Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 109, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.109 EVOLUTION ψ-HILFER FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES JIN LIANG, YUNYI MU, TI-JUN XIAO Abstract. This article concerns evolution equations involving ψ-Hilfer fractional derivative in a Banach space. By using the theory of fractional calculus and ψ-Laplace transform, we firstly derive a definition of mild solutions for these equations. Then we establish theorems for the existence and uniqueness of solutions and the approximate controllability (not the exact controllability) of the ψ-Hilfer fractional differential system under appropriate conditions. We focus on the approximate controllability rather than the exact controllability because the exact controllability cannot be achieved generally for the system in infinite-dimensional spaces. We present a new multidimensional Gronwall-type inequality with multiple singular kernels involv- ing exponential factors, which extends essentially many existing results. We also use the new Gronwall-type inequality to study the dependence of the solution on the order and the initial condition for the fractional integro-differential equations involving ψ-Hilfer fractional derivative. Finally, an example is given to illustrate our main results. 1. Introduction In the past decades, applications of fractional calculus have gradually expanded and cov- ered fields such as fluid mechanics, rheology, viscoelasticity, fractional control systems and con- trollers, electroanalytical chemistry, electrical conductivity in biological systems, fractional models of nerves, fractional regression models, etc. When using fractional order derivatives instead of tra- ditional integer order derivatives to describe problems involving hereditary or memory properties, it often not only simplifies the differential equations but also yields results that are closer to reality. We refer the readers to [1, 2, 5, 6, 7, 8, 9, 11, 10, 12, 16, 19, 20, 22, 23, 24, 25, 26, 32, 33, 35, 36, 39] and the reference therein for theory and applications on fractional calculus. The widely studied fractional derivatives include Riemann-Liouville fractional derivatives and Caputo fractional derivatives, and the solutions of equations containing these two types of de- rivative operators have significantly different properties. In [16], Hilfer combined the Riemann- Liouville fractional derivative and the Caputo fractional derivative to obtain a derivative, which is later referred to as the Hilfer fractional derivative in many literature. This derivative is an inter- polation of the Riemann-Liouville fractional derivative and the Caputo fractional derivative, and he studied differential equations involving this derivative([17]). In [13], the authors studied a class of evolution equations containing Hilfer fractional derivatives in Banach spaces. They introduced the definition of mild solutions for these equations through Laplace transform and the density function, and obtained an existence theorem for mild solutions using non-compactness measures and the Ascoli-Arzela theorem. In [41], the authors introduced ψ-Hilfer fractional derivatives and studied equations involving such derivatives. Compared to traditional derivatives, derivatives containing an arbitrary function ψ(t) are more widely used. The concept of exact controllability was first proposed by Kalman ([18]) in 1963 and gradually became an active research field due to its enormous applications in the field of physics. The application of certain control to a natural or artificial system to influence its behavior to meet predetermined goals is called a control system. The controllability of control systems is one of 2020 Mathematics Subject Classification. 34K37, 34A08, 45G05, 45E05. Key words and phrases. ψ-Hilfer; mild solution; approximate controllability; Gronwall-type inequality. ©2025. This work is licensed under a CC BY 4.0 license. Submitted October 11, 2025. Published November 18, 2025. 