε n ) in such a way that || v n − y ε n || p = dist( v n , K ( w ε n )). Assumption (K ) guarantees that y ε n L p ( I , E ) −−−−−→ n →∞ y ε ∈ K ( w ε ), up to a subsequence. Of course, there is N ∈ Ž such that || v N − v || p ε and || y ε N − y ε || p ε . Now, it is possible to estimatedist( v , K ( w )) || v − v N || p + || v N − y ε N || p + || y ε N − y ε || p + dist( y ε , K ( w )) ε + dist( v N , K ( w ε N )) + ε + h ( K ( w ε ) , K ( w )) ε + h ( K ( w N ) , K ( w ε N )) + h ( K ( w ε ) , K ( w )) ε + L || w N − w ε N || q + L || w − w ε || q ε. Since ε was arbitrary, it follows that v ∈ K ( w ). (cid:3) Lemma 3.
Let X be a compact topological space and Y be a paracompact topological space. Assumethat F : X ⊸ Y is an upper semicontinuous surjective multimap with compact acyclic values and acyclicfibers. Then there is an isomorphism H ∗ ( X ) ≈ H ∗ ( Y ) .Proof. Since X is compact, the product X × Y is a paracompact space. The space Y is regular and themap F is usc so the graph Gr( F ) is a closed subset of X × Y . Thus it is also a paracompact space. Theprojection π : Gr( F ) → X of Gr( F ) onto the domain X is continuous and surjective. It is easy to see that π is a closed map, since F is compact valued and upper semicontinuous. Moreover, the fibers π − ( { x } ) = { x } × F ( x ) are compact acyclic. Hence π is perfect and consequently a proper map. Analogously, theprojection π : Gr( F ) → Y is surjective continuous and the preimage π − ( { y } ) = F − ( { y } ) × { y } is compactacyclic. The map π is also closed, since the domain X is compact. In wiev of Vietoris–Begle mappingtheorem [28, Theorem 6.9.15] it follows that ( π ∗ ) − ◦ π ∗ : H ∗ ( Y ) → H ∗ ( X ) is an isomorphism. (cid:3) Recall that for the sake of convenience we had introduced the letter H to denote the superposition K ◦ N F . Lemma 4.
Let (F ) – (F ) be satisfied. Assume that either E is reflexive and (K ) holds or (K ) – (K ) and (F ) are met. In both cases, the operator H : L p ( I , E ) ⊸ L p ( I , E ) is a compact valued upper semicontin-uous map.Proof. Assume that u n L p ( I , E ) −−−−−→ n →∞ u . Obviously, there exists a subsequence ( u k n ) ∞ n = such that u k n ( t ) E −−−−→ n →∞ u ( t )for a.a. t ∈ I . Let w n ∈ N F ( u n ) and v n ∈ K ( w n ) for n >
1. Observe that the sequence ( w n ) ∞ n = is boundeduniformly q -integrable (or simply bounded for p = ∞ ). RADOSŁAW PIETKUN If E is reflexive, then the map F ( t , · ) is weakly upper semicontinuous and possesses weakly compactvalues a.e. on I . Thus, the sets { w k n ( t ) } ∞ n = are relatively weakly compact for a.a. t ∈ I . The sequence( w k n ) ∞ n = is relatively compact in view of Theorem 1. We may assume, passing again to a subsequence ifnecessary, that w k n L q ( I , E ) −−−−− ⇀ n →∞ w . Condition (K ) implies (cf. Remark 3) that v k n L p ( I , E ) −−−−−→ n →∞ v ∈ K ( w ), up to asubsequence. It is enough to apply Corollary 1 to show that w ∈ N F ( u ). Eventually, v ∈ H ( u ), i.e. theset-valued map H : L p ( I , E ) ⊸ L p ( I , E ) is an upper semicontinuous operator with compact values.If assumption (F ) is met, then the multimap F ( t , · ) is compact valued and upper semicontinuous a.e.on I . In this case the sets { w k n ( t ) } ∞ n = are relatively compact in E for a.a. t ∈ I . Passing to a subsequence ifnecessary, we obtain w k n L q ( I , E ) −−−−− ⇀ n →∞ w . By virtue of Lemma 2 there exists a subsequence (again denoted by)( v k n ) ∞ n = such that v k n L p ( I , E ) −−−−−→ n →∞ v ∈ K ( w ). Lemma 1 implies that w ∈ N F ( u ). This means that v ∈ H ( u ). (cid:3) Lemma 5.
Let U ⊂ L p ( I , E ) and x ∈ U. Assume that F : I × E ⊸ E satisfies (F ) – (F ) . Suppose furtherthat operator H : U ⊸ L p ( I , E ) with uniformly p-integrable range ( or bounded range if p = ∞ ) meetsthe following condition: (6) (cid:16) M ⊂ U and M ⊂ co (cid:0) { x } ∪ H ( M ) (cid:1)(cid:17) = ⇒ M ( t ) is relatively compact for a.a. t ∈ I . Assume also that either E is reflexive and (K ) holds or (K ) – (K ) and (F ) are met. In both the cases,condition (2) is fulfilled.Proof. Assume that M ⊂ U and M ⊂ co (cid:0) { x } ∪ H ( M ) (cid:1) . Consider { v n } ∞ n = ⊂ H ( M ). Let v n ∈ K ( w n ) with w n ∈ N F ( M ). By (6), vertical slices M ( t ) are relatively compact for a.a. t ∈ I .Suppose E is reflexive. Taking into account that { w n ( t ) } ∞ n = ⊂ F ( t , M ( t )) a.e. on I and that F ( t , · ) isweakly upper semicontinuous we see that { w n ( t ) } ∞ n = is relatively weakly compact for a.a. t ∈ I . Since M forms a subset of uniformly p -integrable convex hull co (cid:0) { x } ∪ H ( M ) (cid:1) , the sequence ( w n ) ∞ n = must beuniformly q -integrable. It follows from condition (K ) that the image K (cid:0) { w n } ∞ n = w (cid:1) is compact in L p ( I , E ).Hence the set { v n } ∞ n = is relatively compact. The latter entails the relative compactness of co (cid:0) { x } ∪ H ( M ) (cid:1) and eventually the compactness of the closure M .If conditions (K )–(K ) and (F ) are met, then the map F ( t , · ) is upper semicontinuous, vertical slices { w n ( t ) } ∞ n = are relatively compact a.e. on I and the image K (cid:0) { w n } ∞ n = (cid:1) is relatively compact in the space L p ( I , E ) in view of Lemma 2. Therefore, M is a compact subset of L p ( I , E ). (cid:3) Remark 4.
Clearly, the operator H : U ⊸ L p ( I , E ), which is condensing relative to some monotonenonsingular and regular MNC γ defined on the space L p ( I , E ), satisfies condition (2).The eponymous solvability of operator inclusions of Hammerstein type expresses itself in the follow-ing fixed point principle, formulated in the context of the Bochner space L p ( I , E ). Theorem 6.
Let X be a closed convex subset of the space L p ( I , E ) . Assume that either (i) the space E is reflexive, the operator K : L q ( I , E ) ⊸ X possesses convex or decomposable fibersand satisfies assumption (K ) , the multimap F : I × E ⊸ E meets conditions (F ) – (F ) or (ii) the operator K : L q ( I , E ) ⊸ X possesses compact acyclic values and convex or decomposablefibers and satisfies assumptions (K ) – (K ) , the set-valued map F : I × E ⊸ E meets conditions (F ) – (F ) .Suppose further that there exists a radius R > such that (7) L (cid:18) || b || q + c (cid:16) R + || K (0) || + p (cid:17) pq (cid:19) R OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 9 if p < ∞ and respectively (8) L bR + || K (0) || + ∞ q Rprovided p = ∞ . If the operator H : D L p ( K (0) , R ) ∩ X ⊸ X with uniformly p-integrable range satisfiescondition (6) with x ∈ K (0) , then there exists at least one solution x ∈ D L p ( K (0) , R ) ∩ X of the initialinclusion (1) .Proof.
Fix u ∈ L p ( I , E ). The subset N F ( u ) furnished with the relative weak topology of L q ( I , E ) iscompact (cf. Lemma 1 or Corollary 1). Moreover, ( N F ( u ) , w ) is in fact an acyclic space, given that N F ( u )is always contractible in the weak topology σ ( L q ( I , E ) , L p ( I , E ∗ )) (regardless of whether the values of F are convex or not, because values of the Nemytskiˇı operator are still decomposable). Under assumption(K ) the multimap K : ( N F ( u ) , w ) ⊸ ( H ( u ) , || · || p ) may be regarded as an acyclic operator betweencompact topological space ( N F ( u ) , w ) and a paracompact space ( H ( u ) , || · || p ). The same can be said if weassume that K is Lipschitz with compact acyclic values. Observe that the intersection K − ( { v } ) ∩ N F ( u ) isconvex in case K has convex fibers or decomposable if we assume that the fibers of K are decomposable.Therefore, the fibers of the mutimap under consideration are acyclic. In view of Lemma 3 we are allowedto conclude that the reduced Alexander–Spanier cohomologies ˜ H ∗ (( H ( u ) , || · || p )) are trivial, i.e. the image H ( u ) is acyclic as a subspace of L p ( I , E ).From Lemma 4 follows that the operator H : D L p ( K (0) , R ) ∩ X ⊸ X is compact valued upper semicon-tinuous. Considering what we have established so far, it is apparent that H is an acyclic operator. Lemma5 guarantees that condition (2) is also satisfied.Let p < ∞ and R > u ∈ D L p ( K (0) , R ). Since K (0) is compact(both in the case (i) and in the case (ii)), there is z u ∈ K (0) such that || u − z u || p = dist( u , K (0)). Observethat || N F ( u ) || + q || b || q + c || u || pq p || b || q + c (cid:16) || u − z u || p + || z u || p (cid:17) pq || b || q + c (cid:16) R + || K (0) || + p (cid:17) pq . Whencesup v ∈ H ( u ) dist( v , K (0)) sup w ∈ N F ( u ) h ( K ( w ) , K (0)) sup w ∈ N F ( u ) L || w || q = L || N F ( u ) || + q L (cid:18) || b || q + c (cid:16) R + || K (0) || + p (cid:17) pq (cid:19) . Eventually, H ( D L p ( K (0) , R ) ∩ X ) ⊂ D L p ( K (0) , R ) ∩ X , by (7). The latter entails (3). Indeed, fix any x ∈ K (0), λ ∈ (0 ,
1) and x ∈ D L p ( K (0) , R ). Thensup v ∈ H ( x ) dist((1 − λ ) x + λ v , K (0)) λ sup v ∈ H ( x ) dist( v , K (0)) λ R < R . Thus, x < (1 − λ ) x + λ H ( x ) provided x ∈ ∂ D L p ( K (0) , R ). In analogous manner one can show thatLeray-Schauder boundary condition (3) is satisfied under assumption (8).In view of Theorem 4. we infer that the multifunction H has a fixed point in D L p ( K (0) , R ) ∩ X . (cid:3) Remark 5.
