Analysis of the solvability and stability of the operator-valued Fredholm integral equation in Hölder space

1.   Introduction

Integral equations (IEs) have expansive applications in different areas of science and engineering. There are several problems in science and technology related to IEs. With the help of IEs, we can describe numerous events that arise in real life problems, e.g., problems in the theory of radiative transfer, the theory of neutron transport, and the kinetic theory of gases can be addressed by using the famous quadratic IE of Chandrasekhar type

where $\chi$ is a continuous function defined on ${\bf J};$ see ^[15,19,25]. Many researchers have examined a similar form of the above mentioned IE.

The integral equation:

studied by Banaś and Nalepa ^[5]. They discussed the space of functions with growths tempered by a modulus of continuity; they also proved a sufficient condition for relative compactness. As an example of the mentioned space they discussed Hölder space and some properties regarding the space. At the end, they proved the existence theorem for the Fredholm IE in Hölder space by using the classical Schauder fixed point theorem (FPT) and added an example to illustrate their result.

The following IE of Fredholm type has been studied by Caballero et al. ^[11]

and they added an example to illustrate their result.

The same year, Caballero et al. ^[12] investigated the existence of solutions of the equation

by using the classical Schauder FPT, and they added an example to illustrate their result.

Ersoy and Furkan ^[20] examined the existence of solutions of the equation

in Hölder space by using the classical Schauder FPT. They added some example to illustrate their result.

After that, similar types of equations have been examined by several authors; for references, see ^{[9,10,21,22,33]}.

The objective of this paper is to discuss the existence and uniqueness of a solution, as well as the stability analysis of the nonlinear IE of Fredholm type

for $\zeta_1\in{\bf J}.$ Notice that, Eq (1.6) is more general than many equations considered up to now and we obtain Eqs (1.1)–(1.5) as a special case of Eq (1.6) by using appropriate values. In Section 3, Example 1, we can see that Eq (1.1) is a particular case of Eq (1.6). For $(T_1w)(\tau) = w(\tau)$ and $z(\zeta_1) = 0$ with $a = 0$ and $b = 1$, Eq (1.2) is a particular case of Eq (1.6). If we set $(T_1w)(\tau) = w(r(\tau))$ and $z(\zeta_1) = 0$, then Eq (1.3) becomes the particular case of Eq (1.6). For $(T_1w)(\tau) = \max\limits_{\tau\in [0, r(\tau)]}\lvert w(\tau)\rvert$ and $z(\zeta_1) = 0$, Eq (1.4) is a particular case of Eq (1.6). If we set $z(\zeta_1) = 0$, the Eq (1.5) is a particular case of Eq (1.6). We discuss all of the particular cases as corollaries in Section 3.

Due to immense development in analysis and its application, the branch of nonlinear differential and IEs has motivated the researchers to find new dimensions for its effective analysis. IEs of the type illustrated by Eq (1.6) are often applicable in traffic theory, oscillating magnetic fields and electromagnetic and mathematical physics.

It is worthwhile to mention that more general functional IEs are analyzed by using Darbo's general theorem in Banach algebra ^[7]. An improved version of Darbo's FPT for the product of two operators as applied in conjunction with measures of noncompactness (MNCs), is proved in ^[6] and known as Darbo's general theorem. Darbo's general theorem for Banach algebra is a generalization of many FPTs considered up to now. In ^[8], Banas and Olszoy proved a FPT as applied in conjunction with MNCs, for the product of two operators, and they studied the monotonic solutions of a functional IE of fractional order in Banach algebra. In ^[16], M. Cichoń and Metwali discussed a FPT for the product of nonlinear operators and they extended it to some function spaces which are not necesarily Banach algebras. Recently, K. Cichoń et al. in ^[17], mainly studied the existence of the FPT on some functional problems associated with bilinear operators. In the year of $2022$, M. Cichoń and Metwali ^[18] introduced integral-variation type Hölder space which is also a Banach algebra. They discussed norms and MNCs in the mentioned space and proved the existence theorem for solutions of quadratic IEs by using the Riemann-Liouville fractional operator.

2.   Preliminaries and basic results

Throughout the study, we use the following:

$E$ = a Banach space.

