Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 22, 1-11; http://www.math.u-szeged.hu/ejqtde/

Existence of Solutions for Nonconvex Third Order Differential Inclusions.

Britney Hopkins Department of Mathematics and Statistics University of Arkansas at Little Rock 2801 S. University Little Rock, AR 72204 bjhopkins1@ualr.edu

Abstract This paper proves the existence of solutions for a third order initial value nonconvex differential inclusion. We start with an upper semicontinuous compact valued multifunction F which is contained in a lower semicontinuous convex function ∂V and show that,

x^{(3)}(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x_{0}, x′(0) = y_{0}, x′′(0) = z_{0}.

Keywords: Nonconvex Differential Inclusions AMS Subject Classification: 34G20, 47H20

1 Introduction

The origins of boundary and initial value problems for differential inclusions are in the theory of differential equations and serve as models for a variety of applications including control theory. Existence results for the second order differential inclusion,

x′′ ∈ F (x, x′), x(0) = x_{0}, x′(0) = y_{0},

have been obtained by many authors (see [4], [5] and the references therein). In [5], Lupulescu showed existence for the problem

x′′ ∈ F (x, x′) + f(t, x, x′), x(0) = x_{0}, x′(0) = y_{0}

for the case in which F is an upper semicontinuous compact valued multifunction such the F (x, y) ⊂ ∂V (y) and f is a Carathéodory function.

In this paper, we prove an existence result for the third order differential inclusion,

x^{(3)}(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x_{0}, x′(0) = y_{0}, x′′(0) = z_{0},

where F is an upper semicontinuous compact valued multifunction and F (x, y, z) ⊂ ∂V (z) for some proper lower semicontinuous convex function V. Expounding upon the methods used to establish existence by Lupulescu in [4] and [5], we define a sequence of approximate solutions on a given interval and show that the sequence converges to an actual solution.

2 Preliminaries

Let ℝ^{m} be an m dimensional Euclidean space with an inner product ⟨·, ·⟩ and norm ∥ · ∥.

Let x ∈ ℝ^{m} and r > 0. The open ball centered at x with radius r is defined by

B_{r}(x) = {y ∈ ℝ^{m} : ∥x − y∥ < r},

where \overline{B}_{r}(x) denotes its closure.

For the proper lower semicontinuous convex function V : ℝ^{m} → ℝ, the multifunction ∂V : ℝ^{m} → 2^{ℝ^{m}} defined by

∂V (x) = {γ ∈ R^{m} : V (y) − V (x) ≥ ⟨γ, y − x⟩, ∀y ∈ ℝ^{m}} ,

is the subdifferential of V.

Let L^{2}[a, b] be a Hilbert space with the inner product defined by

⟨x, y⟩ = ∫_{a}^{b} x(t)\overline{y(t)}dt,

where \overline{y(t)} denotes the complex conjugate of y(t), and the norm is defined as

∥x∥ = \sqrt{∫_{a}^{b} |x(t)|^{2}dt}.

Let \overline{co}F (x, y, z) denote the closed convex hull of F and x_{n} ⇉ x denote that x_{n} converges uniformly to x.

We need the following theorems from Aubin and Cellina [1].

Theorem 0.3.4 Consider a sequence of absolutely continuous functions x_{k}(·) from an interval I to a Banach Space X satisfying

(i) for every t ∈ I, x_{k}(t)_{k} is a relatively compact subset of X ; (ii) there exists a positive function c(·) ∈ L^{2}(I) such that, for almost all t ∈ I, ∥x′_{k}(t)∥ ≤ c(t). Then there exists a subsequence, again denoted by x_{k}(·), converging to an absolutely continuous function x(·) from I to X in the sense that

(i) x_{k}(·) converges uniformly to x(·) over compact subsets of I ; (ii) x′_{k}(·) converges weakly to x′_{k}(·) in L^{2}(I, X).

