Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator

Supriya Kumar Paul; Lakshmi Narayan Mishra; Vishnu Narayan Mishra; Dumitru Baleanu

doi:10.1016/j.jksus.2023.102949

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Original article

12 2023

:35;

102949

doi:

10.1016/j.jksus.2023.102949

Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator

Supriya Kumar Paul^a, Lakshmi Narayan Mishra^a,⁎, Vishnu Narayan Mishra^b, Dumitru Baleanu^c,d,e

a Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India

b Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur 484 887, Madhya Pradesh, India

c Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara 09790, Turkey

d Institute of Space Sciences, 077125, Magurele, Ilfov, Romania

e Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut 11022801, Lebanon

⁎Corresponding author. lakshminarayan.mishra@vit.ac.in (Lakshmi Narayan Mishra),

Received: 2023-1-30, Accepted: 2023-10-12,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

This paper investigates the existence, uniqueness and stability of solutions to the nonlinear Volterra–Fredholm integral equations (NVFIE) involving the Erdélyi–Kober (E–K) fractional integral operator. We use the Leray–Schauder alternative and Banach’s fixed point theorem to examine the existence and uniqueness of solutions, and we also explore Hyers–Ulam (H–U) and Hyers–Ulam–Rassias (H–U–R) stability in the space $C ([0, β], R)$ . Furthermore, three solution sets $U_{σ, λ}$ , $U_{θ, 1}$ and $U_{1, 1}$ are constructed for $σ > 0$ , $λ > 0$ , and $θ \in (0, 1)$ , and then we obtain local stability of the solutions with some ideal conditions and by using Schauder fixed point theorem on these three sets, respectively. Also, to achieve the goal, we choose the parameters for the NVFIE as $δ \in (\frac{1}{2}, 1)$ , $ρ \in (0, 1)$ , $γ > 0$ . Three examples are provided to clarify the results.

Keywords

26A33

45G10

45M10

47H10

Erdélyi–Kober fractional integral operator

Hyers–Ulam–Rassias stability

Hyers–Ulam stability

Local stability

Fixed point theorem

1 Introduction

Integral equations evolve spontaneously in various fields of basic sciences and engineering, like mathematical physics, solid state physics, astrophysics, microscopy, chemical reactions, plasma diagnostics, X-ray radiography, semiconductors, fluid flow, mathematical biology, scattering theory, etc. (Ganji, 2006; He, 2005; Liu and Gu, 2001; Rahman, 2007; Wazwaz, 2011; Marzban, 2023b; Marzban and Nezami, 2022; Marzban, 2023a; Marzban and Korooyeh, 2022; Marzban and Ashani, 2020; Rahimkhani and Ordokhani, 2023, 2022 ). Recent years have seen a major increase in interest in the theory of fractional integral equations, which is now a significant field of nonlinear analysis. Although integral equations containing the Erd $\overset{́}{e}$ lyi–Kober (E–K) fractional integral operator are typically used in kinetic theory of gases, to describe the medium with non-integer mass dimension, traffic theory, porous media, viscoelasticity, and electrochemistry (Alamo and Rodrıguez, 1994; Kilbas et al., 2006; Lakshmikantham et al., 2009; Hilfer, 2000; Kiryakova, 1994; Mainardi, 1997 ), it is essential to analyze such integral equations.

Analysis of the existence criteria for the solutions of different kinds of integral equations is an essential part of the study. One can use these requirements to identify the situation under which the problem’s solution exists. The concepts of fixed-point approaches are significant in this sense.

In light of other viewpoints, stability is a crucial consideration for numerical solutions and might be necessary to compare the results and effectiveness of numerical methods. For instance, the papers in Refs. Nwaigwe (2022), Nwaigwe and Benedict (2023), Nwaigwe and Micula (2023), Nwaigwe et al. (2023) deal with numerical solutions of integral equations. Also, different forms of stability analysis have been performed on both differential and integral equations. In this regard, Lyapunov stability has been studied in a wide range of real-world problem settings. Further, exponential and Mittag-Leffler stabilities have been implemented for many topics. In recent times, researchers seem to be steadily more interested in H–U and H–U–R stability. In Refs. Akkouchi (2011), Ali et al. (2019), Amin et al. (2022b, a), Kumam et al. (2017), Morales and Rojas (2011), Subramanian et al. (2022), Paul et al. (2023), some current studies on H–U and H–U–R stabilities are included.

It ought to be noticed that Ma and Pečarić (2008) examined the following integral equation with the E–K fractional integral operator, i.e.,

(1.1)

ζ (y) = g (y) + \frac{λ y^{- ρ s}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{s} ζ (μ) d θ, y > 0 .

An explicit bounds of the solution for Eq. (1.1) is achieved by constructing a generalized weakly singular integral inequality.

Wang et al. (2012) used the Schauder fixed point theorem to study the solvability of the following integral equation with the E–K fractional operator, i.e.,

(1.2)

ζ (y) = g (y) + \frac{b (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H (μ, ζ (μ)) d μ, y \in [0, β], β > 0 .

In Wang et al. (2012), they also discussed the local stability result for the following nonlinear integral equation, i.e., $ζ (y) = g (y) + \frac{b (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H (y, μ, ζ (μ)) d μ, y \in [0, \infty) .$

In $2022$ , Amin et al. (2022b, a) have examined the uniqueness and H–U stability to the solution of the mixed type Volterra–Fredholm fractional integral equations, those are,

(1.3)

ζ (y) = g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y - μ)}^{δ - 1} H_{1} (μ, ζ (μ)) d μ

+ \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β - μ)}^{δ - 1} H_{2} (μ, ζ (μ)) d μ,

and

(1.4)

ζ (y) = g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y - μ)}^{δ - 1} ζ (μ) d μ

+ \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β - μ)}^{δ - 1} ζ (μ) d μ, y \in [0, β], β > 0 .

Motivated by the above literature’s, this paper begins the investigation into the existence, uniqueness, H–U–R stability, H–U stability and local stability to the solutions of the nonlinear Volterra–Fredholm integral equation containing the Erd

\overset{́}{e}

lyi–Kober fractional integral operator, i.e.,

(1.5)

ζ (y) = g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ

+ \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ,

y \in R_{+},

where

δ

ρ

, and

γ

are positive parameters,

R_{+} = [0, \infty)

0 < β < \infty

, the functions

g, b_{1}, b_{2} : R_{+} \to R

and

H_{1}, H_{2} : R_{+} \times R_{+} \times R \to R

are continuous functions. More assumptions will be discussed later.

To establish our proposed results, we use three useful theorems namely, Leray–Schauder alternative, Schauder fixed point theorem and Banach’s fixed point theorem.

This paper is arranged as follows: Notations and supporting information are included in Section 2. In Section 3, theoretical analysis of existence and uniqueness of solutions have been discussed under some suitable conditions. Stability results of solutions have been given under some interesting conditions in Section 4. Three examples are discussed to interpret our established results in Section 5. Conclusions with notions for further research are discussed in Section 6.

2 Notations, definitions and auxiliary facts

Let $A = [0, β]$ , where $0 < β < \infty$ . Let $V = C (A, R)$ be the space of all continuous functions $v (y) : A \to R$ . Let $D = C (R_{+}, R)$ be the space of all continuous functions $d (y) : R_{+} \to R$ , where $R_{+} = [0, \infty)$ and $R$ is the set of real numbers. Then $(V, ‖ . ‖)$ and $(D, ‖ . ‖)$ are the Banach spaces with norm $‖ v ‖ = sup {| v (y) | : y \in [0, β]}$ and $‖ d ‖ = sup {| d (y) | : y \in R_{+}}$ , respectively.

