CUBO A Mathematical Journal Vol.21, No¯ 01, (61–75). April 2019 http: // dx. doi. org/ 10. 4067/ S0719-06462019000100061 On Fractional Integro-differential Equations with State-Dependent Delay and Non-Instantaneous Impulses Khalida Aissani1, Mouffak Benchohra21 and Nadia Benkhettou2 1 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, PO Box 89, 22000, Sidi Bel-Abbès, Algeria. 2 University of Bechar PO Box 417, 08000, Bechar, Algeria benchohra@yahoo.com, aissani − k@yahoo.fr ABSTRACT In this paper, we prove the existence of mild solution of the fractional integro-differential equations with state-dependent delay with not instantaneous impulses. The existence results are obtained under the conditions in respect of Kuratowski’s measure of non- compactness. An example is also given to illustrate the results. RESUMEN En este art́ıculo, demostramos la existencia de soluciones mild de ecuaciones integro- diferenciales fraccionarias con retardo dependiente del estado e impulsos no instantáneos. Los resultados de existencia se obtienen bajo condiciones respecto de la medida de Ku- ratowski de no compacidad. También se entrega un ejemplo para ilustrar los resultados. Keywords and Phrases: Non-instantaneous impulsive conditions, fractional integro-differential equations, Caputo fractional derivative, mild solution, fixed point, state-dependent delay. 2010 AMS Mathematics Subject Classification: 26A33, 34A12, 34A37, 34G20. 1Corresponding author http://dx.doi.org/10.4067/S0719-06462019000100061 62 Khalida Aissani, Mouffak Benchohra and Nadia Benkhettou CUBO 21, 1 (2019) 1 Introduction Fractional differential equations play the crucial and significant role in the field of science and engineering. Most importantly non-integer order differential equations have ability to describe the real behavior and memory effects of the system and processes. For more details about fractional differential equations and its applications refer the monographs of Abbas et al. [1, 2, 3], Baleanu et al. [12], Diethelm [18], Hilfer [24], Kilbas et al. [26], Miller and Ross [32], Samko et al. [37], Tarasov [38], and Zhou [39] and the references therein. Most of the research papers deal with the existence of solutions for differential equations with instantaneous impulsive conditions see [6, 7, 10, 11, 14, 28, 31]. But many times it has seen that certain dynamics of evolution processes cannot describe by instantaneous impulses, For instance: Pharmacotherapy, high or low levels of glucose, this situation can be interpreted as an impulsive action which starts abruptly at certain point of time and continue with a finite time interval. Such type of systems are known as non-instantaneous impulsive systems which are more suitable to study the dynamics of evolution processes [4]. This theory of a new class of impulsive differential equation was initiated by Hernández et al. [23]. Afterwards, Pierri