1 Introduction

In this paper we consider the perturbed Hammerstein integral equation

$$\begin{aligned} u(t)=\gamma (t)H\big (\varphi (u)\big )+\lambda \int _0^1\left( A\left( \int _0^1\big |u(\xi )\big |^q\ \mathrm{d}\xi \right) \right) ^{-1}G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s\nonumber \\ \end{aligned}$$
(1.1)

where \(q\ge 1\) is a constant, each of \(A\ : \ [0,+\infty )\rightarrow {\mathbb {R}}\), \(f\ : \ [0,1]\times [0,+\infty )\rightarrow [0,+\infty )\), \(G\ : \ [0,1]\times [0,1]\rightarrow [0,+\infty )\), \(H \ : \ [0,+\infty )\rightarrow [0,+\infty )\), and \(\gamma \ : \ [0,1]\rightarrow [0,1]\) is continuous, and where

$$\begin{aligned} \varphi (u):=\int _0^1u(s)\ \mathrm{d}\alpha (s); \end{aligned}$$
(1.2)

the integrator \(\alpha \) in (1.2) is of bounded variation on [0, 1] and monotone increasing, the latter assumption so that \(\varphi (u)\le \varphi (w)\) whenever \(u\le w\), which will be important in what follows. Solutions of (1.1) can then be associated to solutions of a boundary value problem, whose boundary conditions will depend on the choice of \(\gamma \) and G. For example, if \(\gamma (t)=1-t\) and

$$\begin{aligned} G(t,s):={\left\{ \begin{array}{ll} t(1-s)\text {, }&{}0\le t\le s\le 1\\ s(1-t)\text {, }&{}0\le s\le t\le 1\end{array}\right. }, \end{aligned}$$

then a solution of (1.1) is a solution of the following nonlocal boundary value problem.

$$\begin{aligned} \begin{aligned} -A\left( \int _0^1\big |u(s)\big |^q\ \mathrm{d}s\right) u''(t)&=\lambda f\big (t,u(t)\big )\text {, }0<t<1\\ u(0)&=H\big (\varphi (u)\big )\\ u(1)&=0 \end{aligned} \end{aligned}$$
(1.3)

Problem (1.3) is an example of a “doubly nonlocal” differential equation in the sense that (1.3) contains two different nonlocal elements.

  1. 1.

    The first nonlocal element is \(\displaystyle A\left( \int _0^1\big |u(s)\big |^q\ \mathrm{d}s\right) \), and it occurs in the differential equation itself. Note that this nonlocal element can be written as \(A\left( \Vert u\Vert _{q}^{q}\right) \) with \(\Vert u\Vert _q\) the \(L^q\) norm of u on [0, 1].

  2. 2.

    The second nonlocal element is \(\varphi (u)\), and it occurs in the boundary condition at \(t=0\). This nonlocal element is a Stieltjes integral, which, therefore, can accommodate many different types of nonlocal boundary conditions by suitably choosing the integrator \(\alpha \). For example, we can accommodate both multipoint-type and integral-type boundary conditions.

Note that the boundary condition at \(t=0\) is not only nonlocal (due to \(\varphi (u)\)) but is also (possibly) nonlinear (due to H).

On the one hand perturbed Hammerstein equations with nonlocal elements of the type \(\varphi (u)\) in (1.2) have been studied extensively in recent years since, as noted, their solutions can be associated to solutions of boundary value problem with nonlocal boundary conditions. For example, one may consult the papers by Anderson [3], Cabada et al. [15], Goodrich [23,24,25], Graef and Webb [37], Infante and Pietramala [41, 42, 44,45,46], Infante et al. [47], Jankowski [48], Karakostas and Tsamatos [49, 50], Karakostas [51], Webb and Infante [55, 56], and Yang [59,60,61,62,63]. Boundary nonlocal elements can arise naturally in mathematical modeling (e.g., beam deflection, chemical reactor theory, and thermodynamics)—see, for example, Cabada et al. [15], Infante and Pietramala [43], and Infante, Pietramala, and Tenuta [47].

On the other hand differential equations with a nonlocal element in the differential equation itself also have been studied extensively. These arise naturally in fractional differential equations since fractional operators are finite convolution operators and thus nonlocal—see, for example, [34, §2], [35, Examples 3, 4, and 5], and [36, Example 5.12]. Nonetheless, more often than not these have fallen into one of two types. The first type has as a model case the equation

$$\begin{aligned} -M\left( \int _0^1\big (u(s)\big )^q\ \mathrm{d}s\right) u''(t)=\lambda f\big (t,u(t)\big )\text {, }t\in (0,1), \end{aligned}$$
(1.4)

where M is some continuous function. This type of equation encompasses as a special case the mean field equation (see Infante [39, (1.2)]), which in its elliptic PDE form is

$$\begin{aligned} -\left( \int _{\Omega }e^u\ \mathrm{d}\varvec{x}\right) \Delta u=\lambda e^u. \end{aligned}$$

The second type has as a model case the equation

$$\begin{aligned} -M\left( \int _0^1\big (u'(s)\big )^q\ \mathrm{d}s\right) u''(t)=\lambda f\big (t,u(t)\big )\text {, }t\in (0,1). \end{aligned}$$
(1.5)

This equation is a particular example of a one-dimensional Kirchhoff equation; higher dimensional Kirchhoff-type equations, which lead to elliptic- and parabolic-type PDEs, have been extensively studied, too. Examples of papers studying equation (1.4) are those by Alves and Covei [4], Aly [2], Bavaud [6], Biler et al. [7], Biler and Nadzieja [8, 9], Caglioti et al. [16], Corrêa [19], Corrêa et al. [20], do Ó et al.  [21], Esposito et al. [22], Goodrich [32], Stańczy [52], Wang et al. [53], Yan and Ma [57], and Yan and Wang [58]. And some examples in the case of equation (1.5) are papers by Afrouzi et al. [1], Azzouz and Bensedik [5], Bouizem et al. [10], Boulaaras [11], Boulaaras et al. [12, 14], Boulaaras and Guefaifia [13], Chung [17], Goodrich [33], and Infante [39, 40].

Recently the author has introduced separately a methodology for these types of nonlocal DEs—i.e., one methodology for nonlocal boundary conditions and another for nonlocal equations such as (1.4). The idea in each case was to consider functions that make the nonlocal element coercive—namely,

$$\begin{aligned} \int _0^1u(s)\ \mathrm{d}\alpha (s)\ge C_0\Vert u\Vert \text { or }\int _0^1u(s)\ \mathrm{d}s\ge C_0\Vert u\Vert , \end{aligned}$$

for some suitably chosen constant \(C_0\in (0,1]\); for example, see [28, (1.9)] and [32, (1.7)]. In addition and respectively, sets of the form

$$\begin{aligned} \left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1u(s)\ \mathrm{d}\alpha (s)<\rho \right\} \text { or }\left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1\big (u(s)\big )^q\ \mathrm{d}s<\rho \right\} , \end{aligned}$$

where \({\mathscr {K}}_0\) is some suitable order cone and \(\rho >0\) is some given number, were used in order to provide direct control over the nonlocal elements. These two ideas used in tandem then allowed for weaker hypotheses on the nonlocal elements in the problem. One of the key ideas is that the boundaries of the above sets are, respectively,

$$\begin{aligned} \left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1u(s)\ \mathrm{d}\alpha (s)=\rho \right\} \text { or }\left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1\big (u(s)\big )^q\ \mathrm{d}s=\rho \right\} . \end{aligned}$$

Thus, for elements of the boundary we have exact control over the nonlocal elements. This fact turns out to be very important in the application of the topological fixed point theory—cf., Lemma 2.12.

