Nonlocal conditions for differential inclusions in the space of functions of bounded variations

Abstract

We discuss the existence of solutions of an abstract differential inclusion, with a right-hand side of bounded variation and subject to a nonlocal initial condition of integral type.

AMS Subject Classification

34A60, 34G20, 26A45, 54C65, 28B20

1 Introduction

Solutions of differential equations with smooth enough coefficients cannot have jump discontinuities, see for instance [1, 2]. The situation is quite different for systems described by differential equations with discontinuous right-hand sides [3]. Examples of such systems are mechanical systems subjected to dry or Coulomb frictions [4], optimal control problems where the control parameters are discontinuous functions of the state [5], impulsive differential equations [6], measure differential equations, pulse frequency modulation systems or models for biological neural nets [7]. For these systems the state variables undergo sudden changes at their points of discontinuity. The mathematical models of many of these systems are described by multivalued differential equations or differential inclusions [8].

Let X be a Banach space with norm |·| _X . Then X is a metric space with the distance d _X defined by

Let I = [0, T] be a compact real interval. We are interested in the study of the following multivalued nonlocal initial value problem

(1)

where F : I × X → X is a multivalued map and g : X → X is continuous.

The investigation of systems subjected to nonlocal conditions started with [9] for partial differential equations and [10] for Sturm-Liouville problems. For more recent work we refer the interested reader to [11] and the references therein.

It is clear that solutions of (1) are solutions of the integral inclusion

(2)

2 Preliminaries

Definition 1 We say that f : I → X is of bounded variation, and we write f ∈ BV (I, X), if

where Π: τ ₀ = 0 < τ ₁ < ⋯ < τ _m = T is any partition of I. The quantity is called the total variation of f.

We shall denote by BV(I, X) the space of all functions of bounded variations on I and with values in X. It is a Banach space with the norm |·| _b given by

In order to discuss the integral inclusion (2) we present some facts from set-valued analysis. Complete details can be found in the books [8, 12, 13]. Let (X, |·| _X ) and (Y, |·| _Y ) be Banach spaces. We shall denote the set of all nonempty subsets of X having property ℓ by ℘ _ℓ (X). For instance, A ∈ ℘ _{c ℓ} (X) means A closed in X, when ℓ = b we have the bounded subsets of X, ℓ = cv for convex subsets, ℓ = cp for compact subsets and ℓ = cp, cv for compact and convex subsets. The domain of a multivalued map ℜ: X → Y is the set domℜ = {z ∈ X; ℜ(z) ≠ ∅}. ℜ is convex (closed) valued if ℜ(z) is convex (closed) for each z ∈ X: ℜ has compact values if ℜ(z) ∈ ℘ _cv(Y) for every z ∈ X; ℜ is bounded on bounded sets if ℜ(A) = ∪ _z∈A ℜ(z) is bounded in Y for all A ∈ ℘ _b (X) (i.e. sup _z∈A {sup{|y| _Y ; y ∈ ℜ(z)}} < ∞): ℜ is called upper semicontinuous (u.s.c.) on X if for each z ∈ X the set ℜ(z) ∈ ℘ _cl (Y) is nonempty, and for each open subset Λ of Y containing ℜ(z), there exists an open neighborhood Π of z such that ℜ(Π) ⊂ Λ. In terms of sequences, ℜ is u.s.c. if for each sequence (z _n ) ⊂ X, z _n → z ₀, and B a closed subset of Y such that ℜ(z _n ) ∩ B ≠ ∅, then ℜ(z ₀) ∩ B ≠ ∅. The set-valued map ℜ is called completely continuous if ℜ(A) is relatively compact in Y for every A ∈ ℘(X). If ℜ is completely continuous with nonempty compact values, then ℜ is u.s.c. if and only if ℜ has a closed graph (i.e. z _n → z, w _n → w, w _n ∈ ℜ(z _n ) ⇒ w ∈ ℜ(z)). When X ⊂ Y then ℜ has a fixed point if there exists z ∈ X such z ∈ ℜ(z). A multivalued map ℜ: J → ℘ _cl (X) is called measurable if for every x ∈ X, the function θ : J → ℝ defined by θ(t) = dist(x, ℜ(t)) = inf{|x - z| _X ; z ∈ ℜ(t)} is measurable. |ℜ(z)| _Y denotes sup{|y| _Y ; y ∈ ℜ(z)}.

