1 Introduction

Variable exponent Lebesgue spaces \(L^{p(\cdot )}\) are a proper tool to study the models with nonstandard growth condition such as elasticity theory [19], electrorheological fluids [32], fluid mechanics [14], differential equations and variational problems [8], image restoration [21], or nonlinear elastic mechanics [34]. The integral provides the model of energy in fluid dynamics [11] in the form

$$ \int _{\omega } \bigl\vert {Du(t)} \bigr\vert ^{p(t)} \,dt, $$

where the exponent is a function of the electric field and Du is a symmetric portion of the gradient of the velocity field.

It is beneficial and useful to analyze and investigate the solutions of many types of (differential, integral, functional, and partial differential) equations in various function spaces using the technique of measures of noncompactness (MNCs) related with fixed point theorems, see [3, 6, 10, 12, 23–26, 31].

That technique has lately been developed in many research papers by constructing new MNCs in numerous function spaces and applying these results to study the solutions of various types of problems in the considered spaces. Recall that the MNCs were demonstrated and defined in the Banach algebras \(C(I)\), \(BC(\mathbb{R}^{+})\) [5], in the space of regular functions [9, 17, 22], in the space of all locally integrable functions \(L^{1}_{\mathrm{loc}}(\mathbb{R}^{+})\) [28], in the space of Lebesgue integrable functions \(L_{1}(\mathbb{R}^{N})\) [7], in the classical Lebesgue spaces \(L^{p}(\mathbb{R}^{N})\) [1, 27], and in the Sobolev space \(W^{k,1}(I)\) [20]. We extend and generalize the above results to the case of \(L^{p(\cdot )}(\mathbb{R}^{+})\)-spaces.

To fulfill this gap, we construct a regular measure of noncompactness in \(L^{p(\cdot )}\)-spaces on unbounded domain \(\mathbb{R}^{+}\) and apply it with the help of Darbo’s fixed point approach in studying the following integral equation in the studied spaces:

$$ x(t)= g(t) + f_{1}\bigl(t,x(t)\bigr)+ \int _{0}^{\infty }K(t,s)f_{2}\bigl(s,x(s) \bigr) \,ds. $$
(1.1)

The importance of studying \(L^{p(\cdot )}\)-spaces is that they are logical extensions of classical constant exponent \(L_{p}\)-spaces, and the investigations of the solutions of the integral equations are naturally studied in these spaces. The \(L^{p(\cdot )}\)-spaces are not rearrangement invariant, the translation operator is not bounded, Young’s convolution inequality does not hold when compared to the theory of Lp-spaces (cf. [13, Sect. 3.6]).

Moreover, that technique is simple to use when examining the solutions of various kinds of problems in \(L^{p(\cdot )}(\mathbb{R}^{+})\), which has a wide range of applications that can be utilized to get beyond the restrictions of Lp-spaces.

2 Notation and auxiliary facts

Let \({\mathbb{R}}=(-\infty ,\infty )\) and \({\mathbb{R}^{+}}=[0,\infty )\). By \(L^{p(\cdot )}(\mathbb{R}^{+}) \) we denote the space of functions \(f(t)\) on \(\mathbb{R}^{+}\) s.t.

$$ I_{p}(f)= \int _{0}^{\infty } \bigl\vert {f(t)} \bigr\vert ^{p(t)} \,dt< \infty , $$

where \(p(t)\) is a measurable function on \(\mathbb{R}^{+}\) with values in \([0, \infty )\) and define \(p- = \operatorname{ess} \inf_{t>0} p(t)\) and \(p+ = \operatorname{ess} \sup_{t >0} p(t)\).

The spaces \(L^{p(\cdot )}(\mathbb{R}^{+})= L^{p(\cdot )}\) are the Banach spaces with the norm

$$ \Vert f \Vert _{p(\cdot )}= \Vert f \Vert _{L^{p(\cdot )}( \mathbb{R}^{+})}= \inf \biggl\{ \lambda >0: I_{p} \biggl( \frac{f}{\lambda} \biggr)\leq 1 \biggr\} , $$
(2.1)

which corresponds to the well-known Luxemburg norm in Orlicz spaces. If \(p(x) = p\) is a constant function, then norm (2.1) coincides with the usual \(L_{p}\)-norm.

The next concepts will be useful and helpful with the framework.

The Hölder inequality is written as follows in \(L^{p(\cdot )}\) [11]: If we assume that f and g are in \(L^{p(\cdot )}\) and \(L^{q(\cdot )}\), respectively, we have \(f\cdot g \in L^{1}( \mathbb{R}^{+})\), where \(\frac{1}{p(\cdot )} + \frac{1}{q(\cdot )} =1\), and

$$ \int _{0}^{\infty } \bigl\vert f(t) g(t) \bigr\vert \,dt \leq k_{p(\cdot )} \Vert f \Vert _{p(\cdot )} \Vert g \Vert _{q(\cdot )}, $$

where \(1\leq p(\cdot ) \leq \infty \) and \(1< k_{p(\cdot )}\leq 4\).

Proposition 2.1

[15, Remark 2.1] Suppose that the sequence \(\{x_{k}\} \subset L^{p(\cdot )} \) converges in norm to \(h \in L^{p(\cdot )} \). Then there exists a subsequence \(\{h_{k_{j}}\}\) and \(g\in L^{p(\cdot )} \) such that the subsequence converges pointwise a.e. to h and, for almost every \(t \in{\mathbb{R}}^{+}\), \(|h_{k_{j}}(t)|\leq g(t)\).

Definition 2.2

Consider that a function \(f(t,x): {\mathbb{R}}^{+} \times {\mathbb{R}} \rightarrow {\mathbb{R}}\) fulfills the Carathéodory conditions i.e. it is continuous in x for almost all \(t \in {\mathbb{R}}^{+}\) and measurable in t for any \(x \in {\mathbb{R}}\). The superposition (Nemytskii) operator \(F_{f}\) can thus be denoted for each measurable function x in the following manner:

$$ F_{f}(x) (t) = f\bigl(t ,x(t)\bigr) ,\quad t\in { \mathbb{R}}^{+} . $$

Lemma 2.3

[18] If \(F_{f}: L^{p(\cdot )} \to L^{q(\cdot )} \), then \(F_{f}\) is continuous and bounded, and there is a constant \(b \geq 0\) and a nonnegative function \(a \in L^{q(\cdot )} \) such that, for \(t \in \mathbb{R}^{+}\) and \(x \in \mathbb{R}\), the following inequality holds:

$$ \bigl\vert f(t,x) \bigr\vert \leq a(t) + b \vert x \vert ^{p(t)/q(t)}. $$
(2.2)

On the other hand, if \(f(t,x)\) satisfies (2.2), then \(F_{f}: L^{p(\cdot )} \to L^{q(\cdot )} \), and thus \(F_{f}\) is continuous and bounded.