Our goal in this paper is to provide a methodology to combine these two types of nonlocal problems. This might seem at first glance to be trivial; after all, it literally simply appears to be combining two already established methodologies. However, it is not so simple. The problem is one of elementary topology. Given two sets A and B, in general, it can\(\underline{{{{\varvec{not}}}}}\) be expected that \(\partial (A\cap B)=\partial A\cap \partial B\). This leads to a fundamental problem since the natural (and, indeed, trivial) way to study jointly these two types of nonlocal problems would be simply to consider

$$\begin{aligned} \left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1u(s)\ \mathrm{d}\alpha (s)<\rho \right\} \cap \left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1\big (u(s)\big )^q\ \mathrm{d}s<\rho \right\} . \end{aligned}$$

But then, generally speaking,

$$\begin{aligned} \begin{aligned}&\partial \left( \left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1u(s)\ \mathrm{d}\alpha (s)<\rho \right\} \cap \left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1\big (u(s)\big )^q\ \mathrm{d}s<\rho \right\} \right) \\&\quad \ne \left\{ u\in {\mathscr {K}}_0\ : \ \int _0^1u(s)\ \mathrm{d}\alpha (s)\text {, }\int _0^1\big (u(s)\big )^q\ \mathrm{d}s=\rho \right\} . \end{aligned} \end{aligned}$$

Due to this problem we must instead study problem (1.1) more carefully. Our methodology consists of using the cone

$$\begin{aligned} \begin{aligned} {\mathscr {K}}&:=\Bigg \{u\in {\mathscr {C}}\big ([0,1]\big )\ :\\&\quad \int _0^1u(s)\ \mathrm{d}\alpha (s)\ge C_0\Vert u\Vert \text {, }\int _0^1u(s)\ \mathrm{d}s\ge C_0\Vert u\Vert \text {, }\min _{t\in [a,b]}u(t)\ge \eta _0\Vert u\Vert \text {, }u\ge 0\Bigg \}, \end{aligned}\nonumber \\ \end{aligned}$$
(1.6)

where \(C_0\), \(\eta _0\in (0,1]\) are constants introduced in Sect. 2. The cone is an amalgamation of the cones used separately to study each type of nonlocal problem. Then using topological fixed point theory we make a dual use of the sets identified above—though individually rather than in intersection. This requires studying carefully the connections, albeit indirect, between the quantities \(\displaystyle \int _0^1u(s)\ \mathrm{d}\alpha (s)\) and \(\displaystyle \int _0^1\big (u(s)\big )^q\ \mathrm{d}s\). Coordinating this is somewhat of a delicate balancing act as the proofs in the next section will reveal.

In the end, as our main results, Theorems 2.13 and 2.16 together with Corollary 2.14, demonstrate, we are able to achieve the same sorts of good features obtainable when studying the two types of problems separately—namely,

  1. 1.

    A need be neither monotone nor strictly positive nor satisfy any global growth condition; and

  2. 2.

    H, likewise, need be neither monotone nor satisfy any either asymptotic or global growth condition.

In particular, Example 2.17, which concludes this paper, demonstrates each of these points. Since condition (1), in particular, is nearly universal among the existing literature, e.g., [4, Condition (2), p. 1], [17, Conditions (M0), (3), (4)], [21, Condition (H1), p. 299], [39, Theorem 2.3], [52, Theorem 2.2], [53, Condition (H1), p. 2], [57, p. 1], and [58, Theorem 4.1, p. 84], it is important to note that we still recover this improvement in spite of the additional complexity created by mixing the two types of nonlocal elements.

2 Main results and an example

As mentioned in Sect. 1 our approach is to use a coordinated pair of open sets in order to apply topological fixed point theory to problem (1.1). In particular, we will consider the open sets \({\widehat{V}}_{\rho }^{q}\), \({\widehat{W}}_{\rho }\subseteq {\mathscr {K}}\) defined as follows; here and throughout \({\mathscr {K}}\) is as in (1.6).

$$\begin{aligned} \begin{aligned} {\widehat{V}}_{\rho }^{q}&:=\left\{ u\in {\mathscr {K}}\ : \ \int _0^1\big (u(s)\big )^q\ \mathrm{d}s<\rho \right\} \\ {\widehat{W}}_{\rho }&:=\left\{ u\in {\mathscr {K}}\ : \ \int _0^1u(s)\ \mathrm{d}\alpha (s)<\rho \right\} \end{aligned} \end{aligned}$$

We note that the open set \({\widehat{V}}_{\rho }^{q}\) has been previously introduced in [32], whereas the open set \({\widehat{W}}_{\rho }\) has been previously used in [26,27,28,29,30,31], for example. However, the dual use of these sets has not been used. In fact, their dual use is nontrivial because as mentioned in Sect. 1 we must carefully analyze the interaction between these two sets.

Going forward it will be useful to make use of some notation. First of all, by \(\varvec{1}\) we denote the function that is identically the constant polynomial 1 on all of \({\mathbb {R}}\)—that is,

$$\begin{aligned} \varvec{1}:=\varvec{1}(x)\equiv 1\text {, }x\in {\mathbb {R}}. \end{aligned}$$

Second of all, for a continuous function \(f\ : \ [0,1]\times [0,+\infty )\rightarrow {\mathbb {R}}\) and for numbers \(0\le a<b\le 1\) and \(0\le c<d<+\infty \) we denote by \(f_{[a,b]\times [c,d]}^{m}\) and \(f_{[a,b]\times [c,d]}^{M}\), respectively, the numbers

$$\begin{aligned} f_{[a,b]\times [c,d]}^{m}:=\min _{(t,y)\in [a,b]\times [c,d]}f(t,y) \end{aligned}$$

and

$$\begin{aligned} f_{[a,b]\times [c,d]}^{M}:=\max _{(t,y)\in [a,b]\times [c,d]}f(t,y). \end{aligned}$$

For a continuous function \(H\ : \ {\mathbb {R}}\rightarrow {\mathbb {R}}\) we will write similarly

$$\begin{aligned} H_{[a,b]}^{m}:=\min _{y\in [a,b]}H(y)\text { and }H_{[a,b]}^{M}:=\max _{y\in [a,b]}H(y), \end{aligned}$$

for any numbers \(-\infty<a<b<+\infty \).

We assume throughout that \({\mathscr {C}}\big ([0,1]\big )\) is equipped with the usual maximum norm denoted by \(\Vert \cdot \Vert \). The coercivity constant, \(C_0\), in the definition of \({\mathscr {K}}\) is defined by

$$\begin{aligned} C_0&:=\min \left\{ \varphi (\varvec{1}),\varphi (\gamma ),\int _0^1\gamma (t)\ \mathrm{d}t,\inf _{s\in S_0}\frac{1}{{\mathscr {G}}(s)}\int _0^1G(t,s)\ \mathrm{d}t,\right. \\&\quad \left. \inf _{s\in S_0}\frac{1}{{\mathscr {G}}(s)}\int _0^1G(t,s)\ \mathrm{d}\alpha (t)\right\} , \end{aligned}$$

where

$$\begin{aligned} {\mathscr {G}}(s):=\max _{t\in [0,1]}G(t,s) \end{aligned}$$
(2.1)

for \(s\in [0,1]\), and \(S_0\subseteq [0,1]\) is a set of full measure on which \({\mathscr {G}}(s)\ne 0\). We also will make the following general assumptions on the functions appearing in integral equation (1.1).

H1::

The functions \(\gamma \ : \ [0,1]\rightarrow [0,\infty )\), \(A\ : \ [0,\infty )\rightarrow {\mathbb {R}}\), \(f\ : \ [0,1]\times [0,\infty )\rightarrow [0,\infty )\), and \(H\ : \ [0,\infty )\rightarrow [0,\infty )\) are continuous. In addition, the function \(\alpha \ : \ [0,1]\rightarrow {\mathbb {R}}\) is of bounded variation on [0, 1] and is monotone increasing.

H2::

The function \(G\ : \ [0,1]\times [0,1]\rightarrow [0,\infty )\) is continuous, and there exist numbers \(0\le a<b\le 1\) and number \(\eta _0\in (0,1]\) such that \(\displaystyle \min _{t\in [a,b]}G(t,s)\ge \eta _0{\mathscr {G}}(s)\), for each \(s\in [0,1]\), where \({\mathscr {G}}\) is defined as in (2.1). Moreover, the set \(S_0\) as described above satisfies \(\big |S_0\big |=1\), where by \(|\cdot |\) we mean the usual Lebesgue measure.

H3::

The function \(\gamma \) satisfies the following three conditions:

1.:

\(\displaystyle \min _{t\in [a,b]}\gamma (t)\ge \eta _0\Vert \gamma \Vert \), where a, b, and \(\eta _0\) are the same numbers as in condition (H2);

2.:

\(0<\Vert \gamma \Vert \le 1\); and

3.:

\(\varphi (\gamma )\ge C_0\Vert \gamma \Vert \) and \(\displaystyle \int _0^1\gamma (s)\ \mathrm{d}s\ge C_0\Vert \gamma \Vert \).

Finally, we will define the operator \(T\ : \ {\mathscr {C}}\big ([0,1]\big )\rightarrow {\mathscr {C}}\big ([0,1]\big )\) by

$$\begin{aligned} (Tu)(t):=\gamma (t)H\big (\varphi (u)\big )+\lambda \int _0^1\left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s. \end{aligned}$$

A fixed point of the operator T will correspond to a solution of integral equation (1.1).

Remark 2.1

Note that conditions (H1) and (H3) ensure that \(\gamma \in {\mathscr {K}}\) since \(\gamma (t)\ge 0\), \(\varphi (\gamma )\ge C_0\Vert \gamma \Vert \), \(\displaystyle \min _{t\in [a,b]}\gamma (t)\ge \eta _0\Vert \gamma \Vert \), and \(\displaystyle \int _0^1\gamma (s)\ \mathrm{d}s\ge C_0\Vert \gamma \Vert \). Therefore, \({\mathscr {K}}\ne \varnothing \).

Remark 2.2

The function \(\gamma (t):=1-t\) satisfies conditions (H3.1), (H3.2), and (H3.3). For this choice of \(\gamma \) solutions of (1.1) correspond to solutions of boundary value problem (1.3).

Remark 2.3

Because we will work within the cone \({\mathscr {K}}\) we will henceforth write \(\big (u(\xi )\big )^q\) instead of \(\big |u(\xi )\big |^q\) when studying problem (1.1)—just as we did in the definition of T above.

We next present a collection of preliminary lemmata. These will be important in the existence theorems for integral equation (1.1). Our first lemma demonstrates that T is a reflexive map on a particular annular subregion of \({\mathscr {K}}\).

Lemma 2.4

Suppose that conditions (H1)–(H3) are satisfied. In addition, assume that \(A(t)>0\) whenever \(t\in \left[ \rho _1,\rho _2\right] \). Then \(T\left( \overline{{\widehat{V}}_{\rho _2}^{q}}\setminus {\widehat{V}}_{\rho _1}^{q}\right) \subseteq {\mathscr {K}}\).

Proof

The proof is similar to a combination of part of the proofs of [28, Theorem 3.1] and [32, Lemma 2.3]. We include the details for completeness.

First note that

$$\begin{aligned} A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) >0 \end{aligned}$$

for each \(u\in \overline{{\widehat{V}}_{\rho _2}^{q}}\setminus {\widehat{V}}_{\rho _1}^{q}\) by virtue of the fact that for any such u it follows that

$$\begin{aligned} \rho _1\le \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \le \rho _2. \end{aligned}$$

Then the assumption in the statement of the lemma establishes the desired claim. So, in particular, the operator T and thus integral equation (1.1) are well defined on the annular region \(\overline{{\widehat{V}}_{\rho _2}^{q}}\setminus {\widehat{V}}_{\rho _1}^{q}\).

We next show that T satisfies the coercivity condition for the functional \(\displaystyle u\mapsto \int _0^1u(s)\ \mathrm{d}s\), namely that

$$\begin{aligned} \int _0^1(Tu)(s)\ \mathrm{d}s\ge C_0\Vert Tu\Vert . \end{aligned}$$

To see that this is true we calculate

$$\begin{aligned} \begin{aligned} \int _0^1(Tu)(t)\ \mathrm{d}t&=H\big (\varphi (u)\big )\int _0^1\gamma (t)\ \mathrm{d}t\\&\quad +\lambda \int _0^1\int _0^1 \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s\ \mathrm{d}t\\&\ge H\big (\varphi (u)\big )\int _0^1\gamma (t)\ \mathrm{d}t\\&\quad +\lambda \int _0^1\left[ \inf _{s\in S_0}\frac{1}{{\mathscr {G}}(s)} \int _0^1G(t,s)\ \mathrm{d}t\right] \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}{\mathscr {G}}(s)f\big (s,u(s)\big )\ \mathrm{d}s\\&\ge H\big (\varphi (u)\big )\int _0^1\gamma (t)\ \mathrm{d}t+C_0\lambda \int _0^1 \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}{\mathscr {G}} (s)f\big (s,u(s)\big )\ \mathrm{d}s\\&\ge C_0\left[ H\big (\varphi (u)\big )\Vert \gamma \Vert +\lambda \int _0^1 \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}{\mathscr {G} }(s)f\big (s,u(s)\big )\ \mathrm{d}s\right] \\&\ge C_0\Vert Tu\Vert , \end{aligned} \end{aligned}$$

using that

$$\begin{aligned} C_0&:=\min \left\{ \int _0^1\gamma (t)\ \mathrm{d}t,\inf _{s\in S_0}\frac{1}{{\mathscr {G}}(s)} \int _0^1G(t,s)\ \mathrm{d}t,\int _0^1\gamma (t)\ \mathrm{d}\alpha (t),\inf _{s\in S_0} \frac{1}{{\mathscr {G}}(s)}\int _0^1G(t,s)\ \mathrm{d}\alpha (t)\right\} . \end{aligned}$$

In a similar manner we can show that T satisfies the coercivity condition for the functional \(\displaystyle u\mapsto \int _0^1u(s)\ \mathrm{d}\alpha (s)\). In particular,

$$\begin{aligned} \begin{aligned} \int _0^1(Tu)(s)\ \mathrm{d}\alpha (s)&=H\big (\varphi (u)\big )\varphi (\gamma )\\&\quad +\lambda \int _0^1\int _0^1 \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}G(t,s)f \big (s,u(s)\big )\ \mathrm{d}s\ \mathrm{d}\alpha (t)\\&\ge H\big (\varphi (u)\big )\varphi (\gamma )+\lambda \int _0^1\left[ \inf _{s\in S_0}\frac{1}{{\mathscr {G}}(s)} \int _0^1G(t,s)\ \mathrm{d}\alpha (t)\right] \\&\quad \times \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}{\mathscr {G}}(s)f\big (s,u(s)\big )\ \mathrm{d}s\\&\ge C_0\left[ H\big (\varphi (u)\big )\Vert \gamma \Vert +\lambda \int _0^1 \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}{\mathscr {G}} (s)f\big (s,u(s)\big )\ \mathrm{d}s\right] \\&\ge C_0\Vert Tu\Vert , \end{aligned} \end{aligned}$$

once again using the definition of \(C_0\).

On the other hand, since both G and \(\gamma \) satisfy, via conditions (H2)–(H3), Harnack-like inequalities, it is straightforward to demonstrate that \(\min _{t\in [a,b]}(Tu)(t)\ge \eta _0\Vert Tu\Vert \). Finally, that \((Tu)(t)\ge 0\) for each \(t\in [0,1]\) follows directly from the definition of T and the nonnegativity of f, G, A, and \(\gamma \). And this completes the proof. \(\square \)

Remark 2.5

If we put \(\gamma (t):=1-t\) and

$$\begin{aligned} G(t,s):={\left\{ \begin{array}{ll} t(1-s)\text {, }&{}0\le t\le s\le 1\\ s(1-t)\text {, }&{}0\le s\le t\le 1\end{array}\right. }, \end{aligned}$$

then we recover problem (1.3)—i.e., inhomogeneous Dirichlet boundary conditions. In this case we see that

$$\begin{aligned} \min \left\{ \int _0^1 1-t\ \mathrm{d}t,\inf _{s\in (0,1)}\frac{1}{{\mathscr {G}}(s)}\int _0^1G(t,s)\ \mathrm{d}t\right\} =\frac{1}{2}, \end{aligned}$$

which matches what was obtained in [32]. Moreover, in this case it is well known that \(\eta _0:=\min \{a,1-b\}\).

We next demonstrate a relationship between the \({\widehat{V}}_{\rho }^q\) and \({\widehat{W}}_{\rho }\) sets. This is crucial for the correct application of the fixed point theorem, Lemma 2.12, later.

Lemma 2.6

For any numbers \(q\ge 1\) and \(0<\rho _1<C_0\rho _2^{\frac{1}{q}}\) it holds that \({\widehat{W}}_{\rho _1}\subset {\widehat{V}}_{\left( \frac{\rho _1}{C_0}\right) ^q}^q\subset {\widehat{V}}_{\rho _2}^{q}\).

Proof

Let \(u\in {\widehat{W}}_{\rho _1}\). Then

$$\begin{aligned} C_0\Vert u\Vert \le \int _0^1u(s)\ \mathrm{d}\alpha (s)<\rho _1 \end{aligned}$$

so that

$$\begin{aligned} \Vert u\Vert <\frac{\rho _1}{C_0}. \end{aligned}$$

At the same time

$$\begin{aligned} \int _0^1\big (y(s)\big )^q\ \mathrm{d}s\le \Vert u\Vert ^q<\left( \frac{\rho _1}{C_0}\right) ^q. \end{aligned}$$

Since, by assumption, we have that \(\displaystyle \rho _1<C_0\rho _2^{\frac{1}{q}}\), it follows that \({\widehat{W}}_{\rho _1}\subset {\widehat{V}}_{\left( \frac{\rho _1}{C_0}\right) ^q}^q\subset {\widehat{V}}_{\rho _2}^{q}\), as was claimed. \(\square \)

Remark 2.7

Notice that the condition \(\displaystyle \rho _1<C_0\rho _2^{\frac{1}{q}}\) depends on initial data only—namely, \(\rho _1\), \(\rho _2\), q, and \(C_0\).

Another relationship between the sets \({\widehat{W}}_{\rho _1}\) and \({\widehat{V}}_{\rho _2}^{q}\) is stated in the next corollary. As with Lemma 2.6 that Corollary 2.8 holds is essential for the correct application of the fixed point theory in the sequel.

Corollary 2.8

For any numbers \(q\ge 1\) and \(0<\rho _1<C_0\rho _2^{\frac{1}{q}}\) it holds that \({\widehat{V}}_{\rho _2}^{q}\setminus \overline{{\widehat{W}}_{\rho _1}}\ne \varnothing \).

Proof

For any \(\rho >0\) define the set \(\Omega _{\rho }\) by

$$\begin{aligned} \Omega _{\rho }:=\big \{u\in {\mathscr {K}}\ : \ \Vert u\Vert <\rho \big \}, \end{aligned}$$

and consider the collection

$$\begin{aligned} {\mathscr {K}}\supseteq \Omega _{\rho _2^{\frac{1}{q}}}\setminus \overline{\Omega _{\frac{\rho _1}{C_0}}}. \end{aligned}$$

Then given any \(u\in \Omega _{\rho _2^{\frac{1}{q}}}\setminus \overline{\Omega _{\frac{\rho _1}{C_0}}}\) we have that

$$\begin{aligned} \frac{\rho _1}{C_0}<\Vert u\Vert <\rho _2^{\frac{1}{q}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \int _0^1u(s)\ \mathrm{d}\alpha (s)\ge C_0\Vert u\Vert >\rho _1, \end{aligned}$$

where we have used the coercivity of the functional \(\displaystyle u\mapsto \int _0^1u(s)\ \mathrm{d}\alpha (s)\). Similarly,

$$\begin{aligned} \int _0^1\big (u(s)\big )^q\ \mathrm{d}s\le \Vert u\Vert ^q<\rho _2. \end{aligned}$$

Thus, we conclude that

$$\begin{aligned} u\in \Omega _{\rho _2^{\frac{1}{q}}}\setminus \overline{\Omega _{\frac{\rho _1}{C_0}}} \Longrightarrow u\in {\widehat{V}}_{\rho _2}^{q}\setminus \overline{{\widehat{W}}_{\rho _1}}. \end{aligned}$$

But since

$$\begin{aligned} \frac{\rho _1}{C_0}<\rho _2^{\frac{1}{q}}, \end{aligned}$$

by assumption, it follows that

$$\begin{aligned} \Omega _{\rho _2^{\frac{1}{q}}}\setminus \overline{\Omega _{\frac{\rho _1}{C_0}}}\ne \varnothing \end{aligned}$$

so that

$$\begin{aligned} {\widehat{V}}_{\rho _2}^{q}\setminus \overline{{\widehat{W}}_{\rho _1}}\ne \varnothing . \end{aligned}$$

And this completes the proof of the corollary. \(\square \)

Our next preliminary lemma demonstrates an important topological condition of the sets \({\widehat{V}}_{\rho }^{q}\) and \({\widehat{W}}_{\rho }\). This lemma is essential for the correct application of the fixed point result Lemma 2.12.

Lemma 2.9

For each \(\rho >0\) each of the sets \({\widehat{V}}_{\rho }^{q}\) and \({\widehat{W}}_{\rho }\) is bounded. Moreover, each set is relatively open in \({\mathscr {K}}\).

Proof

Suppose that \(u\in {\widehat{V}}_{\rho }^{q}\). Then by Jensen’s inequality

$$\begin{aligned} C_0^q\Vert u\Vert ^q\le \int _0^1\big (u(s)\big )^q\ \mathrm{d}s<\rho \end{aligned}$$

so that

$$\begin{aligned} \Vert u\Vert <\frac{\rho ^{\frac{1}{q}}}{C_0}. \end{aligned}$$
(2.2)

So, inequality (2.2) implies that \({\widehat{V}}_{\rho }^{q}\) is bounded. In a similar way, we see that if \(u\in {\widehat{W}}_{\rho }\), then

$$\begin{aligned} C_0\Vert u\Vert \le \int _0^1u(s)\ \mathrm{d}\alpha (s)<\rho . \end{aligned}$$
(2.3)

So, inequality (2.3) implies that \({\widehat{W}}_{\rho }\) is bounded. Finally, that each of these sets is relatively open in \({\mathscr {K}}\) is a simple consequence of the definition of the functionals as well as the fact that \(u\in {\mathscr {C}}\big ([0,1]\big )\). \(\square \)

The next lemma will be used in the existence theorems. It establishes that a certain interval of interest is nonempty.

Lemma 2.10

For each \(\rho _2>0\), \(\eta _0\in (0,1)\), \(0\le a<b\le 1\), \(C_0\in (0,1]\), and \(q\ge 1\), it holds that

$$\begin{aligned} \left[ \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a), \left( \frac{\rho _2}{C_0}\right) ^q\right] \ne \varnothing . \end{aligned}$$

Proof

Note that the interval is nonempty if and only if

$$\begin{aligned} \frac{\eta _0^q}{\left( \varphi (\varvec{1})\right) ^q}(b-a)<C_0^{-q} \end{aligned}$$

or, equivalently,

$$\begin{aligned} \frac{\eta _0}{\varphi (\varvec{1})}(b-a)^{\frac{1}{q}}<C_0^{-1}. \end{aligned}$$
(2.4)

Since \(\varphi (\varvec{1})\ge C_0\), note that

$$\begin{aligned} \frac{\eta _0}{\varphi (\varvec{1})}(b-a)^{\frac{1}{q}} <\frac{\eta _0}{C_0}(b-a)^{\frac{1}{q}}. \end{aligned}$$

Therefore, if

$$\begin{aligned} C_0^{-1}>\frac{\eta _0}{C_0}(b-a)^{\frac{1}{q}}, \end{aligned}$$
(2.5)

then inequality (2.4) will be satisfied. But inequality (2.5) reduces to

$$\begin{aligned} 1>\eta _0(b-a)^{\frac{1}{q}}, \end{aligned}$$

which is always satisfied since \(0<b-a\le 1\). Therefore, inequality (2.4) holds, and so, we conclude that the interval is nonempty, as claimed. \(\square \)

As a consequence of Lemma 2.10 we can prove the following lemma, which concerns a certain inequality involving the coefficient function A.

