Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

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Existence and Uniqueness of Continuous Solutions for Conformable 

Fractional Integro-Differential Equations in Cone Metric Spaces 

 

Kamble Rajratna M.a, Kulkarni Pramod Rb 

aShri Vitthal Rukmini Arts, Commerce and Science College, Sawana, Maharashtra State, India. 

email: kamblerajratna2@gmail.com 

bScience College, Nanded, Maharashtra State, India. 

email: pramodrkul@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

In this paper, by the application of some extensions of Banach's contraction principle in 

complete cone metric space, we have proved the existence and uniqueness of solutions to 

fractional order integro-differential equations of Volterra-Fredholm type which are defined 

in a cone metric space. The fractional order derivative defined in the integro-differential 

equation is the conformable fractional order derivative. The obtained results are used for 

solving a couple of fractional order integro-differential equations of Volterra-Fredholm type. 

Mathematics Subject Classification: 34B05. 

Keywords: Fractional order integro-differential equations, Cone metric space, Contractive 

mapping, ordered Banach space. 

 

1. Introduction 

Numerous scientific and engineering problems involve integral equations. Volterra or Fredholm 

integral equations can be used to solve a wide range of initial and boundary value problems. More 

than any other discipline, the potential theory helped in the development of theory of integral 

equations. Integral equations were also developed using mathematical physics models, including 

water waves, conformal mapping, diffraction issues, and scattering in quantum mechanics. Integral 

equations or integro-differential equations describe a wide range of additional applications in science 

and engineering. Integral equations address a number of topics, including the Volterra population 

growth model, coexisting biological species, the spread of stocked fish in a new lake, heat transport, 

and heat radiation. Integral equations with logarithmic kernels arise in many scientific problems. 

Integral equations are frequently used in electrostatic, low frequency electromagnetic, 

electromagnetic scattering, and acoustic and elastic wave propagation problems. 

In this paper we study the existence and uniqueness of solutions for the conformable fractional order 

Volterra-Fredholm type integro-differentail equations [1, 2, 3] of the form 

𝑑𝛼𝑥(𝑡)

𝑑𝑡
= 𝑓(𝑡) + ∫ 𝑝(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠 + ∫ 𝑞(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠

𝑏

0

𝑡

0

, 𝑡 ∈ 𝐼 = [0, 𝑏]     …       (1) 

              𝑥(0) = 𝑥0                                                                                                                          …        (2) 

mailto:Kamblerajratna2@gmail.com
mailto:pramodrkul@gmail.com


Communications on Applied Nonlinear Analysis 

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where the term 
𝑑𝛼𝑥(𝑡)

𝑑𝑡
 represents the conformable fractional order derivative of fractional order 𝛼 ∈

(0,1). ,  𝑓: 𝐼 → 𝑋, 𝑝, 𝑞: 𝐼 × 𝐼 × 𝑋 → 𝑋  are continuous functions and 𝑥0 is an element of a real Banach 

space 𝑋 , with the norm ‖. ‖. But before investigating the problem, we will take a review of the 

research work done by the mathematicians in the development of the topic. In [4], using the 

Schaefer’s fixed point theorem, Karouni, A. et. al. have established the existence and uniqueness of 

the continuous solution to the nonlinear Fredholm integral equation of the form  

𝑥(𝑡) = 𝑓(𝑡) + ∫ 𝑔(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠,    − ∞ < 𝑎 ≤ 𝑡 ≤ 𝑏 < ∞

𝑏

𝑎

 

where it is assumed that the function 𝑓 is continuous over the interval [𝑎, 𝑏] and bounded over a 

measurable set. In [5], Claudia, A. have proved, using the Picards operator theory, the existence and 

uniqueness to the solution of the Volterra-Fredholm integral equation of the form 

𝑢(𝑥, 𝑡) = 𝑔(𝑥, 𝑡) + ∫ ∫ 𝑘(𝑡, 𝑥, 𝑠, 𝑦, 𝑢(𝑠, 𝑦))𝑑𝑦𝑑𝑠

Ω

𝑡

0

 

where (𝑡, 𝑥) ∈ [0, 𝑇] × Ω ≔ �̅�, 𝑇 > 0, Ω ∈ ℝ𝑚 is bounded and closed. 