Definition 2.1

(Pagnini, 2012; Kilbas et al., 2006). The Erd $\overset{́}{e}$ lyi–Kober fractional integral of a continuous function $ζ : [0, \infty) \to R$ is defined by $I_{ρ}^{s, δ} ζ (y) = \frac{ρ y^{- ρ (δ + s)}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{ρ s + ρ - 1} ζ (μ) d μ,$ with $s \in R$ , and $δ, ρ > 0$ , provided the right side is point-wise defined on $[0, \infty)$ .

Lemma 2.2

(Prudnikov et al., 1981). For $0 < μ < 1$ and $0 < p < q$ , we have $| p^{μ} - q^{μ} | \leq {(p - q)}^{μ}$ .

Lemma 2.3

(Prudnikov et al., 1981). Let $ρ$ , $σ$ , $ν$ and $q$ be positive constants, then $\int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{ν (σ - 1)} μ^{ν (q - 1)} d μ = \frac{y^{α}}{ρ} B (\frac{ν (q - 1) + 1}{ρ}, ν (σ - 1) + 1), y \in R_{+},$ where $α = ν [ρ (σ - 1) + q - 1] + 1$ ,

and $B (r, x) = \int_{0}^{1} θ^{r - 1} {(1 - θ)}^{x - 1} d θ$ , $(R e (r) > 0, R e (x) > 0)$ , is the well-known Beta function.

Theorem 2.4

(Deimling, 1985). If $U$ is a nonempty closed, bounded convex subset of a Banach space $X$ and $f : U \to U$ is completely continuous, then $f$ has a fixed point in $U$ .

Theorem 2.5

Theorem 2.5 Leray–Schauder alternative (Subramanian et al., 2022)

Let $M : E \to E$ be a completely continuous operator. Let $Ω = {y \in E : y = η M (y)$ , for some $0 < η < 1}$ . Then, either the set $Ω$ is unbounded or $M$ has at least one fixed point.

Theorem 2.6

Theorem 2.6 Arzel $\overset{̀}{a}$ -Ascoli theorem (Subramanian et al., 2022)

A subset $H$ in $E ([c, d], R)$ is relatively compact if it is uniformly bounded and equicontinuous on $[c, d]$ .

Theorem 2.7

Theorem 2.7 Banach’s fixed point theorem (Banach, 1922)

Assume that $V$ is a Banach space. Every contraction mapping $M$ defined on $V$ into itself has a unique fixed point in $V$ .

The following Definitions 2.8 and 2.9 are stated in the sense of the papers given in Refs. Akkouchi (2011), Amin et al. (2022b, a), Morales and Rojas (2011), Paul et al. (2023).

Definition 2.8

The Eq. (1.5) has the H–U–R stability, if for each $ζ (y) \in V$ satisfying

(2.6)

| ζ (y) - g (y) - \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ - \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ | \leq w (y),

where

w (y) : A \to R_{+}

\exists

a solution

ζ_{s} (y) \in V

of Eq. (1.5) and a constant

Λ > 0

independent of

ζ

and

ζ_{s}

with

| ζ (y) - ζ_{s} (y) | \leq Λ w (y)

, for all

y \in A

Definition 2.9

We say that the Eq. (1.5) has the H–U stability when w(y) is a constant function in Definition 2.8.

Furthermore, the following definition of the local stability is stated in the sense of the paper given in Ref. Wang et al. (2012).

Definition 2.10

If there exists a solution $ζ (y)$ of Eq. (1.5) such that $lim_{y \to + \infty} | ζ (y) | = 0,$ then the solution of Eq. (1.5) is said to be locally stable.

Now, to prove the main results, we introduce an operator $M$ as

(2.7)

M (ζ (y)) = g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ + \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ .

3 Existence and uniqueness of solutions

We consider the following assumptions for Eq. (1.5):

$g, b_{1}, b_{2}, \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ, \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ \in V$ .
There exist constants $G > 0$ , $B_{1} > 0$ and $B_{2} > 0$ , such that $| g (y) | \leq G$ , $| b_{1} (y) | \leq B_{1}$ and $| b_{2} (y) | \leq B_{2}$ , for all $y \in [0, β]$ .
There exist constants $G_{H_{1}} > 0$ , $G_{H_{2}} > 0$ , such that $| H_{1} (y, μ, ζ_{1}) - H_{1} (y, μ, ζ_{2}) | \leq G_{H_{1}} | ζ_{1} - ζ_{2} |$ and $| H_{2} (y, μ, ζ_{1}) - H_{2} (y, μ, ζ_{2}) | \leq G_{H_{2}} | ζ_{1} - ζ_{2} |$ , for all $ζ_{1}, ζ_{2} \in R$ and $y, μ \in [0, β]$ with $0 \leq μ \leq y$ .
There exist constants $N_{1} > 0$ and $N_{2} > 0$ such that $| H_{1} (y, μ, ζ) | \leq N_{1}$ and $| H_{2} (y, μ, ζ) | \leq N_{2}$ , $\forall$ $ζ \in R$ , $y \in [0, β]$ , $0 \leq μ \leq y$ .

Theorem 3.1

Assume that, assumptions $(A_{1})$ – $(A_{4})$ hold for Eq. (1.5), then with the parameters $δ \in (\frac{1}{2}, 1)$ , $ρ \in (0, 1)$ and $γ > 0$ , Eq. (1.5) has at least one solution defined on $[0, β]$ .

Proof

Assumption $(A_{1})$ ensures that $M : V \to V$ .

Now we will establish this theorem in the following four steps:

Step 1. $M$ is continuous.

Let $ζ \in V$ and ${ζ_{n}}$ be a sequence in $V$ such that ${lim}_{n \to \infty} ‖ ζ_{n} - ζ ‖ = 0$ . Then for $y \in A$ , $| M (ζ_{n} (y)) - M (ζ (y)) | = | \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ_{n} (μ)) d μ + \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ_{n} (μ)) d μ - \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ - \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ | \leq | \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} [H_{1} (y, μ, ζ_{n} (μ)) - H_{1} (y, μ, ζ (μ))] d μ | + | \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} [H_{2} (y, μ, ζ_{n} (μ)) - H_{2} (y, μ, ζ (μ))] d μ | \leq \frac{B_{1}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y, μ, ζ_{n} (μ)) - H_{1} (y, μ, ζ (μ)) | d μ + \frac{B_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y, μ, ζ_{n} (μ)) - H_{2} (y, μ, ζ (μ)) | d μ \leq \frac{B_{1} G_{H_{1}}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | ζ_{n} (μ) - ζ (μ) | d μ + \frac{B_{2} G_{H_{2}}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | ζ_{n} (μ) - ζ (μ) | d μ \leq \frac{B_{1} G_{H_{1}}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ ‖ ζ_{n} - ζ ‖ + \frac{B_{2} G_{H_{2}}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ ‖ ζ_{n} - ζ ‖ \leq \frac{B_{1} G_{H_{1}} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} ‖ ζ_{n} - ζ ‖ + \frac{B_{2} G_{H_{2}} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} ‖ ζ_{n} - ζ ‖ .$ This implies that, $‖ M (ζ_{n}) - M (ζ) ‖ \to 0$ as $n \to \infty$ . So, the operator $M$ is continuous.

Step 2. Bounded sets of $V$ are mapped into bounded sets of $V$ under the mapping $M$ .

Now, for $ζ \in B_{ϵ}$ and for all $y \in A$ , we get $| M (ζ (y)) | = | g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ + \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ | \leq | g (y) | + \frac{| b_{1} (y) |}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y, μ, ζ (μ)) | d μ + \frac{| b_{2} (y) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y, μ, ζ (μ)) | d μ \leq G + \frac{B_{1} N_{1}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ + \frac{B_{2} N_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ \leq G + \frac{B_{1} N_{1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{2} N_{2}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} .$ Thus, $| M (ζ (y)) | \leq G + [\frac{B_{1} N_{1}}{ρ Γ (δ)} + \frac{B_{2} N_{2}}{ρ Γ (δ)}] B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1},$ i.e., $‖ M (ζ) ‖ < \infty .$

Step 3. $M (B_{ϵ})$ is equi-continuous.