$\mathcal{M}_E$ = family of all nonempty and bounded subsets of $E.$

$\mathcal{N}_E$ = subfamily of relatively compact sets of $E$.

$C[\upsilon_1, \upsilon_2]$ = space of continuous functions on $[\upsilon_1, \upsilon_2]$.

$C_\omega[\upsilon_1, \upsilon_2]$ = space of functions with tempered increments on $[\upsilon_1, \upsilon_2]$; see ^[5].

The space satisfying the Hölder condition is an example of $C_\omega[\upsilon_1, \upsilon_2]$, which is provided in Example 2 of ^[5]. Denote $\mathcal{H}_\eta[\upsilon_1, \upsilon_2]$ by the space satisfying the Hölder condition

The least possible constant $K_w> 0$ satisfies the above inequality and is given by

Further, $\mathcal{H}_\eta[\upsilon_1, \upsilon_2]$ is a Banach space with the norm

Also, the inequalities

$0< \eta< \gamma\leq 1,$ hold; see ^[5]. Now, the sufficient condition for relative compactness which has been mentioned in Example 6 of ^[5] in the space $\mathcal{H}_\eta[\upsilon_1, \upsilon_2]$ with $0< \eta\leq 1$ is given as follows. It is noteworthy to mention here that this sufficient condition is one of the important results of this research work.

Theorem 1. ^[5] If $U$ is a bounded subset of $\mathcal{H}_\gamma[\upsilon_1, \upsilon_2],$ then $U$ is relatively compact on $\mathcal{H}_\eta[\upsilon_1, \upsilon_2]$, $0< \eta< \gamma\leq 1$.

Definition 1. ^[3] A MNC is a function $a:\mathcal{M}_E\rightarrow [0, \infty)$ that satisfies the following conditions:

$({\rm{i}})$ The family $\operatorname{ker} a = \{\mathcal{W}\in\mathcal{M}_E:a(\mathcal{W}) = 0\}\ne \emptyset$ and $\operatorname{ker} a\subset\mathcal{N}_E.$

$({\rm{ii}})$ $\mathcal{W}\subset \mathcal{V}\implies a(\mathcal{W})\leq a(\mathcal{V}).$

$({\rm{iii}})$ $a(\bar{\mathcal{W}}) = a(\mathcal{W}).$

$({\rm{iv}})$ $a(\operatorname{Conv}\mathcal{W}) = a(\mathcal{W}).$

$({\rm{v}})$ $a(\lambda_1 \mathcal{W}+(1-\lambda_1)\mathcal{V})\leq\lambda_1 a(\mathcal{W})+(1-\lambda_1) a(\mathcal{V})$ for $\lambda_1\in{\bf J}.$

$({\rm{vi}})$ If $\mathcal{W}_n\in\mathcal{M}_E, \, \mathcal{W}_n = \bar{\mathcal{W}}_n, \, \mathcal{W}_{n+1}\subset \mathcal{W}_n$ $\forall n = 1, 2, \dots$ and $\lim\limits_{n\to\infty}a(\mathcal{W}_n) = 0,$ then $\bigcap\limits_{n = 1}^\infty \mathcal{W}_n\ne \emptyset.$

Also, $a$ is sub-linear whenever it satisfies the following:

$({\rm{vii}})$ $a(\lambda_1 \mathcal{W}) = |{\lambda_1}|a(\mathcal{W})$ for $\lambda_1\in\mathbb{R}.$

$({\rm{viii}})$ $a(\mathcal{W}+\mathcal{V})\leq a(\mathcal{W})+ a(\mathcal{V}).$

Further, $a$ has the maximum property if it satisfies

$({\rm{ix}})$ $a(\mathcal{W}\cup \mathcal{V}) = \max\{a(\mathcal{W}), a(\mathcal{V})\}$.

If ker $a = \mathcal{N}_E,$ then $a$ is called full.

Now, we state Darbo's FPT ^[3], which is a generalization of the Schauder FPT and Banach FPT.

Theorem 2. Let the mapping $G:\Theta\rightarrow \Theta$ be continuous and $\Theta$ be a nonempty, closed, bounded, and convex subset of $E$. If $a (GW)\leq k a(W)$, $k\in [0, 1)$, for any nonempty subset $W$ of $\Theta,$ then $G$ has at least one fixed point in $\Theta$.