Ahmad et al. in [6] have obtained the solutions of the integro-differential equations with non-local 

four point and strip multipoint boundary conditions. Wang et al. [7] have established the conditions 

for the uniqueness and existence of the positive solutions of the fractional integro-differential 

equation 

𝐷𝛼𝑢(𝑡) + 𝑓(𝑡, 𝑢(𝑡), 𝑇𝑢(𝑡), 𝑆𝑢(𝑡)) = 0, 0 < 𝑡 < 1 

under the boundary conditions given by 

𝑢(0) = 𝑢0, 𝑢′(0) =  𝑏1, … , 𝑢(𝑛−3)(0) = 𝑏𝑛−3, 𝑢(𝑛−2)(0) = 𝑏𝑛−2, 𝑢(𝑛−1)(0) = 𝑏𝑛−1  

where 𝑛 − 1 <  𝛼 ≤  𝑛, 0 ≤  µ <  𝑛 − 1, 𝑛 ≥  3, 𝑏𝑖 ≥  0 (𝑖 =  1, 2, … , 𝑛 − 3, 𝑛 − 2, 𝑛 − 1),  

𝐷𝛼 being the Caputo fractional derivative of order 𝛼, 𝑓 is a continuous function from [0,

1]  × ℝ+
3  → ℝ+,  𝑇 and 𝑆 are defined by 

(𝑇𝑥)(𝑡) = ∫ 𝐾(𝑡, 𝑠) 𝑥(𝑠) 𝑑𝑠,     (𝑆𝑥)(𝑡) =

1

0

∫ 𝐻(𝑡, 𝑠) 𝑥(𝑠) 𝑑𝑠   

1

0

 

𝐾∗ = sup 
𝑡∈[0,   1]

∫ 𝐾(𝑡, 𝑠)  𝑑𝑠

𝑡

0

, 𝐻∗ = sup 
𝑡∈[0,   1]

∫ 𝐻(𝑡, 𝑠)  𝑑𝑠

𝑡

0

 

where  𝐾 ∈ 𝐶(𝐷,   ℝ+), 𝐻 ∈ 𝐶([0, 1] ×  [0, 1], ℝ+) 

The authors in [8, 9] have obtained the results stating the existence and uniqueness of the solutions of 

the fractional integro-differential equations under different boundary conditions. In [10], Bragdi, A. 

et al. have obtained the solution of the BVP given by  

𝐷𝛼(𝐷𝛽)𝑢(𝑡) = 𝑓(𝑡, 𝑢(𝑡), 𝜙𝑢(𝑡), 𝜓𝑢(𝑡)) 



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under the boundary conditions given by 𝑢(1) = 𝑢(0) =  𝑢′(0) = 0  where it is assumed that 

 1 <  𝛼 ≤  2, 0 <  𝛽 ≤  1, 𝑓 ∶  𝐼 ×  ℝ3 → ℝ, 𝐼 = [0,   1], the function 𝑓 is continuous and  

𝜙(𝑢)(𝑡) = ∫ 𝛾(𝑡, 𝑠)𝑢(𝑠)  𝑑𝑠,    

𝑡

0

𝜓(𝑢)(𝑡) = ∫ 𝜆(𝑡, 𝑠)𝑢(𝑠)  𝑑𝑠  

𝑡

0

 

𝛾, 𝜆 ∶  𝐼 × 𝐼 →  [0, 1), 𝑠𝑢𝑝 ∫ 𝜆(𝑡, 𝑠)  𝑑𝑠

1

0

< ∞, 𝑠𝑢𝑝 ∫ 𝛾(𝑡, 𝑠)  𝑑𝑠

1

0

< ∞. 

