Existence and uniqueness of solutions of nonlinear Cauchy‐type problems for first‐order fractional differential equations

New properties of the first‐order Riemann–Liouville fractional integrals are proved and first‐order fractional derivatives for the equivalence class of functions in L1(a,b)$$ {L}&amp;#x0005E;1\left(a,b\right) $$ are analyzed. Solutions, normal solutions, and generalized normal solutions of the initial value problems (IVPs) for nonlinear first‐order fractional differential equations (FDEs) with fraction α∈(0,1)$$ \alpha \in \left(0,1\right) $$ are introduced. The equivalences in L1(a,b)$$ {L}&amp;#x0005E;1\left(a,b\right) $$ between the first‐order FDEs and the corresponding (Volterra) integral equations are established based on normal solutions and generalized normal solutions. New results on the existence and uniqueness of generalized normal solutions or nonnegative generalized normal solutions in Lp(a,b)$$ {L}&amp;#x0005E;p\left(a,b\right) $$ of the IVPs are obtained for p∈[1,1/α)$$ p\in \left[1,1/\alpha \right) $$ . These results are applied to study the existence and uniqueness of generalized normal nonnegative solutions in L+p(a,b)$$ {L}_{&amp;#x0002B;}&amp;#x0005E;p\left(a,b\right) $$ of the IVP for the nonlinear first‐order FDE with nonlinearities related to the population models with growth rates of Ricker type.


