An existence theorem for nonlinear functional Volterra integral equations via Petryshyn's fixed point theorem

1.   Introduction

Functional integral equations (FIEs) have many application in mechanical vibrations, kinetic theory of gases, radiative transfer, mathematical physics, control theory, and engineering. The theory of FIEs is speedily growing with the help of different studies of fixed point theory, topology, and non-linear analysis (cf. ^{[2,3,4,7,8,10,11,14,15,18,20,25,31]}).

This article is dedicated to study the following FIEs.

for all $\varphi \in I_b = [0, b]$ in the Banach algebra $C(I_b)$.

Equation (1.1) contains many particular cases see for example ^{[12,19,21,22,23,26]}. Moreover, many previous studies examined the existence of the solutions for different FIEs by Darbo's fixed point theorem in different spaces (cf. ^{[5,6,9,16,17,24,28,30]}). We generalize these results by using Petryshyn's fixed point theorem.

This article is motivated by studying non-linear FIE under a general set of assumptions by using the theory of MNC and Petryshyn's fixed point theorem. Moreover, the bounded condition explains that the "sublinear condition" that has been recognized in literature will not play a meaningful role here. Finally, we present some particular cases and examples that show the utilization of FIEs.

2.   Preliminaries

In this article, let $\mathbb{R}$ be the set of real numbers, $F$ be real Banach space and $B_\rho = B(z, \rho)$ be a closed ball centered at $z$ with radius $\rho.$

Definition 2.1. ^[1] Let $G \in F$ and

is called the Kuratowski MNC.

Definition 2.2. ^[1] The Hausdroff MNC

where, by a finite $\sigma$ net for $G$ in $F$ it involves, as a set $\{z_1, z_2, ..., z_n\}\subset F$ such that the ball $B_{\sigma}(F, z_1), B_{\sigma}(F, z_2), ..., B_{\sigma}(F, z_n)$ over $G.$ Those MNC are commonly related that is

for any bounded set $G \subset F.$

Theorem 2.3. Let $G, \hat{G} \in F$ and $\lambda \in \mathbb{R}$. Then

(ⅰ) $\vartheta(G) = 0$ if and only if $G$ is pre-compact;

(ⅱ) $G \subseteq \hat{G}$ $\implies$ $\vartheta(G) \leq \vartheta(\hat{G})$;

(ⅲ) $\vartheta(\mathit{\mbox{Conv}} G) = \vartheta(G)$;

(ⅳ) $\vartheta(G \cup \hat{G}) = \max \{ \vartheta(G), \vartheta(\hat{G}) \}$;

(ⅴ) $\vartheta(\lambda G) = |\lambda| \vartheta(G)$, where $\lambda G = \{ \lambda z : z \in G \}$;

(ⅵ) $\vartheta(G + \hat{G}) \leq \vartheta(G) + \vartheta(\hat{G}).$

In the sequel, $C[0, b]$ consisting of all real valued continuous function defined on $I_b = [0, b]$ with the usual norm

The space $C[0, b]$ is also the structure of Banach algebra. The modulus of continuity of $z \in C[0, b]$ is defined as

and

Theorem 2.4. ^[19] The Hausdorff MNC is similar to

for all bounded sets $G \subset C[0, b].$

Theorem 2.5. ^[27] Assume $T : F \rightarrow F$ be a continuous mapping of Banach space $F.$ $T$ is called a $k$-set contraction if for all $H \subset F$ with $H$ bounded, $T(H)$ is bounded and

Moreover, if

then $T$ is called densifying or condensing map.

Theorem 2.6. ^[29] Suppose that $T : B_{\rho} \rightarrow F$ be a condensing mapping which satisfying the boundary condition,

then the set of fixed points in $B_{\rho}$ is non-empty. This is called Petryshyn's fixed point theorem.

3.   Main results

Now, we investigate the existence of the Eq (1.1) under the following assumptions;

(1) $q\in C(I_b \times \mathbb{R}\times \mathbb{R}\times \mathbb{R}, \mathbb{R}),$ $f, h \in C(I_b \times \mathbb{R}, \mathbb{R}),$ $p\in C(I_b \times [0, D] \times \mathbb{R}, \mathbb{R}),$ and $\theta : I_b \rightarrow \mathbb{R}^{+},$ $\gamma : [0, D] \rightarrow I_b,$ $\alpha, \beta : I_b \rightarrow I_b,$ are continuous such that $\theta(\varphi) \leq D, \; \; \forall\; \varphi\in I_b, \; D\geq 0.$

(2) There exist non-negative constants $p_i, \; i = 1, \cdots 5$, such that

(3) There exists $\rho > 0$ such that $q$ satisfy the inequality

where

Theorem 3.1. Under the assumptions $(1)$–$(3)$ and if $p_1p_4 + p_2p_5 < 1,$ $\forall z\in I_b$. Then Eq (1.1) has at least one solution in $F = C(I_b)$.

