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# A fixed point theorem for nonautonomous type superposition operators and integrable solutions of a general nonlinear functional integral equation

- Fuli Wang
^{1}Email author

**2014**:487

https://doi.org/10.1186/1029-242X-2014-487

© Wang; licensee Springer. 2014

**Received:**24 April 2014**Accepted:**25 November 2014**Published:**8 December 2014

## Abstract

We first establish a new fixed point theorem for nonautonomous type superposition operators. After that, we prove the existence of integrable solutions for a general nonlinear functional integral equation in an ${L}^{1}$ space on an unbounded interval by using our theorem. Our main tool is the measure of weak noncompactness.

**MSC:**47H30, 47H08.

## Keywords

- superposition operators
- fixed points
- measure of weak noncompactness
- nonlinear functional integral equations

## 1 Introduction

in ${L}^{1}:={L}^{1}({\mathbb{R}}^{+})$, the space of Lebesgue integrable functions on ${\mathbb{R}}^{+}$. Here, $u:{\mathbb{R}}^{+}\times \mathbb{R}\to \mathbb{R}$ and $f:{\mathbb{R}}^{+}\times {\mathbb{R}}^{2}\to \mathbb{R}$ are two given functions, while *k* is a given real function defined on ${\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$.

for two given operators $F:X\times Y\to X$ and $A:\mathcal{D}\subset X\to X$, where *X* and *Y* are two Banach spaces. Our goal in this paper is to study under what conditions Eq. (1.1) is solvable in an ${L}^{1}$ space. To this end, we establish a fixed point theorem for the solvability of Eq. (1.2) in advance.

The organization of this paper is as follows. In Section 2, we gather some notions and preliminary facts, including the concepts and properties of the measure of weak noncompactness, which will be needed in our further considerations. In Section 3, we establish a new fixed point theorem for Eq. (1.2). In Section 4, we prove the existence of integrable solutions for Eq. (1.1) by virtue of the measure of weak noncompactness.

## 2 Preliminaries

**Definition 2.1**Let

*I*be an interval in ℝ. A function $f:I\times \mathbb{R}\to \mathbb{R}$ is said to be a Carathéodory function if:

- (a)
for each fixed $x\in \mathbb{R}$, the function $f(\cdot ,x)$ is Lebesgue measurable in

*I*; - (b)
for almost everywhere (a.e., for short) fixed $t\in I$, the function $f(t,\cdot ):\mathbb{R}\to \mathbb{R}$ is continuous.

Let $m(I)$ be a set of all measurable functions $\psi :I\to \mathbb{R}$. If *f* is a Carathéodory function, then *f* defines a mapping ${\mathcal{N}}_{f}:m(I)\to m(I)$ by $({\mathcal{N}}_{f}\psi )(t)=f(t,\psi (t))$. This mapping is called the superposition operator (or Nemytskii operator) associated to *f*. The theory concerning superposition operators is presented in [1].

For a given measurable function $\psi :I\to \mathbb{R}$, the composite operator ${\mathcal{N}}_{f}\circ \psi (\cdot ):=f(\cdot ,\psi (\cdot ))$ which maps *I* into ℝ is said to be a nonautonomous type superposition operator. By generalizing this concept, the solvability of Eq. (1.2) may be thought the existence of fixed points of the nonautonomous type superposition operator ${\mathcal{N}}_{F}\circ A$ on $\mathcal{D}$.

The following theorem was proved by Krasnosel’skii [2] (see also [3]) in the case when *I* is a bounded interval and has been extended to an unbounded interval by Appell and Zabrejko [1].

**Theorem 2.2** (see [[1], Theorem 3.1, pp.93])

*Let*

*I*

*be a*(

*bounded or unbounded*)

*interval in*ℝ.

*The superposition operator*${\mathcal{N}}_{f}$

*maps*${L}^{1}(I)$

*into*${L}^{1}(I)$

*if and only if there exist a function*${L}_{+}^{1}(I)$

*and a constant*$b>0$

*such that*

*where* ${L}_{+}^{1}(I)$ *denotes a positive cone of the space* ${L}^{1}(I)$.