INTRODUCTION
We study the existence and uniqueness of solutions or nonnegative solutions in L 1 (a, b) of the initial value problems (IVPs) for the first-order fractional differential equations (FDEs): for almost every (a.e.) x ∈ [a, b], (1) subject to the initial condition: lim LAN where w 0 , c 0 ∈ J and  ∶ [a, b] × J → R is a global L p -Carathéodory function, where J = R or J = R + .The notions and symbols used in the Introduction will be given later.The IVPs for linear first-order or higher order FDEs were studied in [1][2][3].
When w 0 = 0, the IVP (1)-( 2) is called a Cauchy-type problem for the Riemann-Liouville (R-L) FDEs; see [4, Section 42, p. 829] and [5,Chapter 3].The uniqueness of solutions of (1)-( 2) with c 0 ∈ R was studied in [6] and [4,Theorem 42.1,p. 830], where the solution space is C[a, b],  is continuous and bounded and is a Lipschitz map with Lipschitz constant L > 0 and in [5,Theorem 3.3,p. 148], where the solution space is L 1 (a, b) and the function  is assumed to satisfy the condition (H):  (x, ) ∈ L 1 (a, b) for  ∈ J.The uniqueness of local solutions of (1)- (2) in L 1 (a, b) was studied in [7, section 5].Such Cauchy-type problems for the first-order or higher order FDEs were studied via the corresponding integral equations by employing fixed point theorems.The equivalences in L 1 (a, b) between FDEs and integral equations were studied and applied before, but the conditions used in the equivalences were often incomplete; see the list of papers in [7, p. 586].This resulted in incomplete understanding of uniqueness of solutions of the IVP (1)- (2).Our results of this paper show that we can only obtain the uniqueness of generalized normal solutions of the IVP (1)-( 2) if one considers solutions in the space L p ([a, b]; J); see Theorem 11.When w 0 ≠ 0, the IVPs or Cauchy-type problems for the nonlinear first-order FDEs have not been studied previously.
In this paper, we consider (1)-( 2) whenever w 0 = 0 or w 0 ≠ 0. The first step is to prove some new properties of the R-L fractional integrals I  a + for  ∈ (0, 1), give rigorous analysis on the first-order fractional derivatives by considering the equivalence class [u] in L 1 (a, b), and provide the conditions under which the functions that are equal to each other a.e. on [a, b] have the equal first-order fractional derivatives a.e. on [a, b].
The second step is to introduce definitions of solutions, normal solutions (i.e., solutions u satisfying I  a + (u − w 0 ) ∈ AC [a, b]), and generalized normal solutions of ( 1) and ( 1)- (2).Introducing the notions of normal solutions and generalized normal solutions is for establishing the equivalences between (1)-( 2) and the corresponding integral equations.Following these definitions, we investigate the relations among these solutions.It is unknown whether each function v in [u] is a solution or normal solution of ( 1)-( 2) even when u is a normal solution.However, we provide conditions on v under which we prove that v is a solution or normal solution.For a generalized normal solution of ( 1)-( 2), we prove that every function in [u] is a generalized normal solution of (1)- (2).
The third step is to establish equivalence results in L 1 (a, b) between ( 1)-( 2) and the integral equation We prove that two important equivalence results.One is that u ∈ L 1 ([a, b]; J) is a normal solution of (1)-( 2) if and only if I  a + (u − w 0 ) is continuous at a and u is a solution of (3).Another is that u is a generalized normal solution of (1)-( 2) if and only if u is a solution of (3).
The fourth step is to employ the Banach contraction principle to obtain the existence and uniqueness of generalized normal solutions or nonnegative generalized normal solutions u of (1)- (2) in L p (a, b) via studying the existence and uniqueness of solutions or nonnegative solutions of the integral equation (3) using the above second equivalence result, where p = ∞ if c 0 = 0 and p ∈ As applications, we consider the uniqueness of nonnegative generalized normal solutions of IVPs of the first-order FDEs with nonlinearities related to the population models with the growth rates of Ricker type.
In Section 2 of this paper, we recall some basic results of the R-L fractional integrals and provide rigorous analysis on the definition of the first-order fractional derivatives.We provide and prove some new properties of the R-L fractional integrals and the first-order fractional derivatives.
In Section 3, we first give a biological interpretation on the first-order integral I 1 a + and the first-order derivative of I 1 a + using the rate of change of total density of population of a single species.Following the biological interpretation on the first-order integral I 1 a + and the first-order derivative of I 1 a + , we give a conjecture on biological interpretations of the first-order R-L fractional integral I  a + and the first-order derivative of I  a + .There were many geometric and physical interpretations on R-L fractional integrals and fractional derivatives, see [8] and the references therein, but there is a lack of biological interpretations on the R-L fractional integrals and fractional derivatives using the rate of change of population of a single species.
In Section 4, we introduce definitions of solutions, normal solutions, and generalized normal solutions of ( 1)-( 2) and study these solutions and the equivalences between (1)-( 2) and (3).Using the equivalence of generalized normal solutions and the Banach contraction principle, we obtain the existence and uniqueness of generalized normal solutions of ( 1)-( 2).
In Section 5, we apply the uniqueness results of generalized normal solutions of ( 1)-( 2) to the IVPs of the first-order FDEs with nonlinearities related to the population models with growth rates of Ricker type.

FRACTIONAL INTEGRAL AND FIRST-ORDER FRACTIONAL DERIVATIVES
Throughout this paper, we always assume  ∈ (0, 1), a, b ∈ R with a < b and use the usual derivative symbol u ′ to denote the first-order derivative of a real-valued function u.Recall that the R-L fractional integral of order  ∈ (0, ∞) of a function u ∈ L 1 (a, b) is defined by a (Volterra) integral operator: where for each n ∈ N, the integral operators I n a + and I n− a + are said to be the nth-order integral operator and the nth-order R-L fractional integral operator with fraction , respectively.

