A family of measures of noncompactness in the space L^p_loc(ℝ^N) and its application to some nonlinear convolution type integral equations

Abstract

The aim of the present paper is to introduce a new family of measures of noncompactness on the Fréchet space L^P_loc(ℝ^N) (1 ≤ p < ∞). Further, we prove a fixed point theorem on the family of measures of noncompactness in L^P_loc(ℝ^N). As an application, we investigate the existence of entire solutions for some classes of nonlinear functional integral equations of convolution type associated with the new family of measures of noncompactness. Finally, we give an illustrative example to verify the effectiveness and applicability of our results.

Keywords:

• Darbo’s fixed point theorem
• Family of measures of noncompactness
• Fixed point
• Integral equations

PUBLIC INTEREST STATEMENT

Metric fixed point theory is a powerful tool for solving several problems in various parts of mathematics and its applications. In particular, the technique of measure of noncompactness is a very useful tool for studying the existing solutions of integral equations and differential equations. In this paper, we introduce a new family of measures of noncompactness in the Frechet space l^p_loc(ℝ^N) and by applying this family of measures of noncompactness, we discuss the existence of solutions for some classes of nonlinear functional integral equations.

1. Introduction and preliminaries

The concept of a measure of noncompactness (MNC) plays a signification role in the nonlinear functional analysis. This notion was initiated by Kuratowski in 1930 (Kuratowski, 1930). In 1955 G. Darbo, using the concept of a measure of noncompactness, proved a theorem guaranteeing the existence of fixed points of the so-called condensing operators (Darbo, 1955). In recent years, a lot of authors such as Aghajani, Allahyari, and Mursaleen (2014), Aghajani, Banaś, and Jalilian (2011), Aghajani, O’Regan, and Shole Haghighi (2015), Arab and Mursaleen (2018), Banaś and O’Regan (2008), Das, Hazarika, Arab, & Mursaleen (2017); Maleknejad, Torabi, and Mollapourasl (2011) and Olszowy (2010, 2012) studied the existence of solutions of integral equations in one or two variables on some spaces. Nonlinear functional integral equations of convolution type play important roles in applied problems, especially numerous branches of mathematical physics such as neutron transportation, radiation, and gas kinetic theory (see e.g., Zabrejko et al., 1968 and the references therein). Equations of this type have been considered in many previous studies Askhabov and Mukhtarove (1987) and Jingqi (1985), which showed that these equations have solutions in some function spaces. As an example, Khosravi et al. defined a new measure of noncompactness on the Banach space $L^p({\mathbb{R}}_{+})$ $(1\le p<\mathrm{\infty })$ and studied the existence of entire solutions for a class of nonlinear functional integral equations of convolution type in Khosravi, Allahyari, and Shole Haghighi (2015).

On the other hand, a locally integrable function is a function which is integrable on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to $L^p$ spaces, but its members are not required to satisfy any growth restriction on their behavior at infinity: in other words, locally integrable functions can grow arbitrarily fast at infinity, but are still manageable in a way similar to ordinary integrable functions. Regarding the above subject, Olszowy introduced a new family of measures of noncompactness on the space $L_{loc}^1({\mathbb{R}}_{+})$ consisting of all real functions locally integrable on ${\mathbb{R}}_{+}$, equipped with a suitable topology. Then, she studied the existence of solutions of a nonlinear Volterra integral equation in the space $L_{loc}^1({\mathbb{R}}_{+})$ (cf. Olszowy, 2014).

In this paper, we define the new family of measures of noncompactness in Fréchet spaces $L_{loc}^p({\mathbb{R}}^N)$ $(1\le p<\mathrm{\infty })$. In addition, we study the existence of solutions for some classes of nonlinear functional integral equations of convolution type (1.1)

(1.1) $u(x)=f(x,u(x))+{\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)(Qu)(y)dy,$(1.1)

by using an extension of Darbo’s fixed point theorem associated with this new family of measures of noncompactness.

In the sequel, we introduce some notations, definitions and preliminary facts which will be needed further on.

Denoted by $\mathbb{R}$ the set of real numbers, ${\mathbb{R}}^N=\left\{(x_1,x_2,\dots ,x_N):x_i\in \mathbb{R}\right\}$ and put ${\mathbb{R}}_{+}=[0,+\mathrm{\infty })$. Let $(E,\parallel .\parallel )$ be a real Banach space with zero element $0$. For a nonempty subset $X$ of $E$, the symbols $\bar{X}$ and $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}X$ will denote the closure and closed convex hull of $X$, respectively. Moreover, let ${\mathfrak{M}}_E$ indicate the family of nonempty and bounded subsets of $E$ and ${\mathfrak{N}}_E$ indicate the family of all nonempty and relatively compact subsets of $E.$

Let $L^p(U)$ $(U\subseteq {\mathbb{R}}^N)$ denote the space of Lebesgue integrable functions on $U$ with the standard norm

$\parallel x{\parallel }_{L^p(U)}={\left({\int }_U|x(t){|}^{p}dt\right)}^{\frac{1}{p}}.$

We say that a function $f:{\mathbb{R}}^N\to \mathbb{R}$ belongs to $L_{loc}^p({\mathbb{R}}^N)$ $(1\le p<\mathrm{\infty })$ if ${\chi }_{K}f\in L^p({\mathbb{R}}^N)$ for every compact set $K\subseteq {\mathbb{R}}^N$. In other words, $f\in L_{loc}^p({\mathbb{R}}^N)$ if only if $f\in L^p([-T,T{]}^N)$ for all $T>0.$ Also if $f\in L_{loc}^p({\mathbb{R}}^N)$, then $f\in L_{loc}^1({\mathbb{R}}^N)$. Recall that ${\mathbb{R}}^N$ is the countable union of compact $N$-$cells$, i.e., ${\mathbb{R}}^N=K_1\cup K_2\cup K_3\cup \dots$ with $K_k=\left\{x\in {\mathbb{R}}^N:\parallel x{\parallel }_{max}\le k\right\},$ where $\parallel x{\parallel }_{max}=max\left\{\left|x_1\left|,\right|x_2\left|,\dots ,\right|x_n\right|\right\}.$ Moreover, any compact subset of ${\mathbb{R}}^N$ is contained in some $K_k$ and so the topology on $L_{loc}^p({\mathbb{R}}^N)$ is given by the countable family of seminorms $\parallel {\chi }_{{[-T,T]}^N}f{\parallel }_p$ for each $T>0$. Then, $L_{loc}^p({\mathbb{R}}^N)$ becomes a Fréchet space with respect to the metric

$d(f,g)=sup\left\{\frac{1}{2^T}min\left\{1,\parallel {\chi }_{{[-T,T]}^N}(f-g){\parallel }_p\right\}:T\in \mathbb{N}\right\}.$

A sequence $\left\{f_n\right\}$ is convergent to $f$ in $L_{loc}^p({\mathbb{R}}^N)$ if and only if for each $T>0$, $\left\{f_n\right\}$ is convergent to $f$ in $L^p([-T,T{]}^N)$.

