A surprising property of nonlocal operators: the deregularising effect of nonlocal elements in convolution differential equations

Christopher S. Goodrich

doi:10.1515/ans-2023-0137

Article Open Access

A surprising property of nonlocal operators: the deregularising effect of nonlocal elements in convolution differential equations

Christopher S. Goodrich

Published/Copyright: June 12, 2024

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From the journal Advanced Nonlinear Studies Volume 24 Issue 4

Abstract

We consider nonlocal differential equations with convolution coefficients of the form

− M ( a * | u | q ) ( 1 ) μ ( t ) u ″ ( t ) = λ f t , u ( t ) , t ∈ ( 0,1 ) ,

where q > 0, subject to given boundary data. The function μ ∈ C [ 0,1 ] modulates the strength of the nonlocal element. We demonstrate that the nonlocality has a strong deregularising effect in the specific sense that nonexistence theorems for this problem are directly affected by the magnitude of the function μ. A specific example illustrates the application of the nonexistence results presented herein.

Keywords and phrases: nonlocal differential equation; nonexistence; finite convolution

2010 Mathematics Subject Classification: Primary: 33B15; 34B10; 34B18; 42A85; 44A35; Secondary: 26A33; 47H30

1 Introduction

Let q > 0 be given. In this paper we consider the one-dimensional nonlocal differential equation

(1.1) − M ( a * | u | q ) ( 1 ) μ ( t ) u ″ ( t ) = λ f t , u ( t ) , t ∈ ( 0,1 ) ,

where λ > 0 is a parameter, μ : [0, 1] → [0, + ∞) is a given a.e. positive C 0 ( 0,1 ) function, and by (a∗b)(t) we mean the finite convolution of two L ¹ functions – i.e.,

( a * b ) ( t ) ≔ ∫ 0 t a ( t − s ) b ( s ) d s , t ≥ 0 .

Problem (1.1) will be subjected to various boundary data, phrased in terms of a Green’s function as is explained toward the beginning of Section 2. Observe that (1.1) is a relative of the classical Kirchhoff PDE

u t t − M ‖ D u ‖ L 2 2 Δ u ( x ) = f x , u ( x ) , x ∈ Ω ⊂ R n ,

in the one-dimensional steady-state setting. Indeed, an important model case of (1.1) occurs when a(t) ≡ 1, for then we obtain (when q ≥ 1)

− M ‖ u ‖ L q q μ ( t ) u ″ ( t ) = λ f t , u ( t ) , t ∈ ( 0,1 ) .

The specific regularity and structural hypotheses on the different functions appearing in (1.1) will be addressed precisely in Section 2. For now we make just two remarks. On the one hand, we will generally assume that the coefficient function M satisfies the following growth regime for some constant c ₂ > 0:

− ∞ < M ( t ) ≤ c 2 t p .

This growth regime is related to p ₁ − p ₂ growth; that is, there exist constants c ₁ > 0, c ₂ > 0, c ₃ ≥ 0, and 1 ≤ p ₁ ≤ p ₂ < + ∞ such that

c 1 t p 1 ≤ M ( t ) ≤ c 2 t p 2 + c 3 , t ≥ 0 .

Its origins lie in regularity theory for elliptic PDEs in which case such growth is a special case of the so-called “p(x)-growth”; see, for example, [1], [2], [3], [4]. We have recently used the concept of p ₁ − p ₂ growth in [5] but in the context of existence theory for nonlocal problems.

On the other hand, the reason for considering the convolutional structure in (1.1) is because finite convolutions provide a natural framework (see [6], [7]) in which to study fractional integrals and derivatives, which are a very commonly studied class of nonlocal operators. Indeed, if we put

a ( t ) ≔ 1 Γ ( α ) t α − 1 , t > 0 ,

where 0 < α < 1, then (a∗u ^q)(t) is the α-th order Riemann–Liouville fractional integral of the function u ^q; see [8], [9] for a detailed account of the fundamentals of the fractional calculus, together with [10], [11], [12], [13], [14], [15] for some applications to boundary value problems and advection-diffusion models. Furthermore, nonlocal operators, whether realised in a convolution form or not, have found applications to thermostats [16], beam deflection [17], chemical reactor theory [18], and nonlocal boundary data [19], [20], [21], [22], [23], [24], [25], [26].

