andJen-ChihYao ValeriObukhovskii SergeiKornev ONASYMPTOTICSOFSOLUTIONSFORACLASSOFFUNCTIONALDIFFERENTIALINCLUSIONS DiscussionesMathematicaeDifferentialInclusions,ControlandOptimization34 ( 2014 ) 219–227doi:10.7151/dmdico.1165

doi:10.7151/dmdico.1165

ON ASYMPTOTICS OF SOLUTIONS FOR A CLASS OF FUNCTIONAL DIFFERENTIAL INCLUSIONS

Sergei Kornev

Faculty of Physics and Mathematics Voronezh State Pedagogical University

394043 Voronezh, Russia e-mail: kornev

vrn@rambler.ru

Valeri Obukhovskii

Faculty of Physics and Mathematics Voronezh State Pedagogical University

394043 Voronezh, Russia e-mail: valerio-ob2000@mail.ru

and Jen-Chih Yao Center for Fundamