CUBO, A Mathematical Journal Vol. 23, no. 02, pp. 225–237, August 2021 DOI: 10.4067/S0719-06462021000200225 Existence results for a multipoint boundary value problem of nonlinear sequential Hadamard fractional differential equations Bashir Ahmad 1 Amjad F. Albideewi 1 Sotiris K. Ntouyas 2,1 Ahmed Alsaedi 1 1 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University,Jeddah 21589, Saudi Arabia. bashirahmad qau@yahoo.com amjad.f.b@hotmail.com aalsaedi@hotmail.com 2 University of Ioannina, 451 10 Ioannina, Greece. sntouyas@uoi.gr ABSTRACT In this paper, existence and uniqueness results are estab- lished for a nonlinear sequential Hadamard fractional dif- ferential equation with multi-point boundary conditions, via Banach and Krasnosel’skĭı’s fixed point theorems and Leray- Schauder nonlinear alternative. An example illustrating the existence of a unique solution is also constructed. RESUMEN En este art́ıculo se establecen resultados de existencia y uni- cidad para una ecuación diferencial fraccional nolineal se- cuencial de Hadamard con condiciones de borde multi-punto, a través de teoremas de punto fijo de Banach y Krasnosel’skĭı y la alternativa nolineal de Leray-Schauder. Se construye un ejemplo ilustrando la existencia de una única solución. Keywords and Phrases: Hadamard fractional integral; Hadamard fractional derivative; multi-point boundary conditions; existence; fixed point theorems. 2020 AMS Mathematics Subject Classification: 34A08, 34A12, 34B15. Accepted: 12 April, 2021 Received: 05 October, 2020 c©2021 B. Ahmad et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000200225 http://orcid.org/0000-0001-5350-2977 http://orcid.org/0000-0001-5793-6756 http://orcid.org/0000-0002-7695-2118 http://orcid.org/0000-0003-3452-8922 226 Bashir Ahmad, Amjad F. Albideewi, Sotiris K. Ntouyas & Ahmed Alsaedi CUBO 23, 2 (2021) 1 Introduction Fractional calculus has been extensively developed during the last few decades as the techniques of this branch of mathematics considerably improved the mathematical modeling of many scien- tific phenomena, for instance, see [16, 17]. In particular, fractional-order nonlocal boundary value problems are found to be of significant interest for many researchers. Much of the literature on this class of problems is based on Riemann-Liouville or Liouville-Caputo type fractional order differen- tial equations. For details, we refer the reader to some recent works [27] and the references cited therein. In addition to Riemann-Liouville and Caputo type derivatives, there is another kind of derivative, which contains logarithmic function of arbitrary exponent in its definition. This deriva- tive is known as Hadamard derivative [14] and its construction is invariant in relation to dilation and is quite suitable for the problems with semi-infinite domain. For example, Lamb-Bateman integral equation is the one containing Hadamard fractional derivatives of order 1/2 [8]. In [11], a modified Lamb-Bateman equation involving Hadamard derivative and fractional Hyper-Bessel- type operators was studied. One can find application details of Hadamard fractional differential equations in the articles [12, 20]. For some recent results on Hadamard type fractional differential equations, for instance, see [2, 4, 5, 10, 18, 19, 21, 22, 23, 25, 26]. In a recent monograph [3], one can find a detailed description of initial/boundary value problems and inequalities involving Hadamard fractional differential equations and inclusions. New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval were studied in [28]. In [6], the