In this case, the operator ${\mathcal{N}}_{f}$ is continuous and bounded in the sense that it maps bounded sets in bounded sets.

The following Scorza-Dragoni theorem explains the structure of the Carathéodory functions on a bounded interval.

**Theorem 2.3** (see [[4], Theorem 3])

*Let* *I* *be a bounded interval of* ℝ, *and let* $f:I\times \mathbb{R}\to \mathbb{R}$ *be a Carathéodory function*. *Then*, *for each* $\epsilon >0$, *there exists a closed subset* ${D}_{\epsilon}$ *of the interval* *I* *such that* $meas(I\mathrm{\setminus}{D}_{\epsilon})<\epsilon $ *and* $f:{D}_{\epsilon}\times \mathbb{R}\to \mathbb{R}$ *is continuous*.

Next, we gather together some notations and preliminary facts of some weak topology feature which will be needed in our further considerations. Let $\mathfrak{B}(X)$ be a collection of all nonempty bounded subsets of a Banach space *X*, and let $\mathfrak{W}(X)$ be a subset of $\mathfrak{B}(X)$ consisting of all weakly compact subsets of *X*. Also, let ${\mathcal{U}}_{r}$ denote a closed ball in *X* centered in 0 and with radius *r*.

In what follows, we accept the following definition [5].

**Definition 2.4**Let

*X*be a Banach space; let

*M*, ${M}_{1}$ and ${M}_{2}$ be in $\mathfrak{B}(X)$. A function $\mu :\mathfrak{B}(X)\to {\mathbb{R}}^{+}$ is said to be a measure of weak noncompactness if it satisfies the following conditions:

- (1)
the family $ker(\mu ):=\{M\in \mathfrak{B}(X):\mu (M)=0\}$ is nonempty and $ker(\mu )$ is contained in the set of relatively weakly compact sets of

*X*; - (2)
${M}_{1}\subset {M}_{2}\Rightarrow \mu ({M}_{1})\le \mu ({M}_{2})$;

- (3)
$\mu (\overline{conv}(M))=\mu (M)$, where $\overline{conv}(M)$ refers to the closed convex hull of

*M*; - (4)
$\mu (\lambda {M}_{1}+(1-\lambda ){M}_{2})\le \lambda \mu ({M}_{1})+(1-\lambda )\mu ({M}_{2})$ for $\lambda \in [0,1]$;

- (5)
if ${({M}_{n})}_{n=1}^{\mathrm{\infty}}$ is a decreasing sequence of nonempty, bounded and weakly closed subsets of

*X*with ${lim}_{n\to \mathrm{\infty}}\mu ({M}_{n})=0$, then ${M}_{\mathrm{\infty}}:={\bigcap}_{n=1}^{\mathrm{\infty}}{M}_{n}$ is nonempty.

The family $ker(\mu )$ described in (1) is called the kernel of the measure of weak noncompactness *μ*. Note that the intersection set ${M}_{\mathrm{\infty}}$ from (5) belongs to $ker(\mu )$ since $\mu ({M}_{\mathrm{\infty}})\le \mu ({M}_{n})$ for every $n\in \mathbb{N}$ and ${lim}_{n\to \mathrm{\infty}}{M}_{n}=0$.

The De Blasi measure of weak noncompactness has some interesting properties. It plays a significant role in nonlinear analysis and has many applications.

*I*is a bounded subinterval of ℝ. In [7], Appell and De Pascale gave to

*ω*the following simple form in spaces:

for all bounded subsets *M* of ${L}^{1}(I)$, where $meas(\cdot )$ denotes the Lebesgue measure.

*M*of the space ${L}^{1}({\mathbb{R}}^{+})$, Banas and Knap [8] constructed a useful measure of weak noncompactness as follows:

Based on the following criterion for weak noncompactness due to Dieudonné [9], it was shown that the function *μ* is a measure of weak noncompactness in the space ${L}^{1}({\mathbb{R}}^{+})$.