Next, let us assume that \((E, \|\cdot \|)\) is a Banach space that has zero element denoted by θ and that \(B_{r}=B(r, \theta )\) denotes a ball with a radius r and a center at θ.

Let \(\boldsymbol{\mathcal{N}}_{E}\) and \(\emptyset \neq \boldsymbol{\mathcal{M}}_{E}\) be, respectively, the subfamily containing all relatively compact sets of E and the family of all nonempty and bounded subsets of E. Convex closed hull and closure of a set Y are denoted by the symbols Conv Y and Y̅, respectively.

Definition 2.4

[4] The function \(\mu :\boldsymbol{\mathcal{M}}_{E} \rightarrow [0, \infty )\) is called a measure of noncompactness (MNC) in E if it fulfills:

  1. (1)

    The family \(\ker \mu = \{ X \in \boldsymbol{\mathcal{M}}_{E} : \mu (X) = 0\}\) is nonempty and \(\ker \mu \subset \boldsymbol{\mathcal{N}}_{E}\).

  2. (2)

    \(Y \subset X \Rightarrow \mu (Y) \leq \mu (X )\).

  3. (3)

    \(\mu ( \overline{Y} ) = \mu (Y)\).

  4. (4)

    \(\mu (Conv Y) = \mu (Y)\).

  5. (5)

    \(\mu (\lambda Y + (1-\lambda )Y) \leq \lambda \mu (Y)+ (1-\lambda ) \mu (Y)\), for \(\lambda \in [0,1]\).

  6. (6)

    If \(\emptyset \neq{Y_{n}} \subset E\) is a sequence of closed and bounded sets, \(Y_{n}=\overline{Y}_{n}\) s.t. \(Y_{n+1} \subset Y_{n}\), \(n = 1, 2, 3, \ldots \) , and \(\lim_{n\rightarrow \infty} \mu (Y_{n}) = 0\), then \(Y_{\infty }= \bigcap_{n=1}^{\infty }Y_{n} \neq \emptyset \).

    We say that an MNC is regular if it additionally satisfies the following axioms:

  7. (7)

    \(\mu (Y \cup X) = \max \{\mu (Y), \mu (X)\}\).

  8. (8)

    \(\mu (Y + X) \leq \mu (Y) + \mu (X)\).

  9. (9)

    \(\mu (\lambda Y) = | \lambda | \mu (Y)\), for \(\lambda \in { \mathbb{R}}\).

  10. (10)

    \(\ker \mu = \boldsymbol{\mathcal{N}}_{E}\).

Examples of MNCs that fit all the axioms listed above are the Kuratowski and Hausdorff MNCs [4].

It will be necessary to use the following Darbo’s fixed point theorem.

Theorem 2.5

[4] Let \(\emptyset \neq Q \subset E\) be a bounded, closed, and convex set and let \(V : Q \rightarrow Q\) be a continuous map** that is a contraction with respect to the MNC μ i.e. there exists \(k \in [0, 1)\) s.t.

$$ \mu \bigl(V(X)\bigr) \leq k\mu (X) $$

for any nonempty \(X \subset E\). Then V has at least one fixed point in Q.

3 Construction of MNC in the space \(L^{p(\cdot )} \)

In this section, we establish a new measure of noncompactness in \(L^{p(\cdot )} \) and investigate its properties. We begin by introducing an extension of the well-known Riesz–Kolmogorov compactness criterion in \(L^{p(\cdot )} \).

Definition 3.1

A subset \(\boldsymbol{\mathcal{F}}\) of \(L^{p(\cdot )} \) is called precompact if its closure is compact.

The Riesz–Kolmogorov compactness criterion in \(L^{p(\cdot )} \) can be stated as follows (cf. [19, Theorem 5] and [16, Theorem 2.1]).

Lemma 3.2

Let \(\boldsymbol{\mathcal{F}}\) be a subset of \(L^{p(\cdot )} \), then the family \(\boldsymbol{\mathcal{F}}\) is precompact in \(L^{p(\cdot )} \) if and only if the following conditions hold:

  1. (i)

    \(\boldsymbol{\mathcal{F}}\) is bounded,

  2. (ii)

    \(\forall _{\epsilon >0}\) \(\exists _{\delta >0}\) \(\forall _{h< \delta}\) \(\forall _{f \in \boldsymbol{\mathcal{F}}}\) \(\|f(\cdot +h)-f( \cdot )\|_{p(\cdot ) } < \epsilon \),

  3. (iii)

    \(\forall _{\epsilon >0}\) \(\exists _{T>0}\) \(\forall _{f \in \boldsymbol{\mathcal{F}}}\) \(\| f\|_{L^{p(\cdot )}[T,\infty )} < \epsilon \).

Theorem 3.3

Assume that \(\emptyset \neq X \subset L^{p(\cdot )} \) is a bounded set. For \(x\in X\) and \(\epsilon >0\), let

$$\begin{aligned}& \omega (x, \epsilon ) = \sup \bigl\{ \bigl\Vert x(\cdot +h)-x(\cdot ) \bigr\Vert _{p( \cdot ) }: \vert h \vert < \epsilon \bigr\} , \\& \omega (X, \epsilon ) = \sup \bigl\{ \omega (x, \epsilon ) : x \in X \bigr\} , \\& \omega (X) = \lim_{\epsilon \to 0}\omega (X, \epsilon ). \end{aligned}$$

Also, let

$$\begin{aligned}& d_{T}(X) = \sup \bigl\{ \Vert x \Vert _{L^{p(\cdot )}[T,\infty )}: x \in X \bigr\} , \\& d(X) = \lim_{T \to \infty}d_{T}(X). \end{aligned}$$

Then \(\mu (X)=\omega (X) +d(X) :\boldsymbol{\mathcal{M}}_{L^{p(\cdot )} } \rightarrow \mathbb{R}^{+}\) represents an MNC in \(L^{p(\cdot )} \).

Proof

First, we begin by demonstrating the validity of axiom (1) of Definition 2.4.