1 2 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 the fundamental concepts in control theory, which is the basis for studying optimal control and estimation. At present, the controllability problem of differential systems described by differential equations has been studied by many scholars. The controllability problem of differential systems in Banach space is often transformed into a fixed point problem of operators, and the condition of compactness is often indispensable when applying the fixed point theorem. It is well known that the exact controllability cannot be achieved generally for control systems in infinite-dimensional spaces. Therefore, in addition to exact controllability, there is also a more widely used concept of approximate controllability in the controllability of differential systems. Exact controllability refers to the ability of a system to achieve the expected precise value under the influence of control functions, while approximate controllability does not require precise achievement, but only requires the system to reach a region near the expected value. For more research on the controllability of differential systems, please refer to [3, 4, 6, 14, 21, 30, 31] and the references therein. Integral inequalities serve as fundamental tools for quantitatively analyzing solutions to differ- ential and integral equations. Among these, the Gronwall-Bellman inequality has demonstrated wide applications in deriving estimates across ordinary differential equations, partial differen- tial equations, stochastic differential equations, and integral-differential equations, as it provides explicit bounds of unknown functions x(t). Given its broad applicability and significance, nu- merous extensions of this classical inequality have been developed. For works on generalized Gronwall-Bellman inequalities, Gronwall-type inequalities, and their applications, we refer readers to [4, 5, 6, 7, 15, 26, 27, 28, 29, 34, 38, 40, 42, 43] and the references therein. The research on evolution differential equations with ψ-Hilfer fractional derivatives in Banach spaces is blank, and this paper fills this gap. The rest of this paper is organized as follows. In section 2, we introduce some notations, recall some basic known results, and derive a defi- nition of mild solutions for the evolution equations involving ψ-Hilfer fractional derivatives. In section 3, we develop the existence and uniqueness theorem of mild solutions for ψ-Hilfer frac- tional system, and discuss the approximate controllability of the control problem in the weighted space C1−α−β(1−α);ψ under suitable conditions. In section 4, we establish a new multidimensional Gronwall-type inequality involving multiple singular kernels and exponential factors, and use it to study the dependence of solution on the order and the initial condition for the fractional integro- differential ψ-Hilfer fractional equations. Finally, in section 5, an example is given to illustrate the main results. 2. Definition of mild solutions We begin this part by setting some notation. Suppose that ψ(t) ∈ C1[0,∞) is strictly increasing and satisfies that ψ(0) = 0 and limt→∞ψ(t) = ∞. Let ζ(t) be the inverse function of the function ψ(t). Denote by X a Banach space with norm ∥·∥. We denote by C(J,X) the space of all X-valued continuous functions on J with the natural norm ∥x∥C(J,X) = supt∈J ∥x(t)∥. Let C1−γ;ψ(J,X) = { x : ψ1−γ(t)x(t) ∈ C(J,X) } (γ ∈ (0, 1)) with the norm ∥x∥C1−γ;ψ = sup { ψ1−γ(t)∥x(t)∥ : t ∈ J } . Obviously, the space C1−γ;ψ(J,X) is a Banach space. Throughout this article, we assume that the semigroup {T (t)}t≥0 is differentiable and uniformly bounded, that is, there is a constant M > 0 such that ∥T (t)∥ ≤M, ∀t ≥ 0. Definition 2.1 ([16]). The fractional integral of order α(0 < α < 1) involving a general function ψ for a function f is defined by Iα;ψf(t) = 1 Γ(α) ∫ t 0 (ψ(t)− ψ(s))α−1f(s)ψ′(s)ds. Remark 2.2. When ψ(t) = t, the fractional integral Iα;ψ becomes the Riemann-Liouville frac- tional integral Iα([16]). EJDE-2025/109 EVOLUTION ψ-HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 3 Definition 2.3. [41] For any 0 < α ≤ 1, 0 ≤ β ≤ 1, ψ is differentiable, the ψ-Hilfer fractional derivative of order α and type β for a function f is defined by Dα,β;ψf(t) = Iβ(1−α);ψ 1 ψ′(t) d dt I(1−β)(1−α);ψf(t). Remark 2.4. When ψ(t) = t, the ψ-Hilfer fractional derivative becomes the Hilfer fractional derivative ([16]). Definition 2.5 ([27]). For f ∈ L1 loc(R+, X) and λ ∈ C, the ψ-Laplace transform of f is defined as Lψ[f ](λ) = f̃(λ) := ∫ ∞ 0 e−λψ(t)f(t)ψ′(t)dt, as long as