As it comes to formulation of su ffi cient conditions for acyclicity of the values of the superpo-sition K ◦ N F (cf. [7, Remark 4.2]), it should be emphasized that condition: for all w , w , w ∈ L q ( I , E ),the equality K ( w ) = K ( w ) implies K (cid:0) w [0 , λ ] + w [ λ , T ] (cid:1) = K (cid:0) w [0 , λ ] + w [ λ , T ] (cid:1) for every λ ∈ I , is much stronger than assumption regarding the decomposability of the fibers of operator K . Similarly, the condition that operator K is a ffi ne is visibly stronger than the fact that K has convexfibers.
4. E xamples
We conclude this paper with examples, which illustrate the wide range of applications of the unifiedtopological approach, developed in the previous section, to integro-di ff erential inclusions. Example 1.
Given an m -accretive operator A : D ( A ) ⊂ E ⊸ E in a Banach space E and a multivaluedperturbation F : I × co D ( A ) ⊸ E we consider the initial value problem:(9) ˙ u ( t ) ∈ − Au ( t ) + F ( t , u ( t )) on I , u (0) = u . If A is m -accretive and U ( · ) x is an integral solution of (9) with F ≡ u (0) = x , then the family { U ( t ) } t > of nonexpansive mappings U ( t ) : D ( A ) → D ( A ) is called the semigroup generated by − A . Theorem 7.
Let E ∗ be strictly convex and A : D ( A ) ⊂ E ⊸ E be m-accretive such that − A generates anequicontinuous semigroup. Assume that F : I × co D ( A ) ⊸ E satisfies (F ) - (F ) together with (F ′ ) there is µ ∈ L ( I , R ) such that || F ( t , x ) || + µ ( t )(1 + | x | ) for all x ∈ E and for a.a. t ∈ Iand (F ) there is a function η ∈ L ( I , R ) such that for all bounded subsets Ω ⊂ E and for a.a. t ∈ I theinequality holds β ( F ( t , Ω )) η ( t ) β ( Ω ) . Then the Cauchy problem (9) has an integral solution for every u ∈ D ( A ) .Proof. As it is well known, we can associate with any w ∈ L ( I , E ) a unique integral solution S ( w ) ∈ C ( I , D ( A )) of the quasi-autonomous problem(10) ˙ u ( t ) ∈ − Au ( t ) + w ( t ) on I , u (0) = u . The mapping S : L ( I , E ) → C ( I , E ) satisfies | S ( w )( t ) − S ( w )( t ) | | S ( w )( s ) − S ( w )( s ) | + Z ts | w ( τ ) − w ( τ ) | d τ for all 0 s t T , which means in particular that S meets condition (K ).Take w , w ∈ S − ( { u } ) and fix λ ∈ (0 , x , y ) ∈ Gr( A ) and 0 s t T the followinginequality holds | u ( t ) − x | − | u ( s ) − x | Z ts h w i ( τ ) − y , u ( τ ) − x i + d τ, where i = i = S ( w ) and S ( w ), respectively. Since E ∗ is strictly convex, thesemi-inner products are indistinguishable, i.e. h x , y i + = h x , y i − . In view of the latter we are allowed towrite down the following estimation: | u ( t ) − x | − | u ( s ) − x | λ Z ts h w ( τ ) − y , u ( τ ) − x i + d τ + (1 − λ ) 2 Z ts h w ( τ ) − y , u ( τ ) − x i + d τ = Z ts λ h w ( τ ) − y , u ( τ ) − x i + + (1 − λ ) h w ( τ ) − y , u ( τ ) − x i + d τ = Z ts h ( λ w + (1 − λ ) w )( τ ) − y , u ( τ ) − x i + d τ. This means that u constitutes a solution to the quasi-autonomous problem˙ u ( t ) ∈ − Au ( t ) + ( λ w + (1 − λ ) w )( t ) on I , u (0) = u . OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 11
In other words u = S ( λ w + (1 − λ ) w ), i.e. the fiber S − ( { u } ) is convex.Consider a compact subset C ⊂ E and a sequence ( w n ) ∞ n = ⊂ L ( I , C ) such that w n L ( I , E ) −−−−− ⇀ n →∞ w . Since − A generates an equicontinuous semigroup and the family { w n } ∞ n = is uniformly integrable, the image S ( { w n } ∞ n = ) ⊂ C ( I , E ) is equicontinuous. This is the thesis of [15, Theorem 2.3]. In view of Pettismeasurability theorem E : = span S ∞ n = ( w n ( I )) is a separable subspace of E . The arguments contained inthe proof of part (b) of [1, Lemmma 4] justify the following estimate β (cid:16) { S ( w n )( t ) } ∞ n = (cid:17) Z t β E (cid:16) { w n ( s ) } ∞ n = (cid:17) d s for t ∈ I . It should be stressed here that the veracity of this formula is completely independent of thegeometrical properties of the dual space E ∗ , such as the uniform convexity assumed by the author of [1].As a consequence, we get that β (cid:16) { S ( w n )( t ) } ∞ n = (cid:17) = t ∈ I . In view of the Arzelà theorem thesequence ( S w n ) ∞ n = must be uniformly convergent to some v . The extra condition regarding the geometryof the dual space E ∗ makes it possible to demonstrate that S ( w ) = v (cf. [29]). Therefore, operator S meets condition (K ).Let R : = sup t ∈ I | U ( t ) x | . To indicate a priori bounds on the solutions of (9) consider w ∈ N F ( S ( w )).It is easy to see that | S ( w )( t ) | | S (0)( t ) | + | S ( w )( t ) − S (0)( t ) | | U ( t ) x | + Z t | w ( s ) | d s R + || µ || + Z t µ ( s ) | S ( w )( s ) | d s for t ∈ I . Hence, || S ( w ) || ( R + || µ || ) e || µ || = : M for any w ∈ Fix( N F ◦ S ), by the Gronwall inequality. Now, the standard trick allows us to assume that || F ( t , x ) || + µ ( t )(1 + M ) = δ ( t ) a.e. on I with δ ∈ L ( I , R + ). Otherwise we may always replace theright-hand side F by F ( · , r ( · )) with r : E → D E (0 , M ) ∩ co D ( A ) being a retraction.Let X : = L ∞ ( I , co D ( A )). Clearly, X is closed and convex in L ∞ ( I , E ). Define H : X ⊸ X to be thesuperposition H : = S ◦ N F , where N F : X ⊸ L ( I , E ) is the Nemytskiˇı operator corresponding to therigh-hand side F . According to the above observations on the growth of F , if w ∈ L ( I , E ) then || S ( w ) − S (0) || | S ( w )(0) − S (0)(0) | + Z T | w ( s ) | d s || δ || = : R . This means, in particular, that H ( ∂ B L ∞ ( S (0) , R ) ∩ X ) ⊂ D L ∞ ( S (0) , R ) ∩ X . Put U : = B L ∞ ( S (0) , R ) ∩ X . Itfollows that operator H : U ⊸ X satisfies boundary condition (3) with x : = S (0) ∈ U .We claim that operator H meets condition (6). Let M ⊂ U be such that M ⊂ co (cid:0) { S (0) } ∪ H ( M ) (cid:1) .Whence, M ( t ) ⊂ co (cid:0) { S (0)( t ) } ∪ H ( M )( t ) (cid:1) a.e. on I . Since N F ( M ) is uniformly integrable, the image S ( N F ( M )) is equicontinuous (cf. [15, Theorem 2.3]). Hence, the mapping I ∋ t β ( H ( M )( t )) ∈ R + must be continuous. Fix t ∈ I . There exists { w tn } ∞ n = ⊂ N F ( M ) such that β (cid:0) { S ( w tn )( t ) } ∞ n = (cid:1) = max { β ( D ) : D ⊂ H ( M )( t ) countable } . Let E be a closed and separable subspace of E such that { w tn ( τ ) } ∞ n = ⊂ E for a.a. τ ∈ I . Under assumption(F ) the following estimate is easily verifiable: β E ( { w tn ( τ ) } ∞ n = ) β ( { w tn ( τ ) } ∞ n = ) β ( F ( τ, { u tn ( τ ) } ∞ n = )) η ( τ ) β ( { u tn ( τ ) } ∞ n = ) η ( τ ) β ( M ( τ )) η ( τ ) β ( H ( M )( τ ))for a.a. τ ∈ I . As we have noticed previously, the proof of [1, Lemmma 4] constitutes a justification forthe estimation β (cid:0) { S ( w tn )( τ ) } ∞ n = (cid:1) Z τ β E (cid:0) { w tn ( s ) } ∞ n = (cid:1) d s . for τ ∈ I . Hence, β ( H ( M )( t )) { β ( D ) : D ⊂ H ( M )( t ) countable } = β (cid:0) { S ( w tn )( t ) } ∞ n = (cid:1) Z t β E (cid:0) { w tn ( s ) } ∞ n = (cid:1) d s Z t η ( s ) β ( H ( M )( s )) d s . Since the time t was chosen arbitrary, we infer that β ( H ( M )( t )) Z t η ( s ) β ( H ( M )( s )) d s on I . Consequently, sup t ∈ I β ( H ( M )( t )) = M ( t ) are relatively compact in E for a.a t ∈ I .Combining the theses of Lemma 4 and 5 with the argument taken from the proof of Theorem 6, weinfer that operator H : U ⊸ X meets all the requirements imposed by Theorem 4. Therefore, H possessesa fixed point u ∈ U ∩ C ( I , E ). Clearly, u is an integral solution to (9). (cid:3) Remark 6.
