Well-Posedness and Stability for a Differential Problem with Hilfer-Hadamard Fractional Derivative

Abstract

Motivated by the Hilfer fractional derivative (which interpolates the Riemann-Liouville derivative and the Caputo derivative), we consider a new type of fractional derivative (which interpolates the Hadamard derivative and its Caputo counterpart). We prove the well-posedness for a basic Cauchy type fractional differential equation involving this kind of derivative. This is established in an appropriate underlying space after proving the equivalence of this problem with a certain corresponding Volterra integral equation.

1. Introduction

In this work, we are concerned with the Hadamard derivative _𝐻𝒟^𝛼_𝑎+𝑓(𝑥)=1Γ()𝑥𝑑1−𝛼𝑑𝑥^𝑥_𝑎𝑥log𝑡^−𝛼𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏. (1) Its Caputo counterpart is ^𝑐_𝐻𝒟^𝛼_𝑎+𝑓=𝒥_𝑎^1−𝛼₊𝑥𝑑𝑑𝑥𝑓, (2) where 𝒥^𝛼_𝑎+𝑓(𝑥)1∶=Γ(𝛼)^𝑥_𝑎𝑥log𝑡^𝛼−1𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏 (3) (see [1–5]). Here, we consider the following fractional derivative _𝐻𝒟_𝑎^𝛼,𝛽₊𝑓(𝑥)=𝒥_𝑎^{𝛽(1−𝛼)}₊⋅_𝐻𝒟_𝑎^{𝛼+𝛽−𝛼𝛽}₊𝑓(𝑥). (4) This type of fractional derivative interpolates the Hadamard fractional derivative (𝛽=0) and the Caputo-Hadamard fractional derivative (𝛽=1). It has been introduced recently in [6]. In introducing this new fractional derivative we were motivated by the Hilfer fractional derivative of order 0<𝛼<1 and type 0≤𝛽≤1 (see [7]) 𝐷₀^𝛼,𝛽₊𝑦(𝑥)=𝐼₀^{𝛽(1−𝛼)}₊𝑑𝐼𝑑𝑥^{0(1−𝛽)(1−𝛼)+}𝑓(𝑥), (5) which interpolates the Riemann-Liouville derivative and the Caputo derivative.

We study the existence and uniqueness of solutions of a basic fractional differential equation _𝐻𝒟_𝑎^𝛼,𝛽₊𝑦(𝑥)=𝑓𝑥,𝑦,𝑥>𝑎>0 (6) with an appropriate initial condition in a suitable underlying space after proving the equivalence of this problem with a corresponding Volterra integral equation. In addition to that, we discuss the stability of solutions for a large and important class of nonlinearities. We find that solutions decay to zero at a logarithmic rate as time goes to infinity. To this end, we prove an inequality (which is important by itself).

The literature is very rich in works on well-posedness for fractional differential equations [8–17] (see also the books [1, 2, 4, 5] and the survey paper [18]) to cite but a few. The Hadamard fractional derivative may be found in the books [1, 2, 4–6]. Differential equations involving such a derivative and others have been treated in [2, 3]. In contrast with the well-posedness, the stability issue and the long time behavior is not well studied [6, 12, 19–23].

The rest of the paper is organized as follows: the next section contains some material needed in our proofs. The different fractional derivatives as well as the new one are defined there. In Section 3, we present our problem and prove an existence and uniqueness result after establishing the equivalence of the differential problem with its corresponding integral equation. Section 4 is devoted to a stability result.

2. Preliminaries

In this section we present some definitions, lemmas, properties, and notation which will be used in our theorems later.

Definition 1 (see [2]). Let Ω=[𝑎,𝑏] (0<𝑎<𝑏<∞) be a finite interval and 0≤𝛾<1, we introduce the weighted space 𝐶_𝛾,log[𝑎,𝑏] of continuous functions 𝑔 on (𝑎,𝑏]𝐶_𝛾,log[]=(]𝑥𝑎,𝑏𝑔∶𝑎,𝑏⟶𝐑∶log𝑎^𝛾𝑔(𝑥)∈𝐶[𝑎,𝑏]. (7) In the space 𝐶_𝛾,log[𝑎,𝑏], we define the norm 𝑔_𝐶𝛾,log=𝑥log𝑎^𝛾𝑔(𝑥)_𝐶,𝐶_0,log[𝑎,𝑏]=𝐶[𝑎,𝑏]. (8)

Definition 2 (see [2]). Let 𝛿=𝑥(𝑑/𝑑𝑥) be the 𝛿-derivative, for 𝑛∈𝐍 we denote by 𝐶^𝑛_𝛿,𝛾[𝑎,𝑏] (0≤𝛾<1) the Banach space of functions 𝑔 which have continuous 𝛿-derivatives on [𝑎,𝑏] up to order 𝑛−1 and have the derivative 𝛿^𝑛𝑔 of order 𝑛 on (𝑎,𝑏] such that 𝛿^𝑛𝑔∈𝐶_𝛾,log[𝑎,𝑏]𝐶^𝑛_𝛿,𝛾[]=𝛿𝑎,𝑏_𝑘𝑔∈𝐶[𝑎,𝑏],𝑘=0,…,𝑛−1,𝛿^𝑛𝑔∈𝐶_𝛾,log[𝑎,𝑏] (9) with the norm 𝑔_{𝐶𝑛𝛿,𝛾}=^𝑛−1_𝑘=0𝛿^𝑘𝑔_𝐶+𝛿^𝑛𝑔_𝐶𝛾,log. (10) When 𝑛=0, we set 𝐶⁰_𝛿,𝛾[𝑎,𝑏]=𝐶_𝛾,log[𝑎,𝑏]. (11)

Definition 3 (see [2]). Let (𝑎,𝑏)(0≤𝑎<𝑏≤∞) be a finite or infinite interval of the half-axis 𝐑⁺ and let 𝛼>0. The Hadamard left-sided fractional integral 𝒥^𝛼_𝑎+𝑓 of order 𝛼>0 is defined by 𝒥^𝛼_𝑎+𝑓(𝑥)1∶=Γ(𝛼)^𝑥_𝑎𝑥log𝑡^𝛼−1𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏, (12) provided that the integral exists. When 𝛼=0, we set 𝒥⁰_𝑎+𝑓=𝑓. (13)

Definition 4 (see [2]). Let (𝑎,𝑏)  (0≤𝑎<𝑏≤∞) be a finite or infinite interval of the half-axis 𝐑⁺ and let 𝛼>0. The Hadamard right-sided fractional integral 𝒥^𝛼_𝑏⁻𝑓 of order 𝛼>0 is defined by 𝒥^𝛼_𝑏⁻𝑓(𝑥)1∶=Γ(𝛼)^𝑏_𝑥𝑡log𝑥^𝛼−1𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏, (14) provided that the integral exists. When 𝛼=0, we set 𝒥⁰_𝑏⁻𝑓=𝑓. (15)

Definition 5 (see [2]). The left-sided Hadamard fractional derivative of order 𝛼 (0≤𝛼<1) on (𝑎,𝑏) is defined by _𝐻𝒟^𝛼_𝑎+𝑓(𝑥)∶=𝛿𝒥_𝑎^1−𝛼₊𝑓(𝑥); (16) that is, _𝐻𝒟^𝛼_𝑎+𝑓(𝑥)=1Γ()𝑥𝑑1−𝛼𝑑𝑥^𝑥_𝑎𝑥log𝑡^−𝛼𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏. (17) In particular, when 𝛼=0 we have _𝐻𝒟⁰_𝑎+𝑓=𝑓. (18)

Definition 6 (see [2]). The right-sided Hadamard fractional derivative of order 𝛼 (0≤𝛼<1) on (𝑎,𝑏) is defined by _𝐻𝒟^𝛼_𝑏⁻𝑓(𝑥)∶=−𝛿𝒥_𝑏^1−𝛼−𝑓(𝑥); (19) that is, _𝐻𝒟^𝛼_𝑏⁻𝑓(𝑥)𝑥𝑑=−1𝑑𝑥Γ(1−𝛼)^𝑏_𝑥𝑡log𝑥^−𝛼𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏. (20) In particular, when 𝛼=0 we have _𝐻𝒟⁰_𝑏⁻𝑓=𝑓. (21)

Definition 7. Let (𝑎,𝑏) be a finite interval of the half-axis 𝐑⁺. The fractional derivative ^𝑐_𝐻𝒟^𝛼_𝑎+𝑓 of order 𝛼 (0<𝛼<1) on (𝑎,𝑏) defined by ^𝑐_𝐻𝒟^𝛼_𝑎+𝑓=𝒥_𝑎^1−𝛼₊𝛿𝑓, (22) where 𝛿=𝑥(𝑑/𝑑𝑥), is called the Hadamard-Caputo fractional derivative of order 𝛼.