Define

It is worth mentioning that, Banaś and Nalepa in ^[4] introduced MNCs for Hölder space and applied them to Hölder space for the first time.

Theorem 3. ^[4] The function ${\bf f}_0:\mathcal{M}_{C_\omega([\upsilon_1, \upsilon_2])}\to [0, \infty)$ is a sublinear MNC with its maximum property in $C_\omega([\upsilon_1, \upsilon_2])$.

Next, let us define a contraction mapping for any normed space $E.$

Definition 2. ^[29] A mapping $F:E\to E$ is said to be a contraction if there is a positive real number $\mathcal{C}_3 < 1$ such that $\|Fw-Fv\|\leq\mathcal{C}_3\|w-v\|,$ $\forall \; w, v\in E.$

Theorem 4. [1,2, Banach FPT] Let $F$ be a contraction mapping on a Banach space $E$; then, $F$ has a unique fixed point in $E.$

3.   Exsistence of solution

To study Eq (1.6), the following assumptions are required:

$({\rm{i}})$ For $y\in \mathcal{H}_\gamma({\bf J})$ there exists $Y_\gamma> 0$ such that

$({\rm{ii}})$ $\varpi:{\bf J}\times{\bf J}\to\mathbb{R}$ is continuous and satisfies the Hölder condition with the exponent $\gamma$. Also,

$({\rm{iii}})$ The function $z\in \mathcal{H}_\gamma({\bf J})$ satisfies the inequality

where $\mathcal{Z}_\gamma > 0.$

$({\rm{iv}})$ $T_1, T_2: \mathcal{H}_\gamma({\bf J})\to C({\bf J})$ are continuous operators on $\mathcal{H}_\gamma({\bf J})$ and there exist increasing functions $h_i: \mathbb{R}_+\to\mathbb{R}_+$ such that

$({\rm{v}})$ $h:{\bf J}\times\mathbb{R}\to \mathbb{R}$ is a continuous function such that there exists an increasing continuous function $H:[0, \infty)\to [0, \infty)$ that satisfy the condition that $H(0) = 0$ and the following holds:

Denote $\bar{\Gamma} = \sup \{\lvert \varpi(\zeta_1, \tau)\rvert: \zeta_1, \tau\in {\bf J}\} \;\mbox{ and}\; \bar{\Omega} = \sup \{\lvert h(\zeta_1, 0)\rvert: \zeta_1\in {\bf J}\}.$

$({\rm{vi}})$There exists an $r_0> 0$ that is a solution to the inequality.

such that $\bar{\Gamma}h_1(r_0)< 1.$

Theorem 5. Under the assumptions (i)–(vi), there is at least one solution of Eq (1.6) in the space $\mathcal{H}_\eta({\bf J})$ with $0 < \eta < \gamma\leq 1$.

Proof. Define an operator $G$ on $\mathcal{H}_\eta({\bf J})$ by

where $w\in \mathcal{H}_\eta({\bf J})$.

Choose $\zeta_1, \zeta_2\in {\bf J}$ with $\zeta_1\neq \zeta_2$. Claim that $Gw\in \mathcal{H}_\eta({\bf J})$. By assumption, we obtain

Hence, we get

Also,

From Eqs (3.2) and (3.3) we obtain the following:

Thus, we get

This proves that the operator $G$ transforms $\mathcal{H}_\eta({\bf J})$ to itself. Since the positive number $r_0$ is the solution of the inequality given in hypothesis $(vi)$, the following inequality holds:

As a result, it follows that $G$ transforms the ball

into itself. Now, we will prove that the operator $G$ is continuous on $B_{r_0}(\subset \mathcal{H}_\eta({\bf J})).$ Suppose that $w\in B_{r_0}$ and $\delta> 0$. Assume that $v\in B_{r_0}$ in such a way that $\left\|w-v\right\|_\eta\leq\delta$ which yields the following:

Hence,

In a similar way,

Hence,

Combining Eqs (3.5) and (3.6), we get

Since $\left\|w\right\|_\eta\leq r_0$, $\left\|v\right\|_\eta\leq r_0$ and $h_i\quad (i = 1, 2)$ is non-decreasing, $h_i(\left\|w\right\|_\eta)\leq h_i(r_0).$ Since $T_i:\mathcal{H}_\gamma({\bf J})\to C({\bf J})$ are continuous operators with respect to the norm $\left\|.\right\|_\eta$, they are also continuous at the point $v\in B_{r_0}.$ Let $\epsilon > 0$ be arbitrary, then there exists $\delta > 0$ such that

for all $w\in B_{r_0}$ with $\left\|w-v\right\|_\eta < \delta.$

Taking into account the continuity of $H$ with $H(0) = 0$, we infer that the operator $F$ is continuous at the point $v\in B_{r_0}.$ This proves that the operator $G$ is continuous on the ball $B_{r_0}$ with respect to the norm $\left\|.\right\|_\eta.$

Assume a non-empty set $W\subseteq B_{r_0}$, for $\epsilon> 0$ and a function $w\in W$. For $\left|\zeta_1-\zeta_2\right|\leq\epsilon$, in the context of Eq (3.2), now we will get

Hence

This implies

Thus, on the basis of assumption $(vi)$ and Darbo's FPT as given by Theorem 2, we deduce that the operator $G$ has at least one fixed point $w\in B_{r_0}\subset \mathcal{H}_\eta({\bf J}).$ Obviously, $w = w(\zeta_1)$ is a solution of Eq (1.6). Now the proof is complete. □

To prove the efficiency of the above result, we consider the following example.

Example 1. Consider the following nonlinear Chandrasekhar integral equation:

where the function $\chi: {\bf J}\to \mathbb{R}$ is continuous and such that $\chi(0) = 0.$ The Eq (3.6) is a special case of Eq (1.6), if we have the following:

and

Let us choose the function $\chi: {\bf J}\to\mathbb{R}$ as $\chi(\tau) = \frac{\tau}{6}.$ Now, $\lvert y(\zeta_1)-y(\zeta_2)\rvert = 0$ implies that $Y_\gamma = 0, |y(0)| = 1.$ and $\lvert z(\zeta_1)-z(\zeta_2)\rvert = 0$ implies that $\mathcal{Z}_\gamma = 0, \, \, |z(0)| = 0.$ It can be shown that the function $k:{\bf J}\times{\bf J}\to\mathbb{R}$ is defined by Eq.(3.8) and is continuous. If $\zeta_1, \zeta_2\in{\bf J}\, \, {{\rm{and}}}\, \, \tau = 0,$ then

If $\zeta_1, \zeta_2\in{\bf J}\, \, {\rm{and}}\, \, \tau\neq 0,$ then

Therefore $\Gamma_1 = \frac{1}{6}.$ Also $||T_1(w)||_\gamma = ||w||_\gamma$ implies that $h_1(r) = r.$ Clearly $H(r) = 0, \, \, h_2(r_0) = 0.$

The first inequality of assumption (vi) has the form

Therefore, $r_1 = 1.98743,$ $r_2 = 2.01272.$ It is easy to see that, the above inequality and the second inequality of assumption $(vi)$ are satisfied for $r\in[1.98743, 2.01272]$. Therefore, by using Theorem 5, we get that Eq (3.6) has at least one solution in $\mathcal{H}_\eta({\bf J})$ for $\eta\in{\bf J}$.

Example 2. Consider the following nonlinear integral equation of Fredholm type

Comparing Eq (3.9) with Eq (1.6), we have

Now,

Hence, we obtain that $Y_1 = \frac{1}{4}$, $\Gamma_1 = \frac{1}{2}$, $Z_1 = 1$ and $H(r) = \frac{r}{2}.$ Also

Therefore, $H(r) = \frac{r}{2}.$ Also,

and

Therefore, $h_1(r) = \frac{r}{2}, h_2(r) = \frac{r}{2}$ we get

Also $T_1$ and $T_2$ are continuous with respect to the $\sup$-norm. The first inequality of assumption $(vi)$ has the following form:

Therefore, $r_1 = \frac{1}{2}$ and $r_2 = 1.$ It is easy to see that, the above inequality and the second inequality of assumption $(vi)$ are satisfied for $r\in[\frac{1}{2}, 1]$. Therefore, by using Theorem 5, Eq (3.9) has at least one solution in Hölder space $\mathcal{H}_\eta({\bf J})$ for $\eta\in{\bf J}$.

