J. Fixed Point Theory Appl. 95:02)81( https://doi.org/10.1007/s11784-018-0534-5 c Springer International Publishing AG,  part of Springer Nature 2018

Journal of Fixed Point Theory and Applications

Topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions JinRong Wang , A. G. Ibrahim

and D. O’Regan

Abstract. In this paper, we study the topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions on a compact interval. We show that the solution set for our problem is nonempty, compact and moreover a Rδ -set. Mathematics Subject Classification. 26A33, 34A60. Keywords. Fractional, non-instantaneous impulsive, evolution inclusions, solution set, Rδ -set.

1. Introduction Existence and stability of solutions to non-instantaneous impulsive differential equations was investigated in [1–10]. Motivated by these contributions, in [11] we show the nonemptiness and compactness of the solution set for the following fractional non-instantaneous impulsive evolution inclusions ⎧ m  ⎪ ⎪ c α ⎪ D u(t) ∈ Au(t) + F (t, u(t)), a.e. t ∈ (ai , bi+1 ] ⊂ J = [0, l], l > 0, ⎪ ,t a i ⎪ ⎪ ⎪ i=0 ⎨ a0 := 0, bm+1 := l, α ∈ (0, 1), ⎪ ⎪ ⎪ ⎪ u(t) = gi (t, u(b− ⎪ i )), t ∈ (bi , ai ] ⊂ [0, l], i = 1, 2, . . . , m, ⎪ ⎪ ⎩ u(0) = u ∈ E, 0 (1) where A is the infinitesimal generator of a C0 -semigroup {T (t) : t ≥ 0} on a Banach space E, c Dα ai ,t denotes the Caputo derivative [12] of order α from the lower limit ai to the upper limit t, F : [0, l] × E → 2E − {φ} is a multifunction, the sequences {ai } and {bi+1 } satisfy bi < ai < bi+1 , i = 0, 1, . . . , m, The authors acknowledge the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Unite Foundation of Guizhou Province ([2015]7640), Graduate ZDKC ([2015]003), and Deanship of Scientific Research of King Faisal University of Saudi Arabia (No. 170060).

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and moreover, gi : [bi , ai ] × E → E, i = 1, 2, . . . , m, and u(b− i ) denotes the left limit of u at bi . Following [11,13,14], we consider the topological structure of the solution set for (1) in this paper. We prove that the solution set for (1) is nonempty and a Rδ -set. The paper is organized as follows. In Sect. 2 we present some preliminary lemmas. In Sect. 3 we first discuss the nonemptiness and compactness of the solution set. Theorems 4.4 and 4.6 are the main results of the paper and we show the Rδ -structure of the solution set.

2. Preliminaries and notation Denote LP (J, E) = {v : v : J → E is Bochner integrable }, P ≥ 1 endowed l 1 with the norm vLP (J,E) = ( 0 v(t)P dt) P . Denote Pck (E) = {B ⊆ E : B is nonempty, convex and compact}. co(B) (respect., co(B)) be the convex hull (respect., convex closed hull in E) of a subset B. Let C (J, E) = {f : f : J → E is continuous } be endowed with the supremum norm. We consider the set of functions P C(J, E) = u : J → − E : u|Ji ∈ C(Ji , E), Ji := (bi , bi+1 ], i = 0, 1, 2, . . . , m and u(b+ i ) and u(bi ) existfor each i = 1, 2, . . . , m . Now P C(J, E) is a Banach space endowed with the Chebyshev P C-norm: uP C(J,E) = max{u(t) : t ∈ J}. Next, we recall the following definitions. Definition 2.1. ([11, Definition 2.5]) A function u ∈ P C(J, E) is called a P C-mild solution of (1) if ⎧

t ⎪ ⎪ K1 (t)u0 + (t − s)α−1 K2 (t − s)f (s)ds, t ∈ [0, b1 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, u(b− )), t ∈ (bi , ai ], i = 1, 2, . . . , m, i u(t) = ⎪ K1 (t − ai )gi (ai , u(b− ⎪ i )) ⎪ ⎪

t ⎪ ⎪ ⎪ ⎪+ ⎩ (t − s)α−1 K2 (t − s)f (s)dθs, t ∈ [ai , bi+1 ], i = 1, 2, . . . , m, ai

(2) ∞ ∞ where K1 (t) = 0 ξα (θ)T (tα θ)dθ, K2 (t) = α 0 θξα (θ)T (tα θ)dθ, ξα (θ) = ∞ 1 1 1 −1− α wα (θ− α ) ≥ 0, and wα (θ) = π1 n=1 (−1)n−1 θ−αn−1 Γ(n n!α+1) αθ sin(nπα), θ ∈ (0, ∞) and f ∈ SFP = {f ∈ LP (J, E): f (t) ∈ F (t, u(t)) a.e. on [ai , bi+1 ], i = 0, 1, 2, . . . , m }, P > α1 . F By the symbol u0 [0, l], we denote the set of mild solutions to (1). The Hausdorff measure of noncompactness on E is defined on bounded subsets as χ (B) = inf{ > 0 : B can be covered by finitely many balls of radius ≤ }. Next, the map χP C : Pb (P C(J, E)) → [0, ∞) is defined as χP C (B) =

max

i=0,1,2,...,m

χi (B|

Ji ),

Ji := [bi , bi+1 ],

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where χi is the Hausdorff measure of noncompactness on the Banach space C(Ji , E) and B|

Ji

= {u∗ : Ji → E : u∗ (t) = u(t), t ∈ Ji and u∗ (bi ) = u(b+ i ), u ∈ B },

i = 0, 1, . . . , m.

Of course B| J0 = {uJ : u ∈ B}. It is easily seen that χP C is the Hausdorff 0 measure of noncompactness on P C(J, E). Definition 2.2. [15] A sequence {fn : n ∈ N} ⊂ L1 (J, E) is said to be semicompact if (i) It is integrable bounded, i.e., there is q ∈ L1 (J, R+ ) such that fn (t) ≤ q(t) a.e. t ∈ J. (ii) The set {fn (t) : n ∈ N} is relatively compact in E a.e. t ∈ J. Lemma 2.3. [16] Every semicompact sequence in L1 (J, E) is weakly compact in L1 (J, E). Definition 2.4. [13,14] A subset Z of a topological space X is called a retract of X if there exists a continuous function r : X → A, such that r(x) = x, for every x ∈ A. We say that X is an absolute retract if and only if for any space Y and for any embedding h : X → Y (i.e., h is a homeomorphism between X and h(X)) the set h(X) is a retract of Y. Definition 2.5. [13,17] A topological space X is called contractible if it is homotopically equivalent to a point x0 ∈ X. That is, there exist a point x0 ∈ X such that the constant map c : X → {x0 } and the identity map IX : X → X are homotopic, i.e., there is a continuous map h : [0, 1] × X → Xsuch that h(0, ·) = IX and h(1, ·) = c(·). The function h is called a homotopy or deformation. Remark 2.6. Note that any convex set is contractible. P Definition 2.7. A sequence {fn }∞ n=1 ⊆ L (J, E)(P ≥ 1) is called P -time semicompact if it is P -time integrably bounded and the set {fn (t) : n ≥ 1} is relatively compact for a.e. t ∈ J.

and S : LP (J, X) → C (J, X) defined by

t (t − s)α−1 K2 (t − s)f (s)ds. Sf (t) =

Corollary 2.8. Let P >

1 α

0 ∞

∞

Then for every P -time semicompact set {fn }n=1 ⊂ LP (J, E) the set {Sfn }n=1 is relatively compact in C (J, E) . Moreover, if (fn )n≥1 converges weakly to f0 in LP (J, E) then Sfn → Sf0 in C (J, E). Proof. The proof is basically a modification of ideas in [18] and can be found in [11, Corollary 2.12].  Lemma 2.9. [16] (Kakutani–Glicksberg–Fan theorem) Let W be a nonempty compact and convex subset of a locally convex topological vector space. If Φ : W → Pcl,cv (W ) is an u.s.c. multi-function, then it has a fixed point.

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Definition 2.10. [13,14] A compact nonempty topological space is called an Rδ −set provided there exists a decreasing sequence {An }of compact absolute retracts such that: X = n≥1 An . Remark 2.11. Any intersection of a decreasing sequence of Rδ −sets is Rδ . Lemma 2.12. [13,19] A nonempty subset A of a complete metric space X is a Rδ -set if and only if it is the intersection of a decreasing sequence {An } of contractible sets with χ(An ) → 0, as n → ∞. Lemma 2.13. ([16, Prop.3.5.1]]) Let W be a closed subset of E and Φ : W → Pk (E) be a closed multi-function which is γ-condensing on every bounded subset of W , where γ is a monotone measure of noncompactness defined on E. If the set of fixed points for Φ is a bounded subset of E then it is compact.