In the rest of the paper we shall assume 𝑎≠0 when considering an interval (𝑎,𝑏).

Lemma 8 (see [2]). Let 𝑛∈𝐍₀={0,1,…} and let 𝜇₁ and 𝜇₂ be real numbers such that 0≤𝜇₁≤𝜇₂<1. (23) The following embeddings hold: 𝐶^𝑛_𝛿[𝑎,𝑏]⟶𝐶^𝑛_𝛿,𝜇1[𝑎,𝑏]⟶𝐶^𝑛_𝛿,𝜇2[𝑎,𝑏], (24) with 𝑓_{𝐶𝑛2𝛿,𝜇}≤𝐾_𝛿𝑓_{𝐶𝑛1𝛿,𝜇},𝐾_𝛿𝑏=min1,log𝑎^{𝜇2−𝜇1},𝑎≠0. (25) In particular, 𝐶[𝑎,𝑏]⟶𝐶_𝜇1,log[𝑎,𝑏]⟶𝐶_𝜇2,log[𝑎,𝑏], (26) with 𝑓_{𝐶𝜇2,log}≤𝑏log𝑎^{𝜇2−𝜇1}𝑓_{𝐶𝜇1,log},𝑎≠0. (27)

Lemma 9 (see [2]). If 𝛼>0, 𝛽>0 and 0<𝑎<𝑏<∞, then 𝒥^𝛼_𝑎+𝑡log𝑎^𝛽−1(𝑥)=Γ𝛽Γ𝑥𝛼+𝛽log𝑎^{𝛽+𝛼−1}_𝐻,𝒟^𝛼_𝑎+𝑡log𝑎^𝛽−1(𝑥)=Γ𝛽Γ𝑥𝛽−𝛼log𝑎^{𝛽−𝛼−1}. (28) In particular, if 𝛽=1 and 𝛼≥0, then the Hadamard fractional derivative of a constant is not equal to zero: _𝐻𝒟^𝛼_𝑎+1(𝑥)=1Γ()𝑥1−𝛼log𝑎^−𝛼, (29) when 0<𝛼<1.

Lemma 10 (see [2]). Let 𝛼>0,  𝛽>0 and 0≤𝜇<1. If 0<𝑎<𝑏<∞, then, for 𝑓∈𝐶_𝜇,log[𝑎,𝑏]𝒥^𝛼_𝑎+𝒥^𝛽_𝑎+𝑓=𝒥_𝑎^𝛼+𝛽₊𝑓 (30) holds at any point 𝑥∈(𝑎,𝑏]. When 𝑓∈𝐶[𝑎,𝑏] this relation is valid at any point 𝑥∈[𝑎,𝑏].

Lemma 11 (see [2]). Let 0<𝛼≤1 and 0<𝑎<𝑏<∞. The equality (_𝐻𝒟^𝛼_𝑎+𝑦)(𝑥)=0 is valid for 𝑦(𝑥)𝑥=𝑐log𝑎^𝛼−1 (31) for any 𝑐∈𝐑.

Theorem 12 (see [2]). Let 0<𝛼<1 and 0<𝑎<𝑏<∞. Also let 𝒥_𝑎^1−𝛼₊𝑓 be the Hadamard fractional integral of order 1−𝛼 of the function 𝑓.
If 𝑓∈𝐶_𝜇,log[𝑎,𝑏](0≤𝜇<1) and 𝒥_𝑎^1−𝛼₊𝑓∈𝐶¹_𝛿,𝜇[𝑎,𝑏], then 𝒥^𝛼_𝑎+𝐻𝒟^𝛼_𝑎+𝑓𝑥)(𝑥)−𝒥(=𝑓^{𝑎1−𝛼+}𝑓(𝑎)Γ(𝛼)𝑥log𝑎^𝛼−1 (32) holds at any point 𝑥∈(𝑎,𝑏]. If 𝑓∈𝐶[𝑎,𝑏] and 𝒥_𝑎^1−𝛼₊𝑓∈𝐶¹_𝛿[𝑎,𝑏], then the relation holds at any point 𝑥∈[𝑎,𝑏].

Lemma 13 (see [2]). Let  0<𝑎<𝑏<∞,𝛼>0 and 0≤𝜇<1.(a)If 𝜇>𝛼>0, then the fractional integration operator 𝒥^𝛼_𝑎+ is bounded from 𝐶_𝜇,log[𝑎,𝑏] into 𝐶_{𝜇−𝛼,log}[𝑎,𝑏]: 𝒥^𝛼_𝑎+𝑓_{𝐶𝜇−𝛼,log}≤𝑘₁𝑓_𝐶𝜇,log, (33) where 𝑘₁=𝑏log𝑎^𝛼Γ1−𝜇Γ.1+𝛼−𝜇 (34) In particular, 𝒥^𝛼_𝑎+ is bounded in 𝐶_𝜇,log[𝑎,𝑏].(b)If 𝜇≤𝛼, then the fractional integration operator 𝒥^𝛼_𝑎+ is bounded from 𝐶_𝜇,log[𝑎,𝑏] into 𝐶[𝑎,𝑏]: 𝒥^𝛼_𝑎+𝑓_𝐶≤𝑘₂𝑓_𝐶𝜇,log, (35) where 𝑘₂=𝑏log𝑎^𝛼−𝜇Γ1−𝜇Γ1+𝛼−𝜇. (36) In particular, 𝒥^𝛼_𝑎+ is bounded in 𝐶_𝜇,log[𝑎,𝑏].

Lemma 14 (see [2]). The Hadamard fractional integration operator 𝒥^𝛼_𝑎+ of order 𝛼(𝛼>0) is a mapping from 𝐶[𝑎,𝑏] to 𝐶[𝑎,𝑏] and 𝒥^𝛼_𝑎+𝑔_{𝐶[𝑎,𝑏]}≤1(𝛼)𝑏𝛼Γlog𝑎^𝛼𝑔_{𝐶[𝑎,𝑏]}, (37) where 𝑔∈𝐶[𝑎,𝑏].

Lemma 15 (see [17]). Let 0≤𝛾<1,0<𝑎<𝑐<𝑏<∞,𝑔∈𝐶_𝛾,log[𝑎,𝑐] and 𝑔∈𝐶[𝑐,𝑏]. Then, 𝑔∈𝐶_𝛾,log[𝑎,𝑏] and 𝑔_{𝐶𝛾,log[𝑎,𝑏]}≤max𝑔_{𝐶𝛾,log[𝑎,𝑐]},𝑏log𝑎^𝛾𝑔_{𝐶[𝑐,𝑏]}. (38)

Lemma 16 (see [24]). Let 𝑓, 𝑔 be two continuous, positive functions defined on [𝑡₀,∞), 𝑡₀≥0, and 𝑤∶[0,∞)→[0,∞) be a continuous monotonic nondecreasing function such that 𝑤(0)=0 and 𝑤(𝑥)>0 for 𝑥>0. If 𝑢 is a positive differentiable function on [𝑡₀,∞) that satisfies 𝑢^′(𝑡)≤𝑓(𝑡)𝑤(𝑢(𝑡))+𝑔(𝑡),𝑡∈𝑡₀,∞, (39) then we have 𝑢(𝑡)≤𝑊⁻¹𝑊𝑢𝑡₀+^𝑡_𝑡0𝑔(𝑠)𝑑𝑠+^𝑡_𝑡0𝑓(𝑠)𝑑𝑠, (40) for the values of 𝑡, for which the right-hand side is well-defined, where 𝑊(𝑟)=^𝑟_𝑟0𝑑𝑠𝑤(𝑠),𝑟>𝑟₀>0. (41)

Theorem 17 (Banach fixed point theorem [2]). Let (𝑈,𝑑) be a non-empty complete metric space, let 0≤𝑤<1, and let 𝑇∶𝑈→𝑈 be a map such that, for every 𝑢,𝑣∈𝑈, the relation 𝑑(𝑇𝑢,𝑇𝑣)≤𝑤𝑑(𝑢,𝑣),0≤𝑤<1 (42) holds. Then, the operator 𝑇 has a unique fixed point 𝑢^∗∈𝑈.
Furthermore, if 𝑇^𝑘(𝑘∈𝐍) is the sequence of operators defined by 𝑇¹=𝑇,𝑇^𝑘=𝑇𝑇^𝑘−1𝑘∈𝐍⧵{1}, (43) then, for any 𝑢₀∈𝑈, the sequence {𝑇^𝑘𝑢₀}^∞_𝑘=1 converges to the above fixed point 𝑢^∗.

