Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the Spaces 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐻_𝛼,𝛿((𝑎,𝑏),𝑋)

Abstract

The spaces 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),ℝ) and 𝐻_𝛼,𝛿((𝑎,𝑏),ℝ) were defined in ((Hüseynov (1981)), pages 271–277). Some singular integral operators on Banach spaces were examined, (Dostanic (2012)), (Dunford (1988), pages 2419–2426 and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in (Babolian (2011), Okayama (2010), and Thomas (1981)) by numerical methods. In this paper, we define the sets 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐻_𝛼,𝛿((𝑎,𝑏),𝑋) by taking an arbitrary Banach space 𝑋 instead of ℝ, and we show that these sets which are different from the spaces given in (Dunford (1988)) and (Plamenevskiy (1965)) are Banach spaces with the norms ‖⋅‖_{𝛼,𝛿,𝛾} and ‖⋅‖_𝛼,𝛿. Besides, the bounded linear integral operators on the spaces 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐻_𝛼,𝛿((𝑎,𝑏),𝑋), some of which are singular, are derived, and the solutions of the linear Fredholm integral equations of the form 𝑓(𝑠)=𝜙(𝑠)+𝜆∫^𝑏_𝑎𝐴(𝑠,𝑡)𝑓(𝑡)𝑑𝑡,𝑓(𝑠)=𝜙(𝑠)+𝜆∫^𝑏_𝑎𝐴(𝑡,𝑠)𝑓(𝑡)𝑑𝑡 and 𝑓(𝑠,𝑡)=𝜙(𝑠,𝑡)+𝜆∫^𝑏_𝑎𝐴(𝑠,𝑡)𝑓(𝑡,𝑠)𝑑𝑡 are investigated in these spaces by analytical methods.

1. Preliminaries, Background, and Notation

The approximate solution of the singular integral equation 𝑢(𝑠)−1𝜋𝑝.𝑣.¹₋₁𝑘(𝑠,𝑡)𝑢(𝑡)(𝑡)𝑡−𝑠𝑑𝑡=𝑓 (1) was obtained in [1], where 𝑘(𝑠,𝑡) is a real-valued kernel, 𝑓(𝑠) is a given function, and 𝑢(𝑠) is the unknown function.

High-order numerical methods for the singular Fredholm integral equations of the form: 𝜆𝑢(𝑡)−𝑝.𝑣.^𝑏_𝑎|𝑡−𝑠|^𝑝−1𝑘(𝑡,𝑠)𝑢(𝑠)𝑑𝑠=𝑔(𝑡),𝑎≤𝑡≤𝑏 (2) were developed, where 𝜆≠0,0<𝑝<1,𝑘 and 𝑔 are given functions, and 𝑢 is the unknown function. Equations of this form often arise in practical applications such as Dirichlet problems, mathematical problems of radiative equilibrium, and radiative heat transfer problems, [2].

The polar kernel of integral equations was introduced in [3, 4]. This singular kernel is in the following form: 𝑘=𝑔𝑥,𝑦𝑥,𝑦𝑥−𝑦^𝛼+ℎ𝑥,𝑦,0<𝛼≤1, (3) where the first term of this kernel is weakly singular and 𝑔 and ℎ are bounded on the square 𝑠=[−1,1]×[−1,1] and 𝑔(𝑥,𝑦)≠0. With 𝑔=1,ℎ=0, we have the special case of the above kernel: 𝑘=1𝑥,𝑦𝑥−𝑦^𝛼,0<𝛼≤1, (4)see [5]. One of the weakly singular integral and integrodifferential equations with this kernel was given in [6–8]. The solution of the singular integral equation of the form: 𝜇(𝑥)𝜙(𝑥)+𝜆(𝑥)𝑝.𝑣.¹₋₁𝜙𝑦𝑦−𝑥^𝛼(𝑥),|𝑥|𝑑𝑦=𝑓<1,0<𝛼≤1 (5) was examined in [5] by numerical methods, where 𝜇(𝑥)≠0,𝜆(𝑥)≠0 and 𝜇(𝑥),𝜆(𝑥),𝑓(𝑥)∈𝐿²[−1,1] are given functions and 𝜙(𝑥) is the unknown function to be determined.

The integral operator defined by 𝐴∶𝐿₂(0,𝑇)→𝐿₂(),(𝑥)=0,𝑇𝐴𝑓^𝑇₀𝑘𝑥−𝑦𝑓𝑦𝑑𝑦 (6) was studied in [9].

An integral equation of the form: 𝑓(𝑠)=𝜙(𝑠)+𝜆^𝑏_𝑎𝐴(𝑠,𝑡)𝑓(𝑡)𝑑𝑡 (7) is called Fredholm integral equation of the second type. Here, [𝑎,𝑏] is a given interval, 𝑓 is a function on [𝑎,𝑏] which is unknown, and 𝜆 is a parameter. The kernel 𝐴 of the equation is a given function on the square [𝑎,𝑏]×[𝑎,𝑏] and 𝜙 is a given function on [𝑎,𝑏].

Now, we may give some required definitions and theorems.

Definition 1 (see [10, page 41]). Let 𝑋 be a Banach space and (𝑆,Σ,𝜇) be a finite measure space. Then 𝑓∶𝑆→𝑋 is called measurable simple function if there exist 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝐸₁,𝐸₂,…,𝐸_𝑛∈Σ such that 𝑓=∑^𝑛_𝑖=1𝑥_𝑖𝜒_𝐸𝑖, where 𝜒_𝐸𝑖(𝑤)=1,𝑤∈𝐸_𝑖0,𝑤∉𝐸_𝑖. (8)

Definition 2 (see [11, page 88]). The Bochner integral of a simple function 𝑓 given by Definition 1 with respect to 𝜇 on 𝐸∈Σ is defined by _𝐸𝑓𝑑𝜇=^𝑛_𝑖=1𝑥_𝑖𝜇𝐸_𝑖∩𝐸. (9)

Definition 3 (see [12, page 201]). Let 𝑋 be a Banach space with the norm ‖⋅‖_𝑋 and (𝑆,Σ,𝜇) be a finite measure space. Then 𝑔∶𝑆→𝑋 is called strongly measurable if there exists a sequence (𝑓_𝑛) of 𝑋-valued simple functions defined on 𝑆 such that lim_𝑛𝑓_𝑛(𝑡)−𝑔(𝑡)_𝑋=0;𝑡∈𝑆−𝑁,𝜇(𝑁)=0. (10)

Theorem 4 (see [11, page 88]). All continuous functions 𝑓∶𝑆→𝑋 are strongly measurable.

Theorem 5 (see [13, page 336]). Let 𝑋 be a Banach space with the norm ‖⋅‖_𝑋 and (𝑆,Σ,𝜇) be a finite measure space. If 𝑔∶𝑆→𝑋 is strongly measurable, then the scalar function ℎ∶𝑆→ℝ,ℎ(𝑡)=𝑔(𝑡)_𝑋 (11) is Σ-measurable.

Definition 6 (see [11, page 88]). Let (𝑆,Σ,𝜇) be a finite measure space and 𝑋 be a Banach space, and then the Bochner integral of a strongly measurable function 𝑔∶𝑆→𝑋 is the strongly limit of the Bochner integral of an approximating sequence (𝑓_𝑛) of simple functions satisfying (10). That is, lim_𝑛𝐸𝑓_𝑛𝑑𝜇=_𝐸𝑔𝑑𝜇. (12)

Theorem 7 (see [12, page 202]). Let (𝑆,Σ,𝜇) be a finite measure space, 𝑋 be a Banach space with the norm ‖⋅‖_𝑋, and 𝑔∶𝑆→𝑋 be a strongly measurable function. If ∫_𝐸𝑔𝑑𝜇 exists, then _𝐸𝑔𝑑𝜇_𝑋≤_𝐸𝑔_𝑋𝑑𝜇. (13)

Theorem 8 (see [12, page 203]). Let (𝑆,Σ,𝜇) be a finite measure space and 𝑋 be a Banach space with the norm ‖⋅‖_𝑋. The Bochner integral ∫_𝐸𝑔𝑑𝜇 exists if and only if 𝑔 is strongly measurable and ∫_𝐸‖𝑔‖_𝑋𝑑𝜇<∞.

Theorem 9 (see [14, page 82]). Let 𝑋 be a real or complex Banach space with the norm ‖⋅‖_𝑋. Let 𝑎,𝑏,𝛼,𝛿 be the real numbers satisfying −∞<𝑎,𝑏<∞,0<𝛼,𝛿<1 and 𝑀 be a nonnegative constant. The set 𝐶^0,𝛼,𝛿((𝑎,𝑏)×(𝑎,𝑏),𝑋) consisting of all functions 𝐴∶(𝑎,𝑏)×(𝑎,𝑏)→𝑋 fulfilling the conditions: ‖𝐴(𝑠,𝑡)‖_𝑋‖𝐴()(≤𝑀,𝑠+Δ𝑠,𝑡+Δ𝑡−𝐴𝑠,𝑡)‖_𝑋≤𝑀|Δ𝑠|^𝛼+|Δ𝑡|^𝛿 (14) for all 𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈(𝑎,𝑏) is a linear space with the algebraic operations: ()()(),()()(𝐴+𝐵)(𝑠,𝑡=𝐴𝑠,𝑡+𝐵𝑠,𝑡𝜆𝐴)(𝑠,𝑡=𝜆𝐴𝑠,𝑡,𝜆∈𝐾,𝐾=ℝor𝐾=ℂ), (15) and 𝐶^0,𝛼,𝛿((𝑎,𝑏)×(𝑎,𝑏),𝑋) is a Banach space with the norm ‖𝐴‖_0,𝛼,𝛿=maxsup_{()𝑠,𝑡∈𝑎,𝑏}‖𝐴(𝑠,𝑡)‖_𝑋,𝑚_𝐴, (16) where 𝑚_𝐴=sup_{()𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈𝑎,𝑏Δ𝑠≠0∨Δ𝑡≠0}‖𝐴(𝑠+Δ𝑠,𝑡+Δ𝑡)−𝐴(𝑠,𝑡)‖_𝑋|Δ𝑠|^𝛼+|Δ𝑡|^𝛿. (17)
Again, let 𝑋 be a real or complex Banach space with the norm ‖⋅‖_𝑋. Let 𝑎,𝑏,𝛼 be the real numbers with −∞<𝑎,𝑏<∞,0<𝛼<1 and 𝑀 also be a nonnegative constant. Let 𝐴 be an 𝑋-valued function defined on (𝑎,𝑏) such that ‖𝐴(𝑠)‖_𝑋‖𝐴()(𝑠≤𝑀,𝑠+Δ𝑠−𝐴)‖_𝑋≤𝑀(Δ𝑠)^𝛼 (18) for all 𝑠,𝑠+Δ𝑠∈(𝑎,𝑏) with Δ𝑠≥0. By 𝐶^0,𝛼((𝑎,𝑏),𝑋), we denote the set of all functions 𝐴 satisfying (18). 𝐶^0,𝛼((𝑎,𝑏),𝑋) is a linear space with the algebraic operations: (𝑠)(𝑠)(𝑠),(𝑠)(𝑠)(𝐴+𝐵)(=𝐴+𝐵𝜆𝐴)(=𝜆𝐴,𝜆∈𝐾,𝐾=ℝor𝐾=ℂ), (19) and 𝐶^0,𝛼((𝑎,𝑏),𝑋) is a Banach space with the norm: ‖𝐴‖_0,𝛼=maxsup_{()𝑠∈𝑎,𝑏}‖𝐴(𝑠)‖_𝑋,sup_{()𝑠+Δ𝑠∈𝑎,𝑏Δ𝑠≠0}‖𝐴(𝑠+Δ𝑠)−𝐴(𝑠)‖_𝑋(Δ𝑠)^𝛼. (20)

𝐶^0,𝛼,𝛿((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐶^0,𝛼((𝑎,𝑏),𝑋) are called a Hölder space. The functions spaces which are similar to 𝐶^0,𝛼,𝛿((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐶^0,𝛼((𝑎,𝑏),𝑋) were investigated in [15, pages 2419–2426], [16, pages 25–51], and [17, pages 18–33]. The class 𝐺₁(ℎ,𝑒) of the functions 𝑘 satisfying the equalities 𝑘𝑥,𝑦=𝑂𝑥−𝑦^−𝑚ℎ𝑥−𝑦 (21) and 𝑘𝑥^′,𝑦−𝑘𝑥^′′ℎ𝑝,𝑦=𝑂^𝑦𝑥^′,𝑥^′′𝑒𝑥^′−𝑥^′′𝑝^𝑚_𝑦𝑥^′,𝑥^′′𝑒𝑝_𝑦𝑥^′−𝑥^′′ (22) was introduced in [18], where functions ℎ,𝑒∶(0,∞)→(0,∞) are increasing, 𝑝_𝑦𝑥^′,𝑥^′′=min𝑦−𝑥^′,𝑦−𝑥^′′ (23) and 𝑚 is a natural number.

Theorem 10 (see [14, page 16]). Let 𝑇∶𝑋→𝑋 be a bounded linear operator mapping a Banach space 𝑋 into itself with ‖𝑇‖<1 and 𝐼∶𝑋→𝑋 denote the identity operator. Then 𝐼−𝑇 has a bounded inverse operator on 𝑋 which is given by the Neumann series: (𝐼−𝑇)⁻¹=^∞_𝑘=0𝑇^𝑘 (24) which satisfies (𝐼−𝑇)⁻¹≤1‖𝑇‖.1− (25) The iterated operators 𝑇^𝑛 are defined by 𝑇⁰=𝐼 and 𝑇^𝑛=𝑇𝑇^𝑛−1. The series in the right of (24) is convergent in the norm on 𝐵(𝑋,𝑋).

2. The Main Results

2.1. The Spaces 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐻_𝛼,𝛿((𝑎,𝑏),𝑋)

In this section, we determine essentially the spaces 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐻_𝛼,𝛿((𝑎,𝑏),𝑋).