Lemma 2.
Let  > 0 and  > −1.Then, If we assume further that  +  ≥ 0, then the above result holds for each x ∈ [a, b].
Let w 0 ∈ R. By Lemma 2, we have and We recall the semigroup property of the R-L fractional integral operator I  a + defined in (4); see [4,9,10].A complete proof can be found in [11,Lemma 3.4].Lemma 3. Let ,  ∈ (0, ∞) satisfy  +  ≥ 1 and u ∈ L 1 (a, b).Then, We give new properties of the integral operator I  a + for  ∈ (0, 1).Recall that two functions in L 1 (a, b) which are equal to each other a.e. on [a, b] are treated as a same function in where We treat the set [u] as a function in L 1 (a, b) in some cases.We use meas(E) to denote the Lebesgue measure of a subset E in R. Noting that I  a + maps L 1 (a, b) to L 1 (a, b), by ( 9), we have and no matter whether u(a) exists or not.The set [a, b]∖E 1 is said to be the natural (or proper) domain of u, where E 1 is the same as in (10).
We define the following operations.
Since I  a + (u − w 0 ) is continuous at a, taking limit on both sides of ( 21) as x → a + , ( By ( 21) and ( 22), we have Since and the first result of (i) holds.Note that ( I  a + w 0 ) (a) = 0.It follows from ( 22) that the second result of (i) holds.(ii) By the hypotheses and the proof of the result (i), we see that (23) holds.Since Corollary 4 shows that the condition Now, we recall the definition of the first-order fractional derivative with fraction  ∈ (0, 1).To better understand the definition, we give the following definitions on differentiability a.e. on [a, b] of a function in u ∈ L 1 (a, b).
It is obvious that u ∈ C 1 a.e.[a, b] if and only if u ∈ L 1 (a, b) and u is differentiable a.e. on (a, b).Moreover, Let u be the Cantor function on [a, b] and [a, b], by (24), there exists This with (10) implies By (18), we have Since u ′ (x 0 ) and v ′ (x 0 ) exist, differentiating both sides of ( 27) at x 0 implies It follows that Since meas(F 1 ∪ F 2 ) = 0, it follows from ( 24) and ( 28) that [a, b], by (24), there exists Then, u(x 0 ) = v(x 0 ) and both u ′ (x 0 ) and v ′ (x 0 ) exist.Therefore, there exists  > 0 such that (x 0 − , Since u, v are continuous on (x 0 − , x 0 + ), we have Taking derivative on both sides of the above equation at  Let u ∈ L 1 (a, b).Then, we have Proof.By conditions (i) and (ii) and a similar proof of Corollary 4 (i), we have By (8), [a, b].By condition (iii) and Theorem 2, and By Theorems 1 and 2, we obtain the relation between the first-order fractional derivatives and the R-L fractional derivatives.

CONJECTURE ON BIOLOGICAL INTERPRETATION OF THE FIRST-ORDER FRACTIONAL INTEGRALS AND DERIVATIVES
We first give a biological interpretation on the first-order integral I 1 a + and its derivative using the rate of change of total density of population of a single species.
Assume that u ∶ [a, b] → [0, ∞) is a function and u ∈ L 1 (a, b), where u(x) denotes the density of a population of a single species at x.Then, for each x ∈ [a, b], the first-order R-L integral represents the total density of the population from a to x and for a.e.x ∈ [a, b], the rate of change at x of the total density I 1 a + u of the population from a to x is Next, following the biological interpretations on the first-order integral I 1 a + and the first-order derivative (37), we give a conjecture on the biological interpretations on the first-order fractional integral I  a + and the first-order derivative of I  a + below.We refer to [8] and the references therein for the geometric and physical interpretation on the R-L fractional integrals and fractional derivatives.
For  ∈ (0, 1), following the biological interpretation on the first-order R-L integral I 1 a + , we conjecture that for a.e.x ∈ [a, b], the first-order fractional integral would represent some type of fractional total density of the population from a to x and for a.e.x ∈ [a, b], the first-order R-L fractional derivative represents the rate of change of such a type of fractional total density of the population at x.However, we have not seen any derivations of the first-order R-L fractional derivative D 1− a + u(x) which shows such type of fractional total density of the population from a to x from mathematical biological models.

