Implicit Caputo fractional q-difference equations with non instantaneous impulses

Differential Equations &Applications With Compliments of the Author DEA-15-12 215–234 Zagreb, Croatia Volume 15, Number 3, August 2023 Sa¨ ıd Abbas, Mouffak Benchohra and Alberto Cabada∗ Implicit Caputo fractional q-difference equations with non instantaneous impulses DIFFERENTIAL EQUATIONS & APPLICATIONS AIMS AND SCOPE Differential Equations and Applications (DEA, Differ. Equ. Appl.) will publish original papers from all branches of ordinary, functional-differential, and partial differential equations. Only papers of the highest quality will be accepted for the publication. The papers which demonstrate novelty or give relations of differential equations with other fields of mathematics and applications are preferred. DEA is published quarterly, in February, May, August and November. SUBMISSION Manuscripts should be submitted electronically in PostScript or Adobe Acrobat PDF format to the DEA Editorial Office through DEA page www.ele-math.com or by e-mail [email protected] The author who submitted the article for publication will be denoted as a corresponding author. He/She manages all communication and correspondence with the DEA regarding the article. Authors may indicate a member of the Editorial Board whom they consider appropriate for the article. However, assignment to that particular editor is not assured. In order to facilitate refereeing, copies of those papers (whether by the author or someone else) which are essential and referred to in the manuscript but are not conveniently accessible, should be enclosed. COPYRIGHT The acceptance of the article automatically implies the copyright transfer to DEA. Manuscripts are accepted for review with the understanding that the same work has not been published (except in the form of an abstract), that it is not under consideration for publication elsewhere, that it will not be submitted to another journal, and that its submission for publication has been approved by all the authors. PREPARATION OF MANUSCRIPT Manuscripts should be written in English. The publisher strongly encourages submission of manuscripts written in TeX or one of its variants La- TeX, AMSTeX or AMSLaTeX. On acceptance of the paper, authors will be asked to send final version of file(s) to the Editorial Office. TITLE PAGE A title page should include the title of the article, author’s name (no degrees) author’s affiliation, e-mail addresses, mailing address of the corresponding author and running head less than 60 characters. Authors should be listed alphabetically. ABSTRACT, KEY WORDS, SUBJECT CLASSIFICATION The manuscript must be accompanied by a brief abstract, no longer than 100-150 words. It should make minimal use of mathematical symbols and displayed formulas. Mathematics Subject Classification (2000) with primary (and secondary) subject classification codes and a list of 4-5 key words must be given. FIGURES Figures should be prepared in a digital form suitable for direct reproduction, grey scale only, preferrably at resolution of 300 dpi or higher, sized to fit a journal page, and in EPS, TIFF or JPEG format. Figures should have a short descriptive title and should be numbered consecutively in order of appearance in the text. The position of each figure in the text should be indicated. REFERENCES Bibliographic references should be listed alphabetically at the end of the article. The author should consult Mathematical Reviews for the standard abbreviations of journal