3. Nonemptiness and compactness of the solution set We list here the following conditions for convenience : (A) A : D(A) ⊆ E → E is a linear closed operator generating a C0 semigroup {T (t) : t ≥ 0} in E. We suppose that there exists a M ≥ 1 such that supt≥0 T (t) ≤ M . Thus, one has K1 (t)u ≤ M u and K2 (t)u ≤ M Γ(α) u for any u ∈ E. (A)∗ The C0 -semigroup {T (t) : t ≥ 0} generated by A is equicontinuous. (HF 1) for every u ∈ E the multivalued function F : J × E → Pck (E) is strongly measurable, and for almost every t ∈ J, u −→ F (t, u) is upper semicontinuous. (HF 2) for any bounded subset Ω there exists a function ϕΩ ∈ L1 (J, R+ ) such that for any u ∈ E F (t, u) ≤ ϕΩ (t),

∀u ∈ Ω

and for a.e.

∗

P

t ∈ J.

(HF 2) there exists a function ϕ ∈ L (J, R ) with P > positive constant c such that for any u ∈ E, F (t, u) ≤ ϕ(t) + cu,

+

1 α

and a

a.e. t ∈ J.

(HF 2)∗∗ there exists a function ϕ ∈ LP (J, R+ ) with P > α1 such that for any u ∈ E, F (t, u) ≤ ϕ(t)(1 + u), a.e. t ∈ J. (HF 3) there exists a function β ∈ LP (J, R+ )(P ≥ 1) satisfying χ(F (t, D)) ≤ β(t)χ(D),

fora.e.

t ∈ J,

for every bounded subset D ⊆ E. (H) gi : [bi , ai ] × E → E, i = 1, 2, . . . , m and gi (·, x) is continuous for every x ∈ E. (H)∗ gi : [bi , ai ] × E → E, i = 1, 2, . . . , m, is uniformly continuous on bounded sets and for any t ∈ [bi , ai ], gi (t, ·) maps any bounded subset of E into a relatively compact subset of E and there exists a positive constant hi such that gi (t, u) ≤ hi u , t ∈ [bi , ai ], u ∈ E.

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We now present some results on nonemptiness and compactness of the solution set. Some of the ideas in [20] are used (however, some of the details on the operators K1 and K2 are quite different). Theorem 3.1. Assume that (A), (HF 1), (HF 2)∗ , (HF3 ) hold. For every u0 ∈ E, the solution set of mild solutions of

c α D0,t u(t) ∈ Au(t) + F (t, u(t)), for a.e. t ∈ [0, b1 ], (3) u(0) = u0 ∈ E, is nonempty and compact. Proof. Notice that (HF 2)∗ implies (HF 2) so from [16, Lemma 5.1.1], the superposition multi operator PF

C(J, E) → P (L1 (J, E)),

:

PF (u) = SF1 (.,u(.)) , generated by F is well defined, and is weakly closed. We define a multi operator R : C(J, E) → P (C(J, E)) as follows: for u ∈ C(J, E), a function y ∈ R(u) if and only if

t y(t) = K1 (t)u0 + (t − s)α−1 K2 (t − s)f (s)ds, t ∈ J, 0

SFP (·,u) .

where f ∈ Note any fixed point for R is a mild solution for (3). We now prove (using [16, Property 3.5.1]) that R has a fixed point. We divide the proof into several steps. Step 1. R is closed with compact values. ∞ Let {un }∞ n=1 , {yn }n=1 be two sequences in C(J, E) such that un → u, yn → y and

t (t − s)α−1 K2 (t − s)fn (s)ds, t ∈ J, yn (t) = K1 (t)u0 + 0

where fn ∈ SFP (.,u) . Since un → u in C(J, E) we can find a positive constant ω such that un C(J,E) ≤ ω. Hypothesis (HF 2)∗ implies fn (t) ≤ ϕ(t) + cω,

a.e.

t ∈ J.

Moreover, hypothesis (HF 3) implies χ{fn (t) : n ≥ 1} ≤ β(t)χ{un (t) : n ≥ 1} = 0,

a.e.

t ∈ J.

It follows that the sequence {fn : n ≥ 1} is P −time semicompact via Definition 2.7, and hence we may assume without loss of generality that fn f0 in LP (J, E). From Corollary 2.8, we have yn → z in C(J, E), where

t (t − s)α−1 K2 (t − s)f0 (s)ds, t ∈ J. z(t) = K1 (t)u0 + 0

It follows from this and the fact that PF is weakly closed that f0 ∈ SFP (.,u) . Therefore, z ∈ R(u). Since yn → z in C(J, E), z = y. This shows that R is closed.

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Let u ∈ C(J, E) and yn ∈ R(u), n ≥ 1. The same argument as above implies that {yn : n ≥ 1} has a convergent subsequence, so R(u) is relatively compact. Argue as above and we see that R(u) is closed. Thus R(u) is compact. Step 2. R is condensing with respect to a nonsingular measure of noncompactness defined on C (J, E) . We define a measure of noncompactness ν on C (J, E) as follows. For each bounded subset Ω of C (J, E) we put ν (Ω) = max (γ(D), modC (D)) ∈ R2 , D∈Δ(Ω)

where Δ (Ω) is the collection of all the countable subsets of Ω, e−Lt χ ({u(t) : u ∈ D}) ,

γ(D) = sup, t∈J

modC (D) is the modulus of equicontinuity of the set of functions D given by the formula modC (D) = lim sup

max u(t1 ) − u(t2 ) ;

δ→0 u∈D |t1 −t2 |≤δ

and L > 0 is a positive real number chosen so that  P1  t 1+ P1 −L(t−s) P ζsup (e β(s)) ds <1 q=2 t∈I

(4)

0

P −1

1

M −1 where ζ = Γ(α) ( PPα−1 ) P lα− P . Note ν is monotone, nonsingular and a regular measure of noncompactness on the space C (J, E) . We show R is ν-condensing. Let Ω be a bounded subset of C (J, E) such that

ν (R(Ω)) ≥ ν (Ω) ;

(5)

here ≥ is taken in the sense of the order in R induced by the cone We need to prove that Ω is relatively compact. Since ν is regular it suffices to prove that ν (Ω) = (0, 0). We show ν (R (Ω)) = (0, 0). Let D = {yn : n ≥ 1} be a countable subset of R(Ω) where the maximum of the left hand side of (5) is achieved. Then there exists a countable set {un : n ≥ 1} of Ω such that yn ∈ R(un ), ∀n ≥ 1 and R2+ .

2

(γ ({yn : n ≥ 1}) ,

modC ({yn : n ≥ 1})) = ν (R (Ω)) .

For every n ≥ 1 and every t ∈ J we have

t (t − s)α−1 K2 (t − s)fn (s)ds, yn (t) = K1 (t)u0 +

(6)

t ∈ J,

0

where fn ∈ SFP (.,u) . From (5) and (6) we obtain (γ ({yn : n ≥ 1}) , modC ({yn : n ≥ 1})) = ν (R (Ω)) ≥ ν (Ω) ≥ (γ ({xn : n ≥ 1}) , modC ({un : n ≥ 1})) . Thus, γ ({yn : n ≥ 1}) ≥ γ ({un : n ≥ 1})

(7)

modC ({yn : n ≥ 1}) ≥ modC ({un : n ≥ 1}) .

(8)

and

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Using (HF 3) we have for a.e. on t ∈ J that χ ({fn (s) : n ≥ 1}) ≤ χ (F (s, {un (s) : n ≥ 1})) ≤ β(s)χ{un (s) : n ≥ 1} ≤ β(s)eLs sup e−Lt χ{un (t) : n ≥ 1} t∈J

Ls

≤ β(s)e γ ({un : n ≥ 1}) . Then applying [Lemma 3.9], we get for t ∈ J that ⎛ t ⎞ P1

1 χ ({yn (t) : n ≥ 1}) ≤ 21+ P ζγ ({un : n ≥ 1}) ⎝ (β(s)eLs )P ds⎠ ,

(9)

0

where ζ =

−1 1 M P −1 PP bα− P Γ(α) ( P α−1 )

. Thus from (4), (7) and (9) we get

γ ({un : n ≥ 1}) ≤ γ ({yn : n ≥ 1}) = sup e−Lt χ ({yn (t) : n ≥ 1}) t∈J

⎛ t ⎞ P1

ζγ ({un : n ≥ 1}) ⎝ (β(s)eLs )P ds⎠ .

