On 2D integro-differential systems. Stability and sensitivity analysis

In the paper a two-dimensional integro-differential system is considered. Using some variational methods we give sufficient conditions for the existence and uniqueness of a solution to the considered system. Moreover, we show that the system is stable and robust.

In the paper, we investigate two-dimensional integro-differential system of the form z xy (x, y) + f 1 (x, y, z(x, y)) with the following boundary conditions where f 1 , f 2 : Q × R n → R, A 1 , A 2 : Q → R n × R n are given functions (for more details see section 3). We shall consider the above system in the space of absolutely continuous functions of two variables. The definition and basic properties of absolutely continuous functions defined on the interval Q are presented in section 2.
In the paper we prove, under assumptions (C1)-(C3) (see section 3), that for any square integrable function v system (7)-(8) possesses a unique solution z v which continuously depends on v and the operator v → z v is differentiable in the Frchet sense, i.e. the considered system is well-posed and robust.
The proof of the main result is based on the global diffeomorphism theorem (Theorem 1). In the final part of the paper we give an example and compare the method used in the paper with the methods based on the contraction principle and the Schauder fixed point theorem.

Preliminaries
We begin with the following theorem on a diffeomorphism between Banach and Hilbert spaces Theorem 1 Let Z be a real Banach space, V be a real Hilbert space, F : Z → V be an operator of C 1 class. If In other words, the operator F is a diffeomorphism between Banach space Z and Hilbert space V .
We recall that functional φ satisfies (PS)-condition if whenever there is a sequence {z n } ⊂ Z with |φ(z n )| ≤ const and φ ′ (z n ) → 0 in Z * , then in the closure of the set {z n : n ∈ N }, there is some pointz where φ ′ (z) = 0 (see [Aubin and Ekeland, 2006]).
From the bounded inverse theorem it follows that for any z ∈ Z there exists a constant α z > 0 such that F ′ (z)h V ≥ α z h z . Therefore it follows easily that the above theorem is equivalent to [Idczak et al., 2012, Theorem 3.1], with f = F . Let us denote by AC(Q, R n ) the space of absolutely continuous vector functions z = (z 1 , z 2 , . . . , z n ) defined on the interval Q. The geometrical definition of the space AC(Q, R) can be found in papers [Berkson and Gillespie, 1984] and [Walczak, 1987]. In this paper we need necessary and sufficient conditions for z : Q → R n to be absolutely continuous on Q i.e. z ∈ AC(Q, R n ). We have the following theorem (see [Berkson andGillespie, 1984,Walczak, 1987]).
Theorem 2 A function z belongs to the space AC(Q, R n ) if and only if there exist functions l ∈ L 1 (Q, R n ), l 1 , l 2 ∈ L 1 ([0, 1], R n ) and a constant c ∈ R n such that Moreover the function z possesses partial derivatives z x , z y , z xy , for a.e. (x, y) ∈ Q and z x (x, y) = y 0 l(x, t)dt + l 1 (x), z y (x, y) = x 0 l(s, y)ds + l 2 (y), z xy (x, y) = l(x, y).
It is easy to check that if the function z satisfies homogeneous boundary conditions, i.e. z(x, 0) = 0 for x ∈ [0, 1] and z(0, y) = 0 for y ∈ [0, 1] then l 1 = 0, l 2 = 0, c = 0 and consequently we can write By AC 2 0 (Q, R n ) we shall denote the space of absolutely continuous functions on the interval Q which satisfy homogeneous boundary conditions z(x, 0) = z(0, , y) = 0 for x, y ∈ [0, 1] and such that z xy ∈ L 2 (Q, R n ). The space AC 2 0 is a Hilbert space with the inner product given by formula In the space AC 2 0 (Q, R n ) we introduce two norms. The first one is a classical norm given by the formula and the second is defined by the integral with exponential weight Exponential norm (12) was introduced by Bielecki in [Bielecki, 1956]. The space AC 2 0 (Q, R n ) with norm (12) will be denoted by AC 2 0,m (Q, R n ). It is easy to notice that e −2m z ≤ z AC 2 0 ,m ≤ z . Thus the norms given by formulas (11) and (12) are equivalent.
Similarly, in the space of square integrable functions on Q we introduce two equivalent norms: The space of square integrable functions with norm (13) will be denoted by L 2 m (Q, R n ).

