Compactness of nonlinear integral operators with discontinuous and with singular kernels

This paper establishes compactness of nonlinear integral operators in the space of continuous functions. One result deals with operators whose kernel can have jumps across a ﬁnite number of curves, which typically arise from the study of ordinary diﬀerential equations with boundary conditions of local or nonlocal type. Several other results deal with operators whose kernels have a singularity, which arise from the study of fractional diﬀerential equations. We motivate the study of these integral equations by discussing some initial value problems for fractional diﬀerential equations of Caputo and Riemann-Liouville type. We prove a compact embedding theorem for fractional integrals in order to give a new treatment for the singular kernel case.


Introduction
In studying existence of solution of differential equations it is often convenient to consider the corresponding problem involving a nonlinear integral operator. The advantage is that integral operators are usually compact.
We will study nonlinear integral operators in the space of continuous functions such as a Fredholm type operator  The important point in this paper is that the kernel G will be allowed to have discontinuities along certain curves. Integral operators such as N F arise in the study of boundary value problems for ordinary and some fractional differential equations, where G is the Green's function and often can have finite jumps. The operator N V occurs when initial value problems are studied, often en route to a boundary value problem. In particular, when it arises from the study of fractional differential equations, the kernel is of the form G(t, s) = (t − s) α−1 and has a singularity when 0 < α < 1.
For a problem such as u (t) +f (t, u(t)) = 0, t ∈ [0, T ] with some boundary conditions (BCs), the problem is often equivalent to finding fixed points of an integral operator Nu(t) = T 0 G(t, s)f (s, u(s)) ds in the space C[0, T ]. The Green's function G is frequently continuous, the integral operator N is compact and continuous (completely continuous) and many theories can be applied to determine existence of unique or of multiple fixed points of N under suitable behaviour of the function f .
There are situations when the kernel G is not continuous at all points but does satisfy Carathéodory conditions, G(t, s) is continuous in t for almost every s and is dominated by an L 1 function of s. Continuity of the function f can also be weakened to Carathéodory conditions. The criterion for compactness in C[0, T ] is the Arzelà-Ascoli theorem, N must map bounded sets into bounded equicontinuous sets. A typical result, often quoted, is Proposition V.3.1 of R.H. Martin's book [11]; the result there was refined in [14,15].
When the problem is u (t) + f (t, u(t), u (t)) = 0, t ∈ [0, T ] with some BCs, the corresponding integral operator N must be studied in the space C 1 [0, T ] so it is required that (Nu) (t) = T 0 ∂ t G(t, s)f (s, u(s), u (s)) ds also defines a compact map. Even in simple cases the kernel ∂ t G(t, s) may not be continuous, a typical situation is when there is a jump across the line s = t, thus it is not continuous in t for almost all s and the previous result is not applicable. A simple example is the problem u + f (t, u, u ) = 0, u(0) = 0, u (1) and ∂ t G is not defined on the diagonal s = t; there is a finite jump discontinuity with both left and right derivatives existing on the diagonal. In a simple case, such as the example just given, it is not hard to carefully prove that the Arzelà-Ascoli theorem works even with the jump. The proof is often omitted, for example [10] study some fourth order problems with third derivative dependence and simply claim the previous result applies, citing [15]. However, this is not applicable to their two problems, the third derivative of the Green's function has a jump on the diagonal s = t; direct use of the Arzelà-Ascoli theorem would work.
However, there are other problems where this is less simple. One goal of this paper is to give a general result which allows for the kernel to have a finite number of discontinuities across certain curves, which will include the simple case of a jump across the 'curve' s = t. One relatively weak hypothesis we use is that For problems involving higher order derivatives, the highest order derivative operator may often have discontinuities in the corresponding partial derivative of the Green's function. López-Somoza and Minhós [9] studied some integral equations of the form with nonlinearities depending on derivatives up to order m. To get compactness in the space C m [0, 1] of m times continuously differentiable functions they used the following hypotheses on K m , the m-th partial derivative with respect to t of the kernel: For every ε > 0 and every fixed τ ∈ [0, 1], there exist a set Z τ ⊂ [0, 1] with measure zero and some δ > 0 such that |t − τ | < δ implies that |K m (t, s) − K m (τ, s)| < ε, for all s ∈ [0, 1] \ Z τ such that s < min{t, τ } or s > max{t, τ }.
Our result will apply to this example with a much simpler looking assumption. The second type of integral operator is with a singular kernel and arises in the study of fractional equations. These often involve the Riemann-Liouville fractional integral For α ≥ 1 the kernel (t − s) α−1 is continuous for s ≤ t, and then we may write and the result of first type applies. The trickier and more interesting case is when 0 < α < 1. A different result is necessary since the kernel for 0 < α < 1 does not satisfy the bound employed in the first type of problem. When α > 1 and f depends on (possibly fractional) derivatives the (fractional) derivative operator has also to be proved compact. Such a derivative operator is then often one having a singular kernel, of the type similar to the case with α < 1.
We motivate the study of these integral equations by discussing some initial value problems for fractional differential equations of Caputo and Riemann-Liouville type. We will give some new concise proofs of compactness of these operators, and more general ones, by proving and utilizing some new properties of fractional integrals, including a compact embedding theorem for fractional integrals, in order to give a new treatment for the singular kernel case.
This approach avoids the longer calculations using the Arzelà-Ascoli theorem that have previously been used on an ad hoc basis for such problems.