Let $ζ \in B_{ϵ}$ and $y_{2}, y_{1} \in A$ with $y_{2} > y_{1}$ . Then $| M (ζ (y_{2})) - M (ζ (y_{1})) | = | g (y_{2}) + \frac{b_{1} (y_{2})}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ + \frac{b_{2} (y_{2})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{2}, μ, ζ (μ)) d μ - g (y_{1}) - \frac{b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ - \frac{b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{1}, μ, ζ (μ)) d μ | \leq | g (y_{2}) - g (y_{1}) | + | \frac{b_{1} (y_{2})}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ + \frac{b_{2} (y_{2})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{2}, μ, ζ (μ)) d μ - \frac{b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ - \frac{b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{1}, μ, ζ (μ)) d μ | \leq | g (y_{2}) - g (y_{1}) | + | \frac{b_{1} (y_{2}) - b_{1} (y_{1}) + b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ - \frac{b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ | + | \frac{b_{2} (y_{2}) - b_{2} (y_{1}) + b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{2}, μ, ζ (μ)) d μ - \frac{b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{1}, μ, ζ (μ)) d μ | \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) | d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} | \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ - \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ | + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) | d μ + \frac{| b_{2} (y_{1}) |}{Γ (δ)} | \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{2}, μ, ζ (μ)) d μ - \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{1}, μ, ζ (μ)) d μ | \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) | N_{1}}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ + \frac{B_{1}}{Γ (δ)} | \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ - \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ | + \frac{B_{1}}{Γ (δ)} | \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ - \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ | + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) | N_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ + \frac{B_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) | N_{1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{1}}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{B_{1}}{Γ (δ)} \int_{0}^{y_{1}} [{(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} - {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1}] μ^{γ} | H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{B_{1}}{Γ (δ)} \int_{y_{1}}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) | N_{2}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) | N_{1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{1}}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{B_{1} N_{1}}{Γ (δ)} \int_{0}^{y_{1}} [{(\frac{1}{y_{1}^{ρ} - μ^{ρ}})}^{1 - δ} - {(\frac{1}{y_{2}^{ρ} - μ^{ρ}})}^{1 - δ}] μ^{γ} d μ + \frac{B_{1} N_{1}}{Γ (δ)} \int_{y_{1}}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) | N_{2}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) | N_{1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{1}}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{B_{1} N_{1}}{Γ (δ)} \int_{0}^{y_{1}} {[\frac{y_{2}^{ρ} - y_{1}^{ρ}}{{(y_{1}^{ρ} - μ^{ρ})}^{2}}]}^{1 - δ} μ^{γ} d μ + \frac{B_{1} N_{1} β^{γ + 1 - ρ}}{ρ δ Γ (δ)} {(y_{2}^{ρ} - y_{1}^{ρ})}^{δ} + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) | N_{2}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) | N_{1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{1}}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{B_{1} N_{1} {(y_{2} - y_{1})}^{ρ (1 - δ)}}{Γ (δ)} \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{2 (δ - 1)} μ^{γ} d μ + \frac{B_{1} N_{1} β^{γ + 1 - ρ}}{ρ Γ (δ + 1)} {(y_{2} - y_{1})}^{ρ δ} + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) | N_{2}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) | N_{1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{1}}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{B_{1} N_{1} {(y_{2} - y_{1})}^{ρ (1 - δ)}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, 2 δ - 1) β^{2 ρ (δ - 1) + γ + 1} + \frac{B_{1} N_{1} β^{γ + 1 - ρ}}{ρ Γ (δ + 1)} {(y_{2} - y_{1})}^{ρ δ} + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) | N_{2}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ .$ Thus, $| M (ζ (y_{2})) - M (ζ (y_{1})) | \to 0$ as $y_{2} \to y_{1}$ .

So, $M (B_{ϵ})$ is equi-continuous.

Hence, combining all the above steps, the operator $M$ is completely continuous by the consequence of Arzel $\overset{̀}{a}$ -Ascoli theorem.

Step 4. Let $Ω = {ζ \in V : ζ = η M (ζ)$ , for some $0 < η < 1}$ .

We need to show that the set $Ω$ is bounded.

Let $ζ (y) \in Ω$ , this indicates that $ζ (y) = η M (ζ (y))$ , for some $0 < η < 1$ .

Then for $y \in A$ , we obtain $| ζ (y) | \leq | g (y) | + \frac{| b_{1} (y) |}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y, μ, ζ (μ)) | d μ + \frac{| b_{2} (y) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y, μ, ζ (μ)) | d μ \leq G + \frac{B_{1} N_{1}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ + \frac{B_{2} N_{2}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ \leq G + \frac{B_{1} N_{1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} + \frac{B_{2} N_{2}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1},$ i.e., $| ζ (y) | \leq G + [\frac{B_{1} N_{1}}{ρ Γ (δ)} + \frac{B_{2} N_{2}}{ρ Γ (δ)}] B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1} < \infty,$ which shows that the set $Ω$ is bounded. Hence, by the Leray–Schauder alternative, $M$ has at least one fixed point, which is a solution of Eq. (1.5). □

Theorem 3.2

Assume that, conditions $(A_{1})$ – $(A_{3})$ hold for Eq. (1.5), and $G = max {G_{H_{1}}, G_{H_{2}}}$ satisfies the relation $0 < G < \frac{ρ Γ (δ)}{(B_{1} + B_{2}) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}$ , then with the parameters $δ \in (\frac{1}{2}, 1)$ , $ρ \in (0, 1)$ and $γ > 0$ , Eq. (1.5) has a unique solution defined on $[0, β]$ .

Proof

Assumption $(A_{1})$ ensures that $M : V \to V$ . Now we need to show that $M$ is a contraction.

Let $ζ_{1}, ζ_{2} \in V$ , then $\forall y \in A$ , we get $| M (ζ_{1} (y)) - M (ζ_{2} (y)) | = | \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ_{1} (μ)) d μ + \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ_{1} (μ)) d μ - \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ_{2} (μ)) d μ - \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ_{2} (μ)) d μ | \leq \frac{| b_{1} (y) |}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y, μ, ζ_{1} (μ)) - H_{1} (y, μ, ζ_{2} (μ)) | d μ + \frac{| b_{2} (y) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y, μ, ζ_{1} (μ)) - H_{2} (y, μ, ζ_{2} (μ)) | d μ \leq \frac{B_{1} G_{H_{1}}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ ‖ ζ_{1} - ζ_{2} ‖ + \frac{B_{2} G_{H_{2}}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ ‖ ζ_{1} - ζ_{2} ‖ \leq \frac{B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ} (\frac{B_{1} G_{H_{1}}}{Γ (δ)} + \frac{B_{2} G_{H_{2}}}{Γ (δ)}) ‖ ζ_{1} - ζ_{2} ‖ \leq \frac{G (B_{1} + B_{2}) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} ‖ ζ_{1} - ζ_{2} ‖ .$ Thus, $‖ M (ζ_{1} (y)) - M (ζ_{2} (y)) ‖ \leq κ ‖ ζ_{1} - ζ_{2} ‖ .$ As by the condition of $G$ , we obtain $κ = \frac{G (B_{1} + B_{2}) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} < 1$ . So, $M$ is a contraction. Hence, by the Banach’s fixed point theorem, $M$ has a unique fixed point, i.e., Eq. (1.5) has a unique solution. □

4 Stability of solutions

We consider the following assumptions for Eq. (1.5):

$g, b_{1}, b_{2}, \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ, \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ \in D$ . Furthermore, the functions $g, b_{1}, b_{2} : R_{+} \to R$ are bounded.
There exist two continuous functions $h_{1}, h_{2} : R_{+} \to R_{+}$ such that $| H_{1} (y, μ, ζ_{1}) - H_{1} (y, μ, ζ_{2}) | \leq h_{1} (y) | ζ_{1} - ζ_{2} |$ and $| H_{2} (y, μ, ζ_{1}) - H_{2} (y, μ, ζ_{2}) | \leq h_{2} (y) | ζ_{1} - ζ_{2} |$ , for all $ζ_{1}, ζ_{2} \in R$ and $y, μ \in R_{+}$ with $μ \leq y$ .
There exist two continuous functions $N_{2}, N_{1} : R_{+} \to R_{+}$ such that $max {| H_{2} (y, μ, 0) : 0 \leq μ \leq y} = N_{2} (y)$ and $max {| H_{1} (y, μ, 0) | : 0 \leq μ \leq y} = N_{1} (y)$ .