authors studied a coupled system of Caputo-Hadamard type sequential fractional differ- ential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. A Caputo-Hadamard fractional turbulent flow model was studied in [24]. However, the Hadamard-type fractional boundary value problems are not sufficiently studied in the mainstream literature. In this paper, motivated by aforementioned work on Hadamard fractional differential equations, we introduce and study a nonlocal multipoint boundary value problem involving a nonlinear sequential Hadamard fractional differential equation to enrich the related literature. Precisely, we investigate the existence criteria for the following problem:        ( HDα + λ HDα−1 ) x(t) = f(t, x(t)), 1 < α ≤ 2, 1 < t < T, x(1) = 0, x(T ) = m ∑ j=1 βj x(tj), (1.1) where HD(·) denotes the Hadamard fractional derivative of order α, f : [1, T ] × R → R is a continuous function, λ ∈ R+, tj, j = 1, 2, . . . , m, are given points with 1 ≤ t1 ≤ . . . ≤ tm < T , and βj are appropriate real numbers. An existence and uniqueness result is proved via Banach’s fixed point theorem and also two existence results are established by using Krasnosel’skĭı’s fixed point theorem and Leray-Schauder nonlinear alternative. CUBO 23, 2 (2021) Multipoint fractional sequential Hadamard BVP 227 The remaining part of the paper is structured as follows: In Section 2 we recall the related back- ground material and establish a lemma regarding a linear variant of the problem (1.1), useful to transform the problem (1.1) into an equivalent fixed point problem. Section 3 contains the main results for the problem (1.1). An example illustrating the existence and uniqueness result is also included. 2 Preliminaries We introduce notations and definitions of fractional calculus. Definition 2.1. ([3, 17]) The Hadamard fractional integral of order q ∈ C, R(q) > 0, for a function g ∈ Lp[a, b], 0 ≤ a ≤ t ≤ b ≤ ∞, is defined as I q a+ g(t) = 1 Γ(q) ∫ t a ( log t s )q−1 g(s) s ds, I q b− g(t) = 1 Γ(q) ∫ b t ( log s t )q−1 g(s) s ds. Definition 2.2. ([3, 17]) Let [a, b] ⊂ R, δ = t d dt and ACnδ [a, b] = {g : [a, b] → R : δn−1(g(t)) ∈ AC[a, b]}. The Hadamard derivative of fractional order q for a function g ∈ ACnδ [a, b] is defined as D q a+ g(t) = δn(I n−q a+ )(t) = 1 Γ(n − q) ( t d dt )n ∫ t a ( log t s )n−q−1 g(s) s ds, D q b− g(t) = (−δ)n(In−q b− )(t) = 1 Γ(n − q) ( −t d dt )n ∫ b t ( log s t )n−q−1 g(s) s ds, where n − 1 < q < n, n = [q] + 1 and [q] denotes the integer part of the real number q and log(·) = loge(·). For more details of the Hadamard fractional integrals and derivatives, we refer the reader to Section 2.7 in the text [17]. Lemma 2.3. Let x ∈ C2δ ([1, T ], R) and g ∈ C([1, T ], R). The (integral) solution of the linear Hadamard fractional boundary value problem:        ( HDα + λ HDα−1 ) x(t) = g(t), 1 < α ≤ 2, 1 < t < T, x(1) = 0, x(T ) = m ∑ j=1 βj x(tj), (2.1) 228 Bashir Ahmad, Amjad F. Albideewi, Sotiris K. Ntouyas & Ahmed Alsaedi CUBO 23, 2 (2021) is given by x(t) = 1 γ ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) { m ∑ j=1 βj t −λ j Γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds − T −λ Γ(α − 1) ∫ T 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds } + t−λ Γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds, (2.2) where it is assumed that γ := T −λ ∫ T 1 sλ−1 (log s)α−2ds − m ∑ j=1 βj t −λ j ∫ tj 1 sλ−1 (log s)α−2ds 6= 0. (2.3) Proof. The linear Hadamard fractional differential equation in (2.1) can be rewritten as HDα−1(tD + λ)x(t) = g(t), t ∈ [1, T ]. (2.4) Applying the Hadamard fractional operator Iα−1 on