et al. [35] continued the work in this field and extend the theory of [23] in a PCα normed Banach space. The existence of solutions for non-instantaneous impulsive fractional differential equations have also been discussed in [8, 19, 27, 29, 34]. Recently, Benchohra et al. [15] investigated the existence and uniqueness of solutions on a compact interval for non-linear fractional integro-differential equations with state-dependent delay and noninstantaneous impulses. Anguraj and Kanjanadevi [9] studied the existence and uniqueness of fractional neutral differential equations with state-dependent delay subject to non-instantaneous impulsive conditions. Motivated by the papers cited above, in this paper, we consider the existence of mild solu- tions for fractional integro-differential equations with state-dependent delay and non instantaneous impulses described by the form C D q t x(t) + Ax(t) = ∫t 0 a(t,s)f(s,xρ(s,xs),x(s))ds, a.e. t ∈ (si, ti+1] ⊂ J,i = 0, . . . ,N, x(t) = hi(t,xρ(t,xt),x(t)), t ∈ (ti,si], i = 1, . . . ,N, x0 = φ ∈ B, (1.1) where CD q t is the Caputo fractional derivative of order 0 < q < 1, A : D(A) ⊂ X → X is the infinitesimal generator of an analytic semigroup {S(t)}t≥0 of uniformly bounded linear operators on X, f : J × B × X −→ X, J = [0,T], T > 0, and ρ : J × B → (−∞,T] are appropriate functions, a : D → R (D = {(t,s) ∈ J × J : t ≥ s}). Here 0 = t0 = s0 < t1 ≤ s1 ≤ t2 < ... < tN−1 ≤ sN ≤ tN ≤ tN+1 = T are pre-fixed numbers, and hi ∈ C((ti,si] × B × X,X), for all i = 1,2, . . . ,N. For CUBO 21, 1 (2019) On Fractional Integro-differential Equations with State-Dependent . . . 63 any continuous function x defined on (−∞,T] and any t ∈ J, we denote by xt the element of B defined by xt(θ) = x(t + θ), θ ∈ (−∞,0]. Here xt represents the history of the state up to the present time t and φ ∈ B to be specified later. 2 Preliminaries Let (X,‖ · ‖) be a real Banach space. C = C(J,X) be the space of all X-valued continuous functions on J. L(X) be the Banach space of all linear and bounded operators on X. L1(J,X) the space of X−valued Bochner integrable functions on J with the norm ‖y‖L1 = ∫T 0 ‖y(t)‖dt. L∞(J,R) is the Banach space of measurable functions which are essentially bounded, normed by ‖y‖L∞ = inf{d > 0 : |y(t)| ≤ d, a.e. t ∈ J}. We need some basic definitions of the fractional calculus theory which are used in this paper. Definition 2.1. Let α > 0 and f : R+ → X be in L1(R+,X). Then the Riemann–Liouville integral is given by: Iαt f(t) = 1 Γ(α) ∫t 0 f(s) (t − s)1−α ds, where Γ(·) is the Euler gamma function. For more details on the Riemann–Liouville fractional derivative, we refer the reader to [17]. Definition 2.2. [36] The Caputo derivative of order α for a function f : [0,+∞) → X can be written as Dαt f(t) = 1 Γ(n − α) ∫t 0 f(n)(s) (t − s)α+1−n ds = In−αf(n)(t), t > 0, n − 1 ≤ α < n. If 0 ≤ α < 1, then Dαt f(t) = 1 Γ(1 − α) ∫t 0 f(1)(s) (t − s)α ds. Obviously, The Caputo derivative of a constant is equal to zero. Definition 2.3. A function f : J×B ×X −→ X is said to be an Carathéodory function if it satisfies : (i) for each t ∈ J the function f(t, ·, ·) : B × X −→ X is continuous; 64 Khalida Aissani, Mouffak Benchohra and Nadia Benkhettou CUBO 21, 1 (2019) (ii) for each (v,w) ∈ B × X the function f(·,v,w) : J → X is measurable . Next we give the concept of a measure of noncompactness [13]. Definition 2.4. Let B be a bounded subset of a Banach space Y. The Kuratowski measure of noncompactness of B is defined as α(B) = inf{d > 0 : B has a finite cover by sets of diameter ≤ d}. We note that this measure of noncompactness satisfies the properties ([13]). Lemma 2.5. 1. If A ⊆ B then α(A) ≤ α(B), 2. α(A) = α(A), where A denotes the closure of A, 3. α(A) = 0 ⇔ A is compact (A is relatively compact), 4. α(λA) = |λ|A, with λ ∈ R, 5. α(A ∪ B) = max{α(A),α(B)}, 6. α(A + B) ≤ α(A) + α(B), where A + B = {x + y : x ∈ A,y ∈ B}, 7. α(A + a) = α(A) for any a ∈ X, 8. α(convA) = α(A), where convA is the closed convex hull of A. For H ⊂ C(J,X), we define ∫t 0 H(s)ds = {∫t 0 u(s)ds : u ∈ H } for t ∈ J, (2.1) where H(s) = {u(s) ∈ X : u ∈ H}. Lemma 2.6. [13] If H ⊂ C(J,X) is a bounded, equicontinuous set, then αC(H) = sup t∈J α(H(t)). (2.2) Lemma 2.7. [21] If {un} ∞ n=1 ⊂ L 1(J,X) and there exists m ∈ L1(J,R+) such that ‖un(t)‖ ≤ m(t), a.e. t ∈ J, then α({un(t)} ∞ n=1) is integrable and α ({∫t 0 un(s)ds }∞ n=1 ) ≤ 2 ∫t 0 α({un(s)} ∞ n=1)ds. (2.3) CUBO 21, 1 (2019) On Fractional Integro-differential Equations with State-Dependent . . . 65 In this paper, we will employ an axiomatic definition for the phase space B which is similar to those introduced by Hale and Kato [20]. Specifically, B will be a linear space of functions mapping (−∞,0] into X endowed with a seminorm ‖ · ‖B, and satisfies the following axioms: (A1) If x : (−∞,T ] −→ X is continuous on J and x0 ∈ B, then xt ∈ B and xt is continuous in t ∈ J and ‖x(t)‖ ≤ C‖xt‖B, (2.4) where C ≥ 0 is a constant. (A2) There exist a continuous function C1(t) > 0 and a locally bounded function C2(t) ≥ 0 in t ≥ 0 such that ‖xt‖B ≤ C1(t) sup s∈[0,t] ‖x(s)‖ + C2(t)‖x0‖B, (2.5) for t ∈ [0,T] and x as in (A1). (A3) The space B is complete. Remark 2.8. Condition (2.4) in (A1) is equivalent to ‖φ(0)‖ ≤ C‖φ‖B, for all φ ∈ B. Example 2.9. The phase space Cr × L p(g,X). Let r ≥ 0,1 ≤ p < ∞, and let g : (−∞,−r) → R be a nonnegative measurable function which satisfies the conditions (g − 5),(g − 6) in the terminology of [25]. Briefly, this means that g is locally integrable and there exists a nonnegative, locally bounded function Λ on (−∞,0], such that g(ξ + θ) ≤ Λ(ξ)g(θ), for all ξ ≤ 0 and θ ∈ (−∞,−r)\Nξ, where Nξ ⊆ (−∞,−r) is a set with Lebesgue measure zero. The space Cr × L p(g,X) consists of all classes of functions ϕ : (−∞,0] → X, such that ϕ is continuous on [−r,0], Lebesgue-measurable, and g‖ϕ‖p on (−∞,−r). The seminorm in ‖.