Ibnelazyz, L. et al.[11] have explored the existence and uniqueness for a nonlinear fractional integro-

differential equations with integral and anti-periodic boundary conditions where the existence is 

proved by means of Krasnoselskii’s fixed point theorem and the uniqueness of solutions is 

established via the Banach’s contraction principle. in [12], Kamble, R., and Kukarni, P. have proved 

the existence and uniqueness of solutions for the following equation  

𝐷𝛼𝐷𝛽𝑥(𝜏) = 𝑓(𝑡, 𝑥(𝜏), 𝜙𝑥(𝜏), 𝜓𝑥(𝜏)), 𝜏𝜖[0,1], 𝑥(0) = 𝑥(1) = 0  

where 0< α ≤ 1, 0 < β ≤ 1, 𝐷𝛼, 𝐷𝛽 are the Caputo fractional derivatives of order 𝛼 , 𝛽,  

𝜆, 𝛿: [0,1] × [0, 1] → [0, +∞),  f : [0, 1] × ℝ3 ⟶ ℝ  is a continuous function, and 

𝜙𝑥(𝜏) = ∫ 𝜆(𝜏, 𝑠)𝑥(𝑠)𝑑𝑠
𝜏

0

, 𝜓𝑥(𝜏) = ∫ 𝛿(𝜏, 𝑠)𝑥(𝑠)𝑑𝑠
𝜏

0

 

𝜙∗ = sup
𝑡∈[0,1]

|∫ 𝜆(𝜏, 𝑠)𝑑𝑠

𝑡

0

| < ∞,  𝜓∗ = sup
𝑡∈[0,1]

|∫ 𝛿(𝜏, 𝑠)𝑑𝑠

𝑡

0

| < ∞,  

2. Some Preliminary Concepts in Fixed Point Theory and Results 

In this section, we take a look on the basic concepts in fixed point theory. 

Definition 2.1: Let 𝑋 be a Banach space over the set of real numbers. A subset 𝐸 of 𝑋 is called a 

cone if 

1) 𝐸 is closed, non-empty with  𝐸 ≠ {0} 

2) 𝑐𝑥 + 𝑑𝑦 ∈ 𝐸 whenever 𝑐, 𝑑 ∈ ℝ and 𝑥, 𝑦 ∈ 𝐸 

3)  𝑥 = 0 whenever both 𝑥, −𝑥 ∈ 𝐸 

Definition 2.2: For a given cone 𝐸 ⊂ 𝑋, 𝑋 being a Banach space over the set of real numbers, the 

relation ≤ defined on 𝐸 by 𝑥 ≤ 𝑦 ⟺  𝑦 − 𝑥 ∈ 𝐸 is a partial ordering relation. The cone 𝐸 is said to 

be normal if there is a real number 𝑘 > 0 such that ‖𝑥‖ ≤ 𝑘‖𝑦‖ ∀ 𝑥, 𝑦 ∈ 𝐸. The least positive 

number 𝑘 satisfying above is called the normal constant of 𝐸. 

Definition 2.3: Let 𝑋 be a non empty real Banach space, 𝐸 be a normal cone in 𝑋 with ≤  as the 

partial ordering relation defined on 𝐸. Then the mapping 𝑑: 𝑋 × 𝑋 → 𝐸 satisfying 

1) 0 ≤ 𝑑(𝑥, 𝑦) ∀ 𝑥, 𝑦 ∈ 𝐸 

2) 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥) ∀ 𝑥, 𝑦 ∈ 𝐸 



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3) 𝑑(𝑥, 𝑦) ≤ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)) ∀ 𝑥, 𝑦 , 𝑧 ∈ 𝐸 

is called a cone metric on 𝑋 and the space (𝑋, 𝑑) is called cone metric space. 

Definition 2.4: Let 𝑋 be an ordered space. A function 𝜑: 𝑋 → 𝑋 is said to a comparison function if 

for every 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≤ 𝑦 ⟹ 𝜑(𝑥) ≤ 𝜑(𝑦), 𝜑(𝑥) ≤ 𝑥 and lim
𝑛→∞

‖𝜑𝑛(𝑥)‖ = 0 for every 𝑥 ∈ 𝑋. 

 A more detailed theory and examples on cone metric spaces and the different versions of the 

fixed points theorems can be obtained in the books by the authors Smart D. and O’Regan D [13, 14]. 

We will take a small review of the research work on cone metric spaces. In [15], the authors have 

proved the following result. 

Lemma 2.1: Let (𝑋, 𝑑) be a complete cone metric space and 𝑃 be a normal cone with normal 

constant 𝐾. Let 𝑓: 𝑋 → 𝑋 be a function such that there exists a comparison function 𝜙: 𝑃 ⟶ 𝑃 such 

that 

𝑑(𝑓(𝑥), 𝑓(𝑦)) ≤ 𝜙(𝑑(𝑥, 𝑦) ∀ 𝑥, 𝑦 ∈ 𝑋 

Then the contraction mapping 𝑓 has a unique fixed point in 𝑋. 