Consider \(X \in \boldsymbol{\mathcal{M}}_{L^{p(\cdot )} }\) such that \(\mu (X)=0\). Let \(\eta >0\) be arbitrary. Since \(\mu (X)=0\), then \(\lim_{\epsilon \to 0} \omega (X, \epsilon )=0\). Therefore, for all \(\eta >0\), there exists \(\delta >0\) such that \(\omega (X, \delta ) <\eta \), and this indicates that

$$ \bigl\Vert x(\cdot +h)- x(\cdot ) \bigr\Vert _{p(\cdot ) }< \eta $$

for all \(x\in X\) and \(h \in \mathbb{R}^{+}\) such that \(|h|<\delta \). Since \(\eta >0\) is arbitrary, we get

$$ \lim_{h \to 0} \bigl\Vert x(\cdot +h)- x(\cdot ) \bigr\Vert _{p(\cdot ) }=0 $$

uniformly in \(x \in X\). Again, kee** in mind that \(\mu (X)=0\), we have

$$ \lim_{T \to \infty} d_{T}(X)=0, $$

and so for \(\epsilon >0\), there exists \(T > 0\) such that

$$ \Vert x \Vert _{L^{p(\cdot )}[T,\infty )} < \epsilon \quad \text{for all } x \in X. $$

Thus, from Lemma 3.2 we infer that the closure of \(X \in L^{p(\cdot )} \) is compact and \(\operatorname{ker} \mu \subset \boldsymbol{\mathcal{N}}_{L^{p(\cdot )} }\). The proof of axiom (2) is clear.

Now, suppose that \(X \in \boldsymbol{\mathcal{M}}_{L^{p(\cdot )} }\) and \((x_{n}) \subset X\) such that \(x_{n} \to x \in \bar{ X}\) in \(L^{p(\cdot )} \). From the definition of \(\omega (X,\epsilon )\) we have

$$ \bigl\Vert x_{n}(\cdot +h)- x(\cdot ) \bigr\Vert _{p(\cdot ) } \leq \omega (X,\epsilon ) $$

for any \(n\in \mathbb{N}\) and \(|h| <\epsilon \). Letting \(n \to \infty \), we get

$$ \bigl\Vert x(\cdot +h)- x(\cdot ) \bigr\Vert _{p(\cdot ) } \leq \omega (X, \epsilon ) \quad \text{for any } \vert h \vert < \epsilon , $$

then

$$ \lim_{\epsilon \to 0} \omega (\bar{X},\epsilon ) \leq \lim _{ \epsilon \to 0} \omega (X,\epsilon ) $$

implies that

$$ \omega (\bar{X}) \leq \omega (X). $$
(3.1)

Similarly, we may demonstrate that \(d(\bar{X}) \leq d(X)\), so from equation (3.1) and axiom (2) we get \(\mu (\bar{X}) \leq \mu (X)\) satisfies axiom (3) of Definition 2.4.

Axioms (4) and (5) can be proved similarly by using the inequality

$$ \bigl\Vert \lambda x + (1-\lambda )x \bigr\Vert _{p(\cdot )} \leq \lambda \Vert x \Vert _{p( \cdot )} + (1-\lambda ) \Vert x \Vert _{p(\cdot )} . $$

To demonstrate axiom (6), let us assume that \(\{X_{n}\}\) is a sequence of closed and nonempty sets from \(\boldsymbol{\mathcal{M}}_{L^{p(\cdot )}} \), where \(X_{ n+1}\) is a subset of \(X_{n}\) for \(n = 1, 2,\ldots \) , and \(\lim_{n\to \infty} \mu (X_{n})=0\). Now, for any \(n\in \mathbb{N}\), take \(x_{n} \in X_{n}\). We claim that \(\boldsymbol{\mathcal{F}}= {\overline{\{x_{n}\}}}\) is a compact set in \(L^{p(\cdot )}\). To prove the claim, we need to check conditions (ii) and (iii) of Lemma 3.2.

Let \(\epsilon >0\) be fixed. Since \(\lim_{n \to \infty} \mu (X_{n})=0\), there exists \(k\in \mathbb{N}\) such that \(\mu (X_{k})< \epsilon \). Hence, we can find \(\delta _{1} >0\) and \(T_{1}>0\) such that

$$ \omega (X_{k}, \delta _{1})< \epsilon \quad \text{and}\quad d_{T_{1}}(X_{k})< \epsilon . $$

Thus, for all \(n \geq k\) and \(|h|\leq \delta _{1}\), we get

$$ \omega (X_{n}, \delta _{1})< \epsilon \quad \text{and}\quad d_{T_{1}}(X_{n})< \epsilon . $$

The set \(\{x_{1}, x_{2},\ldots , x_{k-1}\}\) is compact, hence there exist \(\delta _{2} >0\) and \(T_{2}>0\) such that

$$ \omega (X_{n}, \delta _{2})< \epsilon \quad \text{and} \quad d_{T_{2}}(X_{n})< \epsilon $$

for all \(1 \leq n \leq k\). If we choose \(\delta < \min \{\delta _{1},\delta _{2}\}\) and set \(T = \max \{T_{1}, T_{2}\}\), then

$$ \omega (\boldsymbol{\mathcal{F}})=\lim_{\delta \to 0} \omega ( \boldsymbol{\mathcal{F}}, \delta )=0 \quad \text{and}\quad d(\boldsymbol{ \mathcal{F}})=\lim_{T \to \infty} d_{T}( \boldsymbol{ \mathcal{F})}=0. $$

Consequently, \(\mu (\boldsymbol{\mathcal{F}})= 0\), indicating that \(\boldsymbol{\mathcal{F}}\) is compact. It follows that there exists a subsequence \(\{x_{n_{j}}\}\) and \(x_{0} \in L^{p(\cdot )} \) such that \(x_{n_{j}}\to x_{0}\), and since for all \(x_{n} \in X_{n}\), \(X_{n+1}\subset X_{n}\) and \(X_{n}\) is closed for all \(n \in \mathbb{N}\), we have

$$ x_{0} \in \bigcap_{i=1}^{\infty }X_{n} = X_{\infty}, $$

and this completes the theorem’s proof. □

We now analyze the regularity of μ.

Theorem 3.4

The measure of noncompactness μ given in Theorem 3.3is regular.