Theorem 4.1

Assume that Eq. (1.5) meets all of the requirements of Theorem 3.2. Suppose $ζ (y) \in V$ is such that it satisfies (2.6). Then the Eq. (1.5) has the H–U–R stability.

Proof

According to Theorem 3.2, $\exists$ a unique solution $ζ_{s} \in V$ of Eq. (1.5).

As stated in Definition 2.8, we need to show that $\exists$ a constant $Λ > 0$ such that, $| ζ (y) - ζ_{s} (y) | \leq Λ w (y)$ . Now, $| ζ (y) - ζ_{s} (y) | = | ζ (y) - g (y) - \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ_{s} (μ)) d μ - \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ_{s} (μ)) d μ | \leq | ζ (y) - g (y) - \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ - \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ | + | \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ - \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ_{s} (μ)) d μ | + | \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ - \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ_{s} (μ)) d μ | \leq w (y) + \frac{| b_{1} (y) |}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y, μ, ζ (μ)) - H_{1} (y, μ, ζ_{s} (μ)) | d μ + \frac{| b_{2} (y) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y, μ, ζ (μ)) - H_{2} (y, μ, ζ_{s} (μ)) | d μ \leq w (y) + \frac{B_{1} G_{H_{1}}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ ‖ ζ - ζ_{s} ‖ + \frac{B_{2} G_{H_{2}}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ ‖ ζ - ζ_{s} ‖ \leq w (y) + \frac{B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ} (\frac{B_{1} G_{H_{1}}}{Γ (δ)} + \frac{B_{2} G_{H_{2}}}{Γ (δ)}) ‖ ζ - ζ_{s} ‖ \leq w (y) + \frac{G (B_{1} + B_{2}) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} ‖ ζ - ζ_{s} ‖ .$ Thus, $| ζ (y) - ζ_{s} (y) | \leq ‖ ζ - ζ_{s} ‖ \leq w (y) + \frac{G (B_{1} + B_{2}) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} ‖ ζ - ζ_{s} ‖$ ,

which implies that, $| ζ (y) - ζ_{s} (y) | \leq ‖ ζ - ζ_{s} ‖ \leq Λ w (y)$ , where $Λ = \frac{1}{1 - κ} > 0$ ,

as $κ = \frac{G (B_{1} + B_{2}) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} < 1$ , by the condition of $G$ .

So, the Eq. (1.5) has the H–U–R stability. □

Theorem 4.2

Assume that Eq. (1.5) meets all of the requirements of Theorem 3.2. Let $ζ (y) \in V$ and $ϵ > 0$ such that satisfies $| ζ (y) - g (y) - \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ - \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ | \leq ϵ,$ then the Eq. (1.5) has the H–U stability.

Proof

By using $w (y) = ϵ$ , in Theorem 4.1, we can establish this theorem similarly as well. □

Theorem 4.3

Assume that, there are two constants $σ > 0$ and $λ > 0$ such that $| g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ + \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ | \leq λ y^{- σ}$ holds for Eq. (1.5). Then with the conditions $(S_{1})$ – $(S_{3})$ , and with the parameters $δ \in (\frac{1}{2}, 1)$ , $ρ \in (0, 1)$ and $γ > 0$ , Eq. (1.5) has a solution which is locally stable.

Proof

Define a set $U_{σ, λ} = {ζ (y) : ζ (y) \in D$ and $| ζ (y) | \leq λ y^{- σ}$ , for $y \geq \hat{y} > 0}$ .

It is easy to observe that the set $U_{σ, λ}$ is closed, bounded, and convex subset of $D$ .

Firstly, we show that $M$ maps $U_{σ, λ}$ in $U_{σ, λ}$ .

As by the assumption, we have $| M (ζ (y)) | \leq λ y^{- σ}$ , for $y > 0$ , then $M (U_{σ, λ}) \subset U_{σ, λ}$ .

Now we will establish this theorem in the following two steps:

Step 1. $M$ is continuous.

Let ${ζ_{n}}$ be a sequence in $U_{σ, λ}$ and $ζ \in U_{σ, λ}$ such that ${lim}_{n \to \infty} ‖ ζ_{n} - ζ ‖ = 0$ . Let $ϵ > 0$ be given, $\exists$ $β > 0$ such that $λ y^{- σ} < \frac{ϵ}{2}$ , for $y \geq β$ . Now for $0 < y \leq β$ , $| M (ζ_{n} (y)) - M (ζ (y)) | \leq \frac{| b_{1} (y) |}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y, μ, ζ_{n} (μ)) - H_{1} (y, μ, ζ (μ)) | d μ + \frac{| b_{2} (y) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y, μ, ζ_{n} (μ)) - H_{2} (y, μ, ζ (μ)) | d μ \leq \frac{| b_{1} (y) |}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} h_{1} (y) | ζ_{n} (μ) - ζ (μ) | d μ + \frac{| b_{2} (y) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} h_{2} (y) | ζ_{n} (μ) - ζ (μ) | d μ \leq \frac{sup_{y \in A} (| b_{1} (y) | h_{1} (y))}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ ‖ ζ_{n} - ζ ‖ + \frac{sup_{y \in A} (| b_{2} (y) | h_{2} (y))}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ ‖ ζ_{n} - ζ ‖ \leq \frac{sup_{y \in A} (| b_{1} (y) | h_{1} (y)) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} ‖ ζ_{n} - ζ ‖ + \frac{sup_{y \in A} (| b_{2} (y) | h_{2} (y)) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} ‖ ζ_{n} - ζ ‖ .$ This implies that, $‖ M (ζ_{n}) - M (ζ) ‖ \to 0$ as $n \to \infty$ .

For $y \geq β$ , we get $| M (ζ_{n} (y)) - M (ζ (y)) | \leq | g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ_{n} (μ)) d μ$ $+ \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ_{n} (μ)) d μ$ $- (g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ$ $+ \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ) | \leq 2 λ y^{- σ} < ϵ .$ Thus, for $y > 0$ , $‖ M (ζ_{n}) - M (ζ) ‖ \to 0$ as $n \to \infty$ . So, the operator $M$ is continuous.

Step 2. To prove that $M (U_{σ, λ})$ is equi-continuous.

Let $y_{1}, y_{2} > 0$ with $y_{1} < y_{2}$ . Let $ϵ > 0$ be given, $\exists$ $β > 0$ such that $λ y^{- σ} < \frac{ϵ}{2}$ , for $y > β$ .

