Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 81, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.81 EXISTENCE AND UNIQUENESS OF GENERALIZED NORMAL SOLUTIONS TO FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS AND APPLICATIONS KUNQUAN LAN Abstract. We study the existence and uniqueness of continuous generalized normal solutions to initial value problems of first order fractional differential equations. We use the Banach contraction principle and the Weissinger fixed point theorem to obtain our results. We assume that the absolute values of the nonlinearities have upper bound functions in a subspace of continuous func- tions. As an example, the results are applied to equations with nonlinearities arising in logistic type population models with heterogeneous environments, and to population models of Ricker type. 1. Introduction We study the existence and uniqueness of generalized normal solutions, that is, solutions u satisfying I α a+u ∈ AC[a, b], in C([a, b]; J) of the initial value problems (IVPs) of the first order fractional differential equation (FDE)( D1−α p,a+u ) (x) := ( Iαa+(u− w0) )′ (x) = f(x, u(x)) for each x ∈ (a, b] (1.1) subject to the initial condition (IC) u(a) = w0, (1.2) where α ∈ (0, 1), w0 ∈ J , a, b ∈ R with a < b, Iαa+ is the standard Riemann-Liouville (R-L) fractional integral operator, f : (a, b] × J → R is a continuous function satisfying suitable conditions and J is one of intervals: [c, d] where c, d ∈ R with c < d, R+ or R. We use u′ to denote the first order derivative of a function u. We emphasize that the function f is not assumed to be defined at {a} × J . The symbols used in the introduction will be given later. The first order frac- tional derivatives D1−α p,a+u in (1.1) and higher order fractional derivatives have been studied in [17, 19, 22, 23] and [16, 20], respectively. The FDE (1.1) holds for all x ∈ (a, b] and is different from the following FDE which holds for almost every (a.e.) x ∈ [a, b]:( D1−α p,a+u ) (x) := ( Iαa+(u− w0) )′ (x) = f(x, u(x)) for a.e. x ∈ [a, b]. (1.3) 2020 Mathematics Subject Classification. 34A08, 26A33, 34A12, 45D05. Key words and phrases. First order fractional differential equation; initial value problem; generalized normal solutions; existence and uniqueness. ©2024. This work is licensed under a CC BY 4.0 license. Submitted October 30, 2024. Published December 10, 2024. 1 2 K. LAN EJDE-2024/81 When the value u(a) exists, one can consider the following fractional derivative( D1−α ∗a u ) (x) := ( Iαa+(u− u(a) )′ (x) for a.e. x ∈ (a, b]. The operator D1−α ∗a is called the Caputo fractional operator in [7, Chapter 6] and the derivative D1−α ∗a u is called the modified Caputo derivative in [30]. It is easy to see that (1.3) with (1.2) is equivalent to the IVP of the FDE( D1−α ∗a u ) (x) := ( Iαa+(u− u(a) )′ (x) = f(x, u(x)) for a.e. x ∈ [a, b] (1.4) subject to the IC (1.2). When other ICs such as (1.7) below are considered, the value u(a) is not required to exist, so it is necessary to consider (1.3) instead of (1.4). We refer to [19, 22, 23] for the study on the existence of generalized normal solutions or upper or lower generalized normal solutions of (1.3) where the value u(a) is not required to exist. When f : [a, b]× J → R is an Lp-Carathéodory function for p ∈ ( 1 1−α ,∞), that is, f satisfies the Carathéodory conditions and the absolute value |f | has an upper bound function in Lp +(a, b) on any bounded subset of [a, b] × J , the existence of generalized normal solutions in C([a, b]; J) of (1.3) with (1.2) or (1.4) with (1.2) were studied before via the corresponding integral equation u(x) = w0 + (I 1−α a+ Fu)(x) := Au(x) for each x ∈ [a, b]. (1.5) For example, the minimal and maximal generalized normal solutions in C([a, b]; J) of (1.3) with (1.2) were obtained in [23] by using the monotone iterative techniques for compact maps. When a = 0, the existence of generalized normal solutions of (1.4) with (1.2) were studied in [24, 29] by using the Leray-Schauder theorem for compact maps. Some results on the existence and uniqueness of (1.4) with (1.2) and other ICs can be found in [9, 13, 14, 15, 28]. We refer to [32] for the study on the existence of solutions in C[0, b] of first and higher order FDEs of Bagley-Torvik and Langevin type and to [9, 13, 14, 34, 35] for the study on the existence of global solutions of (1.4) with (1.7). When J = R+ or R and f : [a, b]× J → J satisfies the Carathéodory conditions and a suitable growth condition, the existence and uniqueness of generalized normal solutions in Lp([a, b]; J) of (1.3) subject to the initial condition initial condition lim x→a+ (Iαa+(u− w0))(x) = c0 (1.6) were studied in [19] by using the Banach contraction principle. The conditions on f imply that f is an Lp-Carathéodory function for p = ∞ or p ∈ [1, 1/α). When f : (a, b] × J → J satisfies suitable continuity conditions, the existence and uniqueness of generalized normal solutions in C−α((a, b] : J) of (1.3) with the initital condition (I α a+u)(a) = c0 (1.7) were studied in [22, Section 7]. When w0 = 0, the existence and uniqueness of gen- eralized normal solutions in C[a, b] or C(a, b] of (1.4) subject to the initial condition lim x→a+ (x− a)αu(x) = c0 Γ(1− α) (1.8) were studied, for example, in [3, Theorem 6.2], [6, Theorem 4.3], [11, Lemma 9], [12], [29, Theorem 4.11], [30, Proposition 6.4], [31, section 7], and [37, 38, 39]. In this article, we study the existence and uniqueness of generalized normal solu- tions in C([a, b]; J) of (1.1)-(1.2) via the integral equation (1.5). The nonlinearity f EJDE-2024/81 FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS 3 is required to be defined on (a, b]×J . Since f is only defined on (a, b]×J instead of [a, b]×J , it is unknown under what conditions on f , the operator F in (1.5) is contin- uous and bounded on C([a, b]; J) and the map A in (1.5) is compact from C([a, b]; J) into C[a, b]. Hence, the known fixed point existence theorems for compact maps in Banach spaces such as Schauder fixed point theorem and Leray-Schauder fixed the- orem can not be applied, so we shall employ the Banach contraction principle and Weissinger fixed point theorem to obtain the existence and uniqueness of the gener- alized normal solutions of (1.1)-(1.2) in C([a, b]; J). We shall provide conditions on f such that the map A maps C([a, b]; J) to C([a, b]; J). These new conditions on f show that f is an Lp-Carathéodory function for p ∈ [1, 1 1−α ), so the new results on the existence and uniqueness of (1.1)-(1.2) are different from those considered in [23, 24, 29, 32], where f is defined on [a, b]×J and is an Lp-Carathéodory function for p ∈ ( 1 1−α ,∞). In section 2, we recall basic properties of the generalized R-L fractional integral introduced in [22] and the Banach space C−α(a, b]. In section 3, we study the existence and uniqueness of generalized normal solutions in C([a, b]; J) of (1.1)- (1.2). In section 4, we apply the results obtained in section 3 to study the existence and uniqueness of nonnegative generalized normal solutions of the first order FDEs with nonlinearities arising in logistic type population models with heterogeneous environments, and the population models of Ricker type. 2. Preliminaries Throughout this articler, we assume α ∈ (0, 1) and a, b ∈ R with a < b. For u ∈ L1[a, b], the Riemann-Liouville (R-L) fractional integral Iαa+u is defined by (Iαa+u)(x) := 1 Γ(α) ∫ x a u(y) (x− y)1−α dy (2.1) for x ∈ (a, b] such that the integral in (2.1) exists, where Γ : (0,∞) → R+ is the Gamma function defined by Γ(α) = ∫∞ 0 xα−1e−x dx for α ∈ (0,∞), see [7, p.13] and [27, p.33]. It is well known that Iαa+ maps L1[a, b] to L1[a, b]. Since Iαa+u ∈ L1[a, b], there exists E∗ u ⊂ [a, b] with meas(E∗ u) = 0 such that (a, b] \ E∗ u = {x ∈ (a, b] : (Iαa+u)(x) ∈ R}. (2.2) The set (a, b] \ E∗ u is treated as the natural domain of Iαa+u for each u ∈ L1[a, b]. Definition 2.1. We denote by L1 α(a, b) the set of functions u in L1[a, b] where u satisfies that there exists δu ∈ (0, b− a] such that Iαa+u is well defined on (a, a+ δu] and the limit limx→a+(Iαa+u)(x) exists. The space L1 α(a, b) was introduced in [22, Definition 1] and is a linear space [22, Theorem 3]. Definition 2.2. The operator I α a+ : L1 α(a, b) → L1[a, b] defined by (I α a+u)(x) = { (Iαa+u)(x) for each x ∈ (a, b] \ E∗ u, limx→a+(Iαa+u)(x) if x = a . (2.3) is called the generalized R-L fractional integral of order α. The integral operator I α a+ was introduced in [22, Definition 2]. In many cases, we need (Iαa+u)(x) to be well defined for each x ∈ (a, b]. A proper linear subspace of L1 α(a, b) was introduced in [22, Definition 3], which is stated as follows. 