The theory of fractional differential equations (FDEs) [9] and their applications is a topic of great interest in pure and applied mathematics. The popluarity of FDEs and related problems is largely due to the many applications they have to various branches of science and engineering. The varied applications have yielded many different definitions of fractional derivative and fractional integral, which do not coincide in general. Hilfer [7] introduced the generalized Riemann-Liouville fractional derivative of order µ ~(n-1<µ< n ∈ N) and of type ν ~(0≤ ν ≤ 1), defined by

Da+μ,νyt=Ia+νn−μd dxnIa+1−νn−μyt,

which allows one to interpolate between the Riemann-Liouville derivative Da+μ,0= RLDa+μ and the Caputo derivative Da+μ,1= CDa+μ. Furati et al. [5, 6] considered the basic problems of existence, uniqueness and stability of solutions of the nonlinear Cauchy type problem involving a Hilfer fractional derivative.

Very recently, Sousa and Olivera [15] extended the concept of the Hilfer derivative operator and introduced a new definition of the fractional derivative- namely the Ψ-Hilfer fractional derivative of a function of order µ and of type ν with respect to another function Ψ. They discussed its calculus and derived a class of fractional integrals and fractional derivatives by giving a particular value to the function Ψ. In [16] Sousa and Olivera proved a generalized Gronwall inequality involving thier fractional integral with respect to another function and investigated basic results pertaining to the existence, uniqueness of solutions of, and and data dependence of, the Cauchy type problem involving a Ψ-Hilfer differential operator.

The fundamental problem of Ulam [14] was generalized for the stability of FDEs [18]. Stability of any FDE in the Ulam-Hyers sense is the problem of dealing with the replacement of a given FDE by a fractional differential inequality, and obtaining sufficient conditions about ``When the solutions of the fractional differential inequalities are close to the solutions of given FDE ?". For a Ulam-Hyers stability theory of FDEs and its recent development, one can refer to [1, 2, 3, 17, 18] and the references therein.

Huang et al. [8] investigated HU stability of integer order delay differential equations by the method of successive approximation. Kucche and Sutar [10] extended the idea of [8] and investigated the HU stability of nonlinear delay FDEs with the Caputo derivative. Oliveira and Sousa [4, 17] explored Ulam-Hyers and Ulam-Hyers-Rassias stabilities of Ψ- Hilfer nonlinear fractional differential and integrodifferential equations by means of the fixed point theorem of Banach.

Motivated by the work of [10, 15, 16], in this paper, we consider the Ψ-Hilfer fractional differential equation (Ψ-Hilfer FDE) of the form:

(1.1)HDa+μ,ν; Ψy(t)=f(t,y(t)), t∈[a,b], 0<μ<1, 0≤ν≤1, (1.2)Ia+1−ρ; Ψy(a)=ya∈ℝ, ρ=μ+ν−μν,

where HDa+μ,ν; Ψ(⋅) is the (left-sided) Ψ-Hilfer fractional derivative of order µ and type ν, Ia+1−ρ; Ψ is (left-sided) fractional integral of order 1-Π with respect to another function Ψ, in the Riemann-Liouville sense, and f:[a,b]×ℝ→ℝ is a given function that will be specified latter.

The main objective of this paper is to prove the global existence and uniqueness of solutions to Ψ-Hilfer FDE (1.1)-(1.2). Using the method of successive approximations we investigate Ulam-Hyers (HU) and Ulam-Hyers-Rassias (HUR) stability of (1.1). Utilizing the generalized Gronwall inequality [15] we obtain estimations for the difference between two ϵ-approximated solutions of (1.1)-(1.2). With this we derive the results pertaining to uniqueness and dependence of solutions on the initial conditions.

The Ψ-Hilfer FDE (1.1)-(1.2) is quite general in the sense that for different particular values of the parameters μ, ν and for various specific functions Ψ the derivative operator HDa+μ, ν; Ψ reduces to many well known fractional derivative operators that are recorded in [15]. Among these are the: Riemann-Liouville derivative, Caputo derivative, Hilfer derivative, Katugampola Derivative, Caputo-Katugampola Derivative, Hilfer-Katugampola Derivative, Hadmard Derivative, Caputo-Hadmard Derivative, Hilfer-Hadmard Derivative, Chen fractional derivative, Jumarie derivative, Prabhakar derivative, Erd`elyi-Kober derivative, Riesz derivative, Feller derivative, Weyl derivative, Cassar derivative, and Caputo-Riesz derivative.

Moreover, for Ψ(t)=t and ν=1 the results of the current paper yield results from [10] and for Ψ(t)=t and μ=ν=1 yield results from [8].

The paper is organized as follows. In Section 2, some basic definitions and results concerning the Ψ-Hilfer fractional derivative that are important for the development of the paper are given. Section 3 deals with the existence and uniqueness of solutions of the problem (1.1)-(1.2). Section 4 deals with the HU stability of (1.1) via successive approximations. In Section 5, we study an ϵ-approximate solution of (1.1). In Section 6, we provide an illustrative example.