1+ P1

≤ sup e−Lt 2 t∈J

0

⎛ t ⎞ P1

1 = 21+ P ζγ{un : n ≥ 1}sup ⎝ (β(s)e−L(t−s) )P ds⎠ t∈J

0

= q γ{un : n ≥ 1}. Since q < 1 we have γ ({un : n ≥ 1}) = 0 and consequently γ({yn : n ≥ 1}) = 0. Thus the set {yn : n ≥ 1} is relatively compact in C (J, E) . Hence lim sup max yn (t1 ) − yn (t2 ) = 0. δ→0n≥1|t1 −t2 |<δ

Thus, modC ({yn : n ≥ 1}) = 0, so ν (R (Ω)) = (0, 0) and then ν (Ω) = (0, 0). Step 3. We will apply [16, Corollary 3.3.1]. Thus, we need to show that there is a closed convex bounded set such that R maps it to itself. We introduce the equivalent norm in the space C (J, E) given by u∗ = max e−tN u(t), t∈J

where N is chosen such that

t Mc max (t − s)α−1 e−N (t−s) ds ≤ λ < 1. t∈J Γ(α)

(10)

(11)

0

In the space C (J, E) with the norm defined by (10) we consider the closed ball in C (J, E) : B(0, r) = {u ∈ C (J, E) : uC(J,E) ≤ r},

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where r is a constant chosen so that r≥

M u0  + ςϕLP (J,R+ ) . 1−λ

(12)

Note from (11) the number r is well defined. We show that the multifunction R maps the closed ball B(0, r) into itself. Let u ∈ B(0, r) and y ∈ R (u) . Then there is a f ∈ SFP (.,u) such that for every t ∈ J we have

M −N t t e−N t y(t) ≤ e−N t K1 (t)u0  + e (t − s)α−1 f (s)ds. Γ(α) 0 Using (HF 2)∗ we get e−N t y(t) ≤ e−N t K1 (t)u0 

M −N t t e + (t − s)α−1 (ϕ(s) + cu(s))ds Γ(α) 0 ≤ M u0  + ςϕLP (J,R+ )

t Mc + (t − s)α−1 e−N (t−s) e−N s u(s)ds Γ(α) 0 = M u0  + ςϕLP (J,R+ )

t cM uC(J,E) + (t − s)α−1 ϕ(s)e−N (t−s) ds Γ(α) 0 = M u0  + ςϕLP (J,R+ ) + λr < r. Thus yC(J,E) ≤ r, so y ∈ B(0, r). From [16, Corollary 3.3.1] we have that that R has a fixed point which is a mild solution for (3). Step 4. The set of mild solutions of (3) is a compact subset in C (J, E) . It is sufficient to show that the set of fixed points of R is bounded (see [16, Property 3.5.1]). Let u ∈ R(u) and f ∈ SFP (.,u) be such that

t u(t) = K1 (t)u0 + (t − s)α−1 K2 (t − s)f (s)ds, t ∈ J, 0 ∗

Using (HF 2) and the H¨ older inequality to get for t ∈ J that

t M u(t) ≤ M u0  + (t − s)α−1 (ϕ(s) + cu(s))ds Γ(α) 0

t cM (t − s)α−1 u(s)ds ≤ M u0  + ςϕLP (J,R+ ) + Γ(α) 0

t cM (t − s)α−1 u(s)ds. = M u0  + ςϕLP (J,R+ ) + Γ(α) 0 Apply Gronwall’s inequality and we obtain u(t) ≤ (M u0  + ςϕLP (J,R+ ) )Eα



cM Γ(α)

 , ∀t ∈ J.

Topological structure of the solution set

Therefore,

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 uC(J,E) ≤ (M u0  + ςϕLP (J,R+ ) )Eα

cM Γ(α)

59

 = η.

(13)

From [16, Property 3.5.1] we conclude that the set of mild solutions of (3) is a compact subset in C (J, E) .  Theorem 3.2. Assume (A), (HF1 ), (HF 2)∗ , (HF 3) and (H) hold. Then, the F solution set u0 [0, l] of mild solutions for (1) is nonempty. Proof. We divide the proof into several steps. F Step 1. From Theorem 3.1 the solution set u0 [0, l] of

c α D0,t u(t) ∈ Au(t) + F (t, u(t)), for a.e. t ∈ [0, b1 ], u(0) = u0 ∈ E, is nonempty and a compact subset in C([0, b1 ], E). F Step 2. Fix Z1 ∈ u0 [0, b1 ]. Then there is a f1 ∈ LP ([0, b1 ], E) such that f1 (t) ∈ F (t, Z1 (t)) for a.e. t ∈ [0, b1 ] and

t Z1 (t) = K1 (t)(u0 ) + (t − s)α−1 K2 (t − s)f1 (s)ds, t ∈ [0, b1 ]. 0

Extend the domain of the definition of Z1 to the interval [0, a1 ] by setting Z1 (t) = g1 (t, Z 1 (b− 1 )), t ∈ (b1 , a1 ]. Note (H) so Z1 is continuous on [0, a1 ] F From Theorem 3.1 the solution set g1 (a1 ,Z 1 (b− )) [a1 , b2 ] for 1

c α Da1,t u(t) ∈ Au(t) + F (t, u(t)), for a.e. t ∈ [a1 , b2 ], (14) u(a1 ) = g1 (a1 , Z1 (b− 1 )) = Z1 (a1 ), F is non empty and compact. Let Z2 be a fixed element in g1 (a1 ,Z 1 (b− )) [a1 , b2 ]. 1 Then Z2 : [a1 , b2 ] → E and

t Z2 (t) = K1 (t−a1 )(g1 (a1 , Z1 (b− )))+ (t−s)α−1 K2 (t−s)f2 (s)ds, t ∈ [a1 , b2 ], 1 a1

where f2 ∈ LP ([a1 , b2 ], E) and f2 (t) ∈ F (t, Z2 (t)), a.e. t ∈ [a1 , b2 ]. Note Z2 (a1 ) = g1 (a1 , Z1 (b− 1 )). Extend the domain of the definition of Z2 to the interval [a1 , a2 ] by setting Z2 (t) = g2 (t, Z2 (b− 2 )), t ∈ [b2, a]. Continue this process and in the final step we consider the problem

c α Dam ,t u(t) ∈ Au(t) + F (t, u(t)), for a.e. t ∈ [am , bm + 1], u(am ) = Zm (am ) = gm (am , Zm (b− m )), F and this problem has a solution and the solution set gm (am ,Zm (b− [am , bm + m )) 1] is compact. Let Zm+1 : [am , bm + 1] → E be a fixed element in F [am , bm + 1]. Then gm (am ,Zm (b− m )) Zm+1 (t) = K1 (t − am )(gm (am , Z m (b− m )))

t + (t − s)α−1 K2 (t − s)fm+1 (s)ds, am

t ∈ [am , bm + 1],

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where fm+1 (t) ∈ LP ([am , bm + 1], E) and fm+1 (t) ∈ F (t, Zm+1 (t)), a.e. t ∈ [am , bm + 1]. Note Zm+1 (am ) = gm (am , Zm (b− m )). Define f : J → E as ⎧ f1 (t), t ∈ [0, a1 ], ⎪ ⎪ ⎪ ⎨ f2 (t), t ∈ (a1 , a2 ], f (t) = . .. ⎪ ⎪ ⎪ ⎩ fm+1 (t), t ∈ (am , bm + 1], and define u : J → E as

⎧ Z1 (t), t ∈ [0, b1 ], ⎪ ⎪ ⎪ − ⎪ g ⎪ 1 (t, Z1 (b1 )), t ∈ (b1 , a1 ]. ⎪ ⎪ ⎪ Z2 (t), t ∈ (a1 , b2 ], ⎪ ⎪ ⎪ ⎨ g2 (t, Z2 (b− 2 )), t ∈ (b2 , a2 ], u(t) = . .. ⎪ ⎪ ⎪ ⎪ ⎪ Zm (t), t ∈ (am−1 , bm ], ⎪ ⎪ ⎪ ⎪ (t, Z (b− )), t ∈ (bm , am ], g ⎪ ⎪ ⎩ mm+1 m m (t), t ∈ (am , bm + 1]. Z

i=m Note that f (t) ∈ LP ([0, l], E) and f (t) ∈ F (t, u), a.e. t ∈ i=0 [ai bi+1 ], and u is continuous at ai , i = 0, 1, . . . , m and satisfies the integral equation: t ⎧ α−1 K2 (t − s)f (s)ds, t ∈ [0, b1 ], ⎪ ⎪ K1 (t)u0−+ 0 (t − s) ⎨ gi (t, u(bi )), t ∈ (bi , ai ], = 1, 2, . . . , m, t u(t) = α−1 ⎪ K (t − ai )gi (ai , u(b− K2 (t − s)f (s)ds, ⎪ i )) + ai (t − s) ⎩ 1 t ∈ [ai , bi+1 ], i = 1, 2, . . . , m. Therefore u ∈

F



u0 [0, l].

From [11, Theorem 3.2] we have the following result. Theorem 3.3. If (A)∗ , (HF 1), (HF 2)∗∗ , (HF 3) and (H)∗ hold. Then [0, l] is nonempty and compact if M h + ζϕLP (J,R+ ) < 1, h =

m 

hi .