Note that our result corresponds exactly to the content of [1, Theorem 2], except for theassumption about the geometry of the dual space E ∗ . As we have seen it was enough to assume that E ∗ is strictly convex. The counter-example given in [1] (cf. [1, Example 1]) shows that this geometriccondition on E ∗ cannot be removed. Example 2.
In this example we consider the following semilinear nonlocal Cauchy problem(11) ˙ x ( t ) ∈ Ax ( t ) + F ( t , x ( t )) on I , x (0) = g ( x ) , where the linear operator A : D ( A ) ⊂ E → E is closed densely defined and g : C ( I , E ) → E is continuous.It is well known that if A poses the infinitesimal generator of a C -semigroup { U ( t ) } t > of bounded linearoperators on E , then there are constants M > ω ∈ R such that || U ( t ) || L Me ω t for any t > E in an appropriate way one can achieve that M = E has property S if for all separable subspaces X ⊂ E there existsa separable subspace Y of E with X ⊂ Y and a continuous linear projection P : E → Y of norm 1. Inparticular, weakly compactly generated Banach spaces have property S . Theorem 8.
Let A be an infinitesimal generator of a C -semigroup { U ( t ) } t > . Assume that F : I × E ⊸ Esatisfies (F ) – (F ) together with (F ′ ) and (F ) , whereas g : C ( I , E ) → E is L-Lipschitz and there is k > such that (12) β ( g ( Ω )) k sup t ∈ I β ( Ω ( t )) for every bounded Ω ⊂ C ( I , E ) . Assume also that there exists a constant r > such that (13) 0 < || µ || < M − ln + r + M ( Lr + | g (0) | ) ! with M : = sup t ∈ I || U ( t ) || L . Then the boundary value problem (11) possesses a mild solution in each of thefollowing cases: (i) M ( k + || η || ) < if E is has property S , (ii) M ( k + || η || ) < if E is an arbitrary Banach space.Proof. Define K : L ( I , E ) → C ( I , E ) to be the map which assigns to each f ∈ L ( I , E ) the unique mildsolution of the following problem(14) ˙ x ( t ) = Ax ( t ) + f ( t ) , a.e. on I , x (0) = g ( x ) . OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 13 K is a well defined single-valued mapping. Indeed, suppose x , y are solutions of (14). Then | x ( t ) − y ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U ( t ) g ( x ) + Z t U ( t − s ) f ( s ) d s − U ( t ) g ( y ) − Z t U ( t − s ) f ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M | g ( x ) − g ( y ) | and consequently || x − y || ML || x − y || . From (13) it follows that ML < x = y .Let f , h ∈ L ( I , E ). Then | K ( f )( t ) − K ( h )( t ) | | U ( t ) g ( K ( f )) + U ( t ) g ( K ( h )) | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t U ( t − s ) f ( s ) d s − Z t U ( t − s ) h ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ML || K ( f ) − K ( h ) || + M Z t | f ( s ) − h ( s ) | d s for t ∈ I . Whence || K ( f ) − K ( g ) || M − ML || f − g || , i.e. operator K is Lipschitz.We will show that operator K satisfies also condition (K ). To see this consider a compact subset C ⊂ E and a sequence ( f n ) ∞ n = ⊂ L ( I , C ) such that f n L ( I , E ) −−−−− ⇀ n →∞ f . In the context of [18, Theorem 3.12(c)]and (12) one may estimate β (cid:16) { K ( f n )( t ) } ∞ n = (cid:17) = β ( U ( t ) g ( K ( f n )) + Z t U ( t − s ) f n ( s ) d s ) ∞ n = ! β (cid:16) { U ( t ) g ( K ( f n )) } ∞ n = (cid:17) + β (Z t U ( t − s ) f n ( s ) d s ) ∞ n = ! Mk sup τ ∈ I β (cid:16) { K ( f n )( τ ) } ∞ n = (cid:17) + M Z t β (cid:16) { f n ( s ) } ∞ n = (cid:17) d s Mk sup τ ∈ I β (cid:16) { K ( f n )( τ ) } ∞ n = (cid:17) + Mt β ( C )for t ∈ I . Eventually, sup t ∈ I β (cid:16) { K ( f n )( t ) } ∞ n = (cid:17) Mk sup t ∈ I β (cid:16) { K ( f n )( t ) } ∞ n = (cid:17) , which means that β (cid:16) { K ( f n )( t ) } ∞ n = (cid:17) = t ∈ I .Fix τ ∈ I and ε >
0. There is δ > M || C || + δ < ε . Let s ∈ I be such that | τ − s | < δ . Since { K ( f n )( s ) } ∞ n = is relatively compact, the family { U ( · ) K ( f n )( s ) } ∞ n = is equicontinuous i.e.,sup n > | U ( t − s ) K ( f n )( s ) − U ( τ − s ) K ( f n )( s ) | −−−→ t → τ . Clearly there is δ ∈ (0 , δ ) such that, for any n > | K ( f n )( t ) − K ( f n )( τ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U ( t − s ) K ( f n )( s ) − U ( τ − s ) K ( f n )( s ) + Z ts U ( t − z ) f n ( z ) d z − Z τ s U ( τ − z ) f n ( z ) d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | U ( t − s ) K ( f n )( s ) − U ( τ − s ) K ( f n )( s ) | + N || C || + | t − s | + N || C || + | τ − s | < ε for t ∈ I with | t − τ | < δ . In other words, { K ( f n ) } ∞ n = is equicontinuous. In view of the Arzelà theorem, wesee that (passing to a subsequence if necessary) K ( f n ) C ( I , E ) −−−−→ n →∞ y . Notice that U ( · ) g ( K ( f n )) C ( I , E ) −−−−→ n →∞ U ( · ) g ( y ).Since the linear part of K ( f n ) tends weakly to R · U ( · − s ) f ( s ) d s , one sees that y ( t ) E ↼ −−−− n →∞ K ( f n )( t ) E −−−− ⇀ n →∞ U ( t ) g ( y ) + Z t U ( t − s ) f ( s ) d s , for t ∈ I . Hence, y = K ( f ), i.e. K : L ( I , C ) → C ( I , E ) is strongly continuous. We claim that K has convex fibers. Fix λ ∈ (0 , f , h ∈ K − ( { x } ), then K ( f ) = K ( h ). From this itfollows that R t U ( t − s )( f − h )( s ) d s = t ∈ I . Hence, | K ( λ f + (1 − λ ) h )( t ) − K ( h )( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U ( t ) g ( K ( λ f + (1 − λ ) h )) − U ( t ) g ( K ( h )) + λ Z t U ( t − s )( f − h )( s ) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M | g ( K ( λ f + (1 − λ ) h )) − g ( K ( h )) | ML || K ( λ f + (1 − λ ) h ) − K ( h ) || for t ∈ I . Consequently, K ( λ f + (1 − λ ) h ) = K ( h ) i.e., λ f + (1 − λ ) h ∈ K − ( { x } ).Condition (13) is nothing more than T Z µ ( s ) d s < r Z M ( Lr + | g (0) | ) ds M (1 + s ) . The latter entails existence of an ( K ◦ N F )-invariant subset W ⊂ C ( I , E ), which has a nonempty andconvex interior (see proof of [32, Theorem 3.3]). Therefore, the operator H : Int W ⊸ C ( I , E ) satisfiescondition (3) (cf. Remark 2(ii)).Let Ω ⊂ W be such that Ω ⊂ co (cid:0) { x } ∪ H ( Ω ) (cid:1) for some x ∈ Int W . Let ˜ β be a measure of non-compactness on E generated by the sequential Hausdor ff MNC i.e.,˜ β ( Ω ) : = max { β ( D ) : D ⊂ Ω denumerable } . Clearly, ˜ β is nonsingular and monotone. Fix t ∈ I . Assume that ˜ β ( H ( Ω )( t )) = β (cid:0) { K ( f n )( t ) } ∞ n = (cid:1) for some { K ( f n ) } ∞ n = ⊂ H ( Ω ). By virtue of [18, Theorem 3.12] one gets˜ β ( H ( Ω )( t ) = β (cid:0) { K ( f n )( t ) } ∞ n = (cid:1) = β ( U ( t ) g ( K ( f n )) + Z t U ( t − s ) f n ( s ) d s ) ∞ n = ! M β (cid:16) g (cid:16) { K ( f n ) } ∞ n = (cid:17)(cid:17) + cM Z t β (cid:16) { f n ( s ) } ∞ n = (cid:17) d s Mk sup τ ∈ I β (cid:16) { K ( f n )( τ ) } ∞ n = (cid:17) + cM Z t η ( s ) sup τ ∈ I ˜ β ( Ω ( τ )) d s Mk sup τ ∈ I ˜ β ( H ( Ω )( τ )) + cM Z t η ( s ) d s sup τ ∈ I ˜ β ( H ( Ω )( τ )) , where c = E has property S and c = t was chosen arbitrary,sup t ∈ I ˜ β ( H ( Ω )( t )) M ( k + c || η || ) sup t ∈ I ˜ β ( H ( Ω )( t )) . Therefore, sup t ∈ I ˜ β ( Ω ( t )) =
0. Clearly, ˜ β − ( { } ) is contained in the family of relatively compact subsets ofthe space E . This means that Ω ( t ) is relatively compact for every t ∈ I . In other words, condition (6) issatisfied.In view of Theorem 6 the set-valued operator H : Int W ⊸ C ( I , E ) is admissible. From Theorem 4 itfollows that H has at least one fixed point. This fixed point constitutes a solution of the nonlocal Cauchyproblem (11). (cid:3) Remark 7.