For $y_{1}, y_{2} \in [0, β]$ , let $K = max {| ζ (y) | : y \in A}$ . Then, $| M (ζ (y_{2})) - M (ζ (y_{1})) | = | g (y_{2}) + \frac{b_{1} (y_{2})}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ + \frac{b_{2} (y_{2})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{2}, μ, ζ (μ)) d μ - g (y_{1}) - \frac{b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ - \frac{b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{1}, μ, ζ (μ)) d μ | \leq | g (y_{2}) - g (y_{1}) | + | \frac{b_{1} (y_{2})}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ + \frac{b_{2} (y_{2})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{2}, μ, ζ (μ)) d μ - \frac{b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ - \frac{b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{1}, μ, ζ (μ)) d μ | \leq | g (y_{2}) - g (y_{1}) | + | \frac{b_{1} (y_{2}) - b_{1} (y_{1}) + b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ - \frac{b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ | + | \frac{b_{2} (y_{2}) - b_{2} (y_{1}) + b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{2}, μ, ζ (μ)) d μ - \frac{b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{1}, μ, ζ (μ)) d μ | \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) | d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} | \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ - \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ | + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) | d μ + \frac{| b_{2} (y_{1}) |}{Γ (δ)} | \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{2}, μ, ζ (μ)) d μ - \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y_{1}, μ, ζ (μ)) d μ | \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{2}, μ, 0) + H_{1} (y_{2}, μ, 0) | d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} | \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{2}, μ, ζ (μ)) d μ - \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ | + \frac{| b_{1} (y_{1}) |}{Γ (δ)} | \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ - \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y_{1}, μ, ζ (μ)) d μ | + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{2}, μ, 0) + H_{2} (y_{2}, μ, 0) | d μ + \frac{| b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} (| H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{2}, μ, 0) | + | H_{1} (y_{2}, μ, 0) |) d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{1}} [{(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} - {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1}] μ^{γ} (| H_{1} (y_{1}, μ, ζ (μ)) - H_{1} (y_{1}, μ, 0) | + | H_{1} (y_{1}, μ, 0) |) d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} \int_{y_{1}}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} (| H_{1} (y_{1}, μ, ζ (μ)) - H_{1} (y_{1}, μ, 0) | + | H_{1} (y_{1}, μ, 0) |) d μ + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} (| H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{2}, μ, 0) | + | H_{2} (y_{2}, μ, 0) |) d μ + \frac{| b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{| b_{1} (y_{2}) - b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} (h_{1} (y_{2}) | ζ (μ) | + N_{1} (y_{2})) d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{1}} [{(\frac{1}{y_{1}^{ρ} - μ^{ρ}})}^{1 - δ} - {(\frac{1}{y_{2}^{ρ} - μ^{ρ}})}^{1 - δ}] μ^{γ} (h_{1} (y_{1}) | ζ (μ) | + N_{1} (y_{1})) d μ + \frac{| b_{1} (y_{1}) |}{Γ (δ)} \int_{y_{1}}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} (h_{1} (y_{1}) | ζ (μ) | + N_{1} (y_{1})) d μ + \frac{| b_{2} (y_{2}) - b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} (h_{2} (y_{2}) | ζ (μ) | + N_{2} (y_{2})) d μ + \frac{| b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{sup_{y_{2} \in A} (h_{1} (y_{2}) K + N_{1} (y_{2})) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) | b_{1} (y_{2}) - b_{1} (y_{1}) | + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) | (h_{1} (y_{1}) K + N_{1} (y_{1}))}{Γ (δ)} \int_{0}^{y_{1}} {[\frac{y_{2}^{ρ} - y_{1}^{ρ}}{{(y_{1}^{ρ} - μ^{ρ})}^{2}}]}^{1 - δ} μ^{γ} d μ + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) | (h_{1} (y_{1}) K + N_{1} (y_{1}))}{Γ (δ)} \int_{y_{1}}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} d μ + \frac{sup_{y_{2} \in A} (h_{2} (y_{2}) K + N_{2} (y_{2})) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) | b_{2} (y_{2}) - b_{2} (y_{1}) | + \frac{sup_{y_{1} \in A} | b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ \leq | g (y_{2}) - g (y_{1}) | + \frac{sup_{y_{2} \in A} (h_{1} (y_{2}) K + N_{1} (y_{2})) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) | b_{1} (y_{2}) - b_{1} (y_{1}) | + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) | (h_{1} (y_{1}) K + N_{1} (y_{1}))}{Γ (δ)} \int_{0}^{y_{1}} {(y_{2} - y_{1})}^{ρ (1 - δ)} {(y_{1}^{ρ} - μ^{ρ})}^{2 (δ - 1)} μ^{γ} d μ + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) | (h_{1} (y_{1}) K + N_{1} (y_{1})) β^{γ + 1 - ρ}}{ρ δ Γ (δ)} {(y_{2}^{ρ} - y_{1}^{ρ})}^{δ} + \frac{sup_{y_{2} \in A} (h_{2} (y_{2}) K + N_{2} (y_{2})) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) | b_{2} (y_{2}) - b_{2} (y_{1}) | + \frac{sup_{y_{1} \in A} | b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ,$ i.e., $| M (ζ (y_{2})) - M (ζ (y_{1})) | \leq | g (y_{2}) - g (y_{1}) | + \frac{sup_{y_{2} \in A} (h_{1} (y_{2}) K + N_{1} (y_{2})) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) | b_{1} (y_{2}) - b_{1} (y_{1}) | + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) |}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y_{2}, μ, ζ (μ)) - H_{1} (y_{1}, μ, ζ (μ)) | d μ + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) | (h_{1} (y_{1}) K + N_{1} (y_{1})) β^{2 ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} \times B (\frac{γ + 1}{ρ}, 2 δ - 1) {(y_{2} - y_{1})}^{ρ (1 - δ)} + \frac{sup_{y_{1} \in A} | b_{1} (y_{1}) | (h_{1} (y_{1}) K + N_{1} (y_{1})) β^{γ + 1 - ρ}}{ρ Γ (δ + 1)} {(y_{2} - y_{1})}^{ρ δ} + \frac{sup_{y_{2} \in A} (h_{2} (y_{2}) K + N_{2} (y_{2})) β^{ρ (δ - 1) + γ + 1}}{ρ Γ (δ)} B (\frac{γ + 1}{ρ}, δ) | b_{2} (y_{2}) - b_{2} (y_{1}) | + \frac{sup_{y_{1} \in A} | b_{2} (y_{1}) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y_{2}, μ, ζ (μ)) - H_{2} (y_{1}, μ, ζ (μ)) | d μ .$ Thus, $| M (ζ (y_{2})) - M (ζ (y_{1})) | \to 0$ as $y_{2} \to y_{1}$ .

For $y_{1}, y_{2} > β$ , we have $| M (ζ (y_{2})) - M (ζ (y_{1})) | \leq | g (y_{2}) + \frac{b_{1} (y_{2})}{Γ (δ)} \int_{0}^{y_{2}} {(y_{2}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ$ $+ \frac{b_{2} (y_{2})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ$ $- g (y_{1}) - \frac{b_{1} (y_{1})}{Γ (δ)} \int_{0}^{y_{1}} {(y_{1}^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ$ $- \frac{b_{2} (y_{1})}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ | \leq λ y_{2}^{- σ} + λ y_{1}^{- σ} < ϵ .$ For $0 < y_{1} < β < y_{2}$ , observe that $y_{2} \to y_{1}$ implies $y_{2} \to β$ and $β \to y_{1}$ , then

$| M (ζ (y_{2})) - M (ζ (y_{1})) | \leq | M (ζ (y_{2})) - M (ζ (β)) | + | M (ζ (β)) - M (ζ (y_{1})) | \to 0$ as $y_{2} \to y_{1}$ .

Thus, $| M (ζ (y_{2})) - M (ζ (y_{1})) | \to 0$ as $y_{2} \to y_{1}$ for $y_{2}, y_{1} \in [0, \infty)$ . Therefore, $M (U_{σ, λ})$ is equi-continuous. Subsequently, $M (U_{σ, λ})$ is relatively compact as $M (U_{σ, λ})$ $\subset U_{σ, λ}$ is uniformly bounded. Hence, $M$ is completely continuous on $U_{σ, λ}$ . By Schauder fixed point theorem, $M$ has a fixed point in $U_{σ, λ}$ which is the solution of the Eq. (1.5), and it is easy to see that the solution tends to zero as $y \to \infty$ . Thus, the solution of Eq. (1.5) is locally stable. □

In the next two theorems, we provide another easy checked sufficient conditions for the local stability of the solutions of Eq. (1.5).