Theorem 1.1.4 (the Convergence Theorem) Let F be a proper hemicontinuous map from a Hausdorff locally convex space X to the closed convex subsets of a Banach Space Y. Let I be an interval of ℝ and x_{k}(·) and y_{k}(·) be measurable functions from I to Y respectively satisfying for almost all t in I and for every neighborhood ℵ of 0 in X × Y, there exists a k_{0} = k_{0}(t, ℵ) such that for every k_{0} ≤ k, (x_{k}(t), y_{k}(t)) ∈ graph(F) + ℵ. If,

(i) x_{k}(·) converges almost everywhere to a function x(·) from I to X ;

(ii) y_{k}(·) belongs to L^{2}(I, Y) and converges weakly to y(·) in L^{2}(I, Y),

then, for almost all t ∈ I,

(x(t), y(t)) ∈ graph(F) i.e.y(t) ∈ F (x(t)).

We also need the following lemma from Brezis [2].

Lemma 3.3 Let u ∈ D(V) almost everywhere on [0, T] and suppose g ∈ L^{2}([0, T], ℝ) such that g(t) ∈ ∂V (u(t)) almost everywhere on [0, T]. Then, the function t ⟼ V (u(t)) is absolutely continuous on [0, T].

Also, let t ∈ [0, T] such that u(t) ∈ D(V) and let u and V(u) be differentiable. Then for all t ∈ [0, T]

\frac{d}{dt}V (u(t)) = ⟨h, \frac{du}{dt}(t)⟩ ∀h ∈ ∂V (u(t)).

3 The Main Result

THEOREM: If F : Ω → 2^{ℝ^{m}} and V : ℝ^{m} → ℝ satisfy the assumptions

(A1) Ω ⊂ ℝ^{2m} where Ω is open and F : Ω → 2^{ℝ^{m}} is a compact valued upper semicontinuous multifunction;

(A2) there exists a lower semicontinuous proper convex function V : ℝ^{m} → ℝ such that F (x, y, z) ⊂ ∂V (z) for every (x, y, z) ∈ Ω.

Then, for every (x_{0}, y_{0}, z_{0}) ∈ Ω, there exists a T > 0 and a solution x : [0, T] → ℝ^{m} of

x^{(3)}(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x_{0}, x′(0) = y_{0}, x′′(0) = z_{0}.  (1)

By a solution we are referring to an absolutely continuous function x : [0, T] → ℝ^{m} with absolutely continuous first and second derivatives with the initial values x(0) = x_{0}, x′(0) = y_{0}, x′′(0) = z_{0}, and x^{(3)}(t) ∈ F (x(t), x′(t), x′′(t)), a.e. on [0, T].

PROOF: Suppose (x_{0}, y_{0}, z_{0}) ∈ Ω. Then, K = B̄_{r}(x_{0}, y_{0}, z_{0}) ⊂ Ω for some r > 0 since Ω is open. By assumption (A1)

F (K) = ⋃_{(x,y,z)∈K} F (x, y, z)

is compact. Then there exists an M > 0 such that

sup{∥v∥ : v ∈ F (x, y, z), (x, y, z) ∈ K)} ≤ M.  (2)

Set

T < min \{\frac{r}{M} , (\frac{r}{M} )^{\frac{1}{2}} , (\frac{r}{M} )^{\frac{1}{3}} , \frac{r}{2∥z_{0}∥}, \frac{r}{2∥y_{0}∥}, (\frac{2r}{3∥z_{0}∥})^{\frac{1}{2}} \} .  (3)

Let n, j be integers where 1 ≤ j ≤ n. Set t_{n}^{j} = \frac{jT}{n}. For t ∈ [t_{n}^{j−1}, t_{n}^{j}] define,

x_{n}(t) = x_{n}^{j} + (t − t_{n}^{j}) y_{n}^{j} + \frac{1}{2} (t − t_{n}^{j})^{2} z_{n}^{j} + \frac{1}{6} (t − t_{n}^{j})^{3} v_{n}^{j},  (4)

where x_{n}^{0} = x_{0}, y_{n}^{0} = y_{0}, and z_{n}^{0} = z_{0}.

For 0 ≤ j ≤ n − 1 and v_{n}^{j} ∈ F (x_{n}^{j}, y_{n}^{j}, z_{n}^{j}) , define

\begin{cases}x_{n}^{j+1} & = x_{n}^{j} + (\frac{T}{n} ) y_{n}^{j} + \frac{1}{2} (\frac{T}{n} )^{2} z_{n}^{j} + \frac{1}{6} (\frac{T}{n} )^{3} v_{n}^{j} \\ y_{n}^{j+1} & = y_{n}^{j} + (\frac{T}{n} ) z_{n}^{j} + \frac{1}{2} (\frac{T}{n} )^{2} v_{n}^{j} \\ z_{n}^{j+1} & = z_{n}^{j} + (\frac{T}{n} ) v_{n}^{j}. \end{cases}  (5)

We claim that (x_{n}^{1}, y_{n}^{1}, z_{n}^{1}) ∈ K. Using (5), we have

∥z_{n}^{1} − z_{0}∥ = ∥z_{n}^{0} + (\frac{T}{n} ) v_{n}^{0} − z_{0}∥ ≤ (\frac{T}{n} ) M < r.

As well as,

∥y_{n}^{1} − y_{0}∥ = ∥y_{n}^{0} + (\frac{T}{n} ) z_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} v_{n}^{0} − y_{0}∥

≤ (\frac{T}{n} ) ∥z_{0}∥ + \frac{1}{2} (\frac{T}{n} )^{2} M

< \frac{1}{2}r + \frac{1}{2}r = r.

Also,

∥x_{n}^{1} − x_{0}∥ = ∥x_{n}^{0} + (\frac{T}{n} ) y_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} z_{n}^{0} + \frac{1}{6} (\frac{T}{n} )^{3} v_{n}^{0} − x_{0}∥

≤ (\frac{T}{n} ) ∥y_{0}∥ + \frac{1}{2} (\frac{T}{n} )^{2} ∥z_{0}∥ + \frac{1}{6} (\frac{T}{n} )^{3} M

< T ∥y_{0}∥ + \frac{1}{2}T^{2} ∥z_{0}∥ + \frac{1}{6}T^{3}M

< \frac{1}{2}r + \frac{1}{3}r + \frac{1}{6}r = r.

Hence the claim holds. Now suppose j ≥ 1. We make the assumption that,

\begin{cases}x_{n}^{j} = x_{n}^{0} + j (\frac{T}{n} ) y_{n}^{0} + \frac{1}{2} (\frac{jT}{n} )^{2} z_{n}^{0} + \frac{1}{6} (\frac{T}{n} )^{3} [(3j^{2} − 3j + 1) v_{n}^{0} + (3j^{2} − 9j + 7) v_{n}^{1} \\ + (3j^{2} − 15j + 19) v_{n}^{2} + ⋯ + 7v_{n}^{j−2} + v_{n}^{j−1}], \\ y_{n}^{j} = y_{n}^{0} + j (\frac{T}{n} ) z_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} [(2j − 1)v_{n}^{0} + (2j − 3)v_{n}^{1} + ⋯ + 3v_{n}^{j−2} + v_{n}^{j−1}] , \\ z_{n}^{j} = z_{n}^{0} + (\frac{T}{n} ) [v_{n}^{0} + v_{n}^{1} + ⋯ + v_{n}^{j−1}] . \end{cases}  (6)

To see this, let j = 1. Then,

z_{n}^{1} = z_{n}^{0} + (\frac{T}{n} ) v_{n}^{0} = z_{n}^{0+1},

y_{n}^{1} = y_{n}^{0} + \frac{T}{n} z_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} v_{n}^{0} = y_{n}^{0+1},

x_{n}^{1} = x_{n}^{0} + \frac{T}{n} y_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} z_{n}^{0} + \frac{1}{6} (\frac{T}{n} )^{3} v_{n}^{0} = x_{n}^{0+1}.

Thus, (6) holds when j = 1. Let’s suppose assumption (6) holds for j > 1. Using (5) we see that,

z_{n}^{j+1} = z_{n}^{j} + (\frac{T}{n} ) v_{n}^{j}

= z_{n}^{0} + (\frac{T}{n} ) [v_{n}^{0} + v_{n}^{1} + ⋯ + v_{n}^{j−1}] + (\frac{T}{n} ) v_{j}^{n}

= z_{n}^{0} + (\frac{T}{n} ) [v_{n}^{0} + v_{n}^{1} + ⋯ + v_{n}^{j}] .