If A and B are two subsets of X, equipped with the metric d _X , such that d _X (x, y) = |x - y| _X , the Hausdorff distance between A and B is defined by

Where

It is well known that (℘ _b,cl (X), d _H ) is a metric space and so is (℘ _cp (X), d _H ).

Definition 2 (See [14, 15]) Θ: I → X is of bounded variation (with respect to d _H ) on I if

where the supremum is taken over all partitions Π = {t _i ; i = 1, 2, ..., m} of the interval I.

Definition 3 Let X ^I denote the set of all functions from I into X. The Nemitskii (or superposition) operator corresponding to F : I × X → X is the operator

defined by

Definition 4 The multifunction F : I X → X is of bounded variation if for any function × ∈ BV(I, X) the multivalued map N _F (x): I → X is of bounded variation on I (in the sense of Definition 2) and

Definition 5 Let Δ be a subset of I × X. We say that Δ is measurable if Δ belongs to the σ- algebra generated by all sets of the form J × D where J is Lebesgue measurable in I and D is Borel measurable in X.

Theorem 6 (Generalized Helly selection principle) [[14], Theorem 5.1 p. 812] Let K be a compact subset of the Banach space × and let be a family of maps of uniformly bounded variation from I into K. Then there exists a sequence of maps convergent pointwise on I to a map f : I → K of bounded variation such that .

In the next theorem we shall denote by and ∂U the closure and the boundary of a set U.

Theorem 7 ([[16], Theorem 3.4, p. 34]) Let U be an open subset of a Banach space Z with 0 ∈ U. Let be a single-valued operator and be a multivalued operator such that

• (i) is bounded,

• (ii) A is a contraction with constant k ∈ (0, 1/2),

• (iii) B is u.s.c and compact.

Then either

• (a) the operator inclusion λ x ∈ Ax + Bx has a solution for λ = 1, or

• (b) there is an element u ∈ ∂U such that λ u ∈ Au + Bu for some λ > 1.

3 Main results

In this section we state and prove our main result. We should point out that no semicontinuity property is assumed on the multifunction F, which is usually the case in the literature. We refer the interested reader to the nice collection of papers in [17] and the references therein.

Theorem 8 Assume that the following conditions hold.

(H1) g : X → X is continuous, g(0) = 0 and there exists θ : [0, + ∞) → [0, + ∞) continuous and θ(r) ≤ βr, with β < 1/2 and βT ≠ 1, such that

(H2) F : I × X → ℘ _cp,cv(X) is of bounded variation such that

• (i) (t, x) ↦ F(t, x) is measurable,

• (ii) there exists an integrable function q : I → [0, + ∞) with

  

• (iii) x _k → x as k → ∞ pointwise implies d _H (F(t, x _k ), F(t, x)) → 0, k → ∞.

Then problem (1) has at least one solution in BV(I, X).

Proof. Let . We show that there exists M > 0 such that all possible solutions of (2) in BV(I, X), satisfy

Recall that solutions of (1) satisfy

(3)

Since the multivalued map N _F (x): I → X is of bounded variation it admits a selector f : I → X of bounded variation such that

see [[18], Theorem A, p. 250].

It follows from (3) that

(4)

This implies

The condition on g and (H2) (ii) imply

Hence

This last inequality yields

Since

we obtain

so that

(5)

Inequality (5) and the condition on g imply that

Hence any possible solution x of (2) in BV(I, X), satisfies

Let Π = {t _i ; i = 1, 2, ..., m} be any partition of the interval I, and let x ∈ BV(I, X) be any possible solution of (2). It follows from (4) that

It is easily shown that

Therefore

Letting , we see that

Let

Define two operators

by

and

First, we show that is bounded, i.e. .

Let . Then there exists such that

It follows from (3) that

(H1) implies that the single-valued operator A is a contraction with constant k ∈ (0, 1/2).

Claim 1. The multivalued operator B has compact and convex values. For, since F : I × X → ℘ _cp,cv(X) it follows that NF : X ^I → ℘ _cp,cv(X), i.e. has compact and convex values. This implies that the Aumann integral

has compact and convex values. See for instance [5].

Claim 2. B is completely continuous, i.e. B (Ω) is a relatively compact subset of BV(I, X). Let q ∈ Ω be arbitrary. Then for every f ∈ N _F (q) the function u : I → X defined by

satisfies

If we write

then the operator ϒ: X → X is continuous and

Let (Bx _k ) _k≥1be a sequence in B (Ω). Then the sequence (x _k ) _k≥1is uniformly bounded and is of bounded variation. Theorem 4 shows that there exists a subsequence, which we label the same, and which converges pointwise to y ∈ Ω. We have

Assumption (H2) (iii) implies that

This proves the claim.

Claim 3. B is u.s.c. Since B is completely continuous it is enough to show that its graph is closed. Let {(x _n , y _n )} _n≥1be a sequence in graph(B) and let (x, y) = lim _n→∞ (x _n , y _n ). Then y _n ∈ B(x _n ), i.e , t ∈ I. This implies that

Since x _n → x in X it follows from (H2)(ii) that

which shows that

Hence (x, y) ∈ graph(B), and B has a closed graph.

Finally, alternative (b) in Theorem 5 cannot hold due to (3) and the choice of Ω.

By Theorem 5 the inclusion

has at least one solution in BV(I, X). This completes the proof of the theorem.

For our second result we consider the case when , where ψ : I → ℝ is continuous. Let

From the definition of the function λ we infer that, if ψ* = max _t∈I |ψ(t)|,

Theorem 9 Assume that the following conditions hold

(H3) ψ : I → ℝ is continuous and ψ ₀ ≠ 1,

(H4) F : I × X → ℘ _cp,cv(X) is of bounded variation such that

• (i) (t, x) ↦ F(t, x) is measurable,

• (ii) there exists ω : I × [0, ∞) → (0, ∞) continuous, nondecreasing with respect to its second argument and

  

  (6)

  such that |F(t, x) | _X ≤ ω → (t, |x| _b ).

• (iii) x _k → x pointwise as k → ∞ implies d _H (F (t, x _k ), F (t, x)) → 0 as k → ∞.

Then problem (1) has at least one solution in BV(I, X).

Proof. Since the multivalued map N _F (x): I → X is of bounded variation it admits a selector h : I → X of bounded variation such that

see [[18], Theorem A, p. 250].

Solutions of (2) satisfy

(7)

Substituting the initial condition in (7) we obtain

Since ψ ₀ ≠ 1 it follows that

Thus, solutions of (2) are solutions of

(8)

and vice versa. It follows from (8)

The upper bound on |λ (s)| implies

(9)

which gives

Let Π = {t _i ; i = 1, 2, ..., m} be any partition of the interval I, and let x ∈ BV(I, X) be any possible solution of (2). Then, it follows from (7) that

which leads to

Since , we have

Finally, we see that

(10)

Let

Then (10) yields

(11)

The condition on the function ω implies that there exists ρ* > 0 such that for all ρ > ρ*

(12)

Comparing inequalities (11) and (12) we see that

Let

Then Σ is nonempty, closed, bounded and convex.

Define a multivalued operator

by

(13)

Then solutions of (2) are fixed point of the multivalued operator .

It is clear that . Proceeding as in the above claims we can show that is u.s.c. and is compact. By the Theorem of Bohnenblust and Karlin (see Corollary 11.3 in [8]) has a fixed point in Σ, which is a solution of the inclusion (2), and therefore a solution of (1).