In this section, we recall few definitions, notions and the fundamental results about the fractional integrals of a function with respect to another function [9], and the Ψ-Hilfer fractional operator [15, 16].

Let 0<a<b<∞, Δ=[a,b]⊂ℝ+=[0,∞),0≤ρ<1 and Ψ∈C1(Δ,ℝ) be an increasing function such that Ψ′(x)≠0,∀ x∈Δ. The weighted spaces C1−ρ; Ψ(Δ,ℝ), C1−ρ; Ψρ(Δ,ℝ) and  C1−ρ; Ψμ,ν(Δ,ℝ) of functions are defined as follows:

• (i) C1−ρ; Ψ(Δ,ℝ)=h:(a,b]→ℝ: (Ψ(t)−Ψ(a))1−ρh(t)∈C(Δ,ℝ) , with the norm ||h||C1−ρ;Ψ=maxt∈Δ(Ψ(t)−Ψ(a))1−ρh(t),

• (ii) C1−ρ; Ψρ(Δ,ℝ)=h∈C1−ρ; Ψ(Δ,ℝ):Da+ρh(t)∈C1−ρ;Ψ(Δ,ℝ),

• (iii) C1−ρ; Ψμ,ν(Δ,ℝ)=h∈C1−ρ;Ψ(Δ,ℝ): HDa+μ,νh(t)∈C1−ρ;Ψ(Δ,ℝ).

Definition 2.1.([9, 12])

The Ψ-Riemann fractional integral of order µ > 0 of the function h is given by

Ia+μ; Ψht:=1Γμ∫at LΨ μ (t,η)hηdη,

where

LΨμ(t,η)=Ψ′η Ψt−Ψη μ−1.

Lemma 2.2.

Let µ >0, ν >0 and δ >0. Then

• (i) Ia+μ; ΨIa+ν;Ψh(t)=Ia+μ+ν;Ψh(t),

• (ii) if h(t)=(Ψ(t)−Ψ(a))δ−1, then Ia+μ; Ψh(t)=Γ(δ)Γ(μ+δ)(Ψ(t)−Ψ(a))μ+δ−1.

We need the following results [9, 12] which are useful in the analysis of the paper.

Lemma 2.3.([16])

If µ > 0 and 0 ≤ ρ < 1, then Ia+μ; Ψ is bounded from Cρ; Ψ(Δ,ℝ) to Cρ; Ψ(Δ,ℝ). Also, if ρ ≤ µ, then Ia+μ; Ψ is bounded from Cρ; Ψ(Δ,ℝ) to C(Δ,ℝ).

Definition 2.4.([15])

The Ψ-Hilfer fractional derivative of a function h of order 0 < µ < 1 and type 0 ≤ ν ≤ 1, is defined by

HDa+μ, ν; Ψh(t)=Ia+ν(1−μ);Ψ 1Ψ'td dtIa+(1−ν)(1−μ);Ψh(t).

Lemma 2.5.([15])

If h∈C1(Δ,ℝ), 0 < µ < 1 and 0 ≤ ν ≤ 1 , then

• (i) Ia+μ; ΨHDa+μ,ν;Ψh(t)=h(t)−ΩΨρ(t,a)Ia+(1−ν)(1−μ);Ψh(a), where ΩΨρ(t,a)=(Ψ(t)−Ψ(a)) ρ−1Γ(ρ).

• (ii) HDa+μ, ν; ΨIa+μ; Ψh(t)=h(t).

Definition 2.6.([9])

Let µ > 0, ν > 0. The one parameter Mittag-Leffler function is defined as

Eμ(z)=∑ k=0∞zkΓ(kμ+1),

and the two parameter Mittag-Leffler function is defined as

Eμ,ν(z)=∑ k=0∞zkΓ(kμ+ν).

In this section we derive the existence and uniqueness results of the Cauchy-type problem (1.1)-(1.2) by utilizing the following modified version of contraction principle.

Lemma 3.1.([13])

Let X be a Banach space and let T be an operator which maps the element of X into itself for which T^r is a contraction, where r is a positive integer then T has a unique fixed point.

Theorem 3.2.

Let 0 < µ < 1 and 0≤ ν ≤ 1, and ρ=µ+ν-{µ ν}. Let f:(a,b]×ℝ→ℝ be a function such that f(⋅,y(⋅))∈C1−ρ; Ψ(Δ,ℝ) for any y∈C1−ρ; Ψ(Δ,ℝ), and let f satisfies the Lipschitz condition with respect to second argument

(3.1)|f(t,y1)−f(t,y2)|≤L|y1−y2|,

for all t∈ (a,b] and for all y₁,y₂ ∈ ℛ, where L > 0 is a Lipschitz constant.Then the Cauchy problem (1.1)-(1.2) has a unique solution in C1−ρ; Ψ(Δ,ℝ).