Thus the assumption holds for z_{n}^{j}. Using this and (5) we have,

y_{n}^{j+1} = y_{n}^{j} + (\frac{T}{n} ) z_{n}^{j} + \frac{1}{2} (\frac{T}{n} )^{2} v_{n}^{j}

= y_{n}^{0} + j (\frac{T}{n} ) z_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} [(2j − 1)v_{n}^{0} + (2j − 3)v_{n}^{1} + ⋯ + 3v_{n}^{j−2} + v_{n}^{j−1}]

+ (\frac{T}{n} ) (z_{n}^{0} + \frac{T}{n} [v_{n}^{0} + v_{n}^{1} + ⋯ + v_{n}^{j−1}]) + \frac{1}{2} (\frac{T}{n} )^{2} v_{n}^{j}

= y_{n}^{0} + (j + 1) (\frac{T}{n} ) z_{n}^{0} + (\frac{T}{n} )^{2} [(j − \frac{1}{2}) v_{n}^{0} + (j − \frac{3}{2}) v_{n}^{1} + ⋯ + \frac{1}{2}v_{n}^{j−1} + v_{n}^{1} + … v_{n}^{j−1}]

+ \frac{1}{2} (\frac{T}{n} )^{2} v_{n}^{j}

= y_{n}^{0} + (j + 1) (\frac{T}{n} ) z_{n}^{0} + (\frac{T}{n} )^{2} [(j + \frac{1}{2}) v_{n}^{0} + (j − \frac{1}{2}) v_{n}^{1} + ⋯ + \frac{3}{2}v_{n}^{j−1} + \frac{1}{2}v_{n}^{j}]

= y_{n}^{0} + (j + 1) (\frac{T}{n} ) z_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} [(2j + 1)v_{n}^{0} + (2j − 1)v_{n}^{1} + ⋯ + v_{n}^{j}]

= y_{n}^{0} + (j + 1) (\frac{T}{n} ) z_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} [(2(j + 1) − 1)v_{n}^{0} + (2(j − 1) − 3)v_{n}^{1} + ⋯ + v_{n}^{j}] .

Thus the assumption holds for y_{n}^{j}. Finally, with this and (5) we have,

x_{n}^{j+1} = x_{n}^{j} + (\frac{T}{n} ) y_{n}^{j} + \frac{1}{2} (\frac{T}{n} )^{2} z_{n}^{j} + \frac{1}{6} (\frac{T}{n} )^{3} v_{n}^{j}

= x_{n}^{0} + j (\frac{T}{n} ) y_{n}^{0} + \frac{1}{2} (\frac{jT}{n} )^{2} z_{n}^{0} + \frac{1}{6} (\frac{T}{n} )^{3} [(3j^{2} − 3j + 1)v_{n}^{0} + (3j^{2} − 9j + 7)v_{n}^{1} + ⋯ + v_{n}^{j−1}]

+ (\frac{T}{n} ) (y_{n}^{0} + j (\frac{T}{n} ) z_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} [(2j − 1)v_{n}^{0} + (2j − 3)v_{n}^{1} + ⋯ + 3v_{n}^{j−2} + v_{n}^{j−1}])

+ \frac{1}{2} (\frac{T}{n} )^{2} (z_{n}^{0} + \frac{T}{n} [v_{n}^{0} + ⋯ + v_{n}^{j−1}]) + \frac{1}{6} (\frac{T}{n} )^{3} v_{n}^{j}

= x_{n}^{0} + (j + 1) (\frac{T}{n} ) y_{n}^{0} + \frac{1}{2}(j + 1)^{2} (\frac{T}{n} )^{2} z_{n}^{0}

+ \frac{1}{6} (\frac{T}{n} )^{3} [(3j^{2} − 3j + 1) v_{n}^{0} + (3j^{2} − 9j + 7) v_{n}^{1} + ⋯ + v_{n}^{j−1}]

+ (\frac{T}{n} )^{3} [jv_{n}^{0} + (j − 1)v_{n}^{1} + ⋯ + v_{n}^{j−1}] + \frac{1}{6} (\frac{T}{n} )^{3} v_{n}^{j}