The preceding result may be treated as a refinement of [32, Theorem 3.8]. Specifically, wedid not assume neither equicontinuity of the semigroup { U ( t ) } t > nor that F ( t , · ) must be upper semicon-tinuous. Example 3.
For the third example we consider the following so-called Hammerstein integral inclusion:(15) x ( t ) ∈ h ( t ) + Z T k ( t , s ) F ( s , x ( s )) d s a.e. on I OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 15 with h ∈ L p ( I , E ). We shall assume the following hypotheses about the kernel mapping k : I → L ( E ) :(k ) the function k : I × I → L ( E ) is strongly measurable in a product measure space,(k ) for every t ∈ I , k ( t , · ) ∈ L r ( I , L ( E )) with r ∈ (1 , ∞ ] being the conjugate exponent of q , i.e. q − + r − = ) the function I ∋ t k ( t , · ) ∈ L r ( I , L ( E )) belongs to L p ( I , L r ( I , L ( E ))). Theorem 9.
Let q p < ∞ . Assume that k : I → L ( E ) satisfies (k ) – (k ) , while F : I × E ⊸ Efulfills (F ) – (F ) together with (F ′ ) for every closed separable subspace E of E there exists a function η E ∈ L pqp − q ( I , R ) such that forall bounded subsets Ω ⊂ E and for a.a. t ∈ I the inequality holds β E ( F ( t , Ω ) ∩ E ) η E ( t ) β E ( Ω ) . If there is an R > such that (16) || || k ( t , · ) || r || p (cid:18) || b || q + c (cid:16) R + || h || p (cid:17) pq (cid:19) Rand (17) || || k ( t , · ) || r || p || η E || pqp − q < , then the integral inclusion (15) has at least one p-integrable solution.Proof. Define the external operator K : L q ( I , E ) → L p ( I , E ) in the following way(18) K ( w )( t ) : = h ( t ) + Z T k ( t , s ) w ( s ) d s , t ∈ I . Since, K is a ffi ne, it has convex fibers. It is clear that || K ( w ) − K ( w ) || p || || k ( t , · ) || r || p || w − w || q for any w , w ∈ L q ( I , E ). Thus, K meets condition (K ).To see that (K ) is also satisfied, consider a compact subset C ⊂ E and a sequence ( w n ) ∞ n = ⊂ L q ( I , C )such that w n L q ( I , E ) −−−−− ⇀ n →∞ w . Observe thatsup n > | K ( w n )( t ) | | h ( t ) | + || k ( t , · ) || r sup n > || w n || q a.e. on I , which means that the family { K ( w n ) } ∞ n = is p -integrably bounded. Moreover,sup n > | k ( t , s ) w n ( s ) | || k ( t , s ) || L || C || + a.e. on I and β (cid:16) { k ( t , s ) w n ( s ) } ∞ n = (cid:17) || k ( t , s ) || L β (cid:16) { w n ( s ) } ∞ n = (cid:17) = I . Hence, by [16, Corollary 3.1], we have β (cid:16) { K ( w n )( t ) } ∞ n = (cid:17) = β ( h ( t ) + Z T k ( t , s ) w n ( s ) d s ) ∞ n = ! β (Z T k ( t , s ) w n ( s ) d s ) ∞ n = ! Z T t = t ∈ I . Employing [16, Corollary 3.1] again one obtains: β (Z ts K ( w n )( τ ) d τ ) ∞ n = ! Z ts τ = for every 0 < s < t < T . In other words the sets nR ts K ( w n )( τ ) d τ o ∞ n = are relatively compact in E . On theother hand, the following estimate holds Z T − h | K ( w n )( t + h ) − K ( w n )( t ) | p d t Z T − h Z T || k ( t + h , s ) − k ( t , s ) || L | w n ( s ) | d s ! p d t sup n > || w n || pq Z T − h || k ( t + h , · ) − k ( t , · ) || pr d t . Bearing in mind that the singleton set { t k ( t , · ) } is compact in L p ( I , L r ( I , L ( E ))), the following con-vergence is self-evident sup n > || w n || pq Z T − h || k ( t + h , · ) − k ( t , · ) || pr d t −−−−→ h → + . Therefore, the set { K ( w n ) } ∞ n = is p -equiintegrable. In view of [13, Theorem 2.3.6], K ( w n ) L p ( I , E ) −−−−−→ n →∞ y , up toa subsequence. At the same time y ( t ) E ↼ −−−− n →∞ K ( w n )( t ) E −−−− ⇀ n →∞ h ( t ) + Z T k ( t , s ) w ( s ) d s , for t ∈ I . Eventually, K ( w n ) L p ( I , E ) −−−−−→ n →∞ K ( w ).Let R > H : D L p ( h , R ) ⊸ L p ( I , E ) be defined as usual as H : = K ◦ N F . Take u ∈ D L p ( h , R ) and w ∈ N F ( u ). Then | K ( w )( t ) | | h ( t ) | + Z T || k ( t , s ) || L | w ( s ) | d s | h ( t ) | + Z T || k ( t , s ) || L (cid:16) b ( s ) + c | u ( s ) | pq (cid:17) d s | h ( t ) | + || k ( t , · ) || r (cid:18) || b || q + c || u || pq p (cid:19) | h ( t ) | + || k ( t , · ) || r (cid:18) || b || q + c (cid:16) R + || h || p (cid:17) pq (cid:19) , (19)i.e. the range of the operator H is uniformly p -integrable. Note that the assumption (16) is nothing butcondition (7) formulated in the context of the Hammerstein inclusion (15).Let M ⊂ D L p ( h , R ) be a countable subset of co( { h } ∪ H ( M )). Then there is a subset { v n } ∞ n = ⊂ H ( M )such that M ⊂ co (cid:16) { h } ∪ { v n } ∞ n = (cid:17) . Assume that v n = K ( w n ) and w n ∈ N F ( u n ) with u n ∈ M . In view of thePettis measurability theorem there exists a closed linear separable subspace E of E such that { h ( t ) } ∪ { v n ( t ) } ∞ n = ∪ { w n ( t ) } ∞ n = ∪ (Z T k ( t , s ) w n ( s ) d s ) ∞ n = ∪ M ( t ) ⊂ E for a.a. t ∈ I . Let µ ∈ L p ( I , R ) be such that µ ( t ) : = | h ( t ) | + || k ( t , · ) || r (cid:18) || b || q + c (cid:16) R + || h || p (cid:17) pq (cid:19) . From (19) it follows that | v n ( t ) | µ ( t ) a.e. on I . Since M ( t ) ⊂ co (cid:16) { h ( t ) } ∪ { v n ( t ) } ∞ n = (cid:17) , we infer that | u n ( t ) | µ ( t ) a.e. on I for every n >
1. Eventually,sup n > | w n ( t ) | sup n > || F ( t , u n ( t )) || + b ( t ) + c sup n > | u n ( t ) | pq b ( t ) + c µ ( t ) pq a.e. on I , i.e. the family { w n } ∞ n = is q -integrably bounded. Observe that operator K meets assumptions of [7, Lemma4.3]. Particularly, condition (S1) is satisfied for the kernel ˜ k : I → R + such that ˜ k ( t , s ) : = || k ( t , s ) || L .Therefore,(20) β E (cid:16) { v n ( t ) − h ( t ) } ∞ n = (cid:17) = β E (Z T k ( t , s ) w n ( s ) d s ) ∞ n = ! Z T || k ( t , s ) || L β E (cid:16) { w n ( s ) } ∞ n = (cid:17) d s . OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 17
Under assumption (F ′ ) the following estimate holds: β E (cid:16) { w n ( s ) } ∞ n = (cid:17) β E (cid:16) F (cid:16) s , { u n ( s ) } ∞ n = (cid:17) ∩ E (cid:17) η E ( s ) β E (cid:16) { u n ( s ) } ∞ n = (cid:17) η E ( s ) β E ( M ( s )) η E ( s ) β E (cid:16) { v n ( s ) − h ( s ) } ∞ n = (cid:17) . Using the latter in the context of (20), one gets β E (cid:16) { v n ( · ) − h ( · ) } ∞ n = p || || k ( t , · ) || r || p || η E || pqp − q β E (cid:16) { v n ( · ) − h ( · ) } ∞ n = p . Now, assumption (17) entails β E (cid:16) { v n ( t ) − h ( t ) } ∞ n = (cid:17) = I . This means that vertical slices { v n ( t ) } ∞ n = are relatively compact for a.a. t ∈ I . Consequently, M ( t ) is relatively compact a.e. on I . In this regard,the following condition is met(21) (cid:16) M ⊂ U countable and M ⊂ co (cid:16) { x } ∪ H ( M ) (cid:17)(cid:17) = ⇒ M ( t ) is relatively compact for a.a. t ∈ I . It may be proven in a strictly analogous way to Lemma 5 that (21) entails M ⊂ U , M ⊂ co ( { x } ∪ H ( M ))and M = C with C ⊂ M countable = ⇒ M is compact . Therefore, operator H : D L p ( h , R ) ⊸ L p ( I , E ) satisfies condition (4). Since K is linear continuous, H is acompact convex valued upper semicontinuous map. In view of Theorem 5 there exists a solution of theHammerstein integral inlusion (15), contained in D L p ( h , R ). (cid:3) Remark 8.