Some function spaces
For simplicity we consider functions defined on an arbitrary finite interval [0, T ], which is, by a simple change of variable, equivalent to any finite interval. In this paper all functions are assumed to be measurable, and all integrals are Lebesgue integrals, even if not explicitly stated. The space of functions that are continuous on [0, T ] is denoted by C[0, T ] or sometimes simply C and is a Banach space when endowed with the supremum norm u ∞ := max t∈[0,T ] |u(t)|. For n ∈ N we will write C n = C n [0, T ] to denote those functions u whose n-th derivative u (n) is continuous on [0, T ].
The space of absolutely continuous functions is denoted AC = AC[0, T ]. For n ∈ N, AC n = AC n [0, T ] will denote those functions u whose n-th derivative u (n) is in AC[0, T ], hence u (n+1) (t) exists for a.e. t and is an L 1 function.
A note of caution: some authors denote this space as AC n+1 . The space AC is the appropriate space for the fundamental theorem of the calculus for Lebesgue integrals. In fact, we have the following equivalence. For η ≥ −1, we define a space which allows pointwise singularities at 0, which, as far as we are aware is newly introduced here, and will be utilised in this paper.
we define the norm f η = g ∞ . The spaces with a singularity at zero are L −η where η > 0. For situations that arise from studying fractional derivatives, it is the case L −η with 0 ≤ η < 1 that is important, where η < 1 is required to ensure that certain integrals exist. We recall similar spaces of functions that are continuous except at 0. The space C η is defined by When endowed with the norm f η , C η is a Banach space. The spaces with a singularity at zero are C −η with η > 0. Functions f in C −η with 0 ≤ η < 1 are integrable and are continuous on (0, T ] with lim t→0+ t η f (t) exists. It is an appropriate class for the study of Riemann-Liouville fractional integrals and derivatives, see for example [1]. Note that, for 0 ≤ γ ≤ η < 1, Moreover, for 0 < η < 1, the inclusion L −η ⊂ L p holds when p > 1/η. When 0 < η < 1, the main difference between the space L p with p > 1/η and the space L −η is (roughly) that functions in L −η can be considered as having only one pointwise singularity at 0, functions in L p can have many pointwise singularities. For the functions that arise in applications, these differences rarely occur and the space L −η is often adequate, indeed usually the space C −η suffices. However, we will find the space L −η useful, particularly in Theorem 7.2 below.
The Hölder space denoted C 0,α [0, T ], 0 < α ≤ 1, consists of all functions f such that there is a positive real constant M = M (f ), such that T ] is a Banach space when endowed with the following norm f 0,α := sup Note that there are functions that are Hölder continuous but are not AC, for example a Weierstrass function, and AC functions that are not Hölder continuous, for example We will use the known fact that for 0 < α < 1 the Hölder space C 0,α embeds continuously and compactly into C[0, T ]; since it makes this paper more self contained we include the short proof.
so {u n } is bounded and equicontinuous hence relatively compact in C[0, T ] by the Arzelà-Ascoli theorem.