Proof. The equivalent fractional integral to the initial value problem (1.1)-(1.2) is given by [15]

(3.2)y(t)=ΩΨρ(t,a)ya+Ia+μ; Ψf(t,y(t))=ΩΨρ(t,a)ya+1Γ(μ)∫at L Ψμ (t,η)f(η,y(η))dη,t∈(a,b].

Our aim is to prove that the fractional integral (3.2) has a solution in the weighted space C1−ρ; Ψ(Δ,ℝ). Consider the operator T defined on :C1−ρ; Ψ(Δ,ℝ) by

(3.3)(Ty)(t)=ΩΨρ(t,a)ya+1Γ(μ)∫at L Ψμ (t,η)f(η,y(η))dη.

By Lemma 2.3, it follows that Ia+μ; Ψf(.,y(.))∈C1−ρ; Ψ(Δ,ℝ) Clearly, yaΩΨρ(t,a)∈C1−ρ; Ψ(Δ,ℝ). Therefore, from (3.3), we have Ty∈C1−ρ; Ψ(Δ,ℝ) for any y∈C1−ρ; Ψ(Δ,ℝ) This proves T maps C1−ρ; Ψ(Δ,ℝ) into itself. Note that the fractional integral equation (3.2) can be written as fixed point operator equation

y=Ty, y∈C1−ρ; Ψ(Δ,ℝ).

We prove that the above operator equation has a fixed point which will act as a solution for the problem (1.1)-(1.2). For any t ∈ (a,b], consider the space Ct; Ψ=C1−ρ;Ψ([a,t],ℝ) with the norm defined by,

‖y‖Ct;Ψ=maxw∈[a,t](Ψ(w)−Ψ(a))1−ρy(w).

Using mathematical induction for any y1,y2∈Ct;Ψ and t∈(a,b], we prove that

(3.4)‖Tjy1−Tjy2‖Ct;Ψ≤Γ(ρ)(L(Ψ(t)−Ψ(a))μ )jΓ(jμ+ρ)‖y1−y2‖Ct;Ψ, j∈ℕ.

Let any y1,y2∈Ct;Ψ. Then from the definition of operator T given in (3.3) and using Lipschitz condition on f, we have

‖Ty1−Ty2‖Ct;Ψ=maxw∈[0,t](Ψ(w)−Ψ(a))1−ρTy1(w)−Ty2(w)=maxw∈[0,t](Ψ(w)−Ψ(a))1−ρ1Γ(μ)∫awLΨμ(w,η)f(η,y1(η))−f(η,y2(η))dη≤Lmaxw∈[0,t](Ψ(w)−Ψ(a))1−ρ1Γ(μ)∫awLΨμ(w,η)y1(η)−y(η)dη=Lmaxw∈[0,t](Ψ(t)−Ψ(a))1−ρ1Γ(μ)∫awLΨμ(w,η)(Ψ(η)−Ψ(a))ρ−1×(Ψ(η)−Ψ(a))1−ρy1(η)−y2(η)dη≤L(Ψ(t)−Ψ(a))1−ρΓ(μ)∫atLΨμ(t,η)(Ψ(η)−Ψ(a))ρ−1×maxw∈[0,η](Ψ(w)−Ψ(a))1−ρ(y1(w)−y2(w))dη≤L(Ψ(t)−Ψ(a))1−ρΓ(μ)‖y1−y2‖ct;Ψ∫atLΨμ(t,η)(Ψ(η)−Ψ(a))ρ−1dη≤L‖y1−y2‖Ct;Ψ (Ψ(t)−Ψ(a))1−ρIa+μ; Ψ(Ψ(t)−Ψ(a))ρ−1=(L(Ψ(t)−Ψ(a))μ)Γ(ρ)Γ(μ+ρ)‖y1−y2‖Ct;Ψ.

Thus the inequality (3.4) holds for j=1. Let us suppose that the inequality (3.4) holds for j = r ∈N, i.e. suppose

(3.5)‖Try1−Try2‖Ct;Ψ≤Γ(ρ)(L(Ψ(t)−Ψ(a))μ )rΓ(rμ+ρ)‖y1−y2‖Ct;Ψ

holds. Next, we prove that (3.4) holds for j = r + 1. Let y1,y2∈Ct;Ψ and denote y1*=Try1 and y2*=Try2. Then using the definition of operator T and Lipschitz condition on f, we get