Theorem 11. Let 𝑋 be a real or complex Banach space with the norm ‖⋅‖_𝑋, 𝑎,𝑏,𝛼,𝛿,𝛾 be real numbers with −∞<𝑎,𝑏<∞,0<𝛼,𝛼+𝛾<1,0<𝛾<𝛿<1, and 𝑀 be a nonnegative constant. Then the set 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) of all functions ()×()()()𝐴∶𝑎,𝑏𝑎,𝑏→𝑋,𝑠,𝑡→𝐴𝑠,𝑡 (26) satisfying the inequalities ‖𝐴(𝑠,𝑡)‖_𝑋≤𝑀(𝑡−𝑎)^𝛼,‖Δ𝐴‖_𝑋≤𝑀|Δ𝑠|^𝛿+(Δ𝑡)^𝛾(𝑡−𝑎)^𝛼+𝛾,Δ𝐴=𝐴(𝑠+Δ𝑠,𝑡+Δ𝑡)−𝐴(𝑠,𝑡) (27)(28) for all 𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈(𝑎,𝑏) with Δ𝑡≥0 is a linear space with the usual algebraic operations addition and scalar multiplication defined by ()()(),()()(),𝐴+𝐵)(𝑠,𝑡=𝐴𝑠,𝑡+𝐵𝑠,𝑡𝜆𝐴)(𝑠,𝑡=𝜆𝐴𝑠,𝑡,𝜆∈𝐾,𝐾=ℝ𝑜𝑟𝐾=ℂ (29) and 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) is a Banach space with the norm ‖𝐴‖_{𝛼,𝛿,𝛾}=max𝑝_𝐴,𝑘_𝐴, (30) where 𝑝_𝐴 and 𝑘_𝐴 are defined by 𝑝_𝐴=sup_{()𝑠,𝑡∈𝑎,𝑏}(𝑡−𝑎)^𝛼‖𝐴(𝑠,𝑡)‖_𝑋,𝑘_𝐴=sup_{()𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈𝑎,𝑏Δ𝑠≠0∨Δ𝑡≠0}(𝑡−𝑎)^𝛼+𝛾‖Δ𝐴‖_𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾. (31)(32)

Proof. Let 𝑋 be a Banach space with the norm ‖⋅‖_𝑋. It is known that the set 𝒜={𝐴∣𝐴∶(𝑎,𝑏)×(𝑎,𝑏)→𝑋} is a linear space with operations in 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋). Also, it is obvious that 𝐴+𝐵,𝜆𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋). On the other hand, since 𝐻_{𝛼,𝛿,𝛾})×())={()×()((𝑎,𝑏𝑎,𝑏,𝑋𝐴∣𝐴∶𝑎,𝑏𝑎,𝑏→𝑋,𝐴satisifes(27)and(28)}⊂𝒜, (33)𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) is a linear space.
Furthermore, 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) is a normed space with the norm ‖⋅‖_{𝛼,𝛿,𝛾}. Indeed, consider the following. (N1) It is clear that ‖𝐴‖_{𝛼,𝛿,𝛾}≥0 for all 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋). (N2) If 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and ‖𝐴‖_{𝛼,𝛿,𝛾}=0, then 𝐴(𝑠,𝑡)=0 for all 𝑠,𝑡∈(𝑎,𝑏), since ‖𝐴‖_{𝛼,𝛿,𝛾}=max𝑝_𝐴,𝑘_𝐴=0. (34) So 𝐴=0. On the other hand, if 𝐴=0, it is found that ‖𝐴‖_{𝛼,𝛿,𝛾}=0 by (30), (31) and (32). Hence, the proposition “‖𝐴‖_{𝛼,𝛿,𝛾}=0 if and only if 𝐴=0” is true.(N3) Let 𝜆∈𝐾,(𝐾=ℝor𝐾=ℂ). Since 𝑝_(𝜆𝐴)=sup_{()𝑠,𝑡∈𝑎,𝑏}(𝑡−𝑎)^𝛼‖(𝜆𝐴)(𝑠,𝑡)‖_𝑋=|𝜆|𝑝_𝐴,𝑘_(𝜆𝐴)=sup_{()𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈𝑎,𝑏Δ𝑠≠0∨Δ𝑡≠0}(𝑡−𝑎)^𝛼+𝛾‖Δ(𝜆𝐴)‖_𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾=|𝜆|𝑘_𝐴 (35)by (31) and (32), ‖𝜆𝐴‖_{𝛼,𝛿,𝛾}=max𝑝_(𝜆𝐴),𝑘_(𝜆𝐴)=|𝜆|max𝑝_𝐴,𝑘_𝐴=|𝜆|‖𝐴‖_{𝛼,𝛿,𝛾}. (36) (N4) If 𝐴,𝐵∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋), 𝑝_(𝐴+𝐵)=sup_{()𝑠,𝑡∈𝑎,𝑏}(𝑡−𝑎)^𝛼‖(𝐴+𝐵)(𝑠,𝑡)‖_𝑋≤sup_{()𝑠,𝑡∈𝑎,𝑏}(𝑡−𝑎)^𝛼‖𝐴(𝑠,𝑡)‖_𝑋+sup_{()𝑠,𝑡∈𝑎,𝑏}(𝑡−𝑎)^𝛼‖𝐵(𝑠,𝑡)‖_𝑋=𝑝_𝐴+𝑝_𝐵 (37) by (31) and 𝑘_(𝐴+𝐵)=sup_{()𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈𝑎,𝑏Δ𝑠≠0∨Δ𝑡≠0}(𝑡−𝑎)^𝛼+𝛾‖Δ(𝐴+𝐵)‖_𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾≤sup_{()𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈𝑎,𝑏Δ𝑠≠0∨Δ𝑡≠0}(𝑡−𝑎)^𝛼+𝛾‖Δ𝐴‖_𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾+sup_{()𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈𝑎,𝑏Δ𝑠≠0∨Δ𝑡≠0}(𝑡−𝑎)^𝛼+𝛾‖Δ𝐵‖_𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾=𝑘_𝐴+𝑘_𝐵 (38) by (32). Hence, ‖𝐴+𝐵‖_{𝛼,𝛿,𝛾}=max𝑝_(𝐴+𝐵),𝑘_(𝐴+𝐵)≤max𝑝_𝐴+𝑝_𝐵,𝑘_𝐴+𝑘_𝐵. (39)From the inequality max𝑝_𝐴+𝑝_𝐵,𝑘_𝐴+𝑘_𝐵≤max𝑝_𝐴,𝑘_𝐴+max𝑝_𝐵,𝑘_𝐵 (40) which holds for all real numbers 𝑝_𝐴,𝑝_𝐵,𝑘_𝐴,𝑘_𝐵≥0, we obtain ‖𝐴+𝐵‖_{𝛼,𝛿,𝛾}≤‖𝐴‖_{𝛼,𝛿,𝛾}+‖𝐵‖_{𝛼,𝛿,𝛾}.
As a result, 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) is a normed space with the norm ‖⋅‖_{𝛼,𝛿,𝛾}.
The space 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) is a Banach space with respect to ‖⋅‖_{𝛼,𝛿,𝛾}. To see this we consider an arbitrary Cauchy sequence (𝐴_𝑛) in 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋), and we show that (𝐴_𝑛) converges to a function 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋). Since (𝐴_𝑛) is Cauchy, for every 𝜀>0, there exists 𝑛₀(𝜀)∈ℕ such that 𝐴_𝑛−𝐴_{𝑚𝛼,𝛿,𝛾}<𝜀,𝑚,𝑛>𝑛₀(𝜀). (41) So from (41), sup_{()𝑠,𝑡∈𝑎,𝑏}(𝑡−𝑎)^𝛼𝐴_𝑛(𝑠,𝑡)−𝐴_𝑚(𝑠,𝑡)_𝑋<𝜀,sup_{()𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈𝑎,𝑏Δ𝑠≠0∨Δ𝑡≠0}(𝑡−𝑎)^𝛼+𝛾Δ𝐴_𝑛−𝐴_𝑚𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾<𝜀, (42) where Δ𝐴_𝑛−𝐴_𝑚=𝐴_𝑛−𝐴_𝑚()−𝐴𝑠+Δ𝑠,𝑡+Δ𝑡_𝑛−𝐴_𝑚(𝑠,𝑡). (43) By (42), while Δ𝑠≠0 or Δ𝑡≠0, we have (𝑡−𝑎)^𝛼𝐴_𝑛(𝑠,𝑡)−(𝑡−𝑎)^𝛼𝐴_𝑚(𝑠,𝑡)_𝑋<𝜀,(𝑡−𝑎)^𝛼+𝛾Δ𝐴_𝑛−𝐴_𝑚𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾<𝜀,𝑚,𝑛>𝑛₀(𝜀) (44)(45) for all 𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈(𝑎,𝑏),(Δ𝑡≥0). We see by (44) that ((𝑡−𝑎)^𝛼𝐴_𝑛(𝑠,𝑡)) is Cauchy in 𝑋. Since 𝑋 is complete, there exists a unique 𝑥∈𝑋 such that lim_𝑛→∞(𝑡−𝑎)^𝛼𝐴_𝑛(𝑠,𝑡)=𝑥orlim_𝑛→∞𝐴_𝑛()=1𝑠,𝑡(𝑡−𝑎)^𝛼𝑥. (46) The limit 𝑥∈𝑋 depends on the choice of 𝑠,𝑡∈(𝑎,𝑏). This defines a function: ()×()()(),𝐴∶𝑎,𝑏𝑎,𝑏→𝑋,𝑠,𝑡→𝐴𝑠,𝑡 (47) where lim_𝑛→∞𝐴_𝑛(𝑠,𝑡)=𝐴(𝑠,𝑡). (48) Now, we want to show that lim_𝑛→∞‖𝐴_𝑛−𝐴‖_{𝛼,𝛿,𝛾}=0 and 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋). By (48), the continuity of the norm gives together with (44) and (45) that (𝑡−𝑎)^𝛼𝐴_𝑛(𝑠,𝑡)−𝐴(𝑠,𝑡)_𝑋=(𝑡−𝑎)^𝛼𝐴_𝑛(𝑠,𝑡)−lim_𝑚→∞𝐴_𝑚(𝑠,𝑡)_𝑋=lim_𝑚→∞(𝑡−𝑎)^𝛼𝐴_𝑛(𝑠,𝑡)−𝐴_𝑚(𝑠,𝑡)_𝑋<𝜀,(𝑡−𝑎)^𝛼+𝛾Δ𝐴_𝑛−𝐴_𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾=(𝑡−𝑎)^𝛼+𝛾Δ𝐴_𝑛−lim_𝑚→∞Δ𝐴_𝑚𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾=lim_𝑚→∞(𝑡−𝑎)^𝛼+𝛾Δ𝐴_𝑛−𝐴_𝑚𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾<𝜀 (49)(50) for all 𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈(𝑎,𝑏) with Δ𝑡≥0 such that Δ𝑠≠0 or Δ𝑡≠0. Since ‖𝐴_𝑛−𝐴‖_{𝛼,𝛿,𝛾}<𝜀 for all 𝑛>𝑛₀(𝜀) by (49) and (50), we derive lim_𝑛→∞‖𝐴_𝑛−𝐴‖_{𝛼,𝛿,𝛾}=0. Furthermore, since (𝐴_𝑛) is bounded, there exists the nonnegative constant 𝐶 such that ‖𝐴_𝑛‖_{𝛼,𝛿,𝛾}≤𝐶 which yields sup_{()𝑠,𝑡∈𝑎,𝑏}(𝑡−𝑎)^𝛼𝐴_𝑛(𝑠,𝑡)_𝑋≤𝐶,sup_{()𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈𝑎,𝑏Δ𝑠≠0∨Δ𝑡≠0}(𝑡−𝑎)^𝛼+𝛾Δ𝐴_𝑛𝑋|Δ𝑠|^𝛿+(Δ𝑡)^𝛾≤𝐶 (51) for all 𝑛∈ℕ. Thus, it is obtained by (51) that 𝐴_𝑛(𝑠,𝑡)_𝑋≤𝐶(𝑡−𝑎)^𝛼,𝐴_𝑛(𝑠+Δ𝑠,𝑡+Δ𝑡)−𝐴_𝑛(𝑠,𝑡)_𝑋≤𝐶|Δ𝑠|^𝛿+(Δ𝑡)^𝛾(𝑡−𝑎)^𝛼+𝛾 (52)(53) for all 𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈(𝑎,𝑏) with Δ𝑡≥0 and 𝑛∈ℕ. By (52) and (49), ‖𝐴(𝑠,𝑡)‖_𝑋=𝐴_𝑛(𝑠,𝑡)+𝐴(𝑠,𝑡)−𝐴_𝑛(𝑠,𝑡)_𝑋≤𝐴_𝑛(𝑠,𝑡)_𝑋+𝐴_𝑛(𝑠,𝑡)−𝐴(𝑠,𝑡)_𝑋≤𝐶+𝜀(𝑡−𝑎)^𝛼. (54) By (53) and (50), ‖Δ𝐴‖_𝑋=Δ𝐴+Δ𝐴_𝑛−Δ𝐴_𝑛𝑋≤Δ𝐴_𝑛𝑋+Δ𝐴_𝑛−𝐴_𝑋≤(𝐶+𝜀)|Δ𝑠|^𝛿+(Δ𝑡)^𝛾(𝑡−𝑎)^𝛼+𝛾 (55) for all 𝜀>0 and 𝑛>𝑛₀(𝜀). Therefore, we obtain from (54) and (55) that there exists the nonnegative constant 𝐶 such that ‖𝐴(𝑠,𝑡)‖_𝑋≤𝐶(𝑡−𝑎)^𝛼,‖𝐴(𝑠+Δ𝑠,𝑡+Δ𝑡)−𝐴(𝑠,𝑡)‖_𝑋≤𝐶|Δ𝑠|^𝛿+(Δ𝑡)^𝛾(𝑡−𝑎)^𝛼+𝛾. (56) Hence, 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋). This step completes the proof.

Theorem 12. The inclusion 𝐶^0,𝛿,𝛾((𝑎,𝑏)×(𝑎,𝑏),𝑋)⊂𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) (57) and the inequality ‖𝐴‖_{𝛼,𝛿,𝛾}≤max‖𝐴‖_0,𝛿,𝛾(𝑏−𝑎)^𝛼,‖𝐴‖_0,𝛿,𝛾(𝑏−𝑎)^𝛼+𝛾 (58) hold, where ‖𝐴‖_0,𝛿,𝛾 is defined by (16).