A nonempty subset $X\subset L_{loc}^p({\mathbb{R}}^N)$ is said to be bounded if

$sup\left\{\parallel {\chi }_{{[-T,T]}^N}f{\parallel }_p:f\in X\right\}<\mathrm{\infty }$

for all $T>0$.

Further, let ${\mathfrak{M}}_{L_{loc}^p({\mathbb{R}}^N)}$ denote the family of nonempty and bounded subsets of $L_{loc}^p({\mathbb{R}}^N)$ and ${\mathfrak{N}}_{L_{loc}^p({\mathbb{R}}^N)}$the family of all nonempty and relatively compact subsets of $L_{loc}^p({\mathbb{R}}^N).$

In what follow, we recall the well-known Darbo’s fixed point theorem.

Theorem 1.1. (Darbo Banaś & Goebel, 1980) Let $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$ and let $F:C\to C$ be a continuous mapping. Assume that a constant $k\in [0,1)$ exists such that

$\mu (F(X))\le k\mu (X),$

for any nonempty subset $X$ of $C$, where $\mu$ is a measure of noncompactness defined in $E$. Then $F$ has a fixed point in the set $C$.

Theorem 1.2. (Tychonoff fixed point theorem Agarwal, Meehan, & O’Regan, 2001) Let $E$ be a Hausdorff locally convex linear topological space, $C$ a nonempty convex subset of $E$ and $F:C\to E$ a continuous mapping such that

$F(C)\subseteq A\subseteq C,$

with $A$ compact. Then $F$ has at least one fixed point.

2. Main results

In this section, we will introduce a family of measures of noncompactness in the space $L_{loc}^p({\mathbb{R}}^N)$. Before that, we characterize the construction of compact subsets of $L_{loc}^p({\mathbb{R}}^N)$. First, we quote a useful theorem in (Brezis, 2011).

Theorem 2.1. (Brezis, 2011, Theorem 4.26) (Kolmogorov-M. Riesz-Fréchet). Let $\mathcal{F}$ be a bounded set in $L^p({\mathbb{R}}^N)$ with $1\le p<\mathrm{\infty }$. Assume that

$\underset{\left|h\right|\to 0}{lim}\parallel {\tau }_{h}f-f{\parallel }_p=0uniformlyinf\in \mathcal{F},$

i.e., $\mathrm{\forall }\epsilon >0$ $\mathrm{\exists }\delta >0$ such that $\parallel {\tau }_{h}f-f{\parallel }_p<\epsilon$ $\mathrm{\forall }f\in F,$ $h\in {\mathbb{R}}^N$ with $\left|h\right|<\delta$. Then the closure of $\mathcal{F}{|}_{\mathrm{\Omega }}$ in $L^p(\mathrm{\Omega })$ is compact for any measurable set $\mathrm{\Omega }\subset {\mathbb{R}}^N$ with finite measure.

Here $({\tau }_{h}f)(x)=f(x+h),$ $x\in {\mathbb{R}}^N$, $h\in {\mathbb{R}}^N$ and $\mathcal{F}{|}_{\mathrm{\Omega }}$ denotes the restrictions to $\mathrm{\Omega }$ of the functions in $\mathcal{F}$.

Lemma 2.2 Let $\mathcal{F}$ be a set in $L_{loc}^p({\mathbb{R}}^N)$. $\mathcal{F}$ is totally bounded in $L_{loc}^p({\mathbb{R}}^N)$ if and only if $\mathcal{F}{|}_{{[-T,T]}^N}=\left\{f{|}_{{[-T,T]}^N}:f\in \mathcal{F}\right\}$ is totally bounded in $L^p([-T,T{]}^N)$ for each $T\in \mathbb{N}$.

Proof. First, assume that $\mathcal{F}$ is totally bounded in $L_{loc}^p({\mathbb{R}}^N)$. We prove $\mathcal{F}{|}_{{[-T,T]}^N}$ is totally bounded in $L^p([-T,T{]}^N)$ for all $T\in \mathbb{N}$. For this, fixed $\epsilon >0$ and $T\in N$. Without loss of generality we may assume that $\epsilon <1$. Since $\mathcal{F}$ is totally bounded so there exists $\frac{\epsilon }{2^T}$-cover $U_1,...,U_k$ for $\mathcal{F}$. Take $g_i\in U_i$ for $i=1,...,k$ and let $f\in \mathcal{F}$. Then there exists $1\le j\le k$ such that

$d(f,g_j)=sup\left\{\frac{1}{2^m}min\left\{1,\parallel {\chi }_{{[-m,m]}^N}(f-g_j){\parallel }_p\right\}:m\in N\right\}<\frac{\epsilon }{2^T}.$

So, $min\{1,\parallel {\chi }_{{[-T,T]}^N}(f-g_j){\parallel }_p<\epsilon$ and since $\epsilon <1$ so we have $\parallel {\chi }_{{[-T,T]}^N}(f-g_j){\parallel }_p<\epsilon$. Thus, $\left\{U_1{|}_{{[-T,T]}^N},...,U_k{|}_{{[-T,T]}^N}\right\}$ is an $\epsilon$-cover for $\mathcal{F}{|}_{{[-T,T]}^N}$.

