1 Introduction

This paper is devoted to the study of the existence theorems for integral equations that are generalizations of quadratic equations to the product of n-operators acting on n-different Orlicz spaces, that is, of the form

$$\begin{aligned} x(t)= \prod _{i=1}^n \bigg (g_i(t) + \lambda _i \cdot h_i\big (t,x(t)\big ) \cdot \int _a^b K_i(t,s)f_i(s, x(s))\;ds \bigg ) \end{aligned}$$
(1.1)

for \(n\ge 2\) in Orlicz spaces \(L_\varphi (I),~I=[a,b]\) in three cases when the generating N-functions fulfill \(\Delta ',~\Delta _2,\) and \(\Delta _3\)-conditions. Such a problem requires special attention because we are not dealing with Banach algebras wit respect to the pointwise multiplication, i.e., all relations between the spaces on which the different operators act and between their ranges must be well defined.

We briefly explain why we study problems on Orlicz spaces (so, in particular, with discontinuous solutions) and why n-operators. It is useful to study solutions in Orlicz spaces when we treat operators containing strong nonlinearities in which either the kernels \(K_i\) or the functions \(f_i\) are not with at most polynomial growth (e.g., with exponential growth), then the discontinuous solutions are expected. This is raised by some mathematical models in physics and statistical physics (cf. [6, 24, 28]). Integral equation with exponential nonlinearities

$$\begin{aligned} x(t)+\int _I K(t,s) e^{x(s)}\;ds=0 \end{aligned}$$

has applications in statistical mechanics (cf. [13]). Recall that the existence theorems for quadratic integral equations were investigated in Banach-Orlicz algebras in [16] and in the more general class of Orlicz spaces in [14, 15, 17, 29]. In particular, for \(n=2,~h_i=1\), Eq. (1.1) reduces to the Gripenberg-type equation

$$\begin{aligned} x(t)= k\left( g_1(t) + \int _0^t a_1(t-s) x(s)\; ds \right) \left( g_2(t)+ \int _0^t a_2(\varphi -s)x(s)\; ds \right) , \end{aligned}$$

which has numerous applications in mathematical biology, including models of the spread of illnesses that do not induce permanent immunity (SI models, cf. [21]). Likewise, various types of models for infectious disease depend on possibly discontinuous data functions, we insists on studying discontinuous solutions to these problems. This is in close connection with the purely mathematical motivation for studying integral equations with discontinuous solutions. Quite surprisingly, due to the technical difficulty of the proofs, even classical or quadratic integral equations (\(n=1\) or \(n=2\)) are rarely considered in Orlicz spaces, and if they are, it is generally for N-functions satisfying the so-called \(\Delta _2\)-condition (discussed below). The exceptions are in fact only a few papers such as [2, 3, 17, 31, 32]. A detailed introduction to the theory of integral equations in Orlicz spaces can be found in [25] or [33].

The study of the product of more than two operators is useful, as claimed by Brestovanská and Medveď [11, 12], but this was only for the case of the Banach algebra of continuous functions, and thus with a different method of the proof. Additional assumptions arise when the mutual relations of more than two spaces need to be controlled. The motivations come from the study of product of operators, but here we go beyond the case of Banach algebras. We establish here a mathematical basis for the product of n-integral operators (see also [7, 10, 27]) in n-tuples of spaces and possibly outside the class of Banach algebras.

We will point out where similar motivations were indicated. Let us focus, that in [30] the authors investigated the existence and uniqueness of continuous solutions of the following integral equation

$$\begin{aligned} x(t)=\prod _{i=1}^n \bigg (g_i(t)+ \int _a^t K_i(t,s,x(s))\;ds \bigg ), ~ t \in [a,b]. \end{aligned}$$

The authors in [23] investigated the existence of continuous solutions for the q-fractional integral equation of the form

$$\begin{aligned} x(t)=\prod _{i=1}^n \bigg (f_i(t)+ \frac{g_i(t,x(t))}{\Gamma _q(\alpha _i)}\int _a^t (t-qs)^{\alpha _i -1} u_i(s,x(s))\;ds \bigg ), ~ t \in [0,1]. \end{aligned}$$

Finally, discontinuous solutions were the object studied in the paper [9]. Namely, the existence of integrable solutions was investigated for equation

$$\begin{aligned} x(t)=f(t,x(t))+\prod _{i=1}^n f_i\bigg (t,~ \int _a^t K_i (t,s,x(s))\;ds \bigg ), ~ t>0. \end{aligned}$$

This paper is devoted to extending previous research by studying discontinuous monotone solutions for n-products of integral equations in arbitrary Orlicz spaces. We focus on the assumptions of the studied operators in the Eq. (1.1), which determine the intermediate spaces (which need not be the same spaces) in which our results are in the range space. The Darbo fixed point theorem for the measure of noncompactness is the main tool for proving our results. Although this goes beyond the planned scope of the paper, we should note that our motivations and methods are influenced by papers on products of operators (cf. [4, 8, 18]).

2 Preliminaries

Let \({{\mathbb {R}}}\) be the field of real numbers and \(I=[a,b] \subset {{\mathbb {R}}}\).

A function \(N: {{\mathbb {R}}} \rightarrow [0,+\infty )\) is said to be an N-function if it is even, convex, and continuous with \(\lim _{u\rightarrow 0} N(u)=0\) and \(\lim _{u\rightarrow \infty }N(u)=\infty \) and \(N(x) > 0\) if \(x > 0\) (i.e., \(N(u) = 0 \iff u = 0\)).

For any N-function N, the function \( M: {{\mathbb {R}}} \rightarrow {[0,+\infty })\) defined by \(\sup _{v\ge 0 }\{v |u|- N(v)\}\) is called the complementary function to N and it is known that M is again an N-function. The Orlicz class, denoted by \({\mathbf {{\mathcal {O}}}}_M\), contains measurable functions \(x: I \rightarrow {{\mathbb {R}}}\) for which \(\rho (x;M)= \int _I M(x(t))dt <\infty \). Denote by \(L_M=L_M(I)\) the Orlicz space of all measurable functions \(x: I \rightarrow {{\mathbb {R}}}\) for which

$$\begin{aligned} \Vert x\Vert _M = \inf _{ \epsilon > 0} \left\{ \int _I M \left( \frac{x(s)}{\epsilon }\right) \;ds \le 1\right\} . \end{aligned}$$

Let \(E_M=E_M([a,b])\) is the set of all bounded functions and having absolutely continuous norms from \(L_M\). Orlicz spaces with \(E_M = L_M\) we call regular Orlicz spaces. The special choice \(M_p(u) = \frac{1}{p} |u|^p,~p \in [1,\infty )\) leads to the Lebesgue space \(L_{p}=L_{p}([a,b],{{\mathbb {R}}})\). In this case, it can be easily seen that \(M_p= N_{{\tilde{p}}}\) with \(\frac{1}{p} + \frac{1}{{\tilde{p}}}=1\) for \(p > 1\). In this connection, it is worth recalling that for any N-function \(\psi \) we have \( \psi (u-v) \le \psi (u)- \psi (v) \) and \(\psi (\rho u) \le \rho \psi (u)\) occurs for any \( u,v \in {{\mathbb {R}}}\) and \(\rho \in [0,1]\). Moreover, for the non-trivial N-function M, \(L_\infty \subset L_{M}.\) For further properties of N-functions and Orlicz spaces generated by such functions we refer the reader to [24].

Note that \(E_M \subseteq L_M \subseteq {\mathbf {{\mathcal {O}}}}_M.\) The inclusion \(L_M \subset L_P\) holds if and only if \(\exists ~u_0, a>0\) such that \(P(u) \le a M(u)\) for \(u \ge u_0\). We should now distinguish certain classes of N-functions.

Definition 2.1

[24] The N-function M fulfills \(\Delta _2\)-condition if, \(\exists ~\omega ,~t_0 \ge 0\) such that \(M(2t) \le \omega M(t)\) for \(t \ge t_0\).

For functions M satisfying this condition, we obtain \(E_M = L_M\).

Definition 2.2

[24] The N-function M fulfills \(\Delta '\)-condition if, \(\exists ~l,~t_0 \ge 0\) such that \(M(ts) \le l\cdot M(t)M(s)\) for \(t, s \ge t_0\).