‖Tr+1y1−Tr+1y2‖Ct;Ψ=‖T(Try1)−T(Try2)‖Ct;Ψ=‖Ty1*−Ty2*‖Ct;Ψ=maxw∈[a,t](Ψ(w)−Ψ(a))1−ρTy1*(w)−Ty2*(w)=maxw∈[a,t](Ψ(w)−Ψ(a))1−ρ1Γ(μ)∫awLΨμ(w,η)f(η,y1*(η))−f(η,y2*(η)dη≤Lmaxw∈[a,t](Ψ(w)−Ψ(a))1−ρ1Γ(μ)∫awLΨμ(w,η)(Ψ(η)−Ψ(a))ρ−1×(Ψ(η)−Ψ(a))1−ρy1*(η)−y2*(η)dη≤L(Ψ(t)−Ψ(a))1−ρΓ(μ)∫atLΨμ(t,η)(Ψ(η)−Ψ(a))ρ−1×maxw∈[a,t](Ψ(w)−Ψ(a))1−ρy1*(w)−y2*(w)dη≤L(Ψ(t)−Ψ(a))1−ρΓ(μ)∫atLΨμ(t,η)(Ψ(η)−Ψ(a))ρ−1‖y1*−y2*‖Cη;Ψdη

From (3.5), we have

‖y1*−y2*‖Cs;Ψ=‖Try1−Try2‖Cs;Ψ≤Γ(ρ)(L(Ψ(s)−Ψ(a))μ )rΓ(rμ+ρ)‖y1−y2‖Cs;Ψ.

Therefore,

‖Tr+1y1−Tr+1y2‖Ct;Ψ≤L(Ψ(t)−Ψ(a))1−ρΓ(μ)∫atLΨμ(t,η)(Ψ(η)−Ψ(a))ρ−1×Γ(ρ)(L(Ψ(η)−Ψ(a))μ)rΓ(rμ+ρ)‖y1−y2‖Cη;Ψdη≤Lr+1Γ(ρ)Γ(rμ+ρ)‖y1−y2‖Ct;Ψ×(Ψ(t)−Ψ(a))1−ρ1Γ(μ)∫atLΨμ(t,η)(Ψ(η)−Ψ(a))rμ+ρ−1dη≤Lr+1Γ(ρ)Γ(rμ+ρ)‖y1−y2‖Ct;Ψ(Ψ(t)−Ψ(a))1−ρIa+μ(Ψ(t)−Ψ(a))rμ+ρ−1=Lr+1Γ(ρ)Γ(rμ+ρ)‖y1−y2‖Ct;Ψ(Ψ(t)−Ψ(a))1−ρΓ(rμ+ρ)Γ((r+1)μ+ρ)(Ψ(t)−Ψ(a))(r+1)μ+ρ−1=Γ(ρ)(L(Ψ(t)−Ψ(a))μ)r+1Γ((r+1)μ+ρ)‖y1−y2‖Ct;Ψ.

Thus we have

‖Tr+1y1−Tr+1y2‖Ct;Ψ≤Γ(ρ)(L(Ψ(t)−Ψ(a))μ )r+1Γ((r+1)μ+ρ)‖y1−y2‖Ct;Ψ.

Therefore, by principle of mathematical induction the inequality (3.4) holds for all j ∈ N and for every t in Δ. As a consequence we find on the fundamental interval Δ,

(3.6)‖Tjy1−Tjy2‖C 1−ρ;Ψ(Δ,ℝ)≤Γ(ρ)(L(Ψ(b)−Ψ(a))μ )jΓ(jμ+ρ)‖y1−y2‖C 1−ρ;Ψ(Δ,ℝ).

By definition of two parameter Mittag-Leffler function, we have

Eμ,ρ(L(Ψ(b)−Ψ(a))μ)=∑ j=0∞(L(Ψ(b)−Ψ(a))μ)jΓ(jμ+ρ)

Note that (L(Ψ(b)−Ψ(a))μ)jΓ(jμ+ρ) is the j^th term of the convergent series of real numbers. Therefore,

limj→∞(L(Ψ(b)−Ψ(a))μ)jΓ(jμ+ρ)=0.

Thus we can choose j ∈ N such that

Γ(ρ)(L(Ψ(b)−Ψ(a))μ)jΓ(jμ+ρ)<1,

so that Tj is a contraction.

Therefore, by Lemma 3.1, T has a unique fixed point y* in C1−ρ; Ψ(Δ,ℝ), which is a unique solution of the Cauchy type problem (1.1)-(1.2). □

Remark 3.3.

The existence result proved above is with no restriction on the interval Δ=[a,b], and hence solution y* of (1.1)-(1.2) exists for any a,b (0 < a < b < ∞). Thus the Theorem 3.2 guarantees global unique solution in C1−ρ; Ψ(Δ,ℝ).

To discuss HU and HUR stability of (1.1), we adopt the approach of [11, 18]. For ϵ > 0 and continuous function ϕ:Δ ⇒ [0,∞), we consider the following inequalities:

(4.1)|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϵ, t∈Δ, (4.2)|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϕ(t), t∈Δ, (4.3)|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϵϕ(t), t∈Δ.