Next, assume that $\mathcal{F}{|}_{{[-T,T]}^N}$ is totally bounded for each $T\in \mathbb{N}$. We prove that $\mathcal{F}$ is totally bounded in $L_{loc}^p({\mathbb{R}}^N)$. For this, fixed $\epsilon >0$. We may assume that $\epsilon <1$. Take $T\in \mathbb{N}$ such that $\frac{1}{2^T}<\epsilon$. Also, suppose that ${\left\{U_{\alpha }\right\}}_{\alpha \in I}$ is an $\frac{\epsilon }{2^T}$-cover for $\mathcal{F}$. So, there exists $U_1,...,U_k$ such that $\left\{U_1{|}_{{[-T,T]}^N},...,U_k{|}_{{[-T,T]}^N}\right\}$ is an $\epsilon$-cover for $\mathcal{F}{|}_{{[-T,T]}^N}$. Let $f\in \mathcal{F}$. Thus, there exists $U_j$ $(1\le j\le k)$ such that

$\parallel {\chi }_{{[-T,T]}^N}(f-g_j){\parallel }_p<\epsilon$

for all $g_j\in U_j$. Now, for $m>T$ we have

$\frac{1}{2^m}min\left\{1,\parallel {\chi }_{{[-m,m]}^N}(f-g_j){\parallel }_p\right\}\le \frac{1}{2^m}<\frac{1}{2^T}<\epsilon .$

Also, since $\parallel {\chi }_{{[-T,T]}^N}(f-g_j){\parallel }_p<\epsilon <1$ so, for $m\le T$ we have

$\frac{1}{2^m}min\left\{1,\parallel {\chi }_{{[-m,m]}^N}(f-g_j){\parallel }_p\right\}\le min\left\{1,\parallel {\chi }_{{[-m,m]}^N}(f-g_j){\parallel }_p\right\}\le \parallel {\chi }_{{[-T,T]}^N}(f-g_j){\parallel }_p<\epsilon .$

Hence, we deduce

$d(f,g_j)=sup\left\{\frac{1}{2^m}min\left\{1,\parallel {\chi }_{{[-m,m]}^N}(f-g_j){\parallel }_p\right\}:m\in N\right\}<\epsilon .$

Thus $\left\{U_1,...,U_k\right\}$ is an $\epsilon$-cover for $\mathcal{F}$ and so $\mathcal{F}$ is totally bounded. □

Theorem 2.3. Let $1\le p<\mathrm{\infty }$ and $F\subseteq L_{loc}^p({\mathbb{R}}^N)$. Then $F$ is totally bounded if, and only if, the following hold:

(i) For every $T>0$ there is some $M>0$ so that

${\int }_{{[-T,T]}^N}|f(x){|}^{p}dx\le M$

for all $f\in F.$

(ii) For every $\epsilon >0$ and $T>0$ there is some $\delta >0$ so that

${\left({\int }_{{[-T,T]}^N}|f(x+h)-f(x){|}^{p}dx\right)}^{\frac{1}{p}}<\epsilon$

for all $f\in F$ and $\parallel h{\parallel }_{{\mathbb{R}}^N}<\delta .$

Proof. Assume that $\mathcal{F}$ is totally bounded. For any $T\in \mathbb{N}$, there exists a finite $\frac{1}{2^{T+1}}$-cover $\left\{U_1,U_2,\dots ,U_n\right\}$ for $\mathcal{F}$. Choose $g_i\in U_i$ $(i=1,2,\dots ,n)$. Let $f\in \mathcal{F},$ then there exists $1\le j\le n$ such that $f\in U_j,$ we have

$d(f,g_j)=sup\left\{\frac{1}{2^m}min\left\{1,\parallel {\chi }_{{[-m,m]}^N}(f-g_j){\parallel }_p\right\}:m\in \mathbb{N}\right\}<\frac{1}{2^{T+1}}.$

So

$\parallel {\chi }_{{[-T,T]}^N}(f-g_j){\parallel }_p<\frac{1}{2},$

and

$\parallel {\chi }_{{[-T,T]}^N}(f){\parallel }_p\parallel \le \parallel {\chi }_{{[-T,T]}^N}(g_j){\parallel }_p+\parallel {\chi }_{{[-T,T]}^N}(f-g_j){\parallel }_p<\frac{1}{2}+\underset{i=1,2,\dots ,n}{max}\left\{\parallel {\chi }_{{[-T,T]}^N}(g_i){\parallel }_p\right\}.$

It implies the boundedness of $\mathcal{F}$, thus condition $(i)$ holds. To establish condition $(ii)$, let $\epsilon >0$ and $T\in \mathbb{N}$ be given. Let $\left\{U_1,U_2,...,U_n\right\}$ be an $\frac{\epsilon }{2^T}$-cover of $\mathcal{F}$. If $f\in \mathcal{F},$ then $f\in U_j$ for some $1\le j\le n$. Take $g\in U_j.$ It yields that

(2.1) $\parallel {\chi }_{{[-T,T]}^N}(f-g){\parallel }_p<\epsilon .$(2.1)

Now, since the space $C([-T,T{]}^N)$ is dense in $L^p([-T,T{]}^N)$ thus there exists a continuous function $y\in C([-T,T{]}^N)$ such that

(2.2) $\parallel {\chi }_{{[-T,T]}^N}(y-g){\parallel }_p<\epsilon .$(2.2)

Also, there exists $\delta >0$ such that for all $x,h\in [-T,T{]}^N$ such that $\parallel h{\parallel }_{{\mathbb{R}}^N}\le \delta$ we have

(2.3) $\left|y(x+h)-y(x)\right|<\frac{\epsilon }{{(2T)}^N}.$(2.3)

Therefore, applying (2.1), (2.2) and (2.3) we can write

${\left({\int }_{{[-T,T]}^N}|f(x+h)-f(x){|}^{p}dx\right)}^{\frac{1}{p}}\le {\left({\int }_{{[-T,T]}^N}|f(x+h)-g(x+h){|}^{p}dx\right)}^{\frac{1}{p}}$

$+{\left({\int }_{{[-T,T]}^N}|g(x+h)-y(x+h){|}^{p}dx\right)}^{\frac{1}{p}}$

$+{\left({\int }_{{[-T,T]}^N}|y(x+h)-y(x){|}^{p}dx\right)}^{\frac{1}{p}}$

$+{\left({\int }_{{[-T,T]}^N}|y(x)-g(x){|}^{p}dx\right)}^{\frac{1}{p}}$

$+{\left({\int }_{{[-T,T]}^N}|f(x)-g(x){|}^{p}dx\right)}^{\frac{1}{p}}$

$\le 5\epsilon ,$

thus condition $(ii)$ holds.