**Theorem 2.5**

*A bounded set*

*N*

*is relatively weakly compact in*${L}^{1}({\mathbb{R}}^{+})$

*if and only if the following two conditions are satisfied*:

- (1)
*for any*$\epsilon >0$,*there exists*$\delta >0$*such that if*$meas(D)\le \delta $*then*${\int}_{D}|\psi (t)|\phantom{\rule{0.2em}{0ex}}dt\le \epsilon $*for all*$\psi \in N$, - (2)
*for any*$\epsilon >0$,*there exists*$T>0$*such that*${\int}_{T}^{\mathrm{\infty}}|\psi (t)|\phantom{\rule{0.2em}{0ex}}dt\le \epsilon $*for all*$\psi \in N$.

The nonlinear contractive property of the operators plays some important roles in our subsequent considerations.

**Definition 2.6**Let $\mathcal{D}$ be a subset of the Banach space

*X*. An operator $T:\mathcal{D}\to X$ is said to be nonlinear contractive (or a

*φ*-nonlinear contraction) if there exists a continuous and nondecreasing function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\phi (r)<r$ for $r>0$ such that

for all ${x}_{1},{x}_{2}\in \mathcal{D}$.

**Remark 2.7** If we take $\phi (r)=\lambda r$ with $0\le \lambda <1$, then such a *φ*-nonlinear contraction is also said to be *λ*-contraction.

**Lemma 2.8** *Let* *X* *and* *Y* *be two Banach spaces*, *and let* $\mathcal{D}$ *be a subset of* *Y*. *If* $F:X\times Y\to X$ *is continuous*, *and for any* $y\in \mathcal{D}$ *the operator* $F(\cdot ,y)$ *is a* *φ*-*nonlinear contraction*, *then there exists a continuous map* $J:\mathcal{D}\to X$ *such that* $Jy=F(Jy,y)$ *for any* $y\in \mathcal{D}$.

*Proof* For arbitrary fixed $y\in \mathcal{D}$, the mapping $F(\cdot ,y)$ defined by $x\mapsto F(x,y)$ is a *φ*-nonlinear contraction and maps *X* into *X*, so it has a unique fixed point by [[10], Theorem 1]. Let us denote by $J:\mathcal{D}\to X$ the map which assigns to each $y\in \mathcal{D}$ the unique point in *X* such that $Jy=F(Jy,y)$. Thus, the map *J* is well defined.

Let ${r}_{n}:=\parallel J{y}_{n}-J{y}_{0}\parallel $. Since the operator *F* is continuous, we obtain ${r}_{n}-\phi ({r}_{n})\to 0$ as $n\to \mathrm{\infty}$. The properties of the function *φ* show that ${r}_{n}\to 0$, that is, $J{y}_{n}\to J{y}_{0}$. The continuity of *J* is proved. □

## 3 Fixed point theorem of nonautonomous type superposition operators

**Theorem 3.1**

*Let*

*X*

*and*

*Y*

*be two Banach spaces*,

*and let*$\mathcal{D}$

*be a nonempty subset of*

*X*.

*Suppose that the operators*$A:\mathcal{D}\to Y$

*and*$F:X\times Y\to X$

*satisfy the following*:

- (1)
*A**and**F**is continuous*, - (2)
*for any*$y\in A(\mathcal{D})$,*the operator*$F(\cdot ,y)$*is a**φ*-*nonlinear contraction*, - (3)
*there exists a nonempty*,*compact and convex subset*$\mathcal{P}$*of*$\mathcal{D}$*such that*$x=F(x,Az)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}x\in \mathcal{P}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}z\in \mathcal{P}.$

*Then there is a point* *x* *in* $\mathcal{D}$ *such that* $x=F(x,Ax)$.

*Proof* Let us denote by $J:A(\mathcal{D})\to X$ the map which assigns to each $y\in A(\mathcal{D})$ the unique point in *X* such that $Jy=F(Jy,y)$. From Lemma 2.8, the map *J* is well defined and continuous on $A(\mathcal{D})$.

*A*and

*J*are all continuous, the composite operator $J\circ A$ is continuous on $\mathcal{P}$. Now applying the Schauder fixed point theorem, we conclude that $J\circ A$ has at least one fixed point $x\in \mathcal{P}\subset \mathcal{D}$ such that $(J\circ A)x=x$, which implies that

This completes the proof. □

**Remark 3.2** There are some fixed point theorems, which involve several operators such as the operators $Tx:=Ax+Bx$ in a Banach space, or $Tx:=AxBx+Cx$ in Banach algebras *etc.*, and they may be formulated by Theorem 3.1 in a coincident form.

