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[PUBLICATION ABSTRACT]

Agarwal and Boucherif Advances in Difference Equations 2011, 2011:17 http://www.advancesindifferenceequations.com/content/2011/1/17

RESEARCH Open Access

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

Ravi Agarwal1,2 and Abdelkader Boucherif2*

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 Classification34A60, 34G20, 26A45, 54C65, 28B20

Keywords: Set-valued maps of bounded variation, Differential inclusion, Nonlocal initial condition, Generalized Helly selection principle, Fixed point of multivalued operators

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 dX defined by

dX(x, y) = x

y

* Correspondence: mailto:[email protected]

Web End [email protected]

2Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Box 5046, Dhahran 31261, Saudi ArabiaFull list of author information is available at the end of the article

X, for any x, y X.

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

x (t) F(t, x(t)), t I x(0+) =

T

0 g(x(t))dt,

(1)

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

2011 Agarwal and Boucherif; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Agarwal and Boucherif Advances in Difference Equations 2011, 2011:17 http://www.advancesindifferenceequations.com/content/2011/1/17

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

x(t)

2 Preliminaries

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

VdX(f , I) = sup

i=1dX(f (i), f (i1)) < +,

where : 0 = 0 <1 < ... <m = T is any partition of I. The quantity VdX(f , I) 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

f

b =

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T

0

g(x(t))dt +

t0 F(s, x(s))ds. (2)

m

f

(0+)

X + VdX(f , I), for any f BV(I, X).

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 by (X). For instance, A c (X) means A closed in X, when = b we have the bounded subsets of X, = cv for convex subsets, = cp for compact subsets and = cp, cv for compact and convex subsets. The domain of a multivalued map : X Y is the set dom = {z X; (z) }. is convex (closed) valued if (z) is convex (closed) for each z X: has compact values if (z) cv(Y) for every z X; is bounded on bounded sets if (A) = zA(z) is bounded in Y for all A b(X) (i.e. supzA{sup{|y|Y; y (z)}} <): is called upper semicontinuous (u.s.c.) on X if for each z X the set (z) cl(Y) is nonempty, and for each open subset of Y containing (z), there exists an open neighborhood of z such that () . In terms of sequences, is u.s.c. if for each sequence (zn) X, zn z0, and B a closed subset of Y such that (zn) B , then (z0) B . The set-valued map is called completely continuous if (A) is relatively compact in Y for every A (X). If is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph (i.e. zn z, wn w, wn (zn) w (z)). When X Y then

has a fixed point if there exists z X such z (z). A multivalued map : J cl

(X) is called measurable if for every x X, the function : J defined by (t) = dist(x, (t)) = inf{|x - z|X ; z (t)} is measurable. |(z)|Y denotes sup{|y|Y; y (z)}.

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

dH(A, B) = max{ (A, B) , (B, A)},

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Where

(A, B) = sup

aA

dX(a, B), and dX(a, B) = inf

bB

dX(a, b).

It is well known that (b,cl(X), dH) is a metric space and so is (cp(X), dH).

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

V( , I) = VdH ( , I) = sup

m

i=1dH( (ti), (ti1)

< ,

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

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

NF : XI X,

defined by

NF(x)(t) = F(t, x(t)) for every t I.

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

VdH(F(, x()), I) = VdH(NF(x), I).

Definition 5 Let be a subset of I X. We say that is L Bmeasurable if

belongs to the s- 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 Fbe a family of maps of uniformly

bounded variation from I into K. Then there exists a sequence of maps (fn)n1 Fconvergent pointwise on I to a map f : I K of bounded variation such

that V(f , I) supFV(, I).

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 A : Zbe a single-valued operator and B : cp,cv(Z)be a mul

tivalued operator such that

(i) A() + B() 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 lx Ax + Bx has a solution for l = 1, or(b) there is an element u U such that lu Au + Bu for some l > 1.