Conversely, by using Lemma 2.2, it is enough to show that $\mathcal{F}{|}_{{[-T,T]}^N}$ is totally bounded in $L^p([-T,T{]}^N)$ for every $T\in N$. Now, since $\mathcal{F}{|}_{{[-T,T]}^N}$ satisfies the condition (ii) so by Theorem 2.1 the closure of $\mathcal{F}{|}_{{[-T,T]}^N}$ is compact in $L^p([-T,T{]}^N)$. This completes the proof. □

Now, we are ready to describe a family of measures of noncompactness in the space $L_{loc}^p({\mathbb{R}}^N)$.

Definition 2.4. A family of mappings ${\left\{{\mu }_T\right\}}_{T>0},$ ${\mu }_T:{\mathfrak{M}}_{L_{loc}^p({\mathbb{R}}^N)}\to {\mathbb{R}}_{+}$, is said to be a family of measures of noncompactness in $L_{loc}^p({\mathbb{R}}^N)$ if it fulfils the following conditions:

1. The family $ker\left\{{\mu }_T\right\}=\{X\in {\mathfrak{M}}_{L_{loc}^p({\mathbb{R}}^N)}:{\mu }_T(X)=0$ for $T>0\}$ is nonempty and $ker\left\{{\mu }_T\right\}\subseteq {\mathfrak{N}}_{L_{loc}^p({\mathbb{R}}^N)}$.

2. $X\subset Y\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}{\mu }_T(X)\le {\mu }_T(Y)$ for $T\ge 0$.

3. ${\mu }_T(\bar{X})={\mu }_T(X)$ for $T\ge 0$.

4. ${\mu }_T(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}X)={\mu }_T(X)$ for $T\ge 0$.

5. ${\mu }_T(\lambda X+(1-\lambda )Y)\le \lambda {\mu }_T(X)+(1-\lambda ){\mu }_T(Y)$ for $\lambda \in [0,1]$ and $T\ge 0$.

6. If $\left\{X_n\right\}$ is a sequence of closed chains of ${\mathfrak{M}}_{L_{loc}^p({\mathbb{R}}^N)}$ such that $X_{n+1}\subset X_n$ for $n=1,2,\dots$ and if $\underset{n\to \mathrm{\infty }}{lim}{\mu }_T(X_n)=0$ for each $T\ge 0$, then the intersection set $X_{\mathrm{\infty }}=\underset{n=1}{\overset{\mathrm{\infty }}{\cap }}{X}_n$ is nonempty.

We say that a family of measures of noncompactness is regular, if it additionally satisfies the following conditions:

1. ${\mu }_T(X\cup Y)=max\left\{{\mu }_T(X),{\mu }_T(Y)\right\}$ for $T\ge 0$.

2. ${\mu }_T(X+Y)\le {\mu }_T(X)+{\mu }_T(Y)$ for $T\ge 0$.

3. ${\mu }_T(\lambda X)=\left|\lambda \right|{\mu }_T(X)$ for $T\ge 0$ and $\lambda \in \mathbb{R}$.

4. $ker\left\{{\mu }_T\right\}={\mathfrak{N}}_{L_{loc}^p({\mathbb{R}}^N)}$ for $T\ge 0$.

Theorem 2.5. Suppose $1\le p<\mathrm{\infty }$ and $X$ is a bounded subset of the space $L_{loc}^p({\mathbb{R}}^N).$ For $x\in X$, and $\epsilon >0$ let

${\omega }^T(x,\epsilon )=sup\left\{\parallel {\mathcal{T}}_{h}x-x{\parallel }_{L^p({[-T,T]}^N)}:\parallel h{\parallel }_{{\mathbb{R}}^N}<\epsilon \right\},$

${\omega }^T(X,\epsilon )=sup\left\{{\omega }^T(x,\epsilon ):x\in X\right\},$

${\mu }_T(X)=\underset{\epsilon \to 0}{lim}{\omega }^T(X,\epsilon ),$

where ${\mathcal{T}}_{h}x(t)=x(t+h)$. Then ${\mu }_T:{\mathfrak{M}}_{L_{loc}^p({\mathbb{R}}^N)}\to {\mathbb{R}}_{+}$, $T>0$ defines a regular family of measures of noncompactness on $L_{loc}^p({\mathbb{R}}^N).$

Proof. Assume that $X\in ker\left\{{\mu }_T\right\}$, then ${\mu }_T(X)=0$ for each $T>0$ and therefore $\underset{\epsilon \to 0}{lim}{\omega }^T(X,\epsilon )=0$ for all $T>0$. Thus, for each $\eta >0$, $\delta >0$ exists such that ${\omega }^T(X,\delta )<\eta$. It implies that $\parallel {\mathcal{T}}_{h}x-x{\parallel }_{L^p({[-T,T]}^N)}<\eta ,$ for all $x\in X$ and $h\in {\mathbb{R}}^N$ with $\parallel h{\parallel }_{{\mathbb{R}}^N}<\delta .$ This means that,

${\left({\int }_{{[-T,T]}^N}|x(t+h)-x(t){|}^{p}dt\right)}^{\frac{1}{p}}<\eta ,x\in X,\parallel h{\parallel }_{{\mathbb{R}}^N}<\delta .$

On the other hand, we have

$sup\left\{{({\int }_{{[-T,T]}^N}|x(t){|}^{p}dt)}^{\frac{1}{p}}:x\in X\right\}<\mathrm{\infty }$

for all $T>0$. According to Theorem 2.3, we infer that the closure of $X$ in $L_{loc}^p({\mathbb{R}}^N)$ is compact and $ker\left\{{\mu }_T\right\}\subseteq {\mathfrak{N}}_{L_{loc}^p}({\mathbb{R}}^N).$ Thus $1^{\circ }$ holds.

$2^{\circ }$ is obvious by the definition of ${\mu }_T$.