The main novelty that we investigate in this work is the following. Consider the factor t ↦ μ(t), where μ ∈ C [ 0,1 ] , ( 0 , + ∞ ) , appearing within the nonlocal coefficient M in equation (1.1). This factor can be thought of as modulating the contribution of the nonlocal structure of the equation. For example, if μ ≡ 0, then, assuming that M(0) ≠ 0, (1.1) reduces to the local equation

− M ( 0 ) u ″ ( t ) = λ f t , u ( t ) , t ∈ ( 0,1 ) ,

which we can just as well recast as

u ″ ( t ) = λ ̂ f t , u ( t ) , t ∈ ( 0,1 ) ,

where λ ̂ ≔ M ( 0 ) − 1 λ ; this latter problem, subject to various boundary data, has been studied extensively over the past many decades; see, for example, the classic paper of Erbe and Wang [27]. On the other hand, if μ(t) ≫ 1, then, assuming that (a∗u ^q)(1) ≠ 0, we have that

− M ( a * u q ) ( 1 ) μ ( t ) ︸ ≫ 1 u ″ ( t ) = λ f t , u ( t ) , t ∈ ( 0,1 ) ,

and so, depending upon the behaviour of M, the nonlocal structure of (1.1) is thus amplified. What we demonstrate in this paper (see Section 2 for the precise details) is that the amplification or deamplification of this nonlocal structure can significantly affect existence results for (1.1). And the function μ is a sort-of proxy by which we conduct this analysis. Indeed, it provides a sort of extra “parameter” by which to affect our results.

In particular we prove the following result in this paper. Roughly speaking, and with some degree of simplification to convey the general idea of the result, it is characterised as follows. Note that Figure 1 illustrates a sort-of idealised drawing of the graph of μ referenced both in the theorem and in problem (1.1).

Theorem 2.1: Assume that f satisfies standard growth from below, i.e., f(t, u) > c ₀ u ^r for some c ₀ > 0 and r ≥ 1. Also assume that M satisfies M ( t ) ≤ c 2 t p 2 , where
p 2 = r − 1 q ⟺ q = r − 1 p 2 .

If, for some t ₀ ∈ (0, 1) we have μ t 0 ≈ 0 , then nonexistence of solutions for (1.1) can be guaranteed for all λ > λ ₀, where λ ₀ → 0⁺ as μ t 0 → 0 + .

Figure 1:

Illustration of a possible graph of μ corresponding to Theorem 2.1.

Notice that due to the selection of q in Theorem 2.1 it follows that if a(t) ≡ 1 and r − 1 p 2 ≥ 1 , then Theorem 2.1 applies to the problem

− M ‖ u ‖ L r − 1 p 2 r − 1 p 2 μ ( t ) u ″ ( t ) = λ f t , u ( t ) , t ∈ ( 0,1 ) .

We emphasise that Theorem 2.1 seems to record a property of the nonlocal structure itself. Indeed, as the title of this work suggests, what we demonstrate herein is that the nonlocal structure of (1.1) can be leveraged to cause a strong deregularising effect, wherein nonexistence of solutions is affected directly by the largeness or smallness of the nonlocal effects in the problem. Indeed, Theorem 2.1 can be seen as demonstrating the deregularising effect of the nonlocal element. We are not aware of such a phenomenon being observed before in either Kirchhoff-type or other nonlocal-type boundary value problems. Moreover, it seems rather surprising that if the function μ is very small (but positive), then existence fails for “nearly all” λ (see Example 2.5 for a specific example of this phenomenon).