Proof

Suppose that \(X, Y \in L^{p(\cdot )}\). Since for all \(\epsilon >0\), \(\lambda >0\), \(T>0\) we have

$$\begin{aligned}& \omega (Y \cup X,\epsilon ) \leq \max \bigl\{ \omega (Y,\epsilon ), \omega (X, \epsilon ) \bigr\} , \\& \omega (Y + X,\epsilon ) \leq \omega (Y,\epsilon )+ \omega (X, \epsilon ) , \\& \omega (\lambda Y,\epsilon ) \leq \lambda \omega (Y,\epsilon ) \end{aligned}$$

and

$$\begin{aligned}& \sup_{x \in Y \cup X} \Vert x \Vert _{L^{p(\cdot )}[T,\infty )} \leq \sup \Bigl\{ \sup_{x \in Y } \Vert x \Vert _{L^{p(\cdot )}[T,\infty )}, \sup _{x \in X} \Vert x \Vert _{L^{p(\cdot )}[T,\infty )} \Bigr\} , \\& \sup_{x \in Y+X} \Vert x \Vert _{L^{p(\cdot )}[T,\infty )} \leq \sup _{x \in Y } \Vert x \Vert _{L^{p(\cdot )}[T,\infty )} +\sup _{x \in X} \Vert x \Vert _{L^{p( \cdot )}[T,\infty )} , \\& \sup_{x \in \lambda Y } \Vert x \Vert _{L^{p(\cdot )}[T,\infty )} \leq \lambda \sup _{x \in Y } \Vert x \Vert _{L^{p(\cdot )}[T,\infty )}, \end{aligned}$$

axioms (7), (8), and (9) hold. To show that (10) holds, suppose that \(X \in \boldsymbol{\mathcal{N}}_{L^{p(\cdot )} }\). Thus, the closure of X in \(L^{p(\cdot )} \) is compact, and hence from Lemma 3.2, for any \(\epsilon > 0\), there exists \(T > 0\) such that \(d_{T}(X) < \epsilon \) and also \(\omega (X) < \epsilon \) uniformly in \(x\in X\). From the first conclusion, for all \(\eta >0\), there exists \(\delta >0\) such that \(\|x(\cdot +h)- x(\cdot )\|_{p(\cdot ) }<\eta \) for any \(|h|< \delta \). Then, for all \(x \in X\), we have

$$ \omega (x, \delta ) =\sup \bigl\{ \bigl\Vert x(\cdot +h)- x(\cdot ) \bigr\Vert _{{p( \cdot )} }: \vert h \vert < \delta \bigr\} \leq \epsilon . $$

Therefore,

$$ \omega (X, \delta ) =\sup \bigl\{ \omega (x, \delta ) : x \in X \bigr\} \leq \epsilon , $$

which proves

$$ \lim_{\delta \to 0}\omega (X, \delta )=0 $$
(3.2)

and

$$ \lim_{T \to \infty} d_{T}(X)=0. $$
(3.3)

Now from (3.2) and (3.3), axiom (10) holds. □

4 Applications

In this section, we use the results obtained in Sect. 3 to investigate the existence of solutions for the nonlinear integral equation (1.1).

The operator H can be defined as follows:

$$ x= H (x)= g+ F_{f_{1}}(x)+ Ax, $$

where

$$ Ax= K_{0} \circ F_{f_{2}},\qquad K_{0}x(t)= \int _{0}^{\infty }K(t,s)x(s) \,ds $$

and the superposition operators \(F_{f_{i}}\), \(i,i=1,2\), are the same as in Definition 2.2.

Equation (1.1) will be handled based on the following set of presumptions:

  1. (i)

    \(g(\cdot ) \in L^{p(\cdot )} \).

  2. (ii)

    For \(i=1,2\), suppose that \(f_{i}: {\mathbb{R}}^{+} \times {\mathbb{R}}\rightarrow {\mathbb{R}}\) satisfies Carathéodory conditions and there exist constants \(b_{i}\geq 0\) and functions \(a_{i} \in L^{q(\cdot )}\) such that

    $$ \bigl\vert f_{i}(t,0) \bigr\vert \leq a_{i}(t) $$

    and

    $$ \bigl\vert f_{i}(t,x) -f_{i}(s,y) \bigr\vert \leq \bigl\vert a_{i}(t) -a_{i}(s) \bigr\vert + b_{i} \vert x-y \vert ^{p(s)/q(s)} $$

    for all \(t,s \in {\mathbb{R}}^{+}\) and \(x ,y \in {\mathbb{R}}\).

  3. (iii)

    Let \(K: {\mathbb{R}}^{+} \times{\mathbb{R}}^{+} \to {\mathbb{R}}\) be measurable and there exist functions \(g_{1} \in L^{p(\cdot )}\) and \(g_{2} \in L^{q'(\cdot )}\), \(\frac{1}{p(\cdot )}+\frac{1}{q'(\cdot )}=1\) such that \(|K(t, s)| \leq g_{1} (t)g_{2}(s)\) for all \(t,s \in {\mathbb{R}}^{+}\) and

    $$ \bigl\vert K(t_{1}, s)-K(t_{2}, s) \bigr\vert \leq g_{2}(s) \bigl\vert g_{1}(t_{1}) - g_{1}(t_{2}) \bigr\vert ,\quad t_{1},t_{2},s \in {\mathbb{R}}^{+}. $$
  4. (iv)

    For \(M>1\), let r be a constant satisfying the inequalities

    $$\begin{aligned}& \Vert g \Vert _{p(\cdot )} + \bigl( M \Vert a_{1} \Vert _{q(\cdot )} + b_{1} M r^{(p/q) \pm} \bigr) \\& \quad{} + k_{q(\cdot )} \Vert g_{1} \Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'( \cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )}+ b_{2} r^{(p/q)\pm} \bigr) \leq r \end{aligned}$$

    and

    $$ b_{1} M \cdot (2r)^{(p/q)\pm} \leq 1, $$

    where the + sign occurs when \(r\geq 1\) and the − sign occurs in the case \(r\leq 1\).

Remark 4.1

Under assumption (iii), the linear integral operator \(K_{0}x(t)= \int _{0}^{\infty }K(t,s)x(s) \,ds\) maps \(L^{q(\cdot )}\) into \(L^{p(\cdot )}\).

Proof

By utilizing assumption (iii) and the Hölder inequality, we have

$$\begin{aligned} \bigl\vert K_{0}x(t) \bigr\vert \leq & \int _{0}^{\infty } \bigl\vert K(t,s) \bigr\vert \bigl\vert x(s) \bigr\vert \,ds \\ \leq & \int _{0}^{\infty }g_{1}(t)g_{2}(s) \bigl\vert x(s) \bigr\vert \,ds \\ \leq & k_{q(\cdot )} g_{1}(t) \Vert g_{2} \Vert _{q'(\cdot )} \Vert x \Vert _{q(\cdot )},\quad q'(\cdot )= \frac{q(\cdot )}{q(\cdot )-1}. \end{aligned}$$

Therefore,

$$ \Vert K_{0}x \Vert _{p(\cdot )} \leq k_{q(\cdot )} \Vert g_{1} \Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'(\cdot )} \Vert x \Vert _{q(\cdot )}, $$
(4.1)

then we get our claim. □

Remark 4.2

Under assumption (ii), we have that the superposition operator \(F_{f_{i}}:L^{p(\cdot )} \to L^{q(\cdot )}\) and is continuous.

