On generalization of Petryshyn's fixed point theorem and its application to the product of $n$-nonlinear integral equations

1.   Introduction

Different types of integral equations are crucial to the study of economics, biology, mechanics, mathematical physics, control theory, vehicular traffic, population dynamics and other fields (cf. ^[1,2]).

Recent years have seen some successful attempts to examine the qualitative behavior of solutions for many different types of nonlinear differential or integral equations employing the notion of the measure of noncompactness (M.N.C.) connected to the fixed point approach (F.P.T.) (cf. ^{[3,4,5,6,7,8,9,10]}).

Based on this methodology, we first offer and demonstrate a generalization of Petryshyn's F.P.T. connected with the Hausdorff M.N.C., which is a generalization of numerous F.P.T. types, including Darbo's, Schauder's and traditional Petryshyn's F.P.T.s ^[11]. The benefit of the proposed F.P.T. is that it enables us to skip demonstrating closed, convex and compactness properties of the investigated operators. These enable us to investigate various varieties of differential and integral equations under a weaker and more general set of presumptions.

Second, we employ the presented F.P.T. to solve the product of $n$-nonlinear Volterra integral equations, which are a generalization of the classical and quadratic integral equations of the form

for $n\geq 2$, in the Banach algebra $C(I_a)$.

In particular, for $n = 2, \; f_i(v, z_1, z_2, z_3) = g_i+z_3, \; h_i = l_i(v-s) z_3(s)$, equation (1.1) yields a Gripenberg equation

that has significant applications in biology (SI models, cf. ^[12]).

In ^[13] the authors utilized the F.P.T. approach to establish the existence of $C[a, b]$-solutions of the equation

The authors in ^[14] presented an extension of Darbo F.P.T. in Banach algebra to solve the $q$-integral equation

A generalization of Darbo F.P.T. was used to investigate the existence results for the equation

in ideal spaces (not be Banach algebras) in ^[15] see also ^[16,17,18].

We focus on applying a generalization of Petryshyn's F.P.T. to solve a general form of product-type integral problems in the Banach algebra $C(I_a)$.

2.   Preliminaries

We employ the following symbols in the sequel:

● $\mathbb{E}$: Banach space;

● $\bar{B _{r}}$: A ball of radius $r$ and center at $0$;

● $\partial\bar B_{r}$: Sphere in $E$ with radius $r > 0$ around $0$;

● $C(I_a)$: Space of continuous and real-valued functions on $I_a = [0, a]$;

● (F.P.T.): Fixed point theorem;

● (M.N.C.): Measure of noncompactness.

We recall some theorems & definitions that are required for the sequel.

Definition 2.1. ^[19] Let $Z \subset \mathbb{E}$ be a bounded set, then

is said to be the Kuratowski M.N.C.

Definition 2.2. ^[20] Let $Z \subset \mathbb{E}$ be a bounded set, then

is said to be the Hausdorff M.N.C.

Theorem 2.3. ^[20] For a bounded set $Z\subset \mathbb{E}$, the M.N.C.s $\alpha$ and $\mu$ fulfill

For more information about the properties of the M.N.C. see ^[11,20].

The space $C[0, a]$ yields to a Banach space under the norm $\|z\| = \sup\{|z(v)| : v\in I_a\}$ and we shall write the modulus of continuity of a function $z \in C(I_a)$ as

Theorem 2.4. ^[20] For a bounded set $Z\subset C (I_a)$, the M.N.C. in $C(I_a)$ is denoted by

Definition 2.5. ^[21] Let $P: \mathbb{E} \rightarrow \mathbb{E}$ be a continuous map. $P$ is said to be a contraction map if for all $Z \subset C (I_a)$ be bounded, $P(Z)$ be bounded and

Moreover, $P$ is said to be condensing (densifying) map if

Note that a contraction map yields condensing (densifying) but not vice versa.