Compactness of Fredholm integral operators with discontinuous kernels
We say that a map N : We will extend a result given for linear operators in the book by Kolmogorov and Fomin [6], our proof uses the method from [6], but our new result deals with nonlinear operators and has weaker conditions.
Proof. We will prove that N is bounded and equicontinuous on each bounded set Now we turn to the equicontinuity. Let ε > 0, we have to prove that there exists δ > 0, depending only on ε, such that |Nu( and let As G is open, F is closed, and g is uniformly The second integral |I F | < ε/2 by the choice of ε 2 . These two steps prove that N is a compact map by the Arzelà-Ascoli theorem. Finally we prove that N is continuous. For this, let u n → u in C[0, T ] and let R > 0 be such that By the dominated convergence theorem, Nu n (t) = T 0 g(t, s)f (s, u n (s)) ds → Nu(t) as n → ∞, that is we have pointwise convergence. Since {Nu n } is relatively compact, by a standard argument, the convergence is uniform in t. This completes the proof that N : The result in the book [6, Theorem 1, page 112] is for the linear case when f (s, u(s)) = u(s) and it is assumed that |g(t, s)| is uniformly bounded for all t, s. If f (s, u(s)) is continuous, the substitution operator F u(t) := f (t, u(t)) maps C[0, T ] into itself and the result for linear operators can be used for the nonlinear case, via composition of functions, under stronger restrictions than we have. When f satisfies Carathéodory conditions but is not continuous this is no longer valid. Theorem 3.2 is not true for general curves instead of graphs of functions as the following simple example given in [6] shows.
Then Nu(t) = 1 0 g(t, s)u(s) ds maps the continuous function u ≡ 1 into a discontinuous function.
Here the curve is t = 1/2 which is not a graph; it is not to be confused with s = 1/2 which is an allowed curve. A nonlocal boundary value problem where the corresponding integral operator has a discontinuity on a line s = η is studied in [4].

Some properties of fractional integrals
In the study of fractional integrals and fractional derivatives the Gamma and Beta functions occur naturally and frequently. The Gamma function is, for p > 0, given by which is an improper Riemann integral but is well defined as a Lebesgue integral, and is an extension of the factorial function: Γ(n + 1) = n! for n ∈ N. The Beta function is defined by which is a well defined Lebesgue integral for p > 0, q > 0. It is well known, and proved in calculus texts, The integral I α u is the convolution of the L 1 functions h, u where h(t) = t α−1 /Γ(α), so, by the well known results on convolutions, I α u is defined as an L 1 function, in particular I α u(t) is finite for a.e. t.
If α = 1 this is the usual integration operator which we denote I. We set I 0 u = u. Interchanging the order of integration, using Fubini's theorem, shows that these fractional integral operators satisfy a semigroup property as follows: The proof is given in [12, (2.21)]. A detailed proof is given in [17,Lemma 3.4] where it is also shown that this holds for all t when u ∈ C −γ (γ ∈ [0, 1)) and α + β ≥ γ.
Hardy and Littlewood [3] showed that, for 0 < α < 1 and α > 1/p, the R-L fractional integral operator I α maps L p [0, T ] into the Hölder space C 0,α−1/p . We will prove in Theorem 4.5 below a slightly less general result, but with a simpler argument, that for 0 < γ < α < 1, I α maps L −γ [0, T ] into the Hölder space We first give the following result which is novel and a small improvement on the result in [17], the extension proves useful in Theorems 4.5 and 7.2 below. Theorem 4.3. Let 0 < α < 1 and 0 ≤ γ < 1.
We have When g is continuous, h is continuous by the dominated convergence theorem, which completes the proof.
The promised embedding theorem is as follows.
Theorem 4.5. Let 0 ≤ γ < α < 1. Then Proof. (1) Let 0 ≤ t < t + h ≤ T . We must prove that, for f ∈ L ∞ [0, T ], there exists a constant M , which can depend on f , such that We have The second integral Therefore This proves that . As (2) Let f ∈ L −γ . Then, by the semigroup property of fractional integrals, we have I α f (t) = I α−γ I γ f (t) a.e. We obtain I γ f ∈ L 0 = L ∞ by Theorem 4.3 and, from part (1), I α−γ maps L ∞ into the Hölder space C 0,α−γ [0, T ]. Hence I α f is a continuous function and the last assertion is shown. Remark 4.6. As mentioned above this result was proved for the somewhat more general case when f ∈ L p with p > 1/α by Hardy and Littlewood [3]. When α > 1 the cases α − 1/p ∈ N are a little different, see [3] or [12, Theorem 3.6]. The (two page long) proof for f ∈ L p can also be found in Theorem 2.6 of [2]. A different proof is given in [7, Theorem 3.5]. We believe our new idea of proving the easier part (1) and then deducing part (2) gives a simpler, shorter, proof for this case, which has interest because of equivalences with initial value problems of fractional differential equations where the space C −γ occurs quite naturally. We discovered that the first part of (1), I α is bounded from L ∞ into C 0,α , is given after Corollary 2 in Section 3.1 of [12] in less detail and with some misprints.