The case p = ∞ is a little bit tricky as far as it comes to proving the relative compactness of { K ( w n ) } ∞ n = . The easiest way to avoid such speculations is to impose the following assumption(k ′ ) the function I ∋ t k ( t , · ) ∈ L r ( I , L ( E )) is continuousand to make use of the classical Arzelà criterion. Remark 9.
The above proven result is essentially [7, Corollary 4.5] formulated in the context of the inte-gral inclusion (15). The issue of the existence of continuous solutions to Hammerstein integral inclusionis well established in the literature of the subject. One such result was proved by the author in [27].
Example 4.
The following Volterra integral inclusion(22) x ( t ) ∈ h ( t ) + Z t k ( t , s ) F ( s , x ( s )) d s , a.e. on I is a special case of the problem (15) with k : I → L ( E ) such that k ( t , s ) = t < s . The subsequentexistence result concerning inclusion (22) stems from the application of Theorem 6. Theorem 10.
Let h ∈ L p ( I , E ) . Suppose that all assumptions of Theorem 9 are satisfied with the exceptionof (17) . Then the integral inclusion (22) possesses a p-integrable solution.Proof. In the Volterra case, the mapping K : L q ( I , E ) → L p ( I , E ) should by defined as follows: K ( w )( t ) : = h ( t ) + Z t k ( t , s ) w ( s ) d s , t ∈ I . In order to demonstrate the thesis it is su ffi cient to give reason for condition (21). The proof of thisproperty goes exactly the same as previously until one reaches the estimate (20). Here, we have Z t β E (cid:16) { v n ( s ) − h ( s ) } ∞ n = (cid:17) p d s = Z t β E (Z s k ( s , τ ) w n ( τ ) d τ ) ∞ n = ! p d s Z t Z s || k ( s , τ ) || L β E (cid:16) { w n ( τ ) } ∞ n = (cid:17) d τ ! p d s Z t Z s || k ( s , τ ) || L η E ( τ ) β E (cid:16) { v n ( τ ) − h ( τ ) } ∞ n = (cid:17) d τ ! p d s Z t || k ( s , · ) || pr || η E || p pqp − q Z s β E (cid:16) { v n ( τ ) − h ( τ ) } ∞ n = (cid:17) p d τ d s for every t ∈ I . Hence, β E (cid:16) { v n ( t ) − h ( t ) } ∞ n = (cid:17) = I by the Gronwall inequality. This shows thatcondition (21) is fulfilled in the Volterra case as well. (cid:3) Remark 10.
Of course, modifying the assumptions about the integral kernel k accordingly, it is notdi ffi cult to demonstrate the existence of continuous solutions to Volterra inclusion (cf. [27, Theorem 5]). Corollary 2.
Let p = q < ∞ and h ∈ L p ( I , E ) . Suppose that all assumptions of Theorem 9 aresatisfied with the exception of (16) and (17) . Then the integral inclusion (22) possesses a p-integrablesolution.Proof. Exclusion of assumption (16) means for us necessity of showing that H : D L p ( h , R ) ⊸ L p ( I , E )satisfies boundary condition (3) for some radius R >
0. To this aim, put R : = − p || || k ( t , · ) || r || p (cid:16) || b || p + c || h || p (cid:17) e p p − c p || || k ( t , · ) || r || pp . Assume that u ∈ ∂ D L p ( h , R ). If λ ( u − h ) ∈ H ( u ) − h , then λ | u ( t ) − h ( t ) | || k ( t , · ) || r || b || p + c || k ( t , · ) || r Z t ( | u ( s ) − h ( s ) | + | h ( s ) | ) p d s ! p for a.a. t ∈ I . Whence Z t | u ( s ) − h ( s ) | p d s λ − p p − || || k ( t , · ) || r || pp (cid:16) || b || p + c || h || p (cid:17) p + λ − p p − c p Z t || k ( s , · ) || pr Z s | u ( τ ) − h ( τ ) | p d τ d s for every t ∈ I . The Gronwall inequality implies Z t | u ( s ) − h ( s ) | p d s λ − p p − || || k ( t , · ) || r || pp (cid:16) || b || p + c || h || p (cid:17) p exp λ − p p − c p Z t || k ( s , · ) || pr d s ! for t ∈ I . Eventually, the following estimation holds(23) || u − h || p λ − p || || k ( t , · ) || r || p (cid:16) || b || p + c || h || p (cid:17) e p λ − p p − c p || || k ( t , · ) || r || pp . If λ > || u − h || p < − p || || k ( t , · ) || r || p (cid:16) || b || p + c || h || p (cid:17) e p p − c p || || k ( t , · ) || r || pp , which is in contradiction with the definition of radius R . Therefore λ (cid:3) Example 5.
Let ( H , | · | H ) be a separable Hilbert space and ( E , | · | E ) be a subspace of H carrying the struc-ture of a separable reflexive Banach space, which embeds into H continuously and densly. Identifying H with its dual we obtain E ֒ → H ֒ → E ∗ , with all embeddings being continuous and dense. Such a triple ofspaces is usually called evolution triple ([31, p. 416]). Let us note that since E ֒ → H ֒ → E ∗ continuously, OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 19 there exist constants L , M > | · | H L | · | E and | · | E ∗ M | · | H . Without any loss of generalityand to simplify our calculations we may take L = M = x ( t ) + A ( t , x ( t )) + F ( t , x ( t )) ∋ h ( t ) , a.e. on I : = [0 , , x (0) = x (1) , where A : I × E ⊸ E ∗ , F : I × H ⊸ H and h ∈ L q ( I , E ∗ ).Let 2 p < ∞ and 1 < q E : = L p ( I , E ), E ∗ : = L q ( I , E ∗ ), H : = L p ( I , H ), H ∗ : = L q ( I , H ), hh· , ·ii : = h· , ·i E ∗ × E (the dualitybrackets for the pair ( E ∗ , E )).The symbol W p stands for the Bochner-Sobolev space W p (0 , E , H ) : = { x ∈ E : ˙ x ∈ E ∗ } , where the derivative ˙ x is understood in the sense of vectorial distributions. Denote by N A : E ⊸ E ∗ theNemytskiˇı operator corresponding to the multimap A . By a solution of (24) we mean a function x ∈ W p such that ˙ x ( t ) + g ( t ) + f ( t ) = h ( t ) a.e. on I , x (0) = x (1) with g ∈ N A ( x ) and f ∈ N F ( x ).Our hypothesis on the operator A : I × E ⊸ E ∗ are as follows:(A ) for every ( t , x ) ∈ I × E the set A ( t , x ) is nonempty closed and convex,(A ) the map t A ( t , x ) has a measurable selection for every x ∈ E ,(A ) for a.a. t ∈ I , the operator A ( t , · ) : E ⊸ E ∗ is hemicontinuous (i.e. λ A ( t , x + λ y ) is usc from[0 ,
1] into ( E ∗ , w ) for all x , y ∈ E ),(A ) the map x A ( t , x ) is monotone,(A ) there exists a nonnegative function a ∈ L q ( I , R ) and a constant ˆ c > x ∈ E andfor a.a. t ∈ I , || A ( t , x ) || + E ∗ a ( t ) + ˆ c | x | pq E , (A ) there exists a constant d > x ∈ E and a.e. on I , d | x | pE h A ( t , x ) , x i − : = inf {h y , x i : y ∈ A ( t , x ) } . Theorem 11.