Definition 2.3

[24] The N-function M fulfills \(\Delta _3\)-condition if, \(\exists ~l,~t_0 \ge 0\) such that \(tM(t) \le M(l\cdot t)\) for \(t \ge t_0\).

Interesting examples of functions from these classes can be found in [24]. In the following sections of this paper, we will show how our results change depending on the class of generating functions of the Orlicz spaces considered. Each such class is associated with different growth conditions assumed in our theorems.

For completeness, we will now recall both the definitions of the necessary operators and their properties when acting on Orlicz spaces.

Definition 2.4

[24] Suppose that the function \(f: I \times {{\mathbb {R}}} \rightarrow {{\mathbb {R}}}\) satisfies Carathéodory conditions, i.e., it is measurable in t for any \(x~ \in ~ {{\mathbb {R}}}\) and continuous in x for almost all \(t ~\in ~ I\). Then to any function x(t) that is measurable we can denote the superposition operator \(F_f\) generated by the function f as

$$\begin{aligned} F_f(x)(t)~ =~ f(t, x(t)), \quad t \in I. \end{aligned}$$

The acting property, as well as the boundedness and continuity conditions of this operator on Lebesgue spaces, are well-known, but the case of Orlicz spaces seems to be necessary to recall.

Lemma 2.5

[24, Theorem 17.6] Assume that a function \(f: ~I \times {{\mathbb {R}}}~ \rightarrow ~ {{\mathbb {R}}}\) fulfills Carathéodory conditions. The superposition operator \(F_f\) maps \(E_{M_1} \rightarrow L_{M_2} = E_{M_2}\) is continuous and bounded if and only if

$$\begin{aligned} |f(s,x)| \le a(s) + b M_2^{-1}\big ( M_1(x)\big ), \end{aligned}$$

where \(b\ge 0\) and \(a \in E_{M_2}\) in which the N-function \(M_2(x)\) fulfills the \(\Delta _2\)-condition.

In our study, a certain topology on Orlicz spaces induced from the space of measurable functions, i.e., the topology of convergence in measure, will be relevant.

Let \(S = S(I)\) refers to the set of Lebesgue measurable functions on I and let "meas" refers to the Lebesgue measure in \({{\mathbb {R}}}\). The set S related with the metric

$$\begin{aligned} d(x,y) = \inf _{a > 0}[a + meas\{s: |x(s) - y(s)| \ge a \}] \end{aligned}$$

be a complete space. The convergence in measure on I is equivalent to convergence with respect to d (cf. Proposition 2.14 in [34]). The compactness in that spaces is said to be a “compactness in measure”.

Lemma 2.6

[16] Let \(X\subset L_M\) be a bounded set. Assume that, there is a family of subsets \((\Omega _c)_{0\le c\le b-a}\) of the interval I such that meas \(\Omega _c = c\) for every \(c \in [0, b - a]\) and for every \(x \in X\),

$$\begin{aligned} x(t_1) \ge x(t_2),~~~(t_1 \in \Omega _c,~ t_2 \not \in \Omega _c). \end{aligned}$$

Then X is compact in measure in \(L_M\).

Thus the bounded subsets consisting of a.e. monotone functions are compact in measure. It remains to describe the operators that preserve the monotonicity of the functions.

Lemma 2.7

[26, Theorem 6.2] The operator \(K_{0}x(t) = \int _I K(t,s)x(s) \; ds\) preserve the monotonicity of functions if and only if

$$\begin{aligned} \int _0^b K(t_1,s) \; ds \ge \int _0^b K(t_2,s) \; ds \end{aligned}$$

for \(t_1 < t_2\), \(t_1, t_2 \in I\) and for any \(b \in I\).

It is time to take on the case of the product of n-operators. We need the following lemma:

Lemma 2.8

( [22, Theorem 1]) Let \(n\ge 2\). If \(\varphi \) and \(\varphi _i\) are arbitrary N-functions for \(i =1,2, \cdots n\), then the following statements are equivalent:

  1. 1.

    For every \(u_i \in L_{\varphi _i}\), \(\prod _{i=1}^n u_i \in L_\varphi \).

  2. 2.

    There exists a constant \(l>0\) such that

    $$\begin{aligned} \bigg \Vert \prod _{i=1}^n u_i \bigg \Vert _\varphi \le l \prod _{i=1}^n \Vert u_i \Vert _{\varphi _i}, \end{aligned}$$

    for every \(u_i \in L_{\varphi _i}, i=1,2, \cdots n\).

  3. 3.

    There exists a constant \(C>0\) such that

    $$\begin{aligned} \prod _{i=1}^n \varphi _i^{-1}(t) \le C \varphi ^{-1}(t) \end{aligned}$$

    for every \(t \ge 0\).

  4. 4.

    There exists a constant \(C>0\) such that for all \(t_i \ge 0, i=1, \cdots n,\)

    $$\begin{aligned} \varphi \bigg ( \frac{\prod _{i=1}^n t_i}{C} \bigg ) \le \sum _{i=1} ^n \varphi _i(t_i). \end{aligned}$$

Next, suppose that \((E, \Vert \cdot \Vert )\) be arbitrary Banach space with zero element \(\theta \). We denote by \(B_r\) the closed ball with radius r and center in \(\theta \). The symbol \(B_r(E)\) is to point out the space. Additionally, by \({{\mathcal {M}}}_{E}\) we signify the family of all nonempty and bounded subsets of the Banach space E and by \({{\mathcal {N}}}_{E}\) its subfamily containing all relatively compact subsets. If \(X \subset E\), then \({\bar{X}}\) and conv X demonstrate the closure and convex closure of X, respectively.

Definition 2.9

[5] A mapping \(\mu : {{\mathcal {M}}}_E~ \rightarrow ~ [0,~\infty )\) is said to be a measure of noncompactness in E if it satisfies the following conditions:

  1. (i)

    \(\mu (X) ~=~ 0 ~\Leftrightarrow ~ X ~\in ~{{\mathcal {N}}}_E\).

  2. (ii)

    \(X~ \subset Y ~\Rightarrow ~ \mu (X) ~\le ~\mu (Y ).\)

  3. (iii)

    \(\mu ( {\bar{X}} )~ =~ \mu (conv X) ~= ~\mu (X).\)

  4. (iv)

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

  5. (v)

    \(\mu (X ~+ ~Y) ~\le ~ \mu (X)~ + ~\mu (Y)\).

  6. (vi)

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

  7. (vii)

    If \({X_n}\) is a sequence of nonempty, bounded, closed subsets of E such that \(X_{n+1} ~\subset ~X_n,\) \(n = 1, 2, 3, \cdots \), and \(\lim _{n\rightarrow \infty } \mu (X_n) ~=~ 0\), then the set \(X_\infty ~= ~\bigcap _{n=1}^\infty ~X_n\) is nonempty.

Let us recall a classical example: the Hausdorff measure of noncompactness \(\beta _H(X)\) (cf. [5]) which is defined as follows

$$\begin{aligned} \beta _H(X)= & {} \\{} & {} \inf \{r > 0: \text{ there } \text{ exists } \text{ a } \text{ finite } \text{ subset } \text{ Y } \text{ of } \text{ E } \text{ such } \text{ that } \ X \subset ~ Y~ +~ B_r~\}, \end{aligned}$$

where X is an arbitrary nonempty and bounded subset of E.

Let c refers to the measure of uniform integrability of the set X in \(L_M\) (cf. [34, Definition 3.9] or [20]):

$$\begin{aligned} c(X)~ =~ \limsup _{\varepsilon \rightarrow 0}~ \sup _{mes D \le \varepsilon } \sup _{x \in X} \Vert x\cdot \chi _{D} \Vert _{M}, \end{aligned}$$

where \(\chi _D\) denotes the characteristic function of a measurable subset \(D \subset I\).

Proposition 2.10

[20, Theorem 1] Let \(X\ne \emptyset \) be a bounded and compact in measure subset of regular Orlicz spaces Y. Then

$$\begin{aligned} \beta _H(X) = c(X). \end{aligned}$$

Despite studying the product of multiple operators in place of a single one, our main tool will be:

Theorem 2.11

[5] Let \(Q\ne \emptyset \) be a convex, bounded, and closed subset of E and let \(V: Q ~\rightarrow ~Q\) be a continuous map and there exists a constant \(k ~\in ~ [0, 1)\) such that

$$\begin{aligned} \mu (V(X))~\le ~k\mu (X), \end{aligned}$$

for any \(\emptyset \ne X \subset Q\). Then V has at least one fixed point in the set Q.