names. PROOFS The author will receive proofs only once. Late return of the proofs will delay the article to a later issue. PAGE CHARGES, OFFPRINTS There are no page charges. Authors will receive PDF file of the printed article free of charge. Additional offprints may be ordered from ’DEA’ prior to publication. FORTHCOMING PAPERS Papers accepted and prepared for publication will appear in the forthcoming section of Journal Web page. They are identical in form as final printed papers, except volume, issue and page numbers. DEA is published by Publishing House ELEMENT, Zagreb, Croatia, and Department of Applied Mathematics, FER, University of Zagreb. All correspondence and subscription orders should be addressed to the Editorial Office: Differential Equations & Applications Editorial Office Menceticeva 2, 10000 Zagreb, Croatia www.ele-math.com e-mail: [email protected] Fax: +385 1 6008799 Differential Equations &Applications Volume 15, Number 3 (2023), 215–234 doi:10.7153/dea-2023-15-12 IMPLICIT CAPUTO FRACTIONAL q–DIFFERENCE EQUATIONS WITH NON INSTANTANEOUS IMPULSES SA¨ ID ABBAS,MOUFFAK BENCHOHRA AND ALBERTO CABADA∗ (Communicated by S. K. Ntouyas) Abstract. In the present article, we prove some existence results for a class of implicit Caputo fractional q-difference equations with non instantaneous impulses in Banach spaces. The used techniques rely on the concepts of measure of noncompactness and the use of suitable fixed point theorems. 1. Introduction Fractional differential equations have been applied in various areas. For some fundamental results we refer the reader to [2,3,4,25,34,40], and the references therein. Recently, in [4,10] the authors applied the measure of noncompactness [11,14]tosome classes of functional Riemann-Liouville or Caputo fractional differential equations in Banach spaces. Fractional q-difference equations initiated in the beginning of the 19th century [5,18], and received significant attention in recent years. Motivated from quantum calculus, some interesting results on initial and boundary value problems of q-difference and fractional q-difference equations are given in [8,9,20,26] and references therein. Impulsive differential equations have become more important in recent years in some mathematical models of real phenomena, especially in biological or medical domains, and in control theory, see for example the monographs of Abbas et al. [2,3], Benchohra et al. [15], Graef et al. [21] and papers such as Abbas et al. [26], and the references therein. The study of abstract nonlocal Cauchy problem was initiated by Byszewski [17] in 1991. Evolution equations with nonlocal initial conditions were motivated by physical problems. As a matter of fact, it is demonstrated that the evolution equations with nonlocal initial conditions have better effects in applications than the classical Cauchy problems. For example, it was pointed in [19] that the nonlocal problems are used to represent mathematical models for evolution of various phenomena, such as nonlocal Mathematics subject classification (2020): 26A33, 34A37, 34G20. Keywords and phrases: Fractional q-difference equation, Caputo fractional derivative, measure of noncompactness, non instantaneous impulses, fixed point theorems. Third author is partially supported by Grant PID2020-113275GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” of the “European Union”. ∗Corresponding author. c    ,Zagreb Paper DEA-15-12 215 216 S. ABBAS,M.BENCHOHRA AND A. CABADA neural networks, nonlocal pharmacokinetics, nonlocal