Nonlocal Differential Equations with Convolution Coefficients and Applications to Fractional Calculus

1 Introduction

In this paper, we consider a class of nonlocal differential equations, of which a model case is

where by (b*u)⁢(t) we denote the finite convolution

So, in other words, problem (1.1) means

Thus, we see that the coefficient function A in (1.1) acts as a nonlocal coefficient being as it integrates the product s↦b⁢(1-s)⁢(u⁢(s))q over s∈[0,1]. In fact, (1.1) is but one specific example we consider since our methodology applies to differential equations of orders other than two (including fractional orders) and with a variety of boundary conditions.

Nonlocal differential equations have been studied extensively over the years. In some cases these nonlocal elements occur in the boundary conditions – for example, as studied by Cabada, Infante, and Tojo [12], Goodrich [25, 26, 28, 27], Graef and Webb [41], Infante and Pietramala [45, 46, 47, 48], Infante, Pietramala, and Tenuta [49], Jankowski [50], Karakostas and Tsamatos [51, 52], Webb [62], Webb and Infante [64, 63, 65], and Yang [68, 69, 70]. And in other cases the nonlocal elements occur in the differential equation itself as is the case in problem (1.1). Perhaps one of the best known examples of the latter situation is the class of Kirchhoff-type equations, of which an example in the one-dimensional case is

Kirchhoff-type equations similar to (1.3) and their elliptic PDE analogue have been studied by many authors in recent years – e.g., see papers by Afrouzi, Chung, and Shakeri [1], Azzouz and Bensedik [5], Boulaaras [10], Boulaaras and Guefiafia [11], Chung [15], Goodrich [34], and Infante [43, 44]. A related type of problem is, for q>0 (though usually q≥1),

which has also been extensively studied both in the one-dimensional case as above as well as the elliptic PDE analogue – for example, see the papers by Alves and Covei [2], Aly [3], Bavaud [6], Biler, Krzywicki, and Nadzieja [7], Biler and Nadzieja [8, 9], Caglioti, Lions, Marchioro, and Pulvirenti [14], Corrêa [17], Corrêa, Menezes, and Ferreira [18], do Ó, Lorca, Sánchez, and Ubilla [21], Goodrich [35], Rosier and Rosier [56], Stańczy [59], Wang, Wang, and An [61], Yan and Ma [66], and Yan and Wang [67]. As explained in [43, (1.2)], these types of problems arise in the study of the mean field equation – i.e.,

We note that an approach involving topological fixed point theory is a very popular approach in these types of problems (for example, see [17, 35, 34, 38, 43, 59, 61, 67]), but that other methods such as bifurcation theory [66] and methods involving sub- and super-solutions are also used [10, 11, 18].

One might suspect that the realization of the nonlocal coefficient in (1.1) as a convolution is a gratuitous generalization. But, in fact, there is a very good reason to adopt this very general viewpoint. It turns out that many important operators in applied mathematics can be realized in the form of a convolution-type operator. A principal example is the class of fractional differential operators. Fractional-order equations both in the ODEs and PDEs case have received much attention in recent decades due both to their interesting applications as well as their utility in pure mathematics (e.g., in regularity theory). But a fractional differential operator is straightforwardly a finite convolution. Indeed, if, for example, we choose the kernel b in (1.2) to be defined by

where α>0, then

which is the α-th order Riemann–Liouville fractional integral – see any of the textbooks by Goodrich and Peterson [40], Kilbas, Srivastava, and Trujillo [53], and Podlubny [55] for additional details on the fractional calculus; see also the recent article by Lan [54]. So, for example, the formulation of problem (1.1) allows for nonlocal coefficients in the form of fractional integrals.