3 Main results

We will assume that the particular operators in the product have values in some intermediate spaces \(L_{\varphi _i}, i=1, \cdots n\), and then the product returns to the target space \(L_\varphi \) (i.e., the space of solutions). We will characterize three different cases where the generating N-functions fulfill the conditions \(\Delta ',~\Delta _2\), and \(\Delta _3\). Recall that for different classes of Orlicz spaces we need to consider different growth conditions, so they should be considered separately.

First, we will rewrite Eq. (1.1) in operator form

$$\begin{aligned} x(t)=Hx(t)=\prod _{i=1}^n H_i x(t), \end{aligned}$$

where \( H_i(x) = g_i(t) + U_i(x), \; U_i(x)(t) = \lambda _i \cdot F_{h_i}(x)(t) \cdot A_i(x)(t),\) and \(A_i(x)(t)=K_{0_i} \circ F_{f_i}x(t),~K_{0_i}= \int _a^b K_i(t,s)x(s)\;ds,\) such that \(F_{h_i}, F_{f_i}\) are the superposition operators generated by \(h_i\) and \(f_i\), respectively (as in Definition 2.4).

3.1 The case of \(\Delta '\)-condition

The first result seems to have a lot of technical assumptions, but this is due to the need to check the properties of the different operators, to show the conditions connecting them and finally the interdependencies between spaces, which are not Banach algebras after all due to pointwise multiplication.

Theorem 3.1

Let \(i=1,\cdots n,\) and assume that \(\varphi , \varphi _i,~\phi _i,~\psi _i\) are N-functions and that \(M_i\) and \(N_i\) are complementary N-functions. Furthermore, let us make the following set of assumptions:

  1. (N1)

    There exists a constant \(k > 0\) such that for every \(v_i \in L_{\varphi _i}\), we have \(\Vert \prod _{i=1}^n v_i\Vert _{\varphi } \le k\prod _{i=1}^n \Vert v_i\Vert _{\varphi _i}\).

  2. (N2)

    There exist constants \(k_i^* > 0\) such that for every \(u \in L_{\phi _i}\) and \(w \in L_{\psi _i}\) we have \(\Vert uw\Vert _{\varphi _i} \le k_i^* \Vert u\Vert _{\phi _i} \Vert w\Vert _{\psi _i},~i=1,\cdots n\),

  3. (C1)

    The functions \(g_i \in E_{\varphi _i},~i=1,\cdots n\) are a.e. nondecreasing on I,

  4. (C2)

    The functions \(f_i,h_i: I \times {{\mathbb {R}}} \rightarrow {{\mathbb {R}}}\) satisfy Carathéodory conditions, and \(f_i(t,x),h_i(t,x),\) are assumed to be nondecreasing with respect to both variables t and x separately, for \(i=1,\cdots n\),

  5. (C3)

    \(|f_i(t,x)| \le b_i(t) + R_i(|x|)\) for \(t \in I\) and \(x\in {{\mathbb {R}}}\), where \(b_i \in E_{N_i}\) and \(R_i\) are nonnegative, nondecreasing, continuous function defined on \({{\mathbb {R}}}^+\) for \(~i=1,\cdots n\),

  6. (C4)

    \(|h_i(t,x)| \le a_i(t) + l_i \psi _i^{-1}\bigg (\varphi \big (x\big )\bigg )\) for \(t \in I\) and \(x\in {{\mathbb {R}}}\), where \(a_i \in E_{\psi _i}\) and \(l_i\ge 0\) in which the N-functions \(\psi _i(t)\) satisfy the \(\Delta _2\)-condition for \( i = 1,\cdots ,n\),

  7. (C5)

    Let \(N_i\) fulfill the \(\Delta '\)-condition and suppose that there exist \(\omega , \gamma ,u_0 \ge 0\) for which

    $$\begin{aligned} N_i(\omega (R_i(u))) \le \gamma \varphi (u) \le \gamma M_i(u) ~~\text{ for }~~ u \ge u_0,~i=1,\cdots n. \end{aligned}$$
  8. (K1)

    \(s \rightarrow K_i(t,s) \in L_{M_i}\) for a.e. \(t \in I,~i=1,\cdots n\),

  9. (K2)

    \(K_i \in E_{M_i}(I^2)\) and \(t \rightarrow K_i(t,s) \in E_{\phi _i}\) for a.e. \(s \in I\) with

    $$\begin{aligned} \prod _{i=1}^n \bigg (k_i^* |\lambda _i | l_i \Vert K_i \Vert _{M_i} R_i(1) \bigg ) <\frac{1}{2^{n} k}, \end{aligned}$$
  10. (K3)

    \(\int _a^b K_i(t_1, s) \;ds \ge \int _a^b K_i(t_2, s) \;ds,~i=1,\cdots n\) for \(t_1, t_2 \in I\) with \(t_1 < t_2\).

Then there exist numbers \(\rho _i > 0\) such that for all \(\lambda _i \in {{\mathbb {R}}}\) with \(|\lambda _i | < \rho _i, \ i=1,\cdots n\) such that there exists a solution \(x \in E_\varphi \) of (1.1) which is a.e. nondecreasing on I.

Proof

I. We will show that the operator H is well-defined from the unit ball \(B_1(E_{\varphi }) \rightarrow E_{\varphi }\) and is continuous. First, from assumptions (C2)–(C4) and from Lemma 2.5 the operators \(F_{f_i}: B_1(E_{\varphi }) \rightarrow L_{N_i}\) and \(F_{h_i}: B_1(E_{\varphi }) \rightarrow E_{\psi _i} \) are continuous.

We must now prove that the operators \(U_i: B_1(E_{\varphi }) \rightarrow E_{\varphi _i}, i=1,\cdots n\) are continuous. It suffices to show this property for the operators \(A_i\), according to Lemma 2.8.

Since \(N_i\) are N-functions satisfying \(\Delta '\)-condition and by (C3), we can use [24, Lemma 19.1]. It follows that there are constants \(C_i\) (not dependent on the kernel) such that for any measurable subset T of I and \(x \in L_\varphi ,~ \Vert x\Vert _\varphi \le 1\) we have

$$\begin{aligned} \Vert A_i(x){\chi _T}\Vert _{\phi _i} \le C_i \Vert K_i {\chi _{T\times I}} \Vert _{M_i}. \end{aligned}$$
(3.1)

Now, by the Hölder inequality and the assumption (C3) we obtain

$$\begin{aligned} |K_i(t,s)f_i(s, x(s))| \le \Vert K_i(t,s)\Vert |f_i(s, x(s))| \le \Vert K_i(t,s)\Vert | \left( b_i(s) + R_i(|x(s)|) \right) | \end{aligned}$$

for \(t,s \in I\). Let \(k_i(t) = 2 \Vert K_i(t, \cdot )\Vert _{M_i}\) for \(t \in I\). Since \(K_i \in E_{M_i}(I^2)\), these functions are integrable on I. From assumptions (K1) and (K2) on the kernels \(K_i\) of the operators \(A_i\) (cf. [32]) we obtain that

$$\begin{aligned} \Vert A_i(x)(t) \Vert \le k_i(t) \cdot \left( \Vert b_i\Vert _{N_i} + \Vert R_i(|x(\cdot )|)\Vert _{N_i} \right) {\text{ for } \text{ a.e. } } t \in I. \end{aligned}$$

Hence for any measurable subset T of I and \(x \in E_{\varphi }\)

$$\begin{aligned} \Vert A_i(x)\chi _{T}\Vert _{\phi _i} \le \Vert k_i \chi _{T} \Vert _{\phi _i} \cdot \left( \Vert b_i\Vert _{N_i} + \Vert R_i(|x(\cdot )|)\Vert _{N_i} \right) . \end{aligned}$$

Finally, since \(K_i(t, \cdot ) \in E_{M_i}\) and \(x \in E_{\varphi }\) we have

$$\begin{aligned} \int _T \Vert K_i(t,s)f_i(s,x(s))\Vert \;ds \le 2 \Vert K_i(t,\cdot )\chi _{T} \Vert _{M_i} \cdot \left( \Vert b_i\Vert _{N_i} + \Vert R_i(|x(\cdot )|)\Vert _{N_i} \right) \end{aligned}$$

for a.e. \( t \in I\). That is, we have shown that the operators \(A_i:B_1(E_{\varphi })\rightarrow E_{\phi _i}\).