Proof

From assumption (ii), for \(i=1,2\), we have

| f i ( t , x ) | ≤ | f i ( t , x ) − f i ( t , 0 ) | + | f i ( t , 0 ) | ≤ b i | x | p ( t ) / q ( t ) + | f i ( t , 0 ) | ⟹ | f i ( t , x ) | ≤ a i ( t ) + b i | x | p ( t ) / q ( t ) .

Since \(a_{i} \in L^{q(\cdot )}\), \(i=1,2\), and by utilizing Lemma 2.3, we get our claim. □

Proposition 4.3

[8] Since \(p(t) \leq q(t)\), \(t \in \mathbb{R}^{+}\), the operator \(F_{f_{1}} : L^{p(\cdot )}\to L^{p(\cdot )}\) is well defined, bounded, and continuous and

$$ \bigl\Vert x^{p(\cdot )/q(\cdot )} \bigr\Vert _{q(\cdot )} \leq \Vert x \Vert ^{(p/q) \pm }_{p(\cdot )}. $$

Theorem 4.4

Let assumptions (i)–(iv) be satisfied, then equation (1.1) has at least one solution \(x \in L^{p(\cdot )}(\mathbb{R}^{+})\).

Proof

We will carry out the proof in a number of steps.

Step I. We will show that the operator H is well defined on \(L^{p(\cdot )}\) and is continuous.

At the outset, according to (ii) and Remark 4.2, we have that the operators \(F_{f_{i}}: L^{p(\cdot )} \to L^{q(\cdot )}\), where \(i=1,2 \), and they are continuous. Furthermore, according to Proposition 4.3, the operator \(F_{f_{1}}: L^{p(\cdot )} \to L^{p(\cdot )}\) and is continuous.

Now, we will prove that property for the operator \(A = K_{0} \circ F_{f_{2}}\).

Considering our underlying presumptions and Remark 4.2 with the Hölder inequality, we have

$$\begin{aligned}& \int _{0}^{\infty } \int _{0}^{\infty } \bigl\vert K(t,s)f_{2} \bigl(s, x(s)\bigr) \bigr\vert \,ds \,dt \\& \quad \leq \int _{0}^{\infty } \int _{0}^{\infty } \bigl\vert K(t,s) \bigr\vert \bigl(a_{2}(s) + b_{2} \bigl\vert x(s) \bigr\vert ^{p(s)/q(s)} \bigr) \,ds \,dt \\& \quad \leq \int _{0}^{\infty }k_{q(\cdot )} \bigl\Vert K(t, \cdot ) \bigr\Vert _{q'(\cdot )} \bigl\Vert a_{2} + b_{2} \vert x \vert ^{p(\cdot )/q(\cdot )} \bigr\Vert _{{q(\cdot )}} \,dt \\& \quad \leq k_{q(\cdot )} \bigl( \Vert a_{2} \Vert _{{q(\cdot )}} + b_{2} \bigl\Vert x^{p( \cdot )/q(\cdot )} \bigr\Vert _{{q(\cdot )}} \bigr) \int _{0}^{\infty } \bigl\Vert g_{1}(t) g_{2}(\cdot ) \bigr\Vert _{{q'(\cdot )}} \,dt \\& \quad \leq k_{q(\cdot )} \bigl( \Vert a_{2} \Vert _{{q(\cdot )}} + b_{2} \Vert x \Vert _{{p( \cdot )}} ^{(p/q)\pm} \bigr) k_{p(\cdot )} \Vert g_{1} \cdot \Vert g_{2}\|_{{q'( \cdot )}}\|_{{p(\cdot )}} \Vert 1\|_{{p'(\cdot )}} \\& \quad = k_{q(\cdot )}k_{p(\cdot )} \Vert g_{1} \Vert _{{p(\cdot )}} \Vert g_{2} \Vert _{{q'( \cdot )}} \Vert 1 \Vert _{{p'(\cdot )}} \bigl( \Vert a_{2} \Vert _{{q(\cdot )}} + b_{2} \Vert x \Vert _{{p(\cdot )}} ^{(p/q)\pm} \bigr), \end{aligned}$$
(4.2)

where \(p'(\cdot )= \frac{p(\cdot )}{p(\cdot )-1} \). Then \(A(L^{p(\cdot )} (\mathbb{R}^{+}) ) \subset L_{1}(\mathbb{R}^{+})\).

We will now demonstrate that \(A(L^{p(\cdot )} ) \subset L^{p(\cdot )} \). Using Remark 4.1 and Remark 4.2 again, we derive that, for almost every \(t \in \mathbb{R}^{+}\),

$$\begin{aligned}& \biggl\Vert \int _{0}^{\infty }K(\cdot ,s)f_{2}\bigl(s, x(s)\bigr) \,ds \biggr\Vert _{{p( \cdot )}} \\& \quad \leq \biggl\Vert \int _{0}^{\infty }K(\cdot ,s) \bigl(a_{2}(s) + b_{2} \bigl\vert x(s) \bigr\vert ^{p(s)/q(s)} \bigr) \,ds \biggr\Vert _{{p(\cdot )}} \\& \quad \leq \Vert k_{q(\cdot )} \Vert K(t,\cdot ) \|_{{q'(\cdot )}} \Vert a_{2} + b_{2} \vert x \vert ^{p(\cdot )/q(\cdot )}\|_{{q(\cdot )}} \|_{p( \cdot )} \\& \quad \leq k_{q(\cdot )} \bigl( \Vert a_{2} \Vert _{{q(\cdot )}} + b_{2} \bigl\Vert x^{p( \cdot )/q(\cdot )} \bigr\Vert _{{q(\cdot )}} \bigr) \Vert g_{1}\cdot \Vert g_{2} \|_{{q'( \cdot )}} \|_{{p(\cdot )}} \\& \quad = k_{q(\cdot )} \Vert g_{1} \Vert _{{p(\cdot )}} \Vert g_{2} \Vert _{{q'(\cdot )}} \bigl( \Vert a_{2} \Vert _{{q(\cdot )}} + b_{2} \Vert x \Vert _{{p(\cdot )}}^{(p/q)\pm} \bigr). \end{aligned}$$
(4.3)

Then \(A(L^{p(\cdot )} ) \subset L^{p(\cdot )} \).