Proof. If 𝐴∈𝐶^0,𝛿,𝛾((𝑎,𝑏)×(𝑎,𝑏),𝑋), then by (16) and (17) there exists the constant ‖𝐴‖_0,𝛿,𝛾 satisfying the inequalities: ‖𝐴(𝑠,𝑡)‖_𝑋≤‖𝐴‖_0,𝛿,𝛾(𝑏−𝑎)^𝛼(𝑡−𝑎)^𝛼,‖𝐴(𝑠+Δ𝑠,𝑡+Δ𝑡)−𝐴(𝑠,𝑡)‖_𝑋≤‖𝐴‖_0,𝛿,𝛾(𝑏−𝑎)^𝛼+𝛾|Δ𝑠|^𝛿+(Δ𝑡)^𝛾(𝑡−𝑎)^𝛼+𝛾 (59) for all 𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈(𝑎,𝑏) with Δ𝑡≥0. By taking 𝑀₁=max‖𝐴‖_0,𝛿,𝛾(𝑏−𝑎)^𝛼,‖𝐴‖_0,𝛿,𝛾(𝑏−𝑎)^𝛼+𝛾, (60) it is obtained by (59) that 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and ‖𝐴‖_{𝛼,𝛿,𝛾}≤𝑀₁. That is, 𝐶^0,𝛿,𝛾((𝑎,𝑏)×(𝑎,𝑏),𝑋)⊂𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋), (61) as desired.

Example 13. Let us take ℝ instead of 𝑋 and define the function 𝐴 as 𝐴∶(𝑎,𝑏)×(𝑎,𝑏)→ℝ,𝐴(𝑠,𝑡)=𝑠+𝑡. (62) Then, we have ‖𝐴(𝑠,𝑡)‖_ℝ=|𝐴(𝑠,𝑡)|≤2|𝑏|=𝑀₁,‖𝐴(𝑠+Δ𝑠,𝑡+Δ𝑡)−𝐴(𝑠,𝑡)‖_ℝ=|𝐴()(≤||+||=||𝑠+Δ𝑠,𝑡+Δ𝑡−𝐴𝑠,𝑡)|Δ𝑠Δ𝑡Δ𝑠^𝛿|Δ𝑠|^1−𝛿+|Δ𝑡|^𝛾|Δ𝑡|^1−𝛾≤max(𝑏−𝑎)^1−𝛿,(𝑏−𝑎)^1−𝛾|Δ𝑠|^𝛿+|Δ𝑡|^𝛾=𝑀₂|Δ𝑠|^𝛿+|Δ𝑡|^𝛾 (63) for all 𝑠,𝑡,𝑠+Δ𝑠,𝑡+Δ𝑡∈(𝑎,𝑏). Thus, we obtain by (63) that 𝐴∈𝐶^0,𝛿,𝛾((𝑎,𝑏)×(𝑎,𝑏),𝑋) and ‖𝐴‖_0,𝛿,𝛾≤max{𝑀₁,𝑀₂}=𝑀. From Theorem 12, we conclude that 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and ‖𝐴‖_{𝛼,𝛿,𝛾}≤max𝑀(𝑏−𝑎)^𝛼,𝑀(𝑏−𝑎)^𝛼+𝛾. (64)

Theorem 14. The function 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) is continuous with respect to the Euclidean metric on (𝑎,𝑏)×(𝑎,𝑏)⊂ℝ².

Proof. Let (𝑠₀,𝑡₀)∈(𝑎,𝑏)×(𝑎,𝑏) and 𝑑 be usual metric on ℝ². Then, we wish to show that the function 𝐴∶((𝑎,𝑏)×(𝑎,𝑏),𝑑)→(𝑋,‖⋅‖_𝑋) is continuous. It is clear that lim_{()→𝑠𝑠,𝑡0,𝑡0}𝑑(𝑠,𝑡),𝑠₀,𝑡₀=lim_{()→𝑠𝑠,𝑡0,𝑡0}𝑠−𝑠₀²+𝑡−𝑡₀²=0. (65) Besides, since 0≤𝑠−𝑠₀≤𝑑(𝑠,𝑡),𝑠₀,𝑡₀,0≤𝑡−𝑡₀≤𝑑(𝑠,𝑡),𝑠₀,𝑡₀, (66) we have by (65) that 𝑠→𝑠₀ and 𝑡→𝑡₀. Thus, since the equalities: 𝐴(𝑠,𝑡)−𝐴𝑠₀,𝑡_0𝑋=𝐴𝑠₀+𝑠−𝑠₀,𝑡₀+𝑡−𝑡₀−𝐴𝑠₀,𝑡_0𝑋,𝐴(𝑠,𝑡)−𝐴𝑠₀,𝑡_0𝑋=𝐴𝑠₀,𝑡₀−𝐴(𝑠,𝑡)_𝑋=𝐴𝑠+𝑠₀−𝑠,𝑡+𝑡₀−𝑡−𝐴(𝑠,𝑡)_𝑋 (67)(68) hold, there exists the nonnegative constant 𝑀 by (28) that the inequalities: 0≤𝐴(𝑠,𝑡)−𝐴𝑠₀,𝑡_0𝑋≤𝑀𝑠−𝑠₀^𝛿+𝑡−𝑡₀^𝛾𝑡₀−𝑎^𝛼+𝛾,𝑡≥𝑡₀0≤𝐴(𝑠,𝑡)−𝐴𝑠₀,𝑡_0𝑋≤𝑀𝑠₀−𝑠^𝛿+𝑡₀−𝑡^𝛾(𝑡−𝑎)^𝛼+𝛾,𝑡<𝑡₀ (69) hold from (67) and (68), respectively. By (69), we have lim_{()→𝑠𝑠,𝑡0,𝑡0}𝐴(𝑠,𝑡)−𝐴𝑠₀,𝑡_0𝑋=0. (70) Hence, the function 𝐴∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) is continuous at the arbitrary point (𝑠₀,𝑡₀)∈(𝑎,𝑏)×(𝑎,𝑏) which means that it is continuous on (𝑎,𝑏)×(𝑎,𝑏).

Theorem 15. Let 𝑋 be a real or complex Banach space with the norm ‖⋅‖_𝑋, 𝑎,𝑏,𝛼,𝛿 be real numbers such that −∞<𝑎,𝑏<∞,0<𝛼,𝛿,𝛼+𝛿<1, and 𝑀 be a nonnegative constant. The set 𝐻_𝛼,𝛿((𝑎,𝑏),𝑋) of all functions satisfying ‖𝐴(𝑠)‖_𝑋≤𝑀(𝑠−𝑎)^𝛼,‖𝐴(𝑠+Δ𝑠)−𝐴(𝑠)‖_𝑋≤𝑀(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿 (71) for all 𝑠,𝑡,𝑠+Δ𝑠∈(𝑎,𝑏) with Δ𝑠≥0 is a linear space with the usual algebraic operations addition and scalar multiplication (𝑠)(𝑠)(𝑠),(𝑠)(𝑠)(),𝐴+𝐵)(=𝐴+𝐵𝜆𝐴)(=𝜆𝐴,𝜆∈𝐾,𝐾=ℝ𝑜𝑟𝐾=ℂ (72) and 𝐻_𝛼,𝛿((𝑎,𝑏),𝑋) is a Banach space with the norm: ‖𝐴‖_𝛼,𝛿=maxsup_{()𝑠∈𝑎,𝑏}(𝑠−𝑎)^𝛼‖𝐴(𝑠)‖_𝑋,sup_{()𝑠,𝑠+Δ𝑠∈𝑎,𝑏Δ𝑠≠0}(𝑠−𝑎)^𝛼+𝛿‖Δ𝐴‖_𝑋(Δ𝑠)^𝛿, (73) where Δ𝐴=𝐴(𝑠+Δ𝑠)−𝐴(𝑠).

Theorem 16. The function 𝐴∈𝐻_𝛼,𝛿((𝑎,𝑏),𝑋) is continuous with respect to the Euclidean metric on (𝑎,𝑏)⊂ℝ.

Theorem 17. The inclusion 𝐶^0,𝛿((𝑎,𝑏),𝑋)⊂𝐻_𝛼,𝛿((𝑎,𝑏),𝑋) (74) and the inequality ‖𝐴‖_𝛼,𝛿≤max‖𝐴‖_0,𝛿(𝑏−𝑎)^𝛼,‖𝐴‖_0,𝛿(𝑏−𝑎)^𝛼+𝛿 (75) hold, where ‖𝐴‖_0,𝛿 is the norm in the space 𝐶^0,𝛿((𝑎,𝑏),𝑋).

Since the proofs of Theorems 15–17 are completely similar to that of Theorems 11–14, we leave them to the reader.

Lemma 18. Let 𝑋 be a real or complex Banach algebra with the norm ‖⋅‖_𝑋 and define 𝐵_𝑛 by 𝐵_𝑛(𝑠,𝑡)=𝐴₁(𝑠,𝑡)𝐴₂(𝑠,𝑡)⋯𝐴_𝑛(𝑠,𝑡);𝑠,𝑡∈(𝑎,𝑏), (76) where 𝐴_𝑖∈𝐻_{𝛼𝑖,𝛿𝑖,𝛾})×()),(),𝛼((𝑎,𝑏𝑎,𝑏,𝑋𝑖=1,2,…,𝑛₁+𝛼₂+⋯+𝛼_𝑛+𝛾<1for𝑛≥1. (77) Then, if 𝛿=min𝛿₁,𝛿₂,…,𝛿_𝑛,(),𝐷𝑛≥1₁𝐷=1,_𝑛=max(𝑏−𝑎)^{min{𝛿1,𝛿2,…,𝛿𝑛−1}−𝛿},(𝑏−𝑎)^{𝛿𝑛−𝛿},(𝑛≥2),𝐶_𝑛=𝐷₁𝐷₂⋯𝐷_𝑛, (78)(79) then 𝐵_𝑛∈𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐵_{𝑛𝛼1+𝛼2+⋯+𝛼𝑛,𝛿,𝛾}≤2^𝑛−1𝐶_𝑛𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑛𝛼𝑛,𝛿𝑛,𝛾}. (80)

Proof. We use the induction method. If 𝐴₁∈𝐻_{𝛼1,𝛿1,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋), then 𝛿=min𝛿₁=𝛿₁,𝐵₁(𝑠,𝑡)=𝐴₁(),𝐵𝑠,𝑡₁∈𝐻_{𝛼1,𝛿1,𝛾})×()),𝐵((𝑎,𝑏𝑎,𝑏,𝑋_{1𝛼1,𝛿1,𝛾}=𝐶₁𝐴_{1𝛼1,𝛿1,𝛾}. (81) Thus, Lemma is true for 𝑛=1.
Assume that if 𝐴_𝑖∈𝐻_{𝛼𝑖,𝛿𝑖,𝛾})×()),(),𝛿((𝑎,𝑏𝑎,𝑏,𝑋𝑖=1,2,…,𝑘𝛿=min₁,𝛿₂,…,𝛿_𝑘, (82) then 𝐵_𝑘∈𝐻_{𝛼1+𝛼2+⋯+𝛼𝑘,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐵_{𝑘𝛼1+𝛼2+⋯+𝛼𝑘,𝛿,𝛾}≤2^𝑘−1𝐶_𝑘𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑘𝛼𝑘,𝛿𝑘,𝛾}. (83)
Now, let 𝐴_𝑖∈𝐻_{𝛼𝑖,𝛿𝑖,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋),(𝑖=1,2,…,𝑘+1) and 𝛿=min{𝛿₁,𝛿₂,…,𝛿_𝑘+1}. Then, we must show that 𝐵_𝑘+1∈𝐻_{𝛼1+𝛼2+⋯+𝛼𝑘+1,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and𝐵_{𝑘+1𝛼1+𝛼2+⋯+𝛼𝑘+1,𝛿,𝛾}≤2^𝑘𝐶_𝑘+1𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}. (84)
Then,𝐵_𝑘+1(𝑠,𝑡)=𝐴₁(𝑠,𝑡)𝐴₂(𝑠,𝑡)⋯𝐴_𝑘(𝑠,𝑡)𝐴_𝑘+1(𝑠,𝑡)=𝐵_𝑘(𝑠,𝑡)𝐴_𝑘+1(𝑠,𝑡). (85)

(1) If 𝑏−𝑎≤1, then we have from 0≤|Δ𝑠|<𝑏−𝑎≤1 that |Δ𝑠|^𝛿𝑘+1,|Δ𝑠|^{min{𝛿1,𝛿2,…,𝛿𝑘}}≤|Δ𝑠|^𝛿;𝛿=min𝛿₁,𝛿₂,…,𝛿_𝑘+1. (86) Furthermore, 𝐵_𝑘+1(𝑠,𝑡)_𝑋=𝐵_𝑘(𝑠,𝑡)𝐴_𝑘+1(𝑠,𝑡)_𝑋≤𝐵_𝑘(𝑠,𝑡)_𝑋𝐴_𝑘+1(𝑠,𝑡)_𝑋≤𝐵_{𝑘𝛼1+𝛼2+⋯+𝛼𝑘,𝛿,𝛾}𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}(𝑡−𝑎)^{𝛼1+𝛼2+⋯+𝛼𝑘}(𝑡−𝑎)^𝛼𝑘+1≤2^𝑘−1𝐶_𝑘𝐴_{1𝛼1,𝛿1,𝛾}⋯𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}(𝑡−𝑎)^{𝛼1+𝛼2+⋯+𝛼𝑘+1} (87) for all 𝑠,𝑡∈(𝑎,𝑏). Since 𝐷_𝑘=1 for all 𝑘∈ℕ, 𝐶_𝑘=1. So, we have 𝐵_𝑘+1(𝑠,𝑡)_𝑋≤2^𝑘𝐶_𝑘+1𝐴_{1𝛼1,𝛿1,𝛾}⋯𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}(𝑡−𝑎)^{𝛼1+𝛼2+⋯+𝛼𝑘+1} (88) from (87). Additionally, Δ𝐵_𝑘+1𝑋=𝐵_𝑘(𝑠+Δ𝑠,𝑡+Δ𝑡)𝐴_𝑘+1(𝑠+Δ𝑠,𝑡+Δ𝑡)−𝐵_𝑘(𝑠,𝑡)𝐴_𝑘+1(𝑠,𝑡)_𝑋=Δ𝐵_𝑘𝐴_𝑘+1(𝑠+Δ𝑠,𝑡+Δ𝑡)+𝐵_𝑘(𝑠,𝑡)Δ𝐴_𝑘+1𝑋≤Δ𝐵_𝑘𝑋𝐴_𝑘+1(𝑠+Δ𝑠,𝑡+Δ𝑡)_𝑋+𝐵_𝑘(𝑠,𝑡)_𝑋Δ𝐴_𝑘+1𝑋. (89) By (89) and (83), we get Δ𝐵_𝑘+1𝑋≤𝐵_{𝑘𝛼1+𝛼2+⋯+𝛼𝑘,𝛿,𝛾}𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}|Δ𝑠|^𝛿+(Δ𝑡)^𝛾(𝑡−𝑎)^{𝛼1+𝛼2+⋯+𝛼𝑘+𝛾}(𝑡+Δ𝑡−𝑎)^𝛼𝑘+1+𝐵_{𝑘𝛼1+𝛼2+⋯+𝛼𝑘,𝛿,𝛾}𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}|Δ𝑠|^𝛿𝑘+1+(Δ𝑡)^𝛾(𝑡−𝑎)^{𝛼1+𝛼2+⋯+𝛼𝑘}(𝑡−𝑎)^{𝛼𝑘+1+𝛾}≤2^𝑘−1𝐶_𝑘𝐴_{1𝛼1,𝛿1,𝛾}⋯𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}|Δ𝑠|^𝛿+(Δ𝑡)^𝛾(𝑡−𝑎)^{𝛼1+𝛼2+⋯+𝛼𝑘+1+𝛾}+2^𝑘−1𝐶_𝑘𝐴_{1𝛼1,𝛿1,𝛾}⋯𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}|Δ𝑠|^𝛿𝑘+1+(Δ𝑡)^𝛾(𝑡−𝑎)^{𝛼1+𝛼2+⋯+𝛼𝑘+1+𝛾}, (90) where 𝛿=min{𝛿₁,𝛿₂,…,𝛿_𝑘}. Thus, we conclude that Δ𝐵_𝑘+1𝑋≤2^𝑘𝐶_𝑘+1𝐴_{1𝛼1,𝛿1,𝛾}⋯𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}|Δ𝑠|^𝛿+(Δ𝑡)^𝛾(𝑡−𝑎)^{𝛼1+𝛼2+⋯+𝛼𝑘+1+𝛾} (91) from (86) and (90), where 𝛿=min{𝛿₁,𝛿₂,…,𝛿_𝑘+1}.