Some fractional differential equations and their integral versions
The R-L fractional integral of order α > 0 for u ∈ L 1 [0, T ] is In the important case 0 < α < 1, the kernel (t − s) α−1 is singular on the line s = t. For a real number β ≥ 0 let β (the ceiling function acting on β) denote the smallest integer greater than or equal to β.
Definition 5.1. The Riemann-Liouville (R-L) fractional differential operator of order β is defined when, for n = β , D n−1 (I n−β u) ∈ AC, that is I n−β u ∈ AC n−1 , by D β u := D n I n−β u.
The Caputo differential operator of order β, (or Caputo derivative for short) is defined ([2, Definition 3.2]) when I n−β u ∈ AC n−1 and T n−1 u exists by where T n−1 u is the Taylor polynomial of degree n − 1, T n−1 u(t) := n−1 k=0 where D k are k-th order ordinary derivatives.
Under the given conditions each fractional derivative exists a.e. and involves a fractional integral of order in (0, 1), so a singular kernel is always involved.
Remark 5.2. The Caputo derivative is often defined for u ∈ AC n−1 , by the equation  For the R-L fractional differential equation an equivalence result is as follows. u(t)) a.e. and This is stated in an equivalent form, also for the higher order case, as [5, Lemma 2.5(b)] and is proved in [12,Theorem 2.4]. It is also proved, with an 'initial condition' given in terms of lim t→0+ u(t)t 1−α under some slightly different hypotheses, in [1, Theorems 4.10 and 5.1]. The higher order case is also discussed in [8].
A result used in [1] is as follows.
Lemma 5.5. Let 0 < α < 1 and suppose that u ∈ L 1 . Then A proof is given in [17,Lemma 6.3] and it is proved for α ∈ C with 0 < Re(α) < 1 in [5,Lemma 3.2]. It is also proved in [1, Theorem 6.1] where a converse is also claimed, but the converse part of the proof has a gap. Combining the previous two results we have Proposition 5.6. If u ∈ C α−1 and t → f (t, u(t)) ∈ C −γ for some γ < 1 and satisfies the Volterra integral equation The hypothesis f (t, u(t)) ∈ C −γ implies that I α f ∈ C α−γ ⊂ C α−1 by Theorem 4.3 so all terms make sense.