pollution and nonlocal combustion. Due to nonlocal problems have a wide range of applications in real world applications, evolution equations with nonlocal initial conditions were studied by many authors. The dynamics of many evolving processes are subject to abrupt changes, such as shocks, harvesting and natural disaster. These phenomena involve short term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. Sometimes time abrupt changes may stay for time intervals such impulses are called non-instantaneous impulses. The importance of the study of non-instantaneous impulsive differential equations lies in its diverse fields of applications such as in the theory of stage by stage rocket combustion, maintaining hemodynamical equilibrium etc. A very well known application of non instantaneous impulses is the introduction of insulin in the bloodstream which is abrupt change and the consequent absorption which is a gradual process as it remains active for a finite interval of time. The theory of impulsive differential equations has found enormous applications in realistic mathematical modeling of a wide range of practical situations. It has emerged as an important area of research such as modeling of impulsive problems in physics, population dynamics, ecology, biological systems, biotechnology and so forth. There has been a significant development in impulse theory, in recent years, especially in the area of impulsive differential equations with fixed moments, see for instance, [15,33] and the references therein. In pharmacotherapy, the above instantaneous impulses can not describe the certain dynamics of evolution processes. For example, one considers the hemodynamic equilibrium of a person, the introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous process. From the view point of general theories, Hern´andez and O’Regan [23] initially offered to study a new class of abstract semilinear impulsive differential equations with non instantaneous impulses in a PC-normed Banach space, and Pierri et al. [29]. The existence of solutions for non-instantaneous impulsive fractional and integer order differential equations has also been studied see the book by Agarwal al. [7], and the papers [1,12,13,36,37,38,39]. This paper initiates the study of impulsive implicit fractional q-difference derivative at non instantaneous impulses in finite and infinite dimensional Banach spaces. We first discuss the existence of solutions for the following problem of implicit fractional q-difference equations with non instantaneous impulses ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (C qDr sku)(t)= f(t,u(t),(C qDr sku)(t));t∈Ik,k=0,...,m, u(t)=gk(t,u(t− k));ift∈Jk,k=1,...,m, u(sk)+Q(u)=uk∈R;k=0,...,m, (1.1) where I0:=[0,t1],Jk:=(tk,sk],Ik:=(sk,tk+1];k=1,...,m,f:Ik×R×R→R,gk: Jk×R→R,Q:PC →Rare given continuous functions such that gk(t,u(t− k))|t=sk= uk−Q(u)∈R;k=1,...,m,0=s0<t1⩽s1<t2⩽s2<...⩽sm−1<tm⩽sm<tm+1= T,the set PC is given later, and c qDr skis the Caputo fractional q-difference derivative of order r∈(0,1]. Differ. Equ. Appl. 15, No. 3 (2023), 215–234. 217 Under suitable growth assumptions on the different functions that appear on the equation we prove, by means of a generalization of the classical Banach fixed point theorem, a uniqueness result in the scalar case. Moreover, two existence results of at least one solution are deduced from the Schauder’s and the Schaefer’s fixed point theorems respectively. Next we discuss the existence of solutions for the problem (1.1), when uk∈E,f:Ik×E×E→E,gk:Jk×E→E,Q:PC →Eare given continuous functions such that gk(t,u(t− k))|t=sk=uk∈E;k=1,...,m,and Eis a real Banach space with norm ·. We deduce, from M¨onch’s fixed point theorem, the existence of at least one solution for equations defined on Banach spaces. The paper finalizes with some examples that illustrate the applicability of the obtained results in the scalar as well as the Banach space setting. 