Now, we prove $3^{\circ }$. Suppose that $X\in {\mathfrak{M}}_{L_{loc}^p}({\mathbb{R}}^N)$ and $x\in \bar{X}$. Therefore, a sequence $\left\{x_n\right\}$ in $X$ exists such that $\left\{x_n\right\}$ converges to $x$ in $L_{loc}^p({\mathbb{R}}^N).$ From the definition of ${\omega }^T(x,\epsilon ),$ we have

$\parallel {\mathcal{T}}_h{x}_n-x_n{\parallel }_{L^p({[-T,T]}^N)}\le {\omega }^T(X,\epsilon )$

for any $n\in \mathbb{N}$, $T>0$ and $h\in {\mathbb{R}}^N$ with $\parallel h{\parallel }_{R^N}<\epsilon .$ By letting $n\to \mathrm{\infty }$, we obtain

$\parallel {\mathcal{T}}_{h}x-x{\parallel }_{L^p({[-T,T]}^N)}\le {\omega }^T(X,\epsilon ).$

Therefore,

$\underset{\epsilon \to 0}{lim}{\omega }^T(\bar{X},\epsilon )\le \underset{\epsilon \to 0}{lim}{\omega }^T(X,\epsilon ).$

Consequently, ${\mu }_T(\bar{X})\le {\mu }_T(X)$, from $2^{\circ }$ we infer that ${\mu }_T(\bar{X})={\mu }_T(X).$

The properties $4^{\circ }$ and $5^{\circ }$ are simple consequences of the following inequality

$\parallel \lambda x+(1-\lambda )y{\parallel }_{L^p({[-T,T]}^N)}\le \lambda \parallel x{\parallel }_{L^p({[-T,T]}^N)}+(1-\lambda )\parallel y{\parallel }_{L^p({[-T,T]}^N)},\lambda \in [0,1].$

To prove $6^{\circ }$, assume that $X_n\in {\mathfrak{M}}_{L_{loc}^p({\mathbb{R}}^N)}$, $X_n=\overline{X_n}$, $X_{n+1}\subset X_n$ for $n=1,2,\dots$ and $\underset{n\to \mathrm{\infty }}{lim}{\mu }_T(X_n)=0.$ For any $n\in \mathbb{N}$, take an $x_n\in X_n.$ In the first step, we claim that $F=\overline{\left\{x_n\right\}}$ is a compact set in $L_{loc}^p({\mathbb{R}}^N).$ To establish this claim, we need to check conditions $(i)$ and $(ii)$ of Theorem 2.3. Let $\epsilon >0$ be fixed and take any $T>0$. Since $\underset{n\to \mathrm{\infty }}{lim}{\mu }_T(X_n)=0$, then $k\in \mathbb{N}$ exists such that

${\mu }_T(X_k)<\epsilon .$

Hence, we can find ${\delta }_1>0$ sufficiently small so that

${\omega }^T(X_k,{\delta }_1)<\epsilon .$

Thus, for all $n\ge k$ and $h\in {\mathbb{R}}^N$ with $\parallel h{\parallel }_{R^N}<{\delta }_1$, we can write

$\parallel {\mathcal{T}}_h{x}_n-x_n{\parallel }_{L^p({[-T,T]}^N)}\le {\omega }^T(X_n,{\delta }_1)<\epsilon .$

The set $\left\{x_1,x_2,\dots ,x_{k-1}\right\}$ is compact, hence ${\delta }_2>0$ and $T>0$ exist such that

$\parallel {\mathcal{T}}_h{x}_n-x_n{\parallel }_{L^p({[-T,T]}^N)}<\epsilon$

for $n=1,\dots ,k-1$, $T>0$ and $h\in {\mathbb{R}}^N$ with $\parallel h{\parallel }_{{\mathbb{R}}^N}<{\delta }_2.$ It enforces that

${\left({\int }_{{[-T,T]}^N}|x_n(t+h)-x_n(t){|}^{p}dt\right)}^{\frac{1}{p}}<\epsilon$

for all $n=1,2,\dots ,k-1$, $T>0$ and $\parallel h{\parallel }_{{\mathbb{R}}^N}<{\delta }_2.$

Thus,

${\omega }^T(F,\delta )<\epsilon$

for $\delta <min\left\{{\delta }_1,{\delta }_2\right\}$ and for all $T>0.$

We know that

${\mu }_T(F)=\underset{\delta \to 0}{lim}{\omega }^T(F,\delta ).$

Then, all the hypotheses of Theorem 2.3 are satisfied and so $F$ is compact. Therefore, a subsequence $\left\{x_{n_j}\right\}$ and $x_0\in L_{loc}^p({\mathbb{R}}^N)$ exist such that $\left\{x_{n_j}\right\}$ converges to $x_0$. Since $x_n\in X_n$, $X_n=\overline{X_n}$ and $X_{n+1}\subset X_n$ for all $n\in \mathbb{N}$, we yield

$x_0\in \underset{n=1}{\overset{\mathrm{\infty }}{\cap }}{X}_n=X_{\mathrm{\infty }},$

that finishes the proof of $6^{\circ }$.

The properties $7^{\circ }$-$9^{\circ }$ are obvious. Now, we check that condition ${10}^{\circ }$ holds. Take $X\in {\mathfrak{N}}_{L_{loc}^p({\mathbb{R}}^N)}$. Thus, the closure of $X$ in $L_{loc}^p({\mathbb{R}}^N)$ is compact. By Theorem 2.3, for any $\epsilon >0$ and $T>0$, $\delta >0$ exists such that

${\left({\int }_{{[-T,T]}^N}|x(t+h)-x(t){|}^{p}dt\right)}^{\frac{1}{p}}<\epsilon$

for all $x\in X$, and $h\in {\mathbb{R}}^N$ with $\parallel h{\parallel }_{{\mathbb{R}}^N}<\delta .$

Then for all $x\in X$ we have

${\omega }^T(x,\delta )=sup\left\{\parallel {\mathcal{T}}_{h}x-x{\parallel }_{L^p({[-T,T]}^N)}:\parallel h{\parallel }_{R^N}<\delta \right\}\le \epsilon .$

Therefore,

${\omega }^T(X,\delta )=sup\left\{{\omega }^T(x,\delta ):x\in X\right\}\le \epsilon .$

It in turn implies that

${\mu }_T(X)=\underset{\delta \to 0}{lim}{\omega }^T(X,\delta )=0$

for all $T>0$. Taking into account $1^{\circ }$, we obtain $ker\left\{{\mu }_T\right\}={\mathfrak{N}}_{L_{loc}^p({\mathbb{R}}^N)}.$

The following theorem is a version of Darbo’s fixed point theorem in $L_{loc}^p({\mathbb{R}}^N)$.