Remark 2.6. In $C (I_a)$, the M.N.C. $\mu$ fulfills condition (m) (cf. ^[22]) and its generalization for a finite sequence of bounded sets $\{N_i\}_{i = 1, ..., n}$, $n\geq 2$ (cf. ^[14]) i.e.

3.   Main results

In order to solve Eq (1.1), we first give a fixed point $z \in \bar B_r$ of the problem

where $P_i : \bar B_r \to \mathbb{E}, \; i = 1, \cdots, n, \; n \geq 2$ are known operators.

Definition 2.5 should be rewritten in view of the M.N.C. $\mu$ in $C(I_a)$.

Definition 3.1. The operator $P: C (I_a) \rightarrow C (I_a)$ is said to be a contraction map if for all $Z \subset C (I_a)$ be bounded set, $P(Z)$ be bounded set and

Moreover, $P$ is said to be condensing (densifying) map if

Proof. Since $Z$ and $P(Z)$ are bounded sets in $C (I_a)$ and by using Theorem 2.3, we have

The above inequality with $0 < k < \frac{1}{2}$ finishes the proof. □

Note that a contraction map related to the M.N.C. $\mu$ yields condensing (densifying) with $0 < k < \frac{1}{2}$ but not vice versa.

The following Proposition can be presented and proven by us.

Proposition 3.2. Suppose that the operators $P_i : \bar B_r \to \mathbb{E}, \; i = 1, \cdots n$ and that:

(B1) $P_i$ are continuous on $\bar B_r, \; \; i = 1, \cdots n$.

(B2) There exist $k_i > 0$ such that $P_i$ fulfill:

for arbitrary bounded $Z \subset \mathbb{E}$,

(B3) $\mathbb{K} = \sum_{i = 1}^n k_i \prod_{j = 1, j\neq i}^n \|P_j \bar B_r\| < \frac{1}{2},$

(B4) $P(z) = kz,$ for some $z \in \partial \bar B_r$ then $k\leq 1$,

then the set $Fix(P)$ of fixed points of $P$ in $\bar B_r$ is nonempty.

Proof. Let $\emptyset \neq Z \subset \bar B_r$. By utilizing the above assumptions, we obtain

By using Petryshyn's F.P.T., we have finished. □

Remark 3.3. ● If $n = 1$, Proposition 3.2 reduces to classical Petryshyn's F.P.T. ^[11], which is a generalization of classical Darbo and Schauder F.P.Ts.

● If $n = 2$, Proposition 3.2 reduces to the F.P.T. presented in ^[11,21], which is a generalization of the results presented in ^[22].

● If $n\geq 2$, Proposition 3.2 is a general form of the F.P.T. presented in ^[14,15].

Now, we will apply Proposition 3.2 to check the solvability of Eq. (1.1) under the assumptions:

(A1) Assume that $\alpha_i, \beta_i, \gamma_i: I_a\rightarrow I_a$ and $\varphi_i: I_a\rightarrow R^+$ are continuous s.t. $\varphi_i(v)\leq B,$ for $i = 1, \cdots, n$ and $B\geq0, \; v\in I_a$.

(A2) The functions $h_i\in C(I_a\times [0, B]\times \mathbb{R}, \mathbb{R})$ and $f_i\in C(I_a\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}, \mathbb{R})$, where there exist constants $k_i > 0$, s.t.

(A3) There exists $M_i\geq 0$ and $r_0\geq 0$ such that

where

(A4) $\mathbb{K} = \sum_{i = 1}^n 2k_i \prod_{j = 1, j\neq i}^n \|f_j \| < \frac{1}{2}.$

Theorem 3.4. Under the tacit assumption (A1)–(A4) above, Eq. (1.1) has at least one solution in $C(I_a)$.

Proof. First, let us define the operators $P_i:B_{r_0}\to C(I_a)$, as follows

Next, we will divide the proof into some steps according to Proposition 3.2. Step 1. The operator $P$ is well defined on $C(I_a)$. Obviously from assumptions (A1) and (A2), we have $P: \; C(I_a) \to C(I_a)$.