For example, let $F(x,Ax):=Ax+Bx$ and $\mathcal{D}$ be a nonempty, convex and closed set of a Banach space *X* in Theorem 3.1, where $A:\mathcal{D}\to X$ is compact and continuous, $B:\mathcal{D}\to X$ is a contraction mapping and $Ax+By\in \mathcal{D}$ for all $x,y\in \mathcal{D}$. Then we immediately obtain the celebrated Krasnosel’skii fixed point theorem (see [[11], Theorem 4.4.1, pp.31]), which implies that Theorem 3.1 is a generalization of the Krasnosel’skii fixed point theorem. In fact, if we take $\mathcal{P}=\overline{conv}({(I-B)}^{-1}A(M))$, then $\mathcal{P}$ satisfies the assumption (3) of Theorem 3.1 (see the proof of [[11], Lemma 4.4.2, pp.32]).

*X*in Theorem 3.1, where $B:\mathcal{D}\to X$ is completely continuous, and $A,C:X\to X$ satisfy

It is obvious that $F(\cdot ,y)$ is a *φ*-nonlinear contraction for any $y\in A(\mathcal{D})$ with $\phi (r)=M{\varphi}_{A}(r)+{\varphi}_{C}(r)$, and if we take $\mathcal{P}=\overline{conv}({(\frac{I-C}{A})}^{-1}B(\mathcal{D}))$, it is also easily proved that $\mathcal{P}$ satisfies the assumption (3) of Theorem 3.1. Thus, we obtain the fixed point theorem for the operator $AB+C$ in Banach algebras, which implies that Theorem 3.1 is a generalization of [[12], Theorem 1.5].

## 4 The solvability of general nonlinear integral equations in ${L}^{1}$ space

may all be illustrated as special examples of Eq. (1.1).

Solutions to Eq. (1.1) will be sought in ${L}^{1}:={L}^{1}({\mathbb{R}}^{+})$, the space of Lebesgue integrable functions on ${\mathbb{R}}^{+}$ with values in ℝ, endowed with the standard norm $\parallel x\parallel :={\int}_{0}^{\mathrm{\infty}}|x(t)|\phantom{\rule{0.2em}{0ex}}dt$. Here are some hypotheses on the nonlinear functions involved in Eq. (1.1).

**Assumption 4.1** Assume that

($\mathcal{H}1$) $u:{\mathbb{R}}^{+}\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function, and there exist a function $a\in {L}_{+}^{1}$ and a constant $b>0$ such that $|u(t,x)|\le a(t)+b|x|$;

($\mathcal{H}2$) $k:{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\to \mathbb{R}$ is a Carathéodory function, and $ess{sup}_{s\in {\mathbb{R}}^{+}}{\int}_{s}^{\mathrm{\infty}}|k(t,s)|\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty}$;

($\mathcal{H}3$) $f:{\mathbb{R}}^{+}\times {\mathbb{R}}^{2}\to \mathbb{R}$ is a Carathéodory function; there exist two positive numbers *α*, *β* and a function $g\in {L}_{+}^{1}$ such that $|f(t,x(t),y(t))|\le g(t)+\alpha |x(t)|+\beta |y(t)|$ for a.e. $t\in {\mathbb{R}}^{+}$;

($\mathcal{H}4$) $\alpha +b\beta \parallel K\parallel +\parallel g\parallel \le 1$ if $g\ne 0$, otherwise $\alpha +b\beta \parallel K\parallel <1$, where $\parallel K\parallel $ denotes the norm of the linear Volterra integral operator *K* generated by the function *k*;

**Remark 4.2**First notice that Eq. (1.1) may be written in an abstract form by Eq. (1.2), where

*F*is the superposition operator associated to the function

*f*($F:={\mathcal{N}}_{f}$, the superposition operator of double variables type was proposed by [13]):

*u*with the linear Volterra integral operator defined by

*A*and

*F*are well defined as follows.