the integral on the right hand exists as a Bochner integral. Lemma 2.6. Let 0 < α < 1 and 0 ≤ β ≤ 1. If x ∈ C1−α−β(1−α);ψ(J,X) and x is a solution of the equations ( Dα,β;ψx ) (t) = Ax(t) + g(t), t ∈ J ′, I(1−α)(1−β);ψx(0) = x0, (2.1) then x satisfies the equation x(t) = ( Iβ(1−α)Kα ) (ψ(t))x0 + ∫ t 0 Kα(ψ(t)− ψ(s))g(s)ψ′(s)ds, where Kα(t) = α ∫ ∞ 0 tα−1σξα(σ)T (σt α)dσ, ξq(σ) = 1 q ϖq(σ − 1 q ), σ ∈ (0,∞), ϖq(τ) = 1 π ∞∑ n=1 (−1)n−1τ−qn−1Γ(qn+ 1) n! sin(nπq), τ ∈ (0,∞). Proof. Set ρ(λ) = ∫ ∞ 0 e−λψ(t)g(t)ψ′(t)dt. Applying the ψ-Laplace transform to both sides of the first equation of (2.1), we obtain λαLψ[x](λ)− λβ(α−1)I(1−α)(1−β);ψx(0) = ALψ[x](λ) + ρ(λ). Then (λαI −A)x̃(λ) = λβ(α−1)x0 + ρ(λ); thus ũ(λ) = λβ(α−1)(λαI −A)−1x0 + (λαI −A)−1ρ(λ) = λβ(α−1) ∫ ∞ 0 e−λ αsT (s)x0ds+ ∫ ∞ 0 e−λ αsT (s)ρ(λ)ds. We consider the one-sided stable probability density function in R+ as ϖα(σ) = 1 π ∞∑ n=1 (−1)n−1σ−nα−1Γ(nα+ 1) n! sin(nπα), σ ∈ (0,∞), whose Laplace transform is ∫ ∞ 0 e−λσϖα(σ)dσ = e−λ α , α ∈ (0, 1). (2.2) Then, using (2.2) we obtain∫ ∞ 0 e−λ αsT (s)x0ds = ∫ ∞ 0 ∫ ∞ 0 e−λtσαtα−1ϖα(σ)T (t α)x0dσdt = ∫ ∞ 0 e−λt [ α ∫ ∞ 0 tα−1 σα ϖα(σ)T ( tα σα ) x0dσ ] dt 4 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 = ∫ ∞ 0 e−λψ(t) [ α ∫ ∞ 0 ψα−1(t) σα ϖα(σ)T (ψα(t) σα ) x0dσ ] ψ′(t)dt, and∫ ∞ 0 e−λ αsT (s)ρ(λ)ds = ∫ ∞ 0 ∫ ∞ 0 e−λτσατα−1ϖα(σ)T (τ α) (∫ ∞ 0 e−λtg(ζ(t))dt ) dσdτ = α ∫ ∞ 0 ∫ ∞ 0 e−λϑ ϑα−1 σα ϖα(σ)T (ϑα σα )(∫ ∞ 0 e−λtg(ζ(t))dt ) dϑdσ = α ∫ ∞ 0 (∫ ∞ 0 ∫ τ 0 e−λτ (τ − t)α−1 σα ϖα(σ)T ( (τ − t)α σα ) g(ζ(t))dtdτ ) dσ = ∫ ∞ 0 e−λψ(t) [ α ∫ ψ(t) 0 ∫ ∞ 0 (ψ(t)− s)α−1 σα ϖα(σ)T ( (ψ(t)− s)α σα ) g(ζ(s))dσds ] × ψ′(t)dt = ∫ ∞ 0 e−λψ(t) [ α ∫ t 0 ∫ ∞ 0 (ψ(t)− ψ(s))α−1σξα(σ)T (σ(ψ(t)− ψ(s))α)g(s)ψ′(s)dσds ] ψ′(t)dt. Set Kα(t) = α ∫ ∞ 0 tα−1σξα(σ)T (σt α)dσ. Since the Laplace transform of F(t) := tβ(1−α)−1 Γ(β(1− α)) (0 < β ≤ 1) is L[F(t)](λ) = λβ(α−1), we have that for 0 < β ≤ 1, λβ(α−1) ∫ ∞ 0 e−λ αsT (s)x0ds = L[F(t)](λ)L[Kα(t)](λ)x0 = L [(F ∗Kα)(t)] (λ)x0 = Lψ [( Iβ(1−α)Kα ) (ψ(t)) ] (λ)x0. Observe that∫ ∞ 0 e−λ αsT (s)ρ(λ)ds = ∫ ∞ 0 e−λψ(t) [ ∫ t 0 Kα(ψ(t)− ψ(s))g(s)ψ′(s)ds ] ψ′(t)dt. (2.3) When β = 0, (2.3) also holds if we keep in mind that I0Kα = Kα. Therefore, for 0 < α < 1 and 0 ≤ β ≤ 1, we have x(t) = ( Iβ(1−α)Kα ) (ψ(t))x0 + ∫ t 0 Kα(ψ(t)− ψ(s))g(s)ψ′(s)ds, which completes the proof. □ Remark 2.7. (i) From [11] we have ∥Kα(t)∥ ≤ Mtα−1 Γ(α) , t > 0. (ii) ∥ ( Iβ(1−α)Kα ) (t)∥ ≤ Mtα+β(1−α)−1 Γ(α+ β(1− α)) , t > 0. Proof. ∥ ( Iβ(1−α)Kα ) (t)∥ = ∥∥ 1 Γ(β(1− α)) ∫ t 0 (t− s)β(1−α)−1Kα(s)ds ∥∥ EJDE-2025/109 EVOLUTION ψ-HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 5 ≤ M Γ(α)Γ(β(1− α)) ∫ t 0 (t− s)β(1−α)−1sα−1ds = Mtα+β(1−α)−1 Γ(α+ β(1− α)) . □ 3. Approximate controllability In this section, we consider the approximate controllability of the following ψ-Hilfer fractional control system in a Banach space X:( Dα,β;ψx ) (t) = Ax(t) +Bu(t) + f(t, x(t)), t ∈ J ′, I(1−α)(1−β);ψx(0) = x0, (3.1) where Dα,β;ψ, α ∈ (0, 1), β ∈ [0, 1], is the ψ-Hilfer fractional derivative of order α and type β with the lower limit 0; b > 0 is a constant, J = [0, b], J ′ = (0, b]; Closed unbounded operator A (D(A) ⊆ X) generates a C0 semigroup T (t) on [0,∞); The semilinear function f : J × X → X is a given function to be specified later; x0 ∈ X; The control function u takes its value in V = Lr(J, U) ( r > 1 α ) , and U is a Banach space; B : V → Lr(J,X) is a linear operator. According to Lemma 2.6, we give the following definition. Definition 3.1. A function x ∈ C1−α−β(1−α);ψ(J,X) is called a mild solution of problem (3.1) if it satisfies the integral equation x(t) = ( Iβ(1−α)Kα ) (ψ(t))x0 + ∫ t 0 Kα(ψ(t)− ψ(s))[Bu(s) + f(s, x(s))]ψ′(s)ds. Definition 3.2. Let x(·;u) be a mild solution of problem (3.1) corresponding to the control u(·) ∈ V and the initial value x0 ∈ X. The set Kb(f) := {x(b;u) : u(·) ∈ V } is called the reachable set of problem (3.1) at terminal time b. If Kb(f) = X, problem (3.1) is said to be approximately controllable on J . Before we give the existence and uniqueness lemma of mild solutions of problem (3.1), we pose the following assumptions: (H1) There exist a function µ(·) ∈ Lr(J,R+) and a positive constant ℓ1 such that ∥f(t, x)∥ ≤ µ(t) + ℓ1ψ 1−α−β(1−α)(t)∥x∥, for a.e. t ∈ J and each x ∈ X. (H2) There exists a positive constant ℓ2 such that ∥f(t, x1)− f(t, x2)∥ ≤ ℓ2∥x1 − x2∥, ∀xi ∈ X(i = 1, 2). Lemma 3.3. If (H1), (H2) are satisfied, then for any control function u(·) ∈ V , there exists a unique mild solution for the control problem (3.1) on C1−α−β(1−α);ψ(J,X). Proof. Define the operator T as follows: (Tx)(t) = ( Iβ(1−α)Kα ) (ψ(t))x0 + ∫ t 0 Kα(ψ(t)− ψ(s))[Bu(s) + f(s, x(s))]ψ′(s)ds. (3.2) From our hypotheses, it follows that T maps C1−α−β(1−α);ψ(J,X) into itself. Then, we show that Tn is a contraction mapping on C1−α−β(1−α);ψ(J,X). As a matter of fact, for each x1, x2 ∈ C1−α−β(1−α);ψ(J,X) and t ∈ J , we can obtain ψ1−α−β(1−α)(t)∥(Tx1)(t)− (Tx2)(t)∥ ≤ ψ1−α−β(1−α)(t) ∫ t 0 ∥Kα(ψ(t)− ψ(s))[f(s, x1(s))− f(s, x2(s))]∥ψ′(s)ds ≤ Mℓ2 Γ(α) ψ1−α−β(1−α)(t) ∫ t 0 (ψ(t)− ψ(s))α−1ψα+β(1−α)−1(s) 6 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 × ψ1−α−β(1−α)(s)∥x1(s)− x2(s)∥ψ′(s)ds ≤ Γ(α+ β(1− α))Mℓ2ψ α(t) Γ(2α+ β(1− α)) ∥x1 − x2∥C1−α−β(1−α);ψ . (3.3) Using (3.2), (3.3), and arguing by induction on n, we easily see that ψ1−α−β(1−α)(t)∥(Tnx1)(t)− (Tnx2)(t)∥ ≤ Γ(α+ β(1− α)) (Mℓ2ψ α(t)) n Γ((n+ 1)α+ β(1− α)) ∥x1 − x2∥C1−α−β(1−α);ψ . Hence, we can obtain ∥(Tnx1)(t)− (Tnx2)(t)∥C1−α−β(1−α);ψ ≤ Γ(α+ β(1− α)) (Mℓ2ψ α(t)) n Γ((n+ 1)α+ β(1− α)) ∥x1 − x2∥C1−α−β(1−α);ψ . Because lim n→∞ (Mℓ2ψ α(t)) n Γ((n+ 1)α+ β(1− α)) = 0, there exists a positive integer N such that Γ(α+ β(1− α)) (Mℓ2ψ α(t)) N Γ((N + 1)α+ β(1− α)) < 1. So TN is a contraction mapping on C1−α−β(1−α);ψ(J,X). Then by a well-known extension of the Banach contraction mapping theorem, T has a unique fixed point x(t) on C1−α−β(1−α);ψ(J,X), which is the unique mild solution of problem (3.1). □ Next, we study the approximate controllability of the evolution ψ-Hilfer fractional differential equations (3.1) in the Banach space X. We denote the Nemytskil operator associated with the semilinear function f by Nf : C1−α−β(1−α);ψ(J,X) → Lr(J,X), Nf (x)(t) = f(t, x(t)). The linear bounded operator H : Lr(J,X) → X is defined as Hg = ∫ b 0 Kα(ψ(b)− ψ(s))ψ′(s)g(s)ds, g(·) ∈ Lr(J,X). By Definition 3.2, we easily know that if for any x0 ∈ X and u(·) ∈ V , Kb(f) = X, then problem (3.1) is approximately controllable on J . Therefore, if for any target state ξ ∈ X and each ε > 0, there exists a control function uε(·) ∈ V , such that the mild solution of problem (3.1) satisfies∥∥ξ − ( Iβ(1−α)Kα ) (ψ(b))x0 −HNf (xε)−HBuε ∥∥ < ε, (3.4) where xε(t) = x(t;uε), t ∈ (0, b], then problem (3.1) is approximately controllable on J . To analyze the approximate controllability of problem (3.1), we introduce the assumption (H3) There exists a positive constant ℓ3 such that ∥f(t, x1)− f(t, x2)∥ ≤ ℓ3ψ 1−α−β(1−α)(t)∥x1 − x2∥, ∀xi ∈ X(i = 1, 2), t ∈ J. The following lemma will be used to establish the approximate controllability of problem (3.1). Lemma 3.4. If (H1) and (H3) are satisfied, then ∥x∥C1−α−β(1−α);ψ ≤ σEα ( Mℓ1ψ 1−β(1−α)(b) ) , ∥x− y∥C1−α−β(1−α);ψ ≤ ϕEα ( Mℓ3ψ 1−β(1−α)(b) ) ∥Bu−Bv∥Lr(J,X), where x and y are the unique mild solutions of problem (3.1) with respect to u and v (u, v ∈ V ), respectively, Eα is the Mittag-Leffler function defined by Eα(z) = ∞∑ i=0 zk Γ(kα+ 1) , EJDE-2025/109 EVOLUTION ψ-HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 7 ϕ = Mψ1−1/r(b) Γ(α) ( r − 1 rα− 1 )1−1/r, σ = M Γ(α+ β(1− α)) ∥x0∥+ Mψ1−1/r(b) Γ(α) ( r − 1 rα− 1 )1−1/r ( ∥Bu∥Lr(J,X) + ∥µ∥Lr(J,X) ) . Proof. Since x is the unique mild solution of (3.1) with respect to u ∈ V in C1−α−β(1−α);ψ(J,X), we have x(t) = ( Iβ(1−α)Kα ) (ψ(t))x0 + ∫ t 0 Kα(ψ(t)− ψ(s))[Bu(s) + f(s, x(s))]ψ′(s)ds. For t ∈ J , we have ψ1−α−β(1−α)(t)∥x(t)∥ ≤ ψ1−α−β(1−α)(t)∥ ( Iβ(1−α)Kα ) (ψ(t))x0∥+ ψ1−α−β(1−α)(t) × ∫ t 0 ∥Kα(ψ(t)− ψ(s))ψ′(s)Bu(s)∥ ds + ψ1−α−β(1−α)(t) ∫ t 0 ∥Kα(ψ(t)− ψ(s))ψ′(s)f(s, x(s))∥ ds ≤ M Γ(α+ β(1− α)) ∥x0∥+ M Γ(α) ψ1−α−β(1−α)(t) ∫ t 0 (ψ(t)− ψ(s))1−αψ′(s) × ∥Bu(s)∥ds+ M Γ(α) ψ1−α−β(1−α)(t) ∫ t 0 (ψ(t)− ψ(s))α−1ψ′(s) × [ µ(s) + ℓ1ψ 1−α−β(1−α)(s)∥x(s)∥ ] ds ≤ M Γ(α+ β(1− α)) ∥x0∥+ Mψ1−1/r(b) Γ(α) × ( r − 1 rα− 1 )1−1/r ( ∥Bu∥Lr(J,X) + ∥µ∥Lr(J,X) ) + Mℓ1ψ 1−α−β(1−α)(b) Γ(α) ∫ t 0 (ψ(t)− ψ(s))α−1ψ′(s)ψ1−α−β(1−α)(s)∥x(s)∥ds. Setting M(t) = ψ1−α−β(1−α)(t)∥x(t)∥, in the above inequality we obtain M(t) ≤ σ + Mℓ1ψ 1−α−β(1−α)(b) Γ(α) ∫ t 0 (ψ(t)− ψ(s))α−1ψ′(s)M(s)ds. By [27, Theorem 3], we can obtain M(t) ≤ σEα ( Mℓ1ψ 1−α−β(1−α)(b)ψα(t) ) ≤ σEα ( Mℓ1ψ 1−β(1−α)(b) ) . So ∥x∥C1−α−β(1−α);ψ = sup t∈J