In addition to providing a sort-of unifying framework for differential equations with nonlocal coefficients, we also utilize a topological approach that allows for weaker-than-normal conditions on the coefficient A appearing in problem (1.1). In general, it is usually assumed that A⁢(t)>0 for all t; for example, each of Afrouzi, Chung, and Shakeri [1], Alves and Covei [2], Azzouz and Bensedik [5], Boulaaras [10], Boulaaras and Guefiafia [11], Cabada, Infante and Tojo [13], Chung [15], Corrêa [17], Corrêa, Menezes, and Ferreira [18], do Ó, Lorca, Sánchez, and Ubilla [21], Stańczy [59], Wang, Wang, and An [61], Yan and Ma [66], and Yan and Wang [67] make this assumption. Furthermore, it is often assumed that A, in addition, satisfies some sort of global growth or monotonicity conditions; for example:

1. Both Corrêa [17, (6) and (7)] and Yan and Wang [67, Theorem 4.1, p. 84] impose monotonicity as well as limit-type conditions, whereas both Wang, Wang, and An [61, Condition (H1), p. 2] and Yan and Ma [66, p. 7] impose a monotonicity condition.

2. Stańczy [59, Theorem 2.2] imposes a limit-type condition.

3. do Ó, Lorca, Sánchez, and Ubilla [21, Condition y, p. 299] impose limit-type conditions.

These same types of conditions are almost always imposed in the Kirchhoff case, too – i.e., problem (1.3). The reason for the near universality of these various conditions (especially the positivity condition) is that authors generally have no direct control over the argument of the nonlocal element. Thus, global conditions are generally required to subvert the problem this causes.

Now, it is important to point out that there do exist a handful of papers in which some degree of degeneracy is allowed on the nonlocal coefficient. For example, Ambrosetti and Arcoya [4] considered the following nonlocal elliptic Kirchhoff-type PDE:

Although the nonlocal coefficient function t↦M⁢(t) was required to be positive for 0<t<+∞, M was, nonetheless, allowed to vanish either at t=0 or as t→+∞. Similarly, Santos Júnior and Siciliano [58] considered problem (1.4). As an improvement to the results of [4], they allowed the nonlocal coefficient to vanish along a finite sequence of values and to be positive in-between. More precisely, it was assumed that there exists {tj}j=1N⊆(0,+∞) such that M⁢(tj)=0, j∈{1,2,…,N}, and M⁢(t)>0 for t∈(tj-1,tj) with j∈{1,2,…,N} and t0:=0. The behavior of M was unrestricted for t>tN. Finally, an additional condition involving lower and upper bounds on the quantity

was imposed. Santos Júnior and Siciliano then showed that N positive solutions could be deduced by means of variational methods. Finally, in a very recent paper by Delgado, Morales-Rodrigo, and Santos Júnior [20] the following nonlocal elliptic PDE was considered:

They then assumed that M⁢(t)>0 for t∈(0,+∞)∖{t0}, where t0>0 is some real number. So, once again, the nonlocal coefficient is allowed to vanish – in this case at a single point.

By contrast, we require only pointwise-type conditions on A; see, for example, conditions (2.4) and (2.5) in the statement of Theorem 2.8. Moreover, regarding the three papers mentioned in the preceding paragraph, unlike [4, 20] we allow for A to vanish (or be negative) on a set of infinite measure, and unlike [58] the nonlocal coefficient does not need to be positive on a set, which has 0 as an accumulation point, and, furthermore, it does not need to satisfy a sort-of integral positivity condition. This is all a consequence of using the nonstandard order cone

where C0>0 is a constant defined in Section 2, together with the open set

Thus, we restrict out attention to those elements u of 𝒞⁢([0,1]) that cause the functional

to be coercive with coercivity constant C0. The key fact is that for u∈∂⁡X^ρ it holds that (b*uq)⁢(1)=ρ, which for such elements of 𝒦 gives us direct control over the argument of A. The sets 𝒦 and X^ρ are generalizations, adapted to the convolutional setting, of ideas introduced in [35].

In summary, 𝒦 and X^ρ in tandem give us specific pointwise control over the convolution (b*uq)⁢(1), which in turn facilitates the more relaxed pointwise conditions in our existence theorems. Although the general approach of using these types of nonstandard order cones in nonlocal differential equations was initiated by the author in [30, 31, 32, 33, 35] in the non-convolutional setting, accommodating the convolutional coefficient requires a somewhat different approach (e.g., here we must use reverse Hölder inequalities, which are not required at all in the non-convolutional setting such as studied in [17, 18, 21, 35, 59, 67]; and the existence argument itself proceeds differently due to the presence of the convolution coefficient – once again, different than the approach in other studies). Thus, the case studied here is not merely a trivial extension of the non-convolutional case. Moreover, although the author very recently [36] studied a problem similar to (1.1), in that case 0<q<1, which required some different techniques than used in the case q≥1 treated here.