4 K. LAN EJDE-2024/81 Definition 2.3. We denote by S1 α(a, b) the set of functions u in L1[a, b] where u satisfies that Iαa+u is well defined on (a, b] and the limit limx→a+(Iαa+u)(x) exists. It is known that S1 α(a, b) is a linear space [22, Theorem 4] and it is shown in [22, Propositions 2 and 3] and [22, Corollary 6] that the following inclusions hold. Lp[a,b] ⫋ S1α(a,b) ⫋ L1 α(a,b) ⫋ L1[a,b] for each p ∈ ( 1 α ,∞]. (2.4) For u ∈ L1[a, b] and β ∈ [1,∞), the R-L fractional integral Iβa+u is defined by (Iβa+u)(x) = 1 Γ(β) ∫ x a (x− y)β−1u(y) dy for each x ∈ [a, b]. (2.5) It is known (see [22, Lemma 1 (2)]) that when β ∈ [1,∞), Iβa+u ∈ C[a, b] for each u ∈ L1[a, b]. Hence, for consistency with Definition 2.2, we can rewrite Iβa+ as I β a+ , that is, (Iβa+u) = (I β a+u)(x) := { (Iβa+u)(x) for each x ∈ (a, b], limx→a+(Iβa+u)(x) if x = a. (2.6) Moreover, if u, v ∈ L1[a, b] satisfy u(x) = v(x) for a.e. x ∈ [a, b], then by (2.5) and (2.6), we have for each β ∈ [1,∞), (I β a+u)(x) = (Iβa+u)(x) = (Iβa+v)(x) = (I β a+v)(x) for each x ∈ [a, b]. Lemma 2.4. (1) If γ ∈ [−α,∞), then I α a+(x− a)γ = Γ(1 + γ) Γ(1 + α+ γ) (x− a)α+γ for each x ∈ [a, b]. (2) Let γ ∈ (0,∞) and u ∈ C[a, b]. Then I α a+(I γ a+u)(x) = (I α+γ a+ u)(x) for each x ∈ [a, b]. (3) Assume that u ∈ S1 α(a, b) satisfies I 1−α a+ u ∈ C[a, b]. Then (I2−α a+ u)′(x) = (I 1−α a+ u)(x) for each x ∈ [a, b]. (4) Let w0 ∈ R and u ∈ S1 α(a, b). Then I α a+(u − w0) ∈ AC[a, b] if and only if I α a+u ∈ AC[a, b]. (5) For p ∈ ( 1 α ,∞ ] , I α a+ maps Lp[a, b] to C[a, b] and (I α a+u)(a) = 0 for each u ∈ Lp[a, b]. The results (1)–(5) can be found in Lemma 12, Lemma 14 and Proposition 5, Corollary 5 (2), Theorem 5 (1) in [22], respectively. We define the linear space C−α(a, b] = {u ∈ C(a, b] : lim x→a+ (x− a)αu(x) ∈ R}, (2.7) see [22, (3.21)]. Note that each function in C−α(a, b] is not necessarily defined at a. When a = 0, the set on the right side of (2.7) is denoted by other notations, for example, C−α[a, b] is used in [30], also see [2, Theorem 2], [6, p.612] and [11, p.370]. We introduce the following subset in C−α(a, b]. C0 −α(a, b] = {u ∈ C(a, b] : lim x→a+ (x− a)αu(x) = 0}. (2.8) EJDE-2024/81 FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS 5 It is easy to verify that C−α(a, b] ⊂ C(a, b] ∩ Lp[a, b] for p ∈ [1, 1 α ). (2.9) Lemma 2.5. (i) I α a+ maps C−α(a, b] into C[a, b]. (ii) I α a+ maps C0 −α(a, b] into C[a, b] and (I α a+u)(a) = 0 for each u ∈ C0 −α(a, b]. Proof. The result (i) was proved in [22, Lemma 11 (i)]. By [22, Lemma 11 (ii)] and (2.8), we see that the result (ii) holds. □ 3. Initial value problems of first order FDEs We study the existence and uniqueness of generalized normal solutions or non- negative generalized normal solutions in C([a, b]; J) of the IVPs for the nonlinear first order FDE( D1−α p,a+u ) (x) := ( Iαa+(u− w0) )′ (x) = f(x, u(x)) for each x ∈ (a, b] (3.1) subject to the initial condition u(a) = w0, (3.2) where α ∈ (0, 1), w0 ∈ J and f : (a, b] × J → R is a continuous function, J is one of the intervals J = [c, d], R+ := [0,∞) or R := (−∞,∞) and c, d ∈ R with c < d. We note that the function f is required to be defined only on (a, b]× J and the FDE (3.1) holds for all x ∈ (a, b] instead of for almost every (a.e.) x ∈ [a, b]. Let C([a, b]; J) = {u ∈ C[a, b] : u(x) ∈ J for each x ∈ [a, b]}. (3.3) By (2.4), C([a, b]; J) ⊂ S1 α(a, b). It is known that C([a, b]; [c, d]) is a bounded closed convex subset of C[a, b], C([a, b];R+) = C+[a, b], the cone of nonnegative continuous functions defined on [a, b] and C([a, b];R) = C[a, b]. The existence of nonnegative generalized normal solutions in C([a, b]; J) of the IVPs for the nonlinear first order FDE( D1−α ∗a u ) (x) := ( Iαa+(u− u(a)) )′ (x) = f(x, u(x)) for a.e. x ∈ [a, b] (3.4) subject to (3.2) was studied in [24], where a = 0, J = R+ or J = R and the function f : [a, b]× J → J is an Lp-Carathéodory function for p ∈ ( 1 1−α ,∞). When f(x, u) = xγh(x, u) for x ∈ [0, b]×R+, h is continuous and γ ∈ [0, 1− α), the existence and uniqueness of generalized normal solutions of (3.4) with (3.2) was obtained in [29, Theorem 4.8]. As mentioned in the Introduction, when f is defined on (a, b] × J , (3.1) with other ICs such as (1.7) and (1.8) were studied in [3, 6, 11, 22, 29, 30, 31, 37, 38, 39]. The following definition can be found in [22, Definition 5]. Definition 3.1. A function u ∈ S1 α(a, b) is said to be a generalized normal solution of (3.1)-(3.2) if I α a+u ∈ AC[a, b] and u satisfies (3.1) and (3.2). Definition 3.2. A function f : (a, b] × [c, d] → R is continuous at x0 ∈ (a, b] uniformly with respect to [c, d] if for ε > 0 there exists δ > 0 such that when |x− x0| < δ, |f(x, u)− f(x0, u)| < ε for all u ∈ [c, d]. Definition 3.2 was introduced in [22, Definition 12]. The following result gives its relation with continuity of f . Lemma 3.3. The following assertions are equivalent. 