1 Introduction

In this paper we consider the nonlocal boundary value problem

$$\begin{aligned} -\left[ D_{0^+}^{\nu }y\right] (t)= & {} \lambda f\big (t,y(t)\big )\text {, }0<t<1 \nonumber \\ y^{(i)}(0)= & {} 0\text {, }0\le i\le n-2 \nonumber \\ \left[ D_{0^+}^{\alpha }y\right] (1)= & {} H\big (\varphi (y)\big ), \end{aligned}$$
(1.1)

where \(\nu \in (n-1,n]\) for \(n\in \mathbb {N}_4:=\{4,5,6,\ldots \}\), \(\alpha \in [1,n-2]\), and \(\lambda >0\) is a parameter; here we utilize the Riemann–Liouville derivative. In addition, the maps \(H\ : \ [0,+\infty )\rightarrow [0,+\infty )\) and \(f\ : \ [0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) are continuous. The nonlocal element \(\varphi \ : \ \mathcal {C}([0,1])\rightarrow \mathbb {R}\) is a linear functional, which is realizable as a Stieltjes integral with signed measure, namely

$$\begin{aligned} \varphi (y)=\int _0^1y(t)\ d\alpha (t), \end{aligned}$$
(1.2)

where \(\alpha \in BV([0,1];\mathbb {R})\) is not necessarily monotone increasing—i.e., it may occur that \(\varphi (y)<0\) even though y is nonnegative. As the example will demonstrate in Sect. 3, although it may occur that H is nonlinear, it need not necessarily be so.

The primary contribution of this paper is to introduce a new order cone and attendant open set with which to study fractional differential equation (1.1). In particular, we utilize the cone

$$\begin{aligned} \mathcal {K}:=\left\{ y\in \mathcal {C}([0,1])\ : \ y\ge 0\text {, }\varphi (y)\ge \left( \inf _{s\in S_0}\frac{1}{\mathcal {G}(s)}\int _0^1G(t,s)\ d\alpha (t)\right) \Vert y\Vert \right\} , \end{aligned}$$
(1.3)

where \(\alpha \) is as in (1.2) above, G is the Green’s function, see (2.1), associated to problem (1.1), and \(\mathcal {G}(s):=\max _{t\in [0,1]}G(t,s)\). The set \(S_0\) in (1.3) is a set of full measure on which the infimum is positive. The precise assumption leading to this set is given in Sect. 2. The motivation behind including the coercivity condition in (1.3) is so that we can weaken the growth conditions imposed on both H and f. In particular, we can use asymptotic or even, as we shall see, pointwise conditions on H that, without some sort of coercivity and thus lower control over \(\varphi \), might not be possible to achieve easily.

As previously intimated, in addition to the use of a new cone in fractional nonlocal boundary value problems, we also make use of the coercivity generated by \(\mathcal {K}\) by taking advantage of a new open set in the study of such problems. In particular, we define here the open set \(\widehat{V}_{\rho }\) by

$$\begin{aligned} \widehat{V}_{\rho }:=\left\{ y\in \mathcal {K}\ : \ \varphi (y)<\rho \right\} . \end{aligned}$$
(1.4)

As shall be seen in both Sects. 2 and 3, the use of this open set is only viable since we have that \(\varphi \) is a coercive functional by means of \(\mathcal {K}\). In any case, as shall be shown in Sect. 3, the use of the set \(\widehat{V}_{\rho }\) as defined in (1.4) can produce better existence results. In particular, we can treat cases where H is piecewise linear and no particular growth conditions are imposed on f. Alternatively, we can treat cases where H has only pointwise conditions imposed, which is one interesting and somewhat unusual consequence of our approach. And in each case these conditions are accommodated with \(\lambda >0\) essentially unrestricted.

It should be mentioned at this juncture that the methods we introduce here, namely the use of the new cone \(\mathcal {K}\) and the set \(\widehat{V}_{\rho }\), are applicable to a range of fractional-order boundary value problems. We prefer the specificity and concreteness of (1.1), as it provides an application of the abstract results. But the techniques developed herein should be able to be extended to a variety of settings such as semipositone fractional BVPs, singular fractional BVPs, and fractional-order differential equations equipped with boundary conditions other than those utilized in (1.1).

We conclude the introduction by mentioning some of the relevant literature on both fractional-order and nonlocal boundary value problems and its connection to this present work. Problem (1.1) was studied in both the local and nonlocal boundary condition settings by the author [9, 10]. Many other authors have studied either extensions of (1.1) or similar problems—see, for example, [7, 29, 39, 50, 59]. More generally, there has been substantial research interest in fractional-order differential equations over the past 10 to 15 years, and there are numerous papers within the area. For a representative but certainly not exhaustive sampling, one may consult [1, 2, 4, 8, 20, 30, 41] and the references therein. At the same time, the study of nonlocal boundary value problems has seen intense research recently, and these studies have spanned such topics as nonlinear boundary conditions, linear boundary conditions, semipositone problems, and the construction of various cones to allow for more general linear boundary conditions—see, for example, the contributions by Anderson [3], Cabada, et al. [6], Goodrich [11–15], Graef and Webb [19], Infante [22], Infante and Pietramala [23, 25], Infante, Pietramala, and Minhós [24], Infanate, Pietramala, and Tenuta [26], Jankowski [28], Karakostas and Tsamatos [31, 32], Karakostas [33], Webb and Infante [47], and Yang [53–56]. More generally, the study of various perturbed Hammerstein integral equations, which, as in our work, are typically utilized in the study of boundary value problems, have also been studied by many authors such as Goodrich [17], Lan and Lin [36], Liu and Wu [38], Webb [46], Xu and Yang [51], Yang [57], and Zhao [60]. The articles by Picone [42] and Whyburn [49], while classical, are of interest for their historical value.

It should also be mentioned in particular that the cone we introduce here is inspired by the not dissimilar cones utilized first by Graef et al. [18] and then subsequently by Webb [45] and Ma and Zhong [40]. In addition, the type of open set we introduce here (i.e., one in which a functional is utilized as part of the defining condition) is similar to constructions found in some other works—see, for example, Infante and Maciejewski [27]. However, we have not seen before the particular cone and open set we introduce here, nor used in the specific ways in which we utilize them here.

