Implicit first order differential systems with nonlocal conditions

The present paper is devoted to the existence of solutions for implicit first order differential systems with nonlocal conditions expressed by continuous linear functionals. The lack of complete continuity of the associated integral operators, due to the implicit form of the equations, is overcome by using Krasnoselskii’s fixed point theorem for the sum of two operators. Moreover, a vectorial version of Krasnoselskii’s theorem and the technique based on vector-valued norms and matrices having the spectral radius less than one are likely to allow the system nonlinearities to behave independently as much as possible. In addition, the connection between the support of the nonlocal conditions and the constants from the growth conditions is highlighted.

In the recent years much attention has been given to different types of problems with nonlocal conditions.We refer to the bibliographies of the papers [2-4, 11, 13, 17-19, 25, 31, 32] for references.The motivation is that such problems arise from the mathematical modeling of real processes, such as heat, fluid, chemical or biological flow, where the nonlocal conditions can be seen as feedback controls, see for example [7] and the recent survey paper [28].
One can distinguish between discrete nonlocal conditions, or multi-point boundary conditions, and continuous conditions given by continuous linear functionals.For both types, it is important to take into consideration the interval on which a given condition really acts, that is, the support of that condition.More exactly, the support associated to the condition x (0) = α [x] , where α : C[0, 1] → R is linear, is the minimal closed subinterval [0, t 0 ] of [0, 1]  with the property α [x] = α [y] whenever x = y on [0, t 0 ] .This notion was first introduced in [4] and has become essential for the existence theory of nonlocal problems.Indeed, as shown in [4,19,25], stronger conditions have to be satisfied by the nonlinear terms of the equations on [0, t 0 ], compared to the hypotheses asked on [t 0 , 1].One may assert that the "integral" equation equivalent to a nonlocal problem on the interval [0, 1], is of Fredholm type on the support [0, t 0 ] of the nonlocal condition, and of Volterra type on the remaining interval [t 0 , 1].In the present paper again we shall exploit this idea, even pregnantly.We shall do this, by considering a special norm on C [0, 1], namely Here η < t 0 and θ > 0 are given numbers.As we shall see, the joint role of the parameters η (any fixed number with η < t 0 ) and θ (chosen large enough) is to weaken the assumptions on g 1 (t, x, y) , g 2 (t, x, y) when t ∈ [t 0 , 1].
Note a key property of the functional α in connection with its corresponding support, for every x ∈ C[0, 1], when normally, for any continuous linear functional α : where the notation α is used to denote the norm of the continuous linear functional α.
A standard technique for nonlocal problems, as like for boundary value problems in general, is the reduction of the problem to a fixed point problem for a suitable integral type operator.Then a fixed point theorem guarantees the existence of a solution.Topological fixed point theorems, as well as index theory, are essentially based on compactness, which in case of explicit equations usually holds, while for implicit equations it becomes a problem.In order to overcome this difficulty, one may think to use Krasnoselskii's fixed point theorem for a sum of two operators, a contraction and a completely continuous mapping.Krasnoselskii's theorem [14,15] (see also [10]) has become a basic result of the nonlinear analysis with a large number of applications to nonlinear operator, integral, and differential equations.Combining Banach's contraction principle and Schauder's fixed point theorem, it can be considered a bridge between metrical and topological fixed point theories.There is a rich literature concerning different generalizations and applications of this theorem.Here are some of them: [1,5,6,9,12,20,21,27,30].
Nonlocal problems for implicit differential equations have been, by our knowledge, less investigated.We just mention the paper [16] on the monotone iterative technique for an implicit first order equation subject to a two-point boundary condition, and the paper [8] whose aim is to obtain a Carathéodory solution for an implicit first order equation under a nonlocal condition, via Schauder's fixed point theorem and Kolmogorov's compactness criterion in L 1 .
Since we are interested here in systems of equations, we have opted for a vectorial approach based on the use of vector-valued norms, inverse-positive matrices and of a vectorial version of Krasnoselskii's fixed point theorem for sums of two operators.The vectorial approach allows the system nonlinearities to behave independently as much as possible.
We recall now some basic notions which are involved in our vectorial setting.By a vectorvalued metric on a set X we mean a mapping d : Here by ≤ we mean the natural componentwise order relation of R n , more exactly, if r, s ∈ R n , r = (r 1 , r 2 , . . ., r n ), s = (s 1 , s 2 , . . ., s n ), then by r ≤ s one means that r i ≤ s i for i = 1, 2, . . ., n.
A set X together with a vector-valued metric d is called a generalized metric space.For such a space, the notions of Cauchy sequence, convergence, completeness, open and closed set are similar to those in usual metric spaces.
Similarly, we speak about a vector-valued norm on a linear space X, as being a mapping • : X → R n + with x = 0 only for x = 0; λx = |λ| x for x ∈ X, λ ∈ R, and x + y ≤ x + y for every x, y ∈ X.To any vector-valued norm • one can associate the vectorvalued metric d (x, y) := x − y , and one says that (X, • ) is a generalized Banach space if X is complete with respect to d.
If (X, d) is a generalized metric space and T : X → X is any mapping, we say that T is a generalized contraction (in Perov's sense) provided that a matrix M ∈ M n×n (R + ) exists such that its powers M k tend to the zero matrix 0 as k → ∞, and d(T(x), T(y)) ≤ Md(x, y) for all x, y ∈ X.
Here and throughout the paper, the vectors in R n are seen as column matrices.
There are several characterizations known of the matrices M with M k → 0 as k → ∞ (see [23] and [29, pp. 12, 88]).More exactly, for a matrix M ∈ M n×n (R + ) , the following statements are equivalent: The following almost obvious lemma will be used in the sequel.
whose elements are small enough.
The role of matrices with spectral radius less than one in the study of semilinear operator systems was pointed out in [24], in connection with several abstract results from nonlinear functional analysis.Thus, Banach's contraction principle admits a vectorial version in terms of generalized contractions in Perov's sense.Following Perov's approach, in [30] (see also [22]) the following vectorial version of Krasnoselskii's fixed point theorem for a sum of two operators was obtained.

