CUBO, A Mathematical Journal Vol. 24, no. 02, pp. 273–289, August 2022 DOI: 10.56754/0719-0646.2402.0273 On existence results for hybrid ψ−Caputo multi-fractional differential equations with hybrid conditions Fouad Fredj 1 Hadda Hammouche 1, B 1Mathematics and Applied Sciences Laboratory, Ghardaia University, Ghardaia 47000, Algeria. fouadfredj05@gmail.com fredj.fouad@univ-ghardaia.dz h.hammouche@yahoo.fr B ABSTRACT In this paper, we study the existence and uniqueness results of a fractional hybrid boundary value problem with multi- ple fractional derivatives of ψ−Caputo with different orders. Using a useful generalization of Krasnoselskii’s fixed point theorem, we have established results of at least one solu- tion, while the uniqueness of solution is derived by Banach’s fixed point. The last section is devoted to an example that illustrates the applicability of our results. RESUMEN En este art́ıculo, estudiamos los resultados de existencia y unicidad de un problema de valor en la frontera fraccional h́ıbrido con múltiples derivadas fraccionarias de ψ−Caputo con diferentes órdenes. Usando una generalización útil del teorema del punto fijo de Krasnoselskii, establecemos resul- tados de al menos una solución, mientras que la unicidad de dicha solución se obtiene a partir del punto fijo de Banach. La última sección está dedicada a un ejemplo que ilustra la aplicabilidad de nuestros resultados. Keywords and Phrases: ψ−fractional derivative, fractional differential equation, hybrid conditions, fixed point, existence, uniqueness. 2020 AMS Mathematics Subject Classification: 34A08, 34A12. Accepted: 24 May, 2022 Received: 14 October, 2021 c©2022 F. Fredj et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ https://dx.doi.org/10.56754/0719-0646.2402.0273 https://orcid.org/0000-0001-6088-3210 mailto:h.hammouche@yahoo.fr https://orcid.org/0000-0002-5846-8365 mailto:fouadfredj05@gmail.com mailto:fredj.fouad@univ-ghardaia.dz mailto:h.hammouche@yahoo.fr 274 F. Fredj & H. Hammouche CUBO 24, 2 (2022) 1 Introduction Fractional differential equations have received great attention of many researchers working in differ- ent disciplines of science and technology, especially, since they have found that certain thermal [3], electrochemical [4] and viscoelastic [16] systems are governed by fractional differential equations. Recently some publications show the importance of fractional differential equations in the math- ematical modeling of many real-world phenomena. For example ecological models [10], economic models [20], physics [12], fluid mechanics [21]. There are many studies on fractional differential equations with distinct kinds of fractional derivatives in the literature, such as Riemann-Liouville fractional derivative, Caputo fractional derivative, and Grunwald Letnikov fractional derivative, etc. For example, see [11, 14, 15]. Very recently, a new kind of fractional derivative the ψ−Caputo’s