We shall now demonstrate that the operator A is continuous between indicated spaces.

To accomplish this, let us select the sequence \((x_{n}) \subset L^{p(\cdot )} \) such that

$$ \lim_{n \to \infty} \Vert x_{n} -x \Vert _{p(\cdot )}=0. $$

We will demonstrate that \(A(x_{n}) \to A(x)\) in \(L^{p(\cdot )} \) as \(n \to \infty \). To accomplish this, it is sufficient to demonstrate that any sequence \((x_{n} )\) has a subsequence denoted by \((x_{n_{k}} )\) such that \(A(x_{n_{k}}) \to A(x)\) in \(L^{p(\cdot )} \) as \(k \to \infty \).

Take \((x_{n_{k}} )\subset (x_{n})\). It follows by Proposition 2.1 that there is a subsequence \((x_{n_{k}})\) such that

$$ x_{n_{k}} \to x \quad \text{almost everywhere in } \mathbb{R}^{+} $$
(4.4)

and there exists \(h \in L^{p(\cdot )} \) satisfying

$$ \bigl\vert x_{n_{k}}(t) \bigr\vert \leq h(t) \quad \text{almost every }t \in \mathbb{R}^{+} \text{ for all }k \in \mathbb{N}. $$
(4.5)

Since the function \(K(t,s)f_{2}(s, \cdot ): \mathbb{R} \to \mathbb{R} \) is continuous, from (4.4), we deduce that

$$ K(t,s)f_{2}\bigl(s, x_{n_{k}}(s)\bigr) \to K(t,s)f_{2}\bigl(s, x(s)\bigr) \quad \text{for almost every }t ,s \in \mathbb{R}^{+} . $$

Now, based on presumption (ii) with inequality (4.5), we have for almost every \(t ,s \in \mathbb{R}^{+} \),

$$ \bigl\vert K(t,s)f_{2}\bigl(s, x_{n_{k}}(s)\bigr) \bigr\vert \leq K(t,s) \bigl(a_{2}(s) + b_{2} \bigl\vert h(s) \bigr\vert ^{p(s)/q(s)} \bigr). $$

As before, we can show that for almost every \(t \in \mathbb{R}^{+}\) the function \(t \to K(t,s) (a_{2}(s) + b_{2} |h(s)|^{p(\cdot )/q(\cdot )} )\) lies in \(L_{1}(\mathbb{R}^{+})\).

Consequently, using the Lebesgue dominated theorem, we obtain

$$ \int _{0}^{\infty }K(t,s)f_{2}\bigl(s, x_{n_{k}}(s)\bigr) \,ds \to \int _{0}^{ \infty }K(t,s)f_{2}\bigl(s, x(s) \bigr) \,ds $$

for almost every \(t \in \mathbb{R}^{+}\). Moreover, by using Remark 4.1 and (4.5), we have

$$ \biggl\Vert \int _{0}^{\infty }K(t,s)f_{2}\bigl(s, x_{n_{k}}(s)\bigr) \,ds \biggr\Vert _{{p( \cdot )}} \leq k_{q(\cdot )} \Vert g_{1} \Vert _{{p(\cdot )}} \Vert g_{2} \Vert _{{q'( \cdot )}} \bigl( \Vert a_{2} \Vert _{{q(\cdot )}} + b_{2} \Vert h \Vert _{{p(\cdot )}} ^{(p/q) \pm} \bigr). $$

Consequently, the Lebesgue dominated theorem implies that

$$ \bigl\Vert A(x_{n_{k}}) - A(x) \bigr\Vert _{{p(\cdot )}} \to 0, $$

then we obtain that \(\| A(x_{n}) - A(x)\|_{{p(\cdot )}} \to 0\),

which implies that A maps continuously \(L^{p(\cdot )} \to L^{p(\cdot )} \). Finally, by assumption (i), we have that the operator H maps \(L^{p(\cdot )} \to L^{p(\cdot )} \) and is continuous.

Step II. We will construct an invariant set \(B_{r}\), where r is as in assumption (iv). We shall first show that the operator H is bounded in \(L^{p(\cdot )} \) i.e.

$$ \bigl\Vert H(x) \bigr\Vert _{p(\cdot )} \leq \Vert g \Vert _{p(\cdot )} + \bigl\Vert F_{f_{1}}(x) \bigr\Vert _{p(\cdot )} + \bigl\Vert A(x) \bigr\Vert _{p(\cdot )}. $$

To do this, we use assumptions (ii) and (iii) and the following facts: (i) the \(L^{p(\cdot )} \)-norm is order preserving, (ii) the triangle inequality, (iii) the embedding \(L^{q(\cdot )} \subset L^{p(\cdot )} \) since \(p(\cdot ) \leq q(\cdot )\) (see [11, Theorem 2.45 and Corollary 2.48]), which implies that there exists \(M > 1\) such that, for all \(a_{1} \in L^{q(\cdot )}\), \(\|a_{1} \|_{p(\cdot )} \leq M \|a_{1} \|_{q(\cdot )}\). Then we have

$$\begin{aligned} \bigl\Vert H(x) \bigr\Vert _{p(\cdot )} \leq & \Vert g \Vert _{p(\cdot )} + \bigl\Vert a_{1}+ b_{1} \vert x \vert ^{p( \cdot )/q(\cdot )} \bigr\Vert _{{p(\cdot )}} \\ &{} + k_{q(\cdot )} \Vert g_{1} \Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'( \cdot )} \bigl\Vert F_{f_{2}}(x) \bigr\Vert _{{q(\cdot )}} \\ \leq & \Vert g \Vert _{p(\cdot )} + \bigl( \Vert a_{1} \Vert _{p(\cdot )} + b_{1} \bigl\Vert x^{p(\cdot )/q(\cdot )} \bigr\Vert _{p(\cdot )} \bigr) \\ &{} + k_{q(\cdot )} \Vert g_{1} \Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'( \cdot )} \bigl\Vert a_{2}+ b_{2} \vert x \vert ^{p(\cdot )/q(\cdot )} \bigr\Vert _{{q( \cdot )}} \\ \leq & \Vert g \Vert _{p(\cdot )} + \bigl(M \Vert a_{1} \Vert _{q(\cdot )} + b_{1} M \bigl\Vert x^{p(\cdot )/q(\cdot )} \bigr\Vert _{q(\cdot )} \bigr) \\ &{} + k_{q(\cdot )} \Vert g_{1} \Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'( \cdot )} \bigl( \Vert a_{2} \Vert _{{q(\cdot )}}+ b_{2} \bigl\Vert x^{p(\cdot )/q( \cdot )} \bigr\Vert _{{q(\cdot )}} \bigr) \\ \leq & \Vert g \Vert _{p(\cdot )} + \bigl( M \Vert a_{1} \Vert _{q(\cdot )} + b_{1} M \Vert x \Vert _{{p(\cdot )}}^{(p/q)\pm} \bigr) \\ &{} + k_{q(\cdot )} \Vert g_{1} \Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'( \cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )}+ b_{2} \Vert x \Vert _{{p(\cdot )}}^{(p/q) \pm} \bigr). \end{aligned}$$