All in all, then, in this work we join several of these strands of research by developing a new cone and open set in order to study a nonlocal boundary value problem in the context of fractional differential equations. Furthermore, since \(\nu \) can be an integer in (1.1) and, in fact, the case \(\nu =4\) is important in beam deflection models, the results here also have some interest in the integer-order setting. And, indeed, there have appeared many works on specifically fourth-order BVPs with nonlocal boundary conditions—see, for example, [5, 34, 37, 48, 52, 61]. Thus, our results, while couched in the fractional-order setting, also complement those papers specifically treating the fourth-order setting.

2 Preliminary lemmata and notation

We begin by first stating some notation that will be used throughout the reminder of this paper.

Notation 2.1

For use in the sequel, we appeal to the following notational conventions.

  • Define the map \(\mathcal {G}\ : \ [0,1]\rightarrow [0,+\infty )\) as in Sect. 1—namely, put

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

    where the map \((t,s)\mapsto G(t,s)\) is defined in (2.1) below. Note that for each fixed \(s\in [0,1]\) it holds that \(G(t,s)\le \mathcal {G}(s)\), for all \(t\in [0,1]\).

  • Given a function \(f\ : \ [0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) and a set \(X\subseteq [0,+\infty )\) define the number \(\widetilde{f}_{X}^{M}\) to be

    $$\begin{aligned} \widetilde{f}_{X}^{M}:=\sup _{(t,y)\in [0,1]\times X}f(t,y). \end{aligned}$$

  • Given a number \(\rho >0\) define the open set \(\Omega _{\rho }\) by \(\Omega _{\rho }:=\left\{ y\in \mathcal {K}\ : \ \Vert y\Vert <\rho \right\} \), where \(\mathcal {K}\) is as in (1.3) above.

A couple of basic definitions regarding fractional derivatives and integrals of Riemann–Liouville type are recalled next. For a more detailed exposition on this and a variety of related topics in the continuous fractional calculus, the reader may consult the monograph by Podlubny [43].

Definition 2.2

Let \(\nu >0\) with \(\nu \in \mathbb {R}\). Suppose that \(y\ : \ [a,+\infty )\rightarrow \mathbb {R}\). Then the \(\nu \)th order Riemann–Liouville fractional integral is defined to be

$$\begin{aligned} D_{a^+}^{-\nu }y(t):=\frac{1}{\Gamma (\nu )}\int _a^ty(s)(t-s)^{\nu -1}\ ds, \end{aligned}$$

whenever the right-hand side is defined.

Definition 2.3

For \(y\ : \ [a,+\infty )\rightarrow \mathbb {R}\) and with \(\nu >0\) and \(\nu \in \mathbb {R}\), we define the \(\nu \)th order Riemann–Liouville fractional derivative to be

$$\begin{aligned} D_{a^+}^{\nu }y(t):=\frac{1}{\Gamma (n-\nu )}\frac{d^n}{dt^n}\int _a^t\frac{y(s)}{(t-s)^{\nu +1-n}}\ ds, \end{aligned}$$

where \(n\in \mathbb {N}\) is the unique positive integer satisfying \(n-1\le \nu <n\) and \(t>a\), whenever the right-hand side is defined.

We also need to recall some preliminary lemmata regarding the Green’s function associated to problem (1.1). In particular, we recall the following results, which may be found in a paper by the author [9].

Lemma 2.4

Let \(g\in \mathcal {C}([0,1])\) be given. Then the unique solution to problem \(-D^{\nu }y(t)=g(t)\) together with the boundary conditions

$$\begin{aligned} y^{(i)}(0)= & {} 0\text {, }i\in \{0,1,2,\ldots ,n-2\}\\ \left[ D_{0^+}^{\alpha }y\right] (1)= & {} 0, \end{aligned}$$

where \(\alpha \in [1,n-2]\), is

$$\begin{aligned} y(t)=\int _0^1G(t,s)g(s)\ ds, \end{aligned}$$

where

$$\begin{aligned} G(t,s)={\left\{ \begin{array}{ll} \frac{t^{\nu -1}(1-s)^{\nu -\alpha -1}-(t-s)^{\nu -1}}{\Gamma (\nu )}\text {, }&{}0\le s\le t\le 1\\ \frac{t^{\nu -1}(1-s)^{\nu -\alpha -1}}{\Gamma (\nu )}\text {, }&{}0\le t\le s\le 1\end{array}\right. } \end{aligned}$$
(2.1)

is the Green’s function for this problem.

Lemma 2.5

Let G be as given in the statement of Lemma 2.4. Then we find that:

  1. (1)

    G(t, s) is a continuous function on the unit square \([0,1]\times [0,1]\);

  2. (2)

    \(G(t,s)\ge 0\) for each \((t,s)\in [0,1]\times [0,1]\); and

  3. (3)

    \(\mathcal {G}(s)=G(1,s)\), for each \(s\in [0,1]\).

We state next the structural and regularity conditions that we impose on problem (1.1). Throughout this work we denote by \(\Vert \cdot \Vert \) the usual supremum norm on the space \(\mathcal {C}([0,1])\). In summary, condition (A1) concerns the basic structure of \(\varphi \), conditions (A2)–(A3) concern the growth properties of the maps H and f, condition (A4) concerns the existence of the coercivity constant \(\displaystyle C_0:=\inf _{s\in S_0}\frac{1}{\mathcal {G}(s)}\int _0^1G(t,s)\ d\alpha (t)\), and, finally, condition (A5) is a technical condition that ensures that the cone \(\mathcal {K}\) is neither empty nor trivial. It should be noted that not all of these conditions are used in each existence result—e.g., we are able to weaken or remove conditions (A2) and (A3) by replacing them with other assumptions.

