Int. J. Anal. Appl. (2023), 21:15 Functional Impulsive Fractional Differential Equations Involving the Caputo-Hadamard Derivative and Integral Boundary Conditions Aida Irguedi1,∗, Khadidja Nisse1, Samira Hamani2 1Laboratory of Operators Theory and PDE: Foundations and Applications,Department of Mathematics, Faculty of Exact Sciences, University of El Oued, 39000 El Oued, Algeria 2Laboratoire des Mathématiques Appliqués et Pures, Université de Mostaganem, B.P. 227, 27000, Mostaganem, Algeria ∗Corresponding author: irguedi-aida@univ-eloued.dz Abstract. In this paper, we investigate the existence and uniqueness of solutions for functional impulsive fractional differential equations and integral boundary conditions. Our results are based on some fixed point theorems. Finally, we provide an example to illustrate the validity of our main results. 1. Introduction In this paper, we discuss the existence and uniqueness of solutions to a boundary value problem (BVP for short) for functional impulsive fractional differential equation, in the following form: C HD ry(t) = f (t,yt), t ∈ J = [a,T ],t 6= tk, k = 1, ..,m, (1.1) ∆y |t=tk = Ik(y(t − k )), t = tk, k = 1, ..,m, (1.2) ∆y ′ |t=tk = Ik(y(t − k )), t = tk, k = 1, ..,m, (1.3) y(t) = φ(t), t ∈ [a−τ,a], y ′(T ) = ∫ T a h(s,y(s))ds. (1.4) where CHD r is the Caputo-Hadamard fractional derivative of order 1 < r ≤ 2, a > 0, f : J ×C([a − τ,a],R) →R, h : J×R→R are given continuous functions, φ ∈ C([a−τ,a],R) and Ik, Ik ∈ C(R,R), k = 1, 2, ...,m, a = t0 < t1 < ... < tm < tm+1 = T. For any continuous functions y defined on Received: Jan. 10, 2023. 2020 Mathematics Subject Classification. 26A33; 34A08; 34A37; 47H10 . Key words and phrases. fractional differential equations; Caputo-Hadamard fractional derivative; impulses; integral boundary condition; delay; fixed point theorems. https://doi.org/10.28924/2291-8639-21-2023-15 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-15 2 Int. J. Anal. Appl. (2023), 21:15 J′ = J\{t1, ...,tm}, ∆y|t=tk = y(t + k )−y(t− k ) and ∆y ′|t=tk = y ′(t+ k )−y ′(t− k ) , y(t+ k ), y(t− k ) represent the right and the left limits of y(t) at t = tk , we denote by yt the element of Cτ = C([a−τ,a],R), defined by yt(θ) = y(t + θ) ,θ ∈ [a −τ,a], hence yt(·) represents the history of the state from time t −τ up to the present time t. In the last few decades, the analysis of impulsive boundary value problems has developed. It has also been extremely useful in developing various applied mathematical models of real-world processes in engineering and applied sciences. Tian and Bai [16] discussed some existence results of impulsive boundary value problems involving Caputo’s type fractional derivatives. Results of existence and uniqueness have been developed using fixed-point theorem. Recently, it has been noted that much of the works on this subject are focused on the fractional differential equations of Riemann-Liouville and Caputo types with different conditions such as impulses, time delays, boundary value conditions [1,6–9,14,20,21]. The Hadamard fractional derivative, introduced in 1892, [10] is another type of fractional derivative that appears in the literature alongside the Riemann-Liouville and Caputo derivatives. It differs from the previous ones in that it contains an arbitrary logarithm function, further details can be found in [3–5]. Next, Jarad et al. proposed a