both sides of (2.4), we get ( D + λ t ) x(t) = t−1 ( c1(log t) α−2 + Iα−1g(t) ) , which can be rewritten as D ( tλx(t) ) = c1t λ−1(log t)α−2 + tλ−1Iα−1g(t). (2.5) Integrating (2.5) from 1 to t, we get x(t) = c0t −λ + c1t −λ ∫ t 1 sλ−1 (log s)α−2ds + t−λ ∫ t 1 sλ−1 Iα−1 g(s)ds, (2.6) where ci, (i = 0, 1) are unknown arbitrary constants. Using the initial condition x(1) = 0 in (2.6) implies that c0 = 0, which leads to x(t) = c1t −λ ∫ t 1 sλ−1 (log s)α−2ds + t−λ ∫ t 1 sλ−1 Iα−1 g(s)ds. (2.7) Now using the condition x(T ) = m ∑ j=1 βj x(tj) in (2.7), we have c1T −λ ∫ T 1 sλ−1 (log s)α−2ds + T −λ ∫ T 1 sλ−1 Iα−1 g(s)ds = c1 m ∑ j=1 βj t −λ j ∫ tj 1 sλ−1 (log s)α−2ds + m ∑ j=1 βj t −λ j ∫ tj 1 sλ−1 Iα−1 g(s)ds, which, on solving for c1 together with (2.3), yields c1 = 1 γ [ m ∑ j=1 βj t −λ j ∫ tj 1 sλ−1 Iα−1 g(s)ds − T −λ ∫ T 1 sλ−1 Iα−1 g(s)ds ] . Substituting the above value of c1 in (2.7), we get the desired solution (2.2). The converse of the lemma follows by a direct computation. This completes the proof. CUBO 23, 2 (2021) Multipoint fractional sequential Hadamard BVP 229 The following lemma contains certain estimates that we need in the sequel. Lemma 2.4. For g ∈ C([1, T ], R) with ‖g‖ = sup t∈[1,T ] |g(t)|, we have (i) ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds ∣ ∣ ∣ ∣ ∣ ≤ (log T ) α α(α − 1) ‖g‖. (ii) ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1(log s)α−2ds ∣ ∣ ∣ ∣ ∣ ≤ (log T )α−1 (α − 1) . Proof. Note that ∫ s 1 ( log s r )α−2 1 r dr = (log s)α−1 (α − 1) . Since sλ ≤ tλ for 1 < s < t, then ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds ∣ ∣ ∣ ∣ ∣ ≤ sup t∈[1,T ] ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 g(r) r dr ) ds ∣ ∣ ∣ ∣ ∣ ≤ ‖g‖ sup t∈[1,T ] ∣ ∣ ∣ ∣ ∣ t−λ ∫ t 1 sλ−1 ( (log s)α−1 (α − 1) ) ds ∣ ∣ ∣ ∣ ∣ ≤ ‖g‖(log T ) α α(α − 1) . 3 Existence and uniqueness results Let G = C([1, T ], R) denote the Banach space of all continuous functions from [1, T ] to R endowed with the usual norm ‖x‖ = sup{|x(t)| : t ∈ [1, T ]}, and Cnδ ([1, T ], R) denotes the Banach space of all real valued functions g such that δng ∈ G. Using Lemma 2.3, we can transform the problem (1.1) into a fixed point problem as x = Px, where the operator P : G → G is defined by (Px)(t) = 1 γ ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × { m∑ j=1 βj t −λ j Γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds − T −λ Γ(α − 1) ∫ T 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds } (3.1) + t−λ Γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds, t ∈ [1, T ]. 230 Bashir Ahmad, Amjad F. Albideewi, Sotiris K. Ntouyas & Ahmed Alsaedi CUBO 23, 2 (2021) For computational convenience, we set Λ = (log T )α−1 |γ|(α − 1) [ m∑ j=1 |βj| (log T ) α Γ(α + 1) + (log T )α Γ(α + 1) ] + (log T )α Γ(α + 1) . (3.2) In the next theorem, we prove the uniqueness of solutions for problem (1.1) via Banach’s fixed point theorem. Theorem 3.1. Let f : [1, T ] × R → R be a continuous function and there exists a constant L > 0 such that: (H1) |f(t, x) − f(t, y)| ≤ L|x − y|, ∀t ∈ [1, T ] and x, y ∈ R. Then, problem (1.1) has a unique solution on [1, T ] if LΛ < 1, where Λ is given by (3.2). Proof. Let us define M be finite number given by M = sup t∈[1,T ] |f(t, 0)|, and show that PBr ⊂ Br, where Br = {x ∈ C[1, T ] : ‖x‖ ≤ r} with r ≥ MΛ 1 − LΛ . For x ∈ Br, t ∈ [1, T ], using (H1), we get |f(t, x(t))| = |f(t, x(t)) − f(t, 0) + f(t, 0)| ≤ |f(t, x(t)) − f(t, 0)| + |f(t, 0)| ≤ L‖x‖ + M ≤ Lr + M. Then |P(x)(t)| ≤ sup t∈[1,T ] { 1 |γ| ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × [ m∑ j=1 |βj t −λ j | Γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds + T −λ Γ(α − 1) ∫ T 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds ] + t−λ Γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds } ≤ (Lr + M) [ (log T )α−1 |γ|(α − 1) ( m∑ j=1 |βj| (log T ) α Γ(α + 1) + (log T )α Γ(α + 1) ) + (log T )α Γ(α + 1) ] ≤ Λ(Lr + M) ≤ r. In consequence, ‖Px‖ ≤ r, for any x ∈ Br, which shows that PBr ⊂ Br. Now we prove that the operator P is a contraction. For (x, y) ∈ C([1, T ], R) and for each t ∈ [1, T ], CUBO 23, 2 (2021) Multipoint fractional sequential Hadamard BVP 231 we obtain |(Px)(t) − (Py)(t)| ≤ sup t∈[1,T ] { 1 |γ| ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × [ m∑ j=1 |βj t −λ j | Γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r)) − f(r, y(r))| r dr ) ds + T −λ Γ(α − 1) ∫ T 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r)) − f(r, y(r))| r dr ) ds ] +t−λ ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r)) − f(r, y(r))| r dr ) ds } ≤ L [ (log T )α−1 |γ|(α − 1) ( m∑ j=1 |βj| (log T ) α Γ(α + 1) + (log T )α Γ(α + 1) ) + (log T )α Γ(α + 1) ] ≤ LΛ‖x − y‖. By the given condition LΛ < 1, it follows that the operator P is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (the Banach fixed point theorem). The proof is complete. The following existence result is based on the Leray-Schauder nonlinear alternative. Theorem 3.2 (Nonlinear alternative for single valued maps [13]). Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → C is a continuous, compact (that is, F(U) is a relatively compact subset of C) map. Then either (i) F has a fixed point in U, or (ii) there is a u ∈ ∂U (the boundary of U in C) and ν ∈ (0, 1) with u = νF(u). Theorem 3.3. Let f : [1, T ] × R → R be a continuous function such that the following conditions hold: (H2) There exists a function k ∈ C([1, T ], R+) and a nondecreasing function Ψ : R+ → R+ such that |f(t, x)| ≤ k(t)Ψ(‖x‖) for all (t, x) ∈ [1, T ] × R; (H3) There exists a positive constant S > 0 such that S Ψ(S)‖k‖Λ > 1, where ‖k‖ = sup t∈[1,T ] |k(t)| and Λ is defined by (3.2). Then problem (1.1) has at least one solution on [1, T ]. 232 Bashir Ahmad, Amjad F. Albideewi, Sotiris K. Ntouyas & Ahmed Alsaedi CUBO 23, 2 (2021) Proof. Firstly, we shall show that the operator P defined by (3.1) maps bounded sets into bounded sets in C([1, T ], R). For a number r > 0, let Br = {x ∈ C[1, T ] : ‖x‖ ≤ r} be a bounded set in C([1, T ], R). Then, by assumption (H2), we obtain |(Px)(t)| ≤ sup t∈[1,T ] { 1 |γ| ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × [ m∑ j=1 |βj t −λ j | Γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, y(r))| r dr ) ds + T −λ Γ(α − 1) ∫ T 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds ] + t−λ Γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 |f(r, x(r))| r dr ) ds } ≤ Ψ(‖x‖)‖k‖ [ (log T )α−1 |γ|(α − 1) ( m∑ j=1 |βj| (log T ) α Γ(α + 1) + (log T )α Γ(α + 1) ) + (log T )α Γ(α + 1) ] , and consequently, ‖Px‖ ≤ ΛΨ(r)‖k‖. Next we show that P maps bounded sets into equicontinuous sets of C([1, T ], R). Let τ1, τ2 ∈ [1, T ] with τ1 < τ2 and x ∈ Br. Then, we have |(Px)(τ2) − (Px)(τ1)| ≤ Ψ(r)‖k‖ { 1 |γ| ( |τ−λ1 − τ −λ 2 | ∫ τ1 1 sλ−1 (log s)α−2ds +τ−λ2 ∫ τ2 τ1 sλ−1 (log s)α−2ds ) × [ m∑ j=1 |βj||t −λ j | Γ(α − 1) ∫ τ1 1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds + T −λ Γ(α − 1) ∫ T 1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds ] + |τ−λ1 − τ −λ 2 | Γ(α − 1) ∫ τ1 1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds + τ−λ2 Γ(α − 1) ∫ τ2 τ1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds. Obviously the right-hand side of the above inequality