Let ( E , H , E ∗ ) be an evolution triple such that E ֒ → H compactly. Assume that conditions (A ) – (A ) and (F ) – (F ) are satisfied. Suppose further that the following inequality holds (25) c < d . Then problem (24) has at least one solution. Moreover, these solutions form a compact subset of thespace ( H , || · || H ) .Proof. Let A h : I × E ⊸ E ∗ be defined by A h ( t , x ) : = A ( t , x ) − h ( t ). Evidently, the multimap A h meetsconditions (A )-(A ), with the proviso that || A h ( t , x ) || + E ∗ ( a ( t ) + | h ( t ) | E ∗ ) + ˆ c | x | pq E and h A h ( t , x ) , x i − > d | x | pE − | h ( t ) | E ∗ | x | E . Observe that conditions (F )-(F ) are fulfilled by the multimap A h . Indeed, assumptions (A ), (A ), (A )entail the strong-weak sequential closedness of the graph of A h while weak upper semicontinuity of themap x A h ( t , x ) follows from assumptions (A ) and (A ) (see [13, Proposition 3.2.18]). Therefore, theNemytskiˇı operator N A h : E ⊸ E ∗ is a convex weakly compact valued weakly upper semicontinuous map(cf. Corollary 1). As such it is maximal monotone (cf. [13, Proposition 3.2.19]).Denote by the symbol L : D ( L ) → E ∗ a continuous linear di ff erential operator L : = ddt , with the domain D ( L ) : = n x ∈ W p : x (0) = x (1) o being a closed linear subspace of W p . In view of [31, Proposition 32.10] this operator is maximal mono-tone as well. Therefore, the sum L + N A h : D ( L ) ⊸ E ∗ must be maximal monotone (by [13, Theorem3.2.41]). Since hh Lx , x ii + hh N A h ( x ) , x ii − || x || E > d || x || p E − || h || E ∗ || x || E || x || E = d || x || p − E − || h || E ∗ −−−−−−−→ || x || E → + ∞ + ∞ , the map L + N A h is coercive and the equality ( L + N A h )( D ( L )) = E ∗ follows (cf. [13, Cor.3.2.31]). It isan immediate consequence of definition that the set-valued inverse operator ( L + N A h ) − : E ∗ ⊸ E is alsomaximal monotone, i.e. hh x − y , x − y ii > ∀ ( x , x ) ∈ Gr(( L + N A h ) − ) = ⇒ ( y , y ) ∈ Gr(( L + N A h ) − ) ⊂ E ∗ × D ( L ) . It is easy to realize that (24) is in fact an inclusion of Hammerstein type. Since Lx ∈ − N A h ( x ) + N ( − F ) ( x ) ⇔ x ∈ ( L + N A h ) − ◦ N ( − F ) ( x ) , it is fully understandable that the external operator K : H ∗ ⊸ H should be defined as K : = ( L + N A h ) − .If we set H : H ⊸ H to be H : = K ◦ N ( − F ) , then the value of H at point u is a solution set of the periodicproblem(26) ˙ x ( t ) + A ( t , x ( t )) + F ( t , u ( t )) ∋ h ( t ) , a.e. on Ix (0) = x (1) . Notice that K − ( { x } ) ∩ N ( − F ) ( u ) = (cid:0) L + N A h (cid:1) ( x ) ∩ N ( − F ) ( u ) for any x ∈ H ( u ). This means that operator K possesses convex fibers as a map from N ( − F ) ( u ) onto the image H ( u ), which is exactly what we needto be able to apply Theorem 6.Let w n ⇀ w in H ∗ and x n ∈ K ( w n ) for n >
1. This means that x n ∈ D ( L ) and Lx n + z n = w n for some z n ∈ N A h ( x n ). Since { w n } n > is bounded in H ∗ , we see that || Lx n || E ∗ || z n || E ∗ + || w n || E ∗ || N A h ( x n ) || + E ∗ + || w n || H ∗ || a || q + || h || E ∗ + ˆ c || x n || p − E + sup n > || w n || H ∗ and d || x n || p E − || h || E ∗ || x n || E hh z n , x n ii + hh Lx n , x n ii = hh w n , x n ii || w n || E ∗ || x n || E sup n > || w n || H ∗ || x n || E , i.e. || x n || p − E d − sup n > || w n || H ∗ + || h || E ∗ ! . Thus, the sequence ( x n ) ∞ n = is bounded in W p and we may assume, passing to a subsequence if necessary,that x n W p −−−− ⇀ n →∞ x . Moreover, there is a subsequence (again denoted by) ( z n ) ∞ n = such that z n E ∗ −−−− ⇀ n →∞ z . Since W p embeds into H compactly (see [13, Theorem 2.2.30]), we infer that ( x n ) n > tends strongly, up to asubsequence, to x in the norm of H . Observe that hh ˙ x n , x n − x ii = hh ˙ x , x n − x ii and solim sup n →∞ hh z n , x n − x ii lim sup n →∞ hh− ˙ x n , x n − x ii + lim sup n →∞ hh w n , x n − x ii = lim n →∞ hh− ˙ x , x n − x ii + lim sup n →∞ h w n , x n − x i H ∗ × H + lim sup n →∞ || w n || H ∗ || x n − x || H sup n > || w n || H ∗ lim n →∞ || x n − x || H = . Hence,(27) lim sup n →∞ hh z n , x n ii hh z , x ii . OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 21
Recalling the decisive monotonicity trick one easily sees that z ∈ N A h ( x ). Indeed, by (27) we have hh z − z , x − x ii = hh z , x ii − hh z , x ii − hh z − z , x ii > lim sup n →∞ ( hh z n , x n ii − hh z , x n ii − hh z n − z , x ii ) = lim sup n →∞ hh z n − z , x n − x ii > x , z ) ∈ Gr( N A h ). Since N A h is maximal monotone, we get z ∈ N A h ( x ). The weak convergence x n W p −−−− ⇀ n →∞ x entails x n C ( I , H ) −−−− ⇀ n →∞ x . Thus, x n ( t ) H −−−− ⇀ n →∞ x ( t ) for every t ∈ I . This means that the boundarycondition x (0) = x (1) is satisfied. Since Lx + z E ∗ ↼ −−−− n →∞ Lx n + z n = w n E ∗ −−−− ⇀ n →∞ w , it follows that x ∈ K ( w ). This justifies the claim that the external operator K : H ∗ ⊸ H is a stronglyupper semicontinuous map with compact values. Further, maximal monotonicity of K implies also con-vexity of its values (cf. [13, Proposition 3.2.7]). In other words, operator K satisfies assumption (K ).It is easy to see that operator H maps bounded sets into relatively compact ones. Indeed, let x ∈ H ( u )and w ∈ N ( − F ) ( u ) be such that x ∈ K ( w ). Then || Lx || E ∗ || − N A h ( x ) + w || + E ∗ || N A h ( x ) || + E ∗ + || w || E ∗ || N A h ( x ) || + E ∗ + || w || H ∗ || a || q + || h || E ∗ + ˆ c || x || p − E + || b || q + c || u || p − H . (29)and d || x || p E − || h || E ∗ || x || E hh N A h ( x ) , x ii − hh w , x ii − hh Lx , x ii || w || E ∗ || x || E − (cid:16) || b || q + c || u || p − H (cid:17) || x || E . If u ∈ D H (0 , R ), then(30) || x || p − E d − (cid:16) || b || q + cR p − + || h || E ∗ (cid:17) . Taking into account (29), we arrive at the estimate || Lx || E ∗ || a || q + || h || E ∗ + || b || q + (1 + d − ˆ c ) cR p − + d − ˆ c (cid:16) || b || q + || h || E ∗ (cid:17) . Therefore, the image H ( D H (0 , R )) is bounded in W p . Since the embedding W p ֒ → H is compact, theset H ( D H (0 , R )) must have a compact closure in the space H . This means in particular that operator H : D H (0 , R ) ⊸ H satisfies condition (2).In order to complete the proof we will choose a radius R > u < λ H ( u ) on ∂ D H (0 , R ) for all λ ∈ (0 , R : = || b || q + || h || E ∗ d − c ! qp . This definition is correct, since we have assumed (25). Suppose that u ∈ ∂ D H (0 , R ) and λ u ∈ H ( u ), i.e. L ( λ u ) + w ∈ − N A h ( λ u ) for some w ∈ N F ( u ). One easily sees that d || λ u || p E − || h || E ∗ || λ u || E hh N A h ( λ u ) , λ u ii − hh− L ( λ u ) − w , λ u ii || N F ( u ) || + E ∗ || λ u || E (cid:16) || b || q + c || u || p − H (cid:17) || λ u || E (31)and so d || u || p − H d || u || p − E λ − p (cid:16) || b || q + c || u || p − H + || h || E ∗ (cid:17) . The latter implies that λ
1. Otherwise, the following inequality would have to be satisfied R < || b || q + || h || E ∗ d − c ! qp , which contradicts the definition of radius R . Therefore, operator H : D H (0 , R ) ⊸ H satisfies the bound-ary condition (3). By virtue of Theorem 6 operator H possesses at least one fixed point. Of course, this fixed pointconstitutes a solution of the periodic problem (24).The preceding estimation ensures also that Fix( H ) ⊂ D H (0 , R ). Since H is completely continuous, thefixed point set Fix( H ) is closed and the image of this set H (Fix( H )) possesses a compact closure. Giventhat Fix( H ) ⊂ H (Fix( H )), the solution set of the periodic problem (24) must be compact as a subset ofthe space H . (cid:3) Corollary 3.
Under assumptions of Theorem 11. the solution set of the periodic problem (24) is nonemptyand compact in the norm topology of the space C ( I , H ) .Proof. Suppose that u n ∈ H ( u n ). Since the fixed point set Fix( H ) is bounded in W p (as we have actuallyproved that previously), we may assume, passing to a subsequence if necessary, that u n W p −−−− ⇀ n →∞ u . Let( u k n ) n > be a subsequence strongly convergent to u in H . We know already that the limit point u belongsto Fix( H ). Thus, it is su ffi cient to show that sup t ∈ I | u k n ( t ) − u ( t ) | H → n → ∞ .Let w ∈ N F ( u ) be such that ˙ u ( t ) + w ( t ) − h ( t ) ∈ − A ( t , u ( t )) a.e. on I . Clearly, there are w k n ∈ N F ( u k n )fulfilling w k n H ∗ −−−− ⇀ n →∞ w and ˙ u k n ( t ) + w k n ( t ) − h ( t ) ∈ − A ( t , u k n ( t )) for a.a t ∈ I . Whence h ˙ u k n ( t ) + w k n ( t ) − ˙ u ( t ) − w ( t ) , u k n ( t ) − u ( t ) i , by (A ). Accordingly, h ˙ u k n ( t ) − ˙ u ( t ) , u k n ( t ) − u ( t ) i h w ( t ) − w k n ( t ) , u k n ( t ) − u ( t ) i for a.a. t ∈ I and for every n > u k n ( t ) H −−−−→ n →∞ u ( t ) a.e. on I , at least for a subsequence. Fix t ∈ I such that u k n ( t ) H −→ u ( t ).Engaging the integration by parts formula in W p we get12 (cid:16) | u k n (1) − u (1) | H − (cid:12)(cid:12)(cid:12) u k n ( t ) − u ( t ) | H (cid:17) = Z t h ˙ u k n ( s ) − ˙ u ( s ) , u k n ( s ) − u ( s ) i d s Z t h w ( s ) − w k n ( s ) , u k n ( s ) − u ( s ) i d s = Z t h w ( s ) − w k n ( s ) , u k n ( s ) − u ( s ) i H d s Z | w ( s ) − w k n ( s ) | H | u k n ( s ) − u ( s ) | H d s sup n > || w k n − w || H ∗ ! || u k n − u || H . Thus | u k n (0) − u (0) | H = | u k n (1) − u (1) | H −−−−→ n →∞ . Now, it is clear that0 lim n →∞ sup t ∈ I | u k n ( t ) − u ( t ) | H sup n > || w k n − w || H ∗ ! lim n →∞ || u k n − u || H + lim n →∞ | u k n (0) − u (0) | H = u k n ) n > is norm convergent in C ( I , H ). (cid:3) In many actual parabolic problems the presence of nonmonotone terms depending on lower-orderderivatives forces us to think over the situation where the right-hand side F is defined only on I × E and not on I × H . Application of fixed point approach in this context in conjunction with lightweightcompetence leads to over-restrictive assumptions, as illustrated by the following: Theorem 12.