Theorem 18 (Young’s inequality). If 𝑎 and 𝑏 are nonnegative real numbers and 𝑝 and 𝑞 are positive real numbers such that 1/𝑝+1/𝑞=1 then we have 𝑎𝑎𝑏≤^𝑝𝑝+𝑏^𝑞𝑞. (44) Equality holds if and only if 𝑎^𝑝=𝑏^𝑞.

Finally, we refer the reader to the nice treatments of Hadamard-type fractional calculus in [25, 26]

3. Existence and Uniqueness for an FDE with Hilfer-Hadamard Fractional Derivative

In this section we discuss the existence, uniqueness and the stability of solutions of the Cauchy type problem (46) (below) with Hilfer-Hadamard fractional derivative.

Definition 19 (Hilfer-Hadamard fractional derivative (HHFD)). The left sided fractional derivative of order 𝛼, (0<𝛼<1) and type 0≤𝛽≤1 with respect to 𝑥 is defined by _𝐻𝒟_𝑎^𝛼,𝛽₊𝑓(𝑥)=𝒥_𝑎^{𝛽(1−𝛼)}₊⋅_𝐻𝒟_𝑎^{𝛼+𝛽−𝛼𝛽}₊𝑓(𝑥) (45) for functions for which the expression on the right hand side exists, where _𝐻𝒟_𝑎^{𝛼+𝛽−𝛼𝛽}₊ is the Hadamard fractional derivative (Definition 5).

This new fractional derivative (introduced for the first time in [6]) may be viewed as interpolating the Hadamard fractional derivative and the Hadamard-Caputo fractional derivative. Indeed for 𝛽=0 this derivative (46) reduces to the Hadamard fractional derivative (Definition 5) and when 𝛽=1, we recover the Hadamard-Caputo fractional derivative (Definition 7).

We will study the existence and uniqueness for the Cauchy type problem _𝐻𝒟_𝑎^𝛼,𝛽₊𝑦(𝑥)(𝑥)𝒥=𝑓𝑥,𝑦,𝑥>𝑎>0_{𝑎(1−𝛽)(1−𝛼)+}𝑦(𝑎)=𝑐. (46) We consider the underlying spaces defined by 𝐶^𝛼,𝛽_{1−𝛾,𝜇}[]=𝑎,𝑏𝑦∈𝐶_1−𝛾,log[𝑎,𝑏],_𝐻𝒟_𝑎^𝛼,𝛽₊𝑦∈𝐶_𝜇,log[,𝐶𝑎,𝑏]_{𝛾1−𝛾,log}[]=𝑎,𝑏𝑦∈𝐶_1−𝛾,log[𝑎,𝑏],_𝐻𝒟^𝛾_𝑎+𝑦∈𝐶_1−𝛾,log[𝑎,𝑏], (47)(48) where 𝛾=𝛼+𝛽−𝛼𝛽 and 0≤𝜇<1. It is clear that 0<𝛾<1 for 0<𝛼,𝛽<1.

Here, 𝐶_1−𝛾,log[𝑎,𝑏] and 𝐶_𝜇,log[𝑎,𝑏] are weighted spaces of continuous functions on (𝑎,𝑏] defined by 𝐶_𝛾,log[]=(]𝑥𝑎,𝑏𝑔∶𝑎,𝑏⟶𝐑∶log𝑎^𝛾𝑔(𝑥)∈𝐶[𝑎,𝑏]. (49) Our investigations are based on reducing the fractional differential problem to a Volterra integral equation of the second kind: 𝑦(𝑥)=𝑐Γ𝛾𝑥log𝑎^𝛾−1+1Γ(𝛼)×^𝑥_𝑎𝑥log𝑡^𝛼−1𝑓𝑡,𝑦(𝑡)𝑑𝑡𝑡,𝑥>𝑎, (50) and then using the Banach fixed point theorem.

3.1. Equivalence of the Cauchy Type Problem and the Volterra Integral Equation

Here, we prove the equivalence of the Cauchy type problem (46) and the nonlinear Volterra integral equation (50) in the sense that, if 𝑦∈𝐶^𝛾_1−𝛾,log[𝑎,𝑏] satisfies one of them, then it also satisfies the other one. To establish this result, we assume that the function 𝑓[⋅,𝑦(⋅)] belongs to 𝐶_𝜇,log[𝑎,𝑏] for any 𝑦∈𝐶_𝜇,log[𝑎,𝑏]. We need the following lemma.

Lemma 20. Let 0<𝑎<𝑏<∞, 𝛼>0, 0≤𝜇<1 and 𝑔∈𝐶_𝜇,log[𝑎,𝑏]. If 𝛼>𝜇, then 𝒥^𝛼_𝑎+𝑔 is continuous on [𝑎,𝑏] and 𝒥^𝛼_𝑎+𝑔(𝑎)=lim_𝑥→𝑎+𝒥^𝛼_𝑎+𝑔(𝑥)=0. (51)

Proof. Since 𝑔∈𝐶_𝜇,log[𝑎,𝑏] then (log(𝑥/𝑎))^𝜇𝑔(𝑥) is continuous on [𝑎,𝑏] and on [𝑎,𝑏] we have 𝑥log𝑎^𝜇𝑔(𝑥)≤𝑀, (52) for some positive constant 𝑀. Therefore, 𝒥^𝛼_𝑎+𝑔(𝑥)≤𝑀𝒥^𝛼_𝑎+𝑡log𝑎^−𝜇(𝑥), (53) and by using Lemma 9 (with 𝛽=1−𝜇>0) we have 𝒥^𝛼_𝑎+𝑔(𝑥Γ)≤𝑀1−𝜇Γ𝑥𝛼+1−𝜇log𝑎^𝛼−𝜇. (54) As 𝛼>𝜇, we obtain the result.

Theorem 21. Let 𝛾=𝛼+𝛽−𝛼𝛽 where 0<𝛼<1 and 0≤𝛽≤1. Let 𝑓∶(𝑎,𝑏]×𝐑→𝐑 be a function such that 𝑓(⋅,𝑦(⋅))∈𝐶_𝜇,log[𝑎,𝑏] for any 𝑦∈𝐶_𝜇,log[𝑎,𝑏] with 1−𝛾≤𝜇<1−𝛽(1−𝛼).
If 𝑦∈𝐶^𝛾_1−𝛾,log[𝑎,𝑏], then 𝑦 satisfies the (CFDP) (46) if and only if 𝑦 satisfies the (IE) (50).