F

u0

(15)

i=1

and 4ζβLP (J,R+ ) < 1.

(16)

Moreover, any solution y of (2) satisfies yP C(J,E) ≤ r, where r=

M u0  + ζϕLP (J,R+ ) . 1 − [M h + ζϕLP (J,R+ ) ]

(17)

Topological structure of the solution set

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59

Proof. We sketch the proof. Define a multi operator Φ as follows: let u ∈ P C(J, E), and y ∈ Φ(u) if and only if ⎧

t ⎪ ⎪ K1 (t)u0 + (t − s)α−1 K2 (t − s)f (s)ds, t ∈ [0, b1 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, u(b− t ∈ (bi , ai ], i = 1, 2, . . . , m, i )), y(t) = ⎪ K1 (t − ai )gi (ai , u(b− ⎪ i )) ⎪ ⎪

⎪ t ⎪ ⎪ ⎪ ⎩+ (t − s)α−1 K (t − s)f (s)ds, t ∈ [a , b ], i = 1, 2, . . . , m, 2

ai

i

i+1

(18) where f ∈ SFP (·,u(·)) . Let Br = B(0, r) where r is given in (17). A modification of a standard argument in the literature guarantees that Φ is a closed multifunction from Br to Pck (Br ). Let B1 = conv Φ(Br ), Bn = conv Φ(Bn−1 ), ∞ Bn . In [11] (using a modification of an argument in the n ≥ 2 and B = n=1

literature) we established that Φ : B → Pck (B) is compact. Next let u ∈ B, u ∈ Φ(u) and f ∈ SFP (·,u(·)) such that

u(t) =

⎧

t ⎪ ⎪ K (t)u + (t − s)α−1 K2 (t − s)f (s)ds, t ∈ [0, b1 ], ⎪ 1 0 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, u(b− )), t ∈ (bi , ai ], i = 1, 2, . . . , m, i ⎪ K1 (t − ai )gi (ai , u(b− ⎪ i )) ⎪ ⎪

t ⎪ ⎪ ⎪ ⎪ ⎩+ (t − s)α−1 K2 (t − s)f (s)ds, ai

t ∈ [ai , bi+1 ],

i = 1, 2, . . . , m.

In [11] an easy argument established that uP C(J,E) ≤ r. Now apply Lemma F  2.13 and we deduce that u0 [0, l] is compact in P C(J, E).

4. Topological structure of the solution set Assume F satisfies (HF 2)∗∗ . Consider a multivalued map F : J ×E → Pck (E) defined by  F (t,  u), if u ≤ r,  (19) F (t, u) = ur , if u > r, F t, u where r is defined in (17). It is worth remarking here that the “truncation” method (definition of F) and Lemma 4.1 (in this section) appeared first (to the authors knowledge) in [21]. Since the function

u, if u ≤ r, ωr (u) = ur if u > r, u ,

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is a continuous retraction of E onto the closed subset Br = {u ∈ E : u ≤ r}, it follows that F satisfies (HF 1). Moreover, for a.e. t ∈ J F(t, u) = F (t, u) ≤ ϕ(t)(1 + u) ≤ ϕ(t)(1 + r) := ψ(t), for every x, with u ≤ r, and     ur   ≤ ϕ(t)(1 + r) = ψ(t), F t, F(t, u) =   u  for every x, with u > r. Thus we have for any u ∈ E, F(t, u) ≤ ψ(t),

for a.e.

t ∈ J,

where ψ ∈ LP (J, R+ ). Now we consider ⎧ m  ⎪ c α  ⎪ D u(t) ∈ Au(t) + F (t, u(t)), a.e. t ∈ (ai , bi+1 ] ⊂ J, ⎪ ai ,t ⎪ ⎪ ⎪ i=0 ⎨ a0 := 0, bm+1 := l, ⎪ ⎪ ⎪ ⎪ u(t) = gi (t, u(b− t ∈ (bi , ai ] ⊂ [0, l], i = 1, 2, . . . , m, ⎪ i )), ⎪ ⎩ u(0) = u0 ∈ E.

(20)

Since F and F coincide on Br , the solution set of (1) is equal to the solution set for (20). Consequently, we assume from now on (without loss of generality) that F satisfies the following global integral bounded condition: (HF 2)∗∗∗ there exists a function ψ ∈ LP (J, R+ ), (P > α1 ) such that for any u ∈ E F (t, u) ≤ ψ(t), a.e. t ∈ J. In our next result, we use ideas in [21] (or [14, Theorem 3.5] or [16, Lemma 5.3.4]). Lemma 4.1. Assume that F : J × E → Pck (E) is a multifunction satisfying (HF 1). Then there exists a sequence of multifunctions {Fi }∞ i=1 , Fi : J ×E → Pck (E) such that (i) every multifunction Fi (t, ·) : E → Pck (E) is continuous for a.e. t ∈ J; (ii) F (t, x) ⊆ · · · ⊆ Fi+1 (t, x) ⊆ Fi (t, x) ⊆ · · · ⊆ F1 (t, x) ⊆ coF (t, {y ∈ E : y − x ≤ 31−i }), i ≥ 1; (iii) F (t, x) = i≥1 Fi (t, x); (iv) for every i ≥ 1, there exists a selection zi : J × E → E of Fi such that zi (·, x) is measurable and zi (t, ·) is locally Lipschitz for a.e. t ∈ J. Proof. For any fixed natural number i, cover E by the family of open balls Oi = {B(x, di ),

x ∈ E,

di = 3−i ,

i ≥ 1},

where B(x, di ) = {y ∈ E : y − x ≤ di }. Since E is a metric space, there is a locally finite refinement {Vk : k ∈ I} of the cover Oi . Now let {Pk : k ∈ I} be a locally Lipschitz partition of unity and subordinate to the open cover {Vk : k ∈ I}. This means that Pk : E → [0, 1], k ∈ I, for every point x ∈ E

Topological structure of the solution set

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59

there is aneighborhood of x where all but a finite number of {Pk : k ∈ I} are zero, k∈I Pk (x) = 1 and such that {x ∈ E : Pk (x) = 0} ⊆ Vk . For every k ∈ I, let xk ∈ E be such that Vk ⊆ B(xk , di ) and define  Pk (x)coF (t, B(xk , 2di ). Fi (t, x) = k∈I

To prove (ii) note that pk (x) > 0 implies that x ∈ Vk ⊆ B(xk , di ), hence x ∈ B(xk , 2di ) ⊆ B(x, 3di ). Therefore,  Pk (x)coF (t, B(xk , 2di ) F (t, x) ⊆ k∈I

⊆



Pk (x)coF (t, B(x, 3di )

k∈I

= coF (t, B(x, 3di )



Pk (x)

k∈I

= coF (t, B(x, 3di ). Clearly Fi+1 (t, x) ⊆ Fi (t, x), i ≥ 1. To prove (iii) let (t, x) be a fixed point. Note that F (t, x) ⊆ i≥1 Fi (t, x). Let U be an open convex subset of E and containing F (t, x). From the upper semicontinuity of F (t, ·) at x there is a δ > 0 such that y − x < δ ⇒ F (t, y) ⊆ U . Choose i0 ≥ 1 such that 3di0 < δ. Then coF (t, B(x, 3di0 ) ⊆ U . Consequently  Fi (t, x) ⊆ Fi0 (t, x) ⊆ coF (t, B(x, 3di0 ) ⊆ U . i≥1

Then i≥1 Fi (t, x) ⊆ {U : U ∈ Υ}, where Υ is the family of all open convex subsets containing F (t, x), and hence i≥1 Fi (t, x) ⊆ F (t, x). To prove (iv) we take, for every xk , k ∈ I, a measurable selection hk : J → E of  Fi of the multivalued map Fi (·, xk ) and define zi : J × E → E as  zi (t, x) = k∈I Pk (x)hk (t).

Remark 4.2. Let D be a bounded subset of a Banach space E with a finite ρ-net. That is there are elements x1 , x2 , x3 , . . . , xk of D such that D ⊆ i=k i=1 B(xi , ρ). Let D∗ = B(D, ς) = {y ∈ E : d(y, D) ≤ ς}. Then D∗ ⊆

i=k 

B(xi , ρ + ς).

i=1

In fact, if y ∈ D∗ , then there is a x ∈ D such that d(y, x) ≤ ς. Choose i0 such that d(x, xi0 ) ≤ ρ. Hence, d(y, xi0 ) ≤ d(y, x) + d(x, xi0 ) = ς + ρ. As a result we get χ(D∗ ) ≤ χ(D) + ς.