- (1)It should be noted that assumption ($\mathcal{H}2$) leads to the estimate$\begin{array}{rcl}\parallel {\int}_{0}^{t}k(t,s)\psi (s)\phantom{\rule{0.2em}{0ex}}ds\parallel & =& {\int}_{s}^{\mathrm{\infty}}|k(t,s)|\phantom{\rule{0.2em}{0ex}}dt{\int}_{0}^{\mathrm{\infty}}|\psi (s)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & (ess\underset{s\in {\mathbb{R}}^{+}}{sup}{\int}_{s}^{\mathrm{\infty}}|k(t,s)|\phantom{\rule{0.2em}{0ex}}dt)\parallel \psi \parallel ,\phantom{\rule{1em}{0ex}}\psi \in {L}^{1},\end{array}$
which shows that the linear Volterra integral operator

*K*is continuous from an ${L}^{1}$ space into itself, and $\parallel K\parallel \le ess{sup}_{s\in {\mathbb{R}}^{+}}{\int}_{s}^{\mathrm{\infty}}|k(t,s)|\phantom{\rule{0.2em}{0ex}}dt$. - (2)
Assumption ($\mathcal{H}1$) shows that the superposition operator ${\mathcal{N}}_{u}$ is continuous and maps bounded sets of ${L}^{1}$ into bounded sets of ${L}^{1}$ by Theorem 2.2.

- (3)
Note that $\alpha |x|+\beta |y|$ being an equivalent norm of $(x,y)$ in ${\mathbb{R}}^{2}$, according to the Lucchetti-Patrone theorem (see [14] or [[15], Theorem 1]), assumption ($\mathcal{H}3$) shows that the superposition operator ${\mathcal{N}}_{f}$ is continuous and maps bounded sets of ${L}^{1}\times {L}^{1}$ into bounded sets of ${L}^{1}$.

**Theorem 4.3**

*If Assumption*4.1

*is verified*,

*then the equation*

*that is*, *Eq*. (1.1) *has at least a solution* $x\in {L}^{1}$.

*Proof*It is clear that the solutions of the operator equation $x=F(x,Ax)$ satisfy Eq. (1.1). We will use Theorem 3.1 to prove the present theorem, thus the assumptions of Theorem 3.1 have to be checked. Our proving is divided into several steps.

- (1)
By Remark 4.2, the operators $A:{L}^{1}\to {L}^{1}$, $F:{L}^{1}\times {L}^{1}\to {L}^{1}$ are well defined and continuous, and then the assumption (1) of Theorem 3.1 is fulfilled.

- (2)By ($\mathcal{H}5$), for arbitrary fixed $y=Az$ with $z\in {\mathcal{U}}_{{r}_{0}}$, there exists a continuous and nondecreasing function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that$\begin{array}{rcl}\parallel F({x}_{1},y)-F({x}_{2},y)\parallel & =& {\int}_{0}^{\mathrm{\infty}}|f(t,{x}_{1}(t),y(t))-f(t,{x}_{2}(t),y(t))|\phantom{\rule{0.2em}{0ex}}dt\\ \le & \phi (\parallel {x}_{1}-{x}_{2}\parallel )\end{array}$
for any ${x}_{1},{x}_{2}\in {L}^{1}$, and then the assumption (2) of Theorem 3.1 is fulfilled.