Thus, \(H: L^{p(\cdot )} \to L^{p(\cdot )} \). It follows from our assumption (iv) that the following inequality has a positive solution r such that

$$\begin{aligned} \bigl\Vert H(x) \bigr\Vert _{p(\cdot )} \leq & \Vert g \Vert _{p(\cdot )} + \bigl( M \Vert a_{1} \Vert _{q( \cdot )} + b_{1} M r^{(p/q)\pm} \bigr) \\ &{} + k_{q(\cdot )} \Vert g_{1} \Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'( \cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )}+ b_{2} r^{(p/q)\pm} \bigr) \leq r, \end{aligned}$$

which implies that \(H: \overline{B}_{r} \to \overline{B}_{r}\) is continuous.

Step III. It is obvious that \(\overline{B}_{r}\) is a bounded, closed, and convex set in \(L^{p(\cdot )} \).

Step IV. We will demonstrate that H is a contraction in terms of the MNC μ.

Assume that \(\emptyset \neq X \subset \overline{B}_{r} \) and fixed arbitrary constant \(\epsilon >0\). Then, for arbitrary \(x \in X\) and for \(t, h \in \mathbb{R}\) and \(|h|\leq \epsilon \), we have

$$\begin{aligned}& \bigl\vert (Hx) (t+h) -(Hx) (t) \bigr\vert \\& \quad \leq \bigl\vert g(t+h) -g(t) \bigr\vert + \bigl\vert F_{f_{1}}(x) (t+h)-F_{f_{1}}(x) (t) \bigr\vert + \bigl\vert A(x) (t+h)- A(x) (t) \bigr\vert \\& \quad \leq \bigl\vert g(t+h) -g(t) \bigr\vert + \bigl\vert f_{1} \bigl(t+h,x(t+h) \bigr) -f_{1}\bigl(t,x(t)\bigr) \bigr\vert \\& \qquad {}+ \int _{0}^{\infty } \bigl\vert K(t+h,s)- K(t,s) \bigr\vert \bigl\vert f_{2}\bigl(s,x(s)\bigr) \bigr\vert \,ds \\& \quad \leq \bigl\vert g(t+h) -g(t) \bigr\vert + \bigl( \bigl\vert a_{1}(t+h)-a_{1}(t) \bigr\vert + b_{1} \bigl\vert x(t+h)-x(t) \bigr\vert ^{p(t)/q(t)} \bigr) \\& \qquad {}+ \int _{0}^{\infty } \bigl\vert g_{1}(t+h)- g_{1}(t) \bigr\vert \bigl\vert g_{2}(s) \bigr\vert \bigl\vert a_{2}(s)+ b_{2} \bigl\vert x(s) \bigr\vert ^{p(s)/q(s)} \bigr\vert \,ds \\& \quad \leq \bigl\vert g(t+h) -g(t) \bigr\vert + \bigl( \bigl\vert a_{1}(t+h)-a_{1}(t) \bigr\vert + b_{1} \bigl\vert x(t+h)-x(t) \bigr\vert ^{p(t)/q(t)} \bigr) \\& \qquad {}+ k_{q(\cdot )} \bigl\vert g_{1}(t +h)- g_{1}(t) \bigr\vert \cdot \Vert g_{2} \Vert _{q'(\cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )} + b_{2}r^{(p/q)\pm} \bigr). \end{aligned}$$

Therefore,

$$\begin{aligned}& \bigl\Vert (Hx) (\cdot +h) -(Hx) (\cdot ) \bigr\Vert _{p(\cdot )} \\& \quad \leq \bigl\Vert g(\cdot +h) -g(\cdot ) \bigr\Vert _{p(\cdot )} \\& \qquad {}+ \bigl\Vert a_{1}(\cdot +h)-a_{1}(\cdot ) \bigr\Vert _{p(\cdot )} + b_{1} \bigl\Vert \bigl\vert x(\cdot +h)-x( \cdot ) \bigr\vert ^{p(\cdot )/q(\cdot )} \bigr\Vert _{p(\cdot )} \\& \qquad {}+ \bigl\Vert k_{q(\cdot )} \bigl\vert g_{1}(\cdot +h)- g_{1}(\cdot ) \bigr\vert \cdot \Vert g_{2} \|_{q'(\cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )} + b_{2}r^{(p/q)\pm} \bigr) \bigr\| _{p(\cdot )} \\& \quad \leq \bigl\Vert g(\cdot +h) -g(\cdot ) \bigr\Vert _{p(\cdot )} \\& \qquad {}+ \bigl(M \bigl\Vert a_{1}(\cdot +h)-a_{1}(\cdot ) \bigr\Vert _{q(\cdot )} + M b_{1} \bigl\Vert \bigl\vert x( \cdot +h)-x(\cdot ) \bigr\vert ^{p(\cdot )/q(\cdot )} \bigr\Vert _{q(\cdot )} \bigr) \\& \qquad {}+k_{q(\cdot )} \bigl\Vert g_{1}(\cdot +h)- g_{1}(\cdot ) \bigr\Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'(\cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )} + b_{2}r^{(p/q)\pm} \bigr) \\& \quad \leq \bigl\Vert g(\cdot +h) -g(\cdot ) \bigr\Vert _{p(\cdot )} \\& \qquad {}+ \bigl(M \bigl\Vert a_{1}(\cdot +h)-a_{1}(\cdot ) \bigr\Vert _{q(\cdot )} + M b_{1} \bigl\Vert x(\cdot +h)-x( \cdot ) \bigr\Vert _{p(\cdot )}^{(p/q)\pm} \bigr) \\& \qquad {}+k_{q(\cdot )} \bigl\Vert g_{1}(\cdot +h)- g_{1}(\cdot ) \bigr\Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'(\cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )} + b_{2}r^{(p/q)\pm} \bigr) \\& \quad \leq \bigl\Vert g(\cdot +h) -g(\cdot ) \bigr\Vert _{p(\cdot )} \\& \qquad {}+ \bigl(M \bigl\Vert a_{1}(\cdot +h)-a_{1}(\cdot ) \bigr\Vert _{q(\cdot )} + M b_{1}(2r)^{(p/q) \pm} \bigl\Vert x(\cdot +h)-x(\cdot ) \bigr\Vert _{p(\cdot )} \bigr) \\& \qquad {}+k_{q(\cdot )} \bigl\Vert g_{1}(\cdot +h)- g_{1}(\cdot ) \bigr\Vert _{p(\cdot )} \Vert g_{2} \Vert _{q'(\cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )} + b_{2}r^{(p/q)\pm} \bigr), \end{aligned}$$