Proof. First we prove the necessity. Let 𝑦∈𝐶^𝛾_1−𝛾,log[𝑎,𝑏] be a solution of problem (46). We want to prove that 𝑦 is also a solution of the integral equation (50). By the definition of the space 𝐶^𝛾_1−𝛾,log[𝑎,𝑏] (relation (48)) we have 𝛿𝒥_𝑎^1−𝛾₊𝑦=_𝐻𝒟^𝛾_𝑎+𝑦∈𝐶_1−𝛾,log[𝑎,𝑏]. (55) Moreover, by Lemma 13(b) we have 𝒥_𝑎^1−𝛾₊𝑦∈𝐶[𝑎,𝑏] since 𝑦∈𝐶_1−𝛾,log[𝑎,𝑏]. Then, by Definition 2, we have 𝒥_𝑎^1−𝛾₊𝑦∈𝐶¹_{𝛿,1−𝛾}[𝑎,𝑏]. (56) Thus, we can apply Theorem 12 (with 𝑓 replaced by 𝑦) to get 𝒥^𝛾_𝑎+𝐻𝒟^𝛾_𝑎+𝑦(𝑥)(𝑥)−𝒥=𝑦^{𝑎1−𝛾+}𝑦(𝑎)Γ𝛾𝑥log𝑎^𝛾−1,𝑥∈(𝑎,𝑏], (57) or 𝒥^𝛾_𝑎+𝐻𝒟^𝛾_𝑎+𝑦(𝑥)(𝑥)−𝑐=𝑦Γ𝛾𝑥log𝑎^𝛾−1,𝑥∈(𝑎,𝑏], (58) where 𝑐 comes from the initial condition in (46). By our hypothesis 𝑓[⋅,𝑦(⋅)]∈𝐶_𝜇,log[𝑎,𝑏], since 𝑦∈𝐶_1−𝛾,log[𝑎,𝑏]⊂𝐶_𝜇,log[𝑎,𝑏], Lemma 13(a) and (b) we see that the integral 𝒥^𝛼_𝑎+𝑓[⋅,𝑦(⋅)]∈𝐶_{𝜇−𝛼,log}[𝑎,𝑏] for 𝜇>𝛼 and 𝒥^𝛼_𝑎+𝑓[⋅,𝑦(⋅)]∈𝐶[𝑎,𝑏] for 𝜇≤𝛼. Applying the operator 𝒥^𝛼_𝑎+ to both sides of (46) we get 𝒥^𝛼_𝑎+𝒥_𝑎^{𝛽(1−𝛼)}_+𝐻𝒟^𝛾_𝑎+𝑦(𝑥)=𝒥^𝛼_𝑎+𝑓𝑥,𝑦(𝑥),𝑥∈(𝑎,𝑏]. (59) We can sum up the exponents by Lemma 10 to get 𝒥_𝑎^{𝛼+𝛽(1−𝛼)}_+𝐻𝒟^𝛾_𝑎+𝑦(𝑥)=𝒥^𝛼_𝑎+𝑓𝑥,𝑦(𝑥),𝑥∈(𝑎,𝑏], (60) or 𝒥^𝛾_𝑎+𝐻𝒟^𝛾_𝑎+𝑦(𝑥)=𝒥^𝛼_𝑎+𝑓𝑡,𝑦(𝑡)(𝑥),𝑥∈(𝑎,𝑏]. (61) From (58) and (61) we obtain 𝑦(𝑥)=𝑐Γ𝛾𝑥log𝑎^𝛾−1+𝒥^𝛼_𝑎+𝑓𝑡,𝑦(𝑡)(𝑥), (62) which is (50), and hence the necessity is proved.
Now, we prove the sufficiency. Let 𝑦∈𝐶^𝛾_1−𝛾,log[𝑎,𝑏] satisfy (50), then _𝐻𝒟^𝛾_𝑎+𝑦 exists and _𝐻𝒟^𝛾_𝑎+𝑦∈𝐶_1−𝛾,log[𝑎,𝑏]. Applying the operator _𝐻𝒟^𝛾_𝑎+ to both sides of the last identity we find _𝐻𝒟^𝛾_𝑎+𝑦(𝑥)=𝑐Γ𝛾^𝐻𝒟^𝛾_𝑎+𝑡log𝑎^𝛾−1(𝑥)+_𝐻𝒟^𝛾_𝑎+𝒥^𝛼_𝑎+𝑓𝑡,𝑦(𝑡)(𝑥). (63) By using Lemma 10, Definition 5, Lemma 23 and the hypothesis 𝑓[⋅,𝑦(⋅)]∈𝐶_𝜇,log[𝑎,𝑏], we have _𝐻𝒟^𝛾_𝑎+𝑦(𝑥)=𝛿𝒥_𝑎^1−𝛾₊𝒥^𝛼_𝑎+𝑓(𝑡(𝑥)𝒥𝑡,𝑦)=𝛿_𝑎^{1−𝛽(1−𝛼)}₊𝑓(𝑡𝑥)=𝑡,𝑦)(_𝐻𝒟_𝑎^{𝛽(1−𝛼)}₊𝑓𝑡,𝑦(𝑡)(𝑥),𝑥∈(𝑎,𝑏]. (64) From (64) and the fact that _𝐻𝒟^𝛾_𝑎+𝑦∈𝐶_1−𝛾,log[𝑎,𝑏], we obtain that _𝐻𝒟_𝑎^{𝛽(1−𝛼)}₊𝑓⋅,𝑦(⋅)∈𝐶_1−𝛾,log[𝑎,𝑏]. (65) Next, applying the operator 𝒥_𝑎^{𝛽(1−𝛼)}₊ to both sides of (64) we get 𝒥_𝑎^{𝛽(1−𝛼)}_+𝐻𝒟^𝛾_𝑎+𝑦(𝑥)=𝒥_𝑎^{𝛽(1−𝛼)}_+𝐻𝒟_𝑎^{𝛽(1−𝛼)}₊𝑓𝑡,𝑦(𝑡)(𝑥); (66) that is, 𝒥_𝑎^{𝛽(1−𝛼)}₊𝛿𝒥_𝑎^1−𝛾₊𝑦(𝑥)=𝒥_𝑎^{𝛽(1−𝛼)}_+𝐻𝒟_𝑎^{𝛽(1−𝛼)}₊𝑓𝑡,𝑦(𝑡)(𝑥). (67) By virtue of 𝛿𝒥_𝑎^{1−𝛽(1−𝛼)}₊𝑓(𝑡=𝑡,𝑦)_𝐻𝒟_𝑎^{𝛽(1−𝛼)}₊𝑓⋅,𝑦(⋅)∈𝐶_1−𝛾,log[𝑎,𝑏], (68) and 𝛾>𝛽(1−𝛼) and Definition 2, we have 𝒥_𝑎^{1−𝛽(1−𝛼)}₊𝑓∈𝐶¹_{𝛿;1−𝛾}[𝑎,𝑏] (see the first part of the proof, or Lemma 13(b), for the continuity of 𝒥_𝑎^{1−𝛽(1−𝛼)}₊𝑓 for 𝜇<1−𝛽(1−𝛼)). Then, Theorem 12 allows us to write _𝐻𝒟_𝑎^𝛼,𝛽₊𝑦(𝑥)−𝒥=𝑓𝑥,𝑦^{𝑎1−𝛽(1−𝛼)+}𝑓(𝑎)Γ𝛽(×𝑥1−𝛼)log𝑎^{𝛽(1−𝛼)−1},𝑥∈(𝑎,𝑏]. (69) Lemma 20 implies that 𝒥_𝑎^{1−𝛽(1−𝛼)}₊𝑓(𝑎)=0 (70) because 1−𝛽(1−𝛼)>𝜇. Hence, the relation (69) reduces to _𝐻𝒟_𝑎^𝛼,𝛽₊𝑦(𝑥)=𝑓𝑥,𝑦(𝑥),𝑥>𝑎. (71) Now, we show that the initial condition in (46) also holds. To this end we apply the operator 𝒥_𝑎^1−𝛾₊ to both sides of (50): 𝒥_𝑎^1−𝛾₊𝑦(𝑥)=𝑐Γ𝛾𝒥_𝑎^1−𝛾₊𝑡log𝑎^𝛾−1(𝑥)+𝒥_𝑎^1−𝛾₊𝒥^𝛼_𝑎+𝑓𝑡,𝑦(𝑡)(𝑥) (72) and use the Lemma 8 (with 𝛼 replaced by 1−𝛾 and 𝛽 by 𝛾) and the Lemma 9 to obtain 𝒥_𝑎^1−𝛾₊𝑦(𝑥)=𝑐+𝒥_𝑎^{1−𝛽(1−𝛼)}₊𝑓𝑡,𝑦(𝑡)(𝑥). (73) In (73), taking the limit as 𝑥→𝑎, we obtain 𝒥_𝑎^1−𝛾₊𝑦(𝑎)=𝑐, (74) mentioned above 𝒥_𝑎^{1−𝛽(1−𝛼)}₊𝑓[𝑡,𝑦(𝑡)](𝑎)=0). Therefore, the sufficiency is proved, which completes the proof of Theorem 21.

3.2. Existence and Uniqueness of a Solution

In this section we establish the existence of a unique solution to the Cauchy type problem (46) in the space 𝐶^𝛼,𝛽_{1−𝛾,𝜇}[𝑎,𝑏] defined in (47) above under the conditions of Theorem 21 and an additional Lipschitz condition 𝑓𝑥,𝑦₁−𝑓𝑥,𝑦_{2𝐶1−𝛾,log[𝑎,𝑏]}≤𝐴𝑦₁−𝑦_{2𝐶1−𝛾,log[𝑎,𝑏]} (75) for some positive constant 𝐴 and every 𝑥∈[𝑎,𝑏].

Theorem 22. Let 𝛾=𝛼+𝛽−𝛼𝛽 where (0<𝛼<1,0≤𝛽≤1). Assume that 𝑓∶(𝑎,𝑏]×𝐑→𝐑(𝑎>0) is a function such that 𝑓[⋅,𝑦(⋅)]∈𝐶_𝜇,log[𝑎,𝑏] for any 𝑦∈𝐶_𝜇,log[𝑎,𝑏] with 1−𝛾≤𝜇<1−𝛽(1−𝛼) and is Lipschitz continuous with respect to its second variable. Then, there exists a unique solution 𝑦 for the Cauchy type problem (46) in the space 𝐶^𝛼,𝛽_{1−𝛾,𝜇}[𝑎,𝑏].