Definition 4.1.

Problem (1.1) has HU stability if there exists a real number Cf>0 such that for each ϵ > 0 and for each solution y*∈C1−ρ;Ψ(Δ,ℝ) of the inequation (4.1) there exists a solution y∈C1−ρ; Ψ(Δ,ℝ) of (1.1) with

‖y*−y‖C1−ρ;Ψ(Δ,ℝ)≤Cfϵ.

Definition 4.2.

Problem (1.1) has generalized HU stability if there exists a function Cf∈([0,∞)),[0,∞)) with C_f(0)=0 such that for each solution y*∈C1−ρ;Ψ(Δ,ℝ) of the inequation (4.1) there exists a solution y∈C1−ρ; Ψ(Δ,ℝ) of (1.1) with

‖y*−y‖C1−ρ;Ψ(Δ,ℝ)≤Cf(ϵ).

Definition 4.3.

Problem (1.1) has HUR stability with respect to a function ϕ if there exists a real number C_f,ϕ > 0 such that for each solution y*∈C1−ρ;Ψ(Δ,ℝ) of the inequation (4.3) there exists a solution y∈C1−ρ; Ψ(Δ,ℝ) of (1.1) with

|(Ψ(t)−Ψ(a))1−ρ(y*(t)−y(t))|≤Cf,ϕϵϕ(t), t∈(Δ,ℝ).

Definition 4.4.

Problem (1.1) has generalized HUR stability with respect to a function ϕ if there exists a real number C_f,ϕ>0 such that for each solution y*∈C1−ρ;Ψ(Δ,ℝ) of the inequation (4.2) there exists a solution y∈C1−ρ; Ψ(Δ,ℝ) of (1.1) with

|(Ψ(t)−Ψ(a))1−ρ(y*(t)−y(t))|≤Cf,ϕϕ(t),   t∈Δ.

In the next theorem we will make use of the successive approximation method to prove that the Ψ-Hilfer FDE (1.1) is HU stable.

Theorem 4.5.

Let f:(a,b]×ℝ→ℝ be a function such that f(⋅,y(⋅))∈C1−ρ; Ψ(Δ,ℝ) for any y∈C1−ρ; Ψ(Δ,ℝ) and that satisfies the Lipschitz condition

|f(t,y1)−f(t,y2)|≤L|y1−y2|, t∈(a,b], y1,y2∈ℝ,

where L > 0 is a constant. For every ϵ > 0, if y*∈C1−ρ;Ψ(Δ,ℝ) satisfies

|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϵ, t∈Δ,

then there exists a solution y of equation (1.1) in C1−ρ; Ψ(Δ,ℝ) with Ia+1−ρ; Ψy*(a)=Ia+1−ρ;Ψy(a) such that

‖y*−y‖C1−ρ;Ψ(Δ,ℝ)≤(Eμ(L(Ψ(b)−Ψ(a))μ)−1)L(Ψ(b)−Ψ(a))1−ρϵ, t∈Δ.

Proof. Fix any ϵ > 0, let z∈C1−ρ; Ψ(Δ,ℝ) satisfies

(4.4)|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϵ,  t∈Δ.

Then there exists a function σy*∈C1−ρ;Ψ(Δ,ℝ) ( depending on y* ) such that |σy*(t)|≤ϵ, t∈Δ and

(4.5)HDa+μ,ν;Ψy*(t)=f(t,y*(t))+σy*(t), t∈Δ.

If y*(t) satisfies (4.5) then it satisfies the equivalent fractional integral equation

(4.6)y*(t)=ΩΨρ(t,a)Ia+1−ρ;Ψy*(a)+1Γ(μ)∫at L Ψμ (t,η)f(η,y*(η))dη+1Γ(μ)∫at L Ψμ (t,η)σy*(η)dη, t∈Δ.

Define

(4.7)y0(t)=y*(t),  t∈Δ

and consider the sequence {yn}n=1∞⊆C1−ρ; Ψ(Δ,ℝ) defined by

(4.8)yn(t)=ΩΨρ(t,a)Ia+1−ρ; Ψy*(a)+1Γ(μ)∫at L Ψμ (t,η)f(η,y n−1(η))dη, t∈Δ.

Using mathematical induction firstly we prove that for every t ∈ Δ and yj∈C1−ρ;Ψ[a,t]=Ct;Ψ

(4.9)‖yj−y j−1‖Ct;Ψ≤ϵL(L(Ψ(t)−Ψ(a))μ)jΓ(jμ+1)(Ψ(t)−Ψ(a))1−ρ, j∈ℕ.