  • A1: The functional \(\varphi (y)\) may be written in the form

    $$\begin{aligned} \varphi (y):=\int _{[0,1]}y(t)\ d\alpha (t), \end{aligned}$$

    where \(\alpha \ : \ [0,1]\rightarrow \mathbb {R}\) satisfies \(\alpha \in BV([0,1])\). Moreover, we denote by \(C_1>0\) some finite constant such that

    $$\begin{aligned} \left| \varphi (y)\right| \le C_1\Vert y\Vert , \end{aligned}$$

    for each \(y\in \mathcal {C}([0,1])\).

  • A2: The map \(H\ : \ [0,+\infty )\rightarrow [0,+\infty )\) is continuous, and there exists a number \(\rho _1>0\) such that

    $$\begin{aligned} \frac{H\left( \rho _1\right) }{\rho _1}>\frac{1}{\varphi (\beta )}, \end{aligned}$$

    where the map \(\beta \ : \ [0,1]\rightarrow \mathbb {R}\) is defined by

    $$\begin{aligned} \beta (t):=\frac{\Gamma (\nu -\alpha )}{\Gamma (\nu )}t^{\nu -1}. \end{aligned}$$
    (2.2)

  • A3: The function \(f\ : \ [0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous and satisfies

    $$\begin{aligned} \lim _{y\rightarrow +\infty }\frac{f(t,y)}{y}=0\text {, uniformly for }t\in [0,1]. \end{aligned}$$

  • A4: Assume that the map

    $$\begin{aligned} s\mapsto \frac{1}{\mathcal {G}(s)}\int _0^1G(t,s)\ d\alpha (t) \end{aligned}$$

    is defined for \(s\in S_0\), where \(S_0\subseteq [0,1]\) has full measure (i.e., \(\left| S_0\right| =1\)), and the constant defined by

    $$\begin{aligned} C_0:=\inf _{s\in S_0}\frac{1}{\mathcal {G}(s)}\int _0^1G(t,s)\ d\alpha (t) \end{aligned}$$

    satisfies \(C_0\in \left( 0,C_1\right) \).

  • A5: It holds that

    $$\begin{aligned} \varphi (\beta )\ge C_0\Vert \beta \Vert , \end{aligned}$$

    where \(\beta \) is defined in (2.2)

Remark 2.6

Seeing as condition (A5) implies that \(\beta \in \mathcal {K}\) with \(\Vert \beta \Vert \ne 0\), it follows that this condition ensures both that \(\mathcal {K}\ne \varnothing \) and that \(\mathcal {K}\) is nontrivial; here \(\mathcal {K}\) is as defined earlier in (1.3). Moreover, observe that condition (A5) implies that \(\varphi (\beta )>0\) since \(\Vert \beta \Vert >0\) evidently.

Remark 2.7

As will be shown explicitly by the example in Sect. 3, the range of admissible values for the parameter \(\lambda \), appearing in (1.1), is explicitly computable. Thus, we do not state our results here for some uncomputable, “sufficiently small” parameter \(\lambda \). Moreover, the computation of the coercivity constant \(C_0\), appearing in (A4), is also reasonable to compute as will be shown in Sect. 3.

In order to study problem (1.1) we consider instead the operator \(T\ : \ \mathcal {C}([0,1])\rightarrow \mathcal {C}([0,1])\) defined by

$$\begin{aligned} (Ty)(t):=\beta (t)H(\varphi (y))+\lambda \int _0^1G(t,s)f(s,y(s))\ ds \end{aligned}$$
(2.3)

and then look for solutions of the Hammerstein-type equation \((Ty)(t)=y(t)\). Note that (see [10, Lemma 4.3]) it has been shown that the map \(\beta \) occurring in (2.2) has the property that it is increasing in t, it holds that \(\beta ^{(i)}(0)=0\) for each \(0\le i\le n-2\) with \(i\in \mathbb {N}\), and that \(\left[ D_{0^+}^{\alpha }\beta \right] (1)=1\). These facts combine to show that a solution of the Hammerstein equation is thus a solution of the boundary value problem (1.1).

Ordinarily it is trivial to show that \(T(\mathcal {K})\subseteq \mathcal {K}\). In this case because of the use of a new cone, we provide part of this proof in detail.

Lemma 2.8

Let T be the operator defined in (2.3). Then it holds that \(T(\mathcal {K})\subseteq \mathcal {K}\).

Proof

It is obvious that whenever \(y\in \mathcal {K}\) it holds that \((Ty)(t)\ge 0\), for each \(t\in [0,1]\). Therefore, we focus on the coercivity condition. To this end, let \(y\in \mathcal {K}\) be fixed but arbitrary. Observe that

$$\begin{aligned} \Vert Ty\Vert \le H(\varphi (y))\Vert \beta \Vert +\lambda \int _0^1\mathcal {G}(s)f(s,y(s))\ ds. \end{aligned}$$

Thus, recalling that \(S_0\) is a set of full measure, we write

$$\begin{aligned} \varphi (Ty)= & {} \varphi (\beta )H(\varphi (y))+\lambda \int _0^1\int _0^1G(t,s)f(s,y(s))\ d\alpha (t)\ ds\\= & {} \varphi (\beta )H(\varphi (y))+\lambda \int _{S_0}\left[ \int _0^1G(t,s)\ d\alpha (t)\right] f(s,y(s))\ ds\\= & {} \varphi (\beta )H(\varphi (y))+\lambda \int _{S_0}\left[ \frac{1}{\mathcal {G}(s)}\int _0^1G(t,s)\ d\alpha (t)\right] \mathcal {G}(s)f(s,y(s))\ ds\\\ge & {} \varphi (\beta )H(\varphi (y))+\lambda \int _{S_0}\left[ \inf _{s\in S_0}\frac{1}{\mathcal {G}(s)}\int _0^1G(t,s)\ d\alpha (t)\right] \mathcal {G}(s)f(s,y(s))\ ds\\= & {} \varphi (\beta )H(\varphi (y))+C_0\lambda \int _{S_0}\mathcal {G}(s)f(s,y(s))\ ds\\\ge & {} C_0\left[ H(\varphi (y))\Vert \beta \Vert +\lambda \int _0^1\mathcal {G}(s)f(s,y(s))\ ds\right] \\\ge & {} C_0\Vert Ty\Vert . \end{aligned}$$

As this establishes that \(Ty\in \mathcal {K}\), we conclude that \(T(\mathcal {K})\subseteq \mathcal {K}\), as claimed. \(\square \)

We next discuss the set \(\widehat{V}_{\rho }\), whose definition was given preliminarily in (1.4) in Sect. 1. We first state and prove two elementary lemmata regarding this set. That these hold is essential for the use of \(\widehat{V}_{\rho }\) in the existence arguments.