Proof. First we prove the existence of a unique solution 𝑦 in the space 𝐶_1−𝛾,log[𝑎,𝑏]. According to Theorem 21, it suffices to prove the existence of a unique solution 𝑦∈𝐶_1−𝛾,log[𝑎,𝑏] to the nonlinear Volterra integral equation (50).
Let us select 𝑥₁ in (𝑎,𝑏) such that 𝑤₁𝛾∶=𝐴ΓΓ𝑥𝛼+𝛾log¹𝑎^2𝛼<1, (76) where 𝐴>0 is the Lipschitz constant. We start by proving that a unique solution 𝑦∈𝐶_1−𝛾,log[𝑎,𝑥₁] to (50) exists on the interval (𝑎,𝑥₁]. It is easy to see that the space 𝐶_1−𝛾,log[𝑎,𝑥₁] is a complete metric space when equipped with the distance given by 𝑑𝑦₁,𝑦₂=𝑦₁−𝑦_{2𝐶1−𝛾,log[𝑎,𝑥1]}∶=max_{𝑥∈𝑎,𝑥1}𝑥log𝑎^1−𝛾𝑦₁(𝑥)−𝑦₂(𝑥). (77) The integral equation (50) takes the form 𝑦(𝑥)=𝑇𝑦(𝑥), (78) where 𝑇𝑦(𝑥)=𝑦₀(𝑥)+𝒥^𝛼_𝑎+𝑓𝑡,𝑦(𝑡)(𝑥) (79) with 𝑦₀(𝑥)=𝑐Γ𝛾𝑥log𝑎^𝛾−1. (80) We claim that 𝑇 maps 𝐶_1−𝛾,log[𝑎,𝑥₁] into itself. Indeed, 𝑦₀ given by (80) is clearly in 𝐶_1−𝛾,log[𝑎,𝑥₁]. Also, since 𝑓[⋅,𝑦]∈𝐶_𝜇,log[𝑎,𝑏] for any 𝑦∈𝐶_𝜇,log[𝑎,𝑏] with 𝜇∈𝐑 (0≤𝜇<1), then, by Lemma 13(a) and (b), the integral in the right-hand side of (79) belongs to 𝐶_{𝜇−𝛼,log}[𝑎,𝑏] for 𝜇>𝛼 and to 𝐶[𝑎,𝑏] for 𝜇≤𝛼. Since 𝜇−𝛼<1−𝛾, by Lemma 16 the right-hand side of (79) belongs to 𝐶_1−𝛾,log[𝑎,𝑏].
Our second claim is that 𝑇 is a contraction; that is, 𝑇𝑦₁−𝑇𝑦_{2𝐶1−𝛾,log[𝑎,𝑥1]}≤𝑤₁𝑦₁−𝑦_{2𝐶1−𝛾,log[𝑎,𝑥1]},0<𝑤₁<1. (81) This follows from (79), Lemma 13(a), and the fact that 𝑇𝑦₁−𝑇𝑦_{2𝐶1−𝛾,log[𝑎,𝑥1]}=𝒥^𝛼_𝑎+𝑓𝑡,𝑦₁(𝑡)−𝒥^𝛼_𝑎+𝑓𝑡,𝑦₂(𝑡)_{𝐶1−𝛾,log[𝑎,𝑥1]}≤𝑥log¹𝑎^2𝛼Γ𝛾Γ𝑓𝛼+𝛾𝑡,𝑦₁(𝑡)−𝑓𝑡,𝑦₂(𝑡)_{𝐶1−𝛾,log[𝑎,𝑥1]}𝑥≤𝐴log¹𝑎^2𝛼Γ𝛾Γ𝑦𝛼+𝛾¹(𝑡)−𝑦₂(𝑡)_{𝐶1−𝛾,log[𝑎,𝑥1]}=𝑤₁𝑦₁(𝑡)−𝑦₂(𝑡)_{𝐶1−𝛾,log[𝑎,𝑥1]}. (82) Our assumption (76) allows us to apply the Banach fixed point theorem to obtain a unique solution 𝑦^∗∈𝐶_1−𝛾,log[𝑎,𝑥₁] to (50) on the interval (𝑎,𝑥₁].
This solution 𝑦^∗ is the limit of a convergent sequence 𝑇^𝑚𝑦^∗₀: lim_𝑚→∞𝑇^𝑚𝑦^∗₀−𝑦^∗_{𝐶1−𝛾,log[𝑎,𝑥1]}=0, (83) where 𝑦^∗₀ is any function in 𝐶_1−𝛾,log[𝑎,𝑥₁] and 𝑇^𝑚𝑦^∗₀(𝑥)=𝑇𝑇^𝑚−1𝑦^∗₀(𝑥)=𝑦₀(𝑥)+𝒥^𝛼_𝑎+𝑓𝑡,𝑇^𝑚−1𝑦^∗₀(𝑡)(𝑥),𝑚∈𝐍. (84) Let us take 𝑦^∗₀(𝑥)=𝑦₀(𝑥) with 𝑦₀(𝑥) defined by (80). If we denote by 𝑦_𝑚(𝑥)∶=𝑇^𝑚𝑦^∗₀(𝑥),𝑚∈𝐍 (85) then, clearly, lim_𝑚→∞𝑦_𝑚−𝑦^∗_{𝐶1−𝛾,log[𝑎,𝑥1]}=0. (86) Next, we consider the interval [𝑥₁,𝑏]. From (50) we have 𝑦(𝑥)=𝑐Γ𝛾𝑥log𝑎^𝛾−1+1Γ(𝛼)^𝑥1_𝑎𝑥log𝑡^𝛼−1𝑓𝑡,𝑦(𝑡)𝑡+1𝑑𝑡Γ(𝛼)^𝑥_𝑥1𝑥log𝑡^𝛼−1𝑓𝑡,𝑦(𝑡)𝑡𝑑𝑡.=𝑦₀₁(𝑥)+1Γ(𝛼)^𝑥_𝑥1𝑥log𝑡^𝛼−1𝑓𝑡,𝑦(𝑡)𝑡𝑑𝑡, (87) where 𝑦₀₁(𝑥) is defined by 𝑦₀₁(𝑥)=𝑐Γ𝛾𝑥log𝑎^𝛾−1+1Γ(𝛼)^𝑥1_𝑎𝑥log𝑡^𝛼−1𝑓𝑡,𝑦(𝑡)𝑡𝑑𝑡, (88) and is a known function. We note that 𝑦₀₁∈𝐶[𝑥₁,𝑏].
We want to prove the existence of a unique solution 𝑦∈𝐶[𝑥₁,𝑏] of (50) on the interval [𝑥₁,𝑏]. For this, we also use Banach fixed point theorem for the space 𝐶[𝑥₁,𝑥₂] where 𝑥₂∈(𝑥₁,𝑏] satisfies 𝑤₂=𝐴(𝛼)𝑥𝛼Γlog²𝑥₁^𝛼<1. (89) The space 𝐶[𝑥₁,𝑥₂] is a complete metric space with the distance given by 𝑑𝑦₁,𝑦₂=𝑦₁−𝑦_{2𝐶[𝑥1,𝑥2]}=max_{𝑥𝑥∈1,𝑥2}𝑦₁(𝑥)−𝑦₂(𝑥). (90) The integral equation (87) may be written shortly as 𝑦(𝑥)=𝑇𝑦(𝑥), (91) where the operator (again denoted by 𝑇) is given by 𝑇𝑦(𝑥)=𝑦₀₁(𝑥)+1Γ(𝛼)^𝑥_𝑥1𝑥log𝑡^𝛼−1𝑓𝑡,𝑦(𝑡)𝑡𝑑𝑡. (92) As in the first part of this proof, since 𝑦₀₁∈𝐶[𝑥₁,𝑥₂] and 𝑓[⋅,𝑦(⋅)]∈𝐶_𝜇,log[𝑎,𝑏] for any 𝑦∈𝐶_𝜇,log[𝑎,𝑏], then, 𝑓[⋅,𝑦(⋅)]∈𝐶[𝑥₁,𝑥₂], for any 𝑦∈𝐶[𝑥₁,𝑥₂], then by Lemma 14, we deduce that the integral in the right-hand side of (92) also belongs to 𝐶[𝑥₁,𝑥₂], and hence 𝑇𝑦∈𝐶[𝑥₁,𝑥₂].
Moreover, using the Lipschitz condition and applying the Lemma 14, we find 𝑇𝑦₁−𝑇𝑦_{2𝐶[𝑥1,𝑥2]}=𝒥^𝛼_𝑥⁺₁𝑓𝑡,𝑦₁(𝑡)−𝒥^𝛼_𝑥⁺₁𝑓𝑡,𝑦₂(𝑡)_{𝐶[𝑥1,𝑥2]}≤1(𝛼)𝑥𝛼Γlog²𝑥₁^𝛼𝑓𝑡,𝑦₁(𝑡)−𝑓𝑡,𝑦₂(𝑡)_{𝐶[𝑥1,𝑥2]}≤𝐴(𝛼)𝑥𝛼Γlog²𝑥₁^𝛼𝑦₁(𝑡)−𝑦₂(𝑡)_{𝐶[𝑥1,𝑥2]}=𝑤₂𝑦₁(𝑡)−𝑦₂(𝑡)_{𝐶[𝑥1,𝑥2]}. (93) This, together with our assumption 0<𝑤₂<1, shows that 𝑇 is a contraction and therefore from Theorem 17, there exists a unique solution 𝑦^∗₁∈𝐶[𝑥₁,𝑥₂] to (50) on the interval [𝑥₁,𝑥₂]. Notice that 𝑦^∗₁(𝑥₁)=𝑦^∗(𝑥₁)=𝑦₀₁(𝑥₁). Further, Theorem 17 guarantees that this solution is the limit of a convergent sequence 𝑇^𝑚𝑦^∗₀₁: lim_𝑚→∞𝑇^𝑚𝑦^∗₀₁−𝑦^∗_{1𝐶[𝑥1,𝑥2]}=0, (94) where 𝑦^∗₀₁ is any function in 𝐶[𝑥₁,𝑥₂], which we can pick 𝑦^∗₀₁(𝑥)=𝑦₀₁(𝑥) defined by (88). Therefore, lim_𝑚→∞𝑦_𝑚−𝑦^∗_{1𝐶[𝑥1,𝑥2]}=0, (95) where 𝑦_𝑚(𝑥)=𝑇^𝑚𝑦^∗₀₁(𝑥)=𝑦₀₁(𝑥)+1Γ(𝛼)×^𝑥_𝑥1𝑥log𝑡^𝛼−1𝑓𝑡,𝑇^𝑚−1𝑦^∗₀₁(𝑡)𝑡𝑑𝑡. (96) If 𝑥₂≠𝑏, we consider the interval [𝑥₂,𝑥₃], where 𝑥₃=𝑥₂+ℎ₂, ℎ₂>0 such that 𝑥₃≤𝑏 and 𝑤₃=𝐴(𝛼)𝑥𝛼Γlog³𝑥₂^𝛼<1. (97) Using the same arguments as above, we derive that there exists a unique solution 𝑦^∗₂∈𝐶[𝑥₂,𝑥₃] to (50) on the interval [𝑥₂,𝑥₃]. If 𝑥₃≠𝑏, then we continue the process until we reach a solution 𝑦 to (50), 𝑦(𝑥)=𝑦^∗_𝑘(𝑥), and 𝑦^∗_𝑘∈𝐶[𝑥_𝑘,𝑥_𝑘+1] (𝑘=1,…,𝐿), where 𝑎=𝑥₀<𝑥₁<⋯<𝑥_𝐿+1 and 𝑤_𝑘+1=𝐴(𝛼)𝑥𝛼Γlog^𝑘+1𝑥_𝑘^𝛼<1. (98) Assume that 𝑏−𝑎>Γ(𝛼+𝛾)/𝐴Γ(𝛾) (for otherwise, take 𝑥₁=𝑏). Then, divide the length of the interval [𝑎+Γ(𝛼+𝛾)/𝐴Γ(𝛾),𝑏] by (𝛼Γ(𝛼)/𝐴)^1/𝛼. Let 𝑀 be that quotient. It appears that for 𝐿=[𝑀]+1 and 𝑏 is reached after a finite number of steps, 𝑥_𝐿+1=𝑏. Then, there exists a unique solution 𝑦∈𝐶[𝑥₁,𝑏] to (50) on the interval [𝑥₁,𝑏].
Putting together the solutions in [𝑎,𝑥₁] and [𝑥₁,𝑏] and taking into account the Lemma 15, we obtain that there exists a unique solution 𝑦∈𝐶_1−𝛾,log[𝑎,𝑏] to the Volterra integral equation (50) on the whole interval (𝑎,𝑏]. Hence, 𝑦∈𝐶_1−𝛾,log[𝑎,𝑏] is the unique solution to the Cauchy-type problem (46).
It remains to show that such a unique solution is actually in 𝐶^𝛼,𝛽_{1−𝛾,𝜇}[𝑎,𝑏]. To this end we need to prove that _𝐻𝒟_𝑎^𝛼,𝛽₊𝑦∈𝐶_𝜇,log[𝑎,𝑏]. Let us recall that our 𝑦 is a limit of the sequence 𝑦_𝑚, where 𝑦_𝑚=𝑇^𝑚𝑦^∗₀∈𝐶_1−𝛾,log[𝑎,𝑏]; that is, lim_𝑚→∞𝑦_𝑚−𝑦_{𝐶1−𝛾,log[𝑎,𝑏]}=0, (99) with a certain choice of 𝑦^∗₀(𝑥) on each subinterval [𝑎,𝑥₁],…,[𝑥_𝐿,𝑏]. Indeed, this is a consequence of the construction adopted, the initial values are selected in the space 𝐶_1−𝛾,log[𝑎,𝑏] (see (80) and (88)) and the operator maps this space into itself (see argument right after (80) and (92)). As for the convergence in that space it has been proved in (83) and (95).
If 𝑦₀(𝑥)≠0,then we can take 𝑦^∗₀(𝑥)=𝑦₀(𝑥). Since 𝜇≥1−𝛾, then by (46), the Lipschitz condition and Lemma 8, we have _𝐻𝒟_𝑎^𝛼,𝛽₊𝑦_𝑚−_𝐻𝒟_𝑎^𝛼,𝛽₊𝑦_{𝐶𝜇,log[𝑎,𝑏]}=𝑓𝑥,𝑦_𝑚−𝑓𝑥,𝑦_{𝐶𝜇,log[𝑎,𝑏]}𝑏≤𝐴log𝑎^{𝜇−1+𝛾}𝑦_𝑚−𝑦_{𝐶1−𝛾,log[𝑎,𝑏]}. (100) In virtue of (99) and (100), it follows that lim_{𝑚→∞𝐻}𝒟_𝑎^𝛼,𝛽₊𝑦_𝑚−_𝐻𝒟_𝑎^𝛼,𝛽₊𝑦_{𝐶𝜇,log[𝑎,𝑏]}=0. (101) We entail from this relation that (_𝐻𝒟_𝑎^𝛼,𝛽₊𝑦)∈𝐶_𝜇,log[𝑎,𝑏] if (_𝐻𝒟_𝑎^𝛼,𝛽₊𝑦_𝑚)∈𝐶_𝜇,log[𝑎,𝑏], 𝑚=1,2,… This latter property holds in as much as _𝐻𝒟_𝑎^𝛼,𝛽₊𝑦_𝑚(𝑥)=𝑓[𝑥,𝑦_𝑚−1(𝑥)] and 𝑓[⋅,𝑦(⋅)]∈𝐶_𝜇,log[𝑎,𝑏] for any 𝑦∈𝐶_𝜇,log[𝑎,𝑏].
Consequently, 𝑦∈𝐶^𝛼,𝛽_{1−𝛾,𝜇}[𝑎,𝑏]. This completes the proof of Theorem 22.