Theorem 2.6. Let $C$ be a nonempty, closed and convex subset of a Fréchet space $L_{loc}^p({\mathbb{R}}^N)$ and ${\left\{{\mu }_T\right\}}_{T>0},$ be a family of measures of noncompactness on $L_{loc}^p({\mathbb{R}}^N)$. Let $F:C\to C$ be a continuous operator such that

(2.4) ${\mu }_T(FX)\le k_T{\mu }_T(X),$(2.4)

where $k_T\in [0,1)$ for all $T>0.$ Then $F$ has at least one fixed point in the set $C$.

Proof. By induction, we define a sequence $\left\{C_n\right\}$ by letting $C_0=C$ and $C_n=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(F{C}_{n-1})$, $n\ge 1.$ We have $C_1=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(F{C}_0)\subseteq C_0$, therefore by continuing this process we obtain

$C_0\supseteq C_1\supseteq C_2\supseteq \dots .$

If ${\mu }_T(C_N)=0$ for some integer $N\ge 0$, and for all $T\ge 0$, then $C_N$ is relatively compact. Thus, Theorem 1.2 implies that $F$ has a fixed point. Now, assume $T_1\ge 0$ exists such that ${\mu }_{T_1}(C_n)\ne 0$ for any $n\ge 0$. By (2.4) we have

(2.5) ${\mu }_{T_1}(C_{n+1})={\mu }_{T_1}(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(F{C}_n))={\mu }_{T_1}(F{C}_n)\le k_{T_1}{\mu }_{T_1}(C_n).$(2.5)

Since $k_{T_1}\in [0,1)$, then $\left\{{\mu }_{T_1}(C_n)\right\}$ is a positive decreasing sequence of real numbers. Thus, there is an $r\ge 0$ such that ${\mu }_{T_1}(C_n)\to r$ as $n\to \mathrm{\infty }.$ We show that $r=0$. Suppose, to the contrary that $r=0$. Then from (2.5) we have

$\underset{n\to \mathrm{\infty }}{limsup}{\mu }_{T_1}(C_{n+1})\le \underset{n\to \mathrm{\infty }}{limsup}{k}_{T_1}{\mu }_{T_1}(C_n).$

It enforces that $1\le k_{T_1}$, which is a contradiction. Consequently $r=0$, and hence ${\mu }_{T_1}(C_n)\to 0,$ as $n\to \mathrm{\infty }.$ Using this fact and since the sequence $\left\{C_n\right\}$ is nested, in view of part $6^{\circ }$ of Definition 2.4, we conclude that the set $C_{\mathrm{\infty }}=\underset{n=1}{\overset{\mathrm{\infty }}{\cap }}{C}_n$ is nonempty, closed, convex and $C_{\mathrm{\infty }}\subset C.$ Furthermore, the set $C_{\mathrm{\infty }}$ is invariant under the operator $F,$ and $C_{\mathrm{\infty }}\in ker\left\{{\mu }_T\right\}$. By applying Theorem 1.2, we find that the operator $F$ has a fixed point.

3. Application

To verify the applicability of our results, in the following section, we shall present an existence result for a large class of nonlinear functional integral equations of convolution type on the space $L_{loc}^p({\mathbb{R}}^N)$. We provide an illustrative example to show the effectiveness and applicability of our results.

Definition 3.1. (Aghajani et al., 2015) We say that a function $f:{\mathbb{R}}^N\times {\mathbb{R}}^M\to \mathbb{R}$ satisfies the Carathéodory conditions if:

$(i)$ The function $f(.,u)$ is measurable for any $u\in {\mathbb{R}}^M,$

$(ii)$ The function $f(x,.)$ is continuous for almost all $x\in {\mathbb{R}}^N.$

Theorem 3.2. Assume that the following conditions are satisfied.

$(i)$ $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ satisfies the Caratéodory conditions, and a constant $\lambda \in [0,1)$ and a function $a\in L_{loc}^p({\mathbb{R}}^N)$ exist such that

$\left|f(x,u)-f(y,v)\left|\le \right|a(x)-a(y)\left|+\lambda \right|u-v\right|$

for any $x,y\in {\mathbb{R}}^N$ and almost all $u,v\in \mathbb{R}$.

$(ii)f(.,0)\in L_{loc}^p({\mathbb{R}}^N).$

$(iii)$ The operator $Q$ acts continuously from the space $L_{loc}^p({\mathbb{R}}^N)$ into itself and an increasing function $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ exists such that

$\parallel Qu{\parallel }_{L_{loc}^p({\mathbb{R}}^N)}\le \psi (\parallel u{\parallel }_{L_{loc}^p({\mathbb{R}}^N)})$

for any $u\in L_{loc}^p({\mathbb{R}}^N).$

$(iv)k\in L_{loc}^1({\mathbb{R}}^N).$

$(v)$ A function $r:(0,\mathrm{\infty })\to {\mathbb{R}}_{+}$ exists such that the following relation holds for all $x\in {\mathbb{R}}^N$ and all $T>0$.

$\lambda r(T)+\parallel f(x,0){\parallel }_{L_{loc}^p({\mathbb{R}}^N)}+\psi (r(T))\parallel k{\parallel }_{L_{loc}^1({\mathbb{R}}^N)}\le r(T).$

Then, the nonlinear functional integral equation of convolution type (1.1) has at least one solution in the space $L_{loc}^p({\mathbb{R}}^N)$.

Before giving the proof, we quote the following result, which is a version of Young’s Theorem (Brezis, 2011, Theorem 4.15).