In [16], the authors Ilic, D, and Rakocevic, V. has proved the following result. 

Lemma 2.2: Let (𝑋, 𝑑) be a complete cone metric space and let 𝑃 be a normal cone. Let 𝑓: 𝑋 →  𝑋, 

𝑓2 be continuous, 𝑔: 𝑓 (𝑋) → 𝑋 be such that 𝑔𝑓(𝑋) ⊆ 𝑓2(𝑋), and 𝑓(𝑔(𝑥)) = 𝑔(𝑓(𝑥)) whenever 

both sides are defined. Furthermore, let there exists 𝜆 ∈  (0, 1) such that 𝑑(𝑔𝑥, 𝑔𝑦) ≤ 𝜆 𝑢 for every 

𝑥, 𝑦 ∈  𝑓 (𝑋). Then 𝑓 and 𝑔 have a common unique fixed point 𝑢 in 𝑋. 

3. Fractional Derivative and Conformable Fractional Derivative 

For many centuries, the derivative of non-integer order has been an interesting area of study. 

Riemann-Liouville, Caputo, Hadamard, Grunwald-Letnikov, Marchaud, and Riesz were among the 

fractional derivative types that were introduced [17, 18, 19, 20, 21, 22]. None of these fractional 

derivatives satisfy the properties of the classical integer order derivatives. We are aware that the 

derivative of an integer order constant is zero, and we expect fractional derivatives to be no different. 

This is not the case for most fractional derivatives, except for the Caputo fractional derivative. 

Furthermore, fractional derivatives do not meet the requirements of classical derivatives, including 

the Mean value theorems of Rolle and Lagrange, the Product rule, the Quotient rule, and the Chain 

rule. Many authors, such as R. Khalil [23], Abdljawad et al. [24, 25, 26] etc. have made important 

contributions in this direction. They have established some of the properties that the previous 

fractional derivatives did not meet and defined a few new fractional derivatives. All of these 

conformable derivatives are extensions of the traditional limit form formulation. The following 

definitions are revised from these references.   

Definition 3.1: Let : [0, ∞) → ℝ , be any function of real variable 𝑡. Then the conformable fractional 

derivative of  𝑢  of fractional order 𝛼, 𝛼 ∈ (0,1] at 𝑡 > 0, denoted 𝐷𝛼𝑢(𝑡),  is defined by the limit 

𝐷𝛼𝑢(𝑡) = 𝑙𝑖𝑚
ℎ→0

𝑢(𝑡 + ℎ𝑡(1−𝛼)) − 𝑢(𝑡)

ℎ
      

provided the limit exists. 



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For 𝑛 < 𝛼 ≤ 𝑛 + 1, 𝑛 ∈ 𝑁,  the conformable fractional derivative is defined by   

𝐷𝛼(𝑢)(𝑡) = 𝑙𝑖𝑚
ℎ→0

𝑢⌈𝛼⌉−1(𝑡 + ℎ𝑡(⌈𝛼⌉−𝛼)) − 𝑢⌈𝛼⌉−1(𝑡)

ℎ
 

where ⌈α⌉ is the smallest integer greater than or equal to α provided the limit exist. 

Definition 3.2:  Let 𝛼 ∈ (0, 1] and 𝑡 > 0. The α conformable fractional integral is given by 

𝐼𝛼𝑢(𝑡) = ∫
𝑢(𝑠)

𝑠1−𝛼

𝑡

0

𝑑𝑠. 

Definition 3.3: Let 𝐵 = 𝐶(𝐼, 𝑋) be the Banach space of all continuous functions defined on the set 

𝐼 = [0, 𝑏] into 𝑋, where the norm of an 𝑥 ∈ 𝐵 is defined by ||𝑥||∞ = sup{||𝑥(𝑡)||: 𝑡} ∈ 𝐼 