2. Preliminaries Consider the Banach space C(I):=C(I,E)of continuous functions from I:= [0,T]into Eequipped with the usual norm u∞:=sup t∈I u(t). In the scalar case when E=R,we replace · by |·|.As usual, L1(I)denotes the space of measurable functions v:I→Ewhich are Bochner integrable with the norm v1=I v(t)dt. Let PC =u:I→E:u∈C(∪m k=1(tk,tk+1)),u(t± k)∈R,u(t− k)=u(tk), be the Banach space with the norm uPC =sup t∈I u(t). In the case E=R,we have uPC =sup t∈I |u(t)|. Let us recall some definitions and properties of fractional q-calculus. For a∈R,and 0<q<1,we set [a]q=1−qa 1−q. The qanalogue of the power (a−b)nis (a−b)(0)=1,(a−b)(n)=Πn−1 k=0(a−bqk);a,b∈R,n∈N. In general, (a−b)( α )=a α Π∞ k=0a−bqk a−bqk+ α ;a,b, α ∈R. 218 S. ABBAS,M.BENCHOHRA AND A. CABADA DEFINITION 1. [24]Theq-gamma function is defined by Γq( ξ )=(1−q)( ξ −1) (1−q) ξ −1; ξ ∈R−{0,−1,−2,...} Notice that the q-gamma function satisfies Γq(1+ ξ )=[ ξ ]qΓq( ξ ). DEFINITION 2. [24]Theq-derivative of order n∈Nof a function u:I→Eis defined by (D0 qu)(t)=u(t), (Dqu)(t):=(D1 qu)(t)=u(t)−u(qt) (1−q)t;t=0,(Dqu)(0)=lim t→0(Dqu)(t), and (Dn qu)(t)=(DqDn−1 qu)(t);t∈I,n∈{1,2,...}. Set It:={tqn:n∈N}∪{0}. DEFINITION 3. [24]Theq-integral of a function u:It→Eis defined by (Iqu)(t)=t 0 u(s)dqs= ∞ ∑ n=0 t(1−q)qnf(tqn), provided that the series converges. We note that (DqIqu)(t)=u(t),while if uis continuous at 0,then (IqDqu)(t)=u(t)−u(0). DEFINITION 4. [6] The Riemann-Liouville fractional q-integral of order α ∈ R+:=[0,∞)of a function u:I→Eis defined by (I0 qu)(t)=u(t),and (I α qu)(t)=t 0 (t−qs)( α −1) Γq( α )u(s)dqs;t∈I. LEMMA 1. [30]For α ∈R+:=[0,∞)and λ ∈(−1,∞)we have (I α q(t−a)( λ ))(t)= Γq(1+ λ ) Γ(1+ λ + α )(t−a)( λ + α );0<a<t<T. In particular, (I α q1)(t)= 1 Γq(1+ α )t( α ). DEFINITION 5. [31] The Riemann-Liouville fractional q-derivative of order α ∈ R+of a function u:I→Eis defined by (D0 qu)(t)=u(t),and (D α qu)(t)=(D[ α ] qI[ α ]− α qu)(t);t∈I, where [ α ]is the integer part of α . Differ. Equ. Appl. 15, No. 3 (2023), 215–234. 219 DEFINITION 6. [31] The Caputo fractional q-derivative of order α ∈R+of a function u:I→Eis defined by (CD0 qu)(t)=u(t),and (CD α qu)(t)=(I[ α ]− α qD[ α ] qu)(t);t∈I. REMARK 1. For t∈Ik,k=0,...,m,and r>0,we define qIr skand C qDr tkas (qIr sku)(t)=t sk (t−qs)(r−1) Γq(r)u(s)dqs, and (C qDr tk)u(t)=( qI[r]−r sk C qD[r] tk)u(t). LEMMA 2. [31]Let α ∈R+.Then the following equality holds: (I α q CD α qu)(t)=u(t)− [ α ]−1 ∑ k=0 tk Γq(1+k)(Dk qu)(0). In particular, if α ∈(0,1),then (I α qCD α qu)(t)=u(t)−u(0). From the above lemma (see also [26]), and in order to define the solution for our problem, we deduce the following result. LEMMA 3. Let f :I×E×E→E be continuous. Then the problem ⎧ ⎨ ⎩ (C qDr 0u)(t)= f(t,u(t),(C qDr 0u)(t));t∈I, u(0)=u0, is equivalent to the problem of obtaining the solutions of the integral equation g(t)= f(t,u0+(I α qg)(t),g(t));t∈I and if g ∈C(I),is the solution of this equation, then u(t)=u0+(I α qg)(t);t∈I. LEMMA 4. Let h :I→E be a continuous function. A function u ∈PC is a solution of the fractional integral equation ⎧ ⎨ ⎩ u(t)=uk+( qIr