Corollary 1. If we choose $(T_1w)(\tau) = w(\tau)$ and $z(\zeta_1) = 0,$ then Eq (1.2) as presented in ^[5] is a particular case of Eq (1.6) with $a = 0, b = 1.$

Corollary 2. If we choose $(T_1w)(\tau) = \left\{\max\limits_{\tau\in [0, r(\tau)]}\lvert w(\tau)\rvert\right\}$ and $z(\zeta_1) = 0$ then Eq (1.4) from ^[12] is a particular case of Eq (1.6).

Corollary 3. If we choose $z(\zeta_1) = 0$ then Eq (1.5) from ^[20] is a particular case of Eq (1.6).

4.   Uniqueness of the solution

In this section, we intend to prove the uniqueness of the solution of Eq (1.6) by using the Banach FPT. Here we apply assumptions $(i)$–$(iii)$ of Section 3, along with the following assumptions:

$(iv)^{'}$ $T_i: \mathcal{H}_\gamma({\bf J})\to \mathcal{H}_\gamma({\bf J})$ are continuous operators with respect to the norm $||.||_\eta$ and there exists increasing functions $h_i: \mathbb{R}_+\to\mathbb{R}_+$ such that the following inequality holds:

$(v)^{'}$ $h:{\bf J}\times\mathbb{R}\to \mathbb{R}$ is a continuous function such that there exists $\mathcal{L} > 0$ for which

holds. Also, consider

$(vi)^{'}$ $T_1, T_2: \mathcal{H}_\gamma({\bf J})\to \mathcal{H}_\gamma({\bf J})$ are contraction mappings, more precisely

$(vii)^{'}$ $h_1(R)(2\bar{\Gamma}+\Gamma_\gamma)+R\mathcal{C}_1(2\bar{\Gamma}+\Gamma_\gamma)+\mathcal{L}\mathcal{C}_2(\left|z(0)\right|+\mathcal{Z}_\gamma) < 1.$

Theorem 6. Under the assumptions (i)–(iii) of Section 3 and $(iv)^{'}$–$(vii)^{'}$ of Section 4, the Eq (1.6) has a unique solution in $\mathcal{H}_\eta({\bf J})$, where $0< \eta< \gamma\leq 1$.

Proof. We have defined the operator $G$ in Eq (3.1). To show that $G$ is well-defined. The proof is similar to Theorem 5. We obtain

Next, we shall show the contraction of the operator $G.$ From Theorem 5, we get

We choose $||w||_\eta\leq R,$ where $R\geq 0.$ Thus, $G$ is a contraction mapping. Thus, according to the Banach FPT there exists a unique solution $w$ such that $Gw = w$ of Eq (1.6) under some suitable assumptions. □

We can check the effectiveness of our result by taking the same Example 2 with a minor change in expressions:

Example 3. Consider the nonlinear Fredholm type integral equation

for $\zeta_1\in{\bf J}$.

Comparing Eq (4.2) with Eq (1.6), we have

From Example 2, we obtain that $Y_1 = \frac{1}{6}$, $\Gamma_1 = \frac{1}{2}$, $\bar{\Gamma} = \frac{1}{2},$ $Z_1 = 1$ and $\bar{\Omega} = 0.$

Now,

And

Thus, assumption $(iv)^{'}$ is satisfied with $h_1(R) = 0.32871R$ and $h_2(R) = 0.66666R.$

Again,

Thus, assumption $(v)$ is satisfied with $\mathcal{L} = 0.33333.$

Also,

And

This implies that $T_1$ and $T_2$ satisfy the assumption $(vi)^{'}$ with $\mathcal{C}_1 = 0.32871$ and $\mathcal{C}_2 = 0.66666.$ The inequality in the assumption $(vii)^{'}$

is equivalent to

Thus, the above inequality is true for all $R < 0.78872$ and hence, satisfies all of the assumptions. Hence, Theorem 6 concludes that there exists a unique solution of Eq (4.2).