‖B is defined by ‖ϕ‖B = sup θ∈[−r,0] ‖ϕ(θ)‖ + (∫−r −∞ g(θ)‖ϕ(θ)‖pdθ ) 1 p . The space B = Cr ×L p(g,X) satisfies axioms (A1), (A2), (A3). Moreover, for r = 0 and p = 2, this space coincides with C0 × L 2(g,X),H = 1,M(t) = Λ(−t) 1 2 ,K(t) = 1 + (∫0 −r g(τ)dτ ) 1 2 , for t ≥ 0 (see [25], Theorem 1.3.8 for details). For our purpose we will only need the following fixed point theorems. Theorem 2.10. [5, 33] Let U be a bounded, closed and convex subset of a Banach space, and let N be a continuous mapping of U into itself. If the implication V = convN(V) or V = N(V) ∪ {0} =⇒ α(V) = 0 holds for every subset V of U, then N has a fixed point. 66 Khalida Aissani, Mouffak Benchohra and Nadia Benkhettou CUBO 21, 1 (2019) A continuous map N : D ⊆ E → E is said to be a α-contraction if there exists a constant ν ∈ [0,1) such that α(N(C)) ≤ να(C) for any bounded closed subset C ⊆ D. Theorem 2.11. (Darbo–Sadovskii)[13] Let E be a Banach space. If D ⊆ E is bounded closed and convex, the continuous map N : D → D is a α-contraction, then the map N has at least one fixed point in D. Consider the space PC(J,X) = { x : J → X,x ∈ C ( J ∩ ( ∪Nk=0 (tk,sk] ) ,X ) , and x(t+k ),x(s − k ) exist with, x(s − k ) = x(sk),k = 1, . . . ,N } . Obviously, PC(J,X) is a Banach space with the norm ‖x‖PC = sup t∈J ‖x(t)‖. 3 Existence Results In this section, we prove the existence of mild solution of (1.1). Definition 3.1. A function x : (−∞,T] → X is said to be a mild solution of the equation (1.1) if x0 = φ on (−∞,T],x|[0,T] ∈ PC([0,T],X) and x satisfies x(t) =                                  Q(t)φ(0) + ∫t 0 ∫s 0 R(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds, t ∈ [0,t1], hi(t,xρ(t,xt),x(t)), t ∈ (ti,si], i = 1,2, . . . ,N, Q(t − si)hi(si,xρ(si,xsi ) ,x(si)) + ∫t 0 ∫s 0 R(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds, t ∈ (si,ti+1], (3.1) where Q(t) = ∫ ∞ 0 ξq(σ)S(t qσ)dσ, R(t) = q ∫ ∞ 0 σtq−1ξq(σ)S(t qσ)dσ and ξq is a probability density function defined on (0,∞) such that ξq(σ) = 1 q σ −1−( 1 q ) ̟q(σ − 1 q ) ≥ 0, CUBO 21, 1 (2019) On Fractional Integro-differential Equations with State-Dependent . . . 67 where ̟q(σ) = 1 π ∞∑ k=1 (−1)k−1σ−qk−1 Γ(kq + 1) k! sin(kπq), σ ∈ (0,∞). Remark 3.2. Note that {S(t)}t≥0 is a uniformly bounded i.e there exists a constant M > 0 such that ‖S(t)‖L(X) ≤ M for all t ≥ 0. Remark 3.3. According to [30], direct calculation gives that ‖R(t)‖ ≤ Cq,Mt q−1, t > 0, (3.2) where Cq,M = qM Γ(1 + q) . Set R(ρ−) = {ρ(s,ϕ) : (s,ϕ) ∈ J × B,ρ(s,ϕ) ≤ 0}. We always assume that ρ : J × B → (−∞,T] is continuous. Additionally, we introduce following hypothesis: (Hϕ) The function t → ϕt is continuous from R(ρ−) into B and there exists a continuous and bounded function Lφ : R(ρ−) → (0,∞) such that ‖φt‖B ≤ L φ(t)‖φ‖B for every t ∈ R(ρ −). Remark 3.4. Condition (Hϕ), is frequently verified by the continuous and bounded