Next, we prove that the operators \(A_i: B_1(E_{\varphi }) \rightarrow E_{\phi _i}\) are continuous. Let \(x_n, x_0 \in B_1(E_{\varphi })\) be such that \(\Vert x_n - x_0 \Vert _{\varphi } \rightarrow 0\) as \(n \rightarrow \infty \). Suppose, contrary to our claim, that \(A_i\) are not continuous and \(\Vert A_i(x_n) - A_i(x_0) \Vert _{\phi _i}\) do not converge to zero. Then there exist \(\varepsilon _i > 0\) and a subsequence \((x_{n_{k}})\) such that

$$\begin{aligned} \Vert A_i(x_{n_{k}}) - A_i(x_0) \Vert _{\phi _i} > \varepsilon _i \ {\text{ for } } k = 1,2,... \end{aligned}$$
(3.2)

and this subsequence is a.e. convergent to \(x_0\). Since elements of \((x_n)\) are in the ball, the sequence \((\int _a^b \varphi (|x_n(t)|) dt)\) is bounded. As the space \(E_{\varphi }\) is regular (by definition), the balls are norm-closed in \(L_1(I)\), so the sequence \((\int _a^b |x_n(t)| dt)\) is also bounded.

Moreover, by (C3) and (C5) there exist \(\omega , \gamma ,u_0>0,\) such that ( [24, p. 196]) for any \(i=1, 2,..., n\)

$$\begin{aligned} \Vert R_i(|x(\cdot )|)\Vert _{N_i}= & {} \frac{1}{\omega } \Vert \omega R_i(|x(\cdot )|) \Vert _{N_i} \\\le & {} \frac{1}{\omega } \inf _{r > 0} \left\{ \int N_i( \omega R_i(|x(t)|)/ r ) dt \le 1\right\} \\\le & {} \frac{1}{\omega } \left( 1 + \int _a^b N_i( \omega R_i(|x(t)|) ) dt \right) \\\le & {} \frac{1}{\omega } \left( 1 + N_i(\omega R_i(u_0))\cdot (b-a) + \gamma \int _a^b \varphi (|x(t)|) dt \right) , \end{aligned}$$

whenever \(x \in L_{\varphi }\) with \(\Vert x\Vert _{\varphi } \le 1\). Thus

$$\begin{aligned} \int _T \Vert K_i(t,s)f_i(s,x_n(s))\Vert \;ds\le & {} 2 \Vert K_i(t,\cdot )\chi _{T} \Vert _{M_i} \cdot \left( \Vert b_i\Vert _{N_i} + \Vert R_i(|x_n(\cdot ) |)\Vert _{N_i} \right) \\\le & {} 2 \Vert K_i(t,\cdot )\chi _{T} \Vert _{M_i} \cdot \bigg ( \Vert b_i\Vert _{N_i}\\{} & {} + \left. \frac{1}{\omega } \left[ 1 + N_i(\omega R_i(u_0)) (b-a)+ \gamma \int _a^b \varphi (|x_n(t)|) dt \right] \right) \end{aligned}$$

and then the sequences \((\Vert K_i(t,s)f_i(s,x_n(s))\Vert )\) are equiintegrable on I for a.e. \(t \in I\). From the continuity of \(f_i(t,\cdot )\) we obtain \(\lim _{k \rightarrow \infty } K_i(t,s)f_i(s,x_{n_{k}}(s)) = K_i(t,s)f_i(s,x_{0}(s))\) for a.e. \(s \in I\). Now, applying the Vitali convergence theorem we obtain that

$$\begin{aligned} \lim _{k \rightarrow \infty } A_i(x_{n_{k}})(t) = A_i(x_0)(t) ~~ { \text{ for } \text{ a.e. } } t \in I. \end{aligned}$$

But it follows from Eq. (3.1) that \(A_i(x_{n_{k}})\) are subsets of \(E_{\phi _i}\) and then \(\lim _{k \rightarrow \infty } A_i(x_{n_{k}})(t) = A_i(x_0)(t)\) which contradicts the inequality (3.2).

Then the operators \(A_i\) are continuous between the indicated spaces. It follows from our assumption (C4) that the operators \(F_{h_i}: B_1(E_{\varphi }) \rightarrow E_{\psi _i}\) are continuous, and by (N2) the operators \(U_i: B_1(E_{\varphi }) \rightarrow E_{\varphi _i} \) are continuous. By assumption (C1), the operators \(H_i\) maps \(B_1(E_{\varphi })\) into \(E_{\varphi _i}\) continuously. Finally, we deduce from assumption (N1) that \(H=\prod _{i=1}^n H_i: B_1(E_{\varphi }) \rightarrow E_{\varphi }\) is continuous.

II. We will construct the invariant ball for our operator

$$\begin{aligned} B_1(E_{\varphi })=\{ x \in L_{\varphi }: \Vert x\Vert _\varphi \le 1 \}. \end{aligned}$$

Fix \(\lambda _i \in {{\mathbb {R}}}\) with \(|\lambda _i| < \rho _i\), where

$$\begin{aligned} \rho _i= \frac{1 - \Vert g_i\Vert _{\varphi _i}}{ 2 k_i^* \cdot C_i \cdot \big (\Vert a_i \Vert _{\psi _i}+ l_i \big )\cdot \Vert K_i\Vert _{M_i}}. \end{aligned}$$

Let \(x \in B_1(E_{\varphi })\) be arbitrary and using assumption (C3), formula (3.1) and Proposition 2.8, we have

$$\begin{aligned} \Vert H_i(x)\Vert _{\varphi _i}\le & {} \Vert g_i\Vert _{\varphi _i}+~\Vert U_ix\Vert _{\varphi _i} \\= & {} \Vert g\Vert _{\varphi _i}+~\Vert \lambda _i \cdot F_{h_i}(x) \cdot A_i(x)\Vert _{\varphi _i}\\\le & {} \Vert g_i\Vert _{\varphi _i}+~k_i^* |\lambda _i| \cdot \Vert F_{h_i}(x) \Vert _{\psi _i} \cdot \Vert A_i(x)\Vert _{\phi _i} \\\le & {} \Vert g_i\Vert _{\varphi _i} + k_i^* |\lambda _i| \cdot \Big ( \Vert a_i \Vert _{\psi _i}+l_i \Big \Vert \psi _i^{-1}\big (\varphi \big (x\big )\big )\Big \Vert _{\psi _i} \Big )\\{} & {} \cdot \Big \Vert \int _a^b K_i(\cdot ,s)f_i(s, x(s))\;ds\Big \Vert _{\phi _i}\\\le & {} \Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot |\lambda _i | \cdot C_i \cdot \big (\Vert a_i \Vert _{\psi _i}+ l_i \cdot \Vert x\Vert _{\varphi }\big ) \cdot \Vert K_i\Vert _{M_i} \\\le & {} \Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot \rho _i \cdot C_i \cdot \big (\Vert a_i \Vert _{\psi _i}+ l_i \big )\cdot \Vert K_i\Vert _{M_i}, \end{aligned}$$

whenever \(\Vert x\Vert _{\varphi } \le r\). Therefore, using assumption (N1) we obtain

$$\begin{aligned} \Vert H(x)\Vert _\varphi\le & {} k \prod _{i=1}^n \Vert H_i(x)\Vert _{\varphi _i}\\\le & {} k \prod _{i=1}^n \bigg ( \Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot \rho _i \cdot C_i \cdot \big (\Vert a_i \Vert _{\psi _i}+ l_i \big )\cdot \Vert K_i\Vert _{M_i} \bigg ) \le 1. \end{aligned}$$

Then \(H: B_1(E_{\varphi }) \rightarrow B_1(E_{\varphi })\) is continuous.

III. Let \(Q_1\) is a subset of \(B_1(E_{\varphi })\) consisting of all functions which are a.e. nondecreasing on I. Similarly as claimed in [15] this set is convex, nonempty, closed, and bounded in \(L_{\varphi }\). Moreover, in view of Lemma 2.6, the set \(Q_1\) is compact in measure.

