Application of Pettis integration to delay second order differential inclusions

This paper serves as a corrigendum to the paper titled Application of Pettis integration to delay second order differential inclusions appearing in EJQTDE no. 88, 2012. We present here a corrected version of Theorem 3.1, because Proposition 2.2 is not true. In the above article, Proposition 2.2 is not true since the normed space P 1 E ([0, 1]) is not complete. Consequently, to correct Theorem 3.1 we have to assume that Γ 1 is Pettis uniformly integrable and that Γ 2 is integrably bounded. Then in the proof we can use Proposition 2.2 with L 1 E ([0, 1]) instead of P 1 E ([0, 1]) to conclude the result. This version of Proposition 2.2 can be found in: A. Fryszkowski, Continuous selections for a class of nonconvex mul-The authors thank an anonymous reader for pointing out the mistake.

The paper is organized as follows.After we recall some basic notations and preliminary theorems in section 3 we present our main result.

Notation and Preliminaries
Let (E, • ) be a separable Banach space and E ′ is its topological dual, B(0, ρ) is the closed ball of E of center 0 and radius ρ > 0 and B E is the closed unit ball of E; L([0, 1]) is the σ-algebra of Lebesgue-measurable sets on [0, 1]; λ = dt is the Lebesgue measure on [0,1]; B(E) is the σ-algebra of Borel subsets of E. By L 1 E ([0, 1]) we denote the space of all Lebesgue-Bochner integrable E-valued mappings defined on [0, 1].We denote the topology of uniform convergence on weakly compact convex sets by T w co .Restricted to E ′ , this is the Mackey topology, which is the strongest locally convex topology on E ′ and we denote it by T (E ′ , E).
Let C E ([0, 1]) be the Banach space of all continuous mappings u : [0, 1] → E, endowed with the sup-norm, and C 1 E ([0, 1]) be the Banach space of all continuous mappings u : [0, 1] → E with continuous derivative, equipped with the norm We denote by P 1 E ([0, 1]) the space of all Pettis-integrable E−valued mappings defined on [0, 1].The Pettis norm of any element f ∈ P 1 E ([0, 1]) is defined by The space P 1 E ([0, 1]) endowed with .P e is a normed space.A subset ) is Pettis uniformly integrable ((P U I) for short) if, for every ε > 0, there exists δ > 0 such that for each measurable subset A of [0, 1] we have For more details on the theory of the Pettis integration we can refer the reader to [6], [10], [11] and [15].
A mapping v : [0, 1] → E is said to be scalarly derivable when there exists some mapping v : [0, 1] → E (called the weak derivative of v) such that, for every x ′ ∈ E ′ , the scalar function x ′ , v(•) is derivable and its derivative is equal to x ′ , v(•) .The weak derivative v of v when it exists is the weak second derivative.
By W 2,1 P,E ([0, 1]) we denote the space of all continuous mappings u ∈ C E ([0, 1]) such that their first usual derivatives u are continuous and their second weak derivatives belong to P 1 E ([0, 1]).Recall also that a set ) is said to be decomposable if and only if for every u, v ∈ K and any In the sequel, we need the following lemma that summarizes some properties of some Green type function, see [1] and [3].
Lemma 2.1 Let E be a separable Banach space and let G : [0, 1] × [0, 1] → R be the function defined by Then the following assumptions hold.
(4) The mapping uf is scalarly derivable, that is, there exists a mapping üf : [0, 1] → E such tha, for every Let us mention a useful consequence of Lemma 2.1.
Proposition 2.1 Let E be a separable Banach space and let f : [0, 1] → E be a continuous mapping (respectively a mapping in Then the mapping Proposition 2.2 (See [2]) Let X be a compact space and M : X ⇉ P 1 E ([0, 1]) be a lower semicontinuous multifunction with closed and decomposable values.Then M has a continuous selection.
For the proof of our main result, we also need the following fixed point theorem which is the multivalued analogue of the Shaefer continuation principle.For more details for the fixed point theory we refer the reader to [13].
Theorem 2.1 Let Y be a normed linear space and A : Y ⇉ Y be an upper semicontinuous compact multivalued operator with compact convex values.Suppose that there exists an R > 0 such that the a priori estimate holds.Then A has a fixed point in the ball B(0, R).

Main result
Now, we are able to prove our main existence theorem.
Then the boundary value problem (P r ) has at least one solution in Proof.