tends to zero independently of x ∈ Br as τ2 − τ1 → 0. Therefore, by the Arzelá-Ascoli Theorem, the operator is completely continuous. The result will follow from Theorem 3.2 once it is established that the set of all solutions to equations x = νPx for ν ∈ (0, 1) is bounded. Let x be a solution of problem (1.1). Then, for t ∈ [1, T ], as in the first step, we can find that ‖x‖ = sup t∈[1,T ] {ν(Px)(t)} ≤ ΛΨ(‖x‖)‖k‖, CUBO 23, 2 (2021) Multipoint fractional sequential Hadamard BVP 233 which leads to ‖x‖ ΛΨ(‖x‖)‖k‖ ≤ 1. By condition (H3), there exists S > 0 such that ‖x‖ 6= S. Let us set U = {x ∈ C([1, T ], R) : ‖x‖ < S}. Note that the operator P : U → C([1, T ], R) is continuous and completely continuous. From the choice of U, there is no x ∈ ∂U such that x = νPx for some ν ∈ (0, 1). Consequently, we deduce by Theorem 3.2 that P has a fixed point x ∈ U, which is a solution of problem (1.1). This completes the proof. Our final existence result is based on Krasnosel’skĭı’s fixed point theorem. Theorem 3.4. (Krasnosel’skĭı’s fixed point theorem) Let M be a closed convex and nonempty subset of a Banach space X. Let A,B be the operators such that (i) Ax + By ∈ M whenever x, y ∈ M, (ii) B is a contraction mapping, (iii) A is compact and continuous. Then there exists z ∈ M such that z = Az + Bz. Theorem 3.5. Let f : [1, T ] × R → R be a continuous function satisfying the condition (H1). In addition, we assume that: (H4) |f(t, x)| ≤ µ(t) for all (t, x) ∈ [1, T ] × R, µ ∈ C([1, T ], R+). Then, the boundary value problem (1.1) has at least one solution on [1, T ], provided that L ( Λ − (log T )α Γ(α + 1) ) < 1, (3.3) where Λ is given by (3.2). Proof. Consider Bρ = {x ∈ G : ‖x‖ ≤ ρ}, ‖µ‖ = sup t∈[0,1] |µ(t)|, with ρ ≥ ‖µ‖Λ. Then we define the operators P1 and P2 on Bρ as (P1x)(t) = 1 γ ( t−λ ∫ t 1 sλ−1 (log s)α−2ds ) × { m∑ j=1 βj t −λ j Γ(α − 1) ∫ tj 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds − T −λ Γ(α − 1) ∫ T 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds } , t ∈ [1, T ], (P2x)(t) = t−λ Γ(α − 1) ∫ t 1 sλ−1 ( ∫ s 1 ( log s r )α−2 f(r, x(r)) r dr ) ds, t ∈ [1, T ]. 234 Bashir Ahmad, Amjad F. Albideewi, Sotiris K. Ntouyas & Ahmed Alsaedi CUBO 23, 2 (2021) As in 3.1 we can prove that ‖P1x + P2y‖ ≤ ‖µ‖Λ < ρ, and thus, P1x + P2y ∈ Bρ. By using condition (3.3) it is easy to prove that P1 is a contraction (see also 3.1). Moreover the continuous operator P2 is uniformly bounded, as ‖P2‖ ≤ (log T )α Γ(α + 1) ‖µ‖, and equicontinuous as |(P2x)(τ2) − (P2x)(τ1)| ≤ |τ−λ1 − τ −λ 2 | Γ(α − 1) ∫ τ1 1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds + τ−λ2 Γ(α − 1) ∫ τ2 τ1 sλ−1 ( ∫ s 1 ( log s r )α−2 1 r dr ) ds. Hence, by Arzelá-Ascoli Theorem, P2 is compact on Bρ. Thus all the assumptions of 3.4 are satisfied and the conclusion of 3.4 implies that the boundary value problem (1.1) has at least one solution on [1, T ]. The proof is completed. Example 3.6. Consider the boundary value problem for Hadamard fractional differential equations        ( HD7/4 + 2 HD3/4 ) x(t) = f(t, x(t)), t ∈ [1, e], x(1) = 0, x(e) = 3 ∑ j=1 βj x(tj). (3.4) Here, α = 7/4, λ = 2, T = e, m = 3, β1 = 1/3, β2 = 1/9, β3 = 1/27, t1 = 5/4, t2 = 3/2, t3 = 7/4 and f(t, x) = 1 13 √ t2 + 24 |x| 1 + |x| + 1 t + 2 + log t. Clearly, L = 1/65 as |f(t, x)−f(t, y)| ≤ (1/65)|x−y|. Using the given data, we have |γ| ≈ 0.691358 and Λ ≈ 1.104500. 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Ni, “New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval”, Appl. Math. Lett., vol. 118, ID 107165, 10 pages, 2021. Introduction Preliminaries Existence and uniqueness results