With the primary motivation of this work presented, we conclude this section by placing this work within the broader existing literature on Kirchhoff-type equations. As mentioned at the beginning of this section, the classical Kirchhoff equation is a nonlocal parabolic-type PDE. The steady state version of this equation, that is,

− M ‖ D u ‖ L q q Δ u ( x ) = f x , u ( x ) , x ∈ Ω ⊂ R n ,

together with the related equation

− M ‖ u ‖ L q q Δ u ( x ) = f x , u ( x ) , x ∈ Ω ⊂ R n ,

have been studied extensively during the past couple of decades, both in space dimension n = 1 and n > 1. For example, in the case of the former equation, recent papers include those by Afrouzi, Chung, and Shakeri [28], Azzouz and Bensedik [29], Biagi, Calamai, and Infante [30], Boulaaras [31], Boulaaras and Guefaifia [32], Chung [33], Delgado, Morales-Rodrigo, Santos Júnior, and Suárez [34], Graef, Heidarkhani, and Kong [35], and Infante [36], [37]. On the other hand, in the case of the latter equation, recent papers include those by Alves and Covei [38], Corrêa [39], Corrêa, Menezes, and Ferreira [40], do Ó, Lorca, Sánchez, and Ubilla [41], Goodrich [42], Stańczy [43], Wang, Wang, and An [44], Yan and Ma [45], and Yan and Wang [46].

An important question in the study of nonlocal differential equations, Kirchhoff-type or otherwise, is how to manage the nonlocal structure in the differential equations. Generally, it has been standard to impose conditions on the nonlocal coefficient M that either apply globally or apply asymptotically; moreover, it has almost always been the case (for obvious reasons) that M be positive on its domain. There have been some recent attempts to relax these conditions, see, specifically, both [47] and [48], though these very interesting contributions are not necessarily part of a coherent general theory to weaken the conditions on M. Instead, the author [49], [50], [51], [52], [53], [54], along with Lizama [55], has provided a general theory to minimise the conditions on M (e.g., allowing M to be negative on all but a specified set of small measure), while simultaneously allowing for very general convolution-type coefficients as in (1.1). This was accomplished via topological fixed point theory and the classical fixed point index.

A conspicuous gap, however, in the large literature on Kirchhoff-like problems is nonexistence theorems as they relate to the parameter λ. Shibata [56], [57], [58], [59] has recently provided some interesting results in this direction; the author [60], [61] has also considered nonexistence but using a very different approach from that of Shibata. In this paper, we attempt to continue to fill this gap by focusing on nonexistence. As mentioned earlier, we specifically deduce a surprising nonexistence condition for a class of Kirchhoff-like equations, and this observation does not seem captured by either the results of Shibata or any other results in the literature.

All in all, then, we hope that the results of this paper shed some further light on the abstract mathematical theory of nonlocal differential equations. Especially, that with the proper operator theoretic view, one can leverage the nonlocal structure itself to generate some surprising and mathematically unintuitive results.

2 Nonexistence results for (1.1)

Throughout this paper we will work within the space C [ 0,1 ] , which is upgraded to a Banach space once we equip C [ 0,1 ] with the maximum norm  ‖ u ‖ ∞ ≔ max t ∈ [ 0,1 ] u ( t ) . When (1.1) is subjected to linear, homogeneous boundary conditions and M ( a * | u | q ) ( 1 ) μ ( s ) ≠ 0 , a.e. s ∈ [0, 1], a solution of (1.1) can be expressed as a fixed point of the operator T : C [ 0,1 ] → C [ 0,1 ] defined by

(2.1) ( T u ) ( t ) ≔ λ ∫ 0 1 M ( a * | u | q ) ( 1 ) μ ( s ) − 1 G ( t , s ) f s , u ( s ) d s ,

where the kernel G : [0, 1] × [0, 1] → [0, + ∞) of the operator is known as the Green’s function associated to the problem; the function G encodes the boundary data to which (1.1) is subjected. For example, in the classic case of Dirichlet boundary conditions, i.e., u(0) = 0 = u(1), it is known (see Erbe and Wang [27] for details) that

G ( t , s ) ≔ s ( 1 − t ) , 0 ≤ s ≤ t ≤ 1 t ( 1 − s ) , 0 ≤ t ≤ s ≤ 1 .

Our theory is general enough to permit us to use any boundary data for which G exists, subject to some mild regularity hypotheses, which will be detailed momentarily. Henceforth, by 1 we will denote the constant function 1 : R → { 1 } .