Let us consider the differential inclusion We wish to show that the X-solutions set X Γ of (3.1) is nonempty and convex compact in the Banach space X endowed with the norm .X .Let us recall (see [10]) that the set S P e Γ of all Pettis integrable selections of Γ is nonempty, convex and sequentially compact for the topology of pointwise convergence on L ∞ R ⊗ E ′ and that the multivalued integral Γ is convex and norm compact in E.
In view of Lemma 2.1 and Proposition 2.2, the solutions set X Γ of (3.1) is characterized by EJQTDE, 2012 No. 88, p. 6 Clearly X Γ is convex.Furthermore, for all u ∈ X Γ there is f ∈ S P e Γ such that for t, t ′ ∈ [0, 1] and by Lemma 2.1, The function G is continuous on the compact set [0, 1]× [0, 1], so it is uniformly continuous there.In addition, the set {|δ * (x ′ , Γ(.))| : ).Then, the right-hand side of the above inequalities tends to 0 as t → t ′ .We conclude that the sets X Γ and { u : we get the equicontinuity of X Γ in X.On the other hand, for each t ∈ [−r, 1] and each τ ∈ [0, 1], the sets X Γ (t) = {u(t) : u ∈ X Γ } and { uf (τ ) : u ∈ X Γ } are relatively compact in E because they are included in the norm compact sets 1 0 G(t, s)Γ(s)ds and 1 0 ∂G ∂t (t, s)Γ(s)ds respectively.The Ascoli-Arzelà theorem yields that X Γ is relatively compact in X with respect to .X .We claim that X Γ is closed in (X, .X ).Let (u n ) be a sequence in X Γ converging to ξ ∈ X with respect to .X .Then, for each n, there exists and u n (t) = ϕ(t) for all t ∈ [−r, 0].As S P e Γ is sequentially compact for the topology of pointwize convergence on L ∞ R ⊗ E ′ , we extract from (f n ) a subsequence that we do not relabel and which converges σ In particular, for each x ′ ∈ E ′ and for every t ∈ [0, 1], we have and u(t) = ϕ(t) for all t ∈ [−r, 0].Thus we get ξ = u.This shows the compactness of X Γ in X.
We will prove that, for X Γ endowed with the norm • X , the multifunction Φ admits a continuous selection.It is clear that Φ has nonempty closed decomposable values.According to Proposition 2.2, it sufficient to prove that Φ is lower semicontinuous.Let u 0 ∈ X Γ , v 0 ∈ Φ(u 0 ) and let (u n ) be a sequence in X Γ converging to u 0 in (X, • X ).Since u 0 ∈ X Γ , there exists f 0 ∈ S P e Γ such that and u 0 (t) = ϕ(t) for all t ∈ [−r, 0], and since (u n ) ⊂ X Γ , for each n, there exists f n ∈ S P e Γ such that and u n (t) = ϕ(t) for all t ∈ [−r, 0].For any n ∈ N, H(., u n (.), u n (h(.)), un (.)) is measurable with nonempty closed values, so according to [10, Theorem III.41], the multifunction Λ n defined from [0, 1] into E by Λ n (t) = {w ∈ H(t, u n (t), u n (h(t)), un (t)) : w−v 0 (t) = d(v 0 (t), H(t, u n (t), u n (h(t)), un (t)))}, EJQTDE, 2012 No. 88, p. 8 is also measurable with closed values, and since H(., u n (.), u n (h(.)), un (.)) has compact values, Λ n has nonempty values.In view of the existence theorem of measurable selections (see [10]), there is a measurable mapping v n : [0, 1] → E such that v n (t) ∈ Λ n (t), for all t ∈ [0, 1].This yields v n (t) ∈ H(t, u n (t), u n (h(t)), un (t)) and = 0, the last equality follows from the fact that H is lower semicontinuous with compact values and hence it is h-lower semicontinuous.This shows that (v n ) converges pointwise to v 0 and since H(t, x, y, z) ⊂ Γ 2 (t) for all (t, x, y, z) ∈ [0, 1] × E × E × E, the convergence also holds strongly in As v n (t) ∈ Γ 2 (t) for all n ∈ N and as Γ 2 is scalarly uniformly integrable and hence the set Therefore Φ is lower semicontinous.An application of Proposition 2.2 implies that, for X Γ endowed with the norm • X , there exists a continuous mapping K : X Γ → P 1 E ([0, 1]) such that K(u) ∈ Φ(u) for all u ∈ X Γ , or equivalently, for each u ∈ X Γ the inclusion K(u)(t) ∈ H(t, u(t), u(h(t)), u(t)) holds for a.e.t ∈ [0, 1].