By definition of successive approximations and using (4.6) we have

‖y1−y0‖Ct;Ψ=maxw∈[a,t]|(Ψ(w)−Ψ(a))1−ρy1(w)−y0(w)|=maxw∈[0,t](Ψ(w)−Ψ(a))1−ρΩΨρ(w,a)Ia+1−ρ;Ψz(a)+Ia+μ; Ψf(w,y0(w))−y0(w)=maxw∈[0,t](Ψ(w)−Ψ(a))1−ρΩΨρ(w,a)Ia+1−ρ;Ψz(a)+Ia+μ;Ψf(w,z(w))−z(w)=maxw∈[0,t](Ψ(w)−Ψ(a))1−ρ1Γ(μ)∫awLΨμ(w,η)σz(η)dη≤maxw∈[0,t](Ψ(w)−Ψ(a))1−ρ1Γ(μ)∫awLΨμ(w,η)|σz(η)|dη≤ϵmaxw∈[0,t](Ψ(w)−Ψ(a))1−ρ1Γ(μ)∫awLΨμ(w,η)dη≤ϵΓ(μ+1)(Ψ(t)−Ψ(a))1−ρ(Ψ(t)−Ψ(a))μ=ϵL(L(Ψ(t)−Ψ(a))μ)Γ(μ+1)(Ψ(t)−Ψ(a))1−ρ,

Therefore,

‖y1−y0‖Ct;Ψ≤ϵL(L(Ψ(t)−Ψ(a))μ)Γ(μ+1)(Ψ(t)−Ψ(a)) 1−ρ,

which proves the inequality (4.9) for j = 1. Let us suppose that the inequality (4.9) holds for j=r∈ℕ, we prove it for j=r+1. By definition of successive approximations and Lipschitz condition on f, we obtain

‖yr+1−yr‖Ct;Ψ=maxw∈[0,t](Ψ(w)−Ψ(a))1−ρyr+1(w)−yr(w)=maxw∈[0,t](Ψ(w)−Ψ(a))1−ρIa+μ;Ψf(w,yr(w))−Ia+μ;Ψf(w,yr−1(w))≤Lmaxw∈[0,t](Ψ(w)−Ψ(a))1−ρ1Γ(μ)∫aw LΨμ (w,η)yr(η)−yr−1(η)dη≤L(Ψ(t)−Ψ(a))1−ρΓ(μ)∫atLΨμ(t,η)(Ψ(η)−Ψ(a))ρ−1×maxw∈[0,t](Ψ(w)−Ψ(a))1−ρyr(w)−yr−1(w)dη=L(Ψ(t)−Ψ(a))1−ρΓ(μ)∫atLΨμ(t,η)(Ψ(η)−Ψ(a))ρ−1‖yr−yr−1‖Cη;Ψdη.

Using the inequality (4.9) for j=r, we have

‖yr+1−yr‖Ct;Ψ≤L(Ψ(t)−Ψ(a))1−ρΓ(μ)∫atLΨμ(t,η)(Ψ(η)−Ψ(a))ρ−1×ϵL(L(Ψ(η)−Ψ(a))μ)rΓ(rμ+1)(Ψ(η)−Ψ(a))1−ρdη=ϵLLr+1Γ(rμ+1)(Ψ(t)−Ψ(a))1−ρIa+μ;Ψ(Ψ(t)−Ψ(a))rμ=ϵLLr+1Γ(rμ+1)(Ψ(t)−Ψ(a))1−ρΓ(rμ+1)Γ((r+1)μ+1)(Ψ(t)−Ψ(a))(r+1)μ

Therefore,

‖yr+1−yr‖Ct;Ψ≤ϵL(L(Ψ(t)−Ψ(a))μ)r+1Γ((r+1)μ+1)(Ψ(t)−Ψ(a))1−ρ,

which is the inequality ((4.9)) for j=r+1. Using the principle of mathematical induction the inequality ((4.9)) holds for every j ∈ N and every t ∈ Δ.

Therefore,

‖yj−y j−1‖C1−ρ;Ψ(Δ,ℝ)≤ϵL(L(Ψ(b)−Ψ(a))μ)jΓ(jμ+1)(Ψ(b)−Ψ(a))1−ρ.

Now using this estimation we have

∑ j=1∞‖yj−y j−1‖C 1−ρ;Ψ(Δ,ℝ)≤ϵL(Ψ(b)−Ψ(a))1−ρ∑ j=1∞ (L (Ψ(b)−Ψ(a))μ)jΓ(jμ+1).

Thus we have

(4.10)∑ j=1∞‖yj−y j−1‖C 1−ρ;Ψ(Δ,ℝ)≤ϵL(Ψ(b)−Ψ(a))1−ρEμ(L(Ψ(b)−Ψ(a))μ)−1.

Hence the series

(4.11)y0+∑ j=1∞(yj−y j−1)

converges in the weighted space C1−ρ; Ψ(Δ,ℝ). Let y∈C1−ρ; Ψ(Δ,ℝ) such that

(4.12)y=y0+∑ j=1∞(yj−y j−1).