Define a metric 𝑑: 𝐵 × 𝐵 ⟶ ℝ by 𝑑(𝑥, 𝑦) = (||𝑥 − 𝑦||∞, 𝑎||𝑥 − 𝑦||∞)  ∀𝑥, 𝑦 ∈ 𝐵, 0 < 𝑎 < 1.Then 

it can be verified that (𝐵, 𝑑) is a cone metric space. The function 𝑥 ∈ 𝐵 is called as a solution of the 

initial value problem (1)—(2) if it satisfies the condition 

𝑥(𝑡) = 𝑢0 + ∫ 𝑓(𝑡). 𝑡𝛼−1𝑑𝑡
𝑡

0

+ ∫ 𝑡𝛼−1    [∫ 𝑘(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠
𝑡

0

]
𝑡

0

𝑑𝑡 + ∫ 𝑡𝛼−1
𝑡

0

[∫ ℎ(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠
𝑏

0

] 𝑑𝑡 

4. Main Result   

Now we have enough material to prove the main result. 

Theorem 4.1: The initial value problem (1)—(2) has a unique solution 𝑥 in 𝐼 if the following 

conditions are satisfied 

1) there exist continuous functions 𝑝1, 𝑝2: 𝐼 × 𝐼 → ℝ+ and a comparison function 𝜙: ℝ2 → ℝ2 

satisfying 

(‖𝑘(𝑡, 𝑠, 𝑢) − 𝑘(𝑡, 𝑠, 𝑣)‖, 𝑎‖𝑘(𝑡, 𝑠, 𝑢) − 𝑘(𝑡, 𝑠, 𝑣)‖) ≤ 𝑝1(𝑡, 𝑠)𝜙(𝑑(𝑢, 𝑣)) 

(‖ℎ(𝑡, 𝑠, 𝑢) − ℎ(𝑡, 𝑠, 𝑣)‖, 𝑎‖ℎ(𝑡, 𝑠, 𝑢) − ℎ(𝑡, 𝑠, 𝑣)‖) ≤ 𝑝2(𝑡, 𝑠)𝜙(𝑑(𝑢, 𝑣)) 

where the metric 𝑑: 𝐵 × 𝐵 ⟶ ℝ is defined by 𝑑(𝑥, 𝑦) = (||𝑥 − 𝑦||∞, 𝑎||𝑥 − 𝑦||∞)  ∀𝑥, 𝑦 ∈ 𝐵 

2) 

∫ 𝑡𝛼−1 [∫[𝑝1(𝑡, 𝑠) + 𝑝2(𝑡, 𝑠)]

𝑏

0

𝑑𝑠] 𝑑𝑡 ≤ 1

𝑏

0

 

Proof: We define the operator 𝐹 by 

𝐹𝑥(𝑡) = 𝑥0 + ∫ 𝑓(𝑡). 𝑡𝛼−1𝑑𝑡
𝑡

0

+ ∫ 𝑡𝛼−1
𝑡

0

[∫ 𝑘(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠
𝑡

0

] 𝑑𝑡 + ∫ 𝑡𝛼−1 [∫ ℎ(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠
𝑏

0

]
𝑡

0

𝑑𝑡 

𝐹𝑦(𝑡) = 𝑦0 + ∫ 𝑓(𝑡). 𝑡𝛼−1𝑑𝑡
𝑡

0

+ ∫ 𝑡𝛼−1 [∫ 𝑘(𝑡, 𝑠, 𝑦(𝑠))𝑑𝑠
𝑡

0

] 𝑑𝑡
𝑡

0

+ ∫ 𝑡𝛼−1
𝑡

0

[∫ ℎ(𝑡, 𝑠, 𝑦(𝑠))𝑑𝑠
𝑏

0

] 𝑑𝑡 

Using the conditions 1) and 2), for all 𝑥, 𝑦 ∈ 𝐵, we have  

(‖𝐹𝑥(𝑡) − 𝐹𝑦(𝑡)‖, 𝑎‖𝐹𝑥(𝑡) − 𝐹𝑦(𝑡)‖) 



