Fredholm Property of Nonlocal Problems for Integro-Differential Hyperbolic Systems

The paper concerns nonlocal time-periodic boundary value problems for first-order Volterra integro-differential hyperbolic systems with boundary inputs. The systems are subjected to integral boundary conditions. Under natural regularity assumptions on the data it is shown that the problems display completely non-resonant behaviour and satisfy the Fredholm alternative in the spaces of continuous and time-periodic functions.

The Volterra integral terms in (1.1) are motivated by the aforementioned applications (see, e.g., [13,18]). As it will be seen from our proof of Theorem 1.2, our analysis applies also to the case when these terms are replaced by the Fredholm integral terms.
In general, systems of the type (1.1), (1.3) model a broad range of physical problems such as traffic flows, chemical reactors and heat exchangers [18]. They are also used to describe problems of population dynamics (see, e.g., [3,7,15,20] and references therein) and polymer rheology [4]. Moreover, they appear in the study of optimal boundary control problems [13,16,18,19].
Establishing a Fredholm property is a first step in developing a theory of local smooth continuation [12] and bifurcation [1,2,11] for Fredholm hyperbolic operators, in particular, such tools as Lyapunov-Schmidt reduction. Buono and Eftimie [1] consider autonomous 2×2 nonlocal hyperbolic systems in a single space variable, describing formation and movement of various animal, cell and bacterial aggregations, with some biologically motivated integral terms in the differential equations. One of the main results in [1] is a Fredholm alternative for the linearizations at a steady-state, which enables performing a bifurcation analysis by means of the Lyapunov-Schmidt reduction. Here we continue this line of research, establishing the Fredholm property for a wide range of non-autonomous nonlocal problems for (n × n)-hyperbolic systems, with nonlocalities both in the differential equations and in the boundary conditions. We show that the problem (1.1)-(1.3) demonstrates a completely non-resonant behavior (in other terms, no small divisors occur). More precisely, we prove the Fredholm alternative for (1.1)-(1.3) under the only assumptions that the coefficients in (1.1) and (1.3) are sufficiently smooth and a kind of Levy condition is fulfilled. The proof extends the ideas of [9,10] for proving the Fredholm alternative for first-order one-dimensional hyperbolic systems with reflection boundary conditions, and also the ideas of [8] for proving a smoothing property for boundary value hyperbolic problems. In contrast to [9] and [10], where conditions excluding a resonant behavior are imposed, the present Fredholmness result is unconditional, in this respect.