Theorem 4.4

Assume that, $| g (y) | \leq \frac{1}{3} y^{- θ}$ , $y \in [\hat{y}, + \infty)$ , $\hat{y} > 0$ , $θ \in (0, 1)$ and there are four constants $L_{b_{1}}, L_{b_{2}}, L_{H_{1}}, L_{H_{2}} > 0$ such that $| b_{1} (y) | \leq L_{b_{1}} y^{ρ (1 - δ) - γ - 1}$ , $| H_{1} (y, μ, ζ (μ)) | \leq L_{H_{1}} | ζ |$ , $| b_{2} (y) | \leq L_{b_{2}} y^{- θ}$ ,

$| H_{2} (y, μ, ζ (μ)) | \leq L_{H_{2}} | ζ |$ , where $L_{H_{1}} L_{b_{1}} \leq \frac{ρ Γ (δ)}{3 B (\frac{γ - θ + 1}{ρ}, δ)}$ and $L_{H_{2}} L_{b_{2}} \leq \frac{ρ Γ (δ)}{3 B (\frac{γ - θ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ - θ + 1}}$ .

Then with the conditions $(S_{1})$ – $(S_{3})$ , and with the parameters $δ \in (\frac{1}{2}, 1)$ , $ρ \in (0, 1)$ and $γ > 0$ , Eq. (1.5) has a locally stable solution on $[\hat{y}, + \infty)$ .

Proof

We define a set $U_{θ, 1} = {ζ (y) : ζ (y) \in D$ and $| ζ (y) | \leq y^{- θ}$ , for $y \geq \hat{y} > 0}$ .

It is easy to observe that the set $U_{θ, 1}$ is closed, bounded, and convex subset of $D$ .

Now, we need to show that $M$ maps $U_{θ, 1}$ in $U_{θ, 1}$ .

For $y > 0$ , we get $| M (ζ (y)) | \leq | g (y) | + \frac{| b_{1} (y) |}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y, μ, ζ (μ)) | d μ$ $+ \frac{| b_{2} (y) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y, μ, ζ (μ)) | d μ \leq \frac{1}{3} y^{- θ} + \frac{L_{H_{1}} L_{b_{1}} y^{ρ (1 - δ) - γ - 1}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ - θ} d μ$ $+ \frac{L_{H_{2}} L_{b_{2}} y^{- θ}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ - θ} d μ \leq \frac{1}{3} y^{- θ} + \frac{L_{H_{1}} L_{b_{1}} y^{- θ}}{ρ Γ (δ)} B (\frac{γ - θ + 1}{ρ}, δ)$ $+ \frac{L_{H_{2}} L_{b_{2}} y^{- θ}}{ρ Γ (δ)} B (\frac{γ - θ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ - θ + 1} \leq y^{- θ} .$ Then $M (U_{θ, 1}) \subset U_{θ, 1}$ .

The remaining part of the proof is the same as the proof of Theorem 4.3, it is simple to complete. □

Theorem 4.5

Assume that, $| g (y) | \leq \frac{1}{3} y^{- 1}$ , $y \in [\hat{y}, + \infty)$ , $\hat{y} > 0$ , and there are two constants $N_{b_{1}}, N_{b_{2}} > 0$ such that $| b_{1} (y) | \leq N_{b_{1}} y^{- θ - 1}$ , $| H_{1} (y, μ, ζ (μ)) | \leq e^{- δ μ} | ζ |$ , $| b_{2} (y) | \leq N_{b_{2}} y^{- 1}$ ,

$| H_{2} (y, μ, ζ (μ)) | \leq e^{- δ μ} | ζ |$ , where $N_{b_{1}} = \frac{ρ Γ (δ) {(δ q)}^{\frac{1}{q}}}{3 B (\frac{p (γ - 1) + 1}{ρ}, p (δ - 1) + 1)}$ and $N_{b_{2}} = \frac{ρ Γ (δ) {(δ q)}^{\frac{1}{q}}}{3 B (\frac{p (γ - 1) + 1}{ρ}, p (δ - 1) + 1) β^{θ}}$ ,

$θ = p [ρ (δ - 1) + γ - 1] + 1$ and $\frac{1}{p} + \frac{1}{q} = 1$ .

Then with the conditions $(S_{1}) - (S_{3})$ , and with the parameters $δ \in (\frac{1}{2}, 1)$ , $ρ \in (0, 1)$ and $γ > 0$ , Eq. (1.5) has a locally stable solution on $[\hat{y}, + \infty)$ .

Proof

We define a set $U_{1, 1} = {ζ (y) : ζ (y) \in D$ and $| ζ (y) | \leq y^{- 1}$ , for $y \geq \hat{y} > 0}$ .

It is easy to observe that the set $U_{1, 1}$ is closed, bounded, and convex subset of $D$ .

Now, we need to show that $M$ maps $U_{1, 1}$ in $U_{1, 1}$ .

For $y > 0$ , we obtain $| M (ζ (y)) | \leq | g (y) | + \frac{| b_{1} (y) |}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{1} (y, μ, ζ (μ)) | d μ + \frac{| b_{2} (y) |}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} | H_{2} (y, μ, ζ (μ)) | d μ \leq \frac{1}{3} y^{- 1} + \frac{N_{b_{1}} y^{- θ - 1}}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ - 1} e^{- δ μ} d μ + \frac{N_{b_{2}} y^{- 1}}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ - 1} e^{- δ μ} d μ \leq \frac{1}{3} y^{- 1} + \frac{N_{b_{1}} y^{- θ - 1}}{Γ (δ)} {(\int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{p (δ - 1)} μ^{p (γ - 1)} d μ)}^{\frac{1}{p}} \times {(\int_{0}^{y} e^{- δ q μ} d μ)}^{\frac{1}{q}} + \frac{N_{b_{2}} y^{- 1}}{Γ (δ)} {(\int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{p (δ - 1)} μ^{p (γ - 1)} d μ)}^{\frac{1}{p}} {(\int_{0}^{β} e^{- δ q μ} d μ)}^{\frac{1}{q}},$ i.e., $| M (ζ (y)) | \leq \frac{1}{3} y^{- 1} + \frac{N_{b_{1}} y^{- θ - 1} y^{θ}}{ρ Γ (δ)} B (\frac{p (γ - 1) + 1}{ρ}, p (δ - 1) + 1) \frac{1}{{(δ q)}^{\frac{1}{q}}}$ $+ \frac{N_{b_{2}} y^{- 1} β^{θ}}{ρ Γ (δ)} B (\frac{p (γ - 1) + 1}{ρ}, p (δ - 1) + 1) \frac{1}{{(δ q)}^{\frac{1}{q}}} \leq y^{- 1} .$ Then $M (U_{1, 1}) \subset U_{1, 1}$ .

The remaining part of the proof is the same as the proof of Theorem 4.3, it is simple to complete. □

5 Examples

Three examples are given in this section to interpret our established results.

Example 5.1

Consider the NVFIE with the E–K fractional integral operator as

(5.8)

ζ (y) = y^{2} + y + \frac{sin (y)}{Γ (\frac{4}{5})} \int_{0}^{y} {(y^{\frac{1}{2}} - μ^{\frac{1}{2}})}^{- \frac{1}{5}} μ^{\frac{5}{2}} H_{1} (y, μ, ζ (μ)) d μ + \frac{1}{Γ (\frac{4}{5})} \int_{0}^{1} {(1 - μ^{\frac{1}{2}})}^{- \frac{1}{5}} μ^{\frac{5}{2}} H_{2} (y, μ, ζ (μ)) d μ,

with

H_{1} (y, μ, ζ (μ)) = sin (y μ) sin (ζ (μ))

H_{2} (y, μ, ζ (μ)) = sin (μ) cos (ζ (μ))

y \in [0, 1]

Comparing Eq. (5.8) with Eq. (1.5), we get $δ = \frac{4}{5}$ , $ρ = \frac{1}{2}$ , $γ = \frac{5}{2}$ , $b_{2} (y) = 1$ , $b_{1} (y) = sin (y)$ , $g (y) = y^{2} + y$ , $β = 1$ .