4. Stability

In this section, we consider the weighted Cauchy-type problem _𝐻𝒟_𝑎^𝛼,𝛽₊𝑢(𝑡)((𝑡𝑡=𝑓𝑡,𝑢)),𝑡>𝑎>0,log𝑎^{(1−𝛽)(1−𝛼)}𝑢(𝑡)|_𝑡=𝑎=𝑏, (102) where _𝐻𝒟₊^𝛼,𝛽 is the Hilfer-Hadamard fractional derivative (HHFD) of order 0<𝛼<1 and type 0≤𝛽≤1 and 𝑏∈𝐑^∗ (the set of all real numbers except 0). It is interesting to note that, using an argument similar to the one in the proof of Lemma 3.5 in [2] (see also Lemma 3.2), we can prove that the initial condition in (102) and the one in (46) are equivalent.

We will assume the following hypotheses on the function 𝑓:

(F*) 𝑓(𝑡,𝑢) is a continuous (nonlinear) function on (𝑎,∞)×𝐑 and is such that 𝑓((𝑡≤𝑡𝑡,𝑢))log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚,𝜇≥0,𝑚>1,𝑡≥𝑎, (103) where 𝜑 is a continuous (nonnegative) function on [𝑎,∞).

We first prove the following inequality.

Lemma 23. If 𝜆,𝜈,𝜔>0, then for any 𝑡>𝑎, 𝑎>0 we have 𝑡log𝑎^1−𝜈𝑡_𝑎𝑡log𝑠^𝜈−1𝑠log𝑎^𝜆−1𝑠𝑎^−𝜔𝑑𝑠𝑠≤𝐶𝜔^−𝜆, (104) where 𝐶 is a positive constant independent of 𝑡.