IV. We will now show that H preserves the monotonicity of functions. Take any \(x \in Q_1\), then x is a.e. nondecreasing on I and consequently \(F_{f_i},~F_{h_i}\) are also of the same type by virtue of assumption (C2). Further, \(A_i(x)\) are a.e. nondecreasing on I by assumption (K3). By assumption (N2), the operators \(U_i\) are a.e. nondecreasing on I. Moreover, assumption (C1) allows us to deduce that \(H_i\) are also a.e. nondecreasing on I. Then \(H:Q_1 \rightarrow Q_1\) is continuous.

V. We will prove that H is a contraction with respect to some measure of noncompactness. It is worth recalling that it is now necessary to use assumption (N1) guaranteeing that \(H: E_\varphi \rightarrow E_\varphi \) and our measure of noncompactness should be defined on this space.

Assume that \(X \subset Q_1\) and let the fixed constant \( \varepsilon > 0\) be arbitrary. Then for an arbitrary \(x \in X\) and for a set \(D \subset I\), meas \(D \le \varepsilon \) (\(i= 1,2,..., n\)) we obtain

$$\begin{aligned} \Vert H_i(x)~\chi _D \Vert _{\varphi _i}\le & {} \Vert g_i \chi _D \Vert _{\varphi _i} + \Vert U_i(x) \cdot \chi _D \Vert _{\varphi _i} \\= & {} \Vert g_i \chi _D \Vert _{\varphi _i} + \Vert \lambda _i \cdot F_{h_i}(x) \cdot A_i(x) \chi _D \Vert _{\varphi _i}\\\le & {} \Vert g_i \chi _D \Vert _{\varphi _i} + k_i^* |\lambda _i| \cdot \Vert F_{h_i}(x) \chi _D \Vert _{\psi _i} \cdot \Vert A_i(x) \cdot \chi _D \Vert _{\phi _i} \\\le & {} \Vert g_i \chi _D \Vert _{\varphi _i} + k_i^* |\lambda _i | \cdot \big \Vert \big ( a_i +l_i \psi _i^{-1}\big (\varphi \big (x\big )\big ) \chi _D \big \Vert _{\psi _i} \\{} & {} \times \Big \Vert \int _D K_i(\cdot , s)f_i(s, x(s))\;ds\Big \Vert _{\phi _i}\\\le & {} \Vert g_i \chi _D \Vert _{\varphi _i} + k_i^* |\lambda _i | \Big ( \Vert a_i \chi _D \Vert _{\psi _i}+l_i \Big \Vert \psi _i^{-1}\big (\varphi \big (x\big )\big ) \chi _D \Big \Vert _{\psi _i}\Big )\\{} & {} \times \Big \Vert \int _D | K_i(\cdot ,s) | (b_i(s) + R_i(|x(s)|))\;ds\Big \Vert _{\phi _i} \\\le & {} \Vert g_i \chi _D \Vert _{\varphi _i} + k_i^* |\lambda _i | \big ( \Vert a_i \chi _D \Vert _{\psi _i}+l_i \Vert x \chi _D \Vert _\varphi \big )\\{} & {} \times 2 \Vert K_i \Vert _{M_i}\big (\Vert b_i\chi _D\Vert _{N_i} + R_i(1)\big ). \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert H(x)\cdot \chi _D\Vert _\varphi\le & {} k \prod _{i=1}^n \Vert H_i(x)\Vert _{\varphi _i} \le k \prod _{i=1}^n \Big (\Vert g_i \chi _D \Vert _{\varphi _i} \\{} & {} + 2 k_i^* \Vert K_i \Vert _{M_i} |\lambda _i | \big ( \Vert a_i \chi _D \Vert _{\psi _i}+l_i \Vert x \chi _D \Vert _\varphi \Big ) \big (\Vert b_i\chi _D\Vert _{N_i} + R_i(1)\big ). \end{aligned}$$

Since, \(g_i \in E_{\varphi _i},~a_i \in E_{\psi _i}\), and \(b_i \in E_{N_i}\), we obtain

$$\begin{aligned}{} & {} \lim _{\varepsilon \rightarrow 0}~\{\sup _{mes~D \le \varepsilon } [\sup _{x \in X} \{ \Vert g_i \chi _D \Vert _{\varphi _i} \}]\}=0,~~ \lim _{\varepsilon \rightarrow 0}~\{\sup _{mes~D \le \varepsilon } [\sup _{x \in X} \{ \Vert a_i \chi _D \Vert _{\psi _i} \}]\}=0,\\{} & {} { \text{ and } } \lim _{\varepsilon \rightarrow 0}~\{\sup _{mes~D \le \varepsilon } [\sup _{x \in X} \{ \Vert b_i \chi _D \Vert _{N_i} \}]\}=0. \end{aligned}$$

Therefore, from the definition of c(x), we obtain

$$\begin{aligned} c(H(X)) \le k \prod _{i=1}^n \bigg ( 2k_i^* |\lambda _i | l_i \Vert K_i \Vert _{M_i} R_i(1) \bigg ) c(X). \end{aligned}$$

Since \(\emptyset \ne X \subset Q_1\) is a bounded and compact in measure subset of a regular Orlicz space \(E_\varphi \), we can use Proposition 2.10 and obtain

$$\begin{aligned} \beta _H(H(X)) \le k \prod _{i=1}^n \bigg ( k_i^* |\lambda _i | l_i \Vert K_i \Vert _{M_i} R_i(1) \bigg )2^{n} \beta _H(X). \end{aligned}$$

Assumption (K2) with \(\prod _{i=1}^n \big ( k_i^* |\lambda _i | l_i \Vert K_i \Vert _{M_i} R_i(1) \big ) <\frac{1}{2^{n}k}\) allows us to apply Theorem 2.11, which completes the proof. \(\square \)

3.2 The case of \(\Delta _3\)-condition

In this case,we consider N-functions with the growth in principle faster than the polynomial satisfying \(\Delta _3\)-condition (cf. [24, p. 61]). We denote by \(\vartheta \) the norm of the identity operator from \(L_{\varphi } \rightarrow L^1(I)\), i.e., \(\vartheta =\sup \{ \Vert x\Vert _1: x \in B_1(L_{\varphi }) \}\).

Theorem 3.2

Let \(~i=1,\cdots n,\) and assume, that \(\varphi , \varphi _i,~\phi _i,~\psi _i\) are N-functions and that \(M_i\) and \(N_i\) are complementary N-functions and that (N1), (N2), (C1)-(C4), (K1) and (K3) hold true. In addition, for \(i=1,\cdots n,\) set the following assumptions:

  1. (C6)
    1. 1.

      The N-functions \(N_i\) fulfill the \(\Delta _3\)-condition,

    2. 2.

      \(K_i \in E_{M_i}(I^2)\) and \(t \rightarrow K_i(t,s) \in E_{\phi _i}\) for a.e. \(s \in I\).

    3. 3.

      There exist \(\beta ,~u_0> 0\) such that

      $$\begin{aligned} R_i(u)\le \beta \frac{M_i(u)}{u},~~\text{ for } ~~u\ge u_0. \end{aligned}$$
    4. 4.

      \(\phi _i\) are N-functions satisfying

      $$\begin{aligned} \iint \limits _{I^2} {\phi _i} (M_i( |K_i(t,s)|)) \; dt \; ds < \infty . \end{aligned}$$
  2. (K4)

    Assume that r is a positive number satisfying

    $$\begin{aligned}{} & {} k \prod _{i=1}^n \bigg (\Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot C_i \cdot |\lambda _i| \cdot \Vert K_i\Vert _{\phi _i \circ M_i} \big (\Vert a_i \Vert _{\psi _i}+ l_i \cdot r \big )\nonumber \\{} & {} \times \left( \Vert b_i\Vert _{N_i} + \frac{1}{\omega }(1 + \eta u_0 (b-a) + \eta \vartheta r ) \right) \bigg )\le r \end{aligned}$$
    (3.3)

    and

    $$\begin{aligned}{} & {} \prod _{i=1}^n \bigg ( C_i \cdot k_i^* \cdot |\lambda _i |\cdot l_i \cdot \Vert K_i\Vert _{\phi _i \circ M_i} \cdot R_i(r)\bigg )<\frac{1}{2^nk r^{n-1}}, \\ {}{} & {} \text{ where }~ C_i=(2 + (b-a)(1 + \phi _i (1))). \end{aligned}$$

Then there exist numbers \(\rho _i>0\) such that for all \(\lambda _i \in {{\mathbb {R}}}\) with \(|\lambda _i |<\rho _i,~ i = 1,\cdots ,n\), there exists a solution \(x \in E_\varphi \) of (1.1) which is a.e. nondecreasing on I.