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3 Main results

In this section we state and prove our main result. We should point out that no semi-continuity 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 : [0, + ) [0, + ) continuous and (r) br, with b < 1/2 and bT 1, such that

|g(u) g(v)|X < (|u v|X),

(H2) F : I X cp,cv(X) is of bounded variation such that

(i) (t, x) F(t, x) is L B measurable,

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

|F(t, x)|X q(t) for (t, x) I X,

(iii) xk x as k pointwise implies dH (F(t, xk), F(t, x)) 0, k .

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

Proof. Let Q = sup

tI

t0 q(s)ds. We show that there exists M > 0 such that all possible solutions of (2) in BV(I, X), satisfy

|x|b M.

Recall that solutions of (1) satisfy

x(t)

T

0

T

0

g(x(t))dt +

t0 NF(x)(s)ds. (3)

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

VdX(f , I) VdH(NF(x), I),

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

It follows from (3) that

x(t) =

T

0 g(x(t))dt +

g(x(t))dt +

t0 F(s, x(s))ds =

t

0 f (s)ds, t I. (4)

This implies

x(t)

X

T

0

t0 f (s)ds

X

g(x(t))dt

X

+

T

0

g(x(t))

Xdt +

t 0

f

(s)

Xds.

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

|x(t)|X

T

0

|x(t)|Xdt +

t0 q(s)ds.

Hence

T

0

|x(t)|Xdt T

T

0

|x(t)|Xdt +

T

0

t0 q(s)dsdt.

This last inequality yields

T

0

|x(t)|Xdt

11 T

T

0

t0 q(s)dsdt.

Since

T

0

t0 q(s)dsdt =

T

0

(T s)q(s)ds,

we obtain

T

0

|x(t)|Xdt

11 T

T

0

(T s)q(s)ds,

so that

T

0

|x(t)|Xdt

2T1 T

Q. (5)

Inequality (5) and the condition on g imply that

T

0

|g(x(t))|Xdt

2T 1 T

Q.

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

|x(0+)|X

2T 1 T

Q.

Let = {ti; 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

x(ti) x(ti1) =

ti

ti1 f (s)ds, i = 1, . . . , m.

It is easily shown that

VdX(x, I) VdX(f , I) sup

m

i=1

i i1

q(s)ds

Q.

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Therefore

|x|b

2T 1 T

Q + Q.

Letting M := 1 + T

1 T

Q,, we see that

|x|b M.

Let

:= {x BV(I, X); |x|b < M + 1}.

Define two operators

A : X, B : X by

Ax(t) =

T

0

g(x(t))dt,

and

t0 NF(x) (s) ds.

First, we show that A(

) + B(

) is bounded, i.e.

sup x

Bx(t) =

t0 F(s, x(s))ds =

b; y A(x) + B(x)}} < .

Let y A

{sup{

y

+ B

. Then there exists x such that

y A(x) + B(x).

It follows from (3) that

y

b

M.

(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 : XI cp,cv(X), i.e. has compact and convex values. This implies that the Aumann integral

t0 NF(x) (s) ds

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

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

u(t) =

t0 f (s)ds,

satisfies

u (t) = f (t), u(0 + ) = 0.

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If we write

u = f ,

then the operator : X X is continuous and

B = NF.

Let (Bxk)k1 be a sequence in B (). Then the sequence (xk)k1 is 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 . We have

Bx

k

By

m

i=1

i i1

b

sup

F(s,

xk(s)) F(s, y(s))

Xds

.

Assumption (H2) (iii) implies that

Bx

k

0 as k 0.

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 {(xn, yn)}n1 be a sequence in graph(B) and let (x, y) = limn (xn,

yn). Then yn B(xn), i.e yn (t)

t0 F(s, xn(s))ds, t

By

b

I. This implies that

t0 F(s, x(s))ds +

t0 [F(s, xn(s)) F(s, x(s))] ds.

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

lim

n

yn (t)

yn (t)

t0 F(s, x(s))ds,

which shows that

y B(x).

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 . By Theorem 5 the inclusion

x Ax + Bx,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

T

0

g(x(t))dt =

T

0

(t) x(t)dt, where

: I is continuous. Let

0 =

T

0

s

(t)dt

1 - 0 .