Proof. Define the operator $T : B_{\rho} \rightarrow F,$ where $B_\rho = \{z\in C(I_b): \|z\|\leq \rho\}$ in the following form

Now, we show that $T$ is continuous on $B_{\rho}.$ Choose $\sigma > 0$ and any $z, x \in B_{\rho}$ such that $\| z - x\| < \sigma.$ We get

where

From the uniform continuity of $p(\varphi, s, z)$ on the subset $I_b \times [0, D] \times \mathbb{R},$ we infer that $\omega(p, \sigma) \to 0$ as $\sigma \rightarrow 0.$ Thus, we prove that the operator $T$ is continuous on $B_{\rho}.$

Next, we show that $T$ satisfy the densifying map. Take arbitrary $\sigma > 0$ and $z\in G,$ where $G$ is bounded subset of $F,$ $\varphi_1, \varphi_2 \in I_b$ with $|\varphi_1 - \varphi_2| \leq \sigma,$ we have

where

From above relations, we have

Taking limit as $\sigma \rightarrow 0,$ we get

This provide the following inequality

Hence $T$ is a condensing map. Now, let $z\in \partial B_{\rho}$ and if $Tz = kz$ then $\|Tz\| = k\|z\| = k \rho$ and by $(3)$, then

for all $\varphi \in I_b,$ hence $\|Tz\| \leq \rho$ i.e $k \leq 1.$ This completes the proof.

Corollary 3.2. Assume that

$(1)$ $q\in C(I_b \times \mathbb{R}\times \mathbb{R}\times \mathbb{R}, \mathbb{R}),$ $h \in C(I_b \times \mathbb{R}, \mathbb{R}),$ $p\in C(I_b \times [0, D] \times \mathbb{R}, \mathbb{R}),$ and $\theta : I_b \rightarrow \mathbb{R}^{+},$ $\gamma : [0, D] \rightarrow I_d,$ $\beta : I_b \rightarrow I_b,$ are continuous such that $\theta(\varphi) \leq D, \; \; \forall\; \varphi\in I_b, D\geq 0.$

$(2)$ There exist non-negative constants $p_i, \; i = 1, \cdots 4$ with $p_2 p_4 < 1$ such that

$(3)$ There exists $\rho > 0$ such that $q$ satisfy the inequality

where,

Then

has at least one solution in $C(I_b)$.

Corollary 3.3. Let

$(1)$ $q\in C(I_b \times \mathbb{R}\times \mathbb{R}, \mathbb{R}),$ $f, h \in C(I_b \times \mathbb{R}, \mathbb{R}),$ $p\in C(I_b \times [0, D] \times \mathbb{R}, \mathbb{R}),$ and $\theta : I_b \rightarrow \mathbb{R}^{+},$ $\gamma : [0, D] \rightarrow I_d,$ $\beta : I_b \rightarrow I_b,$ are continuous such that $\theta(\varphi) \leq D, \; \; \forall\; \varphi\in I_b, \; D\geq 0.$

$(2)$ There exist non-negative constants $p_i, i = 1, \dots 4$ with $p_4 + p_1 p_3 < 1$ such that

$(3)$ There exists $\rho > 0$ such that $q$ satisfy the inequality

where,

Then

has at least one solution in $C(I_b)$.

4.   Particular cases and examples

Example 4.1. If $q(\varphi, u, v, w) = 1 + vw, \beta(\varphi) = \theta(\varphi) = \gamma(\varphi) = \varphi, h(\varphi, z(\beta(\varphi))) = z(\varphi)$ and $p(\varphi, s, z) = \frac{\varphi}{\varphi + s}\phi(s)z.$ Then Eq $(1.1)$ has the Chandrasekhar integral equation type in radiative transfer ^[3].

Example 4.2. Let $q(\varphi, u, v, w) = q(\varphi, v, w)$, then the Eq (1.1) takes the form

which is studied in ^[13].

Example 4.3. Let $q(\varphi, u, v, w) = q(\varphi, vw), \; \gamma(\varphi) = \theta(\varphi) = \varphi$, then Eq (1.1) takes the form

which is examined in ^[22].

Example 4.4. Let $q(\varphi, u, v, w) = q(\varphi, u, v, w), \; f(\varphi, z) = h(\varphi, z) = z$, then Eq (1.1) takes the form

which is examined in ^[19].

Example 4.5. Let $q(\varphi, u, v, w) = u + q(\varphi, w, v), \; h(\varphi, z) = z$ and $\gamma(\varphi) = \theta(\varphi) = \alpha(\varphi) = \varphi$, then Eq (1.1) takes the form

which is studied in ^[21].

Example 4.6. Let $q(\varphi, u, v, w) = u + vw$, then Eq (1.1) takes the form

which is studied in ^[28].

Example 4.7. Let the following Volterra non-linear FIE:

Equation (4.1) is special case of Eq (1.1) with

and

where

It is obvious that assumptions (1) and (2) of Theorem 3.1 are satisfied. We need to check that assumption (3) holds true. So, choose $\rho = 3 + \frac{3}{4}e,$ then $H_1 \leq \frac{5e}{8} + \frac{5}{2}$ and

All conditions of Theorem $3.1$ are satisfied, Hence Eq $(4.1)$ has at least one solution in $C[0, 1].$

5.   Conclusions

In this article, we have examined the existence of the solutions of non-linear functional integral equations in Banach algebra by utilizing a strategy, which is distinguishable from different authors technique (see ^{[12,16,17,21,22,23,26]}). The advantage of Theorem 2.6 among the others (Darbo and Schauder fixed point theorems) lies in that in using the theorem, one does not require to confirm the involved operator maps a closed convex subset onto itself.

Acknowledgment

The authors are thankful to the anonymous referees and the editorial board for their thorough reading and valuable guidance on the improvement of the paper.

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.