Proof

I’. In this case, the operator H shall be examined on the whole space \(E_{\varphi }\). By [24, Lemma 15.1 and Theorem 19.2] and the assumption (C6)\(_4\) we obtain

$$\begin{aligned} \Vert A_i(x) \chi _T \Vert _{\phi _i} \le 2 C_i \cdot \Vert K_i \cdot \chi _{T \times I}\Vert _{\phi _i \circ M_i} \left( \Vert b_i\Vert _{N_i} + \Vert R_i(|x(\cdot )|)\Vert _{N_i}\right) \end{aligned}$$
(3.4)

for arbitrary \(x \in L_{\varphi }\) and arbitrary measurable subset T of I.

Note that it follows from assumption (C6)\(_3\) that there exist constants \(\omega , u_0 > 0\) and \(\eta > 1\) such that \(N_i(\omega R_i(u)) \le \eta u\) for \(u \ge u_0\). Thus for \(x \in L_{\varphi }\)

$$\begin{aligned} \Vert R_i(|x(\cdot )|)\Vert _{N_i}\le & {} \frac{1}{\omega }\left( 1 + \int _I N_i(\omega R_i(|x(s)|) \;ds \right) \\\le & {} \frac{1}{\omega }\left( 1 + \eta u_0 (b-a) + \eta \int _I |x(s)| \;ds \right) . \end{aligned}$$

The remaining estimations can be determined as in Theorem 3.1 and then we obtain, that \(A_i: E_{\varphi } \rightarrow E_{\phi _i}\), so by the properties of \(F_{h_i}\) we get \(H_i: E_{\varphi } \rightarrow E_{\varphi _i}\). By assumption (N1), we have \(H=\prod _{i=1}^n H_i: E_{\varphi } \rightarrow E_{\varphi }\) is continuous.

II’. We consider \(B_r(E_{\varphi })\) as the domain of the operator H, where r is as in Eq. (3.3). Let us place

$$\begin{aligned} \rho _i= & {} \Bigg [ 2 k_i^* (2 + (b-a)(1 + \phi _i (1))) \Vert K_i\Vert _{\phi _i \circ M_i}( \Vert a_i \Vert _{\psi _i}+l_i) \\{} & {} \qquad \cdot \left( \Vert b_i\Vert _{N_i} + \frac{1}{\omega }\left( 1 + \eta u_0 (b-a) \right) \right) \Bigg ]^{-1}. \end{aligned}$$

Now, for fixed \(\lambda _i \in {{\mathbb {R}}},~|\lambda _i | < \rho _i,~ i = 1,\cdots ,n\) and \(x \in B_r(E_{\varphi })\) we have

$$\begin{aligned} \Vert H_i(x)\Vert _{\varphi _i}\le & {} \Vert g_i\Vert _{\varphi _i}+~\Vert U_ix\Vert _{\varphi _i} \\= & {} \Vert g_i\Vert _{\varphi _i}+~\Vert \lambda _i \cdot F_{f_i}(x) \cdot A_i(x)\Vert _{\varphi _i} \\\le & {} \Vert g_i\Vert _{\varphi _i}+~k_i^*|\lambda _i|\Vert F_{f_i}(x) \Vert _{\psi _i} \cdot \Vert A_i(x)\Vert _{\phi _i} \\\le & {} \Vert g_i\Vert _{\varphi _i} + k_i^* |\lambda _i | \bigg ( \Vert a_i \Vert _{\psi _i}+l_i \bigg \Vert \psi _i^{-1}\big (\varphi \big (x\big )\big )\bigg \Vert _{\psi _i} \bigg )\\{} & {} \times \bigg \Vert \int _a^b K_i(\cdot ,s)f_i(s, x(s))\;ds \bigg \Vert _{\phi _i} \\\le & {} \Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot C_i \cdot |\lambda _i | \cdot \big (\Vert a_i \Vert _{\psi _i}+ l_i \cdot \Vert x\Vert _{\varphi }\big )\Vert K_i\Vert _{\phi _i \circ M_i}\\{} & {} \times \left( \Vert b_i\Vert _{N_i} +\Vert R_i(|x(\cdot )|)\Vert _{N_i} \right) \\\le & {} \Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot C_i \cdot |\lambda _i | \cdot \big (\Vert a_i \Vert _{\psi _i}+ l_i \cdot \Vert x\Vert _{\varphi }\big )\Vert K_i\Vert _{\phi _i \circ M_i}\\&\qquad \times&\left( \Vert b_i\Vert _{N_i} +\frac{ 1}{\omega }\big (1+ \eta u_0\cdot (b-a) + \eta \Vert x\Vert _1 \big ) \right) \\\le & {} \Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot C_i \cdot |\lambda _i | \big (\Vert a_i \Vert _{\psi _i}+ l_i \cdot \Vert x\Vert _{\varphi }\big ) \Vert K_i\Vert _{\phi _i \circ M_i}\\&\qquad \times&\left( \Vert b_i\Vert _{N_i} +\frac{ 1}{\omega }\big (1+ \eta u_0\cdot (b-a) + \eta \vartheta \Vert x\Vert _{\varphi } \big ) \right) \\\le & {} \Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot C_i \cdot |\lambda _i| \cdot \Vert K_i\Vert _{\phi _i \circ M_i} \big (\Vert a_i \Vert _{\psi _i}+ l_i \cdot r \big ) \\&\qquad \times&\left( \Vert b_i\Vert _{N_i} + \frac{1}{\omega }\big (1 + \eta u_0 (b-a) + \eta \vartheta r \big ) \right) . \end{aligned}$$

Therefore, using assumption (N1), we have

$$\begin{aligned} \Vert H(x)\Vert _{\varphi }\le & {} k \prod _{i=1}^n\Vert H_i(x)\Vert _{\varphi _i} \\\le & {} k \prod _{i=1}^n \bigg (\Vert g_i\Vert _{\varphi _i} + 2 k_i^* \cdot C_i \cdot |\rho _i| \cdot \Vert K_i\Vert _{\phi _i \circ M_i} \big (\Vert a_i \Vert _{\psi _i}+ l_i \cdot r \big )\\&\qquad \times&\left( \Vert b_i\Vert _{N_i} + \frac{1}{\omega }(1 + \eta u_0 (b-a) + \eta \vartheta r ) \right) \bigg )\le r. \end{aligned}$$

Thus, again using assumption (K4), we have that \(H: B_r(E_{\varphi }) \rightarrow B_r(E_{\varphi })\).

The steps III’. and IV’. resemble those of Theorem 3.1 for the subset \(Q_{r} \subset B_{r}(E_{\varphi })\).

V’. Assume that \(\emptyset \ne X \subset Q_r \) and \(\varepsilon > 0\) are arbitrary constant. Then for any \(x \in X\) and for a set \(D \subset I\), with meas \(D \le \varepsilon \) we obtain

$$\begin{aligned} \Vert H_i(x) \cdot \chi _D \Vert _{\varphi _i}\le & {} \Vert g_i \chi _D \Vert _{\varphi _i} + k_i^* \cdot |\lambda _i | \cdot \big (\Vert a_i \chi _D \Vert _{\psi _i}+ l_i \cdot \Vert x \chi _D\Vert _{\varphi }\big )\\{} & {} \times \bigg \Vert \int _D | K_i(\cdot ,s) | (b_i(s) + R_i(|x(s)|))\;ds\bigg \Vert _{\phi _i} \\\le & {} \Vert g_i \chi _D \Vert _{\varphi _i} + ~ 2 \cdot C_i \cdot k_i^* \cdot |\lambda _i | \cdot \big (\Vert a_i \chi _D \Vert _{\psi _i}+ l_i \cdot \Vert x \chi _D\Vert _{\varphi }\big )\\{} & {} \times \Vert K_i \Vert _{\phi _i \circ M_i}(\Vert b_i \chi _D \Vert _{N_i} + R_i(r)). \end{aligned}$$

Since, \( \Vert x \chi _D\Vert _{\varphi }^n \le \Vert x \Vert _{\varphi }^{n-1}\cdot \Vert x \chi _D\Vert _{\varphi }\le r^{n-1} \cdot \Vert x \chi _D\Vert _{\varphi }.\)

Then, as in the previous theorem, we have

$$\begin{aligned} \beta _H(H(X))\le k \prod _{i=1}^n \bigg ( 2 \cdot C_i \cdot k_i^* \cdot |\lambda _i |\cdot l_i \cdot \Vert K_i\Vert _{\phi _i \circ M_i} \cdot R_i(r) \bigg )\cdot r^{n-1} \beta _H(X). \end{aligned}$$

Assumption (K4) together with the estimation \( \prod _{i=1}^n \bigg ( C_i \cdot k_i^* \cdot |\lambda _i |\cdot l_i \cdot \Vert K_i\Vert _{\phi _i \circ M_i} \cdot R_i(r)\bigg )<\frac{1}{2^n k r^{n-1}}\), allow us to use Theorem 2.11, which completes the proof. \(\square \)

3.3 The case of \(\Delta _2\)-condition

In this subsection we deal with the case when the generating N-functions satisfy \(\Delta _2\)-condition. This version of our results includes, as a special case, the case of solutions of the considered equation in Lebesgue spaces \(L_p\), and is also the one most often studied in earlier work on classical integral equations (e.g., [2]).