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= (‖𝑥0 − 𝑦0‖ + ‖∫ 𝑡𝛼−1
𝑡

0

[∫ 𝑘(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠
𝑡

0

] 𝑑𝑡 + ∫ 𝑡𝛼−1
𝑡

0

[∫ ℎ(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠
𝑏

0

] 𝑑𝑡

− ∫ 𝑡𝛼−1
𝑡

0

[∫ 𝑘(𝑡, 𝑠, 𝑦(𝑠))𝑑𝑠
𝑡

0

] 𝑑𝑡 − ∫ 𝑡𝛼−1
𝑡

0

[∫ ℎ(𝑡, 𝑠, 𝑦(𝑠))𝑑𝑠
𝑏

0

] 𝑑𝑡‖ , 𝑎‖𝑥0 − 𝑦0‖

+ 𝑎 ‖∫ 𝑡𝛼−1
𝑡

0

[∫ 𝑘(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠
𝑡

0

] 𝑑𝑡 + ∫ 𝑡𝛼−1
𝑡

0

[∫ ℎ(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠
𝑏

0

] 𝑑𝑡

− ∫ 𝑡𝛼−1
𝑡

0

[∫ 𝑘(𝑡, 𝑠, 𝑦(𝑠))𝑑𝑠
𝑡

0

] 𝑑𝑡 − ∫ 𝑡𝛼−1
𝑡

0

[∫ ℎ(𝑡, 𝑠, 𝑦(𝑠))𝑑𝑠
𝑏

0

] 𝑑𝑡‖) 

≤ (‖𝑥0 − 𝑦0‖ + ∫ 𝑡𝛼−1
𝑡

0

[∫ ‖𝑘(𝑡, 𝑠, 𝑥(𝑠)) − 𝑘(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠
𝑡

0

] 𝑑𝑡

+ ∫ 𝑡𝛼−1 [∫ ‖ℎ(𝑡, 𝑠, 𝑥(𝑠)) − ℎ(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠
𝑏

0

] 𝑑𝑡,
𝑡

0

 𝑎‖𝑥0 − 𝑦0‖

+ 𝑎 ∫ 𝑡𝛼−1
𝑡

0

[∫ ‖𝑘(𝑡, 𝑠, 𝑥(𝑠)) − 𝑘(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠
𝑡

0

] 𝑑𝑡

+ 𝑎 ∫ 𝑡𝛼−1 [∫ ‖ℎ(𝑡, 𝑠, 𝑥(𝑠)) − ℎ(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠
𝑏

0

] 𝑑𝑡 
𝑡

0

) 

≤ (∫ 𝑡𝛼−1
𝑡

0

[∫ ‖𝑘(𝑡, 𝑠, 𝑥(𝑠)) − 𝑘(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠
𝑡

0

] 𝑑𝑡,   𝑎 ∫ 𝑡𝛼−1
𝑡

0

[∫ ‖𝑘(𝑡, 𝑠, 𝑥(𝑠))
𝑡

0

− 𝑘(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠] 𝑑𝑡) 

+ (∫ 𝑡𝛼−1 [∫ ‖ℎ(𝑡, 𝑠, 𝑥(𝑠)) − ℎ(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠
𝑏

0

] 𝑑𝑡,
𝑡

0

 𝑎 ∫ 𝑡𝛼−1 [∫ ‖ℎ(𝑡, 𝑠, 𝑥(𝑠))
𝑏

0

𝑡

0

− ℎ(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠] 𝑑𝑡 ) 

≤ ∫ 𝑡𝛼−1 [∫ 𝑝1(𝑡, 𝑠)𝜙(𝑑(𝑥, 𝑦))𝑑𝑠

𝑡

0

] 𝑑𝑡 + ∫ 𝑡𝛼−1 [∫ 𝑝2(𝑡, 𝑠)𝜙(𝑑(𝑥, 𝑦))𝑑𝑠

𝑏

0

] 𝑑𝑡

𝑡

0

𝑡

0

 

≤ ∫ 𝑡𝛼−1
𝑡

0

[∫ 𝑝1(𝑡, 𝑠)𝜙(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞)𝑑𝑠
𝑡

0

] 𝑑𝑡

+ ∫ 𝑡𝛼−1 [∫ 𝑝2(𝑡, 𝑠)𝜙(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞)𝑑𝑠
𝑏

0

] 𝑑𝑡
𝑡

0

 

 

≤ ∫ 𝑡𝛼−1
𝑏

0

[∫ 𝑝1(𝑡, 𝑠)𝜙(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞)𝑑𝑠
𝑏

0

] 𝑑𝑡

+ ∫ 𝑡𝛼−1 [∫ 𝑝2(𝑡, 𝑠)𝜙(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞)𝑑𝑠
𝑏