Let ( E , H , E ∗ ) be an evolution triple such that E ֒ → H compactly. Assume that the multi-function F : I × E ⊸ H is such that (i) for every ( t , x ) ∈ I × E the set F ( t , x ) is nonempty and convex, (ii) the map F ( · , x ) has a strongly measurable selection for every x ∈ E, OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 23 (iii) the graph
Gr( F ( t , · )) is sequentially closed in ( E , w ) × ( H , w ) for a.a. t ∈ I, (iv) there is a function b ∈ L q ( I , R ) and c > such that for all x ∈ E and for a.a. t ∈ I, || F ( t , x ) || + H b ( t ) + c | x | pq E . If hypotheses (A ) – (A ) hold and h ∈ E ∗ is such that (32) c + || b || q + || h || E ∗ d , c + ˆ c + || a || q + || b || q + || h || E ∗ , then problem (24) has a solution.Proof. Let us use symbols K : H ∗ ⊸ W p and N ( − F ) : W p ⊸ H ∗ to denote the operators we have definedpreviously in the proof of Theorem 11. To tackle the problem of finding solutions of evolution inclusion(24) we shall indicate a fixed point of the superposition H : = K ◦ N ( − F ) .Assume that u n W p −−−− ⇀ n →∞ u and x n ∈ H ( u n ) for n >
1. Let w n ∈ H ∗ be such that w n ∈ N ( − F ) ( u n ) and x n ∈ K ( w n ). Put J : = n t ∈ I : sup n > | u n ( t ) | E = + ∞ o . Since E is reflexive and the sequence ( u n ) ∞ n = isrelatively weakly compact in the Bochner space L ( I , E ), it must be uniformly integrable in view ofDunford-Pettis theorem ([13, Theorem 2.3.24]). If t ∈ J , then for every λ > n ∈ N such that | u n ( t ) | E > λ . Fix λ >
0. The following estimation can be easily verified λ ℓ ( J ) = Z J λ d t Z {| u n | E > λ } λ d t Z {| u n | E > λ }| u n ( t ) | E d t sup n > Z {| u n | E > λ }| u n ( t ) | E d t . Now, if ℓ ( J ) >
0, then lim λ → + ∞ sup n > Z {| u n | E > λ }| u n ( t ) | E d t = lim λ → + ∞ λ ℓ ( J ) = + ∞ , which contradicts the uniform integrability of the sequence ( u n ) ∞ n = . Therefore, ℓ ( t ∈ I : sup n > | u n ( t ) | E < + ∞ )! = ℓ ( I ) . Since the embedding W p ֒ → H is compact, we may assume that u n H −−−−→ n →∞ u . Thus, there exists a subset I of full measure in I such that u k n ( t ) H −−−−→ n →∞ u ( t ) and at the same time sup n > | u k n ( t ) | E < + ∞ for t ∈ I .Fix t ∈ I . For every bounded subsequence ( u l kn ( t )) ∞ n = ) there is a weakly convergent in E subsequence( u m lkn ( t )) ∞ n = . Clearly, u m lkn ( t ) E −−−− ⇀ n →∞ u ( t ) and eventually u k n ( t ) E −−−− ⇀ n →∞ u ( t ). Therefore, u k n ( t ) E −−−− ⇀ n →∞ u ( t )a.e. on I .Using reflexivity of the space H and the fact that F has sublinear growth and x F ( t , x ) is sequentiallyclosed in ( E , w ) × ( H , w ) one can easily show that given a sequence ( x n , y n ) in the graph Gr( F ( t , · ) with x n E −−−− ⇀ n →∞ x , there is a subsequence y k n H −−−− ⇀ n →∞ y ∈ F ( t , x ). This means, in particular, that F ( t , · ) is weaklysequentially upper hemi-continuous multimap for a.a. t ∈ I .Observe that || w k n || H ∗ || b || q + c sup n > || u k n || p − E for every n >
1. Of course, there is a subsequence(again denoted by) ( w k n ) ∞ n = such that − w k n H ∗ −−−− ⇀ n →∞ − w . Hence, by the Convergence Theorem (cf. [3,Lemmma 1]), w ( t ) ∈ − F ( t , u ( t )) a.e. on I .As we have already shown in the proof of Theorem 11, there must be a subsequence ( x l kn ) ∞ n = of thesequence ( x k n ) ∞ n = such that x l kn W p −−−− ⇀ n →∞ x ∈ K ( w ). Since w ∈ N ( − F ) ( u ), it follows that x ∈ H ( u ). The conducted reasoning proves in essence that for every relatively weakly compact C ⊂ W p , the multimapmap H : ( C , w ) ⊸ ( W p , w ) is compact valued upper semicontinuous.Fix u ∈ W p . The subset N ( − F ) ( u ) furnished with the relative weak topology of H ∗ is compact.Moreover, ( N ( − F ) ( u ) , w ) is an acyclic space. The multimap K : ( N ( − F ) ( u ) , w ) ⊸ ( H ( u ) , w ) may be re-garded as an acyclic operator between compact topological space N ( − F ) ( u ) and a paracompact space H ( u )endowed with the relative weak topology of W p . The fibers of this map are formed by intersections( L + N A h )( x ) ∩ N ( − F ) ( u ). Hence, they are convex. Lemma 3 implies that the reduced Alexander–Spaniercohomologies ˜ H ∗ (( H ( u ) , w )) are isomorphic to ˜ H ∗ (( N ( − F ) ( u ) , w )). In other words the set-valued map H : ( W p , w ) ⊸ ( W p , w ) possesses acyclic values.Assumption (32) entails || b || q + || h || E ∗ d − c || a || q + || b || q + || h || E ∗ − ( c + ˆ c ) . Choose a radius R > R ∈ max || b || q + || h || E ∗ d − c ! qp , || a || q + || b || q + || h || E ∗ − ( c + ˆ c ) , . Then(33) d − (cid:16) || b || q + cR p − + || h || E ∗ (cid:17) R p − and(34) || a || q + || b || q + || h || E ∗ + ( c + ˆ c ) R p − R . Take an x ∈ H (cid:16) D W p (0 , R ) (cid:17) . Then || x || p − E R p − , by (30) and (33). The latter combined with (29) and (34)yields || Lx || E ∗ R . Therefore the length of vector x , measured in the equivalent norm max {|| · || E , || L ( · ) || E ∗ } of the space W p , is not greater then R . In other words, H (cid:16) D W p (0 , R ) (cid:17) ⊂ D W p (0 , R ).As we have seen above, the operator H : ( D W p (0 , R ) , w ) ⊸ ( D W p (0 , R ) , w ) is an admissible multimap.Since H (cid:16) D W p (0 , R ) (cid:17) is norm bounded in W p , it is also a compact map. In view of the Dugundji extensiontheorem [9, Theorem 4.1], the ball D W p (0 , R ) constitutes an absolute extensor for the class of metrizablespaces as a convex subset of a locally convex linear space ( W p , w ). Since W p is separable, the space( D W p (0 , R ) , w ) must be metrizable ([11, Proposition 3.107]). From Theorem 3 it follows directly that themultimap H : ( D W p (0 , R ) , w ) ⊸ ( D W p (0 , R ) , w ) must have at least one fixed point. (cid:3) Remark 11.
Generic approach to evolution inclusions governed by operators monotone in the sense ofMinty–Browder essentially comes down to the observation that the sum of densely defined monotonedi ff erential operator and suitably regular, multivalued perturbation is surjective. Proof of Theorem 11illustrates the thesis that the “fixed point method” provides a real alternative to this approach. Example 6.
The last two examples are dedicated to boundary value problems in which the nonlinear partF possesses not necessarily convex values. The first of these concerns solvability of the Hammersteinintegral inclusion (15).
Theorem 13.