We next state the structural and regularity assumptions on the Green’s function G and the kernel a.

H1: The function G : [0, 1] × [0, 1] → [0, + ∞) is continuous, and there exist numbers 0 ≤ c < d ≤ 1 and η ₀ ∈ (0, 1] such that min t ∈ [ c , d ] G ( t , s ) ≥ η 0 G ( s ) , for each s ∈ [0, 1], where G is defined by

G ( s ) ≔ sup t ∈ [ 0,1 ] G ( t , s ) .

H2: The function a : [0, 1] → [0, + ∞) is L 1 ( 0,1 ) and a.e. positive.

Now we provide a nonexistence result, Theorem 2.1, for problem (1.1) subject to the boundary data captured by G. Note that in both the statement and the proof of Theorem 2.1 we use the (slight abuse of) notation

a * G E 0 ( ⋅ , s ) q ( 1 ) ≔ ∫ 0 1 a ( 1 − t ) ∫ E 0 ∩ [ c , d ] G ( t , s ) d s q d t ,

for some measurable set E ₀. In addition, in the statement of Theorem 2.1, we denote by |⋅| the usual Lebesgue measure.

Theorem 2.1.

Assume that conditions (H1)–(H2) hold. In addition, assume that:

The function f : [0, 1] × [0, + ∞) → [0, + ∞) is continuous and satisfies the lower growth condition f(t, u) > c ₀ u ^r, where (t, u) ∈ [0, 1] × [0, + ∞), for some numbers r ≥ 1 and c ₀ > 0.
The function μ : [0, 1] → (0, + ∞) is continuous. Moreover, there exists a measurable, nonempty set E₀ ⊆ [0, 1], where E 0 ∩ [ c , d ] > 0 , together with a number ɛ > 0 such that μ(t) < ɛ for each t ∈ E ₀.
The function M : [ 0 , + ∞ ) → R is continuous and satisfies the upper growth condition
M ( t ) ≤ c 2 t p 2 , t ≥ 0 ,
where p 2 = r − 1 q .

Define the number λ ₀ > 0 by

(2.2) λ 0 ≔ ε p 2 c 0 η 0 r c 2 − 1 1 ( a * 1 ) ( 1 ) − r q a * G E 0 ( ⋅ , s ) q ( 1 ) − 1 q .

Then for each λ > λ ₀ problem (1.1) subject to the boundary data supplied by G has no nontrivial positive solutions. In particular,

lim ε → 0 + λ 0 = 0 .

Proof.

In order to prove this theorem, we will appeal to the general technique introduced in the proof of [60, Theorem 2.1] and then subsequently utilised in [61, Theorem 2.1]. However, we wish to emphasise that, due to the fact that μ is not necessarily a constant map, some important subtleties are introduced here that did not arise in [60], [61]. In fact, the proof is, on the whole, rather different from either [60] or [61].

So, suppose for contradiction that (1.1) has a positive solution, say u ₀. Then there exists ρ ₀ > 0 such that

(2.3) μ ( s ) a * | u 0 | q ( 1 ) ︸ = ρ 0 = ρ 0 μ ( s ) ,

for each s ∈ [0, 1]. Note that ρ ₀ ≠ 0 may be assumed without loss of any generality. Indeed, if ρ ₀ = 0, then

0 = a * | u 0 | q ( 1 ) = ∫ 0 1 a ( 1 − s ) u 0 ( s ) q d s ,

which, since a > 0 a.e. and u ₀ is not zero identically (because, by assumption, it is a nontrivial solution), is not possible. Hence, ρ ₀ > 0. There, thus, are three possibilities for each given number s ∈ [0, 1].

M ρ 0 μ ( s ) > 0
M ρ 0 μ ( s ) = 0
M ρ 0 μ ( s ) < 0

We will complete the proof by considering the following three possibilities, which are exhaustive; note that by the assumed continuity of μ and M, it is impossible to switch from case (A) to case (C) without “passing through” case (B).

We are always in case (A).
There exists at least one point 0 < s < 1 at which we are in case (B).
We are always in case (C).