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF IVPS FOR NONLINEAR FIRST-ORDER FDES
We always assume that  ∈ (0, 1) and J = R or J = R + .We consider the existence and uniqueness of solutions or nonnegative solutions in L 1 (a, b) of the IVPs for the first-order FDEs: subject to the initial condition: lim where When w 0 = 0, the IVP (40)-( 41) is called a Cauchy-type problem for FDEs; see [5, Chapter 3] and [4, Section 42, p. 829].Hence, the nonlinear IVP (40)-( 41) is a generalization of the Cauchy-type problem from w 0 = 0 to w 0 ∈ J.Note that in (40), we do not require the unknown functions u to satisfy that u(a) exists.Therefore, the first-order fractional differential derivative D 1− p,a + u in general is not the Caputo differential derivative D 1− * a + u.Notation: We define an operator F, called the Nemytskii operator, by The existence of nonnegative continuous solutions for the first-order FDEs (40) subject to another initial condition u(a) = c 0 , where u(a) is required to exist, has been studied, for example, in [15] and [17,18].When u(a) exists and w 0 = u(a), the existence of nonnegative continuous solutions in C[0, T] of the IVPs for (40) with [a, b] = [0, T] subject to the initial condition u(a) = c 0 was studied in [15], where ) . The existence and uniqueness of local solutions in C[a, a + h] were studied in [17,Theorem 4.2], where  is continuous and bounded and is a Lipschitz map.Such IVPs do not include the Cauchy-type problems (40)-(41) in general.Below, we provide examples of linear Cauchy-type problems whose solutions may not satisfy u(a) = c 0 and p may not be in We consider solutions in L 1 (a, b) of the IVP of the first-order FDE: subject to the initial condition Example 1.The function u defined in (48) is a solution of ( 46)-(47).
Proof.It is easy to verify that the function u defined in (48) belongs to L 1 (a, b).Applying I  a + to both sides of (48) and using Theorem 1, we have Taking derivative on both sides of the above equation implies By ( 32) and (50), we have and u is a solution of (46).Applying (49) and Theorem 1, we have Since  ∈ (−∞, ), it follows from (49) that By ( 7), we have This with (51) and (52) implies that (47) holds.□ In Example 1, if  1 ≠ 0 and  > 0 or  2 ≠ 0, then u(a) is not defined, so u does not satisfy the initial condition u(a) = c 0 .
Remark 6.By Remark 5, we see that for a solution u ∈ L 1 (a, b) of (40), it is unknown whether each function v in [u] is a solution of (40) because it is unknown whether This with (53) implies and v is a solution of (40).□ As mentioned in Section 2, a sufficient condition for the first-order fractional derivative D 1− p,a + u to exist a.e. on [a, b] is [a, b].Hence, we introduce the notion of a normal solution of the FDE (40).Definition 7. A function u ∈ L 1 ([a, b]; J) is said to be a normal solution of (40) if u is a solution of (40) and satisfies By Remark 6, we see that if u ∈ L 1 (a, b) is a normal solution of (40), then there is no guarantee that each function in [u] is a normal solution of (40).
For a normal solution u of (40), the following result provides sufficient conditions for functions in [u] to be normal solutions.

is a normal solution of (40). If we assume further that u satisfies (41), then v satisfies (41).
Proof.Let u be a normal solution of (40).Then, Since I  a + (v − w 0 ) is continuous at a, taking limit on both sides of (55) and using (55) imply that This with . By (25), we have The result follows from Theorem 5.If u satisfies (41), then by (56), we see that v satisfies (41).□ By Definition 7, a normal solution u of (40) requires that u satisfies the condition which cannot be satisfied in some cases.This leads to introducing the notion of a generalized normal solution of the FDE (40).Definition 8.A function u ∈ L 1 ([a, b]; J) is said to be a generalized normal solution of (40) if u is a solution of (40) and where The following result gives the relations among the solutions, normal solutions, and generalized normal solutions.Its proof follows from the definition of these solutions, and we omit the proof.