(2) If 𝑏−𝑎>1, then 1<𝐷_𝑘+1, and since 𝐶_𝑘+1=𝐶_𝑘𝐷_𝑘+1 for all 𝑘∈ℕ, we have 1≤𝐶_𝑘<𝐶_𝑘+1. That is, (88) is obtained from (87). Besides,(i) if 0≤|Δ𝑠|<1, then (91) is derived from (86) and (90).(ii) If 1≤|Δ𝑠|<𝑏−𝑎, since 0<|Δ𝑠|/(𝑏−𝑎)<1, we get |Δ𝑠|^𝛿𝑘+1≤(𝑏−𝑎)^{𝛿𝑘+1−𝛿}|Δ𝑠|^𝛿,|Δ𝑠|^{min{𝛿1,𝛿2,…,𝛿𝑘}}≤(𝑏−𝑎)^{min{𝛿1,𝛿2,…,𝛿𝑘}−𝛿}|Δ𝑠|^𝛿, (92)by taking |Δ𝑠|/(𝑏−𝑎) instead of |Δ𝑠| in (86). Let 𝐷_𝑘+1=max(𝑏−𝑎)^{min{𝛿1,𝛿2,…,𝛿𝑘}−𝛿},(𝑏−𝑎)^{𝛿𝑘+1−𝛿}. (93) Since 1<𝐷_𝑘+1, (91) is derived from (90). By (1) and (2), we show that (88) and (91) hold. Therefore, 𝐵_𝑘+1∈𝐻_{𝛼1+𝛼2+⋯+𝛼𝑘+1,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐵_{𝑘+1𝛼1+𝛼2+⋯+𝛼𝑘+1,𝛿,𝛾}≤2^𝑘𝐶_𝑘+1𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑘+1𝛼𝑘+1,𝛿𝑘+1,𝛾}. (94) So, Lemma 18 holds for all 𝑛∈ℕ.

2.2. Some Bounded Linear Integral Operators on the Spaces 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐻_𝛼,𝛿((𝑎,𝑏)×(𝑎,𝑏),𝑋)

Hereafter, by 𝐶_𝑛, we mean the constant defined by (79), and, by the integral of the Banach valued functions, we mean the Bochner integral unless stated otherwise.

Theorem 19. Let 𝑋 be a real or complex Banach algebra with the norm ‖⋅‖_𝑋 and 𝐴∈𝐻_{𝛼1,𝛿1,𝛾1}((𝑎,𝑏)×(𝑎,𝑏),𝑋).
(i) Then, the integral operator 𝑇₁∶𝐶^0,𝛼2((𝑎,𝑏),𝑋)→𝐶^0,𝛿1((𝑎,𝑏),𝑋) (95) defined by 𝑇₁𝑓(𝑠)=^𝑏_𝑎𝐴(𝑠,𝑡)𝑓(𝑡)𝑑𝑡;𝑓∈𝐶^0,𝛼2)),()((𝑎,𝑏,𝑋𝑠∈𝑎,𝑏 (96) is bounded with 𝑇₁≤𝑀𝛼₁,𝛾₁‖𝐴‖_{𝛼1,𝛿1,𝛾1}, (97) that is, 𝑇₁∈𝐵(𝐶^0,𝛼2((𝑎,𝑏),𝑋),𝐶^0,𝛿1((𝑎,𝑏),𝑋)).
(ii) The operator 𝑇₂∶𝐻_𝛼2,𝛿2((𝑎,𝑏),𝑋)→𝐻_𝛼1,𝛾1((𝑎,𝑏),𝑋) (98) defined by 𝑇₂𝑓(𝑠)=^𝑏_𝑎𝐴(𝑡,𝑠)𝑓(𝑡)𝑑𝑡;𝑓∈𝐻_𝛼2,𝛿2)),()((𝑎,𝑏,𝑋𝑠∈𝑎,𝑏 (99) is bounded with 𝑇₂≤𝑀𝛼₂‖𝐴‖_{𝛼1,𝛿1,𝛾1}, (100) that is, 𝑇₂∈𝐵(𝐻_𝛼2,𝛿2((𝑎,𝑏),𝑋),𝐻_𝛼1,𝛾1((𝑎,𝑏),𝑋)).
(iii) Suppose that 𝛼₁+𝛼₂+𝛾₁ and 𝛼₁+𝛿₁<1. Then, the operator 𝑇₃∶𝐻_𝛼2,𝛿2((𝑎,𝑏),𝑋)→𝐻_𝛼1,𝛿1((𝑎,𝑏),𝑋) (101) defined by 𝑇₃𝑓(𝑠)=^𝑏_𝑎𝐴(𝑠,𝑡)𝑓(𝑡)𝑑𝑡;𝑓∈𝐻_𝛼2,𝛿2)),()((𝑎,𝑏,𝑋𝑠∈𝑎,𝑏 (102) is bounded with 𝑇₃≤𝑀𝛼₁,𝛼₂,𝛿₁,𝛾₁‖𝐴‖_{𝛼1,𝛿1,𝛾1}, (103) that is, 𝑇₃∈𝐵(𝐻_𝛼2,𝛿2((𝑎,𝑏),𝑋),𝐻_𝛼1,𝛿1((𝑎,𝑏),𝑋)), where the constants 𝑀(𝑐), 𝑀(𝑑,𝑒), and 𝑀(𝑥,𝑦,𝑧,𝑡) are given by 𝑀(𝑐)=(𝑏−𝑎)^1−𝑐,1−𝑐𝑀(𝑑,𝑒)=max{𝑀(𝑑),𝑀(𝑑+𝑒)},𝑀)𝑥,𝑦,𝑧,𝑡=max(𝑏−𝑎^𝑥𝑀𝑥+𝑦,(𝑏−𝑎)^𝑥+𝑧𝑀𝑥+𝑦+𝑡 (104)(105)(106) for all the real numbers 𝑐,𝑑,𝑒,𝑥,𝑦,𝑧,𝑡 such that 𝑐≠1.

Proof. (i) We have 𝐴(𝑠,𝑡)𝑓(𝑡)_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_0,𝛼2(𝑡−𝑎)^𝛼1,[𝐴(𝑠+Δ𝑠,𝑡)−𝐴(𝑠,𝑡)]𝑓(𝑡)_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_0,𝛼2(Δ𝑠)^𝛿1(𝑡−𝑎)^𝛼1+𝛾1 (107) for all 𝐴∈𝐻_{𝛼1,𝛿1,𝛾1}((𝑎,𝑏)×(𝑎,𝑏),𝑋),𝑓∈𝐶^0,𝛼2((𝑎,𝑏),𝑋) and 𝑠,𝑡,𝑠+Δ𝑠∈(𝑎,𝑏) with Δ𝑠≥0. By (107), it is obtained that 𝑇₁𝑓(𝑠)_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_0,𝛼2^𝑏_𝑎𝑑𝑡(𝑡−𝑎)^𝛼1=𝑀𝛼₁‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_0,𝛼2,Δ𝑇₁𝑓_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_0,𝛼2(Δ𝑠)^𝛿1𝑏_𝑎𝑑𝑡(𝑡−𝑎)^𝛼1+𝛾1=𝑀𝛼₁+𝛾₁‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_0,𝛼2(Δ𝑠)^𝛿1 (108) such that Δ𝑇₁𝑓=𝑇₁𝑓(𝑠+Δ𝑠)−𝑇₁𝑓(𝑠). (109) Thus, it is concluded from (108) that 𝑇₁𝑓∈𝐶^0,𝛿1((𝑎,𝑏),𝑋) and 𝑇₁𝑓_𝛿1≤𝑀𝛼₁,𝛾₁‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_0,𝛼2. (110) Besides, the norm of 𝑇₁ holds the inequality 𝑇₁=sup_{𝑓20,𝛼=1}𝑇₁𝑓_𝛿1≤𝑀𝛼₁,𝛾₁‖𝐴‖_{𝛼1,𝛿1,𝛾1} (111) from (110). Also, since 𝑇₁ is linear, it is an element of the space 𝐵(𝐶^0,𝛼2((𝑎,𝑏),𝑋),𝐶^0,𝛿1((𝑎,𝑏),𝑋)).
On the other hand:
(ii) The inequalities 𝐴(𝑡,𝑠)𝑓(𝑡)_𝑋≤‖𝐴(𝑡,𝑠)‖_𝑋𝑓(𝑡)_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(𝑠−𝑎)^𝛼1(𝑡−𝑎)^𝛼2,[𝐴(𝑡,𝑠+Δ𝑠)−𝐴(𝑡,𝑠)]𝑓(𝑡)_𝑋≤‖𝐴(𝑡,𝑠+Δ𝑠)−𝐴(𝑡,𝑠)‖_𝑋𝑓(𝑡)_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(Δ𝑠)^𝛾1(𝑡−𝑎)^𝛼2(𝑠−𝑎)^𝛼1+𝛾1 (112)(113) hold for all 𝐴∈𝐻_{𝛼1,𝛿1,𝛾1}((𝑎,𝑏)×(𝑎,𝑏),𝑋),𝑓∈𝐻_𝛼2,𝛿2((𝑎,𝑏),𝑋),𝑠,𝑡,𝑠+Δ𝑠∈(𝑎,𝑏)   with  Δ𝑠≥0.
So, by (112), 𝑇₂𝑓(𝑠)_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(𝑠−𝑎)^𝛼1𝑏_𝑎𝑑𝑡(𝑡−𝑎)^𝛼2=𝑀𝛼₂‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(𝑠−𝑎)^𝛼1 (114) and by (113), Δ𝑇₂𝑓_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(Δ𝑠)^𝛾1(𝑠−𝑎)^{𝛼1+𝛾1𝑏}_𝑎𝑑𝑡(𝑡−𝑎)^𝛼2=𝑀𝛼₂‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(Δ𝑠)^𝛾1(𝑠−𝑎)^𝛼1+𝛾1, (115) where Δ(𝑇₂𝑓)=(𝑇₂𝑓)(𝑠+Δ𝑠)−(𝑇₂𝑓)(𝑠). By usage of (114) and (115), we get that 𝑇₂𝑓∈𝐻_𝛼1,𝛾1((𝑎,𝑏),𝑋) and 𝑇₂𝑓_𝛼1,𝛾1≤𝑀𝛼₂‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2. (116) Thus, 𝑇₂ is bounded with 𝑇₂=sup_{𝑓𝛼22,𝛿=1}𝑇₂𝑓_𝛼1,𝛾1≤𝑀𝛼₂‖𝐴‖_{𝛼1,𝛿1,𝛾1} (117) from (116). Furthermore 𝑇₂ is linear, and, hence, 𝑇₂∈𝐵(𝐻_𝛼2,𝛿2((𝑎,𝑏),𝑋),𝐻_𝛼1,𝛾1((𝑎,𝑏),𝑋)).
(iii) It is clear that 𝐴(𝑠,𝑡)𝑓(𝑡)_𝑋≤‖𝐴(𝑠,𝑡)‖_𝑋𝑓(𝑡)_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(𝑡−𝑎)^𝛼1+𝛼2,[𝐴(𝑠+Δ𝑠,𝑡)−𝐴(𝑠,𝑡)]𝑓(𝑡)_𝑋≤‖𝐴(𝑠+Δ𝑠,𝑡)−𝐴(𝑠,𝑡)‖_𝑋𝑓(𝑡)_𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(Δ𝑠)^𝛿1(𝑡−𝑎)^{𝛼1+𝛼2+𝛾1} (118) for all 𝑓∈𝐻_𝛼2,𝛿2((𝑎,𝑏),𝑋),𝑠,𝑡,𝑠+Δ𝑠∈(𝑎,𝑏)  with  Δ𝑠≥0.
By (118), 𝑇₃𝑓(𝑠)_𝑋≤^𝑏_𝑎𝐴(𝑠,𝑡)𝑓(𝑡)_𝑋≤‖𝐴‖𝑑𝑡_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2^𝑏_𝑎𝑑𝑡(𝑡−𝑎)^𝛼1+𝛼2≤(𝑏−𝑎)^𝛼1𝑀𝛼₁+𝛼₂‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(𝑠−𝑎)^𝛼1, (119) and by taking Δ𝑇₃𝑓=𝑇₃𝑓(𝑠+Δ𝑠)−𝑇₃𝑓(𝑠),Δ𝑇₃𝑓_𝑋≤^𝑏_𝑎[𝐴(𝑠+Δ𝑠,𝑡)−𝐴(𝑠,𝑡)]𝑓(𝑡)_𝑋≤‖𝐴‖𝑑𝑡_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(Δ𝑠)^𝛿1𝑏_𝑎𝑑𝑡(𝑡−𝑎)^{𝛼1+𝛼2+𝛾1}≤(𝑏−𝑎)^𝛼1+𝛿1𝑀𝛼₁+𝛼₂+𝛾₁‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2(Δ𝑠)^𝛿1(𝑠−𝑎)^𝛼1+𝛿1. (120)(121) Thus, we have by (119) and (121) that 𝑇₃𝑓∈𝐻_𝛼1,𝛿1((𝑎,𝑏),𝑋) and 𝑇₃𝑓_𝛼1,𝛿1≤𝑀𝛼₁,𝛼₂,𝛿₁,𝛾₁‖𝐴‖_{𝛼1,𝛿1,𝛾1}𝑓_𝛼2,𝛿2. (122) So, by (122), 𝑇₃ is a bounded linear operator and also 𝑇₃=sup_{𝑓𝛼22,𝛿=1}𝑇₃𝑓_𝛼1,𝛿1≤𝑀𝛼₁,𝛼₂,𝛿₁,𝛾₁‖𝐴‖_{𝛼1,𝛿1,𝛾1}. (123) Therefore, 𝑇₃∈𝐵(𝐻_𝛼2,𝛿2((𝑎,𝑏),𝑋),𝐻_𝛼1,𝛿1((𝑎,𝑏),𝑋)).