6 K. LAN EJDE-2024/81 (1) f : (a, b]× J → R is continuous (2) f : (a, b]× J → R is continuous at each x ∈ (a, b] uniformly with respect to any interval [c, d] ⊂ J and f(x, ·) : J → R is continuous for each x ∈ (a, b]. Proof. It is sufficient to prove the result when J = [c, d]. (1) implies (2): Since f is continuous, it follows from [22, Lemma 19] that f is continuous at each x ∈ (a, b] uniformly with respect to u ∈ [c, d]. It is obvious that f(x, ·) : [c, d] → R is continuous for each x ∈ (a, b]. (2) impliesg (1): Let (x0, u0) ∈ (a, b]× [c, d] and ε > 0. Since f(x0, ·) : [c, d] → R is continuous, there exists δ0 > 0 such that |f(x0, u)− f(x0, u0)| ≤ ε/2 for all u ∈ [c, d] with |u− u0| < δ0. (3.5) Since f is continuous at x0 uniformly with respect to u ∈ [c, d], by Definition 3.2 there exists δ ∈ (0, δ0) such that when |x− x0| < δ, |f(x, u)− f(x0, u)| < ε/2 for all u ∈ [c, d]. This with (3.5) implies that when |x− x0| < δ and |u− u0| < δ, |f(x, u)− f(x0, u0)| ≤ |f(x, u)− f(x0, u)|+ |f(x0, u)− f(x0, u0)| ≤ ε/2 + ε/2 = ε and f is continuous. □ When J = R+ or R, the condition (2) of Lemma 3.3 first appeared in the condition (H1)J in [22] and was used to prove [22, Lemma 20]. We write by C0 −(1−α),+(a, b] for the set of nonnegative functions in C0 −(1−α)(a, b] defined in (2.8), that is, C0 −(1−α),+(a, b] = {u ∈ C0 −(1−α)(a, b] : u(x) ≥ 0 for all x ∈ (a, b]}. (3.6) Definition 3.4. A function f : (a, b] × J → R is said to be a C0 −(1−α)-function if for any bounded closed interval [c1, d1] in J , there exists g ∈ C0 −(1−α),+(a, b] such that |f(x, u)| ≤ g(x) for all x ∈ (a, b] and u ∈ [c1, d1]. (3.7) A function f : (a, b] × J → R is said to be a C0 −(1−α)-continuous function if f is a C0 −(1−α)-function and is continuous on (a, b]× J . When J = [c, d], then (3.7) holds on [c, d]. In this case, if we want to emphasize the function g in (3.7), then we say that f is a C0 −(1−α),+-function with function g. By (2.9), C−(1−α)(a, b] ⊂ Lp[a, b] for each p ∈ [1, 1 1−α ), Hence, a C0 −(1−α)- functionf on (a, b] × J is an Lp-Carathéodory function for each p ∈ [1, 1 1−α ), that is, f satisfies the Carathéodory conditions and (3.7) holds with g ∈ Lp +[a, b]. See [21, Definition 4.1], [23, Definition 3.3] and [24, Definition 2] for the definition of an Lp-Carathéodory function. We define an operator F on C((a, b]; J), called the Nemytskii operator, by (Fu)(x) = f(x, u(x)) for each x ∈ (a, b]. (3.8) We study generalized normal solutions of (3.1)-(3.2) in C([a, b]; J) via the fixed points of the map A defined by Au(x) = w0 + (I 1−α a+ Fu)(x) for each x ∈ [a, b], (3.9) that is, solutions in C([a, b]; J) of the integral equation u(x) = w0 + (I 1−α a+ Fu)(x) = Au(x) for each x ∈ [a, b]. (3.10) EJDE-2024/81 FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS 7 Definition 3.5. A function u : [a, b] → J is said to be a solution of (3.10) if u ∈ C([a, b]; J) and u satisfies (3.10). Lemma 3.6. (1) If f : (a, b]× J → R is continuous, then F maps C((a, b]; J) to C(a, b]. (2) If f : (a, b] × J → R is a C0 −(1−α)-continuous function, then F maps C([a, b]; J) to C0 −(1−α)(a, b]. (3) If f : (a, b] × J → R is a C0 −(1−α)-continuous function, then the map A maps C([a, b]; J) to C[a, b] and (Au)(a) = w0. Proof. (1) When J = R+ or J = R, Lemma 3.6 (1) was proved in [22, Lemma 20]. We only prove the case J = [c, d]. Let u ∈ C((a, b]; [c, d]) and x0 ∈ (a, b]. Then u(x0) ∈ [c, d]. We consider the following two cases. (i) When u(x0) ∈ (c, d], since f(x0, ·) : [c, d] → R is continuous at u(x0), for ε > 0, there exists ω0 ∈ (0, u(x0)− c] such that when v ∈ [c, d] with |v − u(x0)| < ω0, |f(x0, v)− f(x0, u(x0))| < ε/2. (3.11) Since u is continuous at x0, there exists δ1 ∈ (0,min{x0−a, b−x0}) such that when x ∈ (a, b] with |x− x0| < δ1, |u(x)− u(x0)| < ω0. This with (3.11) implies that when x ∈ (a, b] with |x− x0| < δ1, |f(x0, u(x))− f(x0, u(x0))| < ε/2. (3.12) Since f : (a, b] × J → R is continuous, by (1) implying (2) of Lemma 3.3, f : (a, b] × [c, d] → R is continuous at x0 ∈ (a, b] uniformly with respect to u ∈ [c, d]. Hence, by Definition 3.2 there exists δ0 ∈ (0, δ1) such that when |x− x0| < δ0, |f(x, u(x))− f(x0, u(x))| < ε/2. This with (3.12) implies that when |x− x0| < δ0, |(Fu)(x)− (Fu)(x0)| = |f(x, u(x))− f(x0, u(x0))| ≤ |f(x, u(x))− f(x0, u(x)) + |f(x0, u(x))− f(x0, u(x0))| < ε/2 + ε/2 = ε. It follows that Fu is continuous at x0. (ii) When u(x0) = c, since f(x0, ·) : [c, d] → R is continuous at c, for ε > 0, there exists ω0 ∈ (0, d− c] such that when v ∈ [c, w0 + c], |f(x0, v)− f(x0, u(x0))| = |f(x0, v)− f(x0, c)| < ε/2. The rest of the proof is the same as that after (3.11). (2) Let u ∈ C([a, b]; J). Since f is continuous, by the result (1) we have Fu ∈ C(a, b]. If J = [c, d], then u(x) ∈ J for each x ∈ [a, b]. If J = R+ or J = R, then u(x) ∈ [0, ρ] for each x ∈ [a, b], where ρ = ∥u∥. Since f is a C0 −(1−α)-function, there exists g ∈ C0 −(1−α),+(a, b] such that |(Fu)(x)| = |f(x, u(x))| ≤ g(x) for each x ∈ (a, b]. Multiplying both sides of the above inequality by (x− a)(1−α) implies (x− a)1−α|(Fu)(x)| ≤ (x− a)1−αg(x) for each x ∈ (a, b]. 8 K. LAN EJDE-2024/81 Taking limit superiors on both sides of the above inequality implies lim sup x→a+ (x− a)1−α|(Fu)(x)| ≤ lim sup x→a+ (x− a)1−αg(x), (3.13) see [18] for the theory of limit inferior and superior. Since g ∈ C0 −(1−α),+(a, b], by (2.8) and (3.6) we have lim sup x→a+ (x− a)1−αg(x) = lim x→a+ (x− a)1−αg(x) = 0. This with (3.13) implies lim x→a+ (x− a)1−α|(Fu)(x)| = 0 and Fu ∈ C0 −(1−α)(a, b]. (3) By result (2), F maps C([a, b]; J) to C0 −(1−α)(a, b]. By Lemma 2.5 (i), I 1−α a+ maps C−(1−α)(a, b] into C[a, b]. Since C 0 −α(a, b] ⊂ C−α(a, b], we have for u ∈ C([a, b]; J), I 1−α a+ Fu ∈ C[a, b] and Au = w0+(I 1−α a+ Fu) ∈ C[a, b]. Hence, A maps C([a, b]; J) to C[a, b]. For u ∈ C([a, b]; J), since Fu ∈ C0 −(1−α)(a, b], by Lemma 2.5 (ii), we have (I 1−α a+ Fu)(a) = 0 and (Au)(a) = w0. □ Theorem 3.7. Assume that f : (a, b] × J → R is a C0 −(1−α)-continuous function. Then the following assertions are equivalent. (1) u ∈ C([a, b]; J) is a generalized normal solution of (3.1)-(3.2). (2) u ∈ C([a, b]; J) is a solution of (3.10). Proof. (1) implies (2): Assume that (1) holds. Since u ∈ C([a, b]; J), by Lemma 2.4 (5), we have I α a+(u−w0)(a) = 0. Since I α a+u ∈ AC[a, b], it follows from Lemma 2.4 (4) that I α a+(u− w0) ∈ AC[a, b]. Since I α a+(u− w0)(x) = Iαa+(u− w0)(x) for each x ∈ (a, b], by (3.1), we obtain( I α a+(u− w0) )′ (x) = f(x, u(x)) for each x ∈ (a, b]. (3.14) Since I α a+(u−w0) ∈ AC[a, b], integrating (3.14) from a to x and applying I α a+(u− w0)(a) = 0, we have I α a+(u− w0)(x) = (I 1 a+Fu)(x) for each x ∈ [a, b]. Applying I 1−α a+ to both sides of the above equation and using Lemma 2.4 (2) imply that for each x ∈ [a, b], I 1 a+(u− w0)(x) = I 1−α a+ I α a+(u− w0)(x) = I 1−α a+ (I 1 a+Fu)(x) = (I 2−α a+ Fu)(x). Since I 1−α a+ Fu ∈ C[a, b], by Lemma 2.4 (3) and differentiating both sides of the above equation, we have, for each x ∈ [a, b], u(x)− w0 = ( I 1 a+(u− w0) )′ (x) = ( I 2−α a+ Fu )′ (x) = (I 1−α a+ Fu)(x). and the result (2) holds. EJDE-2024/81 FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS 9 (2) implies (1): Assume that u ∈ C([a, b]; J) is a solution of (3.10). By Lemma 3.6 (3), we have I 1−α a+ Fu ∈ C[a, b]. By Lemma 2.4 (5), I α a+(u−w0) ∈ C[a, b]. Applying I α a+ to both sides of (3.10) and using Lemma 2.4 (2), we have I α a+(u− w0)(x) = I α a+(I 1−α a+ Fu)(x) = (I 1 a+Fu)(x) for each x ∈ [a, b]. (3.15) Since I 1 a+Fu ∈ AC[a, b], it follows from (3.15) that I α a+(u − w0) ∈ AC[a, b]. By Lemma 2.4 (4), we have I α a+u ∈ AC[a, b]. Differentiating both sides of (3.15) implies that ( I α a+(u− w0) )′ (x) = ( I 1 a+Fu )′ (x) = (Fu)(x) for each x ∈ [a, b]. (3.16) Since ( Iαa+(u− w0) )′ (x) = (I α a+(u− w0)) ′(x) for x ∈ (a, b], it follows from (3.16) that( D1−α p,a+u ) (x) = ( Iαa+(u− w0) )′ (x) = f(x, u(x)) for each x ∈ (a, b] and u is a generalized normal solution of (3.1). By Lemma 3.6 (3), u(a) = (Au)(a) = w0 and (3.2) holds. □ Let X be a Banach space. Recall that a map A : D → X is said to be a Lipschitz map with Lipschitz constant L, if ∥Ax−Ay∥ ≤ L∥x− y∥ for x, y ∈ D. A Lipschitz map with Lipschitz constant L < 1 is said to be a contractive map (with contraction constant L). The following Banach contraction principle can be found