functions. For more details see [25]. Remark 3.5. In the rest of this section, C∗1 and C ∗ 2 are the constants C∗1 = sup s∈J C1(s) and C ∗ 2 = sup s∈J C2(s). Lemma 3.6. [22] If x : R → X is a function such that x0 = φ, then ‖xs‖B ≤ (C ∗ 2 + L φ)‖φ‖B + C ∗ 1 sup{|x(θ)|;θ ∈ [0,max{0,s}]}, s ∈ R(ρ −) ∪ J, where Lφ = sup t∈R(ρ−) Lφ(t). Let us introduce the following hypotheses: (H1) f : J × B × X −→ X satisfies the Carathéodory conditions. (H2) There exist functions µ,µ∗ ∈ L1(J,R+) and continuous nondecreasing functions ψ,ψ∗ : R+ → (0,+∞) such that ‖f(t,x,y)‖ ≤ µ(t)ψ(‖x‖B + ‖y‖) , (t,x,y) ∈ J × B × X, ‖hi(t,x,y)‖ ≤ µ ∗ (t)ψ∗ (‖x‖B + ‖y‖) , (t,x,y) ∈ J × B × X, 68 Khalida Aissani, Mouffak Benchohra and Nadia Benkhettou CUBO 21, 1 (2019) (H3) For any bounded sets D1 ⊂ B,D2 ⊂ X, and 0 ≤ s ≤ t ≤ T, there exists an integrable positive function η such that α(R(t − s)f(τ,D1,D2)) ≤ ηt(s,τ) ( α(D2) + sup −∞<θ≤0 α(D1(θ)) ) , where ηt(s,τ) = η(t,s,τ) and sup t∈J ∫t 0 ∫s 0 ηt(s,τ)dτds = η ∗ < ∞. (H4) There exists a constant L > 0 such that, for each bounded sets D1 ⊂ B,D2 ⊂ X, α(hi(τ,D1,D2)) ≤ L ( α(D2) + sup −∞<θ≤0 α(D1(θ)) ) . (H5) For each t ∈ J, a(t,s) is measurable on [0,t] and a(t) = ess sup{|a(t,s)|,0 ≤ s ≤ t} is bounded on J. The map t → at is continuous from J to L∞(J,R), here, at(s) = a(t,s). Set a = sup t∈J a(t). Our first result is based on the Mönch fixed point theorem. Theorem 3.7. Suppose that the assumptions (Hϕ),(H1) − (H5) hold, and if 2ML + 16 a η∗ < 1, (3.3) then the problem (1.1) has at least one mild solution. Proof. Let Y = {u ∈ PC(X) : u(0) = φ(0) = 0} endowed with the uniform convergence topology and define the operator P : Y → Y by P(x)(t) =                                  Q(t)φ(0) + ∫t 0 ∫s 0 R(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds, t ∈ [0,t1], hi(t,xρ(t,xt),x(t)), t ∈ (ti,si], i = 1,2, . . . ,N, Q(t − si)hi(si,xρ(si,xsi ) ,x(si)) + ∫t si ∫s 0 R(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds, t ∈ (si,ti+1], where x : (−∞,T] → X is such that x0 = φ and x = x on J. Let φ : (−∞,T] −→ X be the extension of φ to (−∞,T] such that φ(θ) = φ(0) = 0 on J. Choose r ≥ M‖µ∗‖L1ψ ∗ ( (C∗2 + L φ )‖φ‖B + (C ∗ 1 + 1)r ) +aCq,M‖µ‖L1 Tq q ψ ( (C∗2 + L φ )‖φ‖B + (C ∗ 1 + 1)r ) , and define the set Br = {x ∈ Y : ‖x‖PC ≤ r} , CUBO 21, 1 (2019) On Fractional Integro-differential Equations with State-Dependent . . . 69 then Br is a bounded, closed-convex subset in Y. Step 1: P is continuous. Let {xk}k∈N be a sequence such that x k → x in Br as k → ∞. Case 1. For each t ∈ [0,t1], we have ‖P(xk)(t) − P(x)(t)‖ ≤ ∫t 0 ∫s 0 ‖R(t − s)‖‖a(s,τ)‖‖f(τ,xk ρ(τ,xkτ) ,xk(τ)) − f(τ,xρ(τ,xτ),x(τ))‖dτds ≤ aCq,M ∫t 0 ∫s 0 (t − s)q−1‖f(τ,xk ρ(τ,xkτ) ,xk(τ)) − f(τ,xρ(τ,xτ),x(τ))‖dτds. Case 2. For each t ∈ [ti,si), i = 1,2, . . . ,N, we have ‖P(xk)(t) − P(x)(t)‖ = ‖hi(t,x k ρ(t,xkt ) ,xk(t)) − hi(t,xρ(t,xt),x(t))‖ → 0 k → ∞. Case 3. For each t ∈ (si,ti+1], i = 