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Theorem 4.3. Assume assumptions (A)∗ (HF 1), (HF 2)∗∗∗ , (HF 3), (H)∗ and (15) and 4ζβLP (J,R+ ) < 1

(21)

F hold. Then u0i [0, l] is nonempty for i ≥ 1 and there is a natural number N Fi such that u0 [0, l], i ≥ N is compact. Proof. From (i), (ii) and (iv) of Lemma 4.1 it follows that each Fi satisfies (HF 1), (HF 2)∗∗∗ and (HF 3) (see the proof of [16, Theorem 5.3.1]). Now F Theorem 3.3 guarantees that u0i [0, l] is nonempty for i ≥ 1. To show the ||β||

L (J,R ) < 2 , i ≥ N . second assertion let ε > 0 and choose N such that 3i−1 Let i ≥ N be a fixed natural number. From the proof of [11, Theorem 3.2], F to show that u0i [0, l] is compact it is enough to show that P

+

lim χP C (Bi,n ) = 0,

(22)

n→∞

where Bi,1 = convΦi (Br ), Bni = convΦi (Bi,n−1 ), n ≥ 2, and Φi : P C(J, E) → 2P C(J,E) is defined by: let u ∈ P C(J, E), and a function y ∈ Φi (u) if and only if ⎧

t ⎪ ⎪ K (t)u + (t − s)α−1 K2 (t − s)f (s)ds, t ∈ [0, b1 ], ⎪ 1 0 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, u(b− )), t ∈ (bi , ai ], i = 1, 2, . . . , m, i y(t) = ⎪ K1 (t − ai )gi (ai , u(b− ⎪ i )) ⎪ ⎪

⎪ t ⎪ ⎪ ⎪ ⎩+ (t − s)α−1 K2 (t − s)f (s)ds, t ∈ [ai , bi+1 ], i = 1, 2, . . . , m, ai

and f ∈ SFPi (·,u(·)) . In view of [19, Lemma 2.14] there exists a sequence (yk ), k ≥ 1 in Φi (Bi,n−1 ) such that χP C (Bi,n ) = χP C Φi (Bi,n−1 ) ≤ 2χP C {yk : k ≥ 1} + ε, and as in the proof of [11, Theorem 3.2], χP C (Bi,n ) ≤ 2 sup χ{yk (t) : k ≥ 1} + ε.

(23)

t∈J

Let uk , k ≥ 1 in Bi,n−1 and fk ∈ SFPi (·,uk (·)) such that for every t ∈ J,

yk (t) =

⎧

t ⎪ ⎪ K1 (t)u0 + (t − s)α−1 K2 (t − s)fk (s)ds, t ∈ [0, b1 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, uk (b− t ∈ (bi , ai ], i = 1, 2, . . . , m, i )), ⎪ K1 (t − ai )gi (ai , uk (b− ⎪ i )) ⎪ ⎪

⎪ t ⎪ ⎪ ⎪ ⎩+ (t − s)α−1 K2 (t − s)fk (s)ds, ai

t ∈ [ai , bi+1 ],

i = 1, 2, . . . , m.

Topological structure of the solution set

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59

Notice from (ii) of Lemma 4.1 and Remark 4.2, we have for a.e. t ∈ J that χ({fk (t) : k ≥ 1} ≤ χ{Fi (t, uk (t)) : k ≥ 1} ≤ χcoF {t, B({uk (t) : k ≥ 1}, 31−i )} ≤ β(t)(χ{uk (t) : k ≥ 1} + 31−i ) ≤ β(t)( χ(Bn−1 (t)) + 31−i ) ≤ β(t)(χP C (Bn−1 ) + 31−i ) = γ ∗ (t).

(24)

Using the same arguments as in [11, Theorem 3.2], it follows for t ∈ [0, b1 ] that  t  α−1 χ (t − s) K2 (t − s)fk (s)ds : k ≥ 1 0

≤ ζ(2βLP (J,R+ ) χP C (Bn−1 ) + 2βLP (J,R+ ) 31−i + lq )  PP−1 

αP M P −1 ϕLP (J ,R+ ) (t − s) ds + Γ(α) J 1

≤ ζ(2βLP (J,R+ ) χP C (Bn−1 ) +  + l P ).

(25)

Taking into account that ε is arbitrary, inequality (25) gives for all t ∈ [0, b1 ] that  t  χ (t − s)α−1 K2 (t − s)fk (s)ds : k ≥ 1 ≤ 2ζβLP (J,R+ ) χP C (Bi,n−1 ). 0

Similarly, we can show that if t ∈ [ai , bi+1 ], i = 1, 2, . . . , m, then   t α−1 (t − s) K2 (t − s)fk (s)ds : k ≥ 1 ≤ 2ζβLP (J,R+ ) χP C (Bi,n−1 ). χ si

Then, for every t ∈ J, χ{yk (t) : k ≥ 1} ≤ 2ζβLP (J,R+ ) χP C (Bi,n−1 ). This inequality, (23) and the fact that ε is arbitrary, implies χP C (Bi,n ) ≤ 4ζβLP (J,R+ ) χP C (Bi,n−1 ). From a finite number of steps we get 0 ≤ χP C (Bi,n ) ≤ (4ζβLP (J,R+ ) )n−1 χP C (Bi,1 ),

∀ n ≥ 1.

Since this inequality is true for every n ∈ N, then from (21) and passing to the limit as n → +∞, we obtain (22) and so our result follows.  Theorem 4.4. Under conditions (A)∗ (HF 1), (HF 2)∗∗∗ , (HF 3), (H)∗ we have F  u0

[0, l] =

Fn ∞   n=N u0

[0, l].

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F ∞ F Proof. From (iii) in Lemma 4.1, if u ∈ u0 [0, l] then u ∈ n=N u0n [0, l]. ∞ F Let u ∈ n=N u0n [0, l]. Then there are fn ∈ SFPn (·,u(·)) , n ≥ N such that ⎧

t ⎪ ⎪ K (t)u + (t − s)α−1 K2 (t − s)fn (s)ds, t ∈ [0, b1 ], ⎪ 1 0 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, u(b− )), t ∈ (bi , ai ], i = 1, 2, . . . , m, i u(t) = ⎪ K1 (t − ai )gi (ai , u(b− ⎪ i )) ⎪ ⎪

t ⎪ ⎪ ⎪ ⎪+ ⎩ (t − s)α−1 K2 (t − s)fn (s)ds, t ∈ [ai , bi+1 ], i = 1, 2, . . . , m. ai

(26) We show that the sequence {fn : n ≥ N } is semicompact. Let m > N be a fixed natural number. It follows from Remark 4.2 that for almost t ∈ J, χ({fn (t) : n ≥ m} ≤ χ{Fn (t, u(t)) : n ≥ m} ≤ χcoF {t, B(u(t), 31−m )} ≤ β(t)(χ{B((u(t), 31−m )}. Since 31−m → 0 as m → ∞, we obtain χ{fn (t) : n ≥ N } = 0. Moreover fn (t) ≤ ψ(t) for a.e. t ∈ J. Then {fn : n ≥ N } is semicompact, and hence it is weakly relatively compact due to Lemma 2.3. We may assume, without loss of generality that fn converges weakly to a function f ∈ L1 (J, R+ ). From Mazur’s lemma, for every natural number j ≥ N there is a natural number k0 (j) > j and a sequence of nonnegative real numbers λj,k , k = k0 (j), . . . , j k0 such that k=j λj,k = 1 and the sequence of convex combinations zj = k0 1 k=j λj,k fk , j ≥ 1, converges strongly to f in L (J, E) as j → ∞. From the sequence zn we may select a subsequence, denoted again by zn , convergent to f a.e. Since fn (t) ≤ ψ(t), for a.e. t ∈ J, then {zn } satisfies the same estimate and hence f (t) ≤ ψ(t), for a.e. t ∈ J. Consequently f ∈ LP (J, E). Note that, ⎧

t ⎪ ⎪ K1 (t)u0 + (t − s)α−1 K2 (t − s)zn (s)ds, t ∈ [0, b1 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, u(b− t ∈ (bi , ai ], i = 1, 2, . . . , m, i )), u(t) = ⎪ K1 (t − ai )gi (ai , u(b− ⎪ i )) ⎪ ⎪

⎪ t ⎪ ⎪ ⎪ ⎩+ (t − s)α−1 K2 (t − s)zn (s)ds, t ∈ [ai , bi+1 ], i = 1, 2, . . . , m. ai

From (ii) of Lemma 4.1 we obtain for each i ≥ N , zi (t) ∈ Fn (t, u(t)), for all n satisfying ∞ N ≤ n ≤ i, and therefore for the limiting function f we have f (t) ∈ n=N Fn (t, u(t) for a.e. t ∈ J. Then f (t) ∈ F (t, u(t) for a.e. t ∈ J. Moreover, for every t ∈ J, s ∈ (0, t] and every n ≥ 1 (t − s)α−1 K2 (t − s)fn (s) ≤

M ψ(s) |t − s|α−1 , Γ(α)

ψ(s) ∈ LP ((0, t], R+ ).