derivative, was introduced by Almeida in [1], the main advantage of this derivative is the freedom of choices of the kernels of the derivative by choosing different functions ψ, which gives us some well known fractional derivatives such Caputo, Caputo-Erdelyi-Koper and Caputo Hadamard deriva- tive. For more details on the ψ−Caputo and fractional differential equation involving ψ−Caputo, we refer the reader to a series of papers [1, 2, 7] and the references cited therein. Nowadays, many researchers have shown the interest of quadratic perturbations of nonlinear dif- ferential equations, these kind of differential equations are known under the name of hybrid dif- ferential equations. Some recent works regarding hybrid differential equations can be found in [8, 13, 17, 23] and the references cited therein. Dhage and Lakshmikantham [6] discussed the existence and uniqueness theorems of the solution to the ordinary first-order hybrid differential equation with perturbation of the first type          d dt ( u(t) g(t,u(t)) ) = f(t,u(t)), a.e. t ∈ [t0, t0 + T ], u(t0) = u0, u0 ∈ R, where t0,T ∈ R with T > 0, g : [t0, t0 +T ]×R → R\{0} and f : [t0, t0 +T ]×R → R are continuous functions. By using the fixed point theorem in Banach algebra, the authors obtained the existence results. In [9], Dong et al., established the existence and the uniqueness of solutions for the following implicit fractional differential equation        cDpu(t) = f(t,u(t),c Dpu(t)), t ∈ J := [0,T ], 0 < p ≤ 1, u(0) = u0, where cDp is the Caputo fractional derivative, f : [0,T ]×R×R → R is a given continuous function. Sitho et al. [17] studied existence results for the initial value problems of hybrid fractional sequen- CUBO 24, 2 (2022) Hybrid ψ−Caputo multi-fractional differential equation 275 tial integro-differential equations:                      Dp       Dqu(t) − m ∑ i=1 Iηigi(t,u(t)) h(t,x(t))       = f(t,u(t),Iγx(t)), t ∈ J, u(0) = 0, Dqu(0) = 0, where Dp, Dq denotes the Riemann-Liouville fractional derivative of order p, q respectively and 0 < p,q ≤ 1, Iηi is the Riemann-Liouville fractional integral of order ηi > 0, h ∈ C(J ×R,R\{0}), f ∈ C(J × R2,R) and gi ∈ C(J × R,R) with gi(0,0) = 0, i = 1, . . . ,m. In 2019, Derbazi et al. [8] proved the existence of solutions for the fractional hybrid boundary value problem cDp [ u(t) − g(t,u(t)) h(t,u(t)) ] = f(t,u(t)), t ∈ J, with the fractional hybrid boundary value conditions                  a1 [ u(t) − g(t,u(t)) h(t,u(t)) ] t=0 + b1 [ u(t) − g(t,u(t)) h(t,u(t)) ] t=T = υ1, a2 cDδ [ u(t) − g(t,u(t)) h(t,u(t)) ] t=ξ + b2 cDδ [ u(t) − g(t,u(t)) h(t,u(t)) ] t=T = υ2 , ξ ∈ J, where 1 < p ≤ 2, 0 < δ ≤ 1, ξ ∈ J and a1,a2,b1,b2,υ1,υ2 are real constants. Moreover, two fractional derivatives of Caputo type appeared in the above problem. Motivated by these works, we mainly investigate the existence and uniqueness of solutions for a class of hybrid differential equations of arbitrary fractional