Theorem 3.3

Let \(i=1,...,n,\) and assume that \(\varphi , \varphi _i,~\phi _i,~\psi _i\) are N-functions, \(M_i\) and \(N_i\) are complementary N-functions and that assumptions (N1), (N2), (C1)-(C4), and (K3) hold true. Furthermore, for \(i=1,\cdots ,n,\) set the following assumptions:

  1. (C7)

    Assume that \(\varphi _i\) be N-function and the functions \(N_i\) satisfy \(\Delta _2\)-condition.

    1. 1.

      There are \(\gamma _i ~\ge 0\) such that

      $$\begin{aligned} R_i(u) \le \gamma _i N_i^{-1} \left( \varphi \left( {u}\right) \right) ~~\text{ for }~~u~\ge ~0. \end{aligned}$$
    2. 2.

      \(s \rightarrow K_i(t,s) \in L_{M_i}\) for a.e. \(t \in I \) and \(p_i(t) = \Vert K_i(t,\cdot ) \Vert _{M_i} \in E_{\phi _i}\).

  2. (K5)

    Assume that for some \(q_i > 0\), there exists \(r^*>0\) such that

    $$\begin{aligned}{} & {} \int _{I} \varphi \bigg (\prod _{i=1}^n \bigg [|g_i(t)| + k_i^* |\lambda _i| \bigg ( a_i(t) + l_i \psi _i^{-1}\bigg (\varphi \big (x\big )\bigg )\bigg ) \\{} & {} \cdot |p_i(t)| \bigg ( \Vert b_i\Vert _{N_i}~+~ \gamma _i r^* \bigg )\bigg ]\bigg ) \; dt \le r^* \end{aligned}$$

and

$$\begin{aligned} \prod _{i=1}^n \bigg ( k_i^* \cdot |\lambda _i| \cdot l_i \cdot \Vert p_i \Vert _{\phi _i} \cdot \gamma _i^*\bigg )< \frac{1}{k\cdot r^*{^{2n-1}}}. \end{aligned}$$

Then there are numbers \(\rho _i > 0\) such that for all \(\lambda _i \in {{\mathbb {R}}}\) with \(|\lambda _i | < \rho _i,~i=1,...,n,\) there exists a solution \(x \in E_\varphi \) of (1.1) which is a.e. nondecreasing on I.

Proof

I”. First, from assumptions (C2) and (C3), (cf. [24, Lemma 16.3 and Theorem 16.3] with \(M_1=N_i, M_2 = {\phi _i}\) and \(N_1=M_i \)) it follows that the operators \(A_i\) are continuous mappings from \(B_1(E_{\varphi })\) into \(E_{\phi _i}\). From our assumption (C4) it follows that the operators \(F_{h_i}\) are continuous from \(B_1(E_{\varphi })\) into \(E_{\psi _i}\) and then from (N2) that the operators \(U_i\) are continuous mapping from \(B_1(E_{\varphi })\) into the space \(E_{\varphi _i}\). By assumption (C1), we have that \(H_i:~B_1(E_{\varphi })~\rightarrow ~E_{\varphi _i}\) are continuous. Finally, from assumption (N1) we conclude that the operator \(H:~B_1(E_{\varphi })~\rightarrow ~E_{\varphi }\) is continuous.

II”. We construct an invariant set \(V \subset B_1(E_{\varphi })\) for the operator H as it is bounded in \(L_{\varphi }\). We fix \(\lambda _i \in {{\mathbb {R}}}\) with \(|\lambda _i| < \rho _i\) and let \(\rho _i = \sup Q\), where Q is the set of all positive numbers \(q_i\) for which there exists \(r^*>0\) such that

$$\begin{aligned}{} & {} \int _{I} \varphi \bigg (\prod _{i=1}^n \bigg [|g_i(t)| + k_i^* |\lambda _i| \bigg ( a_i(t) + l_i \psi _i^{-1}\bigg (\varphi \big (x\big )\bigg )\bigg ) \\{} & {} \cdot |p_i(t)| \bigg ( \Vert b_i\Vert _{N_i}~+~ \gamma _i r^* \bigg )\bigg ]\bigg ) \; dt\le r^*. \end{aligned}$$

Let V denote the closure of the set \({{\{ x \in E_{\varphi }: \int _a^b \varphi (|x(s)|) \; ds \le r^* - 1 \} }}\). Obviously, V is not a ball in \(E_{\varphi }\), but \(V \subset B_{r^*}(E_{\varphi })\) (cf. [24, p. 222]). Note that \({\overline{V}}\) is a bounded convex and closed subset of \(E_{\varphi }\).

Let us take any \(x \in V\). Using [24, Theorem 10.5 with \(k=1\)], we obtain that for any \(t \in I\)

$$\begin{aligned} \Vert R_i(|x|) \Vert _{N_i} ~\le ~ \gamma _i \bigg \Vert N_i^{-1} \left( \varphi \left( |x|)\right) \right) \bigg \Vert _{N_i} \le \gamma _i + \gamma _i \int _a^b \varphi \left( {|x(s)|}\right) \; ds \end{aligned}$$
(3.5)

and then by the Hölder inequality and our assumptions we conclude that

$$\begin{aligned} |A_i(x)(t)| \le |p_i(t)| \bigg ( \Vert b_i\Vert _{N_i}~+~ \Vert R_i(|x|) \Vert _{N_i} \bigg ). \end{aligned}$$

Thus for any measurable subset T of I. For arbitrary \(x \in V\) and \(t \in I\), we have

$$\begin{aligned}{} & {} |H_i(x)(t)| \le |g_i(t)| + k_i^* |\lambda _i |\cdot | F_{h_i}(x)| \cdot |A_i(x)(t)| \\\le & {} |g(t)| + k_i^* |\lambda _i| \Big ( a_i(t) + l_i \psi _i^{-1}\Big (\varphi \big (x\big )\Big )\Big ) \cdot |p_i(t)| \bigg ( \Vert b_i\Vert _{N_i}~+~ \Vert R_i(|x|) \Vert _{N_i} \Big )\\\le & {} |g_i(t)| +~ k_i^* |\lambda _i|\bigg ( a_i(t) + l_i \psi _i^{-1}\bigg (\varphi \big (x\big )\bigg )\bigg )\\{} & {} \ \ \ \ \ \times |p_i(t)| \bigg ( \Vert b_i\Vert _{N_i}~+~ \gamma _i + \gamma _i \int _a^b \varphi \left( {|x(s)|}\right) \; ds \bigg )\\\le & {} |g_i(t) | +~ k_i^* |\lambda _i| \bigg ( a_i(t) + l_i \psi _i^{-1}\bigg (\varphi \big (x\big )\bigg )\bigg ) \\{} & {} \ \ \ \ \ \times |p_i(t)| \bigg ( \Vert b_i\Vert _{N_i}~+~ \gamma _i + \gamma _i (r^*-1) \bigg ). \end{aligned}$$

Therefore,

$$\begin{aligned}{} & {} \int _{I} \varphi (H(x)(t))\;dt = \int _{I}\varphi \Big (\prod _{i=1}^n H_i(x)(t)\bigg )\;dt \\{} & {} \le \int _{I} \varphi \Big (\prod _{i=1}^n \Big [|g_i(t)| + k_i^* |\lambda _i| \Big ( a_i(t) + l_i \psi _i^{-1}\Big (\varphi \big (x\big )\Big )\Big ) \\ {}{} & {} \cdot |p_i(t)| \Big ( \Vert b_i\Vert _{N_i} + \gamma _i r^* \Big )\Big ]\Big ) \; dt. \end{aligned}$$

From the definition of \(r^*\) we obtain \(\int _{I} \varphi (H(x)(t)) \; dt \le r^*\), and then \(H(V) \subset V\). As a consequence, \(H({\overline{V}}) \subset \overline{H(V)} \subset {\overline{V}} = V\), from which it follows that \(H: V \rightarrow V\) is continuous on \(V \subset B_{r^*}(E_{\varphi })\).