1,2, . . . ,N, we obtain ‖P(xk)(t) − P(x)(t)‖ ≤ ‖Q(t − si)‖‖hi(si,x k ρ(si,x k si ) ,xk(si)) − hi(si,xρ(si,xsi ) ,x(si))‖ + ∫t si ∫s 0 ‖R(t − s)‖‖a(s,τ)‖‖f(τ,xk ρ(τ,xkτ) ,xk(τ)) − f(τ,xρ(τ,xτ),x(τ))‖dτds ≤ M‖hi(si,x k ρ(si,x k si ) ,xk(si)) − hi(si,xρ(si,xsi ) ,x(si))‖ + aCq,M ∫t si ∫s 0 (t − s)q−1‖f(τ,xk ρ(τ,xkτ) ,xk(τ)) − f(τ,xρ(τ,xτ),x(τ))‖dτds. Since the function hi is continuous and f is of Carathéodory type, we have by the Lebesgue dominated convergence theorem that ‖P(xk)(t) − P(x)(t)‖ → 0 as k → ∞, which shows the operator P is continuous. Step 2: P maps Br into itself. 70 Khalida Aissani, Mouffak Benchohra and Nadia Benkhettou CUBO 21, 1 (2019) Case 1. For all t ∈ [0,t1], we get ‖P(x)(t)‖ ≤ ‖Q(t)φ(0)‖ + ∫t 0 ∫s 0 ‖R(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))‖dτds ≤ MC‖φ‖B + aCq,M ∫t 0 ∫s 0 (t − s)q−1µ(τ)ψ(‖xρ(τ,xτ)‖B + ‖x‖)dτds ≤ MC‖φ‖B + aCq,M ∫t 0 ∫s 0 (t − s)q−1µ(τ) × ψ ( (C∗2 + L φ)‖φ‖B + C ∗ 1r + r ) dτds ≤ MC‖φ‖B + aCq,M‖µ‖L1 Tq q ψ ( (C∗2 + L φ)‖φ‖B + (C ∗ 1 + 1)r ) ≤ r. Case 2. For all t ∈ [ti,si), i = 1,2, . . . ,N, we have ‖P(x)(t)‖ ≤ ‖hi(t,xρ(t,xt),x(t))‖ ≤ µ∗(t)ψ∗ ( ‖xρ(t,xt)‖B + ‖x‖ ) ≤ ‖µ∗‖L1ψ ∗ ( (C∗2 + L φ)‖φ‖B + (C ∗ 1 + 1)r ) ≤ r. Case 3. For all t ∈ (si,ti+1], i = 1,2, . . . ,N, we obtain ‖P(x)(t)‖ ≤ ‖Q(t − si)hi(si,xρ(si,xsi ) ,x(si))‖ + ∫t si ∫s 0 ‖R(t − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))‖dτds, ≤ M‖µ∗‖L1ψ ∗ ( (C∗2 + L φ)‖φ‖B + (C ∗ 1 + 1)r ) + aCq,M‖µ‖L1 Tq q ψ ( (C∗2 + L φ)‖φ‖B + (C ∗ 1 + 1)r ) ≤ r. Step 3: P(Br) is bounded and equicontinuous. Case 1. For each t ∈ [0,t1],0 ≤ τ2 ≤ τ1 ≤ t1, and x ∈ Br. Then we have ‖P(x)(τ1) − P(x)(τ2)‖ ≤ I1 + I2 + I3, where I1 = ‖Q(τ1) − Q(τ2)‖‖φ(0)‖ I2 = ∥ ∥ ∥ ∥ ∫τ2 0 ∫s 0 [R(τ1 − s) − R(τ2 − s)]a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds ∥ ∥ ∥ ∥ I3 = ∥ ∥ ∥ ∥ ∫τ1 τ2 ∫s 0 R(τ1 − s)a(s,τ)f(τ,xρ(τ,xτ),x(τ))dτds ∥ ∥ ∥ ∥ . CUBO 21, 1 (2019) On Fractional Integro-differential Equations with State-Dependent . . . 71 I1 tends to zero as τ2 → τ1, since S(t) is uniformly continuous operator. For I2, using (3.2) and (H2), we have I2 ≤ aψ ( (C∗2 + L φ )‖φ‖B + (C ∗ 1 + 1)r ) ‖µ‖L1 ∫τ2 0 [R(τ1 − s) − R(τ2 − s)]ds ≤ aψ ( (C∗2 + L φ)‖φ‖B + (C ∗ 1 + 1)r ) ‖µ‖L1 × ∫τ2 0 [ q ∫ ∞ 0 σ(τ1 − s) q−1ξq(σ)S((τ1 − s) qσ)dσ −q ∫ ∞ 0 σ(τ2 − s) q−1ξq(σ)S((τ2 − s) qσ)dσ ] ds ≤ aψ ( (C∗2 + L φ )‖φ‖B + (C ∗ 1 + 1)r ) ‖µ‖L1 × [ q ∫τ2 0 ∫ ∞ 0 σ‖[(τ1 − s) q−1 − (τ2 − s) q−1]ξq(σ)S((τ1 − s) qσ) +q ∫τ2 0 ∫ ∞ 0 σ(τ2 − s) q−1ξq(σ)‖S((τ1 − s) qσ) − S((τ2 − s) qσ)‖ ] ≤ aψ ( (C∗2 + L φ)‖φ‖B + (C ∗ 1 + 1)r ) ‖µ‖L1 × [Cq,M ∫τ2 0 ∣ ∣(τ1 − s) q−1 − (τ2 − s) q−1 ∣ ∣ds + q ∫τ2 0 ∫ ∞ 0 σ(τ2 − s) q−1ξq(σ)‖S((τ1 − s) qσ) − S((τ2 − s) qσ)‖dσds]. Clearly, the first term on the right-hand side of the above inequality tends to zero as τ2 → τ1. From the continuity of S(t) in the uniform operator topology for t > 0, The second term on the right-hand side of the above inequality tends to zero as τ2 → τ1. In view