0

] 𝑑𝑡
𝑏

0

 



Communications on Applied Nonlinear Analysis 

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1309 
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≤ 𝜙(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞) ∫ 𝑡𝛼−1 [∫[𝑝1(𝑡, 𝑠) + 𝑝2(𝑡, 𝑠)]

𝑏

0

𝑑𝑠] 𝑑𝑡

𝑏

0

 

≤ 𝜙(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞) 

This implies that 𝑑(𝐹(𝑥), 𝐹(𝑦)) ≤ 𝜙𝑑(𝑥, 𝑦)  ∀ 𝑥, 𝑦 ∈ 𝐵. Now by Lemma 2.1, the operator 𝐹 has a 

unique fixed point in 𝐵. This means that the initial value problem (1)—(2) has a unique solution 𝑥 in 

𝐼. This completes the proof of theorem.            □ 

5. Application of the Result   

In order to support the result proved, now we will present an example. In the initial value problem 

(1)—(2), let 𝑘(𝑡, 𝑠, 𝑥) = 𝑡2𝑠2 +
𝑥𝑠2 

2
,   ℎ(𝑡, 𝑠, 𝑥) = 𝑡2𝑠2 +

𝑡2𝑠2𝑥

2
, 𝑠, 𝑡 ∈ 𝐼 = [0,1], 𝑥 ∈

(𝐶[0,1], ℝ), 0 < 𝛼 < 1. We define the metric 𝑑(𝑥, 𝑦) = (‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞) on (𝐶[0,1], ℝ) and 

𝑎 ≥ 0. Then it is clear that (𝐶[0,1], ℝ ) is a complete cone metric space. 

Now we have 

 (|𝑘(𝑡, 𝑠, 𝑥(𝑠)) − 𝑘(𝑡, 𝑠, 𝑦(𝑠))|, 𝑎|𝑘(𝑡, 𝑠, 𝑥(𝑠)) − 𝑘(𝑡, 𝑠, 𝑦(𝑠))|) 

= (|𝑡2𝑠2 +
𝑥𝑠2

2
− 𝑡2𝑠2 −

𝑦𝑠2 

2
| , 𝑎 |𝑡2𝑠2 +

𝑥𝑠2 

2
− 𝑡2𝑠2 −

𝑦𝑠2 

2
|) 

= (|
𝑥𝑠2 

2
−

𝑦𝑠2 

2
| , 𝑎 |

𝑥𝑠2

2
−

𝑦𝑠2 

2
|) 

=
𝑠2

2
(|𝑥 − 𝑦|, 𝑎|𝑥 − 𝑦|) 

≤
𝑠2 

2
(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞) 

= 𝑝1
∗𝜙∗(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞) 

where 𝑝1
∗ = 𝑠2, which is a continuous function from [0,1] × [0, 1] into ℝ+ and a comparison 

function 𝜙∗(𝑥, 𝑦) =
1

2
(𝑥, 𝑦). Similarly we can prove that  

|ℎ(𝑡, 𝑠, 𝑥(𝑠)) − ℎ(𝑡, 𝑠, 𝑦(𝑠))|, 𝑎|ℎ(𝑡, 𝑠, 𝑥(𝑠)) − ℎ(𝑡, 𝑠, 𝑦(𝑠))| ≤ 𝑝2
∗𝜙∗(‖𝑥 − 𝑦‖∞, 𝑎‖𝑥 − 𝑦‖∞) 

where 𝑝2
∗ = 𝑠2𝑡2 which is a continuous function of [0,1] × [0, 1]into ℝ+. 

Also, we note that  𝑖𝑓 𝛼 =
1

2
 then 

∫ 𝑡𝛼−1 ∫ (𝑝1
∗(𝑡, 𝑠) + 𝑝2

∗(𝑡, 𝑠))𝑑𝑠𝑑𝑡
1

0

1

0

= ∫ 𝑡𝛼−1 ∫ (𝑠2 + 𝑠2𝑡2)𝑑𝑠𝑑𝑡
1

0

1

0

< 1 

With these choices of functions, all the hypothesis in theorem 4.1 are satisfied. Hence the existence 

and uniqueness of the solution is verified. 

 



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 9s (2025) 

 

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