= x_{n}^{0} + (j + 1) (\frac{T}{n} ) y_{n}^{0} + \frac{1}{2}(j + 1)^{2} (\frac{T}{n} )^{2} z_{n}^{0}

+ \frac{1}{6} (\frac{T}{n} )^{3} [(3j^{2} − 3j + 1) v_{n}^{0} + (3j^{2} − 9j + 7) v_{n}^{1} + ⋯ + v_{n}^{j−1}]

+ \frac{1}{6} (\frac{T}{n} )^{3} [6jv_{n}^{0} + (6j − 6)v_{n}^{1} + ⋯ + 6v_{n}^{j−1}] + \frac{1}{6} (\frac{T}{n} )^{3} v_{n}^{j}

= x_{n}^{0} + (j + 1) (\frac{T}{n} ) y_{n}^{0} + \frac{1}{2}(j + 1)^{2} (\frac{T}{n} )^{2} z_{n}^{0}

+ \frac{1}{6} (\frac{T}{n} )^{3} [(3j^{2} + 3j + 1) v_{n}^{0} + (3j^{2} − 3j + 1) v_{n}^{1} + ⋯ + 7v_{n}^{j−1} + v_{n}^{j}] .

Thus the assumption holds for x_{n}^{j}. Using (2), (3) and the relations in (6), we show that (x_{n}^{j}, y_{n}^{j}, z_{n}^{j}) ∈ K.

∥z_{n}^{j} − z_{0}∥ = ∥z_{n}^{0} + (\frac{T}{n} ) [v_{n}^{0} + v_{n}^{1} + ⋯ + v_{n}^{j−1}] − z_{0}∥

≤ j (\frac{T}{n} ) M

≤ T M < r.

And,

∥y_{n}^{j} − y_{0}∥ = ∥y_{n}^{0} + j (\frac{T}{n} ) z_{n}^{0} + \frac{1}{2} (\frac{T}{n} )^{2} [(2j − 1)v_{n}^{0} + (2j − 3)v_{n}^{1} + ⋯ + 3v_{n}^{j−2} + v_{n}^{j−1}] − y_{0}∥

≤ j (\frac{T}{n} ) ∥z_{0}∥ + \frac{1}{2} (\frac{jT}{n} )^{2} M

≤ T ∥z_{0}∥ + \frac{1}{2}T^{2}M

< \frac{1}{2}r + \frac{1}{2}r

= r.

Finally,

∥x_{n}^{j} − x_{0}∥ = ∥x_{n}^{0} + j (\frac{T}{n} ) y_{n}^{0} + \frac{1}{2} (\frac{jT}{n} )^{2} z_{n}^{0}

+\frac{1}{6} (\frac{T}{n} )^{3} [(3j^{2} − 3j + 1) v_{n}^{0} + (3j^{2} − 9j + 7) v_{n}^{1} + ⋯ + v_{n}^{j−1}] − x_{0}∥

≤ \frac{jT}{n} ∥y_{0}∥ + \frac{1}{2} (\frac{jT}{n} )^{2} ∥z_{0}∥ + \frac{1}{6} (\frac{jT}{n} )^{3} M

≤ T ∥y_{0}∥ + \frac{1}{2}T^{2}∥z_{0}∥ + \frac{1}{6}T^{3}M

< \frac{1}{2}r + \frac{1}{2} (\frac{2}{3}) r + \frac{1}{6}r = r.

Thus, (x_{n}^{j}, y_{n}^{j}, z_{n}^{j}) ∈ K = B_{r} (x_{0}, y_{0}, z_{0}) for 1 ≤ j ≤ n. Now, from the definition of x_{n} in (4) we have,

\begin{cases}x′_{n}(t) = y_{n}^{j} + (t − t_{n}^{j}) z_{n}^{j} + \frac{1}{2} (t − t_{n}^{j})^{2} v_{n}^{j}, \\ x′′_{n}(t) = z_{n}^{j} + (t − t_{n}^{j}) v_{n}^{j}, \\ x_{n}^{(3)}(t) = v_{n}^{j}. \end{cases}  (7)

By (2) we have that ∥x_{n}^{(3)}(t)∥ = ∥v_{n}^{j}∥ ≤ M. Similarly, (2) and (3) give the following,

∥x′′_{n}(t)∥ = ∥z_{n}^{j} + (t − t_{n}^{j}) v_{n}^{j}∥

= ∥z_{n}^{0} + (\frac{T}{n} ) [v_{n}^{0} + v_{n}^{1} + ⋯ + v_{n}^{j−1}] + (t − t_{n}^{j}) v_{n}^{j}∥

≤ ∥z_{0}∥ + (\frac{jT}{n} ) M + (\frac{T}{n} ) M

< ∥z_{0}∥ + 2r.