Then the continuity of K2 (t)(t > 0) implies that for every t ∈ J, K2 (t − s)zn (s) → K2 (t − s)f (s), for s ∈ (0, t). Therefore, by passing to the limit as

Topological structure of the solution set

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59

n → ∞ in (26) we obtain from the Lebesgue dominated convergence theorem that ⎧

t ⎪ ⎪ K (t)u + (t − s)α−1 K2 (t − s)f (s)ds, t ∈ [0, b1 ], ⎪ 1 0 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, u(b− )), t ∈ (bi , ai ], i = 1, 2, . . . , m, i u(t) = ⎪ K1 (t − ai )gi (ai , u(b− ⎪ i )) ⎪ ⎪

t ⎪ ⎪ ⎪ ⎪+ ⎩ (t − s)α−1 K (t − s)f (s)ds, t ∈ [a , b ], i = 1, 2, . . . , m, 2

ai

which yields that u ∈

i

i+1

F



u0 [0, l].

Lemma 4.5. Let (M, d) and (Y, ρ) be two metric spaces. If f : (M, d) → (Y, ρ) is locally Lipschitz then it is Lipschitz on every compact subset of M. Proof. Let K be a compact subset of M. For every point x in M there exists an open set Vx containing x on which f satisfies a Lipschitz condition. Because K is compact there exist a finite number of these open sets V1 , V2 , . . . , Vp i=p contained in M such that K ⊆ i=1 Vi and ρ(f (x), f (y)) ≤ Li d(x, y). Since K is compact and f is continuous there is a positive number δ1 such that δ = sup{ρ(f (x), f (y)) : x, y ∈ K}. Again since K is compact there is a positive number (Lebesgue number) λ such that if x, y ∈ K with d(x, y) < λ then x, y ∈ Vi for some i = 1, 2, . . . , p. Set γ = max{max1≤i≤p Li , λδ } Now let x, y ∈ K. If d(x, y) < λ, then there is i = 1, 2, . . . , p. such that x, y ∈ Vi and hence ρ(f (x), f (y)) ≤ Li d(x, y) ≤ γd(x, y). If d(x, y) ≥ λ, then ρ(f (x), f (y)) ≤ δ = The proof is finished.

δ δλ ≤ d(x, y) ≤ γd(x, y). λ λ 

F Theorem 4.6. Under the assumptions of Theorem 4.3, the set u0 [0, l] is a Rδ -set. Fn Proof. We show that u0 [0, l], ∀n ≥ N is contractible. Let n ≥ N be a fixed natural number. From (iv) of Lemma 4.1 there exists zn : J × E → E a selection of Fn such that zn (·, u) is measurable and zn (t, ·) is locally Lipschitz for a.e. t ∈ J. Since the multifunction Fn satisfies (HF 1), (HF 2)∗∗∗ and (HF 3) then zn also satisfies these conditions and from Theorem 3.2, ⎧ m  ⎪ c α ⎪ Dai ,t u(t) = Au(t) + zn (t, u(t)), a.e. t ∈ (ai , bi+1 ] ⊂ J, ⎪ ⎪ ⎪ ⎪ i=0 ⎨ a0 := 0, bm+1 := l > 0, (27) ⎪ ⎪ − ⎪ ⎪ y(t) = gi (t, u(bi )), t ∈ (bi , ai ] ⊂ [0, l], i = 1, 2, . . . , m, ⎪ ⎪ ⎩ u(0) = u0 ∈ E,

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has a mild solution, denoted by y, i.e., ⎧

t ⎪ ⎪ ⎪ K1 (t)u0 + (t − s)α−1 K2 (t − s)zn (s, y(s))ds, t ∈ [0, b1 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ g (t, y(b− t ∈ (bi , ai ], i = 1, 2, . . . , m, ⎪ i )), ⎨ i y(t) = K1 (t − ai )gi (ai , y(b− i )) ⎪ ⎪

t ⎪ ⎪ ⎪ ⎪ + (t − s)α−1 K2 (t − s)zn (s, y(s))ds, t ∈ [ai , bi+1 ], ⎪ ⎪ ⎪ a ⎪ i ⎪ ⎩ i = 1, 2, . . . , m. F F Notice y ∈ u0n [0, l]. We now show that u0n [0, l] is homotopically equivalent F to y, i.e., we will construct a continuous function hn : [0, 1] × u0n [0, l] −→ Fn Fn u0 [0, l] such that hn (0, u) = u and hn (1, u) = y, ∀u ∈ u0 [0, l]. Let Fn (λ, x ) ∈ [0, 1] × u0 [0, l] be a fixed element. Then there is a f ∈ SF1 n (·,x(·)) such that ⎧

t ⎪ ⎪ K (t)u + (t − s)α−1 K2 (t − s)f (s)ds, t ∈ [0, b1 ], ⎪ 1 0 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ gi (t, x (b− t ∈ (bi , ai ], i = 1, 2, . . . , m, i )), x (t) =

t ⎪ ⎪ − ⎪ (bi )) + (t − s)α−1 K2 (t − s)f (s)ds, ⎪ K1 (t − ai )gi (ai , x ⎪ ⎪ ai ⎪ ⎪ ⎩ t ∈ [ai , bi+1 ], i = 1, 2, . . . , m. 1 2 , m+1 , . . . , m+1 Consider a partition for J with the points 0, m+1 m+1 . Let 1 λ ∈ (0, m+1 ]. From Theorem 3.2, ⎧c α D u(t) = Au(t) + zn (t, u(t)), ⎪ ⎨ bm+1 −λ(m+1)(bm+1 −am ),t a.e. t ∈ (bm+1 − λ(m + 1)(bm+1 − am ), bm+1 ], (28) ⎪ ⎩ x(t) = x (t), t ∈ [0, bm+1 − λ(m + 1)(bm+1 − am )]. F has a solution. Observe that any solution of (28) must be in u0n [0, l]. We now show that (28) has a unique solution. Assume that uλ and vλ are two solutions to (28). Then ⎧ x (t), t ∈ [0, bm+1 − λ (m + 1)(bm+1 − am )], ⎪ ⎪ ⎪ ⎪ K ⎨ 1 (t − (bm+1 − λ (m + 1)(bm+1 − am ))) − λ (m + 1)(bm+1 − am )) x (b uλ (t) =  t m+1 ⎪ ⎪ + (t − s)α−1 K2 (t − s)zn (s, uλ (s))ds, ⎪ bm+1 −λ (m+1)(bm+1 −am ) ⎪ ⎩ t ∈ (bm+1 − λ (m + 1)(bm+1 − am ), bm+1 ],

and

⎧ x (t), t ∈ [0, bm+1 − λ (m + 1)(bm+1 − am )] ⎪ ⎪ ⎪ ⎪ K1 (t − (bm+1 − λ (m + 1)(bm+1 − am ))) ⎨ − λ (m + 1)(bm+1 − am )) x (b vλ (t) =  t m+1 ⎪ ⎪ + (t − s)α−1 K2 (t − s)zn (s, vλ (s))ds, ⎪ bm+1 −λ (m+1)(bm+1 −am ) ⎪ ⎩ t ∈ (bm+1 − λ (m + 1)(bm+1 − am ), bm+1 ].

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F Note uλ (t) = vλ (t), t ∈ [0, bm+1 − λ(m + 1)(bm+1 − am )]. Since u0n [0, l] F is compact in P C(J, E), then the set u0n [0, l], t ∈ (bm+1 − λ(m + 1)(bm+1 − am ), bm+1 ] is relatively compact. From Lemma 4.5, zn is Lipschitz in the F second argument on u0n [0, l], t ∈ (bm+1 − λ(m + 1)(bm+1 − am ), bm+1 ] and for any s ∈ (bm+1 − λ(m + 1)(bm+1 − am ), bm+1 ] there is a positive function σ(s) with sups∈J σ(s) = ksup < ∞ such that zn (s, vλ (s)) − zn (s, vλ (s)) ≤ ksup uλ (s) − vλ (s), for s ∈ (bm+1 − λ (m + 1)(bm+1 − am ), bm+1 ]. Thus,

uλ (t)−vλ (t) ≤

ksup M Γ(α)



t

bm+1 −λ (m+1)(bm+1 −am )

(t−s)α−1 uλ (s)−vλ (s)ds.