Lemma 2.11

Suppose that \(u\in \partial {\widehat{W}}_{\rho _2}\) for some number \(\rho _2>0\). If the function A is monotone increasing on the set

$$\begin{aligned} \left[ \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a), \left( \frac{\rho _2}{C_0}\right) ^q\right] , \end{aligned}$$

then

$$\begin{aligned} \left( A\left( \left( \frac{\rho _2}{C_0}\right) ^q\right) \right) ^{-1} \le \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1} \le \left( A\left( \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a)\right) \right) ^{-1}. \end{aligned}$$

Proof

Due to Lemma 2.10 we already know that \(\left[ \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a),\left( \frac{\rho _2}{C_0}\right) ^q\right] \ne \varnothing \). Now, since \(u\in \partial {\widehat{W}}_{\rho _2}\) we can write (using that \(\alpha \) is a monotone increasing integrator)

$$\begin{aligned} \rho _2=\int _0^1u(s)\ \mathrm{d}\alpha (s)\le \varphi (\varvec{1})\Vert u\Vert . \end{aligned}$$
(2.6)

Similarly, it holds that

$$\begin{aligned} C_0\Vert u\Vert \le \int _0^1u(s)\ \mathrm{d}\alpha (s)=\rho _2. \end{aligned}$$
(2.7)

Therefore, inequalities (2.6)–(2.7) imply that

$$\begin{aligned} \frac{\rho _2}{\varphi (\varvec{1})}\le \Vert u\Vert \le \frac{\rho _2}{C_0}. \end{aligned}$$
(2.8)

At the same time, using the fact that u satisfies the Harnack inequality from \({\mathscr {K}}\) together with inequality (2.8) we calculate

$$\begin{aligned} \left( \frac{\rho _2}{C_0}\right) ^q\ge & {} \Vert u\Vert ^q\ge \int _0^1\big (u(\xi ) \big )^q\ \mathrm{d}\xi \ge \int _a^b\big (u(\xi )\big )^q\ \mathrm{d}\xi \ge \eta _0^q\Vert u \Vert ^q(b-a)\nonumber \\\ge & {} \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a). \end{aligned}$$
(2.9)

Thus, inequality (2.9) demonstrates that

$$\begin{aligned} \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \in \left[ \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a),\left( \frac{\rho _2}{C_0}\right) ^q\right] , \end{aligned}$$

which is precisely the interval on which A is monotone increasing. Therefore, we conclude that

$$\begin{aligned} A\left( \left( \frac{\rho _2}{C_0}\right) ^q\right) \ge A\left( \int _0^1 \big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \ge A\left( \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a)\right) \end{aligned}$$

so that

$$\begin{aligned} \left( A\left( \left( \frac{\rho _2}{C_0}\right) ^q\right) \right) ^{-1}\le \left( A \left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}\le \left( A\left( \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})} \right) ^q(b-a)\right) \right) ^{-1}, \end{aligned}$$

as desired. \(\square \)

We conclude our preliminary lemmata with a lemma regarding a fixed point result. For further details on this and related results one may consult, for example, Cianciaruso, Infante, and Pietramala [18, Lemma 2.3], Guo and Lakshmikantham [38], Infante, Pietramala, and Tenuta [47], or Zeidler [64].

Lemma 2.12

Let U be a bounded open set and, with \({\mathcal {K}}\) a cone in a real Banach space \({\mathscr {X}}\), suppose both that \(U_{{\mathscr {K}}}:=U\cap {\mathcal {K}}\supseteq \{0\}\) and that \(\overline{U_{{\mathscr {K}}}}\ne {\mathscr {K}}\). Assume that \(T\ : \ \overline{U_{{\mathscr {K}}}}\rightarrow {\mathscr {K}}\) is a compact map such that \(x\ne Tx\) for each \(x\in \partial U_{{\mathscr {K}}}\). Then the fixed point index \(i_{{\mathscr {K}}}\left( T,U_{{\mathscr {K}}}\right) \) has the following properties.

  1. 1.

    If there exists \(e\in {\mathscr {K}}\setminus \{0\}\) such that \(x\ne Tx+\lambda e\) for each \(x\in \partial U_{{\mathscr {K}}}\) and each \(\lambda >0\), then \(i_{{\mathscr {K}}}\left( T,U_{{\mathscr {K}}}\right) =0\).

  2. 2.

    If \(\mu x\ne Tx\) for each \(x\in \partial U_{{\mathscr {K}}}\) and for each \(\mu \ge 1\), then \(i_{{\mathscr {K}}}\left( T,U_{{\mathscr {K}}}\right) =1\).

  3. 3.

    If \(i_{{\mathscr {K}}}\left( T,U_{{\mathscr {K}}}\right) \ne 0\), then T has a fixed point in \(U_{{\mathscr {K}}}\).

  4. 4.

    Let \(U^1\) be open in X with \(\overline{U_{{\mathscr {K}}}^1}\subseteq U_{{\mathscr {K}}}\). If \(i_{{\mathscr {K}}}\left( T,U_{{\mathscr {K}}}\right) =1\) and \(i_{{\mathscr {K}}}\left( T,U_{{\mathscr {K}}}^{1}\right) =0\), then T has a fixed point in \(U_{{\mathscr {K}}}\setminus \overline{U_{{\mathscr {K}}}^{1}}\). The same result holds if \(i_{{\mathscr {K}}}\left( T,U_{{\mathscr {K}}}\right) =0\) and \(i_{{\mathscr {K}}}\left( T,U_{{\mathscr {K}}}^{1}\right) =1\).

We now present three representative existence results for problem (1.1). The first of these, Theorem 2.13, uses a \({\widehat{W}}_{\rho }\)-type set on the “inner” boundary and a \({\widehat{V}}_{\rho }^{q}\)-type set on the “outer” boundary.

Theorem 2.13

Suppose that conditions (H1)–(H3) are satisfied. In addition, suppose that there exists numbers \(\rho _1\) and \(\rho _2\), where \(\displaystyle 0<\rho _1<C_0\rho _2^{\frac{1}{q}}\), such that

  1. 1.

    A is monotone increasing on \(\displaystyle \left[ \left( \frac{\eta _0\rho _1}{\varphi (\varvec{1})}\right) ^q(b-a),\left( \frac{\rho _1}{C_0}\right) ^q\right] \);

  2. 2.

    \(A(t)>0\) for \(\displaystyle t\in \left[ \left( \frac{\rho _1C_0}{\varphi (\varvec{1})}\right) ^q,\rho _2\right] \);

  3. 3.

    \(\displaystyle H\left( \rho _1\right) \varphi (\gamma )+\lambda \left( A\left( \left( \frac{\rho _1}{C_0}\right) ^q\right) \right) ^{-1}f_{[a,b]\times \left[ \frac{\eta _0\rho _1}{\varphi (\varvec{1})},\frac{\rho _1}{C_0}\right] }^{m}\int _0^1\int _a^bG(t,s)\ \mathrm{d}s\ \mathrm{d}\alpha (t)>\rho _1\); and

  4. 4.

    \(\displaystyle \int _0^1\left[ \gamma (t)H_{\left[ C_0\rho _2^{\frac{1}{q}},\frac{\rho _2^{\frac{1}{q}}\varphi (\varvec{1})}{C_0}\right] }^{M}+\frac{\lambda }{A\big (\rho _2\big )}f_{[0,1]\times \left[ 0,\frac{\rho _2^{\frac{1}{q}}}{C_0}\right] }^{M}\int _0^1G(t,s)\ \mathrm{d}s\right] ^q\ \mathrm{d}t<\rho _2\).

Then problem (1.1) has at least one positive solution, \(u_0\), satisfying the localization

$$\begin{aligned} u_0\in {\widehat{V}}_{\rho _2}^{q}\setminus \overline{{\widehat{W}}_{\rho _1}}. \end{aligned}$$

Proof

As a preliminary observation let us first notice that

$$\begin{aligned} \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}>0 \end{aligned}$$

whenever \(u\in {\widehat{V}}_{\rho _2}^{q}\setminus \overline{{\widehat{W}}_{\rho _1}}\). Indeed, we see that

$$\begin{aligned} \Vert u\Vert \varphi (\varvec{1})>\int _0^1u(s)\ \mathrm{d}\alpha (s)>\rho _1 \end{aligned}$$
(2.10)

since \(u\in {\mathscr {K}}\setminus \overline{{\widehat{W}}_{\rho _1}}\), and that

$$\begin{aligned} \rho _2>\int _0^1\big (u(s)\big )^q\ \mathrm{d}s\ge C_0^q\Vert u\Vert ^q \end{aligned}$$
(2.11)

since \(u\in {\widehat{V}}_{\rho _2}^{q}\). Putting (2.10) and (2.11) together we see that

$$\begin{aligned} \left( \frac{\rho _1C_0}{\varphi (\varvec{1})}\right) ^q\le \int _0^1\big (u(s)\big )^q\ \mathrm{d}s<\rho _2, \end{aligned}$$

which establishes the desired claim due to assumption (2).

We first assume for contradiction the existence of \(u\in \partial {\widehat{W}}_{\rho _1}\) and \(\mu >0\) such that \(u(t)=(Tu)(t)+\mu e(t)\), for \(t\in [0,1]\), with \(e(t)\equiv \varvec{1}\), thereby trying to invoke part (1) of Lemma 2.12. Note that \(\varvec{1}\in {\mathscr {K}}\) by the definition of \(C_0\) and the fact that \(\Vert \varvec{1}\Vert =1\). Then applying \(\varphi \) to both sides of \(u=Tu+\mu e\) yields

$$\begin{aligned} \rho _1&=\varphi (u)\ge \varphi (\gamma )H\big (\varphi (u)\big )\nonumber \\&\quad +\lambda \int _0^1 \int _0^1\left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1} G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s\ \mathrm{d}\alpha (t)\nonumber \\&\ge H\left( \rho _1\right) \varphi (\gamma )\nonumber \\&\quad +\lambda \left( A\left( \left( \frac{\rho _1}{C_0}\right) ^q\right) \right) ^{-1}\int _0^1 \int _0^1G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s\ \mathrm{d}\alpha (t)\nonumber \\&\ge H\left( \rho _1\right) \varphi (\gamma )\nonumber \\&\quad +\lambda \left( A\left( \left( \frac{\rho _1}{C_0}\right) ^q\right) \right) ^{-1}f_{[a,b]\times \left[ \frac{\eta _0\rho _1}{\varphi (\varvec{1})},\frac{\rho _1}{C_0}\right] }^{m} \int _0^1\int _a^bG(t,s)\ \mathrm{d}s\ \mathrm{d}\alpha (t)\nonumber \\&>\rho _1, \end{aligned}$$
(2.12)

where we have used Lemma 2.11 to obtain the lower bound on \(\big (A(\cdot )\big )^{-1}\). We have also used in inequality (2.12) the fact that by (2.8) it follows that

$$\begin{aligned} \frac{\rho _1}{C_0}\ge \Vert u\Vert \ge \min _{t\in [a,b]}u(t)\ge \eta _0\Vert u\Vert \ge \frac{\eta _0\rho _1}{\varphi (\varvec{1})} \end{aligned}$$

so that

$$\begin{aligned} f\big (s,u(s)\big )\ge f_{[a,b]\times \left[ \frac{\eta _0\rho _1}{\varphi (\varvec{1})}, \frac{\rho _1}{C_0}\right] }^{m}. \end{aligned}$$

Since inequality (2.12) is a contradiction, from Lemma 2.12 we conclude that

$$\begin{aligned} i_{{\mathcal {K}}}\left( T,{\widehat{W}}_{\rho _1}\right) =0. \end{aligned}$$
(2.13)

On the other hand, suppose for contradiction the existence of \(u\in \partial {\widehat{V}}_{\rho _2}^{q}\) and \(\mu \ge 1\) such that \(\mu u(t)=(Tu)(t)\) for each \(t\in [0,1]\), thereby trying to invoke part (2) of Lemma 2.12. As a preliminary observation note that since \(u\in \partial {\widehat{V}}_{\rho _2}^{q}\) we obtain from Jensen’s inequality that

$$\begin{aligned} C_0^q\Vert u\Vert ^q\le \int _0^1\big (u(s)\big )^q\ \mathrm{d}s=\rho _2\Longrightarrow \int _0^1u(s)\ \mathrm{d}\alpha (s)\le \Vert u\Vert \varphi (\varvec{1})\le \frac{\rho _2^{\frac{1}{q}}\varphi (\varvec{1})}{C_0}. \end{aligned}$$

In a similar manner we also deduce that

$$\begin{aligned} \Vert u\Vert ^q\ge \int _0^1\big (u(s)\big )^q\ \mathrm{d}s=\rho _2\Longrightarrow \int _0^1u(s)\ \mathrm{d}\alpha (s)\ge C_0\Vert u\Vert \ge C_0\rho _2^{\frac{1}{q}}. \end{aligned}$$

Consequently,

$$\begin{aligned} H\big (\varphi (u)\big )=H\left( \int _0^1u(s)\ \mathrm{d}\alpha (s)\right) \le H_{\left[ C_0\rho _2^{\frac{1}{q}},\frac{\rho _2^{\frac{1}{q}}\varphi (\varvec{1})}{C_0}\right] }^{M}. \end{aligned}$$
(2.14)

Then integrating from \(t=0\) to \(t=1\) both sides of \(\big (\mu u(t)\big )^q=\big ((Tu)(t)\big )^q\) yields

$$\begin{aligned} \rho _2&\le \mu \int _0^1\big (u(t)\big )^q\ \mathrm{d}t\nonumber \\&=\int _0^1\underbrace{\left[ \gamma (t)H\big (\varphi (u)\big )+\lambda \int _0^1 \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1} G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s\right] ^q}_{=\big ((Tu)(t)\big )^q}\ \mathrm{d}t\nonumber \\&\le \int _0^1\left[ \gamma (t)H_{\left[ C_0\rho _2^{\frac{1}{q}}, \frac{\rho _2^{\frac{1}{q}}\varphi (\varvec{1})}{C_0}\right] }^{M} +\lambda \int _0^1\big (A\left( \rho _2\right) \big )^{-1}G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s\right] ^q\ \mathrm{d}t\nonumber \\&\le \int _0^1\left[ \gamma (t)H_{\left[ C_0\rho _2^{\frac{1}{q}}, \frac{\rho _2^{\frac{1}{q}}\varphi (\varvec{1})}{C_0}\right] }^{M} +\frac{\lambda }{A\big (\rho _2\big )}f_{[0,1]\times \left[ 0, \frac{\rho _2^{\frac{1}{q}}}{C_0}\right] }^{M}\int _0^1G(t,s)\ \mathrm{d}s\right] ^q\ \mathrm{d}t\nonumber \\&<\rho _2, \end{aligned}$$
(2.15)

where we have used inequality (2.14). And since inequality (2.15) is a contradiction, we conclude from Lemma 2.12 that

$$\begin{aligned} i_{{\mathcal {K}}}\left( T,{\widehat{V}}_{\rho _2}^{q}\right) =1. \end{aligned}$$
(2.16)

Putting the index calculations (2.13) and (2.16) together we conclude from yet another application of Lemma 2.12 that T has a fixed point, say \(u_0\), satisfying the localization \(\displaystyle u_0\in {\widehat{V}}_{\rho _2}^q\setminus \overline{{\widehat{W}}_{\rho _1}}\). Finally, we observe that

$$\begin{aligned} {\widehat{V}}_{\rho _2}^q\setminus \overline{{\widehat{W}}_{\rho _1}}\ne \varnothing \end{aligned}$$

due to Corollary 2.8, which may be applied since we assumed in the statement of the theorem that \(\displaystyle C_0^{\frac{1}{q}}\rho _2>\rho _1>0\). Therefore, \(u_0\) is a positive solution of integral equation (1.1). \(\square \)

An immediate corollary of Theorem 2.13 is the following. The difference between the two results is twofold. First of all, Corollary 2.14 eliminates the local monotonicity assumption on A. Second of all, the corollary uses a simpler version of condition (3) in Theorem 2.13. Therefore, in practice Corollary 2.14 is likely to be much simpler to apply—a fact illustrated by Example 2.17.

Corollary 2.14

Suppose that conditions (H1)–(H3) are satisfied. In addition, suppose that there exists numbers \(\rho _1\) and \(\rho _2\), where \(\displaystyle 0<\rho _1<C_0\rho _2^{\frac{1}{q}}\), such that

  1. 1.

    \(A(t)>0\) for \(\displaystyle t\in \left[ \left( \frac{\rho _1C_0}{\varphi (\varvec{1})}\right) ^q,\rho _2\right] \);

  2. 2.

    \(\displaystyle \frac{H\left( \rho _1\right) }{\rho _1}>\frac{1}{\varphi (\gamma )}\); and

  3. 3.

    \(\displaystyle \int _0^1\left[ \gamma (t)H_{\left[ C_0\rho _2^{\frac{1}{q}},\frac{\rho _2^{\frac{1}{q}}\varphi (\varvec{1})}{C_0}\right] }^{M}+\frac{\lambda }{A\big (\rho _2\big )}f_{[0,1]\times \left[ 0,\frac{\rho _2^{\frac{1}{q}}}{C_0}\right] }^{M}\int _0^1G(t,s)\ \mathrm{d}s\right] ^q\ \mathrm{d}t<\rho _2\).