The steps III”. and IV”. also resemble in this case those of Theorem 3.1 for the subset \(Q_{r} \subset B_{r}(E_{\varphi })\).

V”. Next, we prove that H is a contraction concerning the measure of noncompactness. Recall that for \( x \in Q_{r^*}\), we have

$$\begin{aligned} \int _I \psi _i\bigg ( \psi _i^{-1}\big [\varphi \big (x(s) \big )\big ] \bigg )\;ds = \int _I \varphi \big (x(s)\big ) \;ds \le \Vert x\Vert _\varphi . \end{aligned}$$

Assume that \(\emptyset \ne X \subset Q_{r^*}\) and \(\varepsilon > 0\) are arbitrary constants. Then for any \(x \in X\) and for a set \(D \subset I\), with meas \(D \le \varepsilon \), we obtain

$$\begin{aligned}{} & {} \Vert H_i(x) \cdot \chi _D \Vert _{\varphi _i} \\ {}{} & {} \quad \le \Vert g_i\cdot \chi _D \Vert _{\varphi _i} + k_i^* \cdot |\lambda _i | \cdot \bigg (\Vert a_i\cdot \chi _D\Vert _{\psi _i} + l_i \Big \Vert \psi _i^{-1}\Big (\varphi \big (x\big )\Big )\cdot \chi _D\Big \Vert _{\psi _i}\Big )\\{} & {} \qquad \times \Big \Vert \int _D | K_i(\cdot ,s) | (b_i(s) + R_i(|x(s)|))\;ds\Big \Vert _{\phi _i} \\{} & {} \quad \le \Vert g_i\cdot \chi _D \Vert _{\varphi _i}~+~ k_i^* \cdot |\lambda _i | \bigg (\Vert a_i\cdot \chi _D\Vert _{\psi _i} + l_i \Vert x \cdot \chi _D \Vert _\varphi \bigg ) \\{} & {} \qquad \times \Vert p_i \Vert _{\phi _i} \bigg ( \Vert b_i \cdot \chi _D \Vert _{N_i} + \gamma _i + \gamma _i \int _a^b \varphi \left( {|x(s)|}\right) \; ds \bigg )\\{} & {} \quad \le \Vert g_i\cdot \chi _D \Vert _{\varphi _i}~+~ k_i^* \cdot |\lambda _i | \bigg (\Vert a_i\cdot \chi _D\Vert _{\psi _i} + l_i \Vert x \cdot \chi _D \Vert _\varphi \bigg ) \\{} & {} \qquad \times \Vert p_i \Vert _{\phi _i} \bigg ( \Vert b_i \cdot \chi _D \Vert _{N_i} + \gamma _i r^*\bigg ). \end{aligned}$$

Hence, we get that

$$\begin{aligned}{} & {} \Vert H(x) \cdot \chi _D \Vert _{\varphi } \le k \prod _{i=1}^n \Vert H_i(x) \cdot \chi _D \Vert _{\varphi _i} \\\le & {} k \prod _{i=1}^n \Big ( \Vert g_i\cdot \chi _D \Vert _{\varphi _i} + k_i^* \cdot |\lambda _i | \Big (\Vert a_i\cdot \chi _D\Vert _{\psi _i} + l_i \Vert x \cdot \chi _D \Vert _\varphi \Big ) \\{} & {} \Vert p_i \Vert _{\phi _i} \Big ( \Vert b_i \cdot \chi _D \Vert _{N_i} + \gamma _i r^*\Big )\Big ). \end{aligned}$$

As in Theorem 3.1, we have

$$\begin{aligned} \beta _H(B(X)) \le k r^*{^n} \prod _{i=1}^n \bigg ( k_i^* \cdot |\lambda _i| \cdot l_i \cdot \Vert p_i \Vert _{\phi _i} \gamma _i \bigg ) {r^*}^{n-1} \beta _H(X). \end{aligned}$$

Since \( \prod _{i=1}^n \bigg ( k_i^* \cdot |\lambda _i| \cdot l_i \cdot \Vert p_i \Vert _{\phi _i} \cdot \gamma _i^*\bigg )< \frac{1}{k \cdot {r^*}^{2n-1}},\) we can use Theorem 2.11, which completes the proof. \(\square \)

4 Remarks and examples

Let us to present some concluding remarks and examples that validate and confirm our results. This should confirm that the objectives of the paper have been achieved.

Remark 4.1

To begin, we give examples of what N-functions are covered by the results of our paper. Suppose that \(N_1, N_2\) are complementary functions for \(M_1, M_2\), respectively. Let \(M_1 (u) = \exp {|u|} - |u| - 1,~N_1 (u) = (1 + |u|) \cdot \ln {(1 + |u|)} - |u|\) and \(M_2 (u) = \frac{u^2}{2} = N_2 (u)\), so \(M_1\) fulfills the \(\Delta _3\)-condition and \(N_1\) fulfills the \(\Delta '\)-condition. Assuming that, we point to the N-function either as \(\Psi (u) = M_2 [N_1 (u)]\) or \(\Psi (u) = N_1 [M_2 (u)]\), then by choosing any kernel K from the space \(L_{\Psi } (I)\) we will apply [24, Theorem 15.4]. Thus \((Kx)(t)=\int _a^b K(t,s)x(s)\;ds: L_{M_1} \rightarrow L_{M_2}\) is continuous and it is helpful to use for our results.

Remark 4.2

A complete study of the acting and continuity conditions for operators of the form \(F_h(x)(t) = l\cdot x(t),~l \ge 0,\) between different Orlicz space gives [24, Theorem 18.2].

Example

If \(n=1,~f_1=1\), then Eq. (1.1) diminishes to the classical Hammerstein integral equations

$$\begin{aligned} x(t)=g(t)+ \int _a^b K(t,s)f(s,x(s))\;ds,~~t\in [a,b], \end{aligned}$$

which were inspected in Orlicz spaces in [2, 3, 24, 32].

Example

The authors in [14, 15, 17] examined the quadratic-Hammerstein integral equation in Orlicz spaces under numerous set of assumptions

$$\begin{aligned} x(t)= g(t) + G(x)(t)\int _a^b K(t,s)f(s,x(s))\;ds, ~t \in [a,b]. \end{aligned}$$

For more results we refer to [16, 29].

Example

Let \(h_i(t,x(t)) = l_i\cdot x(t), l_i\ge 0\), then we have the n-product of quadratic integral equations

$$\begin{aligned} x(t)~=~\prod _{i=1}^n \bigg ( g_i(t) + ~l_i\cdot x(t)\cdot \int _a^b K_i(t,s)f_i(s,x(s))\;ds,\bigg ),~~~t \in [a,b], \end{aligned}$$

which represent a particular case of Eq. (1.1) with appropriate forms of the functions \(g_i\) and \(f_i\). This is closer to the problems we cited regarding the product of operators ( [4, 8, 18]).

Example

In case of \(g_i(t) =1\) and \(h_i(t,x) =x\), Eq. (1.1) is in the form of the n-product of Chandrasekhar equations

$$\begin{aligned} x(t) =\prod _{i=1}^n \bigg ( 1 + \lambda \ x(t) \int _0^1 \frac{t}{t + s}e^{s}(b_i(s)+\log {(1 + |x({s})|^\alpha )}) \; ds \bigg ), \end{aligned}$$

where \(R_i(x)=\log {(1 + |x({s})|^\alpha )}\).

Remark 4.3

Let us note what kind of currently studied issues of this type need further work. In particular, this applies when the integral operators are weakly singular (of fractional order), cf. [23]. Basic results on such operators acting on Orlicz spaces can be found, e.g., in the papers [1, 19], but their versions in the case of equations with products of operators need further study.