GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, 501-508

NONCONVEX DIFFERENTIAL INCLUSIONS WITH NONLINEAR MONOTONE BOUNDARY CONDITIONS

S. A. BRYKALOV

Abstract. Existence results for problems with monotone nonlinear boundary conditions obtained in the previous publications by the author for functional differential equations are transferred to the case of nonconvex differential inclusions with the help of the selection theorem due to A. Bressan and G. Colombo.

The existence of solutions of boundary value problems for differential inclusions with possibly nonconvex right-hand sides was studied in [1–6]. The technique of continuous selections of multifunctions with decomposable values is helpful in these investigations. In particular, this technique allows us to establish a connection between the considered differential inclusion and a functional differential equation. Below we use this connection to transfer some assertions, previously proved by the author for functional differential equations [7–9], to the case of differential inclusions. Thus we obtain solvability results for problems for lower semicontinuous differential inclusions with nonlinear monotone boundary conditions. We essentially employ the selection theorem for multifunctions with decomposable values due to Bressan and Colombo [10]. A systematic account of different aspects of the theory of differential inclusions and the corresponding bibliography can be found in [11–13]. For results on boundary value problems, see e.g., [14, 15] in the case of equations and [16] in the case of convex-valued inclusions, and the references therein.

The following notation is used below. We fix a norm in the n-dimensional space ℝ^{n} and denote it by | · |_{n}. Let C^{k} with an integer k ≥ 0 denote the space of k times continuously differentiable functions (C^{0} is the class of all continuous functions). The space L_{1} consists of all measurable integrable functions. Here and everywhere below we use the Lebesgue measure. CL_{1}^{m}

501 for integer m ≥ 1 contains all x(·) ∈ C^{m−1} such that x^{(m−1)}(·) is absolutely continuous, and consequently x^{(m)}(·) ∈ L_{1}. We put

1991 Mathematics Subject Classification. 34B15, 34A60, 34K10.

Key words and phrases. Boundary value problems, differential inclusions, nonlinear boundary conditions, functional differential equations.

1072-947X/97/1100-0501$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

∥ x(·) ∥_{CL_{1}^{m}} = ∥ x(·) ∥_{C^{m−1}} + ∥ x^{(m)}(·) ∥_{L_{1}}.

Unless otherwise stated explicitly, we assume that all considered functional spaces consist of functions from [a, b] to ℝ^{n}. The interval [a, b] is fixed, −∞ < a < b < +∞.

Let us recall some definitions. For a set A ⊂ [a, b] denote by χ_{A}(·) the characteristic function of A. Thus χ_{A}(t) = 1 if t ∈ A, and χ_{A}(t) = 0 otherwise. A set K ⊂ L_{1} is called decomposable if for any u(·), v(·) ∈ K and any measurable A ⊂ [a, b] the function w(t) = χ_{A}(t)u(t) + χ_{[a,b]\A}(t)v(t) belongs to K. A multifunction G from X to Y, where X, Y are metric spaces, is called lower semicontinuous (l.s.c.) if for any closed E ⊂ Y the set {x ∈ X : G(x) ⊂ E} is closed.

We begin with a description of the required selection technique in the form convenient to us. Let us fix an arbitrary mapping h : CL_{1}^{m} → L_{1}, an arbitrary set V ⊂ CL_{1}^{m}, and a function Θ : [a, b] × [0, ∞) → [0, ∞) such that the function Θ(·, N) is measurable for any real number N ≥ 0. Consider some l.s.c. multifunction G from C^{m−1} to L_{1} with nonempty closed decomposable values. Assume that if y(·) ∈ G(x(·)), then the inequality

| y(t) |_{n}≤ Θ(t, ∥ x(·) ∥_{C^{m−1}})  (1)

holds a.e. on [a, b].

Lemma. Assume that for any continuous g : C^{m−1} → L_{1} which for all x : [a, b] → ℝ^{n}, x(·) ∈ C^{m−1}, satisfies the estimate

| g(x(·))(t) |_{n}≤ Θ(t, ∥ x(·) ∥_{C^{m−1}})  (2)

a.e. on [a, b], there exists some x(·) ∈ V such that h(x(·)) = g(x(·)). Then there exists at least one x(·) ∈ V for which h(x(·)) ∈ G(x(·)).

Remark 1. Inequalities (1), (2) can be replaced by the relations y(t) ∈ U(t, x(·)), g(x(·))(t) ∈ U(t, x(·)) respectively, where U is a fixed multifunction from [a, b] × C^{m−1} to ℝ^{n} with nonempty values.

Proof of Lemma consists in a direct application of the selection theorem of A. Bressan and G. Colombo (Theorem 3 in [10]), due to which the assumptions on G ensure the existence of a continuous selection g_{0} : C^{m−1} → L_{1}, g_{0}(x(·)) ∈ G(x(·)). Because of (1) the mapping g_{0} satisfies (2). According to the conditions of Lemma, one can find a function x(·) ∈ V such that h(x(·)) = g_{0}(x(·)) ∈ G(x(·)), which completes the proof.