As well as,

∥x′_{n}(t)∥ = ∥y_{n}^{j} + (t − t_{n}^{j}) z_{n}^{j} + \frac{1}{2} (t − t_{n}^{j})^{2} v_{n}^{j}∥

≤ ∥y_{0}∥ + (\frac{jT}{n} ) ∥z_{0}∥ + \frac{1}{2} (\frac{jT}{n} )^{2} M + (\frac{T}{n} ) ∥z_{0}∥ + (\frac{jT}{n} )^{2} M + \frac{1}{2} (\frac{T}{n} )^{2} M

≤ ∥y_{0}∥ + T ∥z_{0}∥ + 2T^{2}M + T ∥z_{0}∥

< ∥y_{0}∥ + 3r.

And finally,

∥x_{n}(t)∥ = ∥x_{n}^{j} + (t − t_{n}^{j}) y_{n}^{j} + \frac{1}{2} (t − t_{n}^{j})^{2} z_{n}^{j} + \frac{1}{6} (t − t_{n}^{j})^{3} v_{n}^{j}∥

≤ ∥x_{0}∥ + T ∥y_{0}∥ + \frac{1}{2}T^{2} ∥z_{0}∥ + \frac{1}{6}T^{3}M + (\frac{T}{n} ) (∥y_{0}∥ + T ∥z_{0}∥ + \frac{1}{2}T^{2}M)

+ \frac{1}{2} (\frac{T}{n} )^{2} (∥z_{0}∥ + T M) + \frac{1}{6} (\frac{T}{n} )^{3} M

≤ ∥x_{0}∥ + T ∥y_{0}∥ + \frac{1}{2}T^{2} ∥z_{0}∥ + \frac{1}{6}T^{3}M + T ∥y_{0}∥ + T^{2} ∥z_{0}∥

+ \frac{1}{2}T^{3}M + \frac{1}{2}T^{2}M ∥z_{0}∥ + \frac{1}{2}T^{3}M + \frac{1}{6}T^{3}M

= ∥x_{0}∥ + 2T ∥y_{0}∥ + 2T^{2} ∥z_{0}∥ + T^{3}M + \frac{1}{3}T^{3}M

< ∥x_{0}∥ + r + \frac{4}{3}r + r + \frac{1}{3}r

< ∥x_{0}∥ + 4r.

Since ∥x_{n}^{(3)}(t)∥ ≤ M ∀t ∈ [0, T] the sequence (x_{n}^{(3)}(t)) is bounded in L^{2} ([0, T], R^{m}). Furthermore, suppose ε > 0 and ∀t ∈ [0, T], and ∀τ ∈ [0, T], |t − τ| < \frac{ε}{M} . Then,

∥x′′_{n}(t) − x′′_{n}(τ)∥ ≤ |∫_{τ}^{t} ∥x_{n}^{(3)}(s)∥ds|

≤ |∫_{τ}^{t} Mds|

= M|t − τ|

≤ M (\frac{ε}{M} )

= ε.

Thus, (x′′_{n}) is equicontinuous. Similarly (x′_{n}) and (x_{n}) are equicontinuous. Theorem 0.3.4 in [1] gives the following:

There exists a subsequence, again denoted (x_{n})_{n} that converges to an absolutely continuous function x : [0, T] → R^{m} such that:

(i) (x_{n}) ⇉ x on [0, T],
(ii) (x′_{n}) ⇉ x′ on [0, T],
(iii) (x′′_{n}) ⇉ x′′ on [0, T],
(iv) (x_{n}^{(3)}) converges weakly to x^{3} in L^{2} ([0, T], R^{m}) .