Proposition 4.
(i) If u is a generalized normal solution of (40) and I  a + (u − w 0 )(a) = c 0 , then u is a solution of ( 40)-( 41).(ii) If u is a normal solution of (40), then u is a generalized normal solution of (40).(iii) If u is a generalized normal solution of (40) and I  a + (u − w 0 ) is continuous at a, then u is a normal solution of (40).The following result shows that if u is a generalized normal solution of (40), every function in [u] is a generalized normal solution of (40).
(i) implying (ii).Assume that u is a generalized normal solution of (40).Differentiating both sides of (57), we have Since for each x ∈ (a, b], integrating (60) from a to x implies This with (57) implies Applying I 1− a + to both sides of (62), by Lemmas 2 and 3 and Theorem 1, we obtain for each x ∈ (a, b], Differentiating both sides of (63) and using (5) and Theorem 3, we have and (59) holds.(ii) implying (i).By Lemma 2, we have and by Lemma 3, By (67) and Theorem 1, we have for each x ∈ (a, b], (66) Differentiating both sides of (66) shows that u is a solution of (40).Let The following result shows that if u ∈ L 1 ((a, b); J) is a solution of (67), then every function in [u] is a solution of (67).and Hence, v is a solution of (67).□ Now, we study the relations between solutions of the FDE (40) and the integral equation (67).(2) u is a solution of the integral equation (67). Proof.
(2) implying (1).Assume that u is a solution of the integral equation (67).By a proof similar to (ii) implying (i) of Theorem 8, we have for each x ∈ (a, b], Differentiating both sides of (68) shows that u is a solution of (40).Taking limit on both sides of (68) as x → a + implies lim ) (x) = c 0 and (41) holds.By (68), we have (i) u is a normal solution of ( 40)-( 42).(ii) I  a + (u − w 0 ) is continuous at a and u is a solution of (67). Proof.
(i) implying (ii).Assume that u is a normal solution of (40)-(42).By Proposition 4 (ii), u is a generalized normal solution of the IVP (40)-(41).By Theorem 9, u is a solution of (67).(ii) implying (i).Since u is a solution of (67), by ( 2 Theorem 9 allows us to obtain existence of generalized normal solutions of (40)-(41) in L 1 ([a, b]; J) via studying solutions of (67) in L 1 ([a, b]; J).However, it is unknown that under what conditions on  is the map A defined in (67) compact in L 1 ([a, b]; J), that is, A is continuous and maps bounded subsets of L 1 ([a, b]; J) to precompact subsets in L 1 ([a, b]; J).Hence, some well-known fixed point theorems for compact maps in Banach spaces such as Schauder fixed point theorem and Leray-Schauder fixed point theorems cannot be used to obtain fixed points of the map A.
Fortunately, we can employ the well-known Banach contraction principle which can be found in [19] to study the existence and uniqueness of generalized normal solutions or nonnegative generalized normal solutions of (40)-(41).For completeness, we state the Banach contraction principle below.
Let D be a nonempty subset in a Banach space X.Recall that a map A ∶ D → X is said to be a Lipschitz map with Lipschitz constant L if ||Ax − A|| ≤ L||x − || for x,  ∈ D.
A Lipschitz map with Lipschitz constant L < 1 is said to be a contraction.

Lemma 4.
Assume that A ∶ X → X is a contraction with contraction constant L.Then, the following assertions hold.) .
Then, the following assertions hold.

Definition 3 .
[a, b]; see [9, Definition 2.2], [5, p. 70], [14, p. 88], [4, p. 37], and [11, Definition 4.8].The above R-L fractional derivative of a function has been generalized in[1][2][3] as follows.Let w 0 ∈ R. The first-order fractional derivative with fraction  (relative to w 0 [a, b].The letter p in the symbol D 1− p,a + u is the first letter of polynomial and is used to indicate that the fractional differential operator D 1− p,a + contains a polynomial p 0 (x) = w 0 .The numbers 1 and  in the power 1 −  of D 1− p,a + are used to denote the first-order and proper fraction, respectively.The first-order and higher order fractional derivatives of u ∈ L 1 (a, b) were studied in[1][2][3].A sufficient condition for the first-order fractional derivative D 1− p,a + u to exist a.e. on [a, b] is I  a + (u − w 0 ) ∈ AC[a, b].Definition 3 requires neither the existence of u(a) nor w 0 = u(a) since the number w 0 is independent of the function u.