Then problem (1.1) has at least one positive solution, \(u_0\), satisfying the localization

$$\begin{aligned} u_0\in {\widehat{V}}_{\rho _2}^{q}\setminus \overline{{\widehat{W}}_{\rho _1}}. \end{aligned}$$

Remark 2.15

It is certainly possible in Theorem 2.13 and, thus, in Corollary 2.14 to “reverse” the roles of \(\rho _1\) and \(\rho _2\) in the sense that the conditions (1)–(4) can be rewritten for the case \(\rho _1>\rho _2>0\). We omit the precise statement of this result.

Our third result is an alternative existence result, which complements both Theorem 2.13 and Corollary 2.14. The distinction here is that we use a \({\widehat{W}}_{\rho }\)-type set on both boundaries. This results in a slight alteration of conditions (1), (2), and (4) from Theorem 2.13.

Theorem 2.16

Suppose that conditions (H1)–(H3) are satisfied. In addition, suppose that there exists numbers \(\rho _1\) and \(\rho _2\), where \(0<\rho _1<\rho _2\), such that

  1. 1.

    A is monotone increasing on \(\displaystyle \left[ \left( \frac{\eta _0\rho _1}{\varphi (\varvec{1})}\right) ^q(b-a),\left( \frac{\rho _1}{C_0}\right) ^q\right] \cup \left[ \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a),\left( \frac{\rho _2}{C_0}\right) ^q\right] \);

  2. 2.

    \(A(t)>0\) for \(\displaystyle t\in \left[ \left( \frac{\rho _1C_0}{\varphi (\varvec{1})}\right) ^q,\left( \frac{\rho _2}{C_0}\right) ^q\right] \);

  3. 3.

    \(\displaystyle H\left( \rho _1\right) \varphi (\gamma )+\lambda \left( A\left( \left( \frac{\rho _1}{C_0}\right) ^q\right) \right) ^{-1}f_{[a,b]\times \left[ \frac{\eta _0\rho _1}{\varphi (\varvec{1})},\frac{\rho _1}{C_0}\right] }^{m}\int _0^1\int _a^bG(t,s)\ \mathrm{d}s\ \mathrm{d}\alpha (t)>\rho _1\); and

  4. 4.

    \(\displaystyle H\left( \rho _2\right) \varphi (\gamma )+\lambda \left( A\left( \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a)\right) \right) ^{-1}f_{[0,1]\times \left[ 0,\frac{\rho _2}{C_0}\right] }^{M}\int _0^1\int _0^1G(t,s) \mathrm{d}s\ \mathrm{d}\alpha (t)<\rho _2\).

Then problem (1.1) has at least one positive solution, \(u_0\), satisfying the localization

$$\begin{aligned} u_0\in {\widehat{W}}_{\rho _2}\setminus \overline{{\widehat{W}}_{\rho _1}}. \end{aligned}$$

Proof

Since the proof of this theorem is very similar to the proof of Theorem 2.13, we will only sketch the relevant details. Indeed, really only the first part of the proof changes.

As a preliminary observation let us first notice that

$$\begin{aligned} \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}>0 \end{aligned}$$

whenever \(u\in {\widehat{W}}_{\rho _2}\setminus \overline{{\widehat{W}}_{\rho _1}}\). To argue that this is true we first notice that

$$\begin{aligned} \Vert u\Vert \varphi (\varvec{1})>\int _0^1u(s)\ \mathrm{d}\alpha (s)>\rho _1 \end{aligned}$$
(2.17)

since \(u\in {\mathscr {K}}\setminus \overline{{\widehat{W}}_{\rho _1}}\). At the same time since \(u\in {\widehat{W}}_{\rho _2}\) it follows that

$$\begin{aligned} \rho _2>\int _0^1u(s)\ \mathrm{d}\alpha (s)\ge C_0\Vert u\Vert \end{aligned}$$

so that

$$\begin{aligned} \frac{\rho _1}{\varphi (\varvec{1})}\le \Vert u\Vert \le \frac{\rho _2}{C_0}. \end{aligned}$$
(2.18)

Putting (2.17)–(2.18) together with Jensen’s inequality we see that

$$\begin{aligned} \left( \frac{\rho _1C_0}{\varphi (\varvec{1})}\right) ^q\le C_0^q\Vert u\Vert ^q\le \int _0^1\big (u(s)\big )^q\ \mathrm{d}s\le \Vert u\Vert ^q<\left( \frac{\rho _2}{C_0}\right) ^q, \end{aligned}$$

which establishes the desired claim due to assumption (2). Note that this assumption is only meaningful if

$$\begin{aligned} \frac{\rho _1C_0}{\varphi (\varvec{1})}<\frac{\rho _2}{C_0}. \end{aligned}$$

In other words, it is meaningful only if

$$\begin{aligned} \rho _1<\frac{\varphi (\varvec{1})}{C_0^2}\rho _2. \end{aligned}$$

But now recalling that \(0<C_0\le 1\) and \(\varphi (\varvec{1})\ge C_0\), we deduce that

$$\begin{aligned} \rho _2\le \frac{1}{C_0}\rho _2\le \frac{\varphi (\varvec{1})}{C_0^2}\rho _2. \end{aligned}$$

Since the condition \(\rho _1<\rho _2\) was assumed in the statement of the theorem, we conclude that condition (2) is meaningful.

The first part of the proof is identical to the first part of the proof of Theorem 2.13. On the other hand, we next claim that for each \(u\in \partial {\widehat{W}}_{\rho _2}\) it follows that \(\mu u\ne Tu\) for each \(\mu \ge 1\). So, for contradiction suppose not. Then there is \(u\in \partial {\widehat{W}}_{\rho _2}\) and \(\mu \ge 1\) such that \(\mu u(t)=(Tu)(t)\) for each \(t\in [0,1]\). Recall that since \(u\in \partial {\widehat{W}}_{\rho _2}\) it follows that

$$\begin{aligned} \varphi (u)=\rho _2. \end{aligned}$$

Therefore, integrating from \(t=0\) to \(t=1\) both sides of \(\mu u(t)=(Tu)(t)\) against \(\mathrm{d}\alpha (t)\) we deduce the following estimate:

$$\begin{aligned} \rho _2&=\varphi (u)\nonumber \\&\le \mu \int _0^1u(t)\ \mathrm{d}\alpha (t)\nonumber \\&=\int _0^1\underbrace{\left[ \gamma (t)H\big (\varphi (u)\big )+\lambda \int _0^1 \left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}G(t,s) f\big (s,u(s)\big )\ \mathrm{d}s\right] }_{=(Tu)(t)}\ \mathrm{d}\alpha (t)\nonumber \\&=H\left( \rho _2\right) \varphi (\gamma )\nonumber \\&\quad +\lambda \int _0^1\int _0^1\left( A \left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s\ \mathrm{d}\alpha (t)\nonumber \\&\le H\left( \rho _2\right) \varphi (\gamma )\nonumber \\&\quad +\lambda \int _0^1\int _0^1 \left( A\left( \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a)\right) \right) ^{-1}G(t,s)f\big (s,u(s)\big )\ \mathrm{d}s\ \mathrm{d}\alpha (t)\nonumber \\&\le H\left( \rho _2\right) \varphi (\gamma )\nonumber \\&\quad +\lambda \left( A\left( \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q(b-a)\right) \right) ^{-1}f_{[0,1]\times \left[ 0,\frac{\rho _2}{C_0}\right] }^{M}\int _0^1\int _0^1G(t,s)\ \mathrm{d}s\ \mathrm{d}\alpha (t)\nonumber \\&<\rho _2, \end{aligned}$$
(2.19)

using that

$$\begin{aligned} u\in \partial {\widehat{W}}_{\rho _2}\Longrightarrow \Vert u\Vert \le \frac{\rho _2}{C_0}\Longrightarrow f\big (s,u(s)\big )\le f_{[0,1]\times \left[ 0,\frac{\rho _2}{C_0}\right] }^{M}\text {, }s\in [0,1]. \end{aligned}$$

Note that to obtain the estimate \(\left( A\left( \int _0^1\big (u(\xi )\big )^q\ \mathrm{d}\xi \right) \right) ^{-1}\le \left( A\left( \left( \frac{\eta _0\rho _2}{\varphi (\varvec{1})}\right) ^q (b-a)\right) \right) ^{-1}\), which is used in inequality (2.19), we have used Lemma 2.11. Since inequality (2.19) is a contradiction, we conclude from Lemma 2.12 that \(\displaystyle i_{{\mathscr {K}}}\left( T,{\widehat{W}}_{\rho _2}\right) =1\). Then just as in the proof of Theorem 2.13 we deduce from Lemma 2.12 the existence of at least one positive solution \(u_0\in {\widehat{W}}_{\rho _2}\setminus \overline{{\widehat{W}}_{\rho _1}}\ne \varnothing \) to integral equation (1.1). \(\square \)

We conclude with an example and a remark.