Caputo-type modification of the Hadamard fractional derivative in [15] by the Caputo Hadamard fractional derivative and implemented the fundamental fractional calculus theorem in the Caputo-Hadamard. Recently, some researchers have focused on impulsive differential equations with Hadamard and Caputo-Hadamard derivatives (see [11–13, 17, 18] and the references therein). The rest of the paper is organized as follows. In Section 2, we introduce some notions preliminary and properties on the fractional culcules. In Section 3, we give a supporting lemma describing the solutions of the considered problem and discuss the main findings. Finally, we give an example to illustrate the obtained results. 2. Preliminaries In this section, we introduce notations, definitions and preliminary facts that will be used in the remainder of this paper. By C(J,R) we denote the Banach space of all continuous functions from J into R with the norm ‖y‖∞ = sup{|y(t)| : t ∈ J}. Also Cτ is endowed with the norm ‖φ‖Cτ = sup{‖φ(θ)‖ : a−τ ≤ θ ≤ a}. Let L1(J,R) as the Banach space of Lebesgue integrable functions y : J −→R with the norm ‖y‖L1 = ∫ T a |y(t)|dt. Int. J. Anal. Appl. (2023), 21:15 3 The space AC(J,R) is the space of functions y : J → R that are absolutely continuous. Let δ = t d dt , and then we set ACnδ (J,R) = {y : J −→R, δ n−1y(t) ∈ AC(J,R)}. Definition 2.1. [19] The Hadamard derivative of fractional order r for a Cn−1 function y : [a,T ] →R is defined by HDry(t) = 1 Γ(n− r) (t d dt )n ∫ t a ( log t s )n−r−1 y(s) ds s ,n− 1 < r < n,n = [r] + 1. Definition 2.2. [19] The Hadamard fractional integral of order r for a continuous function y is defined as a function HIry(t) = 1 Γ(r) ∫ t a ( log t s )r−1 y(s) ds s , r > 0, provided the integral exists. Definition 2.3. [19] For an n−times differentiable function y : [a,T ] →R the Caputo type Hadamard derivative of fractional order r is defined as C HD ry(t) = 1 Γ(n− r) ∫ t a ( log t s )n−r−1 δny(s) ds s , n− 1 < r < n, n = [r] + 1, where δ = t d dt and [r] denotes the integer part of the real number r and log(.) = loge(.). Lemma 2.1. [2] Let r ∈R+ and n = [r] = 1. If y(t) ∈ ACnδ (J,R) then Caputo-Hadamard fractional differential equation CDray(t) = 0 has a solution y(t) = n−1∑ k=0 ck ( log t a )k , and the following formula holds: HIra( C HD r ay)(t) = y(t) + n−1∑ k=0 ck ( log t a )k , where ck ∈R, k = 0, 1, 2, ...,n− 1. 3. Existence of Solutions In this section, we will establish the existence and uniqueness of solutions for (1.1)-(1.4). AC′(J,R) = { y : J →R, y ∈ AC2δ ((tk,tk+1],R) and there exist y(t + k ) andy(t− k ), k = 1, ...,m, with, y(t− k ) = y(tk) } , with the norm ‖y‖AC′ = sup{‖y(t)‖ : a ≤ t ≤ T}. 4 Int. J. Anal. Appl. (2023), 21:15 Let B be set defined by B = {y : (a−τ,T ] →R \ y ∈ AC′(J,R) ∩Cτ}, is endowed with the norm ‖y‖B = sup{‖y(t)‖ : t ∈ [a−τ,T ]}. Definition 3.1. A function y ∈ B is said to be a solution of the problem (1.1)-(1.4) if y satisfies the equation CHD ry(t) = f (t,yt) on J′ and the conditions (1.2)-(1.4). We need the following auxiliary lemma to prove the existence and uniqueness of solutions to the problem(1.1)-(1.4). Lemma 3.1. Let 1 < r ≤ 2. Assume that σ,% ∈ AC2(J,R), then the following BVP : H CD ry(t) = σ(t), t ∈ J = [a,T ],t 6= tk, (3.1) ∆y |t=tk = Ik(y(t − k )), t = tk, k = 1, ..,m, (3.2) ∆y ′ |t=tk = Ik(y(t − k )) , t = tk, k = 1, ..,m, (3.3) y(a) = y, y ′(T ) = ∫ T a %(s)ds, (3.4) has the following integral equation: y(t) =   y + c2 ( log t a ) + 1 Γ(r) ∫ t a ( log t s )r−1 σ(s)ds s , if t ∈ [a,t1] y + c2 ( log t a ) + 1 Γ(r) ∫ t tk ( log t s )r−1 σ(s)ds s + 1 Γ(r) ∑k i=1 ∫ ti ti−1 ( log ti s )r−1 σ(s)ds s + ∑k i=1 ( log t ti ) Γ(r−1) ∫ ti ti−1 ( log ti s )r−2 σ(s)ds s + ∑k i=1 Ii (y(t − i )) + ∑k i=1 ti ( log t ti ) Īi (y(t − i )), if t ∈ (tk,tk+1], k = 1, ...,m, where c2 = T ∫T a %(s)ds− [ 1 Γ(r−1) ∫T tm ( log T s )r−2 σ(s) ds s + ∑m i=1 1 Γ(r−1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + ∑m i=1(ti )Īi (y(t − i ) ] Int. J. Anal. Appl. (2023), 21:15 5 proof: Let y be the solution of (3.1)-(3.4). For t ∈ [a,t1]. Using Lemma 2.1, for some constants c1,c2 ∈R, we have y(t) = 1 Γ(r) ∫ t a ( log t s )r−1 σ(s) ds s + c1 + c2 ( log t a ) Acccording the a condition y(a) = y, we deduce that c1 = y and thus y(t) = y + 1 Γ(r) ∫ t a ( log t s )r−1 σ(s) ds s + c2 ( log t a ) , y ′(t) = 1 tΓ(r − 1) ∫ t a ( log t s )r−2 σ(s) ds s + c2 t . If t ∈ (t1,t2], then we have y(t) = 1 Γ(r) ∫ t t1 ( log t s )r−1 σ(s) ds s + d1 + d2 ( log t t1 ) , and y ′(t) = 1 tΓ(r − 1) ∫ t t1 ( log t s )r−2 σ(s) ds s + d2 t . Using the impulses conditions ∆y |t=t1 = y(t + 1 )−y(t − 1 ) = I1(y(t − 1 )) and ∆y ′ |t=t1 = y ′(t+1 )−y ′(t−1 ) = Ī1(y(t − 1 )), we obtain d1 = I1(y(t − 1 )) + y + c2 ( log t1 a ) + 1 Γ(r) ∫ t1 a ( log t1 s )r−1 σ(s) ds s . d2 = t1Ī1(y(t − 1 )) + c2 + 1 Γ(r − 1) ∫ t1 a ( log t1 s )r−2 σ(s) ds s . Thus, for t ∈ (t1,t2] we have y(t) = y + 1 Γ(r) ∫ t t1 ( log t s )r−1 σ(s) ds s + 1 Γ(r) ∫ t1 a ( log t1 s )r−1 σ(s) ds s + ( log t t1 ) Γ(r − 1) ∫ t1 a ( log t1 s )r−2 σ(s) ds s + t1 ( log t t1 ) Ī1(y(t − 1 )) + I1(y(t − 1 )) + c2 ( log t a ) . Continuinq in the same manner, we obtain for t ∈ (tm,T ], y(t) = y + 1 Γ(r) ∫ t tm ( log t s )r−1 σ(s) ds s + 1 Γ(r) m∑ i=1 ∫ ti ti−1 ( log ti s )r−1 σ(s) ds s + m∑ i=1 Ii (y(t − i )) + m∑ i=1 ( log t ti ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + m∑ i=1 ti ( log t ti ) Īi (y(t − i )) + c2 ( log t a ) , and y ′(t) = c2 t + 1 tΓ(r − 1) ∫ t tm ( log t s )r−2 σ(s) ds s + m∑ i=1 1 tΓ(r − 1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + m∑ i=1 ( ti t ) Īi (y(t − i ). 6 Int. J. Anal. Appl. (2023), 21:15 By the application of the boundary condition y ′(T ) = ∫T a %(s)ds, we have y ′(T ) = c2 T + 1 T Γ(r − 1) ∫ T tm ( log T s )r−2 σ(s) ds s + m∑ i=1 1 T Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + m∑ i=1 ( ti T ) Īi (y(t − i )). We obtain the reguired value of the constant c2, where c2 = T ∫T a %(s)ds− [ 1 Γ(r−1) ∫T tm ( log T s )r−2 σ(s) ds s + ∑m i=1 1 Γ(r−1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + ∑m i=1(ti )Īi (y(t − i t)) ] . This completes the proof. Our first result is based on the uniqueness of solutions for problem (1.1)-(1.4) and relies on the Banach fixed point theorem. Theorem 3.1. Assume that : (H1) There exists a constant L1 > 0 such that |f (t,u) − f (t,v)| ≤ L1‖u −v‖Cτ , for each t ∈ J and u,v ∈ Cτ. (H2) There exists a constants L2 > 0 such that |h(t,x) −h(t,y)| ≤ L2|x −y|, for each t ∈ J and x,y ∈R. (H3) For each k = 1, 2, ...,m, there exist l, l∗ > 0 such that |Ik(x) − Ik(y)| ≤ l|x −y|, |Ik(x) − Ik(y)| ≤ l∗|x −y|, for each x,y ∈R. If the condition[ L1 ( m + 1 Γ(r + 1) + 1 + 2m Γ(r) )( log T a )r + L2T (T −a) ( log T a ) + ml + 2ml∗T ( log T a )] < 1, (3.5) then the boundary value problem (1.1)-(1.4) has a unique solution