Then, $| g (y) | = | y^{2} + y | \leq 2$ , $| b_{1} (y) | = | sin (y) | \leq 1$ , $| b_{2} (y) | = 1$ .

As $| H_{2} (y, μ, ζ_{1}) - H_{2} (y, μ, ζ_{2}) | = | sin (μ) cos (ζ_{1}) - sin (μ) cos (ζ_{2}) | \leq | ζ_{1} - ζ_{2} |$ ,

and $| H_{1} (y, μ, ζ_{1}) - H_{1} (y, μ, ζ_{2}) | = | sin (y μ) sin (ζ_{1}) - sin (y μ) sin (ζ_{2}) | \leq | ζ_{1} - ζ_{2} |$ .

Also, $| H_{2} (y, μ, ζ (μ)) | = | sin (μ) cos (ζ (μ)) | \leq 1$ , and $| H_{1} (y, μ, ζ (μ)) | = | sin (y μ) sin (ζ (μ)) | \leq 1$ .

Therefore, assumptions $(A_{1}) - (A_{4})$ are satisfied with $G = 2$ , $B_{1} = 1$ , $B_{2} = 1$ , $G_{H_{1}} = 1$ , $G_{H_{2}} = 1$ , $N_{1} = 1$ , $N_{2} = 1$ .

As a result, all the requirements of Theorem 3.1 are satisfied for Eq. (5.8). Hence, we can say that Eq. (5.8) has at least one solution defined on $[0, 1]$ .

Example 5.2

Consider the NVFIE with the E–K fractional integral operator as

(5.9)

ζ (y) = \frac{y^{2}}{2} + \frac{y}{Γ (\frac{4}{5})} \int_{0}^{y} {(y^{\frac{1}{2}} - μ^{\frac{1}{2}})}^{- \frac{1}{5}} μ^{\frac{5}{2}} H_{1} (y, μ, ζ (μ)) d μ + \frac{cos (y)}{Γ (\frac{4}{5})} \int_{0}^{1} {(1 - μ^{\frac{1}{2}})}^{- \frac{1}{5}} μ^{\frac{5}{2}} H_{2} (y, μ, ζ (μ)) d μ,

with

H_{1} (y, μ, ζ (μ)) = sin (y μ) sin (ζ (μ))

H_{2} (y, μ, ζ (μ)) = \frac{| ζ (μ) |}{{(μ + 8)}^{2} (1 + | ζ (μ) |)}

y \in [0, 1]

Comparing Eq. (5.9) with Eq. (1.5), we get $δ = \frac{4}{5}$ , $ρ = \frac{1}{2}$ , $γ = \frac{5}{2}$ , $g (y) = \frac{y^{2}}{2}$ , $b_{2} (y) = cos (y)$ , $b_{1} (y) = y$ , $β = 1$ .

Then, $| g (y) | = | \frac{y^{2}}{2} | \leq \frac{1}{2}$ , $| b_{2} (y) | = | cos (y) | \leq 1$ , $| b_{1} (y) | = | y | \leq 1$ .

As $| H_{1} (y, μ, ζ_{1}) - H_{1} (y, μ, ζ_{2}) | = | sin (y μ) sin (ζ_{1}) - sin (y μ) sin (ζ_{2}) | \leq | ζ_{1} - ζ_{2} |$ ,

and $| H_{2} (y, μ, ζ_{1}) - H_{2} (y, μ, ζ_{2}) | \leq \frac{1}{64} | ζ_{1} - ζ_{2} |$ .

Therefore, assumptions $(A_{1}) - (A_{3})$ are satisfied with $G = \frac{1}{2}$ , $B_{1} = 1$ , $B_{2} = 1$ , $G_{H_{1}} = 1$ and $G_{H_{2}} = \frac{1}{64}$ . Then $G = max {G_{H_{1}}, G_{H_{2}}} = 1$ and $0 < G < \frac{ρ Γ (δ)}{(B_{1} + B_{2}) B (\frac{γ + 1}{ρ}, δ) β^{ρ (δ - 1) + γ + 1}} = 1.1725$ .

As a result, all the requirements of Theorem 3.2 are satisfied for Eq. (5.9). Hence, we can say that Eq. (5.9) has a unique solution on $[0, 1]$ .

Also for this equation, we can apply Theorems 4.1 and 4.2 to analyze the corresponding H–U–R and H–U stability.

Example 5.3

Consider the NVFIE with the E–K fractional integral operator as

(5.10)

ζ (y) = \frac{y^{- 5}}{4 Γ (\frac{4}{5})} \int_{0}^{y} {(y^{\frac{1}{2}} - μ^{\frac{1}{2}})}^{- \frac{1}{5}} μ^{\frac{5}{2}} H_{1} (y, μ, ζ (μ)) d μ + \frac{y^{- \frac{8}{5}}}{4 Γ (\frac{4}{5})} \int_{0}^{1} {(1 - μ^{\frac{1}{2}})}^{- \frac{1}{5}} μ^{\frac{5}{2}} H_{2} (y, μ, ζ (μ)) d μ,

where

H_{1} (y, μ, ζ (μ)) = sin (μ + ζ (μ))

H_{2} (y, μ, ζ (μ)) = cos (ζ (μ))

y \geq \hat{y} > 0

Comparing Eq. (5.10) with Eq. (1.5), we get $δ = \frac{4}{5}$ , $ρ = \frac{1}{2}$ , $γ = \frac{5}{2}$ , $b_{2} (y) = \frac{y^{- \frac{8}{5}}}{4}$ , $b_{1} (y) = \frac{y^{- 5}}{4}$ , $g (y) = 0$ .

Then, $| g (y) + \frac{b_{1} (y)}{Γ (δ)} \int_{0}^{y} {(y^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{1} (y, μ, ζ (μ)) d μ + \frac{b_{2} (y)}{Γ (δ)} \int_{0}^{β} {(β^{ρ} - μ^{ρ})}^{δ - 1} μ^{γ} H_{2} (y, μ, ζ (μ)) d μ | \leq \frac{y^{- 5}}{4 Γ (\frac{4}{5})} \int_{0}^{y} {(y^{\frac{1}{2}} - μ^{\frac{1}{2}})}^{- \frac{1}{5}} μ^{\frac{5}{2}} d μ + \frac{y^{- \frac{8}{5}}}{4 Γ (\frac{4}{5})} \int_{0}^{1} {(1 - μ^{\frac{1}{2}})}^{- \frac{1}{5}} μ^{\frac{5}{2}} d μ = \frac{y^{- \frac{8}{5}}}{2 Γ (\frac{4}{5})} B (7, \frac{4}{5}) + \frac{y^{- \frac{8}{5}}}{2 Γ (\frac{4}{5})} B (7, \frac{4}{5}) = \frac{y^{- \frac{8}{5}}}{Γ (\frac{4}{5})} B (7, \frac{4}{5}) = \frac{Γ (7)}{Γ (\frac{39}{5})} y^{- \frac{8}{5}} .$ As a result, all the requirements of Theorem 4.3 are satisfied for Eq. (5.10). Hence, we can say that Eq. (5.10) has a solution which is locally stable.

6 Conclusions and future work

In this study, we investigated existence and uniqueness of solutions for the NVFIE given in Eq. (1.5), by using Leray–Schauder alternative and Banach’s fixed point theorem, we also analyzed H–U and H–U–R stability in the space $C ([0, β], R)$ . Moreover, three different solutions sets has been constructed and under some ideal conditions, local stability of solutions has been obtained. Also, to achieve our aim, we have been chosen the parameters as $δ \in (\frac{1}{2}, 1)$ , $ρ \in (0, 1)$ and $γ > 0$ , for Eq. (1.5).

In the future, more results can be investigated, such as local stability results on some different solutions sets by assuming different conditions. Moreover, one can investigate the above results for the quadratic Volterra–Fredholm integral equations involving more generalized integral operator.