Noting that

yn=y0+∑ j=1 n(yj−y j−1)

is the n^th partial sum of the series (4.11), we have

‖yn−y‖C1−ρ;Ψ(Δ,ℝ)→0 as n→∞.

Next, we prove that this limit function y is the solution of fractional integral equation with Ia+1−ρ; Ψy*(a)=Ia+1−ρ;Ψy(a)$. Next, by the definition of successive approximation, for any t ∈ Δ, we have

(Ψ(t)−Ψ(a))1−ρy(t)−ΩΨρ(t,a)Ia+1−ρ; Ψy(a)−1Γ(μ)∫atLΨμ(t,η)f(η,y(η))dη=(Ψ(t)−Ψ(a))1−ρy(t)−ΩΨρ(t,a)Ia+1−ρ; Ψz(a)−1Γ(μ)∫atLΨμ(t,η)f(η,y(η))dη=(Ψ(t)−Ψ(a))1−ρy(t)−yn(t)+1Γ(μ)∫atLΨμ(t,η)f(η,yn−1(η))dη−1Γ(μ)∫atLΨμ(t,η)f(η,y(η))dη≤(Ψ(t)−Ψ(a))1−ρy(t)−yn(t)+(Ψ(t)−Ψ(a))1−ρIa+μ; Ψ{f(t,yn−1(t))−f(t,y(t))}≤y−ynC1−ρ;Ψ[a,b]+L(Ψ(t)−Ψ(a))1−ρ1Γ(μ)∫atLΨμ(t,η)|yn−1(η)−y(η)|dη≤‖y−yn‖C1−ρ;Ψ[a,b]+L‖yn−1−y‖C1−ρ;Ψ[a,b](Ψ(t)−Ψ(a))1−ρIa+μ;Ψ(Ψ(t)−Ψ(a))ρ−1=‖y−yn‖C1−ρ;Ψ[a,b]+LΓρΓ(μ+ρ)(Ψ(t)−Ψ(a))μ‖yn−1−y‖C1−ρ;Ψ[a,b], ∀n∈ℕ.

By taking limit as n → ∞ in the above inequality, for all t ∈ [a,b], we obtain

(Ψ(t)−Ψ(a))1−ρy(t)−ΩΨρ(t,a)Ia+1−ρ;Ψy(a)−1Γ(μ)∫at L Ψμ (t,η)f(η,y(η))dη=0.

Since, (Ψ(t)−Ψ(a))1−ρ≠0 for all t∈Δ, we have

(4.13)y(t)=ΩΨρ(t,a)Ia+1−ρ;Ψy(a)+1Γ(μ)∫at L Ψμ (t,η)f(η,y(η))dη, t∈Δ.

This proves that y is the solution of (1.1)-(1.2) in C1−ρ; Ψ(Δ,ℝ) Further, for the solution y* of inequation (4.4) and the solution y of the equation (1.1), using (4.7) and (4.12), for any t ∈ Δ, we have

|(Ψ(t)−Ψ(a))1−ρ(y*(t)−y(t))|=(Ψ(t)−Ψ(a))1−ρy0(t)−y0(t)+∑j=1∞(yj(t)−yj−1(t))≤∑j=1∞(Ψ(t)−Ψ(a))1−ρ(yj(t)−yj−1(t))≤∑j=1∞‖yj−yj−1‖C1−ρ[a,b]≤ϵL(Ψ(b)−Ψ(a))1−ρ(Eμ(L(Ψ(b)−Ψ(a))μ)−1).

Therefore,

‖y*−y‖C1−ρ;Ψ[a,b]≤(Eμ(L(Ψ(b)−Ψ(a))μ)−1)L(Ψ(b)−Ψ(a))1−ρϵ

This proves the equation (1.1) is HU stable.

Corollary 4.6.

Suppose that the function f satisfies the assumptions of Theorem 4.5. Then, the problem (1.1) is generalized HU stable.

Proof. Set

Ψf(ϵ)=(Eμ(L(Ψ(b)−Ψ(a))μ)−1)L(Ψ(b)−Ψ(a))1−ρϵ,

in the proof of Theorem 4.5. Then Ψ_f(0)=0 and for each y*∈C1−ρ;Ψ(Δ,ℝ) that satisfies the inequality

|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϵ, t∈Δ,

there exists a solution y of equation (1.1) in C1−ρ; Ψ(Δ,ℝ) with Ia+1−ρ; Ψy*(a)=Ia+1−ρ;Ψy(a) such that

‖y*−y‖C1−ρ;Ψ(Δ,ℝ)≤Ψf(ϵ), t∈Δ.

Hence fractional differential equation (1.1) is generalized HU stable.

Theorem 4.7.