for example in [10]. Lemma 3.8. Let K be a nonempty closed convex set in X. Assume that A : K → K is a contractive map with contraction constant L. Then the following assertions hold. (a) A has a unique fixed point u∗ in K. (b) For each u1 in K, the sequence {un+1} defined by un+1 = Aun for each n ∈ N converges to u∗. (c) ∥un − u∗∥ ≤ Ln 1−L∥u1 − u2∥. Another useful fixed point theorem on uniqueness is the Weissinger fixed point theorem. The following definition was introduced in [21, Definition 2.1]. Definition 3.9. Let K be a nonempty closed subset in X. A map A : K → K is said to be a Weissinger map (with a sequence of Lipschitz constants {kn}) if for each n ∈ N, An : K → K is a Lipschitz map with Lipschitz constant kn ∈ (0,∞) and ∑∞ n=1 kn <∞. It is easy to see that if A : K → K is a contractive map with contractive constant L < 1, then A is a Weissinger map with kn = Ln. Lemma 3.10. Let K be a nonempty closed subset in X. Assume that A : K → K is a Weissinger map with Lipschitz constants {kn}. Then A has a unique fixed point u∗ in K and for each n ∈ N, ∥un+1 − u∗∥ ≤ ∞∑ i=n ki∥u2 − u1∥ for each u1 ∈ K, (3.17) 10 K. LAN EJDE-2024/81 where un+1 = Aun for each n ∈ N. Lemma 3.10 was given in [33] whose English version can be found in [7, Theorem D.7] or [8, Theorem 2.3]. A proof in English is given in [21, Lemma 2.4]. By [7, Definition 4.1] and [7, Theorem 4.1], the Mittag-Leffler function E(1−α) satisfies E(1−α)(x) = ∞∑ n=0 xn Γ(1 + (1− α)n) for each x ∈ R. (3.18) It is known that E(1−α)(x) ∈ R for each x ∈ R. Recall that a function f : (a, b]× J → R is said to satisfy a Lipschitz condition in the second variable with function ϕ : (a, b] → R+ if |f(x, u)− f(x, v)| ≤ ϕ(x)|u− v| for each x ∈ (a, b] and all u, v ∈ J. If ϕ(x) ≡ L > 0 for each x ∈ [a, b], then L is said to be a Lipschitz constant. In the next theorem we use the following conditions: (H1) If J = R+ or R, assume that f : (a, b] × J → J is a C0 −(1−α)-function and if J = [c, d], assume that f : (a, b] × [c, d] → R is a C0 −(1−α)-function with function g in C0 −(1−α),+(a, b] satisfying ∥I1−α a+ g∥ ≤ min{w0 − c, d− w0}. (3.19) (H2) f satisfies a Lipschitz condition in the second variable with function ϕ ∈ C−(1−α),+(a, b] and ∥I1−α a+ ϕ∥ < 1. Theorem 3.11. Under assumptions (H1) and (H2), then the following assertions hold. (1) (3.10) has a unique solution u∗ in C([a, b]; J). (2) For each u1 ∈ C([a, b]; J), the sequence {un} defined by un+1 = Aun satis- fies lim n→∞ ∥un − u∗∥ = 0 and ∥un − u∗∥ ≤ ∥I1−α a+ ϕ∥n 1− ∥I1−α a+ ϕ∥ ∥u1 − u2∥ for each n ∈ N. (3) u∗ is a unique generalized normal solution of (3.1)-(3.2) in C([a, b]; J). Proof. Since f : (a, b] × J → R is a C0 −(1−α)-function with function g, by Lemma 3.6 (3), the map A defined in (3.9) maps C([a, b]; J) to C[a, b]. We prove that the map A maps C([a, b]; J) to C([a, b]; J). Indeed, if J = R+ or R, then since f maps (a, b]× J to J , it is easy to see that the map A maps C([a, b]; J) to C([a, b]; J). If J = [c, d], then by Lemma 2.5 (i), I 1−α a+ g ∈ C[a, b]. By (3.19), we have for each x ∈ [a, b], −min{w0 − c, d− w0} ≤ (I 1−α a+ g)(x) ≤ min{w0 − c, d− w0}. This implies that for each x ∈ [a, b], −(w0 − c) ≤ (I 1−α a+ g)(x) and I1−α a+ g(x) ≤ d− w0. (3.20) Let u ∈ C([a, b]; [c, d]). By (3.20), we have that for each x ∈ [a, b], c− w0 ≤ (I 1−α a+ (−g))(x) ≤ (I 1−α a+ Fu)(x) ≤ (I 1−α a+ g)(x) ≤ d− w0. EJDE-2024/81 FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS 11 This with (3.9) implies c ≤ Au(x) ≤ d for each x ∈ [a, b] and Au ∈ C([a, b]; [c, d]). Hence, the map A maps C([a, b]; [c, d]) to C([a, b]; [c, d]). Now we prove that A is a contractive map with Lipschitz constant ∥I1−α a+ ϕ∥ < 1. Indeed, since ϕ ∈ C−(1−α)(a, b], it follows from Lemma 2.5 (i) that I 1−α a+ ϕ ∈ C[a, b]. By (3.10), we have that for u1, u2 ∈ C([a, b]; J), |(Au1)(x)− (Au2)(x)| = |(I1−α a+ (Fu1 − Fu2)(x)| ≤ I 1−α a+ [ϕ|u1 − u2|](x) ≤ ∥I1−α a+ ϕ∥∥u1 − u2∥ for each x ∈ [a, b]. Taking maximum on the above inequality yields ∥Au1 −Au2∥ ≤ ∥I1−α a+ ϕ∥∥u1 − u2∥. Since ∥I1−α a+ ϕ∥ < 1, it follows from Lemma 3.8 that (3.10) has a unique solution u∗ in C([a, b]; J) and the results (1) and (2) hold. By (2) implying (1) of Theorem 3.7, u∗ is a unique generalized normal solution of (3.1)-(3.2) in C([a, b]; [c, d]) and the result (3) holds. □ Let L ∈ (0,∞) and ϕ(x) = L for each x ∈ [a, b], then by Lemma 2.4 (1), (I 1−α a+ ϕ)(x) = L