Let us first suppose that for every s ∈ [0, 1] we are in case (A). Note that in this case u ₀ satisfies the equation Tu ₀ ≡ u ₀, i.e., u ₀ is a fixed point of the operator T introduced in (2.1).

Recall that by assumption M ( t ) ≤ c 2 t p 2 . Consequently, for each 0 ≤ s ≤ 1 we see that

(2.4) M a * | u 0 | q ( 1 ) μ ( s ) − 1 = M ρ 0 μ ( s ) − 1 ≥ c 2 ρ 0 μ ( s ) p 2 − 1 = c 2 − 1 ρ 0 − p 2 1 μ ( s ) p 2 ,

where (2.3) has been used. Therefore, by combining (2.3), (2.4), and using the fact that u ₀ is a positive solution so that T u 0 ≡ u 0 ≡ u 0 ,

(2.5) ρ 0 = a * | u 0 | q ( 1 ) = a * u 0 q ( 1 ) = ∫ 0 1 a ( 1 − t ) ( T u 0 ) q ( t ) d t = ∫ 0 1 a ( 1 − t ) λ ∫ 0 1 M a * | u 0 | q ( 1 ) μ ( s ) − 1 G ( t , s ) f s , u ( s ) d s q d t ≥ λ q ∫ 0 1 a ( 1 − t ) ∫ 0 1 c 2 − 1 ρ 0 − p 2 1 μ ( s ) p 2 G ( t , s ) f s , u ( s ) d s q d t = 1 c 2 ρ 0 p 2 q λ q ∫ 0 1 a ( 1 − t ) ∫ 0 1 1 μ ( s ) p 2 G ( t , s ) f s , u ( s ) d s q d t .

Now, keeping in mind that μ(t) > 0 for all t ∈ [0, 1], recall that, by assumption,

(2.6) μ ( t ) < ε ⟺ 1 μ ( t ) > ε − 1 whenever t ∈ E 0 ⊆ [ 0,1 ] ,

where E 0 ∩ [ c , d ] > 0 so that E 0 > 0 . Therefore, using (2.6) in (2.5) yields the estimate

(2.7) ρ 0 ≥ 1 c 2 ρ 0 p 2 q λ q ∫ 0 1 a ( 1 − t ) ∫ 0 1 1 μ ( s ) p 2 G ( t , s ) f s , u ( s ) d s q d t ≥ 1 c 2 ρ 0 p 2 q λ q ∫ 0 1 a ( 1 − t ) ∫ E 0 1 μ ( s ) p 2 G ( t , s ) f s , u ( s ) d s q d t ≥ 1 c 2 ρ 0 p 2 q ε − p 2 q λ q ∫ 0 1 a ( 1 − t ) ∫ E 0 G ( t , s ) f s , u ( s ) d s q d t .

In order to estimate (2.7) from below, we need some additional preliminary estimates. First of all, since u ₀ ≡ Tu ₀, using (H1) we see that

min t ∈ [ c , d ] u 0 ( t ) = min t ∈ [ c , d ] T u 0 ( t ) = min t ∈ [ c , d ] λ ∫ 0 1 M ρ 0 μ ( s ) − 1 G ( t , s ) f s , u 0 ( s ) d s ≥ λ ∫ 0 1 M ρ 0 μ ( s ) − 1 min t ∈ [ c , d ] G ( t , s ) f s , u 0 ( s ) d s ≥ λ η 0 ∫ 0 1 M ρ 0 μ ( s ) − 1 G ( s ) f s , u 0 ( s ) d s ≥ η 0 ‖ T u 0 ‖ ∞ = η 0 ‖ u 0 ‖ ∞ .