Hereafter, by 𝑀(𝑐),𝑀(𝑑,𝑒), and 𝑀(𝑥,𝑦,𝑧,𝑡) for all the real numbers 𝑐,𝑑,𝑒,𝑥,𝑦,𝑧,𝑡 with 𝑐≠1, we denote the constants given by (104), (105), and (106), respectively.

Theorem 20. Let 𝑋 be a real or complex Banach algebra with the norm ‖⋅‖_𝑋 and 𝐴∈𝐻_{𝛼1,𝛿1,𝛾1}((𝑎,𝑏)×(𝑎,𝑏),𝑋) such that 𝛾₂<𝛿₁. Then, 𝑇 given by (𝑇𝐵)𝑠₁,𝑠₂=^𝑏_𝑎𝐴𝑠₁,𝑡𝐵𝑡,𝑠₂𝑑𝑡;𝐵∈𝐻_{𝛼2,𝛿2,𝛾2}((𝑎,𝑏)×(𝑎,𝑏),𝑋),𝑠₁,𝑠₂∈(𝑎,𝑏) (124) is a linear operator from 𝐻_{𝛼2,𝛿2,𝛾2}((𝑎,𝑏)×(𝑎,𝑏),𝑋) into 𝐻_{𝛼2,𝛿1,𝛾2}((𝑎,𝑏)×(𝑎,𝑏),𝑋), and, also, 𝑇 is bounded with ‖𝑇‖≤𝑁‖𝐴‖_{𝛼1,𝛿1,𝛾1}, (125) where 𝑁=max𝑀𝛼₁,max(𝑏−𝑎)^𝛾2𝑀𝛼₁+𝛾₁,𝑀𝛼₁. (126)

Proof. The function 𝐶_𝑠1,𝑠2 given by 𝐶_𝑠1,𝑠2(𝑡)=𝐴𝑠₁,𝑡𝐵𝑡,𝑠₂,𝑡∈(𝑎,𝑏) (127) is continuous for each 𝑠₁,𝑠₂∈(𝑎,𝑏). Therefore, the function 𝐶_𝑠1,𝑠2∶(𝑎,𝑏)→𝑋 is strongly measurable by Theorem 4, and, hence, the function ‖𝐶_𝑠1,𝑠2‖_𝑋∶(𝑎,𝑏)→ℝ is Lebesgue measurable from Theorem 5. Since 𝐶_𝑠1,𝑠2(𝑡)_𝑋=𝐴𝑠₁,𝑡𝐵𝑡,𝑠_2𝑋≤‖𝐴‖_{𝛼1,𝛿1,𝛾1}‖𝐵‖_{𝛼2,𝛿2,𝛾2}(𝑡−𝑎)^𝛼1𝑠₂−𝑎^𝛼2;𝑠₁,𝑠₂,𝑡∈(𝑎,𝑏), (128) Lebesgue integral ∫^𝑏_𝑎‖𝐶_𝑠1,𝑠2(𝑡)‖_𝑋𝑑𝑡 exists, and, so, Bochner integral in (124) exists by Theorem 8. Now, by taking 𝑁₁=‖𝐴‖_{𝛼1,𝛿1,𝛾1}‖𝐵‖_{𝛼2,𝛿2,𝛾2}, we get (𝑇𝐵)𝑠₁,𝑠_2𝑋≤^𝑏_𝑎𝐶_𝑠1,𝑠2(𝑡)_𝑋≤𝑁𝑑𝑡₁𝑠₂−𝑎^𝛼2𝑏_𝑎𝑑𝑡(𝑡−𝑎)^𝛼1=𝑀𝛼₁𝑁₁𝑠₂−𝑎^𝛼2 (129) from (128). Let define 𝐷_𝑡𝑠₁,𝑠₂=𝐴𝑠₁,𝑡𝐵𝑡,𝑠₂,Δ𝐷_𝑡=𝐷_𝑡𝑠₁+Δ𝑠₁,𝑠₂+Δ𝑠₂−𝐷_𝑡𝑠₁,𝑠₂,Δ_1,𝑡𝐴=𝐴𝑠₁+Δ𝑠₁,𝑡−𝐴𝑠₁,𝑡,Δ_𝑡,2𝐵=𝐵𝑡,𝑠₂+Δ𝑠₂−𝐵𝑡,𝑠₂,Δ(𝑇𝐵)=(𝑇𝐵)𝑠₁+Δ𝑠₁,𝑠₂+Δ𝑠₂−(𝑇𝐵)𝑠₁,𝑠₂, (130)(131)(132) for all Δ𝑠₂≥0 and 𝑡,𝑠₁,𝑠₂,𝑠₁+Δ𝑠₁,𝑠₂+Δ𝑠₂∈(𝑎,𝑏). Since Δ𝐷_𝑡=Δ_1,𝑡𝐴𝐵𝑡,𝑠₂+Δ𝑠₂+𝐴𝑠₁,𝑡Δ_𝑡,2𝐵, (133) we have Δ𝐷_𝑡𝑋≤(𝑏−𝑎)^𝛾2𝑁₁Δ𝑠₁^𝛿1(𝑡−𝑎)^𝛼1+𝛾1𝑠₂−𝑎^𝛼2+𝛾2+𝑁₁Δ𝑠₂^𝛾2(𝑡−𝑎)^𝛼1𝑠₂−𝑎^𝛼2+𝛾2. (134) So, we obtain ‖Δ(𝑇𝐵)‖_𝑋≤∫^𝑏_𝑎‖Δ𝐷_𝑡‖_𝑋𝑑𝑡 which yields ‖Δ(𝑇𝐵)‖_𝑋≤𝑁₂‖𝐴‖_{𝛼1,𝛿1,𝛾1}‖𝐵‖_{𝛼2,𝛿2,𝛾2}Δ𝑠₁^𝛿1+Δ𝑠₂^𝛾2𝑠₂−𝑎^𝛼2+𝛾2, (135) where 𝑁₂=max{(𝑏−𝑎)^𝛾2𝑀(𝛼₁+𝛾₁),𝑀(𝛼₁)}. Since max{𝑀(𝛼₁),𝑁₂}=𝑁, it is found by (129) and (135) that (𝑇𝐵)𝑠₁,𝑠_2𝑋≤𝑁‖𝐴‖_{𝛼1,𝛿1,𝛾1}‖𝐵‖_{𝛼2,𝛿2,𝛾2}𝑠₂−𝑎^𝛼2,‖Δ(𝑇𝐵)‖_𝑋≤𝑁‖𝐴‖_{𝛼1,𝛿1,𝛾1}‖𝐵‖_{𝛼2,𝛿2,𝛾2}Δ𝑠₁^𝛿1+Δ𝑠₂^𝛾2𝑠₂−𝑎^𝛼2+𝛾2. (136) Hence, by (136), 𝑇𝐵∈𝐻_{𝛼2,𝛿1,𝛾2}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and ‖𝑇𝐵‖_{𝛼2,𝛿1,𝛾2}≤𝑁‖𝐴‖_{𝛼1,𝛿1,𝛾1}‖𝐵‖_{𝛼2,𝛿2,𝛾2}. (137) That is, 𝑇 is bounded by (137) and ‖𝑇‖=sup_{‖𝐵‖𝛼222,𝛿,𝛾=1}‖𝑇𝐵‖_{𝛼2,𝛿1,𝛾2}≤𝑁‖𝐴‖_{𝛼1,𝛿1,𝛾1}. (138) Clearly, 𝑇 is linear. So, 𝑇∈𝐵(𝐻_{𝛼2,𝛿2,𝛾2}((𝑎,𝑏),𝑋),𝐻_{𝛼2,𝛿1,𝛾2}((𝑎,𝑏)×(𝑎,𝑏),𝑋)).

2.3. The Solutions of the Linear Fredholm Integral Equations in the Spaces 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and 𝐻_𝛼,𝛿((𝑎,𝑏)×(𝑎,𝑏),𝑋)

We have the following.

Theorem 21. Let 𝑋 be a real or complex Banach algebra and define the function 𝐴 by 𝐴(𝑠,𝑡)=𝐴₁(𝑠,𝑡)𝐴₂(𝑠,𝑡)⋯𝐴_𝑛(𝑠,𝑡);𝑠,𝑡∈(𝑎,𝑏) (139) such that 𝐴_𝑖∈𝐻_{𝛼𝑖,𝛿𝑖,𝛾})×()){},𝛿((𝑎,𝑏𝑎,𝑏,𝑋∀𝑖∈1,2,…,𝑛𝛿=min₁,𝛿₂,…,𝛿_𝑛,𝛼₁+𝛼₂+⋯+𝛼_𝑛+𝛾<1with𝑛∈ℕ. (140) (i) Then, the linear Fredholm integral equation of the form 𝑓(𝑠)=𝜙(𝑠)+𝜆^𝑏_𝑎𝐴(𝑠,𝑡)𝑓(𝑡)𝑑𝑡 (141) has a unique solution 𝑓 in the space 𝐶^0,𝛿((𝑎,𝑏),𝑋) with𝑓_𝛿≤11−|𝜆|2^𝑛−1𝐶_𝑛𝑀𝛼₁+𝛼₂+⋯+𝛼_𝑛,𝛾𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑛𝛼𝑛,𝛿𝑛,𝛾}𝜙_𝛿, (142)where 𝜙∈𝐶^0,𝛿((𝑎,𝑏),𝑋) and 𝜆 is a real or complex parameter satisfying the inequality: |𝜆|<12^𝑛−1𝐶_𝑛𝑀𝛼₁+𝛼₂+⋯+𝛼_𝑛,𝛾𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑛𝛼𝑛,𝛿𝑛,𝛾}. (143)
(ii) The solution of the equation 𝑓(𝑠)=𝜙(𝑠)+𝜆^𝑏_𝑎𝐴(𝑡,𝑠)𝑓(𝑡)𝑑𝑡 (144) uniquely exists in the space 𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋). Besides, 𝑓_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}≤11−|𝜆|2^𝑛−1𝐶_𝑛𝐿𝜙_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}, (145) where 𝜙∈𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋), 𝜆 is a real or complex parameter satisfying the inequality: |𝜆|<12^𝑛−1𝐶_𝑛𝐿,𝐿=𝑀𝛼₁+𝛼₂+⋯+𝛼_𝑛×𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑛𝛼𝑛,𝛿𝑛,𝛾}. (146)
(iii) Suppose that 2𝛼₁+𝛼₂+⋯+𝛼_𝑛+𝛾𝑎𝑛𝑑𝛼₁+𝛼₂+⋯+𝛼_𝑛+𝛿<1. (147) Then, (141) has a unique solution 𝑓 in the space 𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛿}((𝑎,𝑏),𝑋) such that 𝑓_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛿}≤11−|𝜆|2^𝑛−1𝐶_𝑛𝐾𝜙_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛿}, (148) where 𝜙∈𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛿}((𝑎,𝑏),𝑋), 𝐾=𝑀𝛼₁+𝛼₂+⋯+𝛼_𝑛,𝛼₁+𝛼₂+⋯+𝛼_𝑛×𝐴,𝛿,𝛾_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑛𝛼𝑛,𝛿𝑛,𝛾}, (149) and 𝜆 is a real or complex parameter satisfying |𝜆|<12^𝑛−1𝐶_𝑛𝐾. (150)

Proof. (ii) Equation (144) may be rewritten as 𝑓(𝑠)−𝜆^𝑏_𝑎𝐴(𝑡,𝑠)𝑓(𝑡)𝑑𝑡=𝜙(𝑠) (151) or (𝐼−𝜆𝑇)𝑓(𝑠)=𝜙(𝑠),𝑠∈(𝑎,𝑏) (152) which yields (𝐼−𝜆𝑇)𝑓=𝜙, (153) where 𝑇 is defined by 𝑇𝑓(𝑠)=^𝑏_𝑎𝐴(𝑡,𝑠)𝑓(𝑡)𝑑𝑡,𝑓∈𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋), (154) and 𝐼 is an identity operator on 𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋).
By Lemma 18, we have that 𝐴∈𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) and ‖𝐴‖_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛿,𝛾}≤2^𝑛−1𝐶_𝑛𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑛𝛼𝑛,𝛿𝑛,𝛾}. (155) Also, we derive by Theorem 19 that 𝑇∈𝐵(𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋),𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋)) and ‖𝑇‖≤‖𝐴‖_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛿,𝛾}𝑀𝛼₁+⋯+𝛼_𝑛 (156) from (117), and so ‖𝜆𝑇‖≤|𝜆|2^𝑛−1𝐶_𝑛𝑀𝛼₁+⋯+𝛼_𝑛×𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑛𝛼𝑛,𝛿𝑛,𝛾}. (157) Since ‖𝜆𝑇‖<1 by (146), the inverse operator (𝐼−𝜆𝑇)⁻¹∶𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋)→𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋) (158) exists and the norm of 𝐼−𝜆𝑇 satisfies the inequality: (𝐼−𝜆𝑇)⁻¹≤11−|𝜆|2^𝑛−1𝐶_𝑛𝑀𝛼₁+𝛼₂+⋯+𝛼_𝑛𝐴_{1𝛼1,𝛿1,𝛾}𝐴_{2𝛼2,𝛿2,𝛾}⋯𝐴_{𝑛𝛼𝑛,𝛿𝑛,𝛾} (159)from Theorem 10. So, (144) has a unique solution 𝑓 in the space 𝐻_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}((𝑎,𝑏),𝑋) such that 𝑓=(𝐼−𝜆𝑇)⁻¹𝑓𝜙,_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}=(𝐼−𝜆𝑇)⁻¹𝜙_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}≤(𝐼−𝜆𝑇)⁻¹𝜙_{𝛼1+𝛼2+⋯+𝛼𝑛,𝛾}, (160) and this also completes the proof of the second part.
The proof of the first and third parts of Theorem can be completed by the similar way to that of the second part, using Lemma 18, Theorem 19, (111), Lemma 18, Theorem 19, and (123), respectively.