Let f:(a,b]×ℝ→ℝ be a function such that f(⋅,y(⋅))∈C1−ρ; Ψ(Δ,ℝ) for any y∈C1−ρ; Ψ(Δ,ℝ) and that satisfies the Lipschitz condition

|f(t,y1)−f(t,y2)|≤L|y1−y2|, t∈(a,b], y1,y2∈ℝ,

where L > 0 is a constant. For every ϵ > 0, if y*∈C1−ρ; Ψ(Δ,ℝ) satisfies

|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϵϕ(t),  t∈Δ,

where ϕ∈C(Δ,ℝ+) is a non-decreasing function such that

|Ia+μ; Ψϕ(t)|≤λϕ(t), t∈Δ

and ν > 0 is a constant satisfying 0 < ν L < 1. Then, there exists a solution y∈C1−ρ; Ψ(Δ,ℝ) of equation (1.1) with Ia+1−ρ; Ψy*(a)=Ia+1−ρ;Ψy(a) such that

|(Ψ(t)−Ψ(a))1−ρ(y*(t)−y(t))|≤λ1−λL(Ψ(b)−Ψ(a))1−ρϵϕ(t), t∈Δ.

Proof. For every ϵ > 0, let y*∈C1−ρ;Ψ(Δ,ℝ) satisfies

|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϵϕ(t), t∈Δ.

Proceeding as in the proof of Theorem 4.5, there exists a function σy*∈C1−ρ;Ψ(Δ,ℝ) (depending on y*) such that

y*(t)=ΩΨρ(t,a)Ia+1−ρ; Ψy*(a)+Ia+μ;Ψf(t,y*(t))+Ia+μ; Ψσy*(t), t∈Δ,

Further, using mathematical induction, one can prove that the sequence of successive approximations {yn}n=1∞⊆C1−ρ; Ψ(Δ,ℝ) defined by

(4.14)yn(t)=ΩΨρ(t,a)Ia+1−ρ; Ψy*(a)+1Γ(μ)∫at L Ψμ (t,η)f(η,y n−1(η))dη, t∈Δ.

satisfy the inequality

(4.15)‖yj−y j−1‖Ct;Ψ≤ϵL(λL)j(Ψ(t)−Ψ(a))1−ρϕ(t), j∈ℕ.

Using the inequation (4.15), we obatin

∑ j=1∞ yj−y j−1C t;Ψ≤ϵL∑j=1∞ (λL)j(Ψ(t)−Ψ(a))1−ρϕ(t).

Thus

(4.16)∑ j=1∞ yj−y j−1C t;Ψ≤ϵλ1−λL(Ψ(t)−Ψ(a))1−ρϕ(t), t∈Δ.

Following the steps as in the proof of the Theorem 4.5, there exists y∈C1−ρ; Ψ(Δ,ℝ) such that ‖yn−y‖C1−ρ;Ψ(Δ,ℝ)→0 as n→∞. This y is the solution of the problem (1.1)-(1.2) with Ia+1−ρ,Ψy(a)=Ia+1−ρ,Ψy*(a), and we have

y=y0+∑ j=1∞(yj−y j−1).

Further, for the solution y* of inequation and the solution y of the equation (1.1), for any t ∈ Δ,

|(Ψ(t)−Ψ(a))1−ρ(y*(t)−y(t))|=(Ψ(t)−Ψ(a))1−ρy0(t)−y0(t)+∑j=1∞(yj(t)−yj−1(t))≤∑j=1∞(Ψ(t)−Ψ(a))1−ρ(yj(t)−yj−1(t))≤∑j=1∞‖yj−yj−1‖Ct,Ψ=ϵλ1−λL(Ψ(t)−Ψ(a))1−ρϕ(t), t∈Δ.

Thus, we have

|(Ψ(t)−Ψ(a))1−ρ(y*(t)−y(t))|≤λ1−λL(Ψ(b)−Ψ(a))1−ρϵϕ(t), t∈Δ.

This proves the equation (1.1) is HUR stable.

Corollary 4.8.

Suppose that the function f satisfies the assumptions of Theorem 4.7.Then, the problem (1.1) is generalized HUR stable.

Proof. Set ϵ=1 and Cf,ϕ=λ1−λL(Ψ(b)−Ψ(a))1−ρin the proof of Theorem 4.7. Then for each solution y*∈C1−ρ;Ψ(Δ,ℝ) that satisfies the inequality

|HDa+μ,ν; Ψy*(t)−f(t,y*(t))|≤ϕ(t), t∈Δ,

there exists a solution y of equation (1.1) in C1−ρ; Ψ(Δ,ℝ) with Ia+1−ρ; Ψy*(a)=Ia+1−ρ;Ψy(a) such that

|(Ψ(t)−Ψ(a))1−ρ(y*(t)−y(t))|≤Cf,ϕϕ(t), t∈Δ.

Hence the fractional differential equation (1.1) is generalized HUR stable.