ψ1−α−β(1−α)(t)∥x(t)∥ ≤ σEα ( Mℓ1ψ 1−β(1−α)(b) ) . A parallel argument yields ∥x− y∥C1−α−β(1−α);ψ ≤ ϕEα ( Mℓ3ψ 1−β(1−α)(b) ) ∥Bu−Bv∥Lr(J,X). This completes the proof. □ Moreover, to discuss the approximate controllability of problem (3.1), we assume that (H4) For each ε > 0 and δ ∈ Lr(J,X), there exists a control function u ∈ Lr(J, U) such that ∥Hδ −HBu∥ < ε, (3.5) ∥Bu∥Lr(J,X) < C∥δ∥Lr(J,X), (3.6) 8 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 where C is a positive constant independent of δ ∈ Lr(J,X), and Cℓ3Eα ( Mℓ3ψ 1−β(1−α)(b) )Mψ1−1/r(b) Γ(α) ( r − 1 rα− 1 )1−1/r < 1. (3.7) Theorem 3.5. Suppose that the hypotheses of Lemma 3.4 and (H4) hold. Then problem (3.1) is approximately controllable on J . Proof. As the domain D(A) of the operator A is dense in X, we need only prove that D(A) ⊂ Kb(f), i.e., for any ξ ∈ D(A) and each ε > 0, there exits a uε ∈ V , such that ∥ξ − ( Iβ(1−α)Kα ) (ψ(b))x0 −HNf (xε)−HBuε∥ < ε, (3.8) where xε(t) = x(t;uε), t ∈ (0, b]. First, for each x0 ∈ X, due to the differentiability of the C0-semigroup T (t)(t > 0), we know that (Iβ(1−α)Kα)(ψ(b))x0 ∈ D(A). So, for any given ξ ∈ D(A), there exists a function ω ∈ Lr(J,X) such that Hω = ξ − ( Iβ(1−α)Kα ) (ψ(b))x0, for example, we can take ω(t) = [Γ(α)]2(ψ(b)− ψ(t))1−α ψ(b) [ Wα(ψ(b)− ψ(t)) + 2ψ(t) dWα(ψ(b)− ψ(t)) dt ] × [ ξ − (Iβ(1−α)Kα)(ψ(b))x0 ] , t ∈ (0, b), where Wα(t) = t1−αKα(t). Now, we show that there exists a control function uε ∈ V such that the inequality (3.8) holds. Indeed, for any given ε > 0 and u1 ∈ V , by (H4), there exists a u2 ∈ V , such that ∥ξ − ( Iβ(1−α)Kα ) (ψ(b))x0 −HNf (x1)−HBu2∥ < ε 22 , where x1(t) = x(t;u1), t ∈ (0, b]. Denote x2(t) = x(t;u2), t ∈ (0, b]. Using hypothesis (H4) and Lemma 3.4 again, we have that there exists v2 ∈ V such that ∥HBv2 − [HNf (x2)−HNf (x1)]∥ < ε 23 . and ∥Bv2∥Lr(J,X) ≤ C ∥Nf (x2)(·)−Nf (x1)(·)∥ ≤ Cℓ3∥x2 − x1∥C1−α−β(1−α);ψ ≤ Cℓ3Eα ( Mℓ3ψ 1−β(1−α)(b) )Mψ1−1/r(b) Γ(α) ( r − 1 rα− 1 )1−1/r∥Bu2 −Bu1∥Lr(J,X). Let u3(t) = u2(t)− v2(t), u3 ∈ V . Then ∥ξ − ( Iβ(1−α)Kα ) (ψ(b))x0 −HNf (x2)−HBu3∥ ≤ ∥ξ − ( Iβ(1−α)Kα ) (ψ(b))x0 −HNf (x1)−HBu2∥+ ∥HBv2 − [HNf (x2)−HNf (x1)]∥ ≤ ( 1 22 + 1 23 ) ε. By induction we can obtain a sequence {un(·)} ⊂ V satisfying ∥ξ − ( Iβ(1−α)Kα ) (ψ(b))x0 −HNf (xn)−HBun+1∥ < ( 1 22 + · · ·+ 1 2n+1 ) ε, where xn(t) = x(t;un), t ∈ (0, b], and ∥Bun+1 −Bun∥Lr(J,X) ≤ Cℓ3Eα ( Mℓ3ψ 1−β(1−α)(b) )Mψ1−1/r(b) Γ(α) ( r − 1 rα− 1 )1−1/r ∥Bun −Bun−1∥Lr(J,X). EJDE-2025/109 EVOLUTION ψ-HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 9 From (3.7), we see that the sequence {Bun : n ∈ N+} is a Cauchy sequence in the Banach space Lr(J,X). Hence, there exists a function τ(·) ∈ Lr(J,X), such that Bun(·) = τ(·) in Lr(J,X), which implies that for each ε > 0, there exists an integer N > 0, such that ∥HBuN+1 −HBuN∥ ≤ ε 2 . Therefore, ∥ξ − ( Iβ(1−α)Kα ) (ψ(b))x0 −HNf (xN )−HBuN∥ ≤ ∥ξ − ( Iβ(1−α)Kα ) (ψ(b))x0 −HNf (xN )−HBuN+1∥+ ∥HBuN+1 −HBuN∥ ≤ ( 1 22 + · · ·+ 1 2N+1 ) ε+ ε 2 < ε, which yields the approximate controllability of problem (3.1). □ 4. A new Gronwall-type inequality and the dependence of solution on the order and the initial condition First, we present the following multivariate Gronwall-type inequality with multiple different singular kernels involving exponential factors, which generalizes many existing results. Theorem 4.1. Suppose that m, p ∈ N, βki > 0, ai ≥ 0, ψi(t) is an increasing and positive monotone differentiable function on (ai, Ti], ζi(t) is a locally integrable function on (ai, Ti] (1 ≤ i ≤ m, 1 ≤ k ≤ p), a(t1, t2, . . . , tm) is a nonnegative locally integrable function on [a1, T1)× [a2, T2)× · · · × [am, Tm) (some Ti ≤ +∞) and gk(t1, t2, . . . , tm)(1 ≤ k ≤ p) is a nonnegative, nondecreasing continuous function defined on [a1, T1)× [a2, T2)×· · ·× [am, Tm), gk(t1, t2, . . . , tm) ≤ C (constant), and suppose u(t1, t2, . . . , tm) is nonnegative and locally integrable on [a1, T1) × [a2, T2) × · · · × [am, Tm) with u(t1, t2, . . . , tm) ≤ a(t1, t2, . . . , tm) + p∑ k=1 gk(t1, t2, . . . , tm) ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 eζ(si)−ζi(ti)ψ′ i(si) × (ψi(ti)− ψi(si)) βki−1u(s1, s2, . . . , sm)ds1 . . . dsm, on [a1, T1)× [a2, T2)× · · · × [am, Tm). Then u(t1, t2, . . . , tm) ≤ a(t1, t2, . . . , tm) + ∞∑ n=1 ∑ 0≤l1,...,lp≤n l1+···+lp=n ( n l1, . . . , lp ) p∏ k=1 (gk(t1, t2, . . . , tm))lk × ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 eζi(si)−ζi(ti)ψ′ i(si) × [∏p k=1(Γ(βki)) lk Γ( ∑p k=1 lkβki) (ψi(ti)− ψi(si)) ∑p k=1 lkβki−1 ] × a(s1, s2, . . . , sm)ds1 . . . dsm, ai ≤ ti < Ti (1 ≤ i ≤ m), (4.1) where ( n l1, . . . , lp ) = n! l1! . . . lp! , l1 + · · ·+ lp = n. Proof. Define an operator P by (Pu)(t1, t2, . . . , tm) = p∑ k=1 gk(t1, t2, . . . , tm) ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 eζ(si)−ζi(ti)ψ′ i(si) × (ψi(ti)− ψi(si)) βki−1u(s1, s2, . . . , sm)ds1 . . . dsm, 10 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 for each locally integrable function u(t1, t2, . . . , tm). Then the fact u(t1, t2, . . . , tm) ≤ f(t1, t2, . . . , tm) + (Pu)(t1, t2, . . . , tm) implies u(t1, t2, . . . , tm) ≤ n∑ i=0 (P if)(t1, t2, . . . , tm) + (Pn+1u)(t1, t2, . . . , tm), (4.2) for all n ∈ N. Next, we want to prove the following (4.3) by induction: (Pnu)(t1, t2, . . . , tm) ≤ ∑ 0≤l1,...,lp≤n l1+···+lp=n ( n l1, . . . , lp ) p∏ k=1 (gk(t1, t2, . . . , tm))lk × ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 eζi(si)−ζi(ti)ψ′ i(si) [∏p k=1(Γ(βki)) lk Γ( ∑p k=1 lkβki) ] × (ψi(ti)− ψi(si)) ∑p k=1 lkβki−1u(s1, s2, . . . , sm)ds1 . . . dsm, (4.3) for all n ∈ N. Clearly, (4.3) is true for n = 1. Suppose that (4.3) holds for n = l. We want to prove that (4.3) also holds for n = l + 1. In fact, we have (P l+1u)(t) = p∑ j=1 gj(t1, t2, . . . , tm) ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 eζi(si)−ζi(ti)ψ′ i(si) × (ψi(ti)− ψi(si)) βji−1(P lu)(s1, s2, . . . , sm)ds1 . . . dsm ≤ p∑ j=1 ∑ 0≤l1,...,lp≤l l1+···+lp=l ( l l1, . . . , lp ) gj(t1, t2, . . . , tm) ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 eζi(si)−ζi(ti) × ψ′ i(si)(ψi(ti)− ψi(si)) βji−1 p∏ k=1 (gk(s1, s2, . . . , sm))lk ∫ sm am · · · ∫ s2 a2 ∫ s1 a1 × m∏ i=1 eζi(τi)−ζi(si)ψ′ i(τi) [∏p k=1(Γ(βki)) lk Γ( ∑p k=1 lkβki) (ψi(si)− ψi(τi)) ∑p k=1 lkβki−1 ] × u(τ1, τ2, . . . , τm)dτ1 . . . dτmds1 . . . dsm. By exchanging the integration order and noting that the functions gi are nondecreasing, we obtain∫ t1 a1 eζ1(s1)−ζ1(t1)ψ′ 1(s1)(ψ1(t1)− ψ1(s1)) βj1−1 p∏ k=1 (gk(s1, s2, . . . , sm))lk × ∫ s1 a1 eζ1(τ1)−ζ1(s1)ψ′ 1(τ1) [∏p k=1(Γ(βk1)) lk Γ( ∑p k=1 lkβk1) (ψ1(s1)− ψ1(τ1)) ∑p k=1 lkβk1−1 ] × u(τ1, τ2, . . . , τm)dτ1ds1 ≤ p∏ k=1 (gk(t1, s2, . . . , sm))lk [ ∏ k ̸=j(Γ(βk1)) lk ](Γ(βj1)) lj+1 Γ( ∑ k ̸=j lkβk1 + (lj + 1)βj1) × ∫ t1 a1 eζ1(s1)−ζ1(t1)ψ′ 1(s1)(ψ1(t1)− ψ1(s1)) βj1+ ∑p k=1 lkβk1−1u(s1, τ2, . . . , τm)ds1, where we use that∫ t1 τ1 (ψ1(t1)− ψ1(s1)) βj1−1(ψ1(s1)− ψ1(τ1)) ∑p k=1 lkβk1−1ψ′ 1(s)ds = (ψ1(t1)− ψ1(τ1)) βj1+ ∑p k=1 lkβk1−1B(βj1, p∑ k=1 lkβk1) EJDE-2025/109 EVOLUTION ψ-HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 11 and B(βj1, p∑ k=1 lkβk1) = Γ(βj1)Γ( ∑p k=1 lkβk1) Γ( ∑ k ̸=j lkβk1 + (lj + 1)βj1) . Repeating this process for m times, we obtain (P l+1u)(t) ≤ p∑ j=1 ∑ 0≤l1,...,lp≤l l1+···+lp=l ( l l1, . . . , lp )[∏ k ̸=j (gk(t1, s2, . . . , sm))lk ] (gj(t1, s2, . . . , sm))lj+1 × ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 [ ∏ k ̸=j(Γ(βki)) lk ](Γ(βji)) lj+1 Γ( ∑ k ̸=j lkβki + (lj + 1)βji) eζi(si)−ζi(ti)ψ′ i(si) × (ψi(ti)− ψi(si)) ∑ k ̸=j lkβki+(lj+1)βji−1u(s1, s2, . . . , sm)ds1 . . . dsm. Therefore, (P l+1u)(t) = ∑ 0≤l1,...,lp≤l+1 l1+···+lp=l+1 p∑ j=1 ( l l1, . . . , lj−1, lj − 1, lj+1, . . . , lp ) p∏ k=1 (gk(t1, s2, . . . , sm))lk × ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 ∏p k=1(Γ(βki)) lk Γ( ∑p k=1 lkβki) eζi(si)−ζi(ti)ψ′ i(si) × (ψi(ti)− ψi(si)) ∑p k=1 lkβki−1u(s1, s2, . . . , sm)ds1 . . . dsm = ∑ 0≤l1,...,lp≤l+1 l1+···+lp=l+1 ( l + 1 l1, . . . , lp ) p∏ k=1 (gk(t1, s2, . . . , sm))lk × ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 ∏p k=1(Γ(βki)) lk Γ( ∑p k=1 lkβki) eζi(si)−ζi(ti)ψ′ i(si) × (ψi(ti)− ψi(si)) ∑p k=1 lkβki−1u(s1, s2, . . . , sm)ds1 . . . dsm, where we have used m∑ j=1 ( n l1, . . . , lj−1, lj − 1, lj+1, . . . , lm ) = ( n+ 1 l1, . . . , lm ) , in which ∑m j=1 lj = n+ 1, and we take ( n l1, . . . , lm ) = 0 when li = −1 for some i. Thus we show that (4.3) also holds for n = l + 1. Therefore, (4.3) is true. By (4.3) we can see that (Pnu)(t) → 0 as n → ∞. From (4.2) and (4.3), we see that (4.1) holds. The proof is complete □ Remark 4.2. Theorem 4.1 extends essentially many existing results. For example, taking ζi ≡ 0 (1 ≤ i ≤ m) in Theorem 4.1, we obtain the following