Apply the generalized Gronwall inequality (see [22, Theorem 1 and Corollary 2]) and we obtain uλ (t) = vλ (t), t ∈ (bm+1 − λ (m + 1)(bm+1 − am ), bm+1 ]. Thus (28) has a unique solution which is denoted by ⎧ x (t), t ∈ [0, bm+1 − λ(m + 1)(bm+1 − am )], ⎪ ⎪ ⎪ ⎪ K (t − (bm+1 − λ(m + 1)(bm+1 − am ))) ⎪ ⎪ ⎪ 1 ⎪ ⎨ x (bm+1 − λ(m + 1)(bm+1 − am )) xm+1 (t) =

t λ ⎪ ⎪ ⎪ + (t − s)α−1 K2 (t − s)zn (s, xm+1 (s))ds, ⎪ λ ⎪ ⎪ −λ(m+1)(b −a ) b ⎪ m+1 m+1 m ⎪ ⎩ t ∈ (bm+1 − λ(m + 1)(bm+1 − am ), bm+1 ]. (29) 1 2 Next for λ ∈ ( m+1 , m+1 ] consider

⎧c α Db −(m+1)(λ− 1 )(b −a x(t) = Ax(t) + zn (t, x(t)), ⎪ m m m−1 ),t ⎪ m+1 ⎪   ⎪ ⎪ ⎪ 1 ⎪ ⎪ a.e. t ∈ (bm − (m + 1) λ − (bm − am−1 ), bm ] ∪ (am , bm+1 ], ⎨ m+1 ⎪ ⎪ x(t) = gm (t, x(t− m )), t ∈ (bm , am ], ⎪ ⎪   ⎪ ⎪ ⎪ 1 ⎪ ⎩ x(t) = x (t), t ∈ [0, bm−1 − (m + 1) λ − (bm − am−1 )]. m+1

Following the above argument this problem has a unique solution which is denoted by

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  ⎧ 1 ⎪ ⎪ x (t), t ∈ [0, (bm − (m + 1) λ − (bm − am−1 )], ⎪ ⎪ m+1 ⎪ ⎪   ⎪ ⎪ ⎪ 1 ⎪ ⎪ K1 (t − ((bm − (m + 1) λ − (bm − am−1 )) ⎪ ⎪ m+1 ⎪ ⎪ ⎪   ⎪ ⎪ 1 ⎪ ⎪ ×x ((bm − (m + 1) λ − (bm − am−1 )) ⎪ ⎪ ⎪ m +1 ⎪ ⎪ ⎪ t ⎪ ⎨ + (t − s)α−1 K2 (t − s)zn (s, xm m λ (s))ds, xλ (t) = 1 b −(m+1) λ− (b −a ) ( ) m m m−1 ⎪ m+1 ⎪   ⎪ ⎪ ⎪ 1 ⎪ ⎪ t ∈ (bm − (m + 1) λ − (bm − am−1 ), bm ], ⎪ ⎪ m+1 ⎪ ⎪ ⎪ ⎪ − ⎪ ⎪ gm (t, xm t ∈ (bm , am ], λ (bm )), ⎪ ⎪ ⎪ ⎪ m − ⎪ ⎪ K1 (t − am )gm (t, xλ (bm )) ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎩+ (t − s)α−1 K2 (t − s)zn (s, xm t ∈ (am , bm+1 ]. λ (s))ds, am

Notice xm λ belongs to m , 1] let step for λ ∈ ( m+1

Fn u0

[0, l]. We continue this process. In the final

    ⎧ m−1 ⎪ ⎪ x (t), t ∈ 0, b1 − m λ − (b1 − a0 ) , ⎪ ⎪ m ⎪ ⎪ ⎪         ⎪ ⎪ m − 1 m−1 ⎪ ⎪ K1 (t − b1 − m r − (b1 − a0 ) x (b1 − a0 ) ( b1 − m λ − ⎪ ⎪ ⎪ m m ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ (t − s)α−1 K2 (t − s)zn (s, x1r (s))ds, + ⎪ ⎪ ⎪ (b1 −m(r− m−1 )(b1 −a0 ) ⎪ m ⎪ ⎪     ⎨ m−1 x1λ (t) = (b t ∈ b 1 −m λ− 1 − a0 ), b1 , ⎪ m ⎪ ⎪ ⎪ ⎪ m ⎪  ⎪ ⎪ ⎪ gi (t, x1λ (b− t∈ (bi , ai ], ⎪ i )), ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ ⎪ K1 (t − ai )gi (t, x1λ (b− ⎪ i )) ⎪ ⎪ ⎪

⎪ m t ⎪  ⎪ ⎪ ⎪ + (t − s)α−1 K2 (t − s)zn (s, x1λ (s))ds, t ∈ (ai , bi+1 ]. ⎩ ai

i=1

be the unique solution of ⎧c α Db −(m+1)(λ− m )(b −a ),t x(t) = Ax(t) + zn (t, x(t)), ⎪ 1 1 0 ⎪ m+1 ⎪ ⎪     ⎪ m ⎪ ⎪ m ⎪ ⎪ − (m + 1) λ − − a ), b (ai , bi+1 ], a.e. t ∈ b (b ⎪ 1 1 0 1 ⎪ m+1 ⎪ ⎨ i=1 m  ⎪ − ⎪ x(t) = g (t, x(b )), t ∈ (bi , ai ], ⎪ i i ⎪ ⎪ ⎪ i=1 ⎪ ⎪     ⎪ ⎪ m ⎪ ⎪ ⎩ x(t) = x (t), t ∈ 0, b1 − (m + 1) λ − (b1 − a0 ) . m+1

Topological structure of the solution set

Note xm+1 , xm , . . . , x1λ are in λ Fn Fn λ u0 [0, l] → u0 [0, l] as:

Fn u0

Page 21 of 25

[0, l]. Now define hn (·, ·) : [0, 1] ×

⎧ x , if λ = 0,  ⎪ ⎪  ⎪ ⎪ m+1 1 ⎪ x , if λ ∈ 0, ⎪ m+1 , ⎪ ⎨ λ  1 2 , hn (λ, x ) = xm if λ ∈ m+1 , m+1 λ , ⎪ ⎪ ⎪··· ⎪ ⎪   ⎪ ⎪ m ⎩ x1 = y, if λ ∈ ,1 . λ

59

(30)

m+1

Notice that

x1λ (t) =

⎧

t ⎪ ⎪ ⎪ K1 (t) x(0) + (t − s)α−1 K2 (t − s)zn (s, x11 (s))ds, ⎪ ⎪ ⎪ 0 ⎪ ⎪ m ⎪  ⎪ ⎪ ⎪ ⎨ gi (t, x11 (b− )), t ∈ (bi , ai ], i

t ∈ (0, b1 ],

i=1

⎪ ⎪ 1 − ⎪ ⎪ ⎪ K1 (t − ai )gi (t, x1 (bi )) ⎪ ⎪

t ⎪ ⎪ ⎪ ⎪ (t − s)α−1 K2 (t − s)zn (s, x11 (s))ds, ⎪ ⎩+ ai

t∈

m 

(ai , bi+1 ].

i=1

Then x1λ = y. Therefore hn (0, x ) = x  and hn (1, x ) = x1λ = y. We now show Fn hn is continuous. Let x , u  be two elements of u0 [0, l] and λ, μ ∈ [0, 1]. Then ⎧ u , if μ = 0, ⎪ ⎪   ⎪ ⎪ ⎪ 1 ⎪ m+1 ⎪ u , if μ ∈ 0, , ⎪ μ ⎪ m+1 ⎪ ⎪ ⎪   ⎨ 1 2 hn (μ, u ) = , , if μ ∈ , um μ ⎪ m+1 m+1 ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪   ⎪ ⎪ m ⎪ 1 ⎪ ⎩ uμ = y, if μ ∈ ,1 , m+1

(31)

where ⎧ u (t), t ∈ [0, bm+1 − μ(m + 1)(bm+1 − am )], ⎪ ⎪ ⎪ ⎪ K ⎪ 1 (t − (bm+1 − μ(m + 1)(bm+1 − am ))) ⎪ ⎪ ⎪ ⎨ u (bm+1 − μ(m + 1)(bm+1 − am )) um+1 (t) = (32)

t μ ⎪ ⎪ α−1 m+1 ⎪ + (t − s) K2 (t − s)zn (s, uμ (s))ds, ⎪ ⎪ ⎪ bm+1−μ(m+1)(bm+1 −am ) ⎪ ⎪ ⎩ t ∈ (bm+1 − μ(m + 1)(bm+1 − am ), bm+1 ],

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    ⎧ 1 ⎪ ⎪ u (t), t ∈ 0, bm − (m + 1) μ − (bm − am−1 ) , ⎪ ⎪ m+1 ⎪ ⎪     ⎪ ⎪ ⎪ 1 ⎪ ⎪ K1 (t − ( bm − (m + 1) μ − (bm − am−1 ) ⎪ ⎪ m+1 ⎪ ⎪ ⎪   ⎪ ⎪ 1 ⎪ ⎪ ×u ((bm − (m + 1) μ − (bm − am−1 )) ⎪ ⎪ m +1 ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪+ (t − s)α−1 ⎪ ⎪ 1 ⎨ bm −(m+1)(μ− m+1 )(bm −am−1 ) um μ (t) = K2 (t − s)zn (s, um ⎪ μ (s))ds, ⎪ ⎪     ⎪ ⎪ 1 ⎪ ⎪ t ∈ bm − (m + 1) μ − (bm − am−1 ), bm , ⎪ ⎪ ⎪ m+1 ⎪ ⎪ ⎪ m − ⎪ ⎪ ⎪ gm (t, uμ (bm )), t ∈ (bm , am ], ⎪ ⎪

t ⎪ ⎪ ⎪ m − ⎪ ⎪ K (t − a )g (t, u (b )) + (t − s)α−1 1 m m μ m ⎪ ⎪ ⎪ a m ⎪ ⎪ ⎪ ⎪ K2 (t − s)zn (s, um ⎪ μ (s))ds, ⎪ ⎩ t ∈ (am , bm+1 ],