In other words, u ₀, by virtue of being a fixed point of the operator T, satisfies the Harnack-like inequality

min t ∈ [ c , d ] u 0 ( t ) ≥ η 0 ‖ u 0 ‖ ∞ ,

exactly as G does. Second, we recall assumption (1) in the statement of the theorem, namely, that f satisfies

f ( t , u ) > c 0 u r

for each (t, u) ∈ [0, 1] × [0, + ∞). With the preceding two estimates in mind, we can further estimate (2.7) from below by

(2.8) ρ 0 ≥ 1 c 2 ρ 0 p 2 q ε − p 2 q λ q ∫ 0 1 a ( 1 − t ) ∫ E 0 G ( t , s ) f s , u ( s ) d s q d t > 1 c 2 ρ 0 p 2 q ε − p 2 q λ q ∫ 0 1 a ( 1 − t ) ∫ E 0 G ( t , s ) c 0 u ( s ) r d s q d t ≥ 1 c 2 ρ 0 p 2 q ε − p 2 q λ q ∫ 0 1 a ( 1 − t ) ∫ E 0 ∩ [ c , d ] G ( t , s ) ⋅ c 0 η 0 r ‖ u ‖ ∞ r d s q d t = c 0 η 0 r c 2 ρ 0 p 2 q ε − p 2 q ‖ u ‖ ∞ r q λ q ∫ 0 1 a ( 1 − t ) ∫ E 0 ∩ [ c , d ] G ( t , s ) d s q d t ,

where we have used the assumption that

E 0 ∩ [ c , d ] > 0 .

Next, notice that

ρ 0 = a * | u 0 | q ( 1 ) ≤ ‖ u 0 ‖ ∞ q ( a * 1 ) ( 1 )

so that

(2.9) ‖ u ‖ ∞ ≥ ρ 0 ( a * 1 ) ( 1 ) 1 q .

Note that (2.9) is well defined since a(t) > 0, a.e. t ∈ [0, 1]. Thus, substituting (2.9) into inequality (2.8) yields

(2.10) ρ 0 > c 0 η 0 r c 2 ρ 0 p 2 q ε − p 2 q ‖ u ‖ ∞ r q λ q ∫ 0 1 a ( 1 − t ) ∫ E 0 ∩ [ c , d ] G ( t , s ) d s q d t ︸ ≕ a * G E 0 ( ⋅ , s ) q ( 1 ) ≥ c 0 η 0 r c 2 ρ 0 p 2 q ε − p 2 q ρ 0 ( a * 1 ) ( 1 ) 1 q r q λ q a * G E 0 ( ⋅ , s ) q ( 1 ) = c 0 η 0 r c 2 q ρ 0 − p 2 q r + 1 ( a * 1 ) ( 1 ) r ε − p 2 q λ q a * G E 0 ( ⋅ , s ) q ( 1 ) .

Now, a contradiction will be attained from (2.10) if the right-hand side of the inequality exceeds ρ ₀. This happens if and only if

(2.11) λ > ρ 0 p 2 + 1 − r q ε p 2 c 0 η 0 r c 2 − 1 1 ( a * 1 ) ( 1 ) − r q a * G E 0 ( ⋅ , s ) q ( 1 ) − 1 q .

But recall from assumption (3) in the statement of the theorem that

p 2 = r − 1 q ⟺ p 2 + 1 − r q = 0 .

Therefore,

ρ 0 p 2 + 1 − r q = 1 ,

and so, inequality (2.11) is equivalent to

(2.12) λ > ε p 2 c 0 η 0 r c 2 − 1 1 ( a * 1 ) ( 1 ) − r q a * G E 0 ( ⋅ , s ) q ( 1 ) − 1 q .

In other words, if inequality (2.12) is true, then the desired contradiction will be obtained. But as (2.12) is equivalent to λ > λ ₀, which is precisely what was assumed in the statement of the theorem, we conclude from (2.10)– (2.12) that

ρ 0 ≥ c 0 η 0 r c 2 q 1 ( a * 1 ) ( 1 ) r ε − p 2 q λ q a * G E 0 ( ⋅ , s ) q ( 1 ) > ρ 0 ,

which is the desired contradiction. Thus, we conclude that Tu cannot have a nontrivial, positive fixed point in this case. Equivalently, equation (1.1), together with the boundary data inherited from G, has no positive solution.