Let E be a separable Banach space and q p < ∞ . Assume that k : I → L ( E ) satisfies (k ) – (k ) , while F : I × E ⊸ E satisfies the following conditions: (i) for every ( t , x ) ∈ I × E the set F ( t , x ) is nonempty and closed, (ii) the map F ( t , · ) is lower semicontinuous for each fixed t ∈ I, (iii) the map F ( · , · ) is measurable with respect to the product of Lebesgue and Borel σ -fields definedon I and E, respectively, OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 25 (iv) there is a function b ∈ L q ( I , R ) and c > such that for all x ∈ E and for a.a. t ∈ I, || F ( t , x ) || + : = sup {| y | E : y ∈ F ( t , x ) } b ( t ) + c | x | pq , (v) there exists a function η ∈ L pqp − q ( I , R ) such that for all bounded subsets Ω ⊂ E and for a.a. t ∈ Ithe inequality holds β ( F ( t , Ω )) η ( t ) β ( Ω ) . If there is an R > such that (16) holds together with (35) || || k ( t , · ) || r || p || η || pqp − q < , then the integral inclusion (15) has at least one p-integrable solution.Proof. Denote by S ( I , E ) the space of all Lebesgue measurable functions mapping I to E , equiped withthe topology of convergence in measure. Under these assumptions (cf. [22, p. 731]), the Nemytskiˇı oper-ator N F : S ( I , E ) ⊸ S ( I , E ) is lower semicontinuous. Consider a sequence ( u n ) n > such that u n L p ( I , E ) −−−−−→ n →∞ u .In particular, u n −−−−→ n →∞ u in measure. If w is an arbitrary element of N F ( u ), then there exists a se-quence w n ∈ N F ( u n ) such that w n −−−−→ n →∞ w in S ( I , E ). From every subsequence of ( w n ) n > we canextract some subsequence ( w k n ) n > satisfying w k n ( t ) E −−−−→ n →∞ w ( t ) a.e. on I . W.l.o.g we may assume that u k n ( t ) E −−−−→ n →∞ u ( t ) for a.a. t ∈ I . Assumption (iv) means that | w k n ( t ) | b ( t ) + c | u k n ( t ) | pq a.e. on I . Since {| u k n ( · ) | p } ∞ n = is uniformly integrable, the latter implies that the family {| w k n ( · ) | q } ∞ n = is uniformly inte-grable. Secondly, under passage to the limit one sees that | w ( t ) | b ( t ) + c | u ( t ) | pq for a.a. t ∈ I , i.e. w ∈ L q ( I , E ). Hence, the uniform integrability of {| w k n ( · ) − w ( · ) | q } ∞ n = follows. In view of Vitali con-vergence theorem lim n →∞ R I | w k n ( t ) − w ( t ) | q dt =
0. Consequently, w n L q ( I , E ) −−−−−→ n →∞ w . Therefore, the multimap N F : L p ( I , E ) ⊸ L q ( I , E ) meets the definition of lower semicontinuity. It is also clear that this operatorpossesses closed and decomposable values. In view of [2, Theorem 3] there exists a continuous map f : L p ( I , E ) → L q ( I , E ) such that f ( u ) ∈ N F ( u ) for every u ∈ L p ( I , E ).Let K : L q ( I , E ) → L p ( I , E ) be given by (18). Observe that solving the equation u = K ( f ( u )) means tofind a solution of the Hammerstein integral inclusion (15).We have proven previously (see p.16) that the composite map K ◦ N F satisfies the boundary condition(3) provided R > H : D L p ( h , R ) → L p ( I , E ), givenby H : = K ◦ f , will meet condition (3) as well. This operator satisfies also condition (21). It can beshown in a manner strictly analogous to that demonstrated in the proof of Theorem 9. Since the space E is separable, there is no need in particular to pass to the relative MNC β E and inequality (20) andsuccessive ones hold under current assumptions as well. Finally, let us note that H is admissible as aunivalent continuous map. From the direct application of Theorem 4 follows, in this regard, that it mustpossess a fixed point u ∈ D L p ( h , R ). (cid:3) Example 7.
The last example illustrates the application of Rothe-type fixed point argument, as a conclu-sion stemming from Theorem 4, in order to show the existence of solutions of the periodic problem (24)with non-convex perturbation term F . Theorem 14.
Let ( E , H , E ∗ ) be an evolution triple such that E ֒ → H compactly. Assume that the operatorA : I × E ⊸ E ∗ satisfies (A ) – (A ) , while the map F : I × H ⊸ H fulfills conditions (i)–(iv) of Theorem13. Suppose further that the inequality (25) holds. Then problem (24) has at least one solution.Proof. In accordane with what we have shown in the proof of Theorem 13 there must exist a continuousselection f : H → H ∗ of the Nemytskiˇı operator N F . The proof of Theorem 11 leaves no doubt that theexternal operator K : H ∗ ⊸ H , given by K : = ( L + N A h ) − , is an upper semicontinuous multimap with compact convex values. Therefore, the map H : H ⊸ H such that H : = K ◦ f is an admissible one. Proofof Theorem 11 also illustrates the hypothesis that operator H : D H (0 , R ) ⊸ H satisfies the boundarycondition (3), if you adopt that the radius R is not less than (cid:16) || b || q + || h || E ∗ ) / ( d − c ) (cid:17) qp . And finally, theproof of this theorem sets forth reasons for the compactness of the operator H : D H (0 , R ) ⊸ H . Hence,we may again evoke the fixed point result expressed in Theorem 4 to show the existence of a fixed pointof H i.e., a solution of the problem (24). (cid:3) R eferences [1] D. Bothe, Multivalued perturbations of m-accretive di ff erential inclusions , Israel J. Math. 108 (1998), 109-138.[2] A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values , Studia Math. 90 (1988), 69-86.[3] M. Cicho´n, Di ff erential inclusions and abstract control problems , Bull. Austral. Math. Soc. 53 (1996), 109-122.[4] C. Corduneanu, Functional Equations with Causal Operators , in: Stability and Control: Theory, Methods and Applications,vol. 16, Taylor and Francis, London, 2002.[5] J.-F. Couchouron, M. Kamenski,
A unified topological point of view for integro-di ff erential inclusions , in “Di ff erential Inclu-sions and Optimal Control”, Lecture Notes in Nonlinear Analysis, Vol. 2 (1998), 123-137.[6] J.-F. Couchouron, M. Kamenski, An abstract topological point of view and a general averaging principle in the theory ofdi ff erential inclusions , Nonlinear Anal. 42 (2000), 1101-1129.[7] J.-F. Couchouron, R. Precup, Existence principles for inclusions of Hammerstein type involving noncompact acyclic multival-ued maps , Electron. J. Di ff erential Equations Vol. 2002, No. 04 (2002), 1-21.[8] Z. Drici, F. McRae, J. Vasundhara Devi, Monotone iterative technique for periodic boundary value problems with causaloperators , Nonlinear Anal. 64 (6) (2006) 1271-1277.[9] J. Dugundji,
An extension of Tietze’s theorem , Pacific J. Math. 1 (1951), 353-367.[10] N. Dunford, J. Schwartz,
Linear operators: Part I , Interscience, New York, 1957.[11] M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler,
Banach Space Theory: The Basis for Linear and Nonlinear Analysis ,CMS Books in Mathematics, Springer, 2011.[12] G. Fournier, L. Górniewicz,
The Lefschetz fixed point theorem for multi-valued maps of non-metrizable spaces , Fund. Math.92 (1976), 213-222.[13] L. Gasi´nski, N. Papageorgiou,
Nonlinear Analysis , Taylor & Francis Group, Boca Raton, 2005.[14] L. Górniewicz,
Topological fixed point theory of multivalued mappings , Second ed., Springer, Dordrecht, 2006.[15] S. Gutman,
Evolutions governed by m-accretive plus compact operators , Nonlinear Anal. 7 (1983), 707-715.[16] H. P. Heinz,
On the behaviour of measures of noncompactness with respect to di ff erentiation and integration of vector-valuedfunctions , Nonlinear Anal. 7 (1983), 1351-1371.[17] T. Jankowski, Boundary value problems with causal operators , Nonlinear Anal. 68 (12) (2008) 3625-3632.[18] M. Kunze, G. Schlüchtermann,
Strongly generated Banach spaces and measures of noncompactness , Math. Nachr. 191 (1998),197-214.[19] V. Lupulescu,
Causal functional di ff erential equations in Banach spaces , Nonlinear Anal. 69 (12) (2008) 4787-4795.[20] H. Mönch, Boundary value problems for nonlinear ordinary di ff erential equations of second order in Banach spaces , Nonlin-ear Anal. 4 (1980), 985-999.[21] H. Mönch, G.-F. von Harten, On the Cauchy problem for ordinary di ff erential equations in Banach spaces , Arch. Math. 39(1982), 153-160.[22] Y. Nepomnyashchikh, A. Ponosov, The necessity of the Carathéodory conditions for the lower semicontinuity in measure ofthe multivalued Nemytskii operator , Nonlinear Anal. 30 (1997), 727-734.[23] V. Obukhovskii, P. Zecca,
On certain classes of functional inclusions with causal operators in Banach spaces , NonlinearAnal. 74 (2011), 2765-2777.[24] D. O’Regan, R. Precup,
Fixed point theorems for set-valued maps and existence principles for integral inclusions , J. Math.Anal. Appl. 245 (2000), 594-612.[25] R. Pietkun,
Integrated solutions of non-densely defined semilinear di ff erential inclusions: existence, topology and applica-tions , arXiv:1812.01725.[26] R. Pietkun, On some generalizations of the Mönch, Sadovski˘ı and Darbo fixed point theorems , J. Fixed Point Theory Appl.(2018) 20:95. https: // doi.org / / s11784-018-0586-6.[27] R. Pietkun, Structure of the solution set to Volterra integral inclusions and applications , J. Math. Anal. Appl. 403 (2013),643-666.[28] E. Spanier,
Algebraic Topology , McGraw-Hill, New York, 1966.[29] A. Tolstonogov,
Properties of integral solutions of di ff erential inclusions with m-accretive operators , Math. Notes 49 (1991),636-644.[30] A. Ülger, Weak compactness in L ( µ, X ), Proc. Amer. Math. Soc. 113 (1991), 143-149. OLVABILITY OF INCLUSIONS OF HAMMERSTEIN TYPE 27 [31] E. Zeidler,
Nonlinear Functional Analysis and its Applications, II / B. Nonlinear monotone operators , Springer-Verlag, NewYork, 1990.[32] T. Zhu, C. Song, G. Li,
Existence results for abstract semilinear evolution di ff erential inclusions with nonlocal conditions ,Bound. Value Probl. (2013) 2013: 153. https: // doi.org / / oru ´ n , P oland E-mail address ::