Declaration of competing interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

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Introduction

Notations, definitions and auxiliary facts

Existence and uniqueness of solutions

Stability of solutions

Examples

Conclusions and future work

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[1] Akkouchi M., . Hyers-Ulam-Rassias stability of nonlinear Volterra integral equations via a fixed point approach. Acta Univ. Apulensis Math. Inform.. 2011;26:257-266.
[Google Scholar]

[2] Alamo J.A., Rodrıguez J., . Operational calculus for modified Erdélyi-Kober operators. Serdica Bulg. Math. Publ.. 1994;20:351-363.
[Google Scholar]

[3] Ali Z., Zada A., Shah K., . Ulam stability results for the solutions of nonlinear implicit fractional order differential equations. Hacet. J. Math. Stat.. 2019;48(4):1092-1109.
[Google Scholar]

[4] Amin R., Alrabaiah H., Mahariq I., Zeb A., . Theoretical and computational results for mixed type Volterra-Fredholm fractional integral equations. Fractals. 2022;30(1):2240035
[Google Scholar]

[5] Amin R., Senu N., Hafeez M.B., Arshad N.I., Ahmadian A., Salahshour S., Sumelka W., . A computational algorithm for the numerical solution of nonlinear fractional integral equations. Fractals. 2022;30(1):2240030
[Google Scholar]

[6] Banach S., . Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math.. 1922;3:133-181.
[Google Scholar]

[7] Deimling K., . Nonlinear Functional Analysis. Springer-Verlag; 1985.

[8] Ganji D.D., . The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Phys. Lett. A. 2006;355:337-341.
[Google Scholar]

[9] He J.H., . Application of homotopy perturbation method to non linear wave equations. Chaos Solitons Fractls. 2005;26:695-700.
[Google Scholar]

[10] Hilfer R., . Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000.

[11] Kilbas A.A., Srivastava H.M., Trujillo J.J., . Theory and Applications of Fractional Differential Equations. Vol. 204. Elsevier: Amsterdam, The Netherlands; 2006.

[12] Kiryakova V., . Generalized fractional calculus and applications. In: Pitman Research Notes in Math. Vol. 301. N. York: Longman, Harlow - J. Wiley; 1994.
[Google Scholar]

[13] Kumam P., Ali A., Shah K., Khan R.A., . Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations. J. Nonlinear Sci. Appl.. 2017;10:2986-2997.
[Google Scholar]

[14] Lakshmikantham V., Leela S., Devi J.V., . Theory of Fractional Dynamic Systems. Cambridge: Cambridge Academic Publishers; 2009.

[15] Liu G.R., Gu Y.T., . A point interpolation method for tow-dimensional solids. Internat. J. Numer. Methods Engrg.. 2001;50(4):937-951.
[Google Scholar]

[16] Ma Q.H., Pečarić J., . Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations. J. Math. Anal. Appl.. 2008;341(2):894-905.
[Google Scholar]

[17] Mainardi F., . Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996). In: CISM Courses and Lecture Notes. Vol vol. 378. Wien: Springer Verlag; 1997. p. :291-348.
[Google Scholar]

[18] Marzban H.R., . An accurate method for fractional optimal control problems governed by nonlinear multi-delay systems. J. Vib. Control. 2023;29(3–4):820-843.
[Google Scholar]

[19] Marzban H.R., . Optimal control of nonlinear fractional order delay systems governed by Fredholm integral equations based on a new fractional derivative operator. ISA Trans.. 2023;133:233-247.
[Google Scholar]

[20] Marzban H.R., Ashani M.R., . A class of nonlinear optimal control problems governed by Fredholm integro-differential equations with delay. Internat. J. Control. 2020;93(9):2199-2211.
[Google Scholar]

[21] Marzban H.R., Korooyeh S.S., . Optimal control of linear fractional-order delay systems with a piecewise constant order based on a generalized fractional Chebyshev basis. J. Vib. Control 2022
[Google Scholar]

[22] Marzban H.R., Nezami A., . Analysis of nonlinear fractional optimal control systems described by delay Volterra-Fredholm integral equations via a new spectral collocation method. Chaos Solit. Fractals. 2022;162:112499
[Google Scholar]

[23] Morales J., Rojas E.M., . Hyers-Ulam and Hyers-Ulam-Rassias stability of nonlinear integral equations with delay. Int. J. Nonlinear Anal. Appl.. 2011;2(2):1-7.
[Google Scholar]

[24] Nwaigwe C., . Solvability and approximation of nonlinear functional mixed Volterra-Fredholm equation in Banach space. J. Integral Equations Appl.. 2022;34(4):489-500.
[Google Scholar]

[25] Nwaigwe C., Benedict D.N., . Generalized Banach fixed-point theorem and numerical discretization for nonlinear Volterra-Fredholm equations. J. Comput. Appl. Math.. 2023;425:115019
[Google Scholar]

[26] Nwaigwe C., Micula S., . Fast and accurate numerical algorithm with performance assessment for nonlinear functional Volterra equations. Fractal Fract.. 2023;7(4):333.
[Google Scholar]

[27] Nwaigwe C., Weli A., Thanh D.N.H., . Fourth-order ttrapezoid algorithm with four iterative schemes for nonlinear integral equations. Lobachevskii J. Math.. 2023;44(7):2817-2832.
[Google Scholar]

[28] Pagnini G., . Erdélyi-Kober fractional diffusion. Fract. Calc. Appl. Anal.. 2012;15:117-127.
[Google Scholar]

[29] Paul S.K., Mishra L.N., Mishra V.N., Baleanu D., . An effective method for solving nonlinear integral equations involving the Riemann–Liouville fractional operator. AIMS Math.. 2023;8(8):17448-17469.
[Google Scholar]

[30] Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I., 1981. Integrals and Series. In: Elementary Functions [in Russian]. Nauka, Moscow.

[31] Rahimkhani P., Ordokhani Y., . Chelyshkov least squares support vector regression for nonlinear stochastic differential equations by variable fractional Brownian motion. Chaos Solit. Fractals. 2022;163:112570
[Google Scholar]

[32] Rahimkhani P., Ordokhani Y., . Performance of Genocchi wavelet neural networks and least squares support vector regression for solving different kinds of differential equations. Comput. Appl. Math.. 2023;42:71.
[Google Scholar]

[33] Rahman M., . Integral Equations and their Applications. WIT Press; 2007.

[34] Subramanian M., Duraisamy P., Kamaleshwari C., Unyong B., Vadivel R., . Existence and U-H stability results for nonlinear coupled fractional differential equations with boundary conditions involving Riemann–Liouville and Erdélyi-Kober integrals. Fractal Fract.. 2022;6:266.
[Google Scholar]

[35] Wang J., Dong X., Zhou Y., . Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator. Commun. Nonlinear Sci. Numer. Simul.. 2012;17:3129-3139.
[Google Scholar]

[36] Wazwaz A.M., . Linear and Nonlinear Integral Equations: Methods and Applications. Beijing and Springer-Verlag Berlin: Higher Education Press; 2011.

Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator

Abstract

Keywords

26A33

45G10

45M10

47H10

Erdélyi–Kober fractional integral operator

Hyers–Ulam–Rassias stability

Hyers–Ulam stability

Local stability

Fixed point theorem

1 Introduction

2 Notations, definitions and auxiliary facts

Theorem 2.5 Leray–Schauder alternative (Subramanian et al., 2022)

Theorem 2.6 Arzel a ̀ -Ascoli theorem (Subramanian et al., 2022)

Theorem 2.7 Banach’s fixed point theorem (Banach, 1922)

3 Existence and uniqueness of solutions

4 Stability of solutions

5 Examples

6 Conclusions and future work

Declaration of competing interest

References

Suggested read for related articles:

Theorem 2.6 Arzel $\overset{̀}{a}$ -Ascoli theorem (Subramanian et al., 2022)