on [a−τ,T ]. proof : Transform the problem (1.1)-(1.4) into a fixed point problem. Consider the operator F : B → B defined by: (Fy)(t) =   φ(t), if t ∈ (a−τ,a] φ(a) + T ( log t a )∫T a h(s,y(s))ds − ( log t a ) [ 1 Γ(r−1) ∫T tm ( log T s )r−2 f (s,ys) ds s + 1 Γ(r−1) ∑m i=1 ∫ ti ti−1 ( log ti s )r−2 f (s,ys) ds s + ∑m i=1 ti Īi (y(t − i ))] + 1 Γ(r) ∫ t tk ( log t s )r−1 f (s,ys) ds s + 1 Γ(r) ∑k i=1 ∫ ti ti−1 ( log ti s )r−1 f (s,ys) ds s + ∑k i=1 ( log t ti ) Γ(r−1) ∫ ti ti−1 ( log ti s )r−2 f (s,ys) ds s + ∑k i=1 Ii (y(t − i )) + ∑k i=1 ti ( log t ti ) Īi (y(t − i )), if t ∈ [tk,tk+1], k = 1, ...,m. (3.6) Int. J. Anal. Appl. (2023), 21:15 7 Clearly, the fixed point of the operator F are solutions of problem (1.1)-(1.4). Let x,y ∈ B, if t ∈ [a−τ,a], we have |(Fx)(t) − (Fy)(t)| = |φ(t) −φ(t)| = 0 If t ∈ [a,T ], by (H1)-(H3), we have: |(Fx)(t) − (Fy)(t)| ≤ T | ( log t a ) | ∫ T a |h(s,x(s)) −h(s,y(s))|ds + | ( log t a ) | Γ(r − 1) ∫ T tm ( log T s )r−2 |f (s,xs ) − f (s,ys )| ds s + | ( log t a ) | Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 |f (s,xs ) − f (s,ys )| ds s +| ( log t a ) | m∑ i=1 ti|Īi (x(t−i )) − Īi (y(t − i ))| + 1 Γ(r) ∫ t tk ( log t s )r−1 |f (s,xs ) − f (s,ys )| ds s + 1 Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 |f (s,xs ) − f (s,ys )| ds s + k∑ i=1 ( log t ti ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,xs ) − f (s,ys )| ds s + k∑ i=1 |Ii (x(t−i )) − Ii (y(t − i ))| + k∑ i=1 ti ( log t ti ) |Īi (x(t−i )) − Īi (y(t − i ))| ≤ T ( log t a )∫ T a L2|x(s) −y(s)|ds + | ( log t a ) | Γ(r − 1) ∫ T tm ( log T s )r−2 L1‖xs −ys‖Cτ ds s + | ( log t a ) | Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 L1‖xs −ys‖Cτ ds s + | ( log t a ) | m∑ i=1 ti l ∗|x(t−i ) −y(t − i )| + 1 Γ(r) ∫ t tk ( log t s )r−1 L1‖xs −ys‖Cτ ds s + 1 Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 L1‖xs −ys‖Cτ ds s + k∑ i=1 ( log t ti ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 L1‖xs −ys‖Cτ ds s + k∑ i=1 l|x(t−i ) −y(t − i )| + k∑ i=1 ti ( log t ti ) l ∗|x(t−i ) −y(t − i )| ≤ T ( log T a ) (T −a)L2‖x −y‖ + ( log T a ) [ L1‖x −y‖ Γ(r − 1) ∫ T tm ( log T s )r−2 ds s + L1‖x −y‖ Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 ds s + mTl ∗‖x −y‖] + L1‖x −y‖ Γ(r) ∫ t tk ( log t s )r−1 ds s + L1‖x −y‖ Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 ds s + k∑ i=1 L1‖x −y‖ ( log t ti ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 ds s + ml‖x −y‖ + mT ( log T a ) l ∗‖x −y‖ ≤ [ L2(T −a)T ( log T a ) + L1 ( m + 1 Γ(r + 1) + 1 + 2m Γ(r) )( log T a )r + ml + 2mT ( log T a ) l ∗ ] ‖x −y‖ Thus, we have ‖Fx −Fy‖≤ [ L1 ( m + 1 Γ(r + 1) + 1 + 2m Γ(r) )( log T a )r + L2T (T −a) ( log T a ) + ml + 2ml ∗ T ( log T a )] ‖x −y‖. 8 Int. J. Anal. Appl. (2023), 21:15 Consequently by (3.5), F is a contraction, as a conseguence of Banach fixed point theorem, we deduce that F has a fixed point which is a solution of the problem (1.1)-(1.4). This completes the proof. Our second result deals with the existence of solutions for problem (1.1)-(1.4) by applying on Scheafer fixed point theorem. Theorem 3.2. Assume that the following conditions hold : (H4) The function f : J ×Cτ →R is continuous. (H5) The function h : R→R is continuous. (H6) The functions Ik, Ik : R→R are continuous. (H7) There exists a constant N > 0 such that |f (t,y)| ≤ N, for each t ∈ J and y ∈ Cτ. (H8) There exists a constant N∗ > 0 such that |h(t,x)| ≤ N∗ for each x ∈R. (H9) There exist two constants N1 > 0, N2 > 0 such that |Ik(x)| ≤ N1, |Ik(x)| ≤ N2 for each , x ∈R, k = 1, . . . ,m, then the boundary value problem (1.1)-(1.4) has at least one solution on [a−τ,T ], Proof: We shall use Scheafer fixed point theorem to prove that F has a fixed point, defined by 3.6. The proof will be given in several steps. Step 1: F is continuous. Let {yn} be a sequence such that yn → y in B. If t ∈ [a,T ], we have |F (yn)(t) −F (y)(t)| ≤ T ( log t a )∫ T a |h(s,yn(s)) −h(s,y(s))|ds + ( log t a ) Γ(r − 1) ∫ T tm ( log T s )r−2 |f (s,yns) − f (s,ys))| ds s + ( log t a ) Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 |f (s,yns) − f (s,ys))| ds s + ( log t a ) m∑ i=1 ti|Ī(yn(t−i )) − Īi (y(t − i ))| + 1 Γ(r) ∫ t tk ( log t s )r−1 |f (s,yns) − f (s,ys)| ds s + 1 Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 |f (s,yns) − f (s,ys)| ds s + k∑ i=1 ( log t ti ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,yns) − f (s,ys)| ds s + k∑ i=1 |Ii (yn(t−i )) − Ii (y(t − i ))| + k∑ i=1 ti ( log t ti ) |Ī(yn(t−i )) − Īi (y(t − i ))| Int. J. Anal. Appl. (2023), 21:15 9 Since f , h and Ik, Īk, k = 1, . . . ,m, are continuous functions, we have ‖F (yn) −F (y)‖∞ → 0 as n →∞. Step 2: F maps bounded sets into bounded sets in B. Indeed, it is enough to show that for any η∗ > 0, there exists a positive constant L such that for each y ∈ Dη∗ = {y ∈ B : ‖y‖ ≤ η∗}, we have ‖F (y)‖ ≤ L. By (H7), (H8) and (H9), for each t ∈ J, we can obtain |F (y)(t)| ≤ |ϕ(a)| + T ( log t a )∫ T a |h(s,y(s))|ds + | ( log t a ) | Γ(r − 1) ∫ T tm ( log T s )r−2 |f (s,ys)| ds s + | ( log t a ) | Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 |f (s,ys)| ds s + | ( log t a ) | m∑ i=1 ti|Īi (y(t−i ))| + 1 Γ(r) ∫ t tk ( log t s )r−1 |f (s,ys)| ds s + 1 Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 |f (s,ys)| ds s + k∑ i=1 | ( log t ti ) | Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,ys)| ds s + k∑ i=1 |Ii (y(t−i ))| + k∑ i=1 ti| ( log t ti ) ||Īi (y(t−k ))| ≤ |ϕ(a)| + N∗T ( log t a )∫ T a ds + N| ( log t a ) | Γ(r − 1) ∫ T tm ( log T s )r−2 ds s + N| ( log t a ) | Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 ds s + m ( log T a ) TN2 + N Γ(r) ∫ t tk ( log t s )r−1 ds s + N Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 ds s + k∑ i=1 N| ( log t ti ) | Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 ds s + mN1 + mT (log T a )N2 ≤ ‖ϕ‖ + (T −a)T ( log T a ) N∗ + N (1 + 2m) ( log T a )r Γ(r) + N (1 + m) ( log T a )r Γ(r + 1) +mN1 + 2mT ( log T a ) N2. Therefore ‖Fy‖ ≤ ‖ϕ‖ + N [ 1 + m Γ (r + 1) + 1 + 2m Γ(r) ]( log T a )r + mN1 + [(T −a)N∗ + 2mN2] T ( log T a ) := L. Step 3: F maps bounded sets into equicontinuous sets of B . Let τ1,τ2 ∈ J,τ1 < τ2, Dη∗ be a bounded set of B as in Step 2, and let y ∈ Dη∗. Then 10 Int. J. Anal. Appl. (2023), 21:15 |F (y)(τ2) −F (y)(τ1)| ≤ T ( log τ2 τ1 )∫ T a |h(s,y(s))|ds + (log τ2 τ1 ) Γ(r − 1) ∫ T tm ( log T s )r−2 |f (s,ys)| ds s + ( log τ2 τ1 ) Γ(r − 1) m∑ i=i ∫ ti ti−1 ( log ti s )r−2 |f (s,ys)| ds s + ( log τ2 τ1 ) m∑ i=1 ti|Īi (y(t−i ))| + 1 Γ(r) ∫ τ1 tk [( log τ2 s )r−1 − ( log τ1 s )r−1] |f (s,ys)| ds s + 1 Γ(r) ∫ τ2 τ1 ( log τ2 s )r−1 |f (s,ys)| ds s + k∑ i=1 ( log τ2 τ1 ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,ys)| ds s + ( log τ2 τ1 ) k∑ i=1 ti|Īi (y(t−i ))|. As τ1 −→ τ2, the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we can