Theorem 22. Let 𝑋 be a real or complex Banach algebra. Then Fredholm integral equation of the form 𝑓𝑠₁,𝑠₂=𝜙𝑠₁,𝑠₂+𝜆^𝑏_𝑎𝐴𝑠₁,𝑡𝑓𝑡,𝑠₂𝑑𝑡 (161) has a unique solution 𝑓 in 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) with 𝑓_{𝛼,𝛿,𝛾}≤11−|𝜆|𝑁‖𝐴‖_{𝛼,𝛿,𝛾}𝜙_{𝛼,𝛿,𝛾}, (162) where 𝐴,𝜙∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋), 𝑁 is the constant given in Theorem 20, that is, 𝑁=max𝑀(𝛼),max(𝑏−𝑎)^𝛾𝑀𝛼+𝛾,𝑀(𝛼), (163) and 𝜆 is a real or complex parameter fulfilling |𝜆|<1𝑁‖𝐴‖_{𝛼,𝛿,𝛾}. (164)

Proof. Equation (161) can be expressed as 𝑓𝑠₁,𝑠₂−𝜆^𝑏_𝑎𝐴𝑠₁,𝑡𝑓𝑡,𝑠₂𝑑𝑡=𝜙𝑠₁,𝑠₂ (165) or (𝐼−𝜆𝑇)𝑓𝑠₁,𝑠₂=𝜙𝑠₁,𝑠₂;𝑠₁,𝑠₂∈(𝑎,𝑏) (166) which implies (𝐼−𝜆𝑇)𝑓=𝜙, (167) where 𝑇 is defined by 𝑇𝑓𝑠₁,𝑠₂=^𝑏_𝑎𝐴𝑠₁,𝑡𝑓𝑡,𝑠₂𝑑𝑡,𝑓∈𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋), (168) and 𝐼 is an identity operator on 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋).
By Theorem 20, we have that 𝑇 is in the space 𝐵(𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋),𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋)), and ‖𝜆𝑇‖≤|𝜆|𝑁‖𝐴‖_{𝛼,𝛿,𝛾}. (169) Since ‖𝜆𝑇‖<1 by (164), the inverse operator (𝐼−𝜆𝑇)⁻¹∶𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋)→𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) (170) exists and the norm of 𝐼−𝜆𝑇 satisfies the inequality: (𝐼−𝜆𝑇)⁻¹≤1‖‖≤11−𝜆𝑇1−|𝜆|𝑁‖𝐴‖_{𝛼,𝛿,𝛾} (171) from Theorem 10. So, (161) has a unique solution 𝑓 in the space 𝐻_{𝛼,𝛿,𝛾}((𝑎,𝑏)×(𝑎,𝑏),𝑋) such that 𝑓=(𝐼−𝜆𝑇)⁻¹𝑓𝜙,_{𝛼,𝛿,𝛾}=(𝐼−𝜆𝑇)⁻¹𝜙_{𝛼,𝛿,𝛾}≤(𝐼−𝜆𝑇)⁻¹𝜙_{𝛼,𝛿,𝛾} (172) which completes the proof.

2.4. The Bounded Singular Integral Operators on the Space 𝐻_𝛼,𝛿((𝑎,𝑏),𝑋)

Lemma 23. Let 𝐴∈𝐻_𝛼,𝛿((𝑎,𝑏),𝑋), 𝑠∈𝐼_𝑝=(𝑎,𝑎+𝑝(𝑏−𝑎)/(𝑝+2)), and 0≤Δ𝑠<(𝑠−𝑎)/𝑝 with 𝑝∈(1,∞). Then the improper integral given by Γ(𝐴)(𝑠)=𝑝.𝑣.^𝑏_𝑎𝐴(𝑡)𝑡−𝑠𝑑𝑡,𝑠∈𝐼_𝑝 (173) exists, and there exists the nonnegative constant 𝑀(𝛼,𝛿,𝑝) depending only on 𝛼, 𝛿, and 𝑝 such that ‖Γ(𝐴)(𝑠)‖_𝑋≤𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(𝑠−𝑎)^𝛼,‖Γ(𝐴)(𝑠+Δ𝑠)−Γ(𝐴)(𝑠)‖_𝑋≤𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿 (174) hold for all 𝑠,𝑠+Δ𝑠∈(𝑎,𝑏), where ‖𝐴‖_𝛼,𝛿 is defined by (73).

Hereafter, by 𝐼_𝑝, we mean the interval 𝐼_𝑝=(𝑎,𝑎+𝑝(𝑏−𝑎)/(𝑝+2)) with 1<𝑝<∞.