The preceding argument is valid provided that case (A) holds for each s ∈ [0, 1]. Now suppose that, for an alleged positive solution u ₀, for some s ∈ [0, 1] we are in case (B). Now, the operator representation, u ₀ ≡ Tu ₀, is invalid in this case since the nonlocal coefficient is zero for at least one t, say t ₀. Nonetheless, u ₀, by definition, must satisfy the differential equation. Since u ₀ must satisfy the differential equation at t = t ₀, it follows that

0 = − M a * | u 0 | q ( 1 ) μ t 0 ︸ = 0 u 0 ″ t 0 = λ f t 0 , u 0 t 0 > λ c 0 u 0 t 0 r ,

from which it follows that

(2.13) u 0 t 0 < 0 .

Since u ₀ is an alleged positive solution of (1.1), inequality (2.13) is a contradiction. Consequently, we deduce that if we are ever in case (B), then a contradiction follows. Thus, this regime cannot occur.

Finally, suppose that case (C) holds at each s ∈ [0, 1]. Note that in this case, the identity Tu ₀ ≡ u ₀ is valid. Therefore, keeping in mind that u ₀ is a positive solution, we deduce that

0 ≤ u 0 ( t ) = λ ∫ F 0 M ( a * u ) ( 1 ) μ ( s ) − 1 ︸ < 0 G ( t , s ) f s , u ( s ) ︸ > 0 d s < 0 ,

which is a contradiction with the assumption that u ₀ is a positive solution. Therefore, this case is dispatched, too.

All in all, then, in either regime we are led to a contradiction. Therefore, we conclude that, under the hypotheses of the theorem, whenever λ > λ ₀, problem (1.1) cannot have a positive solution when subject to the boundary data inherited by G. And this completes the proof of the theorem.□

Figure 2 illustrates the surface defined by the graph of the function p 2 : [ 1 , + ∞ ) × ( 0 , + ∞ ) → R defined by

p 2 ( r , q ) ≔ r − 1 q ;

that is, the surface indicates how p ₂ must change relative to a given pair (r, q) in order to achieve

p 2 + 1 − r q = 0 .

$Figure 2: Illustration of the graph of p 2 ( r , q ) = r − 1 q ${p}_{2}\left(r,q\right)=\frac{r-1}{q}$ for 1 ≤ r ≤ 3 and 1 ≤ q ≤ 3.$

Figure 2:

Illustration of the graph of p 2 ( r , q ) = r − 1 q for 1 ≤ r ≤ 3 and 1 ≤ q ≤ 3.

Thus, Theorem 2.1 applies when the parameter triple q , r , p 2 lives on the surface in Figure 2. If we further assume both that

r − 1 p 2 ≥ 1

and that

a ≡ 1 ,

then we can obtain the following corollary, which provides a nonexistence result for problem (1.1) in case the nonlocal element is interpreted as a L r − 1 p 2 -norm.

Corollary 2.2.

Assume that the hypotheses in the statement of Theorem 2.1 hold. In addition, assume that

r − 1 p 2 ≥ 1 .

Define the number λ ₀ > 0 as in (2.2). Then for each λ > λ ₀ the nonlocal equation

− M ‖ u ‖ L r − 1 p 2 r − 1 p 2 μ ( t ) u ″ ( t ) = λ f t , u ( t ) , t ∈ ( 0,1 )

subject to the boundary data suppled by G has no nontrivial positive solutions.

Remark 2.3.

Let us consider the import of Theorem 2.1. The theorem asserts that if the relationship p 2 = r − 1 q holds, then as ɛ → 0⁺, the nonlocal equation will have no positive solutions for larger and larger ranges of the parameter λ. In other words, if μ is very close to zero on a set of positive measure, then this can have a strong deregularising effect on the problem inasmuch as existence of solution becomes more difficult to achieve.

Remark 2.4.

We wish to point out that compared to [60, Theorem 2.1], which was also a nonexistence result, our results here are somewhat different. First of all, the results presented here do not require the stronger regularity hypothesis of [60, H2.3]. In addition, while here we assume the upper growth condition on M, which was not assumed in [60, Theorem 2.1], this growth assumption allows us to clarify the effect of μ on the nonexistence of solution – something that was not addressed in either [60] or, for that matter, in the recent papers of Shibata [56], [57], which also concern nonexistence of solution to nonlocal ODEs.