Step 2. We will demonstrate that the operators $P, \; P_i, i = 1, \cdots, n$ are continuous on the ball $B_{r_0}$.

Take arbitrary $z, y\in B_{r_0}$ and $\varepsilon > 0$ s.t. $\| z-y\| \leq\varepsilon$, then for $v \in I_a$, we obtain

where $\omega(h_i, \varepsilon) = \sup\big\{|h_i(v, s, z)-h_i(v, s, y)|: v\in I_a, s\in [0, B], z, y\in [-r_0, r_0], \|z-y\|\leq \varepsilon \big\}.$

From assumption (A2), the functions $h_i = h_i(v, s, z)$ are uniformly continuous on $[0, a]\times[0, B]\times \mathbb{R}$, we indicate that $\omega(h_i, \varepsilon)\rightarrow 0$ as $\varepsilon \rightarrow 0$. Thus, the operators $P_i, i = 1, \cdots, n$ are continuous on $B_{r_0}$ and consequently, the operator $P = \prod_{i = 1}^n P_i$ is continuous on $B_{r_0}$.

Step 3. We will demonstrate that the operator $P$ fulfills the densifying condition in view of $\mu$.

Take arbitrary $\rho > 0$ and $z \in M \subset C(I_a)$ is bounded set and for $v_1, v_2 \in I_a$ s.t. $v_1\leq v_2$ with $v_2-v_1\leq\rho$, we obtain

where

and

From the above relations we get

Let $\rho\to0$, we get

This yields the following estimation:

Therefore,

From assumption (A4), we get $P$ is a condensing map with $\mathbb{K} < \frac{1}{2}.$

Step 4. We will demonstrate assumption (B4) of Proposition 3.2.

Suppose $z\in \partial \bar B_{r_0}$. If $Tz = kz$ then we get $kr_0 = k\|z\| = \|Pz\|$ and by (H3) we have

for all $v\in I_a$, hence $\|Pz\|\leq r_0$, so this shows $k\leq 1.$

Step 5. The proof is completed when Proposition 3.2 is applied. □

4.   Applications and examples

To demonstrate the value of our results, we provide a few examples and instances of integral equations.

● If $n = 2, \; f_1(v, \Omega_1, \Omega_2, \Omega_3) = f(v, \Omega_1)+p(v, \Omega_1, \Omega_3), \; \alpha_1(v) = \varphi_1(v) = v, f_2(v, \Omega_1, \Omega_2, \Omega_3) = q(v, \Omega_1, \Omega_3), \; \varphi_2(v) = a$, then we have

which was inspected in ^[23].

● For $n = 2, \; f_i(v, \Omega_1, \Omega_2, \Omega_3) = p_i(v, \Omega_1, \Omega_3), \; \gamma_1(v) = \gamma_2(v) = \varphi_1(v) = v, \; \varphi_2(v) = a$, we have

which was inspected in ^[24,25].

● If $n = 2, \; f_I(v, \Omega_1, \Omega_2, \Omega_3) = p_i(v, \Omega_1, \Omega_3), \; \varphi_2(v) = 1$ then we get

which was inspected in ^[26,27].

● If $n = 2, \; f_1(v, \Omega_1, \Omega_2, \Omega_{3_1}) = a(v)\cdot \Omega_{3_1}, \; f_2(v, \Omega_1, \Omega_2, \Omega_{3_2}) = \Omega_{3_1}\cdot \Omega_{3_2}, \; \alpha_i(v) = \varphi_1(v) = \gamma_i(v) = \gamma_2(v) = v, \varphi_2(v) = a$, then we get

which was inspected in ^[28].