- (3)If there is $x\in {L}^{1}$ such that $x(t)=f(t,x(t),Az(t))$ for $z\in {\mathcal{U}}_{{r}_{0}}$, then by ($\mathcal{H}3$) we have$\left|f(t,x(t),Az(t))\right|\le g(t)+\alpha |x(t)|+\beta |Az(t)|=g(t)+\alpha |x(t)|+\beta |(K\circ {\mathcal{N}}_{u}\circ z)(t)|.$It follows that$\parallel x\parallel ={\int}_{0}^{\mathrm{\infty}}\left|f(t,x(t),Az(t))\right|\phantom{\rule{0.2em}{0ex}}dt\le \parallel g\parallel +\alpha \parallel x\parallel +\beta \parallel K\parallel (\parallel a\parallel +b\parallel z\parallel ),$that is,$\begin{array}{rcl}\parallel x\parallel & \le & {(1-\alpha )}^{-1}(\parallel g\parallel +\beta \parallel K\parallel (\parallel a\parallel +b\parallel z\parallel ))\\ \le & {(1-\alpha )}^{-1}(\parallel g\parallel +\beta \parallel K\parallel \parallel a\parallel +b\beta \parallel K\parallel {r}_{0})\le {r}_{0}\end{array}$
since $\parallel g\parallel +\beta \parallel K\parallel \parallel a\parallel \le {r}_{0}(1-\alpha -b\beta \parallel K\parallel )$ by ($\mathcal{H}5$). This shows that the nonautonomous type superposition operator ${\mathcal{N}}_{F}\circ A$ maps ${\mathcal{U}}_{{r}_{0}}$ into itself.

- (4)Let ${\mathcal{P}}_{0}:={\mathcal{U}}_{{r}_{0}}$, and let${\mathcal{P}}_{n}:=\overline{conv}\{x\in {L}^{1}:x(t)=f(t,x(t),{\int}_{0}^{t}k(t,s)u(s,z(s))\phantom{\rule{0.2em}{0ex}}ds),z\in {\mathcal{P}}_{n-1}\},\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.$

Then ${\mathcal{P}}_{n}$ ($n=0,1,2,\dots $) are all nonempty closed convex, and then they are weakly closed. Moreover, we have ${\mathcal{P}}_{1}\subset {\mathcal{U}}_{{r}_{0}}={\mathcal{P}}_{0}$ from step (3), and by the induction we may infer that ${\mathcal{P}}_{n}\subset {\mathcal{P}}_{n-1}$ for all $n\in \mathbb{N}$.

*D*of ${\mathbb{R}}^{+}$ such that $meas(D)\le \epsilon $, we know that if there exist $z\in {\mathcal{P}}_{n-1}$ and $x\in {\mathcal{P}}_{n}$ such that

where $\lambda :={(1-\alpha )}^{-1}b\beta \parallel K\parallel <1$ by ($\mathcal{H}4$).

- (5)In this final step, let us prove that $\mathcal{P}$ is compact. To this end, for any sequence ${({x}_{n})}_{n\in \mathbb{N}}\subset \mathcal{P}$ with${x}_{n}(t)=f(t,{x}_{n}(t),{\int}_{0}^{t}k(t,s)u(s,{z}_{n}(s))\phantom{\rule{0.2em}{0ex}}ds),\phantom{\rule{1em}{0ex}}{({z}_{n})}_{n\in \mathbb{N}}\subset \mathcal{P},$(4.6)

we shall show that ${({x}_{n})}_{n\in \mathbb{N}}$ possesses a convergent subsequence in ${L}^{1}$.

for all $m,n\in \mathbb{N}$.

*k*and

*u*are continuous on ${D}_{\epsilon}\times [0,T]$ and ${D}_{\epsilon}\times \mathbb{R}$, respectively, where $meas([0,T]\mathrm{\setminus}{D}_{\epsilon})\le \epsilon $. By taking ${t}_{1},{t}_{2}\in {D}_{\epsilon}$ with ${t}_{1}<{t}_{2}$, we have

where $\tilde{k}:=max\{|k(t,s)|:(t,s)\in {D}_{\epsilon}\times [0,T]\}$, and ${\omega}^{T}(k,|{t}_{2}-{t}_{1}|)$ denotes the modulus of continuity of the function *k* on the set ${D}_{\epsilon}\times [0,T]$.

Since ${({z}_{n})}_{n\in \mathbb{N}}\subset \mathcal{P}$ is relatively weakly compact and the set consisting of one element is also weakly compact, by Theorem 2.5 we infer that the terms ${\int}_{{t}_{1}}^{{t}_{2}}|{z}_{n}(s)|\phantom{\rule{0.2em}{0ex}}ds$ and ${\int}_{{t}_{1}}^{{t}_{2}}a(s)\phantom{\rule{0.2em}{0ex}}ds$ in (4.8) may all be arbitrarily small provided that the number $|{t}_{1}-{t}_{2}|$ is small enough. Thus, we obtain that the sequence ${({y}_{n}(t))}_{n\in \mathbb{N}}$ is equicontinuous on $\mathcal{C}({D}_{\epsilon})$ (the space of all continuous functions defined on ${D}_{\epsilon}$).

which implies the sequence ${({y}_{n}(t))}_{n\in \mathbb{N}}$ is uniformly bounded on $\mathcal{C}({D}_{\epsilon})$.