Basic assumptions and lemmas
On the functions defining system (7) we assume that for (x, y) ∈ Q and |z| ≤ ̺.
In the following lemma we prove some estimates for functions from the space Remark 1 The norms · L 2 m and · AC 2 0,m are defined by (12) and (13) respectively.
Proof Let z be an arbitrary function from the space AC 2 0 (Q, R n ). By (10), (12) and (13) we get Integrating by parts we obtain successively |z xy (s, y)| 2 ds dy Thus we proved inequality (14). By the above and applying the CauchySchwarz inequality we get Let us prove the next estimation. By (9) we have Integrating by parts as in (18), we get The proof of (17) is similar. ⊓ ⊔ Denote by F : AC 2 0 (Q, R n ) → L 2 (Q, R n ) the operator: We will prove that the norm of F is coercive.
(a) the sequence of functions {z k } tends uniformly to z 0 on the interval Q; (b) the sequence {g k } tends to g 0 for (x, y) ∈ Q a.e.
Proof We first prove that the weak convergence of the sequence {z k } to z 0 in the space AC 2 0 (Q, R n ) implies the uniform convergence of the sequence {z k } to z 0 on the interval Q. By the definition of the inner product (see (10)) the weak convergence of the sequence {z k } to z 0 in the space AC 2 0 (Q, R n ) is equivalent to the weak convergence of mixed second order derivatives {z k xy } to z 0 xy in the space L 2 (Q, R n ). Without loss of generality we can assume that z 0 = 0. Suppose that z k does not converge uniformly to z 0 = 0 while it converges to 0 weakly in AC 2 0 (Q, R n ). Therefore there exists ε 0 > 0 such that for any n ∈ N there is a point (x n , y n ) ∈ Q such that |z n (x n , y n )| > ε 0 .
The sequence {(x n , y n )} ⊂ Q is compact. Passing if necessary to a subsequence we can assume, that (x n , y n ) tends to some (x,ỹ) ∈ Q. Denote by χ n the characteristic function of the interval {(x, y) ∈ Q : 0 ≤ x < x n , 0 ≤ y < y n } and byχ the characteristic function of the interval It is easy to notice that χ n tends toχ on Q a.e. This implies the following inequalities Since z n xy tends to zero weakly in L 2 (Q, R n ) the last limit is equal zero. Therefore lim n→∞ |z n (x n , y n )| ≤ lim where C > 0 is some constant such that z n xy ≤ C. Consequently, lim n→∞ |z n (x n , y n )| = 0. This contradicts our assumption (22). Thus z k tends to z 0 uniformly on Q.

Main result and example
Let us consider a functional ϕ : AC 2 0 → R given by the formula where F is the operator defined by (3.6) and v is a fixed function from the space L 2 (Q, R n ). We begin by proving some lemmas.
Proof Let {z k } ⊂ AC 2 0 be an arbitrary (PS)-sequence for the functional ϕ. By Lemma 2 ϕ is coercive. It implies that the sequence {z k } is weakly compact in AC 2 0 . Passing if necessary to a subsequence we can assume, that z k tends to some z 0 weakly in AC 2 0 . We claim that {z k } is compact with respect to the norm topology of the space AC 2 0 . Thanks to assumptions (C1)-(C2) it is easy to check that the functional ϕ is Frchet differentiable and where the sequence {g k } ⊂ L 2 (Q, R n ) is given by formula (21). Let us put h k − z k − z 0 , k = 1, 2, .... From (29) it follows that where z k xy (x, y) + g k (x, y) dxdy z 0 xy (x, y) + g 0 (x, y) dxdy. By the Cauchy-Schwarz inequality we have the following estimation Since z k xy − z 0 xy converges weakly to zero in L 2 (Q, R n ), therefore there exists a constant C > 0 such that By Lemma 2 and Lebesgue dominated convergence theorem it follows that V 1 z k → 0 as k → ∞. We have proved that z k (x, y) tends to z 0 (x, y) uniformly on Q (see Lemma 2). Therefore, it is easy to notice that V 2 z k and V 3 z k converge to zero as k → ∞.
Let us consider the functional V 4 . By (24) we have z k xy (x, y) + g k (x, y) dxdy Using the Cauchy-Schwarz inequality and Lemma 2 it is easy to show that V 4 z k → 0 as k → ∞. Similar considerations can be applied to V 5 z k . Thus lim k→∞ Now, let us observe that lim k→∞ ϕ ′ z k z k − z 0 = 0 because {z k } is the (PS)-sequence for the functional ϕ and the sequence z k − z 0 is bounded. Moreover, lim k→∞ ϕ ′ z 0 z k − z 0 = 0 since z k tends weakly to z 0 in AC 2 0 . Combining these equalities and (30) we conclude that lim This gives us the desired conclusion that the functional ϕ given by (28) satisfies (PS)-condition.