Remark 3.3. Under hypothesis $(iv)$, the linear operator $G:L_{loc}^p({\mathbb{R}}^N)\to L_{loc}^p({\mathbb{R}}^N)$ defined by

$(Gu)(x)={\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)u(y)dy,(x=(x_1,\dots ,x_N))$

is a continuous operator, and moreover, we have

$\parallel Gu{\parallel }_{L_{loc}^p({\mathbb{R}}^N)}\le \parallel k{\parallel }_{L_{loc}^1({\mathbb{R}}^N)}\parallel u{\parallel }_{L_{loc}^p({\mathbb{R}}^N)}.$

Proof. First, we define the operator $F:L_{loc}^p({\mathbb{R}}^N)\to L_{loc}^p({\mathbb{R}}^N)$ by formula

$F(u)(x)=f(x,u(x))+{\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)(Qu)(y)dy,$

where $x=(x_1,\dots ,x_N)$. By considering the Caratéodory conditions, we infer that $Fu$ is measurable for any $u\in L_{loc}^p({\mathbb{R}}^N).$ Now, we prove that $Fu\in L_{loc}^p({\mathbb{R}}^N)$ for any $u\in L_{loc}^p({\mathbb{R}}^N).$ For this purpose, one only needs to prove that $Fu\in L^p([-T,T{]}^N)$ for all $T>0.$

Let us fix $T>0$. By using condition $(i)$ we have

$\left|F(u)(x)\right|=\left|f(x,u(x))+{\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)(Qu)(y)dy\right|$

$\le \left|f(x,u(x))-f(x,0)\left|+\right|f(x,0)\left|+\right|{\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)(Qu)(y)dy\right|$

$\le \lambda \left|u(x)\right|+\left|f(x,0)\left|+\right|{\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)(Qu)(y)dy\right|$

for any $u\in L_{loc}^p({\mathbb{R}}^N)$ and for a.e. $x\in [-T,T{]}^N.$ Thus, we get

${\left({\int }_{{[-T,T]}^N}|F(u)(x){|}^{p}dx\right)}^{\frac{1}{p}}\le \lambda {\left({\int }_{{[-T,T]}^N}|u(x){|}^{p}dx\right)}^{\frac{1}{p}}+{\left({\int }_{{[-T,T]}^N}|f(x,0){|}^{p}dx\right)}^{\frac{1}{p}}$

$+{\left({\int }_{{[-T,T]}^N}|{\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)(Qu)(y)dy{|}^{p}dx\right)}^{\frac{1}{p}}.$

According to Remark 3.3 and $(iii)$ we deduce

$\parallel Fu{\parallel }_{L^p({[-T,T]}^N)}\le \lambda \parallel u{\parallel }_{L^p({[-T,T]}^N)}+\parallel f(x,0){\parallel }_{L^p({[-T,T]}^N)}$

$+\psi (\parallel u{\parallel }_{L^p({[-T,T]}^N)})\parallel k{\parallel }_{L^1({[-T,T]}^N)}.$

Hence, $Fu\in L_{loc}^p({\mathbb{R}}^N)$ and $F$ is well defined. We consider the subset $B$ of $L_{loc}^p({\mathbb{R}}^N)$ given by

$B=\{u\in L_{loc}^p({\mathbb{R}}^N):\parallel u{\parallel }_{L^p({[-T,T]}^N)}\le r(T)\mathrm{f}\mathrm{o}\mathrm{r}T>0\}.$

Then, the subset $B$ is nonempty, convex, closed and bounded in $L_{loc}^p({\mathbb{R}}^N)$, and condition $(v)$ ensures that $F$ transforms $B$ into itself. Furthermore, $F$ is continuous in $L_{loc}^p({\mathbb{R}}^N)$, because $f(x,.)$, $k$ and $Q$ are continuous for a.e. $x\in [-T,T{]}^N.$

In order to finish the proof, we show that $Fu$ satisfies in Theorem 2.6.

Let us take a nonempty and bounded subset $X$ of the set $B$ and an arbitrary positive number $T>0$. Let $u\in X$ and $x,h\in [-T,T{]}^N,$ with $\parallel h{\parallel }_{R^N}<\epsilon$ for arbitrarily small positive number $\epsilon$. We obtain

$\left|F(u)(x+h)-F(u)(x)\right|$

$=|f(x+h,u(x+h))+{\int }_{[-x_1-h_1,x_1+h_1]\times \dots \times [-x_N-h_N,x_N+h_N]}k(x+h-y)(Qu)(y)dy$

$-\left(f(x,u(x))+{\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)(Qu)(y)dy\right)|$

$\le \left|f(x+h,u(x+h))-f(x+h,u(x))\left|+\right|f(x+h,u(x))-f(x,u(x))\right|$

$+|{\int }_{[-x_1-h_1,x_1+h_1]\times \dots \times [-x_N-h_N,x_N+h_N]}k(x+h-y)(Qu)(y)dy$

$-{\int }_{[-x_1,x_1]\times \dots \times [-x_N,x_N]}k(x-y)(Qu)(y)dy|$

$\le \lambda \left|u(x+h)-u(x)\right|+\left|a(x+h)-a(x)\right|+\left|{\int }_{{[-T,T]}^N}(k(x+h-y)-k(x-y))(Qu)(y)dy\right|.$

Therefore, in view of Young’s Theorem we get

${\left({\int }_{{[-T,T]}^N}|F(u)(x+h)-F(u)(x){|}^{p}dx\right)}^{\frac{1}{p}}$

$\le \lambda {\left({\int }_{{[-T,T]}^N}|u(x+h)-u(x){|}^{p}dx\right)}^{\frac{1}{p}}+{\left({\int }_{{[-T,T]}^N}|a(x+h)-a(x){|}^{p}dx\right)}^{\frac{1}{p}}$

$+{\left({\int }_{{[-T,T]}^N}{\left|{\int }_{{[-T,T]}^N}\left|(k(x+h-y)-k(x-y))(Qu)(y)\right|dy\right|}^{p}dx\right)}^{\frac{1}{p}}$

$\le \lambda {\omega }^T(u,\epsilon )+{\omega }^T(a,\epsilon )+\parallel Qu{\parallel }_{L^p({[-T,T]}^N)}\parallel {\mathcal{T}}_{h}k-k{\parallel }_{L^1({[-T,T]}^N)}.$