inequality in [27], while taking ζi ≡ 0 (1 ≤ i ≤ m) and p = 1 in Theorem 4.1, we obtain the inequality in [26]; taking m = p = 1, a = 0, ζ ≡ 0, and ψ(t) = t in Theorem 4.1, we obtain the inequality in [43]; and taking m = p = 1, g(t) ≡ b, a = 0, ζ ≡ 0, and ψ(t) = t in Theorem 4.1, we obtain the inequality in [15]. Corollary 4.3. Suppose that m, p ∈ N, βki > 0, ai ≥ 0, ψi(t) is an increasing and posi- tive monotone differentiable function on (ai, Ti] (1 ≤ i ≤ m, 1 ≤ k ≤ p), a(t1, t2, . . . , tm) is a nonnegative locally integrable function on [a1, T1) × [a2, T2) × · · · × [am, Tm) (some Ti ≤ +∞) and gk(t1, t2, . . . , tm)(1 ≤ k ≤ p) is a nonnegative, nondecreasing continuous function defined on 12 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 [a1, T1)× [a2, T2)× · · · × [am, Tm), gk(t1, t2, . . . , tm) ≤ C (constant), and suppose u(t1, t2, . . . , tm) is nonnegative and locally integrable on [a1, T1)× [a2, T2)× · · · × [am, Tm) with u(t1, t2, . . . , tm) ≤ a(t1, t2, . . . , tm) + p∑ k=1 gk(t1, t2, . . . , tm) ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 ψ′ i(si)(ψi(ti)− ψi(si)) βki−1 × u(s1, s2, . . . , sm)ds1 . . . dsm, on [a1, T1)× [a2, T2)× · · · × [am, Tm). Then u(t1, t2, . . . , tm) ≤ a(t1, t2, . . . , tm) + ∞∑ n=1 ∑ 0≤l1,...,lp≤n l1+···+lp=n ( n l1, . . . , lp ) p∏ k=1 (gk(t1, t2, . . . , tm))lk × ∫ tm am · · · ∫ t2 a2 ∫ t1 a1 m∏ i=1 ψ′ i(si) [∏p k=1(Γ(βki)) lk Γ( ∑p k=1 lkβki) (ψi(ti)− ψi(si)) ∑p k=1 lkβki−1 ] × a(s1, s2, . . . , sm)ds1 . . . dsm, ai ≤ ti < Ti (1 ≤ i ≤ m), where ( n l1, . . . , lp ) = n! l1! . . . lp! , l1 + · · ·+ lp = n. Using Theorem 4.1, we obtain the following result of the dependence of the solution on the order and the initial condition for the fractional Cauchy problem. Theorem 4.4. Let f : [0, b] ×X ×X → X be a bounded function and satisfy the Lipschitz-type condition with respect to the second and third variable, i.e., for any t ∈ [0, b], xi, yi ∈ X (i = 1, 2), there exist a L > 0 and an integrable function ζ(t) ≤ 0, a.e. t ∈ [0, b] such that ∥f(t, x1, x2)− f(t, y1, y2)| ≤ Leζ(t) (∥x1 − y1∥+ ∥x2 − y2∥) . Assume that x(t) and y(t) are the solutions of the following initial-value integro-differential equa- tions (4.4) and (4.5), respectively:( Dα,β;ψx ) (t) = Ax(t) + f ( t, x(t), ∫ t 0 ρ(t, s)x(s)ds ) , 0 < α < 1, 0 ≤ β ≤ 1, t ∈ (0, b], I1−γ;ψx(0) = x0, α ≤ γ = α+ β − αβ < 1, (4.4) and( Dα′,β′;ψy ) (t) = Ay(t) + f ( t, y(t), ∫ t 0 ρ(t, s)y(s)ds ) , 0 < α′ < 1, 0 ≤ β′ ≤ 1, t ∈ (0, b], I1−γ ′;ψy(0) = y0, α′ ≤ γ′ = α′ + β′ − α′β′ < 1, (4.5) where b > 0, and ρ(t, s) = eζ(s)−ζ(t)(ψ(t)− ψ(s))r−1ψ′(s) (r ∈ (0, 1]). Then, for t ∈ (0, b], ∥x(t)− y(t)∥ ≤ A(t) + ∞∑ n=1 (LM)n ∞∑ k=0 Ckn (Γ(r))n−k (Γ(α′))k(Γ(α′ + r))n−k × ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))nα ′+(n−k)r−1A(s)ψ′(s)ds, where A(t) = ∥ ( Iβ(1−α)Kα ) (ψ(t))x0 − ( Iβ(1−α)Kα ) (ψ(t))y0∥ + ∫ t 0 ∥Kα(ψ(t)− ψ(s))−Kα′(ψ(t)− ψ(s))∥ψ′(s)ds · ∥f∥, and ∥f∥ = sup(t,x1,x2)∈(0,b]×X×X∥f(t, x1, x2)∥. EJDE-2025/109 EVOLUTION ψ-HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 13 Proof. By Lemma 2.6, the solution of (4.4) satisfies x(t) = ( Iβ(1−α)Kα ) (ψ(t))x0 + ∫ t 0 Kα(ψ(t)− ψ(s))f ( s, x(s), ∫ s 0 ρ(s, τ)x(τ)dτ ) ψ′(s)ds. Also byLemma 2.6, the solution of (4.5) satisfies y(t) = ( Iβ ′(1−α′)Kα′ ) (ψ(t))y0 + ∫ t 0 Kα′(ψ(t)− ψ(s))f ( s, y(s), ∫ s 0 ρ(s, τ)y(τ)dτ ) ψ′(s)ds. Then we have ∥x(t)− y(t)∥ ≤ ∥ ( Iβ(1−α)Kα ) (ψ(t))x0 − ( Iβ ′(1−α′)Kα′ ) (ψ(t))y0∥ + ∫ t 0 ∥Kα(ψ(t)− ψ(s))−Kα′(ψ(t)− ψ(s))∥ ∥∥f(s, x(s),∫ s 0 ρ(s, τ)x(τ)dτ )∥∥ψ′(s)ds + ∫ t 0 ∥Kα′(ψ(t)− ψ(s))∥ ∥∥f(s, x(s),∫ s 0 ρ(s, τ)x(τ)dτ ) − f ( s, y(s), ∫ s 0 ρ(s, τ)y(τ)dτ ) ψ′(s)ds ∥∥ ≤ A(t) + LM Γ(α′) ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))α ′−1∥x(s)− y(s)∥ψ′(s)ds + LM Γ(α′) ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))α ′−1 × [ ∫ s 0 eζ(τ)−ζ(s)(ψ(s)− ψ(τ))r−1∥x(τ)− y(τ)∥ψ′(τ)dτ ] ψ′(s)ds ≤ A(t) + LM Γ(α′) ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))α ′−1∥x(s)− y(s)∥ψ′(s)ds + LMΓ(r) Γ(α′ + r) ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))α ′+r−1∥x(s)− y(s)∥ψ′(s)ds, where A(t) = ∥ ( Iβ(1−α)Kα ) (ψ(t))x0 − ( Iβ ′(1−α′)Kα′ ) (ψ(t))y0∥ + ∫ t 0 ∥Kα(ψ(t)− ψ(s))−Kα′(ψ(t)− ψ(s))∥ψ′(s)ds · ∥f∥. An application of Theorem 4.1 (with m = 1 and p = 2) yields ∥x(t)− y(t)∥ ≤ A(t) + ∞∑ n=1 (LM)n ∞∑ k=0 Ckn (Γ(r))n−k (Γ(α′))k(Γ(α′ + r))n−k × ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))nα ′+(n−k)r−1A(s)ψ′(s)ds, where Cmn = n! m!