order of the form cDp;ψ       cDq;ψu(t) − m ∑ i=1 Iηi;ψgi(t,u(t)) h(t,u(t))       = f       t,u(t),c Dp;ψ       cDq;ψu(t) − m ∑ i=1 I ηi;ψgi(t,u(t)) h(t,u(t))             , t ∈ J, (1.1) 276 F. Fredj & H. Hammouche CUBO 24, 2 (2022) endowed with the hybrid fractional integral boundary conditions                                                                      u(0) = 0, cDq;ψu(0) = 0, a1       cDq;ψu(t) − m ∑ i=1 Iηi;ψgi(t,u(t)) h(t,u(t))       t=0 + b1       cDq;ψu(t) − m ∑ i=1 Iηi;ψgi(t,u(t)) h(t,u(t))       t=T = υ1, a2 cD δ;ψ       cDq;ψu(t) − m ∑ i=1 Iηi;ψgi(t,u(t)) h(t,u(t))       t=ξ + b2 cD δ;ψ       cDq;ψu(t) − m ∑ i=1 Iηi;ψgi(t,u(t)) h(t,u(t))       t=T = υ2 , ξ ∈ J, (1.2) where J := [0,T ], Dp;ψ, Dq;ψ and Dδ;ψ denote the ψ−Caputo fractional derivative of order 2 < p ≤ 3 and 0 < q,δ ≤ 1 respectively, Iηi;ψ is the ψ−Riemann-Liouville fractional integral of order ηi > 0, h ∈ C(J×R,R\{0}), f ∈ C(J ×R2,R) and gi ∈ C(J×R,R) with gi(0,0) = 0, i = 1, . . . ,m, a1, a2,b1, b2, υ1, υ2 are real constants such that b1 6= 0 and 2 ( a2Ψ 2−δ 0 (ξ) + b2Ψ 2−δ 0 (T) ) − Ψ10(T)(2 − δ) ( a2Ψ 1−δ 0 (ξ) + b2Ψ 1−δ 0 (T) ) 6= 0. The rest of the paper is arranged as follows. Section 2 gives some background material needed in this paper, such as fractional differential equations and fixed point theorems. Section 3 treats the main results concerning the existence and uniqueness results of the solution for the given problem (1.1)-(1.2) by employing hybrid fixed point theorem for a sum of two operators in Banach algebra space and Banach’s fixed point. In the last section, an example is presented to illustrate our results. 2 Preliminaries In this section, we introduce some preliminaries and lemmas that will be used throughout this paper. We will prove an auxiliary lemma, which plays a key role in defining a fixed point problem associated with the given problem. Let ψ : J → R an increasing function satisfying ψ′(t) 6= 0 for all t ∈ J. For the sake of simplicity, we set Ψr0(t) = (ψ(t) − ψ(0))r. Definition 2.1 ([2]). The ψ−Riemann-Liouville fractional integral of order (p > 0) of an integrable CUBO 24, 2 (2022) Hybrid ψ−Caputo multi-fractional differential equation 277 function g : [0,∞) → R is defined by Ip;ψg(t) = 1 Γ(p) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))p−1g(s)ds, 0 < s < t. Definition 2.2 ([2]). The ψ−Caputo fractional derivative of order p (n − 1 < p < n ∈ N) of a function g ∈ Cn[0,∞) is defined by cDp;ψg(t) = 1 Γ(p − n) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))p−n−1Dnψ g(s)ds, 0 < s < t, where n = [p] + 1 and Dnψ = ( 1 ψ′(t) d dt )n . In case, if 2 < p ≤ 3, we have cDp;ψg(t) = 1 Γ(p − 3) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))p−4D3ψg(s)ds, 0 < s < t. Lemma 2.3 ([2]). Let p > 0. The following hold • If g ∈ C(J,R), then cDp;ψIp;ψg(t) = g(t), t ∈ J. • If g ∈ Cn(J,R), n − 1 < p < n, then I p;ψc D p;ψ g(t) = g(t) − n−1 ∑ k=0 ckΨ k 0(t), t ∈ J, where ck = Dkψg(0) k! . Lemma 2.4. Let 2 < p < 3, 0 < q < 1. For any functions F ∈ C(J,R), H ∈ C(J,R \ {0}) and Gi ∈ C(J,R) with Gi(0) = 0, i = 1, . . . ,m, the following linear fractional boundary value problem Dp;ψ       cDq;ψu(t) − m ∑ i=1 Iηi;ψGi(t) H(t)       = F(t), 2 < p ≤ 3, 0 < q ≤ 1, t ∈ J, (2.1) 278 F. Fredj & H. Hammouche CUBO 24, 2 (2022) supplemented with the following conditions                                                                      u(0) = 0, cDq;ψu(0) = 0, a1       cDq;ψu(t) − m ∑ i=1 Iηi;ψGi(t) H(t)       t=0 + b1       cDq;ψu(t) − m ∑ i=1 Iηi;ψGi(t) H(t)       t=T = υ1, a2 cD δ;ψ       cDq;ψu(t) − m ∑ i=1 Iηi;ψGi(t) H(t)       t=ξ + b2 cD δ;ψ       cDq;ψu(t) − m ∑ i=1 I ηi;ψGi(t) H(t)       t=T = υ2, ξ ∈ J, (2.2) has a unique solution, which is given by u(t) = Iq;ψ ( H(s)Ip;ψF(s) ) (t) + m ∑ i=1 I ηi+q;ψGi(s)(t) + Iq;ψ ( H(s) ( Ψ10(s)Ω3 − Ψ20(s)Ω2 )( υ1 b1 − Ip;ψF(s) ) ) (t) + Ω1 ( υ2 − a2Ip−δ;ψF(ξ) − b2Ip−δ;ψF(T) ) Iq;ψ ( H(s) ( Ψ20(s) − Ψ10(T)Ψ10(s) ) ) (t), (2.3) where Ω1 = Γ(3 − δ) 2 ( a2Ψ 2−δ 0 (ξ) + b2Ψ 2−δ 0 (T) ) − Ψ1 0 (T)(2 − δ) ( a2Ψ 1−δ 0 (ξ) + b2Ψ 1−δ 0 (T) ), Ω2 = a2Ψ 1−δ 0 (ξ) + b2Ψ 1−δ 0 (T) Γ(2 − δ)Ω1 , Ω3 = 1 + Ω2Ψ 1 0(T). Proof. Applying the ψ−Caputo fractional integral of order p to both sides of equation in (2.1) and using Lemma 2.3, we get cDq;ψu(t) − m ∑ i=1 I ηi;ψGi(t) H(t) = Ip;ψF(t) + c0 + c1Ψ 1 0(t) + c2Ψ 2 0(t), (2.4) where c0,c1,c2 ∈ R . Next, applying the ψ−Caputo fractional integral of order q to both sides (2.4), we get CUBO 24, 2 (2022) Hybrid ψ−Caputo multi-fractional differential equation 279 u(t) = Iq;ψ ( H(s)Ip;ψF(s) ) (t) + m ∑ i=1 Iηi+q;ψGi(s)(t) + Iq;ψ ( H(s) ( c0 + c1Ψ 1 0(s) + c2Ψ 2 0(s) )) (t) + c3, c3 ∈ R. (2.5) With the help of conditions u(0) = 0 and cDq;ψu(0) = 0, we find, c3 = 0 and c0 = 0 respectively. Applying the boundary conditions (2.2), and from (2.4), we obtain c1Ψ 1 0(T) + c2Ψ 2 0(T) = υ1 b1 − Ip;ψF(T), and c1 Γ(2 − δ) ( a2Ψ 1−δ 0 (ξ) + b2Ψ 1−δ 0 (T) ) + 2c2 Γ(3 − δ) ( a2Ψ 2−δ 0 (ξ) + b2Ψ 2−δ 0 (T) ) = υ2 − a2Ip−δ;ψF(ξ) − b2Ip−δ;ψF(T). Solving the resulting equations for c1 and c2, we find that c1 = ( υ1 b1 − Ip;ψF(T) ) Ω3 − ( υ2 − a2Ip−δ;ψF(ξ) − b2Ip−δ;ψF(T) ) Ω1Ψ 1 0(T), c2 = ( υ2 − a2Ip−δ;ψF(ξ) − b2Ip−δ;ψF(T) ) Ω1 − ( υ1 b1 − Ip;ψF(T) ) Ω2. Inserting c1 and c2 in (2.5), which leads to the solution system (2.3). Let E = C(J,R) be the Banach space of continuous real-valued functions defined on J. We define in E a norm ‖ · ‖ by ‖u‖ = sup t∈J |u(t)|, and a multiplication by (uv)(t) = u(t)v(t), ∀t ∈ J. Clearly E is a Banach algebra with above defined supremum norm and multiplication. Lemma 2.5 ([5]). Let S be a nonempty, convex, closed, and bounded set such that S ⊆ E, and let A : E → E and B : S → E be two operators which satisfy the following: (1) A is contraction, (2) B is completely continuous, and (3) u = Au + Bv, for all v ∈ S ⇒ u ∈ S. Then the operator equation u = Au + Bu has at least one solution in S. Theorem 2.6 ([18]). Let S be a non-empty closed convex subset of a Banach space E, then any contraction mapping A of S into itself has a unique fixed point. 