Proof. We can write Γ(𝐴)(𝑠)=_𝑎^{𝑠−((𝑠−𝑎)/𝑝)}ℎ_𝑠(𝑡)𝐴(𝑡)+𝑑𝑡_{)𝑠+((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠(𝑡)𝐴(𝑡)+𝑑𝑡_{𝑏()𝑠+𝑠−𝑎/𝑝}ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡 (175) for all 𝑠∈𝐼_𝑝. Here, ℎ_𝑥 is defined for each 𝑥∈(𝑎,𝑏) by ℎ_𝑥(𝑡)=1/(𝑡−𝑥),𝑡∈(𝑎,𝑏) and 𝑡≠𝑥. Since the functions 𝐴∈𝐻_𝛼,𝛿((𝑎,𝑏),𝑋) and ℎ_𝑥 are continuous for each 𝑥∈(𝑎,𝑏), the function 𝐵_𝑥 defined by 𝐵_𝑥(𝑡)=ℎ_𝑥(𝑡)𝐴(𝑡) is continuous, and, so, it is strongly measurable by Theorem 4, where 𝑡∈(𝑎,𝑏) and 𝑡≠𝑥.
Since ℎ_𝑠(𝑡)𝐴(𝑡)_𝑋≤ℎ_𝑠(𝑡)‖𝐴(𝑡)‖_𝑋≤ℎ_𝑠(𝑡)‖𝐴‖_𝛼,𝛿(𝑡−𝑎)^𝛼 (176) and 𝑠−𝑡>(𝑠−𝑎)/𝑝>0 for all 𝑡∈(𝑎,𝑠−(𝑠−𝑎)/𝑝), we get 1/(𝑠−𝑡)<𝑝/(𝑠−𝑎). So, _𝑎^{𝑠−((𝑠−𝑎)/𝑝)}𝑑𝑡(𝑡−𝑎)^𝛼||≤𝑝𝑡−𝑠𝑠−𝑎_𝑎^{𝑠−((𝑠−𝑎)/𝑝)}𝑑𝑡(𝑡−𝑎)^𝛼=𝑝𝑠−𝑎₀^{(𝑝−1)(𝑠−𝑎)/𝑝}𝑢^−𝛼=𝑝𝑑𝑢_𝛼𝑝−1^1−𝛼(1−𝛼)(𝑠−𝑎)^𝛼. (177) By (176), _𝑎^{𝑠−((𝑠−𝑎)/𝑝)}ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡_𝑋≤_𝑎^{𝑠−((𝑠−𝑎)/𝑝)}ℎ_𝑠(𝑡)𝐴(𝑡)_𝑋≤𝑝𝑑𝑡_𝛼𝑝−1^1−𝛼‖𝐴‖_𝛼,𝛿(1−𝛼)(𝑠−𝑎)^𝛼. (178)
On the other hand, since (𝑠−𝑎)/𝑝>0, we get ∫^{𝑠+(𝑠−𝑎)/𝑝}_{𝑠−(𝑠−𝑎)/𝑝}𝑑𝑡/(𝑡−𝑠)=0. Thus, ₎^{𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠(𝑡)𝐴(𝑡)=𝑑𝑡_{)𝑠+((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠(𝑡𝐴(𝑡)(𝑠=)[−𝐴)]𝑑𝑡_{𝑠()𝑠−𝑠−𝑎/𝑝}ℎ_𝑠(𝑡𝐴(𝑡)(𝑠+)[−𝐴)]𝑑𝑡_{𝑠𝑠+((𝑠−𝑎)/𝑝)}ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]𝑑𝑡. (179)
Since 𝑡∈(𝑠−(𝑠−𝑎)/𝑝,𝑠), we obtain 𝑠−𝑡>0, 𝑡−𝑎>(𝑝−1)(𝑠−𝑎)/𝑝>0 and 1/(𝑡−𝑎)<𝑝/(𝑝−1)(𝑠−𝑎). So, we have by (28) that ℎ_𝑠(𝑡)‖𝐴(𝑡)−𝐴(𝑠)‖_𝑋=ℎ_𝑠(𝑡)‖𝐴(𝑠)−𝐴(𝑡)‖_𝑋=−ℎ_𝑠(𝑡)‖𝐴(𝑡+𝑠−𝑡)−𝐴(𝑡)‖_𝑋≤‖𝐴‖_𝛼,𝛿(𝑠−𝑡)^𝛿−1(𝑡−𝑎)^𝛼+𝛿≤𝑝^𝛼+𝛿‖𝐴‖_𝛼,𝛿(𝑠−𝑡)^𝛿−1𝑝−1^𝛼+𝛿(𝑠−𝑎)^𝛼+𝛿. (180) By (180) and ^𝑠_{()𝑠−(𝑠−𝑎/𝑝)}(𝑠−𝑡)^𝛿−1𝑑𝑡=₀^{(𝑠−𝑎)/𝑝}𝑢^𝛿−1()𝑑𝑢=𝑠−𝑎^𝛿𝛿𝑝^𝛿, (181) we have ^𝑠_{()𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)‖𝐴(𝑡)−𝐴(𝑠)‖_𝑋𝑝𝑑𝑡≤^𝛼‖𝐴‖_𝛼,𝛿𝑝−1^𝛼+𝛿𝛿(𝑠−𝑎)^𝛼. (182) Similarly, since 𝑡−𝑠>0 for all 𝑡∈(𝑠,𝑠+(𝑠−𝑎)/𝑝), we get ℎ_𝑠(𝑡)‖𝐴(𝑡)−𝐴(𝑠)‖_𝑋=ℎ_𝑠(𝑡)‖𝐴(𝑠+𝑡−𝑠)−𝐴(𝑠)‖_𝑋≤‖𝐴‖_𝛼,𝛿(𝑡−𝑠)^𝛿−1(𝑠−𝑎)^𝛼+𝛿 (183) from (28). Also, by _𝑠^{𝑠+((𝑠−𝑎)/𝑝)}(𝑡−𝑠)^𝛿−1𝑑𝑡=₀^{(𝑠−𝑎)/𝑝}𝑢^𝛿−1()𝑑𝑢=𝑠−𝑎^𝛿𝛿𝑝^𝛿 (184) and (183), we derive _𝑠^{𝑠+((𝑠−𝑎)/𝑝)}ℎ_𝑠(𝑡)‖𝐴(𝑡)−𝐴(𝑠)‖_𝑋‖𝐴‖𝑑𝑡≤^𝛼,𝛿𝛿𝑝^𝛿(𝑠−𝑎)^𝛼. (185) Thus, by taking 𝐼=∫^{𝑠+(𝑠−𝑎)/𝑝}_{𝑠−(𝑠−𝑎)/𝑝}ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡, ‖𝐼‖≤^𝑠_{()𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)‖𝐴(𝑡)−𝐴(𝑠)‖_𝑋+𝑑𝑡_{𝑠𝑠+((𝑠−𝑎)/𝑝)}ℎ_𝑠(𝑡)‖𝐴(𝑡)−𝐴(𝑠)‖_𝑋≤𝑝𝑑𝑡_𝛼𝑝−1^𝛼+𝛿𝛿+1𝛿𝑝^𝛿‖𝐴‖_𝛼,𝛿(𝑠−𝑎)^𝛼 (186) from (182) and (185).
From the inequality 𝑡−𝑎>𝑠+((𝑠−𝑎)/𝑝)−𝑎=(𝑝+1)(𝑠−𝑎)/𝑝 which is satisfied for all 𝑡∈(𝑠+(𝑠−𝑎)/𝑝,𝑏), we get 𝑡−𝑠=𝑡−𝑎−(𝑠−𝑎)>𝑡−𝑎−𝑝(𝑡−𝑎)/(𝑝+1)=(𝑡−𝑎)/(𝑝+1)>0, and, thus, 1/(𝑡−𝑠)<(𝑝+1)/(𝑡−𝑎). Hence, ℎ_𝑠(𝑡)𝐴(𝑡)_𝑋≤‖𝐴‖_𝛼,𝛿(𝑡−𝑎)^𝛼||≤‖𝐴‖𝑡−𝑠𝑝+1_𝛼,𝛿(𝑡−𝑎)^𝛼+1. (187) From ^𝑏_{()𝑠+(𝑠−𝑎/𝑝)}𝑑𝑡(𝑡−𝑎)^𝛼+1=𝑝^𝛼𝛼𝑝+1^𝛼(𝑠−𝑎)^𝛼−1𝛼(𝑏−𝑎)^𝛼≤𝑝^𝛼𝛼𝑝+1^𝛼(𝑠−𝑎)^𝛼 (188) and (187), it is obtained that ^𝑏_{()𝑠+(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡_𝑋≤^𝑏_{()𝑠+(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)𝐴(𝑡)_𝑋≤𝑝𝑑𝑡𝑝+1^𝛼‖𝐴‖_𝛼,𝛿𝛼𝑝+1^𝛼(𝑠−𝑎)^𝛼. (189) By (178), (186), and (189), we have ‖Γ(𝐴)(𝑠)‖_𝑋≤_𝑎^{𝑠−((𝑠−𝑎)/𝑝)}ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡_𝑋+₍₎^{𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−𝑠−𝑎/𝑝}ℎ_𝑠(𝑡)𝐴(𝑡)d𝑡_𝑋+^𝑏_{)𝑠+(𝑠−𝑎/𝑝}ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡_𝑋≤𝑀₁𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(𝑠−𝑎)^𝛼 (190) such that 𝑀₁=𝑝𝛼,𝛿,𝑝^𝛼𝑝−1^1−𝛼+𝑝1−𝛼_𝛼𝑝−1^𝛼+𝛿𝛿+1𝛿𝑝^𝛿+𝑝+1𝑝^𝛼𝛼𝑝+1^𝛼. (191)
Furthermore, since 1()=1𝑡−𝑠−Δ𝑠)(𝑡−𝑠1Δ𝑠−1𝑡−𝑠−Δ𝑠,𝑡−𝑠0<Δ𝑠<𝑠−𝑎𝑝, (192) we can write Γ(𝐴)(𝐴𝑠)=)(𝑠+Δ𝑠−Γ)(_𝑏𝑎ℎ_𝑠+Δ𝑠(𝑡)𝐴(𝑡)𝑑𝑡−^𝑏_𝑎ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡=Δ𝑠_𝑎^{𝑠−((𝑠−𝑎)/𝑝)}ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)𝐴(𝑡)+𝑑𝑡_{)𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)𝐴(𝑡)−ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡+Δ𝑠^𝑏_{()𝑠+Δ𝑠+(𝑠−𝑎/𝑝)}ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡. (193) Thus, ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)𝐴(𝑡)_𝑋≤‖𝐴‖_𝛼,𝛿(𝑡−𝑎)^𝛼||≤𝑝𝑡−𝑠−Δ𝑠||𝑡−𝑠₂‖𝐴‖_𝛼,𝛿(𝑠−𝑎)²(𝑡−𝑎)^𝛼, (194) since |𝑡−𝑠−Δ𝑠|>|𝑡−𝑠|>(𝑠−𝑎)/𝑝>0 and 1/|𝑡−𝑠−Δ𝑠|<1/|𝑡−𝑠|<𝑝/(𝑠−𝑎) for all 𝑡∈(𝑎,𝑠−(𝑠−𝑎)/𝑝). Besides, by _𝑎^{𝑠−((𝑠−𝑎)/𝑝)}𝑑𝑡(𝑡−𝑎)^𝛼=₀^{((𝑝−1)(𝑠−𝑎))/𝑝}𝑢^−𝛼=𝑑𝑢𝑝−1^1−𝛼(1−𝛼)𝑝^1−𝛼(𝑠−𝑎)^𝛼−1,Δ𝑠=(Δ𝑠)^𝛿(Δ𝑠)^1−𝛿≤(Δ𝑠)^𝛿(𝑠−𝑎)^1−𝛿 (195) and (194), we get Δ𝑠_𝑎^{𝑠−((𝑠−𝑎)/𝑝)}ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡_𝑋≤𝑝^1+𝛼𝑝−1^1−𝛼‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(1−𝛼)(𝑠−𝑎)^𝛼+𝛿. (196)
Since 𝑡−𝑎>𝑡−𝑠>𝑡−𝑠−Δ𝑠>0 for all 𝑡∈(𝑠+Δ𝑠+(𝑠−𝑎)/𝑝,𝑏), the inequality 1/(𝑡−𝑎)<1/(𝑡−𝑠)<1/(𝑡−𝑠−Δ𝑠) holds, and, so, ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)𝐴(𝑡)_𝑋≤‖𝐴‖_𝛼,𝛿(𝑡−𝑎)^𝛼()≤‖𝐴‖𝑡−𝑠−Δ𝑠)(𝑡−𝑠_𝛼,𝛿(𝑡−𝑠−Δ𝑠)^2+𝛼. (197) By ^𝑏_{)𝑠+Δ𝑠+(𝑠−𝑎/𝑝}𝑑𝑡(𝑡−𝑠−Δ𝑠)^2+𝛼=₍₎^{𝑏−𝑠−Δ𝑠}_{𝑠−𝑎/𝑝}𝑢^−2−𝛼=𝑝𝑑𝑢_1+𝛼(1+𝛼)(𝑠−𝑎)^1+𝛼−1(1+𝛼)(𝑏−𝑠−Δ𝑠)^1+𝛼≤𝑝^1+𝛼(1+𝛼)(𝑠−𝑎)^1+𝛼 (198) and (195), we obtain Δ𝑠^𝑏_{()𝑠+Δ𝑠+(𝑠−𝑎/𝑝)}ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡_𝑋≤𝑝^1+𝛼‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(1+𝛼)(𝑠−𝑎)^𝛼+𝛿 (199) from (197). Then,₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)𝐴(𝑡)−ℎ_𝑠(𝑡)𝐴(𝑡=)𝑑𝑡_{()𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠+Δ𝑠(𝑡𝐴(𝑡)(−)[−𝐴𝑠+Δ𝑠)]𝑑𝑡_{()𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)[𝐴(t)(𝑠()−𝐴)]𝑑𝑡+𝐴𝑠+Δ𝑠₍₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠+Δ𝑠(𝑡)(𝑠)𝑑𝑡−𝐴₍₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)𝑑𝑡. (200) Also, we have ₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)=𝑑𝑡_{)𝑠+Δ𝑠−((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)+𝑑𝑡_{)𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)𝑠+Δ𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)=𝑑𝑡_{)𝑠+Δ𝑠−((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)𝑑𝑡, (201) since ∫^{𝑠+Δ𝑠+(𝑠−𝑎)/𝑝}_{𝑠+Δ𝑠−(𝑠−𝑎)/𝑝}𝑑𝑡/(𝑡−𝑠−Δ𝑠)=∫^{𝑢+(𝑠−𝑎)/𝑝}_{𝑢−(𝑠−𝑎)/𝑝}𝑑𝑡/(𝑡−𝑢)=0. Therefore, we get by (195) that ^𝑠+Δs₎^{+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)𝑑𝑡≤₎^{𝑠+Δ𝑠−((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡=)𝑑𝑡_{)𝑠+Δ𝑠−((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝}−ℎ_𝑠+Δ𝑠(𝑡)≤𝑝𝑑𝑡𝑠−𝑎₍₎^{𝑠+Δ𝑠−((𝑠−𝑎)/𝑝)}_{𝑠−𝑠−𝑎/𝑝}𝑑𝑡≤𝑝Δ𝑠≤𝑝()𝑠−𝑎Δ𝑠^𝛿(𝑠−𝑎)^𝛿, (202) since 𝑠+Δ𝑠−𝑡>(𝑠−𝑎)/𝑝>0 which implies that 1/(𝑠+Δ𝑠−𝑡)<𝑝/(𝑠−𝑎) for all 𝑡∈(𝑠−(𝑠−𝑎)/𝑝,𝑠+Δ𝑠−(𝑠−𝑎)/𝑝). So, 𝐴(𝑠+Δ𝑠)₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)𝑑𝑡_𝑋≤𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠+Δ𝑠−𝑎)^𝛼(𝑠−𝑎)^𝛿≤𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿. (203) Since ∫^{𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−((𝑠−𝑎)/𝑝)}𝑑𝑡/(𝑡−𝑠)=0 and 𝑡−𝑠>(𝑠−𝑎)/𝑝>0 which yields 1/(𝑡−𝑠)<𝑝/(𝑠−𝑎) for all 𝑡∈(𝑠+(𝑠−𝑎)/𝑝,𝑠+Δ𝑠+(𝑠−𝑎)/𝑝), we obtain ₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠(𝑡)=𝑑𝑡_{()𝑠+((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)𝑑𝑡+₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠+(𝑠−𝑎/𝑝}ℎ_𝑠(𝑡)=𝑑𝑡_{)𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)𝑠+(𝑠−𝑎/𝑝}ℎ_𝑠(𝑡)≤𝑝𝑑𝑡𝑠−𝑎₍₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠+(𝑠−𝑎/𝑝)}𝑑𝑡=𝑝Δ𝑠≤𝑝()𝑠−𝑎Δ𝑠^𝛿(𝑠−𝑎)^𝛿 (204) from (195). Hence, −𝐴(𝑠)₍₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)𝑑𝑡_𝑋=‖𝐴(𝑠)‖_𝑋()^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)𝑝‖𝐴‖𝑑𝑡≤^𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿. (205) By (192), ₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡𝐴(𝑡)(−)[−𝐴𝑠+Δ𝑠)]𝑑𝑡_{()𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]𝑑𝑡=Δ𝑠₍₎^{𝑠−(Δ𝑠/2)}_{𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡𝐴(𝑡)(𝑠−)[−𝐴)]𝑑𝑡_{()𝑠−(Δ𝑠/2)𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠+Δ𝑠(𝑡𝐴()(𝑠+)[𝑠+Δ𝑠−𝐴)]𝑑𝑡_{()𝑠+(3Δ𝑠/2)𝑠−Δ𝑠/2}ℎ_𝑠+Δ𝑠(𝑡𝐴(𝑡)(−)[−𝐴𝑠+Δ𝑠)]𝑑𝑡_{()𝑠+(3Δ𝑠/2)𝑠−Δ𝑠/2}ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]𝑑𝑡+Δ𝑠₍₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠+3Δ𝑠/2}ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡𝐴(𝑡)(−)[−𝐴𝑠+Δ𝑠)]𝑑𝑡_{()𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)𝑠+3Δ𝑠/2}ℎ_𝑠(𝑡)[𝐴(𝑠+Δ𝑠)−𝐴(𝑠)]𝑑𝑡. (206) Since 𝑠−𝑡+Δ𝑠>𝑠−𝑡>0 and 𝑡−𝑎>(𝑝−1)(𝑠−𝑎)/𝑝>0 which imply that 1/(𝑠−𝑡+Δ𝑠)<1/(𝑠−𝑡) and 1/(𝑡−𝑎)<𝑝/(𝑝−1)(𝑠−𝑎) for all 𝑡∈(𝑠−((𝑠−𝑎)/𝑝),𝑠−(Δ𝑠/2)), it is found that ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]_𝑋=ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)[𝐴(𝑡+𝑠−𝑡)−𝐴(𝑡)]_𝑋≤‖𝐴‖_𝛼,𝛿(𝑠−𝑡)^𝛿−2(𝑡−𝑎)^𝛼+𝛿≤𝑝^𝛼+𝛿‖𝐴‖_𝛼,𝛿(𝑠−𝑡)^𝛿−2𝑝−1^𝛼+𝛿(𝑠−𝑎)^𝛼+𝛿. (207) By ₍₎^{𝑠−(Δ𝑠/2)}_{𝑠−(𝑠−𝑎/𝑝)}(𝑠−𝑡)^𝛿−2𝑑𝑡=^{(𝑠−𝑎)/𝑝}_Δ𝑠/2𝑢^𝛿−2=()𝑑𝑢Δ𝑠^𝛿−1(1−𝛿)2^𝛿−1−(𝑠−𝑎)^𝛿−1(1−𝛿)𝑝^𝛿−1≤(Δ𝑠)^𝛿−1(1−𝛿)2^𝛿−1, (208) and (207), Δ𝑠₍₎^{𝑠−(Δ𝑠/2)}_{𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]𝑑𝑡_𝑋≤𝑝^𝛼+𝛿‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿𝑝−1^𝛼+𝛿(1−𝛿)2^𝛿−1(𝑠−𝑎)^𝛼+𝛿. (209) Since ₍₎^{𝑠+(3Δ𝑠/2)}_{𝑠−Δ𝑠/2}ℎ_𝑠+Δ𝑠(𝑡𝐴(𝑡)(=)[−𝐴𝑠+Δ𝑠)]𝑑𝑡_{()𝑠+Δ𝑠𝑠−Δ𝑠/2}ℎ_𝑠+Δ𝑠(𝑡𝐴(𝑡)(+)[−𝐴𝑠+Δ𝑠)]𝑑𝑡_{𝑠+(3Δ𝑠/2)𝑠+Δ𝑠}ℎ_𝑠+Δ𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠+Δ𝑠)]𝑑𝑡 (210) and 𝑠+Δ𝑠−𝑡>0, 𝑡−𝑎>𝑠−𝑎−(Δ𝑠/2)>𝑠−𝑎−((𝑠−𝑎)/2𝑝)=(2𝑝−1)(𝑠−𝑎)/2𝑝 which yields 1/(𝑡−𝑎)<2𝑝/(2𝑝−1)(𝑠−𝑎) for all 𝑡∈(𝑠−(Δ𝑠/2),𝑠+Δ𝑠), we get ℎ_𝑠+Δ𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠+Δ𝑠)]_𝑋=ℎ_𝑠+Δ𝑠(𝑡)‖𝐴(𝑡+𝑠+Δ𝑠−𝑡)−𝐴(𝑡)‖_𝑋≤‖𝐴‖_𝛼,𝛿(𝑠+Δ𝑠−𝑡)^𝛿−1(𝑡−𝑎)^𝛼+𝛿≤2𝑝^𝛼+𝛿‖𝐴‖_𝛼,𝛿(𝑠+Δ𝑠−𝑡)^𝛿−12𝑝−1^𝛼+𝛿(𝑠−𝑎)^𝛼+𝛿. (211) By ₍₎^𝑠+Δ𝑠_{𝑠−Δ𝑠/2}(𝑠+Δ𝑠−𝑡)^𝛿−1𝑑𝑡=₀^3Δ𝑠/2𝑢^𝛿−13𝑑𝑢=^𝛿(Δ𝑠)^𝛿𝛿2^𝛿 (212) and (211), ₍₎^𝑠+Δ𝑠_{𝑠−Δ𝑠/2}ℎ_𝑠+Δ𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠+Δ𝑠)]𝑑𝑡_𝑋≤2𝑝^𝛼+𝛿3^𝛿‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿2𝑝−1^𝛼+𝛿𝛿2^𝛿(𝑠−𝑎)^𝛼+𝛿. (213)
Since 𝑡−𝑠−Δ𝑠>0 for all 𝑡∈(𝑠+Δ𝑠,𝑠+(3Δ𝑠/2)), we derive ℎ_𝑠+Δ𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠+Δ𝑠)]_𝑋=ℎ_𝑠+Δ𝑠(𝑡)[𝐴(𝑠+Δ𝑠+𝑡−𝑠−Δ𝑠)−𝐴(𝑠+Δ𝑠)]_𝑋≤‖𝐴‖_𝛼,𝛿(𝑡−𝑠−Δ𝑠)^𝛿−1(𝑠+Δ𝑠−𝑎)^𝛼+𝛿≤‖𝐴‖_𝛼,𝛿(𝑡−𝑠−Δ𝑠)^𝛿−1(𝑠−𝑎)^𝛼+𝛿. (214) By ^{𝑠+(3Δ𝑠/2)}_𝑠+Δ𝑠(𝑡−𝑠−Δ𝑠)^𝛿−1𝑑𝑡=₀^Δ𝑠/2𝑢^𝛿−1()𝑑𝑢=Δ𝑠^𝛿𝛿2^𝛿 (215) and (214), ^{𝑠+(3Δ𝑠/2)}_𝑠+Δ𝑠ℎ_𝑠+Δ𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠+Δ𝑠)]𝑑𝑡_𝑋≤‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿𝛿2^𝛿(𝑠−𝑎)^𝛼+𝛿. (216)
So, we derive ₍₎^{𝑠+(3Δ𝑠/2)}_{𝑠−Δ𝑠/2}ℎ_𝑠+Δ𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠+Δ𝑠)]𝑑𝑡_𝑋≤2𝑝^𝛼+𝛿3^𝛿2𝑝−1^𝛼+𝛿𝛿2^𝛿+1𝛿2^𝛿‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿 (217) from (213) and (216). ₍₎^{𝑠+(3Δ𝑠/2)}_{𝑠−Δ𝑠/2}ℎ_𝑠(𝑡𝐴(𝑡)(𝑠=)[−𝐴)]𝑑𝑡_{𝑠()𝑠−Δ𝑠/2}ℎ_𝑠(𝑡𝐴(𝑡)(𝑠+)[−𝐴)]𝑑𝑡_{𝑠𝑠+(3Δ𝑠/2)}ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]𝑑𝑡. (218) Since 𝑡−𝑠<0 and 𝑡−𝑎>(2𝑝−1)(𝑠−𝑎)/2𝑝, which implies 1/(𝑡−𝑎)<2𝑝/(2𝑝−1)(𝑠−𝑎) for all 𝑡∈(𝑠−(Δ𝑠/2),𝑠), ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]_𝑋=−ℎ_𝑠(t)‖𝐴(𝑠)−𝐴(𝑡)‖_𝑋=−ℎ_𝑠(𝑡)‖𝐴(𝑡+𝑠−𝑡)−𝐴(𝑡)‖_𝑋≤‖𝐴‖_𝛼,𝛿(𝑠−𝑡)^𝛿−1(𝑡−𝑎)^𝛼+𝛿≤2𝑝^𝛼+𝛿‖𝐴‖_𝛼,𝛿(𝑠−𝑡)^𝛿−12𝑝−1^𝛼+𝛿(𝑠−𝑎)^𝛼+𝛿. (219)
Since 𝑡−𝑠>0 for all 𝑡∈(𝑠,𝑠+(3Δ𝑠/2)), ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]_𝑋=ℎ_𝑠(𝑡)‖𝐴(𝑠+𝑡−𝑠)−𝐴(𝑠)‖_𝑋≤‖𝐴‖_𝛼,𝛿(𝑡−𝑠)^𝛿−1(𝑠−𝑎)^𝛼+𝛿. (220) By ^𝑠_{()𝑠−Δ𝑠/2}(𝑠−𝑡)^𝛿−1=𝑑𝑡_0Δ𝑠/2𝑢^𝛿−1()𝑑𝑢=Δ𝑠^𝛿𝛿2^𝛿_{𝑠𝑠+(3Δ𝑠/2)},(𝑡−𝑠)^𝛿−1𝑑𝑡=₀^3Δ𝑠/2𝑢^𝛿−13𝑑𝑢=^𝛿(Δ𝑠)^𝛿𝛿2^𝛿, (221) equations (219) and (220), it is obtained that ^𝑠_{()𝑠−Δ𝑠/2}ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]𝑑𝑡_𝑋≤2𝑝^𝛼+𝛿‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿2𝑝−1^𝛼+𝛿𝛿2^𝛿(𝑠−𝑎)^𝛼+𝛿_{𝑠𝑠+(3Δ𝑠/2)},ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]𝑑𝑡_𝑋≤3^𝛿‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿𝛿2^𝛿(𝑠−𝑎)^𝛼+𝛿. (222) By (222), ₍₎^{𝑠+(3Δ𝑠/2)}_{𝑠−Δ𝑠/2}ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠)]𝑑𝑡_𝑋≤2𝑝^𝛼+𝛿2𝑝−1^𝛼+𝛿𝛿2^𝛿+3^𝛿𝛿2^𝛿‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿. (223)
Since 𝑡−𝑠>𝑡−𝑠−Δ𝑠>0 and 𝑠+Δ𝑠−𝑎>𝑠−𝑎>0 which yield 1/(𝑡−𝑠)<1/(𝑡−𝑠−Δ𝑠) and 1/(𝑠+Δ𝑠−𝑎)<1/(𝑠−𝑎), ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠+Δ𝑠)]_𝑋=ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)[𝐴(𝑠+Δ𝑠+𝑡−𝑠−Δ𝑠)−𝐴(𝑠+Δ𝑠)]_𝑋≤‖𝐴‖_𝛼,𝛿(𝑡−𝑠−Δ𝑠)^𝛿−2(𝑠+Δ𝑠−𝑎)^𝛼+𝛿≤‖𝐴‖_𝛼,𝛿(𝑡−𝑠−Δ𝑠)^𝛿−2(𝑠−𝑎)^𝛼+𝛿. (224) By ₍₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠+3Δ𝑠/2}(𝑡−𝑠−Δ𝑠)^𝛿−2𝑑𝑡=^{(𝑠−𝑎)/𝑝}_Δ𝑠/2𝑢^𝛿−2=()𝑑𝑢Δ𝑠^𝛿−1(1−𝛿)2^𝛿−1−(𝑠−𝑎)^𝛿−1(1−𝛿)𝑝^𝛿−1≤(Δ𝑠)^𝛿−1(1−𝛿)2^𝛿−1 (225) and (224), Δ𝑠₍₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠+3Δ𝑠/2}ℎ_𝑠+Δ𝑠(𝑡)ℎ_𝑠(𝑡)[𝐴(𝑡)−𝐴(𝑠+Δ𝑠)]𝑑𝑡_𝑋≤‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(1−𝛿)2^𝛿−1(𝑠−𝑎)^𝛼+𝛿,−₍₎^{𝑠−(Δ𝑠/2)}_{𝑠−(𝑠−𝑎/𝑝)}ℎ_𝑠+Δ𝑠(𝑡)[𝐴(𝑠+Δ𝑠)−𝐴(𝑠)]𝑑𝑡−₍₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠+3Δ𝑠/2}ℎ_𝑠(𝑡𝐴()(𝑠=[𝐴(𝑠)()[𝑠+Δ𝑠−𝐴)]𝑑𝑡−𝐴𝑠+Δ𝑠)]₍₎^−3Δ𝑠/2_{−(𝑠−𝑎/𝑝)−Δ𝑠}𝑑𝑢𝑢−^{Δ𝑠+((𝑠−𝑎)/𝑝)}_3Δ𝑠/2𝑑𝑢𝑢=[𝐴(𝑠)−𝐴(𝑠+Δ𝑠)]^{Δ𝑠+((𝑠−𝑎)/𝑝)}_3Δ𝑠/2𝑑𝑢𝑢−^{Δ𝑠+((𝑠−𝑎)/𝑝)}_3Δ𝑠/2𝑑𝑢𝑢=0. (226) Thus, by taking 𝑀₂𝑝𝛼,𝛿,𝑝=2𝑝+^𝛼+𝛿𝑝−1^𝛼+𝛿(1−𝛿)2^𝛿−1+2𝑝^𝛼+𝛿3^𝛿2𝑝−1^𝛼+𝛿𝛿2^𝛿+1𝛿2^𝛿+2𝑝^𝛼+𝛿2𝑝−1^𝛼+𝛿𝛿2^𝛿+3^𝛿𝛿2^𝛿+1(1−𝛿)2^𝛿−1, (227) we obtain ₎^{𝑠+Δ𝑠+((𝑠−𝑎)/𝑝)}_{𝑠−(𝑠−𝑎/𝑝}ℎ_𝑠+Δ𝑠(𝑡)𝐴(𝑡)−ℎ_𝑠(𝑡)𝐴(𝑡)𝑑𝑡_𝑋≤𝑀₂𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿 (228) from (203), (205), (209), (217), (223), (226). By (196), (199), and (228), we find ‖Γ(𝐴)(𝑠+Δ𝑠)−Γ(𝐴)(𝑠)‖_𝑋≤𝑀₂𝛼,𝛿,𝑝+𝑀₃𝛼,𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿 (229) such that 𝑀₃=𝑝𝛼,𝑝^1+𝛼(1+𝛼)𝑝−1^1−𝛼+1−𝛼().1−𝛼)(1+𝛼 (230) By (190) and (229), ‖Γ(𝐴)(𝑠)‖_𝑋≤𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(𝑠−𝑎)^𝛼,‖Γ(𝐴)(𝑠+Δ𝑠)−Γ(𝐴)(𝑠)‖_𝑋≤𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿 (231) with 𝑀𝛼,𝛿,𝑝=max𝑀₁𝛼,𝛿,𝑝,𝑀₂𝛼,𝛿,𝑝+𝑀₃𝛼,𝑝 (232) which terminates the proof.