From the definition of the function l we infer that, if * = maxtI |(t)|,

(s)

(t)dt and (s) =

T

2T1 0

for any s I.

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Theorem 9 Assume that the following conditions hold

(H3) : I is continuous and 0 1,

(H4) F : I X cp,cv(X) is of bounded variation such that

(i) (t, x) F(t, x) is L B measurable,

(ii) there exists : I [0, ) (0, ) continuous, nondecreasing with respect to its second argument and

lim sup

1

1 0 + 2T 1 0

T

0

(s, ) ds < 1, (6)

such that |F(t, x) |X (t, |x|b).

(iii) xk x pointwise as k implies dH (F (t, xk), F (t, x)) 0 as k .

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

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

VdX(h, I) VdH(NF(x), I),

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

Solutions of (2) satisfy

x(t) = x(0+) +

t0 h(s)ds, h NF(x). (7)

Substituting the initial condition in (7) we obtain

x(t) =

T

0

(t) x(t)dt +

t0 h(s)ds, h NF(x)

Since 0 1 it follows that

x(t) =

T

0

(t) 1 0

t0 h(s)dsdt +

t0 h(s)ds, h NF(x).

Thus, solutions of (2) are solutions of

x(t) =

T

0

(s)h(s)ds +

t0 h(s)ds, h NF(x), (8)

and vice versa. It follows from (8)

x(t)

X

T

0

| (s)| (s, |x|b) ds +

t0 (s, |x|b) ds.

The upper bound on |l (s)| implies

x(t)

X

2T1 0

T

0

(s, |x|b) ds +

t0 (s, |x|b) ds, (9)

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which gives

x(0+)

X

2T1 0

T

0

(s, |x|b) ds.

Let = {ti; 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

x(ti) x(ti1) =

ti ti1

h(s)ds, i = 1, . . . , m,

which leads to

VdX(x, I) VdX(h, I)

T

0

(s, |x|b) ds.

Since |x|b =

x(0+)

X + VdX(x, I), we have

|x|b

2T1 0

T

0

(s, |x|b) ds +

T

0

(s, |x|b) ds.

Finally, we see that

|x|b

1 0 + 2T

1 0

T

0

(s, |x|b) ds. (10)

Let

0 = |x|b.

Then (10) yields

1

1 0

1 0 + 2T1 0

T

0

(s, 0) ds. (11)

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

1

1 0 + 2T 1 0

T

0

(s, ) ds < 1. (12)

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

0 = |x|b .

Let

= {x BV(I, X); |x|b }.

Then is nonempty, closed, bounded and convex.

Define a multivalued operator

[Digamma] : BV(I, X) cp, cv(X),

Agarwal and Boucherif Advances in Difference Equations 2011, 2011:17 http://www.advancesindifferenceequations.com/content/2011/1/17

by

t0 NF(x)(s)ds. (13)

Then solutions of (2) are fixed point of the multivalued operator [Digamma] : cp, cv(X). It is clear that [Digamma] ( ) . Proceeding as in the above claims we can show that [Digamma] is

u.s.c. and [Digamma] ( ) is compact. By the Theorem of Bohnenblust and Karlin (see Corollary11.3 in [8]) [Digamma] has a fixed point in , which is a solution of the inclusion (2), and therefore a solution of (1).

Acknowledgements

The authors are grateful to King Fahd University of Petroleum and Minerals for its constant support. The authors would like to thank an anonymous referee for his/her comments.

Author details

1Department of Mathematics, Florida Institute of Technology, Melbourne, FL, USA 2Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Box 5046, Dhahran 31261, Saudi Arabia

Authors contributions

Both authors have read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 7 February 2011 Accepted: 24 June 2011 Published: 24 June 2011

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doi:10.1186/1687-1847-2011-17Cite this article as: Agarwal and Boucherif: Nonlocal conditions for differential inclusions in the space of functions of bounded variations. Advances in Difference Equations 2011 2011:17.

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[Digamma]x (t) =

T

0

(s)NF(x)(s)ds +