Step 3. We transform the problem (P) into a fixed point inclusion in the Banach space X Γ .By Lemma 2.1 and Proposition 2.2, the existence of solutions of (P) is equivalent to the problem of finding u ∈ X Γ such that EJQTDE, 2012 No. 88, p. 9 where Then, the integral inclusion (3.4) is equivalent to the operator inclusion Let us show that S P e F has nonempty values.Indeed, for any Lebesgue measurable mappings u, w : [0, 1] → E and v : [−r, 1] → E, there is a Lebesgue-measurable selection s ∈ S P e Γ 1 such that s(t) ∈ F (t, u(t), v(h(t)), w(t)) a.e.Indeed, there exist sequences (u n ), (v n ) and (w n ) of simple E-valued mappings which converge pointwise to u, v and w respectively, for E endowed with the norm topology.Notice that the multifunctions F (., u n (.), v n (h(.)), w n (.)) are Lebesgue-measurable.In view of the existence theorem of measurable selection (see [10]), for each n, there is a Lebesgue-measurable selection s n of F (., u n (.), v n (h(.)), w n (.)).As s n (t) ∈ F (t, u n (t), v n (h(t)), w n (t)) ⊂ Γ 1 (t), for all t ∈ [0, 1] and as S P e Γ 1 is sequentially weakly compact in P 1 E ([0, 1]), by Eberlein-Smulian theorem, we may extract from ( This last equality shows that for each ) k∈N be a dense sequence for the Mackey topology τ (E ′ , E).Let k ∈ N be fixed.Applying the Banach-Mazur's theorem trick to ( e * k , s ′ n (.) ) n provides a sequence (z n ), z n ∈ co{ e * k , s ′ m (.) : m n} such that (z n ) converges pointwise a.e to e * k , s(.) .Using this fact and the pointwise convergence of the sequences (u n ), (v n ) and (w n ), the upper semicontinuity of F(t, ., ., .) and the compacity of its values, it is not difficult to check that s(t) ∈ F (t, u(t), v(h(t)), w(t)) a.e.a.e, and then the operator A is well defined.Using Lemma 2.1 and the assumption ϕ(0) = 0, it is clear that A has its values in X Γ .Now, we will show that the multivalued operator A satisfy all the conditions of Theorem 2.1.Clearly Au is convex for each u ∈ X Γ .First, we prove that A has compact values in X Γ .Since X Γ is compact, it suffices to see that A has closed values in X Γ .For each u ∈ X Γ , let (v n ) be a sequence in Au converging to v ∈ X Γ .Then by (3.5), for every n there exists f n ∈ S P e F (u) ⊂ S P e Γ 1 such that where  ∂G ∂t (., s)g(s)ds, for E endowed with the strong topology using as above, the weak convergence of (g n ) and the norm compactness of the set-valued integral and v(t) = ϕ(t) for all t ∈ [−r, 0].Since g = f + K(u) and f ∈ S P e F (u), we get v ∈ A. This says that Au is compact in X Γ .
Next, we show that A is a compact operator, that is, A maps bounded sets into relatively compact sets in X Γ .Let S be a bounded set in X Γ .We have A(S) ⊂ X Γ .But X Γ is compact in X, then A(S) is relatively compact in X and hence A is compact. where Γ is sequentially compact for the topology of pointwise convergence on L ∞ R ⊗ E ′ , we may extract from (g n ) a subsequence (that we do note relabel) converging σ , un (t)).Since u n − u X → 0 and F (t, ., ., .) is upper semicontinuous on E × E × E with convex compact values, repeating the arguments given above, we conclude that f (t) = g(t) − K(u)(t) ∈ F (t, u(t), u(h(t)), u(t)).Equivalently, f ∈ S P e F (u). On the other hand, it is not difficult to see that the sequence (v n (.)) = ( Taking the above inequalities into account, we obtain Hence by Theorem 2.1, we conclude that A has a fixed point u in the ball B(0, R), what, in turn, means that this point is a solution in X Γ to the problem (P).That is, ü(t) ∈ F (t, u(t), u(h(t)), u(t)) + K(u)(t), a.e.t ∈ [0, 1] and u(t) = ϕ(t) for all t ∈ [−r, 0].Since K(u)(t) ∈ H(t, u(t), u(h(t)), u(t)), we get that u is a solution in X Γ to our boundary value problem (P r ) and the proof of the theorem is complete.
For closed subsets A and B of E, the excess of A over B is defined by e(A, B) = sup a∈B d(a, B) = sup a∈A ( inf b∈B a − b ), and the support function δ * (•, A) associated with A is defined on E ′ by δ * (x ′ , A) = sup a∈A x ′ , a .