By using the above estimate we have

${\omega }^T(FX,\epsilon )\le \lambda {\omega }^T(X,\epsilon )+{\omega }^T(a,\epsilon )+\psi (\parallel u{\parallel }_{L^p({[-T,T]}^N)})\parallel {\mathcal{T}}_{h}k-k{\parallel }_{L^1({[-T,T]}^N)}.$

Since $\left\{a\right\}$ is a compact set in $L_{loc}^p({\mathbb{R}}^N)$ and $\left\{k\right\}$ is a compact set in $L_{loc}^1({\mathbb{R}}^N)$, we have ${\omega }^T(a,\epsilon )\to 0$, $\parallel {\mathcal{T}}_{h}k-k{\parallel }_{L_{loc}^1({\mathbb{R}}^N)}\to 0$ as $\epsilon \to 0.$ Then we obtain

(3.1) ${\mu }_T(FX)\le {\lambda }_T{\mu }_T(X),$(3.1)

where ${\lambda }_T=\lambda$ for each $T>0$. By (3.1) and Theorem 2.6 we infer that, the operator $F$ has a fixed point $u$ in $B$ and therefore the nonlinear functional integral equation of convolution type (1.1) has at least one solution in the space $L_{loc}^p({\mathbb{R}}^N).$□

Example 3.4. Consider the following functional integral Equation (3.2)

(3.2) $u(x)=\frac{cosu(x)}{\parallel x\parallel +5}+{\int }_{[-x_1,x_1]\times [-x_2,x_2]}{e}^{2(x_1-y_1)}(x_2-y_2+1{)}^{2}cos(\left|x_1-y_1\left|)sin(\right|2x_2-3y_2\left|)ln(\right|u(x)\right|+1)dy,$(3.2)

where $x=(x_1,x_2)\in {\mathbb{R}}^2$ and $\parallel x\parallel$ is the Euclidean norm. We study the solvability of integral Equation (3.2) on the space $L_{loc}^p({\mathbb{R}}^2)$ for $p>2.$ Observe that Equation (3.2) is a special case of the Equation (1.1) when

$f(x,u)=\frac{cosu(x)}{\parallel x\parallel +5},$

$k(x)=e^{2x_1}(x_2+1{)}^{2}cos(\left|x_1\left|)sin(\right|2x_2\right|),(x=(x_1,x_2))$

$Qu(x)=ln(\left|u(x)\right|+1).$

In this example, hypothesis $(i)$ of Theorem 3.2 holds with $a(x)=\frac{1}{\parallel x\parallel +5}$ and $\lambda =\frac{1}{5}$. Indeed, we have

$\left|f(x,u)-f(y,v)\left|=\right|\frac{cosu}{\parallel x\parallel +5}-\frac{cosv}{\parallel y\parallel +5}\right|$

$\le \left|\frac{1}{\parallel x\parallel +5}-\frac{1}{\parallel y\parallel +5}\right|\left|cosv\right|+\frac{1}{\parallel x\parallel +5}\left|cosu-cosv\right|$

$\le \left|\frac{1}{\parallel x\parallel +5}-\frac{1}{\parallel y\parallel +5}\right|+\frac{1}{5}\left|u-v\right|.$

It is easy to see that $f(.,0)$ satisfies assumption $(ii)$. In fact, for each $T>0$ we can write

$\underset{-T}{\overset{T}{\int }}\underset{-T}{\overset{T}{\int }}\left|\frac{1}{\parallel x\parallel +5}{|}^{p}d{x}_{1}d{x}_2\le \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\right|\frac{1}{\parallel x\parallel +5}{|}^{p}d{x}_{1}d{x}_2$

$=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{r}{{(r+5)}^p}drd\theta$

$=\frac{2\pi }{(p^2-3p+2)5^{p-2}}$

for all $p>2$. Thus, we have $\parallel f(.,0){\parallel }_{L_{loc}^p({\mathbb{R}}^2)}\le (\frac{2\pi }{(p^2-3p+2)5^{p-2}}{)}^{\frac{1}{p}}$ and so $f(.,0)\in L_{loc}^p({\mathbb{R}}^2)$.

Furthermore, $Qu(x)=ln(\left|u(x)\right|+1)$ satisfies hypothesis $(iii)$ with $\psi (t)=t$. Obviously, the function $Qu(x)=ln(\left|u(x)\right|+1)$ is increasing and $ln(\left|u(x)\right|+1)<\left|u(x)\right|$. Next, we show that $k$ satisfies assumption $(iv)$. Indeed, we have

$\underset{-T}{\overset{T}{\int }}\underset{-T}{\overset{T}{\int }}\left|{(x_2+1)}^{2}cos(\left|x_1\left|)sin(\right|2x_2\right|)e^{2x_1}\right|d{x}_{1}d{x}_2$

$\le \underset{-T}{\overset{T}{\int }}\underset{-T}{\overset{T}{\int }}(x_2+1{)}^2{e}^{2x_1}d{x}_{1}d{x}_2=\frac{1}{3}(T^3+3T)(\frac{e^{4T}-1}{e^{2T}}).$

Finally, the inequality from assumption $(v)$, has the form

$\frac{1}{5}r(T)+(\frac{2\pi }{(p^2-3p+2)5^{p-2}}{)}^{\frac{1}{p}}+\frac{1}{3}(T^3+3T)(\frac{e^{4T}-1}{e^{2T}})r(T)\le r(T),$

so we can take

$r(T)=\frac{{(\frac{2\pi }{(p^2-3p+2)5^{p-2}})}^{\frac{1}{p}}}{\frac{4}{5}-\frac{1}{3}(T^3+3T)(\frac{e^{4T}-1}{e^{2T}})}\mathrm{f}\mathrm{o}\mathrm{r}p>2.$

Consequently, all the conditions of Theorem 3.2 are satisfied, and thus functional integral equation of convolution type (3.2) has at least one solution in the space $L_{loc}^p({\mathbb{R}}^2)$ if $p>2$.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Hojjatollah Amiri Kayvanloo

The authors have studied the existence of solutions for some classes of nonlinear functional integral equations of convolution type by using measure of noncompactness. In particular, they introduced a new family of measures of noncompactness on the Frechet space l^p_loc(ℝ^N) Also, by this family of measures of noncompactness, they investigated the existence of solutions for some classes of nonlinear functional integral equations. The results of their article extend and improve previously known results.