Proof. Let us denote by 𝐼(𝑡) the left-hand side of the inequality in the Lemma. We consider the change of variable, 𝜉=(log(𝑠/𝑎))/(log(𝑡/𝑎)) then 𝑠/𝑎=(𝑡/𝑎)^𝜉 and log(𝑡/𝑠)=(1−𝜉)(log(𝑡/𝑎)). It follows that 𝐼(𝑡)=𝑡log𝑎^𝜆1₀(1−𝜉)^𝜈−1𝜉^𝜆−1𝑡𝑎^−𝜔𝜉𝑑𝜉, (105) or 𝐼(𝑡)=𝑡log𝑎^𝜆1₀(1−𝜉)^𝜈−1𝜉^𝜆−1𝑡exp−𝜔𝜉log𝑎𝑑𝜉. (106) Observe that, for 𝜉≥1 and [𝜆]+1≥𝜆 we have 𝜉^[𝜆]+1≥𝜉^𝜆. Also since 𝜆+2≥[𝜆]+2 and the Gamma function is increasing in [2,∞) we have Γ(𝜆+2)≥Γ([𝜆]+2) or 1/Γ([𝜆]+2)≥1/Γ(𝜆+2). Moreover, 𝑒^𝜉≥(𝜉^[𝜆]+1/Γ([𝜆]+2)) (Γ([𝜆]+2)=([𝜆]+1)!), and hence 𝑒^𝜉≥𝜉^[𝜆]+1Γ𝜆])≥𝜉([+2Γ𝜆]^𝜆)≥𝜉([+2Γ(^𝜆),𝜆+2 (107) or 𝑒^−𝜉≤Γ(𝜆+2)𝜉^𝜆. (108) Therefore, for 0≤𝜉<1/2 we get (1−𝜉)^𝜈−1≤max1,2^1−𝜈. (109) For 1/2<𝜉≤1 and 𝑡 such that 𝜔𝜉log(𝑡/𝑎)≥1 we have 𝑡exp−𝜔𝜉log𝑎≤Γ(𝜆+2)𝜔𝜉log(𝑡/𝑎)^𝜆≤𝜔^−𝜆𝜉Γ(𝜆+2)≤2𝜔^−𝜆Γ(𝜆+2). (110) This means that 𝑡log𝑎^𝜆(1−𝜉)^𝜈−1𝜉^𝜆−1𝑡exp−𝜔𝜉log𝑎≤max1,2^1−𝜈𝑡log𝑎^𝜆𝜉^𝜆−1𝑡×exp−𝜔𝜉log𝑎1,0≤𝜉<2,2(1−𝜉)^𝜈−1Γ(𝜆+2)𝜔^−𝜆,12<𝜉≤1. (111) Consequently, 𝐼(𝑡)≤max1,2^1−𝜈𝑡log𝑎^𝜆×₀^1/2𝜉^𝜆−1𝑡exp−𝜔𝜉log𝑎𝑑𝜉+2𝜔^−𝜆Γ(𝜆+2)¹_1/2(1−𝜉)^𝜈−1𝑑𝜉. (112) Let 𝑢=𝜔𝜉log(𝑡/𝑎), we see that 𝐼(𝑡)≤max1,2^1−𝜈𝑡log𝑎^𝜆×^∞₀𝑢𝜔log(𝑡/𝑎)^𝜆−1𝑒^−𝑢𝑑𝑢𝜔log(𝑡/𝑎)+2𝜔^−𝜆Γ(−()𝜆+2)1−𝜉^𝜈𝜈¹_𝜉=1/2. (113) Thus, 𝐼(𝑡)≤max1,2^1−𝜈𝜔^−𝜆Γ(𝜆)+2^1−𝜈𝜔^−𝜆Γ(𝜆+2)𝜈. (114) As a result, 𝐼(𝑡)≤max{1,2^1−𝜈}Γ(𝜆)(1+𝜆(𝜆+1)/𝜈)𝜔^−𝜆.
For 0<𝜂<1, 𝑒^𝜂≥1 it is clear that Γ(𝜆+2)𝑒^𝜂≥1≥𝜂^𝜆 (115) holds and we proceed in the same manner to conclude that for 𝑡 such that 0<𝜔𝜉log(𝑡/𝑎)<1𝑡log𝑎^1−𝜈𝑡_𝑎𝑡log𝑠^𝜈−1𝑠log𝑎^𝜆−1𝑠𝑎^−𝜔𝑑𝑠𝑠≤𝐶𝜔^−𝜆, (116) where 𝐶=max{1,2^1−𝜈}Γ(𝜆)(1+𝜆(𝜆+1)/𝜈). The proof is complete.

Let 𝑝 and 𝑞 be conjugate exponents; that is, 𝑝𝑞=𝑝+𝑞, and let 𝜆₁∶=1+𝑝[𝜇−(1−𝛾)𝑚] and 𝜆₂∶=1+𝑝(𝛼−1), where 𝛾∶=𝛼+𝛽−𝛼𝛽. If 𝜇−(𝑚−1)(1−𝛾)>0 and 𝑞>1/𝛼, then 𝜆₁>0 and 𝜆₂>0. We denote by 𝐋^∗ the positive real number 𝐋^∗Γ(𝛼)∶=2^{𝑚+(𝛼−1)}|𝑏|^𝑚−1𝑚(2𝑎)^𝑚𝑚−1^1/𝑞×𝑝−1^𝜆1Γ𝜆₁1+𝜆₁/𝜆₂^𝑚/𝑝. (117)

Theorem 24. Suppose that 𝑓 satisfies (𝐅^∗) and 𝜇>(𝑚−1)(1−𝛾). If 𝜑(𝑡)_𝑞^𝑚−1𝜑(𝑡)𝑡log𝑎^{−𝑚𝛽(1−𝛼)}_𝑞<𝐋^∗ (118) for some 𝑞>1/𝛼, then, for any solution of Problem (102), there exists a positive constant 𝐶 such that |𝑢(𝑡)|≤𝐶(log(𝑡/𝑎))^𝛾−1, 𝑡>𝑎, 𝑎>0, where 𝛾=𝛼+𝛽−𝛼𝛽.

