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+⋯}