(33)

· · · and

    ⎧ m−1 ⎪ ⎪ x(t) = u  (t), t ∈ 0, b − m μ − − a ) . (b 1 1 0 ⎪ ⎪ m ⎪ ⎪ ⎪     ⎪ ⎪ ⎪ ⎪ K1 (t − b1 − m r − m − 1 (b1 − a0 ) ⎪ ⎪ ⎪ m ⎪ ⎪     ⎪ ⎪ m − 1 ⎪ ⎪ ⎪ u  ( b − m μ − − a ) (b 1 1 0 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ t ⎪ ⎨ + (t − s)α−1 K2 (t − s)zn (s, u1μ (s))ds, m−1 u1μ (t) = (34) b1 −m(r− m )(b1 −a0 ) ⎪ ⎪   ⎪ ⎪ m−1 ⎪ ⎪ ⎪ t ∈ (b1 − m μ − (b1 − a0 ), b1 ], ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ gi (t, u1μ (t− ⎪ i )), t ∈ (bi , ai ], i = 1, 2, . . . , m, ⎪ ⎪ ⎪ ⎪ ⎪ K1 (t − ai )gi (t, u1μ (b− i )) ⎪ ⎪ ⎪

t ⎪ m ⎪  ⎪ ⎪ ⎪+ (t − s)α−1 K2 (t − s)zn (s, u1μ (s))ds, t ∈ (ai , bi+1 ]. ⎩ ai

i=1

If λ = μ = 0, then from (30) and (31), hn (λ, x ) = x , and hn (μ, x ) = u , 1 ) = hn (λ, u ). Let λ, μ ∈ (0, m+1 ]. Then from so when u →x  we get hn (λ, x (29) and (32) (and let μ → λ) we have

Topological structure of the solution set

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59

lim um+1 (t) − xm+1 (t) μ λ ⎧ u−x P C(J,E) , t ∈ [0, bm+1 − λ(m + 1)(bm+1 − am )], ⎪ ⎪  ⎪ ⎪ ⎪ M  u−x P C(J,E) ⎪ ⎪ ⎪ ⎪

t ⎨ M + (t − s)α−1 zn (s, um+1 (s)) ≤ λ Γ(α) ⎪ tm+1 −λ (m+1)(bm+1 −am ) ⎪ ⎪ ⎪ ⎪ ⎪ (s))ds, − zn (s, xm+1 ⎪ λ ⎪ ⎪ ⎩ t ∈ (b m+1 − λ(m + 1)(bm+1 − am ), bm+1 ].

μ→λ

Since for any s ∈ J, zn (s, ·) is locally Lipschitz at xm+1 , we can find λ ) of a τ (s) > 0 with sups∈J τ (s) := ιsup < ∞ and a neighborhood N (xm+1 λ such that xm+1 λ (s)) − zn (s, xm+1 (s)) ≤ ιsup um+1 (s) − xm+1 (s) zn (s, um+1 λ λ λ λ ∈ N (xm+1 ). Therefore, if t ∈ (bm+1 − λ(m + 1)(bm+1 − whenever um+1 λ λ am ), bm+1 ], then um+1 (t) − xm+1 (t) λ λ = lim um+1 (t) − xm+1 (t)P C(J,E) μ λ μ→λ

M ιsup t ≤ M  u−x P C(J,E) + (t − s)α−1 um+1 (s) − xm+1 (s)ds, λ λ Γ(α) 0 ∈ N (xm+1 ). whenever um+1 λ λ Apply Gronwall’s inequality (see [22, Theorem 1 and Corollary 2]) and we have (t) − xm+1 (t) ≤ M  u−x P C(J,E) Eα (M ιsup lα ), um+1 λ λ

t ∈ J,

which implies that lim

μ→λ, u→ x

um+1 (t) − xm+1 (t) = 0, μ λ

∀ t ∈ J,

where Eα denotes the Mittag–Leffler function. Similarly we see that lim

μ→λ, u→ x

ukμ (t) − xkλ (t) = 0,

k = m, m − 1, . . . , 1.

F Then, the set u0n [0, l] is contractible. Now, from Theorems 4.3 and 4.4 F ∞ F F and (ii) of Lemma 4.1 we have that u0 [0, l] = n=N u0i [0, l], u0i [0, l](i ≥ Fi Fi −1 N ) is compact and [0, l]. Therefore from Lemma 2.12, u0 [0, l] ⊆ u0 F  u0 [0, l] is an Rδ -set. Acknowledgements The authors thank the referees for carefully reading the manuscript and for their valuable comments.

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References [1] Hern´ andez, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641–1649 (2013) [2] Hern´ andez, E., Pierri, M., O’Regan, D.: On abstract differential equations with non instantaneous impulses. Topol. Methods Nonlinear Anal. 46, 1067–1085 (2015) [3] Abbas, S., Benchohra, M., Darwish, M.A.: New stability results for partial fractional differential inclusions with not instantaneous impulses. Frac. Calc. Appl. Anal. 18, 172–191 (2015) [4] Abbas, S., Benchohra, M.: Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses. Appl. Math. Comput. 257, 190–198 (2015) [5] Wang, J., Feˇckan, M.: A general class of impulsive evolution equations. Topol. Methods Nonlinear Anal. 46, 915–933 (2015) [6] Wang, J., Feˇckan, M., Tian, Y.: Stability analysis for a general class of noninstantaneous impulsive differential equations. Mediter. J. Math 14(46), 1–21 (2017) [7] Wang, J.: Stability of noninstantaneous impulsive evolution equations. Appl. Math. Lett. 73, 157–162 (2017) [8] Agarwal, R., O’Regan, D., Hristova, S.: Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses. Appl. Math. Comput. 298, 45–56 (2017) [9] Agarwal, R., Hristova, S., O’Regan, D.: Noninstantaneous impulses in Caputo fractional differential equations and practical stability via Lyapunov functions. J. Franklin Inst. 354, 3097–3119 (2017) [10] Agarwal, R., O’Regan, D., Hristova, S.: Non-instantaneous impulses in Caputo fractional differential equations. Frac. Calc. Appl. Anal. 20, 595–622 (2017) [11] Wang, J., Ibrahim, A.G., O’Regan, D.: Nonemptyness and compactness of the solution set for fractional semi linear evolution inclusions with noninstantaneous impulses. Submitted to Electron. J. Differ. Equ. (2017) [12] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) [13] G´ orniewicz, L.: Homological methods in fixed point theory of multivalued maps. Dissert. Math. 129, 1–71 (1976) [14] Gabor, G., Grudzka, A.: Structure of the solution set to impulsive functional differential inclusions on the half-line. Nonlinear Differ. Equ. Appl. 19, 609–627 (2012) [15] Aubin, J.P., Frankoeska, H.: Set-valued Analysis. Birkh¨ auser, Boston (1990) [16] Kamenskii, M., Obukhowskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Walter de Gruyter, Berlin (2001) [17] Hyman, D.H.: On decreasing sequence of compact absolute retract. Fund. Math. 64, 91–97 (1969) [18] Liu, Z., Zeng, B.: Existence and conrollability for fractional evolution inclusions of Clark’s subdifferential type. Appl. Math. Comput. 257, 178–189 (2015) [19] Bothe, D.: Multivalued perturbation of m-accerative differential inclusions. Israel J. Math. 108, 109–138 (1998)

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[20] Djebali, S., G´ orniewicz, L., Ouahab, A.: Solution Sets for Differential Equations and Inclusions. De Gruyter Series in Nonlinear Analysis and Applications, vol. 18. Walter De Gruyter, Berlin, Germany (2012) [21] G´ orniewicz, L.: On the solution sets of differential inclusions. J. Math. Anal. Appl. 113, 235–244 (1986) [22] Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007) JinRong Wang Department of Mathematics Guizhou University Guiyang 550025 Guizhou People’s Republic of China and School of Mathematical Sciences Qufu Normal University Qufu 273165 Shandong People’s Republic of China e-mail: [email protected] A. G. Ibrahim Department of Mathematics, Faculty of Science King Faisal University Al-Ahasa 31982 Saudi Arabia e-mail: [email protected] D. O’Regan School of Mathematics, Statistics and Applied Mathematics National University of Ireland Galway Ireland e-mail: [email protected]