We can apply Lemma to a concrete problem in the following way. Take for V the set of all functions that satisfy the boundary conditions. Suppose that we know an existence result for a boundary value problem h(x(·)) = g(x(·)), x(·) ∈ V. Here we deal with a functional differential equation with an arbitrary continuous mapping g which satisfies the growth restrictions. Based on this result and the lemma, we can immediately obtain an existence theorem for the boundary value problem h(x(·)) ∈ G(x(·)), x(·) ∈ V. That is, we can consider a differential inclusion with the possibly nonconvex righthand side G satisfying the growth conditions.

In particular, the differential inclusion h(x(·)) ∈ G(x(·)) may have the form

x^{(m)}(t) + a_{m−1}(t)x^{(m−1)}(t) + … + a_{0}(t)x(t) ∈
∈ F (t, x(t), …, x^{(m−1)}(t)), t ∈ [a, b],  (3)

where a_{0}(t), …, a_{m−1}(t) are n×n matrices, F is defined on [a, b]×ℝ^{mn} and its values are closed sets in ℝ^{n}. In that case h(x(·))(t) coincides with the left-hand side of (3), and G(x(·)) is the set of all measurable selections of the multifunction t → F (t, x(t), …, x^{(m−1)}(t)). Conditions on F can be given that ensure the necessary properties of the corresponding multifunction G; see Proposition 2.1 in [17]. Namely, in addition to the upper estimate that guarantees (1), one should impose on F properties of measurability and of lower semicontinuity in the arguments x(t), …, x^{(m−1)}(t). In particular, these assumptions hold for Hausdorff continuous F.

Now, let us apply the above reasoning to the case of nonlinear monotone boundary conditions. Consider a boundary value problem

x^{(m)}(·) ∈ G(x(·));  (4)
B_{ki}(x_{i}^{(k)}(·), x(·)) = 0, k = 0, …, m − 1, i = 1, …, n.  (5)

Here x : [a, b] → ℝ^{n}, x(·) ∈ CL_{1}^{m} is the unknown function. By x_{i}^{(k)}(·) we denote the coordinate number i of the derivative x^{(k)}(·). The multifunction G from C^{m−1} to L_{1} is lower semicontinuous and has nonempty closed decomposable values. For some fixed integrable function ξ(t) ≥ 0, for any x(·) ∈ C^{m−1} and y(·) ∈ G(x(·)), the inequality

| y(t) |_{n}≤ ξ(t)  (6)

holds a.e. on [a, b]. The mappings B_{ki} : C^{0} × CL_{1}^{m} → ℝ are continuous. (Here C^{0} = C^{0}([a, b], ℝ) is the space of scalar functions, whereas the other employed functional spaces have values in ℝ^{n}.)

Theorem 1. Let the nonlinear functionals B_{ki} be monotone with respect to the first argument in the following sense. For any u(·), v(·), z(·), if u(t) ≤ v(t) for all t ∈ [a, b], then B_{ki}(u(·), z(·)) ≤ B_{ki}(v(·), z(·)). Fix nondecreasing functions Ω_{ki} : [0, ∞) → [0, ∞). For any indices k, i and function z(·), let there exist some u(·) such that B_{ki}(u(·), z(·)) = 0 and

∥ u(·) ∥_{C^{0}} ≤ Ω_{ki}(∥ z^{(k+1)}(·) ∥_{CL_{1}^{m−k−1}})  (7)

(for k = m − 1 we take CL_{1}^{0} = L_{1}). Then problem (4), (5) has at least one solution.

Remark 2. In fact, a more general assertion is valid. For any fixed u(·), z(·), let the function B_{ki}(u(·) + c, z(·)) be nondecreasing with respect to the real argument c. Fix mappings Υ_{ki} : [0, ∞) × [0, ∞) → [0, ∞) which are nondecreasing in each of their two arguments. For any k, i, u(·), z(·), let there exist a number c such that the equality B_{ki}(u(·) + c, z(·)) = 0 and the estimate

| c |≤ Υ_{ki}(∥ u(·) ∥_{C^{0}}, ∥ z^{(k+1)}(·) ∥_{CL_{1}^{m−k−1}})

hold. Then problem (4), (5) has at least one solution.

It is easy to check that the requirement for monotonicity of B_{ki}(u(·) + c, z(·)) with respect to the number c is weaker than the assumption of monotonicity of B_{ki}(u(·), z(·)) with respect to the function u(·).

Remark 3. In the assumptions of Theorem 1, let the functions Ω_{ki}(M) = p_{ki}M + q_{ki} be linear. Then estimate (6) can be replaced by the following weaker requirement. For some fixed integrable ζ(t) ≥ 0 and nondecreasing Φ : [0, ∞) → [0, ∞) such that

lim inf_{M→∞} M^{−1}Φ(M) = 0  (8)

for any x(·) ∈ C^{m−1} and y(·) ∈ G(x(·)), the inequality

| y(t) |_{n}≤ ζ(t) + Φ(∥ x(·) ∥_{C^{m−1}})  (9)

holds a.e. on [a, b].