Theorem 1.2 ([30]
).Let (X, • ) be a generalized Banach space, D a nonempty closed bounded convex subset of X and T : D → X such that: (i) T = G + H with G : D → X completely continuous and H : D → X a generalized contraction, i.e. there exists a matrix M ∈ M n×n (R + ) with ρ (M) < 1, such that H(x) − H(y) ≤ M x − y for all x, y ∈ D; (ii) G(x) + H(y) ∈ D for all x, y ∈ D.
Then T has at least one fixed point in D.
The proofs of the vectorial versions of the Banach and Krasnoselskii theorems follow the same ideas as for the original results.However, for applications to systems, these versions allow nonlinearities to behave independently one to each other, and differently with respect to the system variables.

Main result
In order to obtain the equivalent integral form of the problem (1.1), denote Then, using the nonlocal conditions we obtain Also define Then the problem (1.1) is equivalent to the system Note that we look for solutions with x, y The system (2.1) appears as a fixed point problem for the operator where T 1 , T 2 are given by We can rewrite (2.2) in a vectorial form, as a sum of two operators, namely where We shall assume that the nonlocal conditions expressed by the functionals α, β have the same support [0, t 0 ] , and that the growth of g 1 (t, u, v) , g 2 (t, u, v) with respect to u and v is at most linear, on each of the two subintervals [0, t 0 ] and [t 0 , 1] , that is (2.3) for all (u, v) ∈ R 2 , and the functions h 1 , h 2 satisfy the Lipschitz conditions for all (u, v) , (u, v) ∈ R 2 and t ∈ [0, 1].Here, for i = 1, 2, a i , b i , c i > 0 and A i , B i , C i , a i , b i are nonnegative numbers.Denote and consider the matrices With those notations, we can state and prove our main existence result.
Theorem 2.1.Assume that g 1 , g 2 satisfy (2.3) and h 1 , h 2 satisfy (2.4).If the spectral radius of the matrix M 0 + M 1 is less than one, then problem (2.1) has at least one solution (x, Proof.We shall apply the vectorial version of Krasnoselskii's fixed point theorem to the space X = C [0, 1] , R 2 , endowed with the vector-valued norm • C([0,1],R 2 ) defined by Step 1.The operator G is completely continuous.This follows from the continuity of g 1 , g 2 and the fact that the terms t 0 u(s) ds, t 0 v(s) ds guarantee the equicontinuity in the Arzelà-Ascoli theorem.
Step 3. In what follows, we look for a nonempty, bounded, closed and convex subset D of Taking the supremum, we have Furthermore, for t ∈ [t 0 , 1], we have that O. Bolojan and R. Precup Dividing by e θ(t−η) and taking the supremum when t ∈ [t 0 , 1], we obtain (2.11) Now we can take advantage from the special choice of the norm |•| C θ [t 0 ,1] , more exactly from the choice of η < t 0 , to assume (choosing large enough θ > 0) that and this way to eliminate from the first part of our estimation the growth constants A 1 , B 1 , C 1 .Indeed, (2.11) and (2.12) give Now, (2.10) and (2.13) imply that (2.15) The inequalities (2.14) and (2.15) can be put under the vectorial form On the other hand, from (2.9), we deduce that for every (u, v) ∈ C [0, 1] , R 2 , where Now we look for the set According to the estimations (2.16) and (2.17), the condition < 1 and the entries of M 2 are as small as desired for large enough θ > 0, from Lemma 1.1, we can choose θ such that Then, according to the property (d) of matrices with spectral radius less than one, the inequality (2.18) is equivalent to This proves the existence of radii R 1 , R 2 ≥ 0 for which the inwardness condition G (D) + H (D) ⊂ D is satisfied.Thus Theorem 1.2 applies and guarantees the existence in D of at least one fixed point for T.
Remark 2.2.It is worth to underline the exclusive contribution to the matrix M 0 + M 1 of the growth constants a 1 , a 2 , b 1 , b 2 corresponding to the support interval [0, t 0 ] , in contrast to the constants A 1 , A 2 , B 1 , B 2 which are not involved in any conditions.
Remark 2.3.In view of (1.2), the spectral radius of the matrix M 0 + M 1 is less than one if the following inequality holds: We note that Theorem 2.1 can be easily extended to general n-dimensional systems.In that case, the assumption about the spectral radius of the corresponding matrix of order n can be checked using computer algebra programs such as Maple and Mathematica.
We conclude this paper by an example illustrating our main result.
O. Bolojan and R. Precup where a, b, c, d, m, n, p, q, f 1 , f 2 ∈ C[0, 1].In this case, and we have t 0 = 1/2 and g 1 , g 2 satisfy (2.3) with Also, h 1 , h 2 satisfy (2.4) with In addition, For this example Therefore, according to Theorem 2.1 and Remark 2.3, if then the problem (2.19) has at least one solution.
Here are three particular cases: ( where I stands for the unit matrix of the same order as M);(c) the eigenvalues of M are located inside the unit disc of the complex plane, i.e. ρ (M) < 1, where ρ (M) is the spectral radius of M; (d) I − M is nonsingular and inverse-positive, i.e. (I − M) −1 has nonnegative entries.Let us note that for a square matrix M ∈ M 2×2 (R + ) of order 2, M = a b c d , one has ρ (M) < 1 if and only if