Proof. Let us consider the Volterra integral equation 𝑢(𝑡)𝑡=𝑏log𝑎^𝛾−1+1Γ(𝛼)^𝑡_𝑎𝑡log𝑠^𝛼−1𝑓[𝑠,𝑢(𝑠)]𝑑𝑠𝑠,𝑡>𝑎>0 (119) associated to problem (102). Multiplying both sides of (119) by (log(𝑡/𝑎))^1−𝛾 and using the assumption (𝐅^∗) on 𝑓 we get 𝑡log𝑎^1−𝛾|𝑢(𝑡≤|𝑏|+())|log𝑡/𝑎^1−𝛾Γ(𝛼)×^𝑡_𝑎𝑡log𝑠^𝛼−1𝑠log𝑎^𝜇𝜑(𝑠)|𝑢(𝑠)|^𝑚𝑑𝑠𝑠. (120) Let 𝑣(𝑡) denote the left-hand side of (120). The insertion of the term 𝑠log𝑎^{(1−𝛾)𝑚}𝑠log𝑎^{−(1−𝛾)𝑚} (121) inside the integral gives 𝑣(𝑡)≤|𝑏|+log(𝑡/𝑎)^1−𝛾Γ(𝛼)×^𝑡_𝑎𝑡log𝑠^𝛼−1𝑠log𝑎^{𝜇−(1−𝛾)𝑚}×𝜑(𝑠)𝑣^𝑚(𝑠)𝑑𝑠𝑠,𝑡>𝑎. (122) Now, the Hölder inequality with exponents 𝑝 and 𝑞 yields ^𝑡_𝑎𝑡log𝑠^𝛼−1𝑠log𝑎^{𝜇−(1−𝛾)𝑚}𝜑(𝑠)𝑣^𝑚(𝑠)𝑑𝑠𝑠≤^𝑡_𝑎𝑡log𝑠^{𝑝(𝛼−1)}𝑠log𝑎^{𝑝(𝜇−(1−𝛾)𝑚)}𝑑𝑠𝑠^𝑝1/𝑝×^𝑡_𝑎𝜑^𝑞(𝑠)𝑣^𝑞𝑚(𝑠)𝑑𝑠^1/𝑞≤𝑎^{−1/𝑞𝑡}_𝑎𝑡log𝑠^{𝑝(𝛼−1)}𝑠log𝑎^{𝑝(𝜇−(1−𝛾)𝑚)}𝑠𝑎^{−(𝑝−1)}𝑑𝑠𝑠^1/𝑝×^𝑡_𝑎𝜑^𝑞(𝑠)𝑣^𝑞𝑚(𝑠)𝑑𝑠^1/𝑞,𝑡>𝑎. (123) Since 𝜆₁−1=𝑝[𝜇−(1−𝛾)𝑚], 𝜆₂−1=𝑝(𝛼−1) and 𝜆₁, 𝜆₂, 𝑝−1>0, we may apply Lemma 9 (with 𝜈 replaced by 𝜆₂, 𝜆 replaced by 𝜆₁ and 𝜔 replaced by 𝑝−1) to get ^𝑡_𝑎𝑡log𝑠^𝛼−1𝑠log𝑎^{𝜇−(1−𝛾)𝑚}𝜑(𝑠)𝑣^𝑚(𝑠)𝑑𝑠𝑠≤𝑎^−1/𝑞𝐶₁𝑡log𝑎^𝛼−1×^𝑡_𝑎𝜑^𝑞(𝑠)𝑣^𝑞𝑚(𝑠)𝑑𝑠^1/𝑞,𝑡>𝑎, (124) where 𝐶₁ is the constant appearing in Lemma 9 corresponding to the present exponents. That is, 𝐶₁=2^{𝑝(𝛼−1)}Γ𝜆₁𝜆1+¹𝜆₁+1𝜆₂𝑝−1^{−𝜆11/𝑝}. (125) Combining (122) and (124) we entail that 𝑣(𝑡)≤|𝑏|+𝐶₁𝑡log𝑎^{−𝛽(1−𝛼)}×^𝑡_𝑎𝜑^𝑞(𝑠)𝑣^𝑞𝑚(𝑠)𝑑𝑠^1/𝑞,𝑡>𝑎, (126) where 𝐶₁=𝑎^−1/𝑞(𝐶₁/Γ(𝛼)). Multiplying both sides of (126) by (log(𝑡/𝑎))^{𝛽(1−𝛼)}, we obtain 𝑡log𝑎^{𝛽(1−𝛼)}𝑣(𝑡)≤|𝑏|𝑡log𝑎^{𝛽(1−𝛼)}+𝐶₁^𝑡_𝑎𝜑^𝑞(𝑠)𝑣^𝑞𝑚(𝑠)𝑑𝑠^1/𝑞,𝑡>𝑎. (127) Let 𝑧(𝑡) denote the left-hand side of (127). The insertion of the term 𝑠log𝑎^{−𝑞𝑚𝛽(1−𝛼)}𝑠log𝑎^{𝑞𝑚𝛽(1−𝛼)} (128) inside the integral gives 𝑧(𝑡)≤|𝑏|𝑡log𝑎^{𝛽(1−𝛼)}+𝐶₁^𝑡_𝑎𝜑^𝑞(𝑠)𝑠log𝑎^{−𝑞𝑚𝛽(1−𝛼)}𝑧^𝑞𝑚(𝑠)𝑑𝑠^1/𝑞. (129) Raising both sides of (129) to the power 𝑞 we get 𝑧^𝑞(𝑡)≤2^𝑞−1|𝑏|^𝑞𝑡log𝑎^{𝑞𝛽(1−𝛼)}+𝐶^𝑞₁^𝑡_𝑎𝜑^𝑞(𝑠)𝑠log𝑎^{−𝑞𝑚𝛽(1−𝛼)}𝑧^𝑞𝑚(𝑠)𝑑𝑠. (130) Let us set 𝑤(𝑡)=𝐶^𝑞₁^𝑡_𝑎𝜑^𝑞(𝑠)𝑠log𝑎^{−𝑞𝑚𝛽(1−𝛼)}𝑧^𝑞𝑚(𝑠)𝑑𝑠,𝑡>𝑎. (131) Then, clearly 𝑤(𝑎)=0, and by differentiation 𝑤^′(𝑡)=𝐶^𝑞₁𝜑^𝑞(𝑡)𝑡log𝑎^{−𝑞𝑚𝛽(1−𝛼)}𝑧^𝑞𝑚(𝑡),𝑡>𝑎. (132) Moreover, it is clear that 𝑤 is a continuous, nonnegative and nondecreasing function in [𝑎,∞).
Now, we would like to estimate the right hand side of (132) in term of 𝑤(𝑡). From (130) and (131) we entail that 𝑧^𝑞(𝑡)≤2^𝑞−1|𝑏|^𝑞𝑡log𝑎^{𝑞𝛽(1−𝛼)}+𝑤(𝑡),𝑡>𝑎. (133) Raising both sides of (133) to the power 𝑚 we get 𝑧^𝑞𝑚(𝑡)≤2^𝑚𝑞−1|𝑏|^𝑚𝑞𝑡log𝑎^{𝑚𝑞𝛽(1−𝛼)}+𝑤^𝑚(𝑡),𝑡>𝑎. (134) The substitution of (134) in (132) yields 𝑤^′(𝑡)≤2^𝑚𝑞−1𝐶^𝑞₁𝜑^𝑞(𝑡)𝑡log𝑎^{−𝑞𝑚𝛽(1−𝛼)}×|𝑏|^𝑚𝑞𝑡log𝑎^{𝑚𝑞𝛽(1−𝛼)}+𝑤^𝑚(𝑡)≤2^𝑚𝑞−1|𝑏|^𝑚𝑞𝐶^𝑞₁𝜑^𝑞(𝑡)+2^𝑚𝑞−1𝐶^𝑞₁×𝑡log𝑎^{−𝑞𝑚𝛽(1−𝛼)}𝜑^𝑞(𝑡)𝑤^𝑚(𝑡). (135) Applying Lemma 16 (with 𝑤(𝑢)=𝑢^𝑚) we infer that 𝑤(𝑡)≤𝐺⁻¹𝐺𝑤(𝑎)+2^𝑚𝑞−1|𝑏|^𝑚𝑞𝐶^𝑞₁^𝑡_𝑎𝜑^𝑞(𝑠)𝑑𝑠+2^𝑚𝑞−1𝐶^𝑞₁^𝑡_𝑎𝑠log𝑎^{−𝑞𝑚𝛽(1−𝛼)}𝜑^𝑞(𝑠)𝑑𝑠. (136) Let us set 𝑙(𝑡)=2^𝑚𝑞−1|𝑏|^𝑚𝑞𝐶^𝑞₁^𝑡_𝑎𝜑^𝑞(𝑠)𝑘(𝑡)𝑑𝑠,=2^𝑚𝑞−1𝐶^𝑞₁^𝑡_𝑎𝑠log𝑎^{−𝑞𝑚𝛽(1−𝛼)}𝜑^𝑞(𝑠)𝑑𝑠, (137) then 𝑤(𝑡)≤𝐺⁻¹[𝐺(𝑙(𝑡))+𝑘(𝑡)], (138) where we have used the fact that 𝑤(𝑎)=0. Since 𝐺(𝑟)=∫^𝑟_𝑟0(𝑑𝑠/𝑠^𝑚), 𝑟>0, 𝑟₀>0, then 𝐺(𝑟)=(𝑟^1−𝑚/(1−𝑚))−(𝑟₀^1−𝑚/(1−𝑚)) and 𝐺⁻¹(𝑦)=[𝑟₀^1−𝑚−(𝑚−1)𝑦]^{−1/(𝑚−1)}. That is, 𝑤(𝑡)≤𝐺⁻¹𝑙(𝑡)^1−𝑚−𝑙𝑡1−𝑚^01−𝑚(𝑡≤𝑙𝑡1−𝑚+𝑘)_01−𝑚−()𝑙(𝑡)𝑚−1^1−𝑚−𝑙𝑡1−𝑚^01−𝑚(𝑡)1−𝑚+𝑘^{−1/(𝑚−1)}≤𝑙(𝑡)^1−𝑚−(𝑚−1)𝑘(𝑡)^{−1/(𝑚−1)}. (139) As long as 𝑙(𝑡)^𝑚−1𝑘(𝑡)<1.𝑚−1 (140) In particular, if (‖𝜑(𝑡)‖_𝑞)^𝑚−1‖𝜑(𝑡)(log(𝑡/𝑎))^{−𝑚𝛽(1−𝛼)}‖_𝑞<𝐋^∗/𝟐, then 𝑤(𝑡)≤𝐾₁ for some positive constant 𝐾₁ for all 𝑡>𝑎, and thus, from (129), we see that 𝑧(𝑡)≤|𝑏|𝑡log𝑎^{𝛽(1−𝛼)}+𝐾₁^1/𝑞, (141) and then 𝑣(𝑡)≤|𝑏|+𝐾₁^1/𝑞𝑡log𝑎^{−𝛽(1−𝛼)}≤𝐶,𝑡≥𝑡₀>𝑎 (142) for some positive constant 𝐶. This yields that |𝑢(𝑡)|≤𝐶(log(𝑡/𝑎))^𝛾−1 for 𝑡≥𝑡₀>𝑎. The proof is complete.

Acknowledgment

The authors are very grateful for the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through Project no. 101003.