Compactness of integral operator arising from Caputo IVP
We now prove the complete continuity of the integral operator that arises from the initial value problem for the Caputo differential equation as in Proposition 5.3.
For t = 0, we have I α (F u n )(0) = 0 = I α (F u)(0). For each (fixed) t > 0, we have (t − s) α−1 (F u n )(s) → (t − s) α−1 (F u)(s) for s = 0 and s = t, thus, by the dominated convergence theorem, we conclude that  → (t, u(t), u {1} (t), . . . , u {n−1} (t)) is continuous. For example, let 0 < β < α < 1 and 0 ≤ γ < α − β. Let X is a Banach space, for example Su [13] proves the R-L case, [18,Proposition 6.7] proves the Caputo case. If f : [0, T ] × R 2 → R is continuous and we write (F u)(t) := t −γ f (t, u(t), D β * u(t), then, for u in a bounded subset of X, {I α F u} and {I α−β F u} belong to compact subsets of C[0, T ], which proves the corresponding operators N and D β * N are compact from X into X. Further details of this case are given in Theorem 8.3. Remark 6.3. For N as in Theorem 6.1, it is proved in [16,Theorem 4.8] that N is completely continuous using a much longer argument directly with the Arzelà-Ascoli theorem, which is only used here, Lemma 2.1, to prove the Hölder space embeds compactly into the space of continuous functions. Our new proof combines some useful properties of the Riemann-Liouville fractional integral.
Recently Lan [7] proved, with longer, different methods, that the fractional integral operator I α is compact from L p to C when p > 1/α, which can also be used to deduce the compactness result of Theorem 6.1.

Integral operator arising from a R-L fractional derivative IVP
Recall that, for 0 < α < 1, from Proposition 5.6, if u ∈ C α−1 then implies that D α u(t) = f (t, u(t)) and lim t→0+ u(t)t 1−α = u 0 . Note that u is a solution in C α−1 of (7.1) if and only if v(t) Therefore we will give conditions so that defines a completely continuous operator in C[0, T ]. For this, writing f (t, v) = f (t, t α−1 v), it is enough to prove that the following fractional integral operator is continuous and compact in C[0, T ].
This is similar to the L q -Carathéodory condition but no continuity condition in the u variable is imposed. We believe the next result is new. It is precisely what is needed for proving an existence result.
Proof. Let {v n } be bounded in C[0, T ], say v n ∞ ≤ R. Write f n (t) =f (t, v n (t)) and g n (t) := g(t, v n (t)), then there exists g R ∈ L ∞ such that g n (t) ≤ g R (t) for all t ∈ [0, T ] and all n. The operator N can be written as N v n = I α f n . Firstly we suppose that f n ≥ 0. Then we have f n (t) ≤ t −γ g n (t), so there exists λ n,t ∈ [0, 1] such that f n (t) = λ n,t t −γ g n (t) and . By the first part, Then then v n ∞ ≤ R for some R and g(s, v n (s)) ≤ g R (s) for all s ∈ [0, T ] and all n. Also f (s, s α−1 v n (s)) → f (s, s α−1 v(s)) for each s > 0. For each fixed t > 0, as which is an L 1 function of s ∈ [0, t], by the dominated convergence theorem we see that N v n (t) →Nv(t), that is we have pointwise convergence. Since {Nv n } belongs to a compact subset of Then the integral operator Proof. We have Clearly g ∈ L ∞ loc , thus the result follows from Theorem 7.2.
Remark 7.4. I believe the cases β = 1 are new. The case β < 1 could be deduced from the case β = 1 since |u| β ≤ 1 + |u|. For the special case β = 1, a slightly more general result is essentially proved in [19,Lemma 4.1] where it is assumed that |f (t, u)| ≤ k(t) + l(t)|u| where k ∈ L p and t α−1 l(t) ∈ L p for some p > 1/α. The longer proof in [19] does not use any of the properties of fractional integrals that we have used, it is essentially using the Arzelà-Ascoli theorem. For the case when β = 1 there are Gronwall type inequalities [16,19] and, using these and the complete continuity of the integral operator, global existence of solutions of initial value problems of fractional equations can be proved, as is done in [16,19].