Lemma 2.9

For each fixed \(\rho >0\) it holds that

$$\begin{aligned} \Omega _{\frac{\rho }{C_1}}\subseteq \widehat{V}_{\rho }\subseteq \Omega _{\frac{\rho }{C_0}}. \end{aligned}$$
(2.4)

Proof

That (2.4) holds is both a consequence of the linearity of \(\varphi \) and the coercivity condition. To see this, first fix \(y\in \widehat{V}_{\rho }\). Then we note that

$$\begin{aligned} \rho >\varphi (y)\ge C_0\Vert y\Vert . \end{aligned}$$

In light of the coercivity condition above, we see that \(y\in \widehat{V}_{\rho }\) implies that

$$\begin{aligned} \Vert y\Vert \le \frac{\rho }{C_0}, \end{aligned}$$

and this establishes that

$$\begin{aligned} \widehat{V}_{\rho }\subseteq \Omega _{\frac{\rho }{C_0}}. \end{aligned}$$

On the other hand, fix \(y\in \Omega _{\frac{\rho }{C_1}}\). Then it follows that

$$\begin{aligned} \varphi (y)\le C_1\Vert y\Vert <\rho , \end{aligned}$$

whence \(y\in \widehat{V}_{\rho }\) so that \(\widehat{V}_{\rho }\supseteq \Omega _{\frac{\rho }{C_1}}\).

Finally, let us note that it does, in fact, hold that

$$\begin{aligned} \Omega _{\frac{\rho }{C_1}}\subsetneq \Omega _{\frac{\rho }{C_0}}, \end{aligned}$$

for each \(\rho >0\). This follows from the observation that \(C_0<C_1\). Thus, the inclusion is well defined, and this completes the proof. \(\square \)

Lemma 2.10

For each fixed \(\rho \in (0,+\infty )\) the set \(\widehat{V}_{\rho }\) defined in (1.4) is a (relatively) open set in \(\mathcal {K}\) and, furthermore, is bounded.

Proof

Since this is an obvious consequence of the continuity of the linear functional \(\varphi \), inclusion (2.4), and the strict inequality in the definition of \(\widehat{V}_{\rho }\), we omit the formal proof of this fact. \(\square \)

Remark 2.11

We note that the set \(\widehat{V}_{\rho }\) is similar in spirit to the set commonly denoted \(V_{\rho }\) in the literature, which is defined, for an appropriate order cone \(\mathcal {K}_0\) and fixed \((a,b)\Subset (0,1)\), by

$$\begin{aligned} V_{\rho }:=\left\{ y\in \mathcal {K}_0\ : \ \min _{t\in [a,b]}y(t)<\rho \right\} , \end{aligned}$$

and which was originally utilized by Lan [35]. In fact, the reason we have denoted our new set in (1.4) by \(\widehat{V}_{\rho }\) is since it is clearly inspired by Lan’s construction, \(V_{\rho }\), above. Moreover, as mentioned in Sect. 1 sets similar to \(\widehat{V}_{\rho }\) have been utilized in other works—e.g., [27].

Remark 2.12

The dual use of the new cone \(\mathcal {K}\) together with the new open set \(\widehat{V}_{\rho }\) will make the proof of the existence results very simple. This is one of the advantages of the dual use of these two constructions.

We conclude by stating the fixed point theorem, which we use in the existence arguments in Sect. 3. In particular, our approach here is index theoretic. To this end, we recall a basic result in this direction, and one may consult Infante et al. [26], Guo and Lakshmikantham [21], or Zeidler [58] for further details on these types of results.

Lemma 2.13

Let D be a bounded open set and, with \(\mathcal {K}\) a cone in a Banach space \(\mathscr {X}\), suppose both that \(D\cap \mathcal {K}\ne \emptyset \) and that \(\overline{D}\cap \mathcal {K}\ne \mathcal {K}\). Let \(D_1\) be open in \(\mathscr {X}\) with \(\overline{D}_1\subseteq D\cap \mathcal {K}\). Assume that \(T\ : \ \overline{D}\cap \mathcal {K}\rightarrow \mathcal {K}\) is a compact map such that \(Tx\ne x\) for \(x\in \mathcal {K}\cap \partial D\). If \(i_{\mathcal {K}}\left( T,D\cap \mathcal {K}\right) =1\) and \(i_{\mathcal {K}}\left( T,D_1\cap \mathcal {K}\right) =0\), then T has a fixed point in \((D\cap \mathcal {K}){\setminus }\left( \overline{D_1\cap \mathcal {K}}\right) \). Moreover, the same result holds if \(i_{\mathcal {K}}(T,D\cap \mathcal {K})=0\) and \(i_{\mathcal {K}}\left( T,D_1\cap \mathcal {K}\right) =1\).

3 Main result and example

We begin by stating and proving the existence results for the Hammerstein-type equation \((Ty)(t)=y(t)\) and thus for problem (1.1).

Theorem 3.1

Assume that conditions (A1)–(A2) and (A4)–(A5) hold. Fix \(\lambda >0\) and assume that there exists a number \(\rho _2>\frac{C_1}{C_0}\rho _1\) such that

$$\begin{aligned} \frac{H\left( \rho _2\right) }{\rho _2}\varphi (\beta )+\frac{\lambda }{\rho _2}\widetilde{f}_{\left[ 0,\frac{\rho _2}{C_0}\right] }^{M}\int _0^1\int _0^1G(t,s)\ d\alpha (t)\ ds<1. \end{aligned}$$

Then problem (1.1) has at least one positive solution.

Proof

Having showed that \(T(\mathcal {K})\subseteq \mathcal {K}\) in Lemma 2.4 and in observation of the fact that T is completely continuous, we focus on the actual index calculations. To this end, we begin by noting that by condition (A2) there exists \(\rho _1>0\) such that

$$\begin{aligned} \frac{H\left( \rho _1\right) }{\rho _1}>\frac{1}{\varphi (\beta )}. \end{aligned}$$
(3.1)

We show now for each \(y\in \partial \widehat{V}_{\rho _1}\) and each \(\mu \ge 0\) that \(y\ne Ty+\mu e\), where we put \(e(t):=\beta (t)\). Recall here that \(\beta \in \mathcal {K}\) by condition (A5), and, moreover, that \(\Vert \beta \Vert \ne 0\); thus, \(\beta \) represents a valid selection for the map \(t\mapsto e(t)\). So, suppose for contradiction the existence of fixed \(y\in \partial \widehat{V}_{\rho _1}\) and \(\mu \ge 0\) such that \(y=Ty+\mu e\). Then applying \(\varphi \) to both sides of the equality \(y=Ty+\mu e\) and using the fact that \(\varphi (\mu e)\ge 0\), we obtain that

$$\begin{aligned} \rho _1= & {} \varphi (y)\\\ge & {} \varphi (\beta )H(\varphi (y))+\lambda \int _0^1\int _0^1G(t,s)f(s,y(s))\ d\alpha (t)\ ds\\\ge & {} \varphi (\beta )H(\varphi (y))=\varphi (\beta )H\left( \rho _1\right) , \end{aligned}$$

whence

$$\begin{aligned} \frac{H\left( \rho _1\right) }{\rho _1}\le \frac{1}{\varphi (\beta )}. \end{aligned}$$

Since this is a contradiction to inequality (3.1), we deduce that

$$\begin{aligned} i_{\mathcal {K}}\left( T,\widehat{V}_{\rho _1}\right) =0. \end{aligned}$$
(3.2)

On the other hand, we now show that for \(\rho _2>\frac{C_1}{C_0}\rho _1\), where \(\rho _2\) is the number given in the statement of the theorem, we have

$$\begin{aligned} i_{\mathcal {K}}\left( T,\widehat{V}_{\rho _2}\right) =1. \end{aligned}$$
(3.3)

In order to prove (3.3) we show that for each \(\mu \ge 1\) and each \(y\in \widehat{V}_{\rho _2}\) it holds that \(Ty\ne \mu y\). Therefore, suppose not. Then we have \(\varphi (Ty)=\mu \varphi (y)\ge \varphi (y)\) for some \(y\in \partial \widehat{V}_{\rho _2}\). Consequently, we may write

$$\begin{aligned} \varphi (y)\le \varphi (\beta )H(\varphi (y))+\lambda \int _0^1\int _0^1G(t,s)f(s,y(s))\ d\alpha (t)\ ds, \end{aligned}$$
(3.4)

which, since \(\varphi (y)=\rho _2\), implies that

$$\begin{aligned} 1\le & {} \frac{H\left( \rho _2\right) }{\rho _2}\varphi (\beta )+\frac{\lambda }{\rho _2}\int _0^1\int _0^1G(t,s)f(s,y(s))\ d\alpha (t)\ ds \nonumber \\\le & {} \frac{H\left( \rho _2\right) }{\rho _2}\varphi (\beta )+\frac{\lambda }{\rho _2}\int _0^1\int _0^1G(t,s)\widetilde{f}_{\left[ 0,\frac{\rho _2}{C_0}\right] }^{M}\ d\alpha (t)\ ds \nonumber \\< & {} 1, \end{aligned}$$
(3.5)

which is a contradiction, and so, establishes (3.3). Observe that to establish (3.5) we are tacitly using the fact that since \(s\mapsto \mathcal {G}(s)\) is a nonnegative map, it follows from condition (A4) that

$$\begin{aligned} \int _0^1G(t,s)\ d\alpha (t)>0, \end{aligned}$$

for a.e. \(s\in [0,1]\), seeing as \(\big |[0,1]{\setminus } S_0\big |=0\). We are also appealing to Lemma 2.9 so that since \(y\in \overline{\widehat{V}}_{\rho _2}\), it thus follows that \(y\in \Omega _{\frac{\rho _2}{C_0}}\), whence

$$\begin{aligned} 0\le f(s,y(s))\le \widetilde{f}_{\left[ 0,\frac{\rho _2}{C_0}\right] }^{M}, \end{aligned}$$

for each \(s\in [0,1]\).

All in all, then, combining (3.2)–(3.3) we conclude the existence of a map

$$\begin{aligned} y_0\in \widehat{V}_{\rho _2}{\setminus }\overline{\widehat{V}}_{\rho _1} \end{aligned}$$

such that \(Ty_0=y_0\). Note that

$$\begin{aligned} \widehat{V}_{\rho _2}{\setminus }\overline{\widehat{V}}_{\rho _1}\ne \varnothing \end{aligned}$$
(3.6)

since from Lemma 2.9 we see that \(\widehat{V}_{\rho _1}\subseteq \Omega _{\frac{\rho _1}{C_0}}\) and that \(\widehat{V}_{\rho _2}\supseteq \Omega _{\frac{\rho _2}{C_1}}\). Thus, we see that if \(\rho _2>\frac{C_1}{C_0}\rho _1\), then it follows that

$$\begin{aligned} \widehat{V}_{\rho _2}\supseteq \Omega _{\frac{\rho _2}{C_1}}\supsetneq \overline{\Omega }_{\frac{\rho _1}{C_0}}\supset \overline{\widehat{V}}_{\rho _1}\supsetneq \widehat{V}_{\rho _1}, \end{aligned}$$

which establishes (3.6). Since this map \(y_0\) solves (1.1) and satisfies \(\Vert y_0\Vert \ne 0\), the proof is thus complete. \(\square \)

Remark 3.2

As the proof of Theorem 3.1 demonstrates, the solution to (1.1) guaranteed by Theorem 3.1 satisfies the norm localization

$$\begin{aligned} 0<\frac{\rho _1}{C_1}<\Vert y_0\Vert <\frac{\rho _2}{C_0}, \end{aligned}$$

where here we appeal to Lemma 2.9, noting especially that

$$\begin{aligned} \mathcal {K}{\setminus }\overline{\widehat{V}}_{\rho _1}\subseteq \mathcal {K}{\setminus }\overline{\Omega }_{\frac{\rho _1}{C_1}} \end{aligned}$$
(3.7)

and

$$\begin{aligned} \widehat{V}_{\rho _2}\subseteq \Omega _{\frac{\rho _2}{C_0}}, \end{aligned}$$
(3.8)

whereupon combining inclusions (3.7)–(3.8) we obtain

$$\begin{aligned} \widehat{V}_{\rho _2}{\setminus }\overline{\widehat{V}}_{\rho _1}\subseteq \Omega _{\frac{\rho _2}{C_0}}{\setminus }\overline{\Omega }_{\frac{\rho _1}{C_1}}, \end{aligned}$$

which provides the desired localization.

Next we state two selected corollaries of Theorem 3.1. Since the proofs are mostly obvious modifications of the proof of Theorem 3.1, we omit parts of them. In particular, the first of these, Corollary 3.3, demonstrates that we may replace the pointwise condition in (A2) with a limit condition; the upshot of this is that we no longer have to require that \(\rho _2>\frac{C_1}{C_0}\rho _1\). On the other hand, the second of these, Corollary 3.4, demonstrates that if we assume condition (A3) in addition to (A2) and a limit condition on the behavior of \(\frac{H(z)}{z}\) as \(z\rightarrow +\infty \), then we can obtain an existence result that is applicable no matter the value of \(\lambda >0\).

Corollary 3.3

Suppose that conditions (A1) and (A4)–(A5) hold. Fix \(\lambda >0\) and assume that there exists a number \(\rho _2>0\) such that

$$\begin{aligned} \frac{H\left( \rho _2\right) }{\rho _2}\varphi (\beta )+\frac{\lambda }{\rho _2}\widetilde{f}_{\left[ 0,\frac{\rho _2}{C_0}\right] }^{M}\int _0^1\int _0^1G(t,s)\ d\alpha (t)\ ds<1. \end{aligned}$$

If, in addition, it holds that

$$\begin{aligned} \liminf _{z\rightarrow 0^+}\frac{H(z)}{z}>\frac{1}{\varphi (\beta )}, \end{aligned}$$

then problem (1.1) has at least one positive solution.

Proof

Omitted. \(\square \)

Corollary 3.4

Suppose that conditions (A1)–(A5) hold. Let \(\lambda \in (0,+\infty )\) be fixed but arbitrary. If, in addition, it holds that

$$\begin{aligned} \limsup _{z\rightarrow +\infty }\frac{H(z)}{z}<\frac{1}{\varphi (\beta )}, \end{aligned}$$

then problem (1.1) has at least one positive solution.

Proof

The first part of the proof is identical to the first part of the proof of Theorem 3.1. For the second part, by the assumption in the statement of the corollary we may fix a number \(\varepsilon >0\) sufficiently small such that for each \(z\ge z_0:=z_0(\varepsilon )\) it holds that

$$\begin{aligned} \frac{H\left( z\right) }{z}<\frac{1}{\varphi (\beta )}-\varepsilon . \end{aligned}$$

Note that without loss of generality we may assume that \(\varepsilon \) satisfies the inequality

$$\begin{aligned} \varepsilon <\frac{1}{\varphi (\beta )}, \end{aligned}$$

where here we use the assumption from (A5) that \(\varphi (\beta )>0\). It then follows that there exists a number \(\rho _2>0\) sufficiently large, which may be assumed without loss to satisfy \(\rho _2>\max \left\{ z_0,\frac{C_1}{C_0}\rho _1\right\} \), with \(\rho _1\) as before, such that each of

$$\begin{aligned} \frac{\widetilde{f}_{\left[ 0,\rho _2\right] }^{M}}{\rho _2}<\varepsilon \varphi (\beta )\left( \lambda \int _0^1\int _0^1G(t,s)\ d\alpha (t)\ ds\right) ^{-1} \end{aligned}$$
(3.9)

and

$$\begin{aligned} \frac{H\left( \rho _2\right) }{\rho _2}<\frac{1}{\varphi (\beta )}-\varepsilon \end{aligned}$$
(3.10)

holds; note that (3.9) is well defined due to condition (A4). Here to obtain inequality (3.9) we are using [16, Lemma 3.2]—see also [44, Lemma 2.8]; this, in particular, allows us to assert that if condition (A3) is in force, then it follows that \(\lim \nolimits _{\rho \rightarrow +\infty }\frac{\widetilde{f}_{[0,\rho ]}^{M}}{\rho }=0\). Then putting estimates (3.9)–(3.10) into inequality (3.4), for any \(y\in \partial \widehat{V}_{\rho _2}\) we obtain

$$\begin{aligned} \rho _2<&\varphi (\beta )\left( \frac{1}{\varphi (\beta )}-\varepsilon \right) \rho _2+\lambda \rho _2\int _0^1\int _0^1\frac{\widetilde{f}_{\left[ 0,\frac{\rho _2}{C_0}\right] }^{M}}{\rho _2}G(t,s)\ d\alpha (t)\ ds\\< & {} \varphi (\beta )\left( \frac{1}{\varphi (\beta )}-\varepsilon \right) \rho _2+\lambda \rho _2\left( \lambda \int _0^1\int _0^1G(t,s)\ d\alpha (t)\ ds\right) ^{-1}\\&\times \,\varepsilon \varphi (\beta )\int _0^1\int _0^1G(t,s)\ d\alpha (t)\ ds, \end{aligned}$$

from which it follows that

$$\begin{aligned} 1<1-\varepsilon \varphi (\beta )+\varepsilon \varphi (\beta )=1, \end{aligned}$$

which is a contradiction, and so, (3.3) holds. Hence, as in the proof of Theorem 3.1 we conclude that problem (1.1) has at least one positive solution. \(\square \)

Remark 3.5

Note that Corollary 3.4 applies for any value of \(\lambda >0\)—c.f., [6, Theorem 3.1] and [16, Theorem 3.3]. In some sense, both Theorem 3.1 and Corollary 3.3 also allow for an unrestricted \(\lambda \), although, in general, the larger \(\lambda \) is, the smaller the quantity \(\widetilde{f}_{\left[ 0,\frac{\rho _2}{C_0}\right] }^{M}\) will have to be.

We conclude this section and the paper with an example to illustrate, in particular, the computation and application of the coercivity constant \(C_0\) as well as the application of the existence theorems.

Example 3.6

In this example we consider the nonlocal element

$$\begin{aligned} \varphi (y):=\frac{1}{2}y\left( \frac{1}{4}\right) -\frac{1}{20}y\left( \frac{1}{10}\right) . \end{aligned}$$

We complete our calculations in the first place with arbitrary \(\alpha \in [1,n-2]\) and \(\nu \in (3,+\infty )\) with \(n-1<\nu \le n\). In this general case we find that

$$\begin{aligned}&\frac{1}{\mathcal {G}(s)}\int _0^1G(t,s)\ d\alpha (t) \nonumber \\&\quad \quad ={\left\{ \begin{array}{ll}\frac{\left[ \frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}\right] (1-s)^{\nu -\alpha -1}-\frac{1}{2}\left( \frac{1}{4}-s\right) ^{\nu -1}+\frac{1}{20}\left( \frac{1}{10}-s\right) ^{\nu -1}}{(1-s)^{\nu -\alpha -1}-(1-s)^{\nu -1}}\text {, }&{}0<s<\frac{1}{10}\\ \frac{\frac{1}{2}\left[ \left( \frac{1}{4}\right) ^{\nu -1}(1-s)^{\nu -\alpha -1}-\left( \frac{1}{4}-s\right) ^{\nu -1}\right] -\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}(1-s)^{\nu -\alpha -1}}{(1-s)^{\nu -\alpha -1}-(1-s)^{\nu -1}}\text {, }&{}\frac{1}{10}\le s<\frac{1}{4}\\ \frac{\left[ \frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}\right] (1-s)^{\nu -\alpha -1}}{(1-s)^{\nu -\alpha -1}-(1-s)^{\nu -1}}\text {, }&{}\frac{1}{4}\le s<1\end{array}\right. }. \end{aligned}$$
(3.11)

Observe from (3.11) that \(S_0:=(0,1)\) here, and so, \(S_0\) is indeed a set of full measure.

To compute the value of

$$\begin{aligned} C_0:=\inf _{s\in S_0}\frac{1}{\mathcal {G}(s)}\int _0^1G(t,s)\ d\alpha (t) \end{aligned}$$

some elementary calculus is required. These calculations, in part, rely on the observation that there exists a number \(N_0\in \mathbb {N}\) such that \(\nu -\alpha -1-N_0\le 0\) but \(\nu -1-N_0>0\). (This invokes the fact that \(\alpha \ge 1\).) Thus, \(N_0\) applications of L’Hôpital’s rule yield

$$\begin{aligned}&\lim _{s\rightarrow 1^-}\frac{\displaystyle \left[ \frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}\right] (1-s)^{\nu -\alpha -1}}{(1-s)^{\nu -\alpha -1}-(1-s)^{\nu -1}}\\&\quad \quad \overset{L'H}{=}\lim _{s\rightarrow 1^-}\frac{\displaystyle \left[ \frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}\right] (-1)^{N_0}(1-s)^{\nu -\alpha -N_0-1}\displaystyle \prod _{j=1}^{N_0}\left( \nu -\alpha -j\right) }{\left[ (-1)^{N_0}(1-s)^{\nu -\alpha -1-N_0}\displaystyle \prod _{j=1}^{N_0}\left( \nu -\alpha -j\right) \right] -\left[ (-1)^{N_0}(1-s)^{\nu -1-N_0}\displaystyle \prod _{j=1}^{N_0}(\nu -j)\right] }\\&\quad \quad =\lim _{s\rightarrow 1^-}\frac{\displaystyle \frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}}{1-\displaystyle \frac{(1-s)^{\nu -1-N_0}}{(1-s)^{\nu -\alpha -N_0-1}}\cdot \frac{\displaystyle \prod _{j=1}^{N_0}(\nu -j)}{\displaystyle \prod _{j=1}^{N_0}(\nu -\alpha -j)}}\\&\quad \quad =\frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}. \end{aligned}$$

We also calculate by means of a single application of L’Hôpital’s rule that

$$\begin{aligned}&\lim _{s\rightarrow 0^+}\frac{\displaystyle \left[ \frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}\right] (1-s)^{\nu -\alpha -1}-\frac{1}{2}\left( \frac{1}{4}-s\right) ^{\nu -1}+\frac{1}{20}\left( \frac{1}{10}-s\right) ^{\nu -1}}{(1-s)^{\nu -\alpha -1}-(1-s)^{\nu -1}}\\&\quad \overset{L'H}{=}\frac{1}{\alpha }\Bigg \{\frac{1}{2}\left[ \left( \frac{1}{4}\right) ^{\nu -2}(\nu -1)-\left( \frac{1}{4}\right) ^{\nu -1}(\nu -\alpha -1)\right] \\&\qquad \qquad \qquad +\frac{1}{20}\left[ \left( \frac{1}{10}\right) ^{\nu -1}(\nu -\alpha -1)-\left( \frac{1}{10}\right) ^{\nu -2}(\nu -1)\right] \Bigg \}. \end{aligned}$$

Then essentially routine but tedious computations, whose details we omit, demonstrate that for each \(\nu \) and each admissible \(\alpha \) we have that

$$\begin{aligned} C_0:=C_0(\nu )=\frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}>0. \end{aligned}$$

The table provided below can then be generated from the map \(\nu \mapsto C_0(\nu )\), and it summarizes how the coercivity constant changes as we alter the order, \(\nu \), of the fractional differential equation; here we have approximated \(C_0\) to three decimal places of accuracy.