Example 4.1. Consider the integral equation in $C[0, 1]$

Equation (4.1) is a particular form of Eq (1.1) such that:

● $f_1(v, z(\alpha_1(v)), z(\beta_1(v)), W_1) = \frac{v^2}{15(1+v^2)}\sin(|z(v)|)+\frac{1}{2}\ln(1+|z(\sqrt{v})|)+\frac{1}{4}W_1, \; \; W_1 = \int_{0}^{\sqrt{v}}\frac{ s\sin(z(\sqrt{s}))}{1+s+e^{v}}ds,$

● $f_2(v, z(\alpha_2(v)), z(\beta_2(v)), W_2) = \frac{e^{-v}(z(v)+2z(1-v))}{6+v}+\frac{1}{8+v}W_2, \quad W_2 = \int_{0}^{\frac{1}{2}v}\frac{v\big(1+ \arctan(\frac{z(s^2)}{1+z(s^2)})\big)}{2+s}ds,$

● $f_3(v, z(\alpha_2(v)), z(\beta_2(v)), W_3) = \frac{v^4e^{-v}z(\frac{1}{2}v)}{3}+\frac{1}{2+\ln(1+s)+e^v}W_3, \quad W_3 = \int_{0}^{v^3}\frac{se^{-2t}z(s)}{2+|\cos(z(s))|}ds.$

It can be seen that

So we can choose

and so the conditions (A1) and (A2) hold. Moreover, for $\|z\|\leq r_0,$ we get

This shows that $r_0 \leq 2.1104$. Also, for $r_0 \in [0, 0.64368]\subset [0, 2.1104]$ we have

Therefore, assumptions (A1)–(A4) be fulfilled and Theorem 3.4 indicates the solution of (4.1) in $C[0, 1]$.

Example 4.2. Consider the integral equation in $C[0, 1]$

● Here $\alpha_1(v) = v^3, \; \alpha_2(v) = \sqrt{v}, \; \; \; \beta_1(v) = \beta_2(v) = \sqrt{v}, \gamma_1(v) = \gamma_2(v) = \sqrt{v}, , \varphi_1(v) = \sqrt{v}, \varphi_2(v) = \sqrt{v},$

● $f_1(v, z(\alpha_1(v)), z(\beta_1(v)), W_1) = \frac{1}{2}v e^{-v}+\frac{\cos(z(v^3))}{8}+\frac{1}{8}\sin(\frac{z(\sqrt{v})}{2+v})+\frac{1}{8+v^2}W_1,$

● $f_2(v, z(\alpha_2(v)), z(\beta_2(v)), W_2) = \frac{1}{9}v\cos(z(v))+\frac{tz(\sqrt{v})}{9(1+z(\sqrt{v}))}+\frac{1}{9(e^v+3t^4)}W_2,$

● $W_1 = \int_{0}^{v}\frac{(\sqrt{1+2|z(\sqrt{s})|}+ts^2)\cos(s)}{4+3\sqrt{s}}ds, \quad W_2 = \int_{0}^{\sqrt{v}}\frac{(1+\cos(\sqrt{s}))(\sqrt{1+2|z(\sqrt{s})|})}{1+st\ln(1+s)} ds.$

It can be seen that

So we can choose

and so the conditions (A1) and (A2) hold. Moreover, for $\|z\|\leq r_0,$ we get

This shows that $r_0 \geq 0.41410$. Also, for $r_0 \in [0.41410, 9.3765]$ we have

Therefore, assumptions (A1)–(A4) be fulfilled and Theorem 3.4 indicates the solution of (4.1) in $C[0, 1]$.

5.   Conclusions and perspective

In this article, a generalization of Petryshyn F.P.T. and the MNC idea were used to analyze the solutions for products of $n$-nonlinear integral equations in the Banach algebra $C(I_a)$. The presented F.P.T. is a generalization of Darbo, Schauder and the classical Petryshyn F.P.T. Examples are provided to demonstrate the usefulness of our findings. The upcoming work in this field will consider different Banach algebras, including $AC, C^1$ or $BV$ spaces.

Use of AI tools declaration

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

Conflict of interest

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

Acknowledgments

The researchers would like to acknowledge the Deanship of Scientific Research, Taif University for funding this work.