Γ(2− α) (x− a)1−α for each x ∈ [a, b] and ∥I1−α a+ ϕ∥ = L(b− a)1−α Γ(2− α) . (3.21) If L ≥ Γ(2−α) (b−a)1−α , then by (3.21) we have ∥I1−α a+ ϕ∥ ≥ 1, so Theorem 3.11 can not be applied. The following result can be used to deal with the case L ≥ Γ(2−α) (b−a)1−α . Theorem 3.12. Assume that condition (H1) holds and f satisfies a Lipschitz con- dition in the second variable with a Lipschitz constant L > 0. Then the following assertions hold. (1) (3.10) has a unique solution u∗ in C([a, b]; J). (2) For each u1 ∈ C([a, b]; J), the sequence {un} defined by un+1 = Aun for each n ∈ N satisfies ∥un+1 − u∗∥ ≤ ∞∑ i=n ki∥u2 − u1∥ for each n ∈ N, where kn = Ln(b− a)n(1−α) Γ(1 + n(1− α)) for each n ∈ N. (3.22) (3) u∗ is a unique generalized normal solution of (3.1)-(3.2) in C([a, b]; J). Proof. By the proof of Theorem 3.11, we see that the map A defined in (3.9) maps C([a, b]; J) to C([a, b]; J). Since f satisfies a Lipschitz condition with Lipschitz constant L, by (3.10), we have for u1, u2 ∈ C([a, b]; J) and x ∈ [a, b], |Au1(x)−Au2(x)| ≤ (I 1−α a+ (Fu1 − Fu2))(x) ≤ (I 1−α a+ L|u1 − u2|)(x). 12 K. LAN EJDE-2024/81 This with Lemma 2.4 (2) implies that for each x ∈ [a, b], |A2u1(x)−A2u2(x)| ≤ |A(Au1)(x)−A(Au2)(x)| ≤ ( I 1−α a+ L|Au1 −Au2| ) (x) ≤ ( I 1−α a+ L [ (I 1−α a+ L|u1 − u2|) ]) (x) = L2 ( I 1−α a+ ( I 1−α a+ |u1 − u2| )) (x) = L2 ( I 2(1−α) a+ |u1 − u2| ) (x). Repeating the process and using Lemmas 2.4 (1) and (2) imply for n ∈ N, |Anu1(x)−Anu2(x)| ≤ Ln ( I n(1−α) a+ |u1 − u2| ) (x) ≤ Ln∥u1 − u2∥(I n(1−α) a+ 1)(x) = Ln∥u1 − u2∥ ( I n(1−α) a+ (x− a)0 ) = Ln∥u1 − u2∥ Γ(1)(x− a)n(1−α) Γ(1 + n(1− α)) = Ln(x− a)n(1−α) Γ(1 + n(1− α)) ∥u1 − u2∥ ≤ Ln(b− a)n(1−α) Γ(1 + n(1− α)) ∥u1 − u2∥ = kn∥u1 − u2∥ for each x ∈ [a, b]. (3.23) By (3.22) and (3.18), we have ∞∑ n=0 kn = ∞∑ n=0 [L(b− a)1−α]n Γ(1 + n(1− α)) = E1−α(L(b− a)1−α) <∞. This with (3.23) implies that A is a Weissinger map with Lipschitz constants {kn}. Results (1) and (2) follow from Lemma 3.10. By (2) implying (1) of Theorem 3.7, u∗ is a unique generalized normal solution of (3.1)-(3.2) in C([a, b]; J) and the result (3) holds. □ 4. Examples Motivated by some population models with heterogeneous environments, we give two examples of the IVPs of the first order FDEs with nonlinearities arising from population models to exhibit the applications of Theorem 3.11. The first example is to study the existence and uniqueness of nonnegative gen- eralized normal solutions of first order FDEs of the form( D1−α p,a+u ) (x) := ( Iαa+(u− w0) )′ (x) = λu(x) [ ψ1(x)− ψ2(x)u r(x) ] (4.1) for each x ∈ (a, b] subject to the IC u(a) = w0, where w0 ∈ (0,∞), λ > 0, r > 0 and ψ1, ψ2 : (a, b] → R are continuous functions. The nonlinearities arise from logistic type population models with heterogeneous environments and the population models of Ricker type of a single species. We remark that the derivative D1−α p,a+u in (4.1) possibly has biological interpretations but not as growth rates in population models. We refer to [4, 5, 21] for the interpretations of the various terms on the right-side of (4.1). EJDE-2024/81 FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS 13 Example 4.1. Let ρ > w0 > 0 and r > 0. If ψ1, ψ2 ∈ C0 −(1−α)(a, b] \ {0}. Then (4.1) has a unique nonnegative generalized normal solution from [a, b] to [0, ρ] for each λ ∈ (0,min{λ1(ρ), λ2(ρ)}), where λ1(ρ) = 1 ∥I1−α a+ |ψ1|∥+ (1 + r)ρr∥I1−α a+ |ψ2|∥ , (4.2) λ2(ρ) = min{w0, ρ− w0} ρ∥I1−α a+ |ψ1|∥+ ρr∥I1−α a+ |ψ2|∥ . (4.3) Proof. Since ψ1, ψ2 ∈ C0 −(1−α)(a, b] \ {0}, ∥I1−α a+ |ψ1|∥ ̸= 0 and ∥I1−α a+ |ψ2|∥ ̸= 0. Hence, λ1(ρ) in (4.2) and λ2(ρ) in (4.3) are well defined. Let ρ > w0 and λ > 0. We define a function f : (a, b]× [0, ρ] → R by f(x, u) = λu [ ψ1(x)− ψ2(x)u r ] . (4.4) Since ψ1, ψ2 ∈ C(a, b], f : (a, b] × [0, ρ] → R is continuous. We define a function ϕ : (a, b] → R by ϕ(x) = λ [ |ψ1(x)|+ (1 + r)|ψ2(x)|ρr ] . (4.5) Since ψ1, ψ2 ∈ C0 −(1−α)(a, b], we have ϕ ∈ C0 −(1−α)(a, b] and for each x ∈ (a, b],∣∣∣∂f(x, u) ∂u ∣∣∣ = ∣∣∣λ[ψ1(x)− (1 + r)ψ2(x)u r ]∣∣∣ ≤ ϕ(x) for each u ∈ [c, d]. For each x ∈ (a, b] and u, v ∈ [0, ρ] with v < u, there exists ξ ∈ (u, v) such that |f(x, u)− f(x, v)| = ∣∣∂f(x, ξ) ∂u (u− v) ∣∣ ≤ ϕ(x)|u− v|. Hence, f satisfies a Lipschitz condition in the second variable with the function ϕ ∈ C0 −(1−α)(a, b]. By (4.5), we have (I 1−α a+ ϕ)(x)∥ = λ [ (I 1−α a+ |ψ1|)(x) + (1 + r)ρr(I 1−α a+ |ψ2|)(x) ] and