Example 2.17

We will demonstrate the application of Corollary 2.14 to a problem of the form (1.3). In particular, suppose that we choose \(A(t):=\sin {t}\), \(\displaystyle \varphi (u):=\frac{1}{2}u\left( \frac{1}{3}\right) +\frac{1}{50}u\left( \frac{1}{10}\right) \), \(\gamma (t):=1-t\), \(H(t):=\frac{9}{10}\sqrt{t}\), and \(q:=2\) so that (1.3) becomes

$$\begin{aligned} -\sin {\left( \int _0^1\big (u(s)\big )^2\ \mathrm{d}s\right) }u''(t)= & {} \lambda f\big (t,u(t)\big )\text {, }0<t<1\nonumber \\ u(0)= & {} \frac{9}{10}\sqrt{\frac{1}{2}u \left( \frac{1}{3}\right) +\frac{1}{50}u\left( \frac{1}{10}\right) }\nonumber \\ u(1)= & {} 0. \end{aligned}$$
(2.20)

Note that since \(\varphi \) is a multipoint-type nonlocal element with positive coefficients, it follows that the Stieltjes integrator, \(\alpha \), associated to it will be monotone increasing.

For the Green’s function associated to the Dirichlet problem it is know that \(\eta _0=\min \{a,1-b\}\). If we choose here \(\displaystyle a:=\frac{1}{4}\) and \(\displaystyle b:=\frac{3}{4}\), then \(\displaystyle \eta _0=\frac{1}{4}\). In addition, we calculate the following.

$$\begin{aligned} \begin{aligned} \int _0^1\gamma (t)\ \mathrm{d}t&=\frac{1}{2}\\ \varphi (\varvec{1})&=\frac{1}{2}+\frac{1}{50}=\frac{13}{25}\\ \varphi (\gamma )&=\frac{1}{2}\left[ 1-\frac{1}{2}\right] +\frac{1}{50} \left[ 1-\frac{1}{10}\right] =\frac{67}{250}. \end{aligned} \end{aligned}$$

At the same time since \({\mathscr {G}}(s)=s(1-s)\) we also calculate

$$\begin{aligned} \inf _{s\in (0,1)}\frac{1}{s(1-s)}\int _0^1G(t,s)\ \mathrm{d}t= & {} \inf _{s\in (0,1)}\frac{1}{s(1-s)}\underbrace{\left[ \int _0^st(1-s)\ \mathrm{d}t+\int _s^1s(1-t)\ \mathrm{d}t\right] }_{\frac{1}{2}s(1-s)}\\= & {} \frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\inf _{s\in (0,1)}\frac{1}{s(1-s)}\int _0^1G(t,s)\ \mathrm{d}\alpha (t)\\&\quad =\inf _{s\in (0,1)}\frac{1}{s(1-s)}\left[ \frac{1}{2}G \left( \frac{1}{3},s\right) +\frac{1}{50}G\left( \frac{1}{10},s\right) \right] \\&\quad =\inf _{s\in (0,1)}\frac{1}{s(1-s)}{\left\{ \begin{array}{ll}\frac{1}{3}s +\frac{9}{500}s\text {, }&{}0<s<\frac{1}{10}\\ \frac{1}{3}s +\frac{1}{500}(1-s)\text {, }&{}\frac{1}{10}\le s<\frac{1}{3}\\ \frac{1}{6}(1-s)+\frac{1}{500}(1-s)\text {, }&{}\frac{1}{3}\le s<1\end{array}\right. }\\&\quad =\inf _{s\in (0,1)}{\left\{ \begin{array}{ll}\frac{527}{1500(1-s)}\text {, } &{}0<s<\frac{1}{10}\\ \frac{497s+3}{1500s(1-s)}\text {, } &{}\frac{1}{10}\le s<\frac{1}{3}\\ \frac{253}{1500s}\text {, }&{}\frac{1}{3}\le s<1\end{array}\right. }\\&\quad =\frac{253}{1500}. \end{aligned} \end{aligned}$$

Therefore, we conclude that

$$\begin{aligned} C_0:=\min \left\{ \frac{1}{2},\frac{13}{25},\frac{67}{250},\frac{1}{2}, \frac{253}{1500}\right\} =\frac{253}{1500}. \end{aligned}$$

With these preliminary calculations completed we now examine conditions (1)–(3) in the statement of Corollary 2.14. Note that

$$\begin{aligned} \left[ \left( \frac{\rho _1C_0}{\varphi (\varvec{1})}\right) ^q,\rho _2\right] =\left[ \left( \frac{253}{780}\right) ^2\rho _1^2,\rho _2\right] \approx \big [0.105\rho _1^2,\rho _2\big ]. \end{aligned}$$

Now choose

$$\begin{aligned} \rho _1:=\frac{1}{20}\;\;\text { and }\;\;\rho _2:=\frac{\pi }{2}. \end{aligned}$$

Then \(A(t)=\sin {t}>0\) on \(\displaystyle \left[ \left( \frac{253}{780}\right) ^2\rho _1^2,\rho _2\right] \). So, condition (1) of Corollary 2.14 is satisfied. Moreover, since

$$\begin{aligned} \rho _1=\frac{1}{20}<\frac{253}{1500}\sqrt{\frac{\pi }{2}}=C_0\rho _2^{\frac{1}{2}}, \end{aligned}$$

it follows that the condition \(\rho _1<C_0\rho _2^{\frac{1}{q}}\) is also satisfied. In addition, condition (2) of the corollary is satisfied since

$$\begin{aligned} \frac{H \left( \rho _1\right) }{\rho _1}= \frac{H\left( \frac{1}{20}\right) }{\frac{1}{20}}=18\sqrt{\frac{1}{20}}> \frac{250}{67}=\frac{1}{\varphi (\gamma )}. \end{aligned}$$

Now suppose that both f and \(\lambda \) satisfies the inequality

$$\begin{aligned} \int _0^1\left[ \frac{9}{10}\sqrt{\frac{780}{253}\sqrt{\frac{\pi }{2}}}(1- t)+\lambda f_{[0,1]\times \left[ 0, \frac{1500}{253}\sqrt{\frac{\pi }{2}}\right] }^{M}\int _0^1G(t,s)\ ds\right] ^2\ dt<\sqrt{\frac{\pi }{2}}.\nonumber \\ \end{aligned}$$
(2.21)

Then condition (3) of the corollary will be satisfied. Therefore, provided that f and \(\lambda \) are such that inequality (2.21) holds, then by Corollary 2.14 problem (2.20) has at least one positive solution, \(u_0\), satisfying the localization \(\displaystyle u_0\in {\widehat{V}}_{\frac{\pi }{2}}^{2}\setminus \overline{{\widehat{W}}_{\frac{1}{20}}}\).

Remark 2.18

Note in Example 2.17 that the coefficient function A is not nonnegative on \({\mathbb {R}}\)—nor is it strictly positive on \({\mathbb {R}}\). It is also not monotone on \({\mathbb {R}}\). As explained in Sect. 1 these are all typical conditions on nonlocal functions in the existing literature. By using the nonstandard cone \({\mathscr {K}}\) together with the nonstandard open sets \({\widehat{V}}_{\rho }^{q}\) and \({\widehat{W}}_{\rho }\) we are able to avoid those more restrictive conditions on the coefficient function A. Yet at the same time we are able to recover the pointwise-type conditions on the coefficient A as well as the function H in the nonlocal boundary condition. And in this way, as explained in Sect. 1, we are able to merge the good features of the different methodologies in [27,28,29, 32, 36].