5.   Hyers-Ulam stability analysis

A functional equation is said to be stable if, for every approximate solution, there is an exact solution near it. In 1940, Ulam ^[35] raised the stability problem of functional equations: Under what condition does there exist a linear mapping near an approximately linear mapping? After 1 year, this problem was solved by Hyers ^[26] for approximately additive mappings in Banach space. In 1978, Rassias ^[34] generalized Hyers result by proving the existence of unique linear mappings near approximate additive mappings. Since then, the stability of functional equations has been extensively investigated by a number of authors. Apart from functional equations, this concept of stability can also be applied to differential equations, integral equations, and integrodifferential equations. In 2007, Jung ^[27] investigated the Hyers-Ulam (H-U) stability of the following Volterra Integral Equation (VIE) using the idea of the fixed point method discussed by Cadariu and Radu ^[13],

In 2009, Li and Hua ^[28] proved the H-U stability of the Banach FPT for a polynomial equation defined on a finite interval. In the same year, Castro and Ramos ^[14] studied the H-U stability of a VIE on an infinite interval and the H-U stability on a finite interval by using the Banach FPT in generalized metric space. In 2010, Gachpazan and Baghani ^[23] generalized the previous result for a finite interval using the successive approximation method for H-U stability of the nonhomogeneous VIE. In 2011, Morales and Rojas ^[30], generalized the work of Gachpazan and Baghani regarding the H-U stability for nonhomogeneous VIEs with delay on a finite interval by using the iterative method. Apart from this, many research papers on the stability of these kinds of integral equations have been examined by several authors; see ^[24,31,32].

We have not found any investigation into any type of integral equation in Hölder space, so we were motivated by these articles to discuss the H-U-R stability of Eq (1.6) in Hölder space. Let us define H-U stability for Eq (1.6).

Definition 3. ^[27] We say that Eq (1.6) has H-U stability if there exists $K\geq 0$ with the following property: For every $\epsilon > 0, \, \, w$ satisfying

then there exists some $v$ satisfying Eq (1.6) such that

We call such $K$ a H-U stability constant.

In this section, we are going to prove that Eq (1.6) has H-U stability. In the previous section, we showed that $G$ is a contraction mapping and Eq (1.6) has a unique solution by using the Banach FPT. This leads to the following theorem:

Theorem 7. Under the assumption of Theorem 6, the equation $(G-I)w = 0,$ defined by Eq (1.6), has H-U stability, that is for $\epsilon\geq 0,$ if

then there exists a unique $v\in \mathcal{H}_\eta({\bf J})$ satisfying

with

for some $K\geq 0.$

Proof. We have defined the operator $G$ in Eq (3.1). According to Theorem 6, operator $G$ is a contraction mapping with a unique solution $w\in H_\eta({\bf J})$ to Eq (1.6). For every $\epsilon > 0,$ if $\|Gw-w\|_\eta\leq\epsilon,$ then

Using Eq (4.1), we obtain

where $K = \frac{1}{1-\mathcal{C}_3}$ and $\mathcal{C}_3 < 1.$ This completes the proof. □

6.   Conclusions

In this work, we discussed the existence of a solution for the Fredholm type IE in the space $\mathcal{H}_\eta({\bf J})$ based on the Darbo FPT. Using the Banach FPT, we proved the uniqueness of the solution in the same space. We have provided examples to verify the effectiveness and applicability of the existence and uniqueness results. Further, another example has been exhibited to show that the Chandrasekhar IE is a special case of our proposed equation. At the end, we checked the stability of the solution by performing H-U stability.

Equation (1.6) can be solved in different types of Banach spaces by using the FPT with suitable assumptions. These kinds of IEs are often applicable to the kinetic theory of gases, traffic theory, oscillating magnetic theory, neutron transport theory, radiative transfer theory and electromagnetic and mathematical physics.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The research of the first author (Manalisha Bhujel) is supported by SERB, DST, New Delhi, India under the INSPIRE code DST/INSPIRE Fellowship/2019/IF190364.

Moreover, we would like to thank the academic editor and anonymous reviewers for taking the necessary time and effort to review the manuscript. We sincerely appreciate all of their valuable comments and suggestions, which have helped us to improve the quality of the manuscript.

Conflict of interest

The authors declare that they do not have any conflict of interest.

Prof. Dimplekumar Chalishajar is the Guest Editor of special issue "Recent advances in differential and partial differential equations and its applications" for AIMS Mathematics. Prof. Dimplekumar Chalishajar was not involved in the editorial review and the decision to publish this article.