Finally, the outline of the remainder of the paper is as follows. In Section 2, we collect some preliminary lemmata concerning, especially, some topological properties of the set X^ρ introduced earlier. We then state and prove two general existence theorems for problem (1.1). Then, in Section 3, we apply our results to two specific boundary value problems. The first is problem (1.1) subject to Dirichlet boundary conditions and in which the convolution reduces to a fractional integral. The second is a fractional differential equation with fractional integral coefficient together with boundary conditions that are related to the (n-1,1)-conjugate problem – see either Davis and Henderson [19] or Eloe and Henderson [22] for some background on the (n-k,k)-conjugate problem. These two examples thus demonstrate how the abstract theory can be applied to the theory of fractional-order boundary value problems in new ways.

2 Preliminary Lemmata and Existence Theory

In order to study problem (1.1) with the possibility of a variety of boundary conditions, we will consider the operator T:𝒞⁢([0,1])→𝒞⁢([0,1]) defined by

where it will always be assumed that q≥1. Then, depending upon the choice of G, a fixed point of T corresponds to the solution of a specific differential equation with specified boundary data. For example, if

then a fixed point of T will be a solution of the following nonlocal second-order ODE subject to Dirichlet boundary conditions:

For the given kernel G we will denote by 𝒢 the function defined by

We already introduced the finite convolution notation in (1.2). Another useful notation, which will occur frequently in this paper, is to denote by 𝟏 the function on [0,1] that is identically 1, i.e.,

In addition, given a continuous function

and numbers 0≤a<b≤1 and 0≤c<d<+∞, we will denote by f[a,b]×[c,d]m and f[a,b]×[c,d]M, respectively, the numbers

and

Equip the space 𝒞⁢([0,1]) with the usual maximum norm, which we denote by ∥⋅∥. As suggested in Section 1, we will consider solutions of the operator equation T⁢u=u from the positive cone

in which the coercivity constant C0 is defined by

where S0⊆[0,1] is a set of full measure dependent upon the choice of the kernel G appearing in T; see condition (H3) below. Furthermore, in what follows and as also mentioned in Section 1, it will be useful to make use of the following set in conjunction with the cone 𝒦:

Finally, the following conditions will be imposed on the functions in the operator T:

1. The functions A:[0,∞)→ℝ and f:[0,1]×[0,∞)→[0,∞) are continuous.

2. The function b:[0,1]→[0,∞) is L1⁢([0,1]) and has the property that

   1. b ⁢ ( t ) ≠ 0 for a.e. t∈[0,1];

   2. ( b * 𝟏 ) ⁢ ( 1 ) > 0 .

3. The function G:[0,1]×[0,1]→[0,∞) is continuous. Furthermore, there exists a set S0⊆[0,1] of full measure such that

   C 0 := min ⁡ { ( b * 𝟏 ) ⁢ ( 1 ) , inf s ∈ S 0 ⁡ 1 𝒢 ⁢ ( s ) ⁢ ( b * G ⁢ ( ⋅ , s ) ) ⁢ ( 1 ) } ∈ ( 0 , + ∞ ) .

Our first preliminary lemma demonstrates that T is a reflexive operator on the closure of a relevant solid annular subset of 𝒦. This is essential for the proper application of the fixed point theorem, Lemma 2.7, later in this section.

The following topological property of X^ρ will be essential for the proper application of the topological fixed point theory.

We next characterize a relationship between an element of ∂⁡X^ρ and its norm.

Our final preliminary lemma is a fixed point theorem, which we will use in the existence theorem. For further details on this and related results one may consult, for example, Cianciaruso, Infante, and Pietramala [16, Lemma 2.3], Guo and Lakshmikantham [42], Infante, Pietramala, and Tenuta [49], or Zeidler [71].

Now we state and prove our main existence result.