where

$$\begin{aligned} \bigl\Vert x(\cdot +h)-x(\cdot ) \bigr\Vert _{p(\cdot )}^{(p/q)\pm} =& \bigl\Vert x( \cdot +h)-x(\cdot ) \bigr\Vert _{p(\cdot )}^{{(p/q)\pm}-1} \bigl\Vert x(\cdot +h)-x( \cdot ) \bigr\Vert _{p(\cdot )} \\ \leq & (2r)^{(p/q)\pm} \bigl\Vert x(\cdot +h)-x(\cdot ) \bigr\Vert _{p(\cdot )}. \end{aligned}$$

Thus we obtain

$$\begin{aligned} \omega (HX, \epsilon ) \leq & \omega (g, \epsilon ) + \bigl(M \omega (a_{1}, \epsilon ) +b_{1} M\cdot (2r)^{(p/q)\pm} \omega (x, \epsilon ) \bigr) \\ &{}+ k_{q(\cdot )} \Vert g_{2} \Vert _{q'(\cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )} + b_{2}r^{(p/q) \pm} \bigr)\omega (g_{1}, \epsilon ). \end{aligned}$$

Also we have \(\omega (g, \epsilon )\to 0\), \(\omega (a_{1}, \epsilon )\to 0\), and \(\omega (g_{1}, \epsilon ) \to 0\) as \(\epsilon \to 0\). Then we obtain

$$ \omega (HX) \leq b_{1} M \cdot (2r)^{(p/q)\pm} \cdot \omega (X). $$
(4.6)

Next, let us fix an arbitrary number \(T > 0\). Then, using our hypotheses, for an arbitrary function \(x\in X\) we have

$$\begin{aligned} \Vert HX \Vert _{L^{p(\cdot )}[T, \infty )} \leq & \Vert g \Vert _{L^{p(\cdot )}[T, \infty )} \\ &{} + \bigl(M \Vert a_{1} \Vert _{L^{q(\cdot )}[T, \infty )} + b_{1} M (2r)^{(p/q) \pm} \Vert x \Vert _{L^{p(\cdot )}[T, \infty )} \bigr) \\ & {} + k_{q(\cdot )}\cdot \Vert g_{1} \Vert _{L^{p(\cdot )}[T, \infty )} \Vert g_{2} \Vert _{q'(\cdot )} \bigl( \Vert a_{2} \Vert _{q(\cdot )} + b_{2}r^{(p/q)\pm} \bigr). \end{aligned}$$

Also we have \(\|g\|_{L^{p(\cdot )}[T, \infty )} \to 0\), \(\|a_{1}\|_{L^{q(\cdot )}[T, \infty )} \to 0 \), and \(\|g_{1}\|_{L^{p(\cdot )}[T, \infty )}\to 0\) as \(T \to \infty \), we have

$$ d (HX) \leq b_{1} M \cdot (2r)^{(p/q)\pm} \cdot d (X). $$
(4.7)

Combining (4.6) and (4.7), we have

$$ \mu (HX) \leq b_{1} M \cdot (2r)^{(p/q)\pm} \cdot \mu (X). $$

From the above inequality and by recalling all the above established properties, we can apply Theorem 2.5, which finishes the proof. □

5 Particular cases and examples

Let us recall some examples and particular cases of our outcomes that were covered and studied in earlier publications.

Example

The tools of the weak MNC related to the fixed point theorem for contractions in the space \(L^{1}(0,1)\) were utilized for studying the existence of integrable solutions of the Hammerstein and Urysohn integral equations (cf. [2])

$$\begin{aligned}& x(t) = g(t)+ \int _{0}^{1} k(t,s)f\bigl(s,x(s)\bigr) \,ds, \\& x(t) = g(t)+ \int _{0}^{1} u\bigl(t,s,x(s)\bigr) \,ds. \end{aligned}$$

Example

The authors in [28] constructed a new MNC in the space \(L_{\mathrm{loc}}^{1}(\mathbb{R}^{+})\) of all real functions locally integrable on \(\mathbb{R}^{+}\), and they utilized that method together with a family of MNCs to investigate the existence of solutions of the nonlinear Volterra integral equation

$$ x(s) = f \biggl(s, \int _{0}^{s} u\bigl(s,t,x(t)\bigr) \,dt \biggr). $$

Example

The authors in [1] examined a set of nonlinear functional integral equations using a new MNC on \(L_{p}(\mathbb{R}^{N}) (1 \leq p < \infty )\) together with Darbo’s fixed point theorem

$$ x(s)= g\bigl(s,x(s)\bigr) + \int _{\mathbb{R}^{N}} k(s,t) (Qx) (t) \,dt, $$

where Q is an operator map** the space \(L_{p}(\mathbb{R}^{N})\) into itself.

Example

The Urysohn integral equations were examined in Orlicz spaces \(L_{\varphi}(I)\) in [29, 30]

$$ x(t)=g(t)+ \int _{I} u\bigl(t,s,x(s)\bigr) \,ds,\quad t\in I. $$

The case of Hammerstein integral equations were also discussed in Orlicz spaces in [33].

Example

The existence and uniqueness of the solutions of Volterra integral equations with Carathéodory functions having diverse growth behaviors

$$ x(s)= g\bigl(s,x(s)\bigr) + \lambda \int _{a}^{b} u\bigl(s,t,x(t)\bigr) \,dt,\quad s \in (a,b), \lambda \in \mathbb{R} $$

were studied in [8] in \(L^{p(\cdot )}(a,b)\) by using degree theory and fixed point approach.

Example

In [20], the authors presented a new MNC in the Sobolev spaces \(W^{k,1}(I)\) and used it to investigate the existence of solutions of the integral-differential equation

$$ x(t)=p(t)+q(t)x(t)+ \int _{I} k(t,s)g \biggl(t,x(s), \frac{\partial u}{\partial t_{1}}(s),\ldots , \frac{\partial u}{\partial t_{n}}(s), Tu(s) \biggr) \,ds. $$