skh)(t);t∈Ik,k=0,...,m, u(t)=gk(t,u(t− k));t∈Jk,k=1,...,m, (2.1) 220 S. ABBAS,M.BENCHOHRA AND A. CABADA if and only if u is a solution of the following problem ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (C qDr tku)(t)=h(t);t∈Ik,k=0,...,m, u(t)=gk(t,u(t− k));t∈Jk,k=1,...,m, u(sk)=uk. (2.2) From the above Lemmas, we arrive at the following one: LEMMA 5. Let f :Ik×E×E→E;k=0,...,m,be a continuous function. Then problem (1.1)is equivalent to the problem of solving the equation g(t)= f(t,uk−Q(u)+(qIr skg)(t),g(t)),t∈Ik,(2.3) together with u(t)=gk(t,u(t− k));if t ∈Jk,k=1,...,m. Moreover, if g ∈C(Ik);k=0,...,m,is the solution of this equation, then ⎧ ⎨ ⎩ u(t)=uk−Q(u)+(qIr skg)(t);t∈Ik,k=0,...,m, u(t)=gk(t,u(t− k));t∈Jk,k=1,...,m. Let MXdenote the class of all bounded subsets of a metric space X. DEFINITION 7. Let Xbe a complete metric space. A map μ :MX→[0,∞)is called a measure of noncompactness on Xif it satisfies the following properties for all B,B1,B2∈MX. (a) μ (B)=0 if and only if Bis precompact (Regularity), (b) μ (B)= μ (B)(Invariance under closure), (c) μ (B1∪B2)=max{ μ (B1), μ (B2)}(Semi-additivity). DEFINITION 8. [14]Let(X,·X)be a Banach space and let ΩXbe the family of bounded subsets of E.The Kuratowski measure of noncompactness is the map μ : ΩX→[0,∞)defined by μ (M)=inf{ ε >0:M⊂∪ m j=1Mj,diam(Mj)⩽ ε } where M∈ΩE,and diam(Mj)= sup μ , ν ∈Mj  μ − ν X;j=1,···,m. The Kuratowski measure of noncompactness satisfies the following properties: (1) μ (M)=0⇔Mis compact (Mis relatively compact). (2) μ (M)= μ (M). Differ. Equ. Appl. 15, No. 3 (2023), 215–234. 221 (3) M1⊂M2⇒ μ (M1)⩽ μ (M2). (4) μ (M1+M2)⩽ μ (M1)+ μ (M2). (5) μ (cM)=|c| μ (M),c∈R. (6) μ (convM)= μ (M). For our purpose we will need the following fixed point theorems: THEOREM 1. [16,27]Let (X,d)be a complete metric space and T :X→Xbe a mapping such that d(T(x),T(y)) ⩽kd(x,y), where k ∈[0,1). Then T has a unique fixed point in X. THEOREM 2. (Schauder’s fixed point theorem [35]) Let X be a Banach space, D be a bounded closed convex subset of X and T :D→D be a compact and continuous map. Then T has at least one fixed point in D. THEOREM 3. (Schaefer’s fixed point theorem [22]) Let X be a Banach space and N:X→X be a completely continuous operator. If the set E={u∈X:u= λ N(u);for some λ ∈(0,1)} is bounded, then N has fixed points. THEOREM 4. (Monch’s fixed point theorem [28]) Let D be a bounded, closed and convex subset of a Banach space such that 0∈D,and let N be a continuous mapping of D into itself. If the implication V=conv(N(V)) or V =N(V)∪{0}⇒V is compact,(2.4) holds for every subset V of D,then N has a fixed point. 3. Existence results in the scalar case In this section, we present some results concerning the existence of solutions for the problem (1.1). DEFINITION 9. By a solution of problem (1.1)wemeanafunctionu∈PC that satisfies the condition u(sk)=uk;k=0,...,m,and the equations (C qDr tku)(t)= f(t,u(t), (C qDr tku)(t)) on Ik;k=0,...,m,and u(t)=gk(t,u(t− k)) on Jk;k=1,...,m. The following hypotheses will be used in the sequel: (H01)The functions Q,f,gk;k=1,...,m,are continuous. 