By the Convergence Theorem, theorem 1.4.1 in [1], we have that

x^{(3)}(t) ∈ \overline{co}F (x(t), x′(t), x′′(t)) ⊂ ∂V (x′′(t)) a.e., t ∈ [0, T].

Also, by the above and lemma 3.3 in [2],

\frac{d}{dt}V (x′′(t)) = ⟨x^{(3)}(t), x^{(3)}(t)⟩ = ∥x^{(3)}(t)∥^{2} .

Since, ∫_{0}^{T} \frac{d}{dt}V (x′′(t)) dt = ∫_{0}^{T} ∥x^{(3)}(t)∥^{2} dt we have,

V (x′′(T)) − V (x′′(0)) = ∫_{0}^{T} ∥x^{(3)}(t)∥^{2} dt.  (8)

However, by (7) we also have x_{n}^{(3)}(t) = v_{n}^{j} ∈ F (x_{n}^{j}, y_{n}^{j}, z_{n}^{j}) ⊂ ∂V (x′′_{n}(t_{n}^{j})) , ∀t ∈ [t_{n}^{j−1}, t_{n}^{j}]. Which, from the definition of subdifferential, gives the following,

V (x′′_{n} (t_{n}^{j})) − V (x′′_{n} (t_{n}^{j−1})) ≥ ⟨x_{n}^{(3)}(t), x′′_{n} (t_{n}^{j}) − x′′_{n} (t_{n}^{j−1})⟩

= ⟨x_{n}^{(3)}(t), ∫_{t_{n}^{j−1}}^{t_{n}^{j}} x_{n}^{(3)}(s)ds⟩

= ∫_{t_{n}^{j−1}}^{t_{n}^{j}} ⟨x_{n}^{(3)}(t), x_{n}^{(3)}(t)⟩ dt

= ∫_{t_{n}^{j−1}}^{t_{n}^{j}} ∥x_{n}^{(3)}(t)∥^{2} dt.

Combining the above inequalities with (8), we get the following inequality,

V (x′′_{n}(T)) − V (z_{0}) ≥ ∫_{0}^{T} ∥x_{n}^{(3)}(t)∥^{2} dt.

If we let n approach infinity, we have

V (x′′(T)) − V (z_{0}) ≥ lim sup_{n→∞} ∫_{0}^{T} ∥x_{n}^{(3)}(t)∥^{2} dt
∫_{0}^{T} ∥x^{(3)}(t)∥^{2} dt ≥ lim sup_{n→∞} ∫_{0}^{T} ∥x_{n}^{(3)}(t)∥^{2} dt
∥x^{(3)}(t)∥^{2} ≥ lim sup_{n→∞} ∥x_{n}^{(3)}(t)∥^{2} .

However, the weak lower semicontinuity of the norm gives,

∥x^{(3)}(t)∥^{2} ≤ lim inf_{n→∞} ∥x_{n}^{(3)}(t)∥^{2} .

Thus we have,

∥x^{(3)}(t)∥^{2} = lim_{n→∞} ∥x_{n}^{(3)}(t)∥^{2} .

Hence the sequence (x_{n}^{(3)}) → x^{(3)} pointwise. By assumption (A1), F is closed, implying

x^{(3)}(t) ∈ F (x(t), x′(t), x′′(t)), a.e. t ∈ [0, T].

References

[1] J. P. Aubin and A. Cellina; Differential inclusions, Berlin, Springer-Verlag, 1984.

[2] H. Brezis; Operateurs Maximaux Monotones et Semigroups de Contractions Dans Les Espaces de Hilbert, Amsterdam, North-Holland, 1973.

[3] K. Deimling; Multivalued Differential Equations, Walter de Gruyter, Berlin, New York, 1992.

[4] V. Lupulescu; Existence of Solutions for Nonconvex Functional Differential Inclusions, Electronic Journal of Differential Equations, Vol. 2004(2004), No. 141, pp.1-6.

[5] V. Lupulescu; Existence of Soutions for Nonconvex Second Order Differential Inclusions, Applied Mathematics E-Notes, 3(2003), 115-123.

(Received September 16, 2005)