Order α ∈ (0, 1) when f depends on fractional derivatives
We consider Caputo fractional equations when the nonlinearity depends on fractional derivatives. Let 0 < β < α < 1 and f be continuous. Consider the problem: If u is a sufficiently regular solution of (8.1) then u is a solution of the integral equation However, the two problems are not necessarily equivalent. Equivalence depends on how 'solution' of each problem is defined. An appropriate Banach space is Define the operator N : X → X by The following is readily proved.
Proposition 8.2. Let f be continuous. Then u ∈ X is a solution of (8.1) if and only if u ∈ X is a solution of (8.2), that is u is a fixed point of N in X.
To obtain fixed points it is useful to have the following result. Proof. For u ∈ X let (F u)(t) := f (t, u(t), D β * u(t)), then t → (F u)(t) is continuous and, since u 0 plays no part, we want to prove that maps into a compact subset of C[0, T ]. We also want D β * N to do the same. Since the Caputo derivative of a constant is zero we have where we used the semigroup property.
Thus we need to prove that both I α F and I α−β F map into compact subsets of C[0, T ]. In fact, these results follow immediately from Remark 6.2. Continuity of these maps is proved as in the proof of Theorem 6.1.

Order α ∈ (1, 2) when f depends on fractional derivatives
We now consider the problem for 0 < γ ≤ β ≤ 1, 0 < α ≤ α + β − γ < 1 and α + β > 1 and f is continuous. We do not impose an ordering between α and β. This is the integral equation version of the initial value problem of the sequential fractional differential problem Note that this is not the same as having D α+β * u on the left since fractional derivatives do not commute or satisfy a semigroup property in general, see Examples 2.6 and 2.7 in Diethelm's book [2]. Informally we can 'solve' (9.2) as follows: Equations (9.1) and (9.3) should be studied in a space where at least u and D β * u are continuous such as the space X = {u ∈ C[0, T ], D β * u ∈ C[0, T ]} as in the previous section, but we will not study existence of solutions here. We only discuss the compactness requirements in this space. For u ∈ X, t → (F u)(t) is continuous. We want both I α+β (F u) and I α (F u) to be completely continuous in C[0, T ]. We have We can write this as I α+β (F u)(t) = Since α + β > 1 this is a special case of Theorem 3.2. Furthermore, I α (F u) is completely continuous in C[0, T ] by Remark 6.2.

Comments on boundary value problems
For Caputo fractional differential equations one can impose many types of boundary condition and, under appropriate conditions, find a Green's function. For examples, for a problem of order α ∈ (1, 2) such as if we write (F u)(t) = f (t, u(t)), the first step is to say solutions are of the form u(t) = u 0 + tu 1 + I α (F u)(t) and solve the algebraic equations obtained from the boundary conditions to determine u 0 , u 1 . Since it is necessary that I α (F u) is a compact operator it is useful to know when this is the case; our results give this. In particular, if it is shown that u 0 , u 1 belong to uniformly bounded sets then the operator is compact. For example, for the case , if f (t, u) = t −γ g(t, u) with g continuous, I α (F u)(t) will belong to a compact set so the endpoint I α (F u)(1) will be uniformly bounded and thus all terms are completely continuous. When f depends on a fractional derivative, the 'derivative' operator must also be considered as discussed in Section 9 above. With more complicated problems and more general boundary conditions this procedure might be difficult to implement.
For boundary value problems for R-L fractional differential equations with α ∈ (1, 2) such as D α u(t) = f (t, u(t)), the first step is to claim that solutions are of the form u(t) = c 1 t α−2 + c 2 t α−1 + I α f . This is only correct if solutions are sought in an appropriate space such as either L 1 or C α−2 . Using the space C α−2 the integral operator is similar to the ones studied in the Caputo case and Theorems we have proved will be applicable. The space L 1 has a different compactness criterion. If solutions are sought in the space C[0, T ] then the term c 1 t α−2 cannot occur and the only consistent boundary condition at 0 is u(0) = 0. With the boundary condition u(0) = 0 many types of boundary condition can be considered at t = 1; the integral operator is again similar to the ones studied in the Caputo case.

Conclusion
We have established several compactness results which can be applied in many situations, more than we have described. We hope this can replace many longer calculations using the Arzelà-Ascoli theorem.