of (H2), we have I3 ≤ a Cq,M ∫τ1 τ2 ∫s 0 (τ1 − s) q−1‖f(τ,xρ(τ,xτ),x(τ))‖dτds ≤ a Cq,Mψ ( (C∗2 + L φ)‖φ‖B + (C ∗ 1 + 1)r ) ‖µ‖L1 ∫τ1 τ2 (τ1 − s) q−1ds. As τ2 → τ1, I3 tends to zero. Case 2. For each t ∈ [ti,si), i = 1,2, . . . ,N,ti ≤ τ2 ≤ τ1 ≤ si, and x ∈ Br. Then we have ‖P(x)(τ1) − P(x)(τ2)‖ = ‖hi(τ1,xρ(τ1,xτ1 ) ,x(τ1)) − hi(τ2,xρ(τ2,xτ2 ) ,x(τ2))‖ → 0 as τ2 → τ1. Case 3. For each t ∈ (si,ti+1], i = 1,2, . . . ,N,si ≤ τ2 ≤ τ1 ≤ ti+1, and x ∈ Br. Then we have ‖P(x)(τ1) − P(x)(τ2)‖ ≤ ‖Q(τ1 − si) − Q(τ2 − si)‖‖hi(si,xρ(si,xsi ) ,x(si))‖ + I1 + I2 + I3. Since S(t) is uniformly continuous operator, so lim τ2→τ1 ‖Q(τ1 − si) − Q(τ2 − si)‖ = 0, i = 1, . . . ,N. 72 Khalida Aissani, Mouffak Benchohra and Nadia Benkhettou CUBO 21, 1 (2019) Consequently lim τ2→τ1 ‖P(x)(τ1) − P(x)(τ2)‖ = 0. Thus, P(Br) is equicontinuous. Now let V be a subset of Br such that V ⊂ conv(P(V) ∪ {0}). Moreover, for any ε > 0 and bounded set D, we can take a sequence {vn} ∞ n=1 ⊂ D such that α(D) ≤ 2α({vn}) +ε ([16], P. 125). Thus, for {vn} ∞ n=1 ⊂ V, and using lemmas 2.5-2.7 and (H3), we have, for t ∈ [0,t1], α(PV) ≤ 2α({Pvn}) + ε = 2sup t∈J α({Pvn(t)}) + ε = 2sup t∈J α ({∫t 0 R(t − s) ∫s 0 a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds }) + ε ≤ 4sup t∈J ∫t 0 α ({ R(t − s) ∫s 0 a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds }) + ε ≤ 8sup t∈J ∫t 0 ∫s 0 α({R(t − s)a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds}) + ε ≤ 8 asup t∈J ∫t 0 ∫s 0 α({R(t − s)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds}) + ε ≤ 8 a sup t∈J ∫t 0 ∫s 0 ηt(s,τ) [ α(vn(τ)) + sup −∞<θ≤0 α(vn(θ + τ)) ] dτds + ε ≤ 8 a sup t∈J ∫t 0 ∫s 0 ηt(s,τ) [ α(vn) + sup 0<µ≤τ α(vn(µ)) ] dτds + ε ≤ 16 a α(vn) sup t∈J ∫t 0 ∫s 0 ηt(s,τ)dτds + ε ≤ 16 a η∗α(V) + ε. For any t ∈ [ti,si), i = 1,2, . . . ,N, we get α(PV) = α ( hi(t,xρ(t,xt),x(t)) ) ≤ L ( α(vn(t)) + sup −∞<θ≤0 α(vn(θ + t)) ) ≤ L ( α(vn) + sup 0<µ≤τ α(vn(µ)) ) ≤ 2Lα(vn) ≤ 2Lα(V). CUBO 21, 1 (2019) On Fractional Integro-differential Equations with State-Dependent . . . 73 In the same way, for any t ∈ (si,ti+1], i = 1,2, . . . ,N, we obtain α(PV) ≤ 2α({Pvn}) + ε = 2sup t∈J α({Pvn(t)}) + ε = 2sup t∈J α ( Q(t − si)hi(si,xρ(si,xsi ) ,x(si)) ) + 2sup t∈J α ({∫t si R(t − s) ∫s 0 a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds }) + ε ≤ 2MLα(vn) + 4sup t∈J ∫t si α ({ R(t − s) ∫s 0 a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds }) + ε ≤ 2MLα(vn) + 8sup t∈J ∫t si ∫s 0 α({R(t − s)a(s,τ)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds}) + ε ≤ 2MLα(vn) + 8 asup t∈J ∫t si ∫s 0 α({R(t − s)f(τ,yτ + vnτ,y(τ) + vn(τ))dτds}) + ε ≤ 2MLα(vn) + 8 asup t∈J ∫t si ∫s 0 ηt(s,τ) [ α(vn(τ)) + sup −∞<θ≤0 α(vn(θ + τ)) ] dτds + ε ≤ 2MLα(vn) + 8 a sup t∈J ∫t si ∫s 0 ηt(s,τ) [ α(vn) + sup 0<µ≤τ α(vn(µ)) ] dτds + ε ≤ 2MLα(vn) + 16 a α(vn) sup t∈J ∫t 0 ∫s 0 ηt(s,τ)dτds + ε ≤ 2MLα(V) + 16 a η∗α(V) + ε ≤ (2ML + 16 a η∗)α(V) + ε. Therefore, in view of