⊓ ⊔
Next, we prove the following

Lemma 5
If the functions f 1 ,f 2 , A 1 , A 2 satisfy assumptions (C1)-(C3) then for any v ∈ L 2 (Q, R n ) there exists a unique solution h v ∈ AC 2 0 to the system where the operator F : AC 2 0 → L 2 (Q, R n ) is given by (19) and z 0 ∈ AC 2 0 is an arbitrary function.
Proof Let us put where g ∈ L 2 (Q, R n ). Substituting the above into (31) we obtain Let us denote byH the operator defined bỹ We will restrict our investigation of the operatorH to the space L 2 m (Q, R n ). We prove that for sufficiently large m > 0 the mappingH is contracting with respect to the norm · L 2 m defined by (13). Under assumptions (C2) and (C3), there exists a constant d > 0 such that  Integrating by parts twice, in much the same way as in the proof of inequality (18), we obtain Hence for sufficiently large m, i.e. m > 2 √ d, the operatorH is contracting and, consequently, has a unique fixed point. It means that, there exists exactly one point g 0 ∈ L 2 (Q, R n ) such that g 0 =Hg 0 . By (32) we get Hg 0 = v and it follows easily that a function h v given by is a solution of (31) for fixed v ∈ L 2 (Q, R n ).

⊓ ⊔
We are now in a position to show the main result of the work.
Theorem 3 If the functions f 1 ,f 2 , A 1 , A 2 satisfy assumptions (C1)-(C3) then for any v ∈ L 2 (Q, R n ) the integro-differential system (7)-(8) has a unique solution z v ∈ AC 2 0 . The solution z v continuously depends on v with respect to the norm topology in the spaces L 2 (Q, R n ) and AC 2 0 . Moreover, the operator Proof If follows from Lemmas 4 and 5 that the operator F given by (19) meets assumptions of Theorem 1. Thus system (7)-(8) has a solution z v which satisfies the requirements of our theorem.

⊓ ⊔
We now give an example of integro-differential system of the form (7)-(8) which satisfies assumptions of Theorem 3. For simplicity we put n = 1.
Example 1 Consider 2D integro-differential system z xy (x, y) + w 1 (x, y) z 3 (x, y) 1 + z 2 (x, y) + ψ 1 (z(x, y) where w 1 , w 2 , A 1 , A 2 are some polynomials, v ∈ L 2 (Q, R) and ψ 1 , ψ 2 are some C 1 −class functions with unbounded derivatives. For example one can take ψ 1 (z) = cos z k and ψ 2 (z) = sin z l , where k, l > 1. This simple and theoretical example allows us to emphasize the difference between our work and some other methods of nonlinear analysis. It is easy to see that system (33) satisfies assumptions (C1)-(C3). Hence by Theorem 3 for any v ∈ L 2 (Q, R) there exists a solution z v ∈ AC 2 0 to the system (33) with the following properties: 1. the solution z v is unique, 2. z v continuously depends on v with respect to the norm topology of the spaces L 2 (Q, R n ) and AC 2 0 , i.e. system (33) is stable, 3. the operator L 2 (Q, R) ∋ v → z v ∈ AC 2 0 is differentiable in Frchet sense, i.e. system (33) is robust.
Let us notice that the functions f 1 (x, y, z) = w 1 (x, y) z 3 1+z 2 + ψ 1 (z) and f 2 (x, y, z) = w 2 (x, y) z−1 1+z 2 + ψ 2 (z) are not Lipschitz functions (sin z l and cos z l with k, l ≥ 1 have "fast variation" when |z| → ∞) and consequently we cannot apply the Banach contraction principle. In this case the Schauder fixed point theory may be applicable. But even using sophisticated fixed point theorems we get only the existence of a solution to system (33) and can hardly say anything related to properties (1)-(3).

Concluding remarks
In the paper two-dimensional integro-differential system was investigated. The main result of this work is theorem 3 on the stability and robustness of a solution to considered system (7)-(8). As far as we know 2D integro-differential systems have not been studied before. One-dimensional integro-differential systems described by ordinary differential operators were examined in many works (see monogrph [Lakshmikantham, 1995] and references therein). It is important to notice that integro-differential operators can be used in mathematical modeling of systems with "memory", i.e. systems where the state at each moment t depends on its behavior on some interval [t 0 , t). In our opinion 2D integro-differential systems have the potential to play a similar role.