by (4.2) and λ < λ1(ρ), ∥I1−α a+ ϕ∥ ≤ λ [ ∥I1−α a+ |ψ1|∥+ (1 + r)ρr∥I1−α a+ |ψ2|∥ ] < 1. By (4.4), we have for (x, u) ∈ (a, b]× [0, ρ], |f(x, u)| ≤ λu [ |ψ1(x)|+ |ψ2(x)|ur ] ≤ λρ [ |ψ1(x)|+ |ψ2(x)|ρr ] = g(x). Since ψ1, ψ2 ∈ C0 −(1−α)(a, b], g ∈ C0 −(1−α)(a, b]. Since λ ∈ (0, λ2(ρ)), we have ∥I1−α a+ g∥ ≤ λρ [ (I 1−α a+ |ψ1|)(x) + (I 1−α a+ |ψ2|)(x)ρr ] ≤ λρ [ ∥I1−α a+ |ψ1|∥+ ∥I1−α a+ |ψ2|∥ρr ] ≤ λ2(ρ)ρ [ ∥I1−α a+ |ψ1|∥+ ∥I1−α a+ |ψ2|∥ρr ] ≤ min{w0, ρ− w0}. The result follows from Theorem 3.11 with J = [0, ρ]. □ The second example is to study the existence and uniqueness of nonnegative generalized normal solutions in C[a, b] of the IVP for the nonlinear first order FDE( D1−α p,a+u ) (x) = ( Iαa+(u− w0) )′ (x) = ξ(x) + φ(x)u(x)er−δu(x)− m 1+ku(x) (4.6) for each x ∈ (a, b] subject to the IC u(a) = w0, where w0,m ∈ R+, r, δ, k > 0 and ξ, φ : (a, b] → R+ are continuous functions. 14 K. LAN EJDE-2024/81 The FDE (4.6) is motivated by the population models with the growth rates of Ricker type governed by difference equations and first order ordinary differential equations in [1, 25, 26, 36]. The existence and uniqueness of generalized normal solutions in Lp([a, b]; J) of the IVP (4.6) were studied in [19, Theorem 12], where ξ ∈ Lp +[a, b] and ϕ ∈ L∞[a, b]. Example 4.2. Assume that ξ, φ ∈ C0 −(1−α),+(a, b] and ϕ satisfies ∥I1−α a+ φ∥ < e−r ( 1 + m 4 )−1 . (4.7) Then (4.6) has a unique generalized normal solution u∗ in C+[a, b]. Proof. We define a function f : [a, b]× R+ → R+ by f(x, u) = ξ(x) + φ(x)h(u), (4.8) where h : R+ → R+ is defined by h(u) = uer−δu− m 1+ku . (4.9) Since ξ, φ ∈ C(a, b], f : (a, b]× R+ → R+ is continuous. It is easy to see that f(x, u) = ξ(x) + φ(x)uer−δu− m 1+ku ≤ gρ(x) for each x ∈ (a, b] and u ∈ [0, ρ], where gρ(x) = ξ(x) + φ(x)uer for each x ∈ (a, b]. Since ξ, φ ∈ C0 −(1−α),+(a, b], we obtain gρ ∈ C0 −(1−α),+(a, b]. Hence, f : (a, b] × R+ → R+ is a C0 −(1−α)-continuous function. By (4.9), we have h′(u) = er−δu− m 1+ku [ 1− δu+ mku (1 + ku)2 ] for each u ∈ R+. It follows that for each u ∈ R+, h′(u) ≤ er−δu|1− δu|+ er mku (1 + ku)2 ≤ er [ e−δu|1− δu|+ m 4 ] . (4.10) We show that e−δu|1− δu| ≤ 1 for each u ∈ R+. (4.11) Indeed, (4.11) holds if and only if −eδu ≤ 1− δu ≤ eδu for each u ∈ R+. (4.12) Since eδu ≥ 1 + δu for each u ∈ R+, we have eδu + 1− δu ≥ 1 + δu+ 1− δu = 2 ≥ 0 for each u ∈ R+ and the first inequality of (4.12) holds. It is obvious that the second inequality of (4.12) holds. By (4.10) and (4.11), we have h′(u) ≤ er [ e−δu|1− δu|+ mk 4k ] ≤ er [ 1 + m 4 ] (4.13) By (4.8) and (4.13), for u, v ∈ R+, there exists µ ∈ [u, v] such that for each x ∈ (a, b], |f(x, u)− f(x, v)| = φ(x)|h(u)− h(v)| = φ(x)h′(µ)|u− v| ≤ ϕ(x)|u− v|, where ϕ(x) = φ(x)er ( 1 + m 4 ) for each x ∈ (a, b]. (4.14) EJDE-2024/81 FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS 15 Since φ ∈ C0 −(1−α),+(a, b], we obtain ϕ ∈ C0 −(1−α),+(a, b]. Hence, f : (a, b] × R+ → R+ satisfies a Lipschitz condition in the second variable with function ϕ ∈ C−(1−α)(a, b]. By (4.7) and (4.14), we have ∥I1−α a+ ϕ∥ = er ( 1 + m 4 ) ∥I1−α a+ φ∥ < 1. The results follow from Theorem 3.11 with J = R+. □ Acknowledgments. The author was supported in part by the Natural Sciences and Engineering Research Council of Canada, Grant/Award Number: RGPIN- 2023-04024. Also, the author wants to thank the referees for reading this paper carefully and providing valued comments. References [1] A. S. Ackleh, P. L. 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Li; Stability analysis of a population model with maturation delay and Ricker birth function, Abst. Appl. Anal. 2014, Article ID 136707. [37] T. Zhu; Weakly singular integral inequalities and global solutions for fractional differential equations of Riemann-Liouville type, Mediterr. J. Math. 18 (2021), no. 5, Paper No. 184, 17pp. [38] T. Zhu; Attractivity of solutions of Riemann-Liouville fractional differential equations, Elec- tron. J. Qual. Theory Differ. Equ., (2022), Paper No. 52, 12pp. [39] T. Zhu; Global attractivity for fractional differential equations of Riemann-Liouville type, Fract. Calc. Appl. Anal., 26 (2023), 2264-2280. Kunquan Lan Department of Mathematics, Toronto Metropolitan University, Toronto, Ontario, M5B 2K3, Canada Email address: klan@torontomu.ca 1. Introduction 2. Preliminaries 3. Initial value problems of first order FDEs 4. Examples Acknowledgments References