(n−m)! is a binomial coefficient. Thus, the proof is complete. □ Remark 4.5. In Theorem 4.4, we do not require the function ζ(t) to be nondecreasing on [0, b], so using Corollary 4.3 without exponential factors cannot lead to the conclusion of Theorem 4.4. Corollary 4.6. Under the hypotheses of Theorem 4.4, if α = α′ and β = β′, then ∥x(t)− y(t)∥ ≤ 1 Γ(α+ β(1− α)) [ M(ψ(t))α+β(1−α)−1 + ∞∑ n=1 LnMn+1 ∞∑ k=0 Ckn × (Γ(r))n−k (Γ(α))k(Γ(α+ r))n−k ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))nα+(n−k)r−1 14 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 × (ψ(s))α+β(1−α)−1ψ′(s)ds ] ∥x0 − y0∥, for t ∈ (0, b]. Proof. If α = α′ and β = β′, then A(t) = ∥ ( Iβ(1−α)Kα ) (ψ(t))(x0 − y0)∥ ≤ M(ψ(t))α+β(1−α)−1 Γ(α+ β(1− α)) ∥x0 − y0∥. By Theorem 4.4, we have ∥x(t)− y(t)∥ ≤ A(t) + ∞∑ n=1 (LM)n ∞∑ k=0 Ckn (Γ(r))n−k (Γ(α))k(Γ(α+ r))n−k × ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))nα+(n−k)r−1A(s)ψ′(s)ds = 1 Γ(α+ β(1− α)) [ M(ψ(t))α+β(1−α)−1 + ∞∑ n=1 LnMn+1 ∞∑ k=0 Ckn × (Γ(r))n−k (Γ(α))k(Γ(α+ r))n−k ∫ t 0 eζ(s)−ζ(t)(ψ(t)− ψ(s))nα+(n−k)r−1 × (ψ(s))α+β(1−α)−1ψ′(s)ds ] ∥x0 − y0∥, for t ∈ (0, b]. The proof is complete. □ 5. An example In this section, we give an example to show the applicability of the results obtained in previous sections. Let X = {u(t) : u(t) ∈ L2[0, π], u(t) is a real function} and U = X. We define the inner product and norm on X respectively, for u1, u2 ∈ X, by ⟨u1, u2⟩ = ∫ π 0 u1(t)u2(t)dt, ∥u1∥X = (∫ π 0 u21(t)dt )1/2 . We define the operator A : D(A) ⊂ X → X by D(A) := {v ∈ X : v′′ ∈ X, v(0) = v(π) = 0}, Au = ∂2u ∂x2 . From [34] we know that −A has eigenvalues of the form n2 (n ∈ N+), and the corresponding normalized eigenfunctions are given by en = √ 2 π sin(nx) (n ∈ N+). Moreover, A generates a compact analytic semigroup {T (t)}t≥0 on X, and T (t)u = ∞∑ n=1 e−n 2t⟨u, en⟩en. We can verify that ∥T (t)∥ ≤ e−t for all t ≥ 0, and take M = 1. Furthermore, by [37] we know that {T (t)}t≥0 is continuous in the uniform operator topology for t > 0. For each u(·) ∈ V = L2(J, U), we have u(t) = ∞∑ n=1 un(t)en, un(t) =< u(t), en > . The operator B is defined by Bu(t) = ∞∑ n=1 vn(t)en, where vn(t) = { 0, t ∈ [0, 1− 1 n3 ], un(t), t ∈ (1− 1 n3 , 1], where n ∈ N+. Then we can see that ∥Bu(·)∥ ≤ ∥u(·)∥, which means that B ∈ L ( V,L2(J,X) ) . EJDE-2025/109 EVOLUTION ψ-HILFER FRACTIONAL DIFFERENTIAL EQUATIONS 15 We consider the following fractional differential control problem involving ψ-Hilfer fractional derivative: (Dα,β;ψx)(t) = Ax(t) + f(t, x(t)) +Bu(t), t ∈ J ′ = (0, 1], I(1−α)(1−β);ψx(0) = x0, (5.1) where α = 4 5 , β = 3 4 , x0 ∈ X, ψ(t) = t3, and f(t, x) = Lt3/20 sin(x), t ∈ (0, 1]. Next, we verify that hypothesis (H4) holds. To do this, for each h(·) ∈ L2(J,X), let l = ∫ 1 0 Kα(ψ(1)− ψ(s))ψ′(s)h(s)ds = ∞∑ n=1 lnen, where ln = ⟨l, en⟩. We can take un(t) = 3n3 1− e−ψ(3) lne −n3(1−ψ(t)), 1− 1 n3 ≤ t ≤ 1, ln = ∫ 1 1− 1 n3 ∫ ∞ 0 (1− ψ(t))− 1 5σξ 4 5 e−n 3σ(1−ψ(t)) 4 5 u(t)ψ′(t)dσdt. And define u(t) := ∞∑ n=1 ũn(t)en, where ũn(t) = { 0, t ∈ [0, 1− 1 n3 ], un(t), t ∈ (1− 1 n3 , 1], for n ∈ N+. So, for each given function h(·) ∈ L2(J,X), there exists u(·) ∈ V such that∫ 1 0 Kα(1− ψ(s))ψ′(s)Bu(s)ds = ∫ 1 0 Kα(1− ψ(s))ψ′(s)h(s)ds, which implies that condition (3.5) of (H4) holds. Moreover, we can obtain ∥Bu(·)∥2 = ∞∑ n=1 ∫ 1 1− 1 n3 |u(t)|2dt = ( 1− e−ψ(3) )−1 ∞∑ n=1 3n3l2n = 4 3 ( 1− e−ψ(3) )−1 ∞∑ n=1 ( 1− e−ψ(3)n 3 )∫ 1 0 |hn(t)|2dt ≤ 4 3 ( 1− e−ψ(3) )−1 |h(·)|2. Hence, condition (3.6) of (H4) is also satisfied, and if 4 √ 15 9(1− e−ψ(3))Γ( 45 ) ℓ3E 4 5 (ℓ3) < 1, then problem (5.1) is approximately controllable on J . Acknowledgments. 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Jin Liang School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China Email address: jinliang@sjtu.edu.cn Yunyi Mu (corresponding author) School of Arts and Sciences, Shanghai Dianji University, Shanghai 201306, China Email address: muyy@sdju.edu.cn Ti-Jun Xiao Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China Email address: tjxiao@fudan.edu.cn 1. Introduction 2. Definition of mild solutions 3. Approximate controllability 4. A new Gronwall-type inequality and the dependence of solution on the order and the initial condition 5. An example Acknowledgments References