Lemma 2.5, we have α(V) ≤ α(PV) ≤ (2ML + 16 a η∗)α(V) + ε, since ε is arbitrary we obtain that α(V) ≤ (2ML + 16 a η∗)α(V). This means that α(V) (1 − (2ML + 16 a η∗)) ≤ 0. By (3.3) it follows that α(V) = 0. In view of the Ascoli-Arzelà theorem, V is relatively compact in Br. Applying now Theorem 2.10, we conclude that P has a fixed point which is a solution of the problem (1.1). The second result is established using the Darbo’s fixed point theorem. 74 Khalida Aissani, Mouffak Benchohra and Nadia Benkhettou CUBO 21, 1 (2019) Theorem 3.8. Assume that (H1)−(H5) are satisfied, then the problem (1.1) has at least one mild solution. Proof. In what follows we show that the operator P : Y → Y is a strict set contraction. We know that P : Y → Y is bounded and continuous, we need to prove that there exists a constant 0 ≤ ν < 1 such that α(PV) ≤ να(V) for V ⊂ Br. Using the same method as the proof of Theorem 3.7, for t ∈ [0,T], we have α(PV) ≤ (2ML + 16 a η∗)α(V) + ε, since ε is arbitrary we obtain that α(PV) ≤ να(V). Hence P is a set contraction. According to Theorem 2.11 the operator P has at least one fixed point which is obviously a mild solution of the problem (2.4). This completes the proof. 4 An Example We consider the fractional integro-differential equations with state-dependent delay and non- instantaneous impulses of the form ∂ q t ∂tq v(t,ζ) + ∂2 ∂ζ2 v(t,ζ) = ∫t 0 (t − s)2 ∫s −∞ γ(τ − s)v(τ − ρ1(s)ρ2(|v(s,ζ)|),ζ)dτds + ∫t 0 (t − s)2 cos |v(s,ζ)|ds, (t,x) ∈ N ∈ ∪ni=1[si,ti+1] × [0,π], v(t,0) = v(t,π) = 0, t ∈ [0,T], v(τ,ζ) = v0(θ,ζ), θ ∈ (−∞,0],x ∈ [0,π] v(t,ζ) = Hi(t,v(t − ρ1(t)ρ2(|v(t,ζ)|),ζ),ζ), (t,x) ∈ (ti,si] × [0,π], i = 1,2, . . . ,N, (4.1) where 0 < q < 1,0 = t0 = s0 < t1 ≤ s1 ≤ t2 < ... < tN−1 ≤ sN ≤ tN ≤ tN+1 = T are prefixed real numbers and the functions γ : R → R,ρi : [0,+∞) → [0,+∞), i = 1,2 are continuous functions. Let X = L2([0,π]) and define the operator A : D(A) ⊂ X → X by Aω = ω′′ with domain D(A) = {ω ∈ E : ω,ω′ are absolutely continuous, ω′′ ∈ E,ω(0) = ω(π) = 0}. Then Aω = ∞∑ n=1 n2(ω,ωn)ωn, ω ∈ D(A), CUBO 21, 1 (2019) On Fractional Integro-differential Equations with State-Dependent . . . 75 where ωn(x) = √ 2 π sin(nx),n ∈ N is the orthogonal set of eigenvectors of A. It is well known that A is the infinitesimal generator of an analytic semigroup {S(t)}t≥0 in X and is given by S(t)ω = ∞∑ n=1 e−n 2 t(ω,ωn)ωn, ∀ω ∈ X, and every t > 0. From these expressions, it follows that {S(t)}t≥0 is a uniformly bounded compact semigroup on X. For the phase space, we choose B = C0 × L 2(g,X), see Example 2.9 for details. Set x(t)(ζ) = v(t,ζ), φ(θ)(ζ) = v0(θ,ζ), a(t,s) = (t − s)2 f(t,ϕ,x(t))(ζ) = ∫0 −∞ γ(t)ϕ(t,ζ)ds + cos |x(t)(ζ)|, hi(t,ϕ,x(t))(ζ) = Hi(t,v(t − ρ1(t)ρ2(|x(t)|),ζ),ζ) ρ(t,ϕ) = t − ρ1(t)ρ2(|ϕ(0)|). Under the above conditions, we can represent the problem (4.1) by the abstract problem (1.1). 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