Theorem 4 .
a special case of the fractional differential operator D 1− p,a + given in Definition 3. If u(a) exists and w 0 = u(a), then D 1− p,a + u(x) = D 1− * a + u(x) ∶= (I  a + (u − u(a))) ′ (x) for a.e.x ∈ [a, b].The operator D 1− * a + is called the Caputo differential operator of order 1 − , which requires u(a) to exist; see [9, Definition 3.2] and [5, Section 2.4, p. 90].Note that the fractional derivatives D 1− p,a + u and D 1− a + u do not require the existence of u(a).When u(a) = 0, D 1− * a + u = D 1− a + u, so if u(a) = 0, the Caputo differential operator D 1− * a + is a special case of the R-L fractional differential operator D 1− a + .When u(a) ≠ 0, by (32) with w 0 = u(a), the Caputo differential operator D 1− * a + is not the R-L fractional differential operator D 1− a + .Hence, the class of functions u having Caputo differential derivatives D 1− * a + u and the class of functions u having R-L fractional derivatives D 1− a + u have overlap but do not contain each other.However, the class of functions u which have fractional derivatives D 1− p,a + u contains both classes of the functions which have the R-L fractional derivatives D 1− a + u or the Caputo differential derivatives D 1− * a + u.Remark 5. When D 1− p,a + u(x) exists for a.e.x ∈ [a, b], it is unknown whether D 1− p,a + v(x) exists for a.e.x ∈ [a, b] for each v ∈ [u].The following result provides conditions on u or v such that the first-order fractional derivatives of u and v are equal to each other for a.e.x ∈ [a, b].Let u ∈ L 1 (a, b) and v ∈ [u] satisfy that D 1− p,a + u(x) exists for a.e.x ∈ [a, b].Assume that one of the following conditions holds.
exists for a.e x ∈ [a, b], differentiating both sides of the above equation implies (35).Assume that (ii) holds.Let w 1 = I  a + (v − w 0 ) and w 2 = I  a + (u − w 0 ).By the hypothesis, w 1 , w 2 ∈ C 1 a.e.[a, b] and since v ∈ [u], we have w 1 (x) = w 2 (x) for a.e.x ∈ [a, b].It follows from Theorem 3 that (a, b) and L p ([a, b]; R + ) = L p + (a, b), the cone of nonnegative functions in L p (a, b).To introduce definitions of solutions in L 1 ([a, b]; J) for both (40) and (40)-(41), we first introduce the notion of a global L p -Carathéodory function.Recall that a function  ∶ [a, b]×J → R is said to satisfy the Carathéodory conditions (on [a, b]×J
x) for each x ∈ [a, b].Then,  ∈ AC[a, b].It follows from (66) that I  a + (u − w 0 ) ∈ AC[a, b].It follows from Definition 8 that u is a generalized normal solution of (40).□ Let c 0 ∈ R. We study solutions of (40)-(41) in L 1 (a, b) via the following integral equation u

Theorem 11 .
(a) A has a unique fixed point p in X.(b) For each u 1 in X, the sequence {u n+1 } defined by u n+1 = Au n for each n ∈ N converges to p. (c) ||u n − p|| ≤ L n 1−L ||u 1 − u 2 ||.Recall that a function  ∶ [a, b] × J → R is said to satisfy a Lipschitz condition (in the second variable) with a Lipschitz constant l > 0 if | (x, u) −  (x, v)| ≤ l|u − v| for a.e.x ∈ [a, b] and all u, v ∈ J. (69) Assume that  ∶ [a, b]×J → R satisfies that the map A in (67) maps L p ([a, b]; J) to L p ([a, b]; J).For each u 1 ∈ L p ([a, b]; J), we define a sequence {u n } by u n+1 = Au n for each n ∈ N. Let w 0 , c 0 ∈ J and p ∈ J(c 0 ).Assume that  ∶ [a, b] × J → J satisfies the Carathéodory conditions and the following conditions.(i) There exist  ∈ L p + (a, b) and  > 0 such that | (x, u)| ≤ (x) + u for a.e.x ∈ (a, b) and each u ∈ J. (72) (ii)  satisfies a Lipschitz condition with a Lipschitz constant l ∈ [ 0, Γ(2−) (b−a) 1−