Since the map *J*, which signs each $y\in A(\mathcal{P})$ the unique point $x\in \mathcal{P}$ such that $x(t)=f(t,x(t),y(t))$, is well defined and uniformly continuous on ${D}_{\epsilon}\times [-\overline{Y},\overline{Y}]$, by Lemma 2.8 we infer that the sequence ${({x}_{n}(t))}_{n\in \mathbb{N}}$ with ${x}_{n}(t)=f(t,{x}_{n}(t),{y}_{n}(t))$ is uniformly bounded and equicontinuous on $\mathcal{C}({D}_{\epsilon})$. Hence, by applying the Arzéla-Ascoli theorem, we obtain that the set $\{{x}_{n}:n\in \mathbb{N}\}$ forms a relatively compact set in $\mathcal{C}({D}_{\epsilon})$.

Note that our reasoning does not depend on the choice of *ε*. Thus we can construct a sequence ${({D}_{1/k})}_{k\in \mathbb{N}}$ of closed subsets of the interval $[0,T]$ such that $meas([0,T]\mathrm{\setminus}{D}_{1/k})\to 0$ as $k\to \mathrm{\infty}$, and the sequence ${({x}_{n})}_{n\in \mathbb{N}}$ is relatively compact in every space $\mathcal{C}({D}_{1/k})$. Passing to subsequences if necessary, we can assume that ${({x}_{n})}_{n\in \mathbb{N}}$ is a Cauchy sequence in each space $\mathcal{C}({D}_{1/k})$ for $k\in \mathbb{N}$.

which shows that ${({x}_{n})}_{n\in \mathbb{N}}$ is a Cauchy sequence in an ${L}^{1}$ space. Thus, the sequence ${({x}_{n})}_{n\in \mathbb{N}}\subset \mathcal{P}$ has a convergent subsequence, which implies that the closed set $\mathcal{P}$ is compact.

This shows that the assumptions of Theorem 3.1 are all fulfilled, which completes the proof. □

**Remark 4.4** The techniques of the proof of Theorem 4.3 based on Carathéodory conditions and the Scorza-Dragoni theorem were already used in [16–19] for proving the solvability of Eq. (4.1), (4.2), *etc.*

Finally, we provide an example, which is not included in Eq. (4.1)-(4.3), and which may be treated by our Theorem 4.3.

**Example 4.5**Consider the following nonlinear integral equation:

for $t\in {\mathbb{R}}^{+}$. In order to show that such an equation admits a solution in ${L}^{1}$, we are going to check that the conditions of Theorem 4.3 are satisfied.

*u*,

*k*and

*f*are all Carathéodory functions. Taking $a(t)={(1+t)}^{-2}$ and $b=1$, we have

*u*satisfies (${\mathcal{H}}_{1}$). Taking $g(t)=t{e}^{-2t}$, $\alpha =1/4$ and $\beta =1/2$, we have

*f*satisfies (${\mathcal{H}}_{3}$). By a simple calculation, we obtain that

which shows that (${\mathcal{H}}_{2}$) and (${\mathcal{H}}_{4}$) are satisfied.

for all $y\in {L}^{1}$. So (${\mathcal{H}}_{5}$) is satisfied for $\phi (r):=\frac{1}{2}r$.

Since the assumptions (${\mathcal{H}}_{1}$)-(${\mathcal{H}}_{5}$) are all satisfied, we apply Theorem 4.3 to derive the existence of a solution to Eq. (4.12).

## Declarations

### Acknowledgements

The author is grateful to the referees for the careful reading of the manuscript. The remarks motivated the author to make some valuable improvements.

## Authors’ Affiliations

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