conclude that F : B → B is completly continuous. Step 4: A priori bounds. Now it remains to show that the set ε = {y ∈ B → B : y = λF (y) f or some 0 < λ < 1} is bounded. Let y ∈ ε, then y = λF (y) for some 0 < λ < 1. Thus, for each t ∈ J we have (Fy)(t) = λϕ(a) + λT ( log t a )∫ T a h(s,y(s))ds − λ ( log t a ) Γ(r − 1) ∫ T tm ( log T s )r−2 f (s,ys) ds s − λ ( log t a ) Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 f (s,ys) ds s −λ ( log t a ) m∑ i=1 ti Īi (y(t − i )) + λ Γ(r) ∫ t tk ( log t s )r−1 f (s,ys) ds s + λ Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 f (s,ys) ds s +λ k∑ i=1 ( log t ti ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 f (s,ys) ds s + λ k∑ i=1 Ii (y(t − i )) +λ k∑ i=1 ti ( log t ti ) Īi (y(t − i )) For each t ∈ J, by (H7)-(H9), we have ‖Fy‖ ≤ ‖ϕ‖ + T (T −a) ( log T a ) N∗ + N [ 1 + m Γ(r + 1) + 1 + 2m Γ(r) ]( log T a )r + mN1 + 2mT ( log T a ) N2. This shows that the set ε is bounded. As a consequence of Schaefer’s fixed point theorem, we deduce that F has a fixed point which is a solution of the problem (1.1)-(1.4). By applying the of Leray-Schauder nonlinear alternative type. Theorem 3.3. Assume that (H4)-(H6) and the following conditions hold : Int. J. Anal. Appl. (2023), 21:15 11 (H10) There exist φf ∈ C(J,R+) and ψ : [0,∞) → [0,∞) continuous and non-decreasing such that |f (t,u)| ≤ φf (t)ψ(|u|), f or all t ∈ J, u ∈ Cτ. (H11) There exist φh ∈ L(J,R+) and ψ∗ : [0,∞) → [0,∞) continuous and non-decreasing such that |h(t,v)| ≤ φh(t)ψ∗(|v|), f or all t ∈ J, v ∈R. (H12) There exist ψ̄∗, ψ̄∗∗ : [0,∞) → [0,∞) continuous and non-decreasing such that |Ik(v)| ≤ ψ̄∗(|v|), |Īk(v)| ≤ ψ̄∗∗(|v|), f or all v ∈R, k = 1, . . . ,m. (H13) There exists a number M > 0 such that M ‖ϕ‖ + T ( log T a ) ψ∗(M)‖φh‖L1 + φψ(M) ( 1+m Γ(r+1) + 1+2m Γ(r) )( log T a )r + mψ̄∗(M) + 2mT ( log T a ) ψ̄∗∗(M) ≥ 1, where φ = sup{φf (t) : t ∈ J}, then the problem(1.1)-(1.4) has at least one solution on [a−τ,T ]. Proof: Consider the operator F defined as in 3.6. It can be easily shown that F is continuous and completely continuous. For λ ∈ [0, 1] and each t ∈ J, let y(t) = λ(Fy)(t). Then from(H10)-(H12), we have |(Fy)(t)| ≤ |ϕ(a)| + T ( log t a )∫ T a |h(s,y(s))|ds + ( log t a ) Γ(r − 1) ∫ T tm ( log T s )r−2 |f (s,ys )| ds s + ( log t a ) Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 |f (s,ys )| ds s + ( log t a ) m∑ i=1 ti|Īi (y(t−i ))| + 1 Γ(r) ∫ t tk ( log t s )r−1 |f (s,ys )| ds s + 1 Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 |f (s,ys )| ds s + k∑ i=1 ( log t ti ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,ys )| ds s + k∑ i=1 |Ii (y(t−i ))| + k∑ i=1 ti ( log t ti ) |Īi (y(t−i ))| ≤ |ϕ(a)| + T ( log t a )∫ T a φh(s)ψ ∗ (|y(s)|)ds + ( log t a ) Γ(r − 1) ∫ T tm ( log T s )r−2 φf (s)ψ(|ys|) ds s + ( log t a ) Γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 φf (s)ψ(|ys|) ds s + ( log t a ) m∑ i=1 tiψ̄∗∗(|y(t−i )|) + 1 Γ(r) ∫ t tk ( log t s )r−1 φf (s)ψ(|ys|) ds s + 1 Γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 φf (s)ψ(|ys|) ds s + k∑ i=1 ( log t ti ) Γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 φf (s)ψ(|ys|) ds s + k∑ i=1 ψ̄∗(|y(t−i )|) + k∑ i=1 ti ( log t ti ) ψ̄∗∗(|y(t−i )|) ≤ ‖ϕ‖ + T ( log T a ) ψ ∗ (‖y‖) ∫ T a φh(s)ds + ( log T a )r Γ(r) φψ(‖y‖) + m ( log T a )r Γ(r) φψ(‖y‖) 12 Int. J. Anal. Appl. (2023), 21:15 +mT ( log T a ) ψ̄∗∗(‖y‖) + ( log T a )r Γ(r + 1) φψ(‖y‖) + m ( log T a )r Γ(r + 1) φψ(‖y‖) + m ( log T a )r Γ(r) φψ(‖y‖) + mψ̄∗(‖y‖) + mT ( log t a ) ψ̄∗∗(‖y‖) ≤‖ϕ‖ + T ( log T a ) ψ ∗ (‖y‖)‖φh‖L1 + φψ(‖y‖) ( (1 + m) Γ(r + 1) + (1 + 2m) Γ(r) )( log T a )r +mψ̄∗(‖y‖) + 2mT ( log T a ) ψ̄∗∗(‖y‖) Thus ‖y‖ ‖ϕ‖ + T ( log T a ) ψ∗(‖y‖)‖φh‖L1 + φψ(‖y‖) ( 1+m Γ(r+1) + 1+2m Γ(r) )( log T a )r + mψ̄∗(‖y‖) + 2mT ( log T a ) ψ̄∗∗(‖y‖) ≤ 1. Then by condition (H13), there exists M such that ‖y‖ 6= M. Let U = {y ∈ B : ‖y‖≤ M}. The operator F : U → B is continuous and completely continuous. From the choice of U, there is no y ∈ ∂U such that y = λF (y) for some λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray-Schauder type, we deduce that F has a fixed point y ∈ U which is a solution of the problem (1.1)-(1.4). This completes the proof. 4. Example Let consider the following problem: C HD 3 2 y(t) = et (et + 5)2 |yt| (1 + |yt|) , t ∈ [1, 2],t 6= 4 3 , (4.1) ∆y( 4 3 ) = |y( 4 3 − )| 15 + |y( 4 3 − )| , (4.2) ∆y ′( 4 3 ) = |y( 4 3 − )| 17 + |y( 4 3 − )| , (4.3) y(t) = φ(t), t ∈ [1 −τ, 1], y ′(2) = ∫ 2 1 |y(s)| 13 + |y(s)| ds. (4.4) Set f (t,yt) = et (et + 5)2 |yt| (1 + |yt|) , (t,y) ∈ J ×C([1 −τ, 1],R), h(t,y(t)) = ∫ 2 1 |y(s)| 13 + |y(s)| ds, (t,y) ∈ J ×R, I(y) = |y| 15 + |y| , I(y) = |y| 17 + |y| , y ∈R. Int. J. Anal. Appl. (2023), 21:15 13 Hence the hypotheses (H1)-(H3) holds, with L1 = 1 36 ,L2 = 1 13 , l = 1 15 , l∗ = 1 17 . We shall check that condition (3.5). With r = 3 2 ,m = 1,t1 = 4 3 ,T = 2,a = 1. Further [ L2T (T −a) ( log T a ) + L1 ( m + 1 Γ(r + 1) + 1 + 2m Γ(r) )( log T a )r + ml + 2ml∗T ( log T a )] = [ 2 13 (log 2) + 1 36 ( 2 Γ ( 5 2 ) + 3 Γ( 3 2 ) ) (log 2) 3 2 + ml + 4 17 (log 2) ] = 0.414779517 < 1. Note that Γ( 3 2 ) = 1 2 √ π, Γ( 5 2 ) = 3 4 √ π. Then all hypotheses of Theorem (3.1) are fulfilled, and consequently the boundary value problem (4.1)-(4.4) has a unique solution on [1, 2]. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] R.P. Agarwal, M. Benchohra, S. Hamani, A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions, Acta Appl. Math. 109 (2008), 973–1033. https://doi.org/10. 1007/s10440-008-9356-6. [2] Y. Adjabi, F. Jarad, D. Baleanu, T. Abdeljawad, On Cauchy Problems With Caputo Hadamard Fractional Deriva- tives, J. Comput. Anal. Appl. 21 (2016), 661-681. http://hdl.handle.net/20.500.12416/1783. [3] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Compositions of Hadamard-Type Fractional Integration Operators and the Semigroup Property, J. Math. Anal. Appl. 269 (2002), 387–400. https://doi.org/10.1016/s0022-247x(02) 00049-5. [4] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Mellin Transform Analysis and Integration by Parts for Hadamard-Type Fractional Integrals, J. Math. Anal. Appl. 270 (2002), 1–15. https://doi.org/10.1016/s0022-247x(02)00066-5. [5] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Fractional Calculus in the Mellin Setting and Hadamard-Type Fractional Integrals, J. Math. Anal. Appl. 269 (2002), 1–27. https://doi.org/10.1016/s0022-247x(02)00001-x. [6] Z. Cui, P. Yu, Z. Mao, Existence of Solutions for Nonlocal Boundary Value Problems of Nonlinear Fractional Differential Equations, Adv. Dyn. Syst. Appl. 7 (2012), 31-40. [7] Y.K. Chang, A. Anguraj, M. Mallika Arjunan, Existence Results for Impulsive Neutral Functional Differential Equa- tions With Infinite Delay, Nonlinear Anal.: Hybrid Syst. 2 (2008), 209–218. https://doi.org/10.1016/j.nahs. 2007.10.001. [8] J. Cao, Y. Luo, G. Liu, Some Results for Impulsive Fractional Differential Inclusions With Infinite Delay and Sectorial Operators in Banach Spaces, Appl. Math. Comput. 