We conclude with an example, which illustrates the application of Corollary 2.2.

Example 2.5.

To illustrate the fundamental import of the nonexistence theorem without unnecessary complications, let us take a ≡1. Suppose that M and f satisfy, respectively, the bounds

M ( t ) ≤ t

and

f ( t , u ) > u 2 .

That is, we choose p 2 = 1 2 and r = 2 so that

q = 2 − 1 1 2 = 2 ≥ 1 .

In addition, put

μ ( t ) ≔ ε 1 t − 1 2 2 + ε 0 ,

for some ɛ ₀, ɛ ₁ > 0. Finally, define G by

G ( t , s ) ≔ s ( 1 − t ) , 0 ≤ s ≤ t ≤ 1 , t ( 1 − s ) , 0 ≤ t ≤ s ≤ 1 .

Note that on account of the selection of p ₂, r, and G, it follows that we are considering the problem

(2.14) − M ‖ u ‖ L 2 2 μ ( t ) u ″ ( t ) = λ f t , u ( t ) , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = 0 .

Now, taking c ≔ 1 4 and d ≔ 3 4 , one can calculate that

∫ 0 1 ∫ 1 4 3 4 G ( t , s ) d s 2 d t = 29 30720 .

In addition, it is known (see Erbe and Wang [27], for example) that for the Green’s function selected as above

η 0 ≔ min { c , 1 − d } = min 1 4 , 3 4 = 1 4 .

Also note that if we take E 0 ≔ 1 4 , 3 4 ≡ [ c , d ] , then

μ ( t ) ≤ 1 16 ε 1 + ε 0 , whenever t ∈ E 0 ,

where 1 16 ε 1 + ε 0 is the number ɛ in the statement of Corollary 2.2. Therefore, we conclude from Corollary 2.2 that boundary value problem (2.14) has no nontrivial positive solutions whenever

λ > λ 0 ≔ 16 30720 29 1 16 ε 1 + ε 0 .

In particular, since

16 30720 29 1 16 ε 1 + ε 0 → 0 as ε i → 0 , i ∈ { 0,1 } ,

it follows that the parameter interval 0 , λ 0 for which (2.14) may have positive solutions satisfies m ( 0 , λ 0 ) → 0 , where m denotes the usual Lebesgue measure.

Remark 2.6.

We wish to emphasise some of the advantages of our approach in this work relative to other nonexistence results in the literature.

We note that although Example 2.5 utilised Dirichlet boundary data, such boundary data is not a requirement of our theory. Indeed, Theorem 2.1 permits any boundary data such that the associated Green’s function, G, satisfies condition (H1).
Further to the previous point, some methodologies are inherently restrictive as to the boundary data considered – see, for example [62], as but one of many examples.
As compared to the recent nonexistence-type results of Shibata [56], [57], [58], [59], which were mentioned in Section 1, our results here treat nonlocal equations within a rather more general framework since we allow for the convolutional structure in the nonlocal coefficient of (1.1). In fact, other than our own nonexistence results [60], [61] we are not aware of any such results in this way. But neither [60] nor [61] studies the effect of any time-varying modulation within the nonlocal coefficient.

All in all, then, not only are the nonexistence results presented herein new on account of the modulating map μ, the general methodology is also rather more flexible than some competing approaches.

Corresponding author: Christopher S. Goodrich, School of Mathematics and Statistics, UNSW Sydney, Sydney, NSW 2052, Australia, E-mail: c.goodrich@unsw.edu.au

I dedicate this paper in loving memory of Maddie Goodrich (16 March 2002–16 March 2020).

Acknowledgments

I gratefully acknowledge both the careful reading and the helpful suggestions by the three anonymous referees.

Research ethics: Not applicable.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The author states no conflict of interest.
Research funding: None declared.
Data availability: Not applicable.

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Received: 2023-12-16

Accepted: 2024-04-08

Published Online: 2024-06-12

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Keywords for this article

nonlocal differential equation; nonexistence; finite convolution

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