Our result
By C n,2π we denote the vector space of all 2π-periodic in t and continuous maps u : Similarly, C 1 n,2π denotes the Banach space of all u ∈ C n,2π such that ∂ x u, ∂ t u ∈ C n,2π , with the norm For simplicity, we skip subscript n if n = 1 and write C 2π and C 1 2π for C 1,2π and C 1 1,2π , respectively.
We make the following natural assumptions on the coefficients of (1.1) and (1.3): a j ∈ C 1 2π and b jk , ∂ t b jk , g jk , h jk , r jk , ∂ t r jk ∈ C 2π for all j ≤ n and k ≤ n, and for all 1 ≤ j = k ≤ n there existsb jk ∈ C 2π such that ∂ tbjk ∈ C 2π and b jk =b jk (a k − a j ). (1.6) The assumption (1.5) is standard and means the non-degeneracy of the hyperbolic system (1.1). The assumption (1.6) is a kind of the well-known Levy condition appearing in various aspects of the hyperbolic theory, for instance, for proving the spectrum-determined growth condition for semiflows generated by initial value problems for hyperbolic systems [5,14,17]. It plays also a crucial role in the Fredholm analysis of hyperbolic PDEs (see Example 1.3 below). Given j ≤ n, x ∈ [0, 1], and t ∈ R, the j-th characteristic of (1.1) is defined as the solution ξ ∈ [0, 1] → ω j (ξ, x, t) ∈ R of the initial value problem , ω j (x, x, t) = t. (1.7) To shorten notation, we will write ω j (ξ) = ω j (ξ, x, t). In what follows we will use the equalities where by ∂ i here and below we denote the partial derivative with respect to the i-th argument. Set and Integration along the characteristic curves brings the system (1.1)-(1.3) to the integral form This is a linear inhomogeneous ordinary differential equation for the function u j (·, ω j (·, x, t)), and the variation of constants formula (with initial condition at x j ) gives Inserting the boundary conditions (1.3) and using the notation (1.10), we get (1.11), as desired.
This problem is a particular case of (1.1)-(1.3) and satisfies all assumptions of Theorem 1.2 with the exception of (1.6). It is straightforward to check that are infinitely many linearly independent solutions to the problem (1.12)-(1.14) and, therefore, the kernel of the operator of (1.12)-(1.14) is infinite dimensional. Thus, the conclusion of Theorem 1.2 is not true without (1.6).
Then the system (1.11) can be written as the operator equation Note that Theorem 1.2 says exactly that the operator I −R−B−G−H : C n,2π → C n,2π is Fredholm of index zero. Nikolsky's criterion [6, Theorem XIII.5.2] says that an operator I + K on a Banach space is Fredholm of index zero whenever K 2 is compact. It is interesting to note that the compactness of K 2 and the identity I − K 2 = (I + K)(I − K) imply that the operator I − K is a parametrix of the operator I + K (see [21]).
We, therefore, have to show that the operator K 2 : C n,2π → C n,2π for K 2 = (R + B + G+H) 2 is compact. Since the operators R, B, G, and H are bounded and the composition of a bounded and a compact operator is compact, it is enough to show that the operators H, G, R 2 , RB, B 2 , BR : C n,2π → C n,2π are compact. (2.4) We start with the compactness of H. By C 2π (R) we denote the space of all continuous and 2π-time-periodic maps v : R → R. Fix arbitrary j ≤ n and k ≤ n and define the operator H jk ∈ L(C 2π (R), C 2π ) by (2.5) It suffices to show the compactness of H jk . Change the variable ξ to z = ω j (ξ) and denote the inverse map by ξ =ω j (z) =ω j (z, x, t). Afterwards (2.5) reads By the regularity assumption (1.4), the functions ω j (x j ),ω j (z), d j (ξ, x, t), h jk (x, z), and a j (x, z) are continuous in all their arguments and 2π-periodic in t and, hence, are uniformly continuous in x and t. Then the equicontinuity property of (H jk v)(x, t) for v over a bounded subset of C 2π (R) straightforwardly follows. Using the Arzela-Ascoli precompactness criterion, we conclude that H jk and, hence, H are compact. Now we consider the operator G. Changing the variable ξ to z = ω j (ξ, x, t) in (2.2), we get Similarly to the above, the functions ω j (x j ),ω j (z), d j (ω j (z), x, t), and a j (ω j (z), z) are 2πperiodic in t and uniformly continuous in x and t. This entails the equicontinuity property for (Gu) j (x, t) for u over a bounded subset of C n,2π . The compactness of G again follows from the Arzela-Ascoli theorem.
We further proceed with the compactness of R 2 . For j ≤ n and k ≤ n define operators R jk ∈ L(C 2π ) by Fix arbitrary j ≤ n, k ≤ n, and i ≤ n. We prove the compactness of the operator R jk R ki ; the compactness of all other operators contributing into the R 2 will follow from the same argument. Introduce operators P j , Q jk : C 2π → C 2π by Then we have We aim at showing the compactness of P j Q jk P k , as this and the boundedness of Q ki will entail the compactness of R jk R ki . The operator P j Q jk P k reads r jk (ξ, ω j (x j , ξ, t))c k (x k , ξ, ω j (x j , ξ, t)) 1 0 w(η, ω k (x k , ξ, t)) dηdξ. (2.10) Changing the variable ξ to z = ω k (x k , ξ, t), we get (2.12) Similarly to the above, the compactness of P j Q jk P k now immediately follows from the regularity assumption (1.4) and the Arzela-Ascoli theorem. Now we treat the operator ×b kl (ξ, ω k (ξ, η, ω j (x j )))u l (ξ, ω k (ξ, η, ω j (x j ))) dξdη for an arbitrary fixed j ≤ n. After changing the order of integration we get the equality ×b kl (ξ, ω k (ξ, η, ω j (x j )))u l (ξ, ω k (ξ, η, ω j (x j ))) dηdξ.
Then we change the variable η to z = ω k (ξ, η, ω j (x j )). Since the inverse is given by where ∂ 3ωk (ω j (x j ), ξ, z) is given by (2.12). The functions ω j (ξ, x, t) and the kernels of the integral operators in (2.13) are continuous and t-periodic functions and, hence, are uniformly continuous functions in x and t. This means that we are again in the conditions of the Arzela-Ascoli theorem, as desired.
We proceed to show that B 2 : C n,2π → C n,2π is compact. By the Arcela-Ascoli theorem, C 1 n,2π is compactly embedded into C n,2π . Then the desired compactness property will follow if we show that B 2 maps continuously C n,2π into C 1 n,2π . (2.14) By using the equalities (1.8), (1.9), and (2.1), the partial derivatives ∂ x B 2 u, ∂ t B 2 u exist and are continuous for each u ∈ C 1 n,2π . Since C 1 n,2π is dense in C n,2π , the desired condition (2.14) will follow from the bound To prove (2.15), for given j ≤ n and u ∈ C 1 n,2π , let us consider the following representation for (B 2 u) j (x, t) obtained after the application of the Fubini's theorem: for all j ≤ n, ϕ ∈ C 1 (R), x, ξ ∈ [0, 1], and t ∈ R, one can easily check that for all j ≤ n and u ∈ C 1 n,2π .
The desired estimate (2.18) now easily follows from the assumptions (1.4)-(1.6). Returning back to (2.4), it remains to prove that the operator BR : C n,2π → C n,2π is compact. By the definitions of B and R, ×r kl (η, ω k (x k , ξ, ω j (ξ)))u l (η, ω k (x k , ξ, ω j (ξ))) dξdη, j ≤ n. (2.21) The integral operators in (2.21) are similar to those in (2.16) and, therefore, the proof of the compactness of BR follows along the same line as the proof of the compactness of B 2 . The proof of Theorem 1.2 is complete.