273 (2016), 237–257. https://doi.org/10.1016/ j.amc.2015.09.072. [9] J. Dabas, A. Chauhan, Existence and Uniqueness of Mild Solution for an Impulsive Neutral Fractional Integro- Differential Equation With Infinite Delay, Math. Computer Model. 57 (2013), 754–763. https://doi.org/10. 1016/j.mcm.2012.09.001. [10] J. Hadamard, Essai Sur l’Etude des Fonctions Donnees par Leur Developpement de Taylor, J. Math. Pure Appl. 8 (1892), 101-186. https://doi.org/10.1007/s10440-008-9356-6 https://doi.org/10.1007/s10440-008-9356-6 http://hdl.handle.net/20.500.12416/1783 https://doi.org/10.1016/s0022-247x(02)00049-5 https://doi.org/10.1016/s0022-247x(02)00049-5 https://doi.org/10.1016/s0022-247x(02)00066-5 https://doi.org/10.1016/j.nahs.2007.10.001 https://doi.org/10.1016/j.nahs.2007.10.001 https://doi.org/10.1016/j.amc.2015.09.072 https://doi.org/10.1016/j.amc.2015.09.072 https://doi.org/10.1016/j.mcm.2012.09.001 https://doi.org/10.1016/j.mcm.2012.09.001 14 Int. J. Anal. Appl. (2023), 21:15 [11] S. Hamani, A. Hammou, J. Henderson, Impulsive Fractional Differential Equations Involving The Hadamard Frac- tional Derivative, Commun. Appl. Nonlinear Anal. 24 (2017), 48-58. [12] A. Hammou, S. Hamani, J. Henderson, Impulsive Hadamard Fractional Differential Equations in Banach Spaces, Commun. Appl. Nonlinear Anal. 28 (2018), 52-62. [13] A. Hammou, S. Hamani, J. Henderson Initial Value Problems for Impulsive Caputo-Hadamard Fractional Differential Inclusions, Commun. Appl. Nonlinear Anal. 22 (2019), 17-35. [14] S. Heidarkhani, A. Salari, G. Caristi, Infinitely Many Solutions for Impulsive Nonlinear Fractional Boundary Value Problems, Adv. Differ. Equ. 2016 (2016), 196. https://doi.org/10.1186/s13662-016-0919-y. [15] F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-Type Modification of the Hadamard Fractional Derivatives, Adv. Differ. Equ. 2012 (2012), 142. https://doi.org/10.1186/1687-1847-2012-142. [16] Y. Tian, Z. Bai, Impulsive Boundary Value Problem for Differential Equations with Fractional Order, Differ Equ. Dyn. Syst. 21 (2012), 253–260. https://doi.org/10.1007/s12591-012-0150-6. [17] P. Thiramanus, S.K. Ntouyas, J. Tariboon, Existence and Uniqueness Results for Hadamard-Type Fractional Differ- ential Equations with Nonlocal Fractional Integral Boundary Conditions, Abstr. Appl. Anal. 2014 (2014), 902054. https://doi.org/10.1155/2014/902054. [18] A. Nain, R. Vats, A. Kumar, Caputo-Hadamard Fractional Differential Equation With Impulsive Boundary Condi- tions, J. Math. Model. 9 (2021), 93-106. https://doi.org/10.22124/jmm.2020.16449.1447. [19] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North- Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. [20] S. Abbas, M. Benchohra, J.R. Graef, J. Henderson, Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018. [21] S. Abbas, M. Benchohra, G.M. N’Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-4036-9. https://doi.org/10.1186/s13662-016-0919-y https://doi.org/10.1186/1687-1847-2012-142 https://doi.org/10.1007/s12591-012-0150-6 https://doi.org/10.22124/jmm.2020.16449.1447 https://doi.org/10.1007/978-1-4614-4036-9 1. Introduction 2. Preliminaries 3. Existence of Solutions 4. Example References