228 S. ABBAS,M.BENCHOHRA AND A. CABADA with c∗=max k=1,...,msup t∈Jk {c(t)}<1,and Q(u)⩽L(1+uPC); for each u∈PC. (H4)For each bounded set D⊂PC we have μ (Q(D)) ⩽Lsup t∈I μ (D(t)), where D(t)={u(t):u∈D};t∈I,and for each bounded and measurable set B⊂Ewe have μ (f(t,B,C qDr tkB)) ⩽pk(t) μ (B);t∈Ik,k=0,...,m, where C qDr tkB={C qDr tkw:w∈B},and μ (gk(t,B)⩽ck(t) μ (B);t∈Jk,k=0,...,m. Set p∗=max k=0,...,msup t∈J {pk(t)}. THEOREM 7. Assume that the hypotheses (H1),(H2),(H3)and (H4)hold. If the conditions φ 1 ρ r Γq(r+1)+ φ 2<1,(4.1) p∗1+ ρ r Γq(1+r)<1, L⎛ ⎝1+ ρ rp∗ Γq(1+r) 1 1−p∗1+ ρ r Γq(1+r)⎞ ⎠<1 and L+p∗ ρ r Γq(1+r)<1 (4.2) are satisfied, then problem (1.1)has at least one solution defined on I. Proof. Consider the operator N:PC →PC defined in (3.3). Let R>0,such that R=max k=1,2,...,m ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ c∗ 1−c∗, |uk|+L+ ρ rp∗ Γq(1+r) 1+|uk|+L 1−p∗1+ ρ r Γq(1+r) 1−L1+ ρ rp∗ Γq(1+r) 1 1−p∗1+ ρ r Γq(1+r) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ , and consider the ball BR:=B(0,R)={w∈PC,wPC ⩽R}. Differ. Equ. Appl. 15, No. 3 (2023), 215–234. 229 Let u∈PC and t∈Ik.Then (Nu)(t)=     uk−Q(u)+t sk (t−qs)(r−1) Γq(r)g(s)dqs     , where g∈C(Ik)is the unique solution of the equation g(t)= f(t,uk−Q(u)+(qIr skg)(t),g(t)). The uniqueness of such functions is deduced from (H2)and condition (4.1), as in the proof of Theorem 5. We shall show that the operator N:BR→BRsatisfies all the assumptions of Theorem 4. As in the proof of Theorem 6, we can show that N(BR)⊂BR,N:BR→BRis continuous, and N(BR)is bounded and equicontinuous. We still have to prove that The implication (2.4)holds. Set ν :=maxc∗,L+p∗ ρ r Γq(1+r). Let Vbe a subset of BRsuch that V⊂N(V)∪{0},Vis bounded and equicontinuous and therefore the function t→ v(t)= μ (V(t)) is continuous on J.By (H4)and the properties of the measure μ ,for each t∈Ik,we have v(t)⩽ μ ((NV)(t)∪{0}) ⩽ μ ((NV)(t)) = μ (uk−Q(v)+(qIr skg)(t)), where g∈C(Ik);k=0,...,m,is the unique solution of the equation g(t)= f(t,uk−Q(v)+(qIr skg)(t),g(t)). Then, we obtain v(t)⩽ μ Lv(t)+t sk (t−qs)(r−1)p(s) Γq(r)v(s)dqs ⩽L μ (V(t)) + t sk (t−qs)(r−1)p(s) Γq(r) μ (V(s))dqs ⩽L+p∗ ρ r Γq(1+r)vPC. Thus, for each t∈Ik,we get v(t)⩽ ν vPC. 230 S. ABBAS,M.BENCHOHRA AND A. CABADA Also, for each t∈Jk;k=1,...,m,we obtain v(t)⩽ μ ((NV)(t)∪{0}) ⩽ μ ((NV)(t)) ⩽c∗ μ (V(s)) ⩽c∗vPC. Thus, we obtain vPC ⩽ ν vPC. Hence, we get vPC =0,that is v(t)= β (V(t)) = 0,for each t∈I,and then V(t)is relatively compact in E.In view of the Ascoli-Arzel`a theorem, Vis relatively compact in BR.Applying now Theorem 4, we conclude that Nhas a fixed point which is a solution of problem (1.1).  5. Examples EXAMPLE 1. Consider the problem of implicit impulsive q-fractional differential equation of the form ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (C qDr tku)(t)= f(t,u(t),(C qDr tku)(t));t∈Ik,k=0,1,2, u(t)=gk(t,u(t− k));t∈Jk,k=1,2, u(0)=Q(u), (5.1) where I=[0,1],I0=[0,1 5],J1=(1 5,2 5],I1=(2 5,3 5],J2=(3 5,4 5],I2=(4 5,1],r∈(0,1], f(t,u(t),(C qDr tku)(t)) = Γq(1+r)t2(1+u(t))+C qDr tku(t) et+5(2+|u(t)|);t∈[0,1], Q(u)=1+uPC 3e5, and gk(t,u(t− k)) = 1+u(t− k) 3et+5;k=1,...,m. Clearly, the function fis continuous. For each t∈Ik,we have |f(t,u(t),(C qDr tku)(t))|⩽Γq(1+r)t2e−(t+5)(1+|u(t)|+|(C qDr tku)(t)|). |Q(u)|⩽1 3e5 and |gk(t,u)|⩽1 3e5(1+|u|). Differ. Equ. Appl. 15, No. 3 (2023), 215–234. 231 Hence, the hypotheses (H02)and (H03)is satisfied with φ 1= φ 2=l∗=e−5Γq(1+r), and c∗=L=1 3e5. Also, we can verify that conditions (3.1), l∗1+ ρ r Γq(1+r)<1, and L⎛ ⎝1+ ρ rl∗ Γq(1+r) 1 1−l∗1+ ρ r Γq(1+r)⎞ ⎠<1, are satisfied. Hence all conditions of Theorem 6are satisfied. It follows that the problem (5.1) has at least one solution. EXAMPLE 2. Let E=l1=u=(u1,u2,...,un,...), ∞ ∑ n=1 |un|<∞ be the Banach space with the norm uE= ∞ ∑ n=1 |un|. Consider the problem of implicit impulsive q-fractional differential equation of the form ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (C qDr tku)(t)= f(t,u(t),(C qDr tku)(t));t∈Jk,k=0,...,m, u(t)=gk(t,u(t));k=1,...,m, u(0)=Q(u), (5.2) where I=[0,1],r∈(0,1],u=(u1,u2,...,un,...), f=(f1,f2,..., fn,...), C qDr tku=( C qDr tku1,C qDr tku2,..., C qDr tkun,...);k=0,...,m, fn(t,u(t),(C qDr tku)(t)) = Γq(1+r)t2e−7+1 et+5(2−n+un(t)+C qDr tkun(t));t∈[0,1], gk(t,u(t− k)) = 1+u(t− k)E 3et+5;k=1,...,m, and Q(u)=1+uPC 3e5. For each u∈Eand t∈[0,1],we have f(t,u(t),(C qDr tku)(t))E⩽Γq(1+r)t2e−7+1 et+5(1+uE+C qDr tkuE), 232 S. ABBAS,M.BENCHOHRA AND A. CABADA gk(t,u)E⩽1+uE 3e5, and Q(u)E⩽1+uPC 3e5. Hence, the hypotheses (H3),(H4)are satisfied with φ 1= φ 2=p∗=2e−5Γq(1+r), and L=c∗=1 3e5. We assume, for instance, that the number of impulses m=3,and r=1 2.Then by simple computations, we can show that all conditions of Theorem 7are satisfied. consequently, problem (5.2) has at least one solution on [0,1]. 6. Conclusion In the present research, we have provided some existence results of solutions of implicit fractional q-difference equations with Caputo fractional derivative and non instantaneous impulses. The fixed-point approach was used with the concept of measure of noncompactness. Illustrative examples are presented. REFERENCES [1] S. ABBAS AND M. BENCHOHRA,Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput. 257 (2015), 190–198. [2] S. ABBAS,M.BENCHOHRA,J.R.GRAEF AND J. HENDERSON,Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018. [3] S. ABBAS,M.BENCHOHRA AND G. M. N’GU´ ER´ EKATA,Topics in Fractional Differential Equations, Springer, New York, 2012. [4] S. ABBAS,M.BENCHOHRA AND G. M. N’GU´ ER´ EKATA,Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015. [5] C.R.A DAMS,On the linear ordinary q-difference equation, Annals Math. 30 (1928), 195–205. [6] R. AGARWAL,Certain fractional q-integrals and q-derivatives, Proc. Camb. Philos. Soc. 66 (1969), 365–370. [7] R.P.A GARWAL,S.HRISTOVA,D.O’REGAN,Non-Instantaneous Impulses in Differential Equations, Springer, New York, 2017. [8] B. AHMAD,Boundary value problem for nonlinear third order q-difference equations, Electron. J. Differential Equations 2011 (2011), no. 94, pp 1–7. [9] B. AHMAD,S.K.NTOUYAS AND L. K. PURNARAS,Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations, Adv. Difference Equ. 2012, 2012:140. [10] J. C. ALV ´ AREZ,Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces, Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid 79 (1985), 53–66. [11] J. M. AYERBEE TOLEDANO,T.DOM´ INGUEZ BENAVIDES AND G. L´ OPEZ ACEDO,Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory, Advances and Applications, vol. 99, Birkh¨auser, Basel, Boston, Berlin, 1997. [12] L. BAI,J.J.NIETO,Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett. 73 (2017), 44–48. [13] L. BAI,J.J.NIETO,X.WANG,Variational approach to non-instantaneous impulsive nonlinear differential equations, J. Nonlinear Sci. Appl. 10 (2017), 2440–2448. Differ. Equ. Appl. 15, No. 3 (2023), 215–234. 233 [14] J. BANA` S AND K. GOEBEL,Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980. [15] M. BENCHOHRA,J.HENDERSON AND S. K. NTOUYAS,Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, vol. 2, New York, 2006. [16] F. BROWDER,On the convergence of successive approximations for nonlinear functional equations, Indag. Math. 30 (1968), 27–35. [17] L. BYSZEWSKI,Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494–505. [18] R. D. CARMICHAEL,The general theory of linear q-difference equations, American J. Math. 34 (1912), 147–168. [19] K. DENG,Exponential decay of solutions of semilinearparabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179 (1993), 630–637. [20] S. ETEMAD,S.K.NTOUYAS AND B. AHMAD,Existence theory for a fractional q-integro-difference equation with q-integral boundary conditions of different orders, Mathematics, 7659 (2019), 1–15. [21] J. R. GRAEF,J.HENDERSON AND A. OUAHAB,Impulsive Differential Inclusions. A Fixed Point Approch, De Gruyter, Berlin/Boston, 2013. [22] J. HENDERSON,C.TISDELL,Topological transversality and boundary value problems on time scales, J. Math. Anal. Appl. 289 (2004), 110–125. [23] E. HERN ´ ANDEZ AND D. O’REGAN,On a new class of abstract impulsive differential equations,Proc. Amer. Math. Soc. 141 (2013), 1641–1649. [24] V. KAC AND P. CHEUNG,Quantum Calculus, Springer, New York, 2002. [25] A. A. KILBAS,H.M.SR I VA S TAVA A N D J. J. TRUJILLO,Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006. [26] N. LALEDJ,A.SALIM,J.E.LAZREG,S.ABBAS,B.AHMAD AND M. BENCHOHRA,On implicit fractional q-difference equations: Analysis and stability, Math. Methods Appl. Sci. 45 (2022), no. 17, 10775–10797. [27] J. MATKOWSKI,Integrable solutions of functional equations, Dissertationes Math. 127 (1975), 1–68. [28] H. M ¨ ONCH,Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4(1980), 985–999. [29] M. PIERRI,D.O’REGAN,V.ROLNIK,Existence of solutions for semi-linear abstract differential equations with not instantaneous, Appl. Math. Comput. 219 (2013), 6743–6749. [30] P. M. RAJKOVIC,S.D.MARINKOVIC AND M. S. STANKOVIC,Fractional integrals and derivatives in q-calculus, Appl. Anal. Discrete Math., 1(2007), 311–323. [31] P. M. RAJKOVIC,S.D.MARINKOVIC AND M. S. STANKOVIC,On q-analogues of Caputo derivative and Mittag-Leffler function, Fract. Calc. Appl. Anal., 10 (2007), 359–373. [32] I. RUS,A.PETRUSEL,G.PETRUSEL,Fixed Point Theory, Cluj University Press, Cluj, 2008. [33] A. M. SAMOILENKO,N.A.PERESTYUK,Impulsive Differential Equations, World Scientific, Singapore, 1995. [34] V. E. TARASOV,Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010. [35] J. M. AYERBE TOLEDANO,T.DOMINGUEZ BENAVIDES AND G. LOPEZ ACEDO,Measures of Noncompactness in Metric Fixed Point Theory, Birkhauser, Basel, 1997. [36] J. WANG,A.G.IBRAHIM,D.O’REGAN,Topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions, J. Fixed Point Theory Appl. 20 (2018), no. 2, Art. 59, 25 pp. [37] J. WANG AND X. LI,Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput. 46 (2014), 321–334. [38] D. YANG,J.WANG,D.O’REGAN,A class of nonlinear non-instantaneous impulsive differential equations involving parameters and fractional order, Appl. Math. Comput. 321 (2018), 654–671. 234 S. ABBAS,M.BENCHOHRA AND A. CABADA [39] X. ZHANG,Y.LI,P.CHEN,Existence of extremal mild solutions for the initial value problem of evolution equations with non-instantaneous impulses, J. Fixed Point Theo. Appl. 19 (2017), 3013– 3027. [40] Y. ZHOU,Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. (Received December 5, 2022) Sa¨ıd Abbas Department of Electronics Tahar Moulay University of Sa¨ıda P.O. Box 138, EN-Nasr, 20000 Sa¨ıda, Algeria e-mail: [email protected] Mouffak Benchohra Laboratory of Mathematics Djillali Liabes University of Sidi Bel-Abb`es P.O. Box 89, Sidi Bel-Abb`es 22000, Algeria e-mail: [email protected] Alberto Cabada CITMAga, 15782, Santiago de Compostela, Galicia, Spain and Departamento de Estatistica An´alise Matem´atica e Optimizaci´on Facultade de Matem´aticas Universidade de Santiago de Compostela Santiago de Compostela, Spain e-mail: [email protected] Differential Equations & Applications www.ele-math.com [email protected]