\(\nu \)

3.01

3.05

3.5

3.8

4

5.5

\(C_0\)

0.030

0.029

0.015

0.010

0.008

0.001

Note, in particular, that \(C_0(\nu )>0\), for each \(\nu >3\), and, moreover, that

$$\begin{aligned} \lim _{\nu \rightarrow +\infty }\left( \frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}\right) =\lim _{\nu \rightarrow +\infty }\left( 2^{1-2\nu }-2^{-\nu -1}5^{-\nu }\right) =0. \end{aligned}$$

In addition, note that

$$\begin{aligned} \varphi (\beta )=\frac{\Gamma (\nu -\alpha )}{\Gamma (\nu )}\left[ \frac{1}{2}\left( \frac{1}{4}\right) ^{\nu -1}-\frac{1}{20}\left( \frac{1}{10}\right) ^{\nu -1}\right] =\frac{\Gamma (\nu -\alpha )}{\Gamma (\nu )}C_0(\nu ). \end{aligned}$$

In observance of the fact that \(\displaystyle \Vert \beta \Vert =\frac{\Gamma (\nu -\alpha )}{\Gamma (\nu )}\), it follows that

$$\begin{aligned} \varphi (\beta )\ge C_0\Vert \beta \Vert =\frac{\Gamma (\nu -\alpha )}{\Gamma (\nu )}C_0(\nu ). \end{aligned}$$

Thus, condition (A5) will be satisfied for any admissible choice of \(\alpha \) and \(\nu \) for the functional \(\varphi \) given in this example. Since the quantity \(\displaystyle \frac{1}{\varphi (\beta )}\) occurs in the application of the existence results, we present in the following table values of \(\displaystyle \frac{1}{\varphi (\beta )}\) for selected choices of \(\alpha \) and \(\nu \).

\(\nu \)

3.01

3.01

3.01

3.8

3.8

4

\(\alpha \)

1

1.5

1.95

1

1.5

1

\(\displaystyle \frac{1}{\varphi (\beta )}\)

5.449

75.064

68.699

273.721

393.319

386.473

Finally, to see how to apply the existence theorems, let us suppose that \(\nu =3.01\) and \(\alpha =1\). Let us also define the map H by

$$\begin{aligned} H(z):={\left\{ \begin{array}{ll} 6z\text {, }&{}z<2\\ 5(z-2)+12\text {, }&{}z\ge 2\end{array}\right. }. \end{aligned}$$

Note that H is a piecewise linear map. Then we conclude that the boundary value problem

$$\begin{aligned} -\left[ D_{0^+}^{3.01}y\right] (t)= & {} \lambda f(t,y(t))\text {, }0<t<1 \nonumber \\ y(0)= & {} y'(0)=y''(0)=0 \nonumber \\ y'(1)= & {} H(\varphi (y)) \end{aligned}$$
(3.12)

has at least one positive solution:

  • for any \(\lambda \in (0,+\infty )\) if condition (A3) holds; or

  • for each given \(\lambda \in (0,+\infty )\) such that there exists a number \(\rho _2>0\) satisfying

    $$\begin{aligned} \frac{H\left( \rho _2\right) }{\rho _2}\varphi (\beta )+\frac{\lambda }{\rho _2}\widetilde{f}_{\left[ 0,\rho _2\left( \frac{1}{2}\left( \frac{1}{4}\right) ^{2.01}-\frac{1}{20}\left( \frac{1}{10}\right) ^{2.01}\right) ^{-1}\right] }^{M}\int _0^1\int _0^1G(t,s)\ d\alpha (t)\ ds<1, \end{aligned}$$

where these conclusions hold, respectively, by Corollaries 3.4 and 3.3 since we note that

$$\begin{aligned} 5=\lim _{z\rightarrow +\infty }\frac{H(z)}{z}<\frac{1}{\varphi (\beta )}<\lim _{z\rightarrow 0^+}\frac{H(z)}{z}=6. \end{aligned}$$

By altering the form of H and utilizing the above tables, we can apply the existence results for other choices of \(\alpha \) and \(\nu \). Finally, we note that the boundary condition at \(t=1\) can be written in the form

$$\begin{aligned} y'(1)={\left\{ \begin{array}{ll}3y\left( \frac{1}{4}\right) -\frac{3}{10}y\left( \frac{1}{10}\right) \text {, }&{}0\le \frac{1}{2}y\left( \frac{1}{4}\right) -\frac{1}{20}y\left( \frac{1}{10}\right) <2\\ \frac{5}{2}y\left( \frac{1}{4}\right) -\frac{1}{4}y\left( \frac{1}{10}\right) +2\text {, }&{}2\le \frac{1}{2}y\left( \frac{1}{4}\right) -\frac{1}{20}y\left( \frac{1}{10}\right) \end{array}\right. }. \end{aligned}$$

Remark 3.7

The localization from Remark 3.2 assures us that the solution, say \(y_0\), of (3.12) must satisfy

$$\begin{aligned} 0<\frac{20}{11}\rho _1<\Vert y_0\Vert <\rho _2\left[ \frac{1}{2}\left( \frac{1}{4}\right) ^{2.01}-\frac{1}{20}\left( \frac{1}{10}\right) ^{2.01}\right] ^{-1}\approx 32.969\rho _2, \end{aligned}$$

where the numbers \(\rho _1\) and \(\rho _2\) would depend upon which existence result was used as well as the choice of f and H.

Remark 3.8

Note that in Example 3.6 we see that the strength of the coercivity condition is maximized as \(\nu \rightarrow 3^+\) and weakened as \(\nu \) increases away from 3. This can affect the applicability of the existence results since, in general, a larger value of \(C_0\) will impose less of a restriction on f in Theorem 3.1 and Corollary 3.3, for example.