Hereafter, we assume unless stated otherwise that 𝑀(𝛼,𝛿,𝑝) is defined by (232).

Theorem 24. The operator 𝑇 defined by 𝑇(𝐴)(𝑠)=𝑝.𝑣.^𝑏_𝑎𝐴(𝑡)𝑡−𝑠𝑑𝑡;𝐴∈𝐻_𝛼,𝛿((𝑎,𝑏),𝑋),𝑠∈𝐼_𝑝 (233) is the singular integral operator in the space 𝐵(𝐻_𝛼,𝛿((𝑎,𝑏),𝑋),𝐻_𝛼,𝛿(𝐼_𝑝,𝑋)), and ‖𝑇‖ satisfies the inequality ‖𝑇‖≤2𝑝^𝛿𝑀(𝛼,𝛿,𝑝).

Proof. If 0≤Δ𝑠<(𝑠−𝑎)/𝑝, we get by Lemma 23 that ‖𝑇(𝐴)(𝑠)‖_𝑋≤𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(𝑠−𝑎)^𝛼,‖𝑇(𝐴)(𝑠+Δ𝑠)−𝑇(𝐴)(𝑠)‖_𝑋≤𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿 (234)(235) hold for all 𝐴∈𝐻_𝛼,𝛿((𝑎,𝑏),𝑋), 𝑠,𝑠+Δ𝑠∈𝐼_𝑝. Furthermore, if Δ𝑠≥(𝑠−𝑎)/𝑝>0, we obtain by (234) and 1≤1𝑠+Δ𝑠−𝑎,1𝑠−𝑎(Δ𝑠)^𝛿≤𝑝^𝛿(𝑠−𝑎)^𝛿 (236) that ‖Δ𝑇(𝐴)‖_𝑋≤‖𝑇(𝐴)(𝑠+Δ𝑠)‖_𝑋+‖𝑇(𝐴)(𝑠)‖_𝑋≤𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(𝑠+Δ𝑠−𝑎)^𝛼+𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(𝑠−𝑎)^𝛼≤2𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼(Δ𝑠)^𝛿≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿, (237) which holds for all 𝑠,𝑠+Δ𝑠∈𝐼_𝑝, where Δ𝑇(𝐴)=𝑇(𝐴)(𝑠+Δ𝑠)−𝑇(𝐴)(𝑠). Thus, by (235) and (237), ‖𝑇(𝐴)(𝑠+Δ𝑠)−𝑇(𝐴)(𝑠)‖_𝑋≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(Δ𝑠)^𝛿(𝑠−𝑎)^𝛼+𝛿 (238) and by (234), ‖𝑇(𝐴)(𝑠)‖_𝑋≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(𝑠−𝑎)^𝛼. (239) Since 𝑇(𝐴)∈𝐻_𝛼,𝛿(𝐼_𝑝,𝑋) and ‖𝑇(𝐴)‖_𝛼,𝛿≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿 (240) from (238) and (239), we get ‖𝑇‖=sup_{‖𝐴‖𝛼,𝛿=1}‖𝑇(𝐴)‖_𝛼,𝛿≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝. (241) Also, 𝑇 is linear. So, 𝑇∈𝐵(𝐻_𝛼,𝛿((𝑎,𝑏),𝑋),𝐻_𝛼,𝛿(𝐼_𝑝,𝑋)). This also concludes the proof.

Theorem 25. The operator 𝑇 defined by 𝑇(𝐴)(𝑠)=𝑝.𝑣.^𝑏_𝑎𝐾(𝑠,𝑡)𝐴(𝑡)𝑑𝑡,𝐴∈𝐻_𝛼,𝛿((𝑎,𝑏),𝑋),𝑠∈(𝑎,𝑏) (242) is the singular integral operator in the space 𝐵(𝐻_𝛼,𝛿((𝑎,𝑏),𝑋),𝐿_𝑞((𝑎,𝑏),𝑋)) such that ‖𝑇‖≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝𝑀𝛼𝑞^1/𝑞, (243) where 1≤𝑞<∞,𝛼<1/𝑞 and 𝐾∶(𝑎,𝑏)×(𝑎,𝑏)→ℝ is given by 𝐾()=1𝑠,𝑡𝑡−𝑠,𝑠∈𝐼_𝑝,𝑡∈(𝑎,𝑏),𝑠≠𝑡0,𝑠∉𝐼_𝑝,𝑡∈(𝑎,𝑏). (244)

Proof. Equation (242) may be rewritten as 𝑇(𝐴𝑠)=∫)(𝑝.𝑣.^𝑏_𝑎𝐴(𝑡)𝑡−𝑠𝑑𝑡,𝑠∈𝐼_𝑝0,𝑠∉𝐼_𝑝. (245) From (239) and (245), ‖𝑇(𝐴)(𝑠)‖_𝑋≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿(𝑠−𝑎)^𝛼 (246) for all 𝑠∈(𝑎,𝑏). Thus, we have by (246) that ^𝑏_𝑎‖𝑇(𝐴)(𝑠)‖^𝑞_𝑋𝑑𝑠≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿^𝑞𝑏_𝑎𝑑𝑠(𝑠−𝑎)^𝛼𝑞=2𝑝^𝛿𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿^𝑞𝑀𝛼𝑞. (247) So, it is obvious by (247) that the inequality ‖𝑇(𝐴)‖_{𝐿𝑞((𝑎,𝑏),𝑋)}≤2𝑝^𝛿𝑀𝛼,𝛿,𝑝‖𝐴‖_𝛼,𝛿𝑀𝛼𝑞^1/𝑞 (248) holds. From (248), we obtain ‖𝑇‖≤2𝑝^𝛿𝑀(𝛼,𝛿,𝑝)[𝑀(𝛼𝑞)]^1/𝑞 and since 𝑇 is linear, 𝑇∈𝐵(𝐻_𝛼,𝛿((𝑎,𝑏),𝑋),𝐿_𝑞((𝑎,𝑏),𝑋)), and this completes the proof of theorem.

Acknowledgment

The authors would like to thank the referee for his/her careful reading and helpful comments that have improved the quality of the paper.