In connection with Remark 3, let us specify a simple case in which functions Ω_{ki} can be chosen not only linear, but even constant. This is the case where the mappings B_{ki}(u(·), z(·)) = B_{ki}(u(·)) do not depend on the second argument.

Proof of Theorem 1. Put

h(x(·))(t) = x^{(m)}(t), Θ(t, N) = ξ(t)

and take for V the set of all x(·) ∈ CL_{1}^{m} that satisfy the boundary conditions (5). The application of Lemma reduces Theorem 1 to an existence result for the case of functional differential equations, which can be found in [7, 8].

In the same way, Remarks 2 and 3 are reduced to existence results of [7, 8]. The only difference is that to prove Remark 3 we put Θ(t, N) = ζ(t)+Φ(N).

To demonstrate the application of Theorem 1 and Remark 3, let us employ the following boundary value problem with nonlinear integral boundary conditions:

x^{(m)}(·) ∈ G(x(·));  (10)
∫_{a}^{b} φ_{k}(x^{(k)}(τ))dτ = 0, k = 0, …, m − 1.  (11)

The unknown function x(·) ∈ CL_{1}^{m}([a, b], ℝ) is scalar, G is a lower semicontinuous multifunction from C^{m−1}([a, b], ℝ) to L_{1}([a, b], ℝ) with nonempty closed decomposable values. We fix an integrable function ζ(t) ≥ 0 and numbers ε ∈ (0, 1), A ≥ 0. Assume that the estimate

| y(t) |≤ ζ(t) + A∥ x(·) ∥_{C^{m−1}}^{1−ε}

holds for any x(·) ∈ C^{m−1}([a, b], ℝ), y(·) ∈ G(x(·)) and almost all t ∈ [a, b]. For any index k, the function φ_{k} : ℝ → ℝ is continuous, nondecreasing, and takes the value zero at least at one point.

Corollary. Problem (10), (11) admits a solution.

Proof of Corollary is based on Theorem 1 and Remark 3. We take

B_{k}(u(·), z(·)) = ∫_{a}^{b} φ_{k}(u(τ))dτ

omitting the index i in B_{ki} as n = 1. The required continuity and monotonicity in u(·) hold. To ensure that B_{k}(u(·), z(·)) = 0, it suffices to put u(t) ≡ c_{k}, where c_{k} is a constant such that φ_{k}(c_{k}) = 0. Thus in (7) the functions Ω_{k} ≡| c_{k} | can be taken constant, and Remark 3 is applicable. We assume in (9) Φ(M) = AM^{1−ε}, which makes (8) valid. Problem (10), (11) coincides with (4), (5) and is solvable.

In [7], [8] one can find another example of nonlinear integral boundary conditions of form (5), which contain some odd-degree moments of the graph of the solution and are related to the center of gravity of this graph. Now, based on [9], we will give an example with a minimum and a maximum in the boundary conditions.

Consider the problem

ẍ(·) ∈ G(x(·)),  (12)
min_{t} x(t) = α,  (13)
max_{t} x(t) = β.  (14)

The solution x is a scalar function belonging to the space CL_{1}^{2}([a, b], ℝ). The numbers α, β are fixed. Conditions (13), (14) specify the minimal and maximal values of the unknown function on the interval [a, b]. The lower semicontinuous multifunction G from C^{1}([a, b], ℝ) to L_{1}([a, b], ℝ) has nonempty closed decomposable values. Assume also that for fixed M ≥ 0 and any x(·), y(·), if y(·) ∈ G(x(·)), then | y(t) |≤ M a.e. on [a, b].

Theorem 2. If β − α > \frac{M}{8} (b − a)^{2}, then problem (12)–(14) is solvable.

The constant \frac{M}{8} (b−a)^{2} in Theorem 2 is the best possible one and cannot be replaced by a smaller one, see Remark 1 in [9].

One can derive Theorem 2 from Theorem 1 after some transformation of conditions (13), (14). The transformation of (13), (14) was described in detail in [9], where it was applied to the case of ordinary differential equations. The reasoning in the case of differential inclusion (12) is quite similar. The paper [9] also gives some properties of solutions concerned with the points of a minimum and a maximum. Actually, in the proof in [9], two solutions with different properties appear.

Acknowledgement

The author is grateful to Professor F. S. De Blasi and Professor G. Pianigiani for the stimulating discussions that initiated the present publication.

This research was partly financed by the Russian Foundation for Fundamental Research, Grant 94-01-00310. The research described in this publication was made possible in part by Grant NMS000 from the International Science Foundation.

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(Received 18.12.95)

Author’s address: Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences ul. Kovalevskoi 16 620066 Ekaterinburg Russia