280 F. Fredj & H. Hammouche CUBO 24, 2 (2022) 3 Main result In this section, we derive conditions for the existence and uniqueness of a solution for the problem (1.1)-(1.2). The following assumptions are necessary in obtaining the main results. (H1) The functions h ∈ C(J × R,R \ {0}), and f ∈ C(J × R2,R) are continuous, and there exist bounded functions L,M : J → [0,∞), such that |h(t,u(t)) − h(t,v(t))| ≤ L(t)|u(t) − v(t)|, and |f(t,u(t),v(t)) − f(t,u(t),v(t))| ≤ M(t) ( |u(t) − u(t)| + |v(t) − v(t)| ) , for t ∈ J and u,v,u,v ∈ R. (H2) There exist functions ϑ,χ,ϕi ∈ C(J, [0,∞)) such that |f(t,u(t),v(t))| ≤ ϑ(t) for each t,u ∈ J × R, |h(t,u(t))| ≤ χ(t) for each t,u ∈ J × R, |gi(t,u(t))| ≤ ϕi(t) for each t,u ∈ J × R, i = 1, . . . ,m, for t ∈ J and u ∈ R. (H3) The functions gi ∈ C(J × R,R) are continuous, and there exist bounded functions Ki : J → (0,∞), such that |gi(t,u(t)) − gi(t,v(t))| ≤ Ki(t)|u(t) − v(t)|. We set L∗ = supt∈J |L(t)|, M∗ = supt∈J |M(t)|, χ∗ = supt∈J |χ(t)|, ϑ∗ = supt∈J |ϑ(t)| and ϕ∗i = supt∈J |ϕi(t)|, K∗i = supt∈J |Ki(t)|, i = 1,2, . . . ,m. 3.1 Existence of solutions In this subsection, we prove the existence of a solution for the problem (1.1)–(1.2) by applying a generalization of Krasnoselskii’s fixed point theorem. Theorem 3.1. Assume that hypotheses (H1)–(H2) hold and if Λ = Ψ p 0(T) Γ(p + 1) ( χ∗M∗ 1 − M∗ + ϑ∗L∗ )( Ψ q 0(T) Γ(q + 1) + |Ω3|Ψq+10 (T) Γ(q + 2) + 2|Ω2|Ψq+20 (T) Γ(q + 3) ) + |Ω1|(q + 4) Ψ q+2 0 (T) Γ(q + 3) ( |υ2|L∗ + |a2|Ψp−δ0 (ξ) + |b2|Ψ p−δ 0 (T) Γ(p − δ + 1) × ( χ∗M∗ 1 − M∗ + ϑ∗L∗ )) + |υ1|L∗ |b1| ( |Ω3|Ψq+10 (T) Γ(q + 2) + 2|Ω2|Ψq+20 (T) Γ(q + 3) ) < 1. (3.1) Then the problem (1.1)–(1.2) has at least one solution on J. CUBO 24, 2 (2022) Hybrid ψ−Caputo multi-fractional differential equation 281 Proof. First, we choose r > 0 such that r ≥χ∗ϑ∗ Ψ p+q 0 (T) Γ(p + 1)Γ(q + 1) + χ∗ ( |Ω3|Ψq+10 (T) Γ(q + 2) + 2|Ω2|Ψq+20 (T) Γ(q + 3) ) (|υ1| |b1| + Ψ p 0 (T) Γ(p + 1) ϑ ∗ ) + χ∗|Ω1| (q + 4)Ψ q+2 0 (T) Γ(q + 3) ( |υ2| + ϑ∗ |a2|Ψp−δ0 (ξ) + |b2|Ψ p−δ 0 (T) Γ(p − δ + 1) ) + n ∑ i=1 ϕ∗i Ψ ηi+q 0 (T) Γ(ηi + q + 1) . Set Br = {u ∈ E : ‖u‖ ≤ r}. Clearly Br is a closed, convex and bounded subset of the Banach space E. Let u(t) be a solution of the problem (1.1)–(1.2). Define Fu(t) := f       t,u(t),c Dp;ψ       cDq;ψu(t) − m ∑ i=1 I ηi;ψgi(t,u(t)) h(t,u(t))             . Then cDp;ψ       cDq;ψu(t) − m ∑ i=1 I ηi;ψgi(t,u(t)) h(t,u(t))       = Fu(t), supplemented with the conditions (1.2), then by Lemma 2.4, we get u(t) = Iq;ψ ( h(s,u(s))Ip;ψFu(s) ) (t) + m ∑ i=1 I ηi+q;ψgi(s,u(s))(t)+ + Iq;ψ ( h(s,u(s)) ( Ψ10(s)Ω3 − Ψ20(s)Ω2 )( υ1 b1 − Ip;ψFu(s) ) ) (t) + Ω1 ( υ2 − a2Ip−δ;ψFu(ξ) − b2Ip−δ;ψFu(T) ) Iq;ψ ( h(s,u(s)) ( Ψ20(s) − Ψ10(T)Ψ10(s) ) ) (t), Let us define three operators Cp,Cp−δ : E → E and D : E → E such that Cpu(t) = 1 Γ(p) ∫ t 0 ψ ′(s)(ψ(t) − ψ(s))p−1Fu(s)ds, t ∈ J, Cp−δu(t) = 1 Γ(p − δ) ∫ t 0 ψ ′(s)(ψ(t) − ψ(s))p−δ−1Fu(s)ds, t ∈ J, and Du(t) = h(t,u(t)), t ∈ J. Then, using assumptions (H1)–(H2) , we have |Cpu(t) − Cpv(t)| ≤ 1 Γ(p) ∫ t 0 ψ ′(s)(ψ(t) − ψ(s))p−1|Fu(s) − Fv(s)|ds, (3.2) 282 F. Fredj & H. Hammouche CUBO 24, 2 (2022) and |Fu(t) − Fv(t)| ≤ |f(t,u(t),Fu(t)) − f(t,v(t),Fv(t))| ≤ M(t) ( |u(t) − v(t)| + |Fu(t) − Fv(t)| ) ≤ M(t) 1 − M(t) ‖u(·) − v(·)‖. (3.3) By replacing (3.3) in (3.2), we obtain |Cpu(t) − Cpv(t)| ≤ M∗Ψ p 0(T) (1 − M∗)Γ(p + 1) ‖u(·) − v(·)‖, and |Du(t) − Dv(t)| ≤ L∗‖u(·) − v(·)‖, |Cpu(t)| ≤ Ψ p 0(T) Γ(p + 1) ϑ∗, and |Du(t)| ≤ χ∗. Now we define two more operators A : E → E and B : Br → E such that Au(t) = Iq;ψ ( Du(s)Cpu(s) ) (t) + Iq;ψ ( Du(s) ( Ψ10(s)Ω3 − Ψ20(s)Ω2 )( υ1 b1 − Cpu(s) ) ) (t) + Ω1 ( υ2 − a2Cp−δu(ξ) − b2Cp−δu(T) ) Iq;ψ ( Du(s) ( Ψ20(s) − Ψ10(T)Ψ10(s) ) ) (t), and Bu(t) = m ∑ i=1 Iηi+q;ψgi(s,u(s))(t). We need to show that the two operators A and B satisfy all conditions of Lemma 2.5. This can be achieved in the following steps. Step 1. First we show that A is a contraction mapping. Let u(t),v(t) ∈ Br, then we have |Au(t) − Av(t)| ≤ Iq;ψ ( ∣ ∣Du(s)Cpu(s) − Dv(s)Cpv(s) ∣ ∣ ( 1 + ∣ ∣Ψ10(s)Ω3 − Ψ20(s)Ω2 ∣ ∣ ) ) (t) + Iq;ψ ( |υ1| |b1| ∣ ∣Ψ10(s)Ω3 − Ψ20(s)Ω2 ∣ ∣ ∣ ∣Du(s) − Dv(s) ∣ ∣ ) (t) + |Ω1|Iq;ψ ( ∣ ∣Ψ20(s) − Ψ10(T)Ψ10(s) ∣ ∣ ( |υ2| ∣ ∣Du(s) − Dv(s) ∣ ∣ + |a2| ∣ ∣Du(s)Cp−δu(ξ) − Dv(s)Cp−δv(ξ) ∣ ∣ + |b2| ∣ ∣Du(s)Cp−δu(T) − Dv(s)Cp−δv(T) ∣ ∣ ) ) (t) CUBO 24, 2 (2022) Hybrid ψ−Caputo multi-fractional differential equation 283 ≤ Iq;ψ ( ( ∣ ∣Du(s) ∣ ∣ ∣ ∣Cpu(s) − Cpv(s) ∣ ∣ + ∣ ∣Cpv(s)||Du(s) − Dv(s) ∣ ∣ ) × ( 1 + ∣ ∣Ψ10(s)Ω3 − Ψ20(s)Ω2 ∣ ∣ ) ) (t) + Iq;ψ ( |υ1| |b1| ∣ ∣Ψ10(s)Ω3 − Ψ20(s)Ω2 ∣ ∣ ∣ ∣Du(s) − Dv(s) ∣ ∣ ) (t) + |Ω1|Iq;ψ ( ∣ ∣Ψ20(s) − Ψ10(T)Ψ10(s) ∣ ∣ ( ∣ ∣Du(s) − Dv(s) ∣ ∣ ( |υ2| + |a2| ∣ ∣Cp−δv(ξ) ∣ ∣ + |b2| ∣ ∣Cp−δv(T) ∣ ∣ ) + ∣ ∣Du(s) ∣ ∣ ( |a2| ∣ ∣Cp−δu(ξ) − Cp−δv(ξ) ∣ ∣ + |b2| ∣ ∣Cp−δu(T) − Cp−δv(T) ∣ ∣ ) )) (t) Using the hypotheses (H1)–(H2) and taking the supremum over t, we get ‖Au(·) − Av(·)‖ ≤ Λ‖u(·) − v(·)‖. (3.4) Therefore from (3.1), we conclude that the operator A is a contraction mapping. Step 2. Next, we prove that the operator B satisfies condition (2) of Lemma 2.5, that is, the operator B is compact and continuous on Br. Therefore first, we show that the operator B is continuous on Br. Let un(t) be a sequence of functions in Br converging to a function u(t) ∈ Br. Then, by the Lebesgue dominant convergence theorem, for all t ∈ J, we have lim n→∞ Bun(t) = lim n→∞ m ∑ i=1 1 Γ(ηi + q) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))ηi+q−1gi(s,un(s))ds = m ∑ i=1 1 Γ(ηi + q) ∫ t 0 ψ ′(s)(ψ(t) − ψ(s))ηi+q−1 lim n→∞ gi(s,un(s))ds = m ∑ i=1 1 Γ(ηi + q) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))ηi+q−1gi(s,u(s))ds. Hence limn→∞ Bun(t) = Bu(t). Thus B is a continuous operator on Br. Further, we show that the operator B is uniformly bounded on Br. For any u ∈ Br, we have ‖Bu(·)‖ ≤ sup t∈J { m ∑ i=1 1 Γ(ηi + q) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))ηi+q−1|gi(s,u(s))|ds } ≤ m ∑ i=1 Ψ ηi+q 0 (T) Γ(ηi + q + 1) ϕ∗i ≤ r. Therefore Bu(t) ≤ r, for all t ∈ J, which shows that B is uniformly bounded on Br. Now, we show that the operator B is equi-continuous. Let t1, t2 ∈ J with t1 > t2. Then for any 284 F. Fredj & H. Hammouche CUBO 24, 2 (2022) u(t) ∈ Br, we have |Bu(t1) − Bu(t2)| ≤ m ∑ i=1 1 Γ(ηi + q) ∣ ∣ ∣ ∣ ∫ t2 0 ψ′(s) ( (ψ(t1) − ψ(s))ηi+q−1 − (ψ(t2) − ψ(s))ηi+q−1 ) gi(s,u(s))ds ∣ ∣ ∣ ∣ + m ∑ i=1 1 Γ(ηi + q) ∣ ∣ ∣ ∣ ∫ t1 t2 ψ ′(s)(ψ(t1) − ψ(s))ηi+q−1gi(s,u(s))ds ∣ ∣ ∣ ∣ ≤ m ∑ i=1 ϕ∗i Γ(ηi + q + 1) ( 2|ψ(t1) − ψ(t2)|ηi+q + ∣ ∣Ψ ηi+q 0 (t2) − Ψ ηi+q 0 (t1) ∣ ∣ ) . As t2 → t1, so the right-hand side tends to zero. Thus B is equi-continuous. Therefore, it follows from the Arzelá–Ascoli theorem that B is a compact operator on Br. We conclude that B is completely continuous. Step 3. It remains to verify the condition (3) of Lemma 2.5. For any v ∈ Br, we have ‖u(·)‖ = ‖Au(·) + Bv(·)‖ ≤ ‖Au(·)‖ + ‖Bv(·)‖ ≤ sup t∈J {∣ ∣ ∣ ∣ Iq;ψ ( Du(s)Cpu(s) ) (t) + Iq;ψ ( Du(s) ( Ψ10(s)Ω3 − Ψ20(s)Ω2 )( υ1 b1 − Cpu(s) ) ) (t) + Ω1 ( υ2 − a2Cp−δu(ξ) − b2Cp−δu(T) ) Iq;ψ ( Du(s) ( Ψ20(s) − Ψ10(T)Ψ10(s) ) ) (t) ∣ ∣ ∣ ∣ } + sup t∈J { m ∑ i=1 Iηi+q;ψ ∣ ∣gi(s,v(s)) ∣ ∣(t) } ≤ χ∗ϑ∗ Ψ p+q 0 (T) Γ(p + 1)Γ(q + 1) + χ∗ ( |Ω3|Ψq+10 (T) Γ(q + 2) + 2|Ω2|Ψq+20 (T) Γ(q + 3) )( |υ1| |b1| + Ψ p 0(T) Γ(p + 1) ϑ∗ ) + χ∗|Ω1| (q + 4)Ψ q+2 0 (T) Γ(q + 3) ( |υ2| + ϑ∗ |a2|Ψp−δ0 (ξ) + |b2|Ψ p−δ 0 (T) Γ(p − δ + 1) ) + n ∑ i=1 ϕ∗i Ψ ηi+q 0 (T) Γ(ηi + q + 1) . Which implies, from the choice of r that ‖u‖ ≤ r, and so u ∈ Br. Hence all conditions of Lemma 2.5 are satisfied. Therefore, the operator equation u(t) = Au(t) + Bu(t) has at least one solution in Br. Consequently, the problem (1.1)–(1.2) has at least on solution on J. Thus the proof is completed. 3.2 Uniqueness of solutions In the next result, we apply the Banach fixed theorem to prove the uniqueness of solutions for the problem (1.1)–(1.2). CUBO 24, 2 (2022) Hybrid ψ−Caputo multi-fractional differential equation 285 Theorem 3.2. Assume that the hypotheses(H1)–(H3) together with the inequality Λ + m ∑ i=1 K∗i Ψ ηi+q 0 (T) Γ(ηi + q) < 1. are satisfied, then the problem (1.1)–(1.2) has an unique solution. Proof. According to Lemma 2.4, we define the operator Q : E → E by Qu(t) = Au(t) + Bu(t). First, we show that Q(Br) ⊂ Br. As in the previous proof (step 3) of Theorem 3.1, we can obtain for u ∈ Br and t ∈ J ‖Qu(·)‖ ≤ χ∗ϑ∗ Ψ p+q 0 (T) Γ(p + 1)Γ(q + 1) + χ∗ ( |Ω3|Ψq+10 (T) Γ(q + 2) + 2|Ω2|Ψq+20 (T) Γ(q + 3) )( |υ1| |b1| + Ψ p 0(T) Γ(p + 1) ϑ∗ ) + χ∗|Ω1| (q + 4)Ψ q+2 0 (T) Γ(q + 3) ( |υ2| + ϑ∗ |a2|Ψp−δ0 (ξ) + |b2|Ψ p−δ 0 (T) Γ(p − δ + 1) ) + n ∑ i=1 ϕ ∗ i Ψ ηi+q 0 (T) Γ(ηi + q + 1) ≤ r. This shows that Q(Br) ⊂ Br. Next, we prove that the operator Q is a contractive operator. For u,v ∈ Br ‖Qu(·) − Qv(·)‖ ≤ ‖Au(·) − Av(·)‖ + ‖Bu(·) − Bv(·)‖, and ‖Bu(·) − Bv(·)‖ ≤ sup t∈J { m ∑ i=1 1 Γ(ηi + q) ∫ t 0 ψ′(s)(ψ(t) − ψ(s))ηi+q−1 ∣ ∣gi(s,u(s)) − gi(s,v(s)) ∣ ∣ds } ≤ m ∑ i=1 K∗i Ψ ηi+q 0 (T) Γ(ηi + q + 1) ‖u(·) − v(·)‖. (3.5) From (3.4) and (3.5), we get ‖Qu(·) − Qv(·)‖ ≤ ( Λ + m ∑ i=1 K∗i Ψ ηi+q 0 (T) Γ(ηi + q + 1) ) ‖u(·) − v(·)‖. This implies that the operator Q is a contractive operator. Consequently, by Theorem 3.2, we conclude that Q has an unique fixed point, which is a solution of the problem (1.1)–(1.2). This completes the proof. 286 F. Fredj & H. Hammouche CUBO 24, 2 (2022) 4 Example Consider the following fractional hybrid differential equation                                                                        cD 5 2 ;t       cD 3 4 ;tu(t) − 3 ∑ i=1 I ηi;tgi(t,u(t)) h(t,u(t))       = f       t,u(t),c D 5 2 ;t       cD 3 4 ;tu(t) − 3 ∑ i=1 I ηi;tgi(t,u(t)) h(t,u(t))             , u(0) = 0, cD 3 4 ;tu(0) = 0, 2       cD 3 4 ;tu(t) − 3 ∑ i=1 Iηi;tgi(t,u(t)) h(t,u(t))       t=0 + 2 7       cD 3 4 ;tu(t) − 3 ∑ i=1 Iηi;tgi(t,u(t)) h(t,u(t))       t=1 = 7 2 , 7 13 cD 4 5 ;t       cD 3 4 ;tu(t) − 3 ∑ i=1 Iηi;tgi(t,u(t)) h(t,u(t))       t= 4 5 + 1 2 cD 4 5 ;t       cD 3 4 ;tu(t) − 3 ∑ i=1 Iηi;tgi(t,u(t)) h(t,u(t))       t=1 = 2 , (4.1) where 3 ∑ i=1 I ηi;tgi(t,u(t))(s) = I 1 3 ;t ( sin2 x(s) 8(s + 1)2 ) (t) + I 3 2 ;t ( 1 2π √ 81 + s2 |x(s)| 2 + |x(s)| ) (t) + I 7 3 ;t ( sinx(s) 3π √ 49 + s2 ) (t), h(t,u(t)) = e−3t cosu(t) 2t + 40 + 1 80 (t3 + 1), and f       t,u(t),c D 5 2 ;t       cD 3 4 ;tu(t) − 3 ∑ i=1 Iηi;tgi(t,u(t)) h(t,u(t))             = 1 60 √ t + 81       |x(t)| 3 + |x(t)| − arctan       cD 5 2 ;t       cD 3 4 ;tu(t) − 3 ∑ i=1 I ηi;tgi(t,u(t)) h(t,u(t))                   . CUBO 24, 2 (2022) Hybrid ψ−Caputo multi-fractional differential equation 287 Here T = 1,p = 5 2 ,q = 3 4 ,m = 3,η1 = 1 3 ,η2 = 3 2 ,η3 = 7 3 ,δ = 4 5 ,a1 = 2,a2 = 7 13 ,b1 = 2 7 , b2 = 1 2 ,υ1 = 7 2 ,υ2 = 2,ξ = 4 5 , g1 = sin2 x(t) 8(t + 1)2 , g2 = 1 2π √ 81 + t2 |x(t)| 2 + |x(t)| , g3 = sinx(t) 3π √ 49 + t2 . The hypotheses (H1), (H2) and (H4) are satisfied with the following positives functions: L(t) = e−3 2t + 40 , M(t) = ϑ(t) = 1 60 √ t + 81 , ϕ1(t) = K1(t) = 1 8(t + 1)2 , ϕ2(t) = K2(t) = 1 2π √ 81 + t2 , ϕ3(t) = K3(t) = 1 3π √ 49 + t2 and χ(t) = e−3 2t + 40 + 1 80 (t3 +1), which gives us L∗ = 1 40 , M∗ = ϑ∗ = 1 540 ,χ∗ = 3 80 , ϕ∗1 = K ∗ 1 = 1 8 ,ϕ∗2 = K ∗ 2 = 1 18π , ϕ∗3 = K ∗ 3 = 1 21π . With the given data, we find that Ω1 ≃ 1.81820508, Ω2 ≃ 0.60797139, Ω3 ≃ 1.60797139, and Λ ≃ 0.48820986 < 1. By Theorem 3.1, the problem (4.1) has a solution on [0,1]. Also, we have Λ + 3 ∑ i=1 K∗i Ψ ηi+q 0 (1) Γ(ηi + 7 4 ) ≃ 0.61782704 < 1. 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Zhao, S. Sun, Z. Han and Q. Li, “Theory of fractional hybrid differential equations”, Comput. Math. Appl., vol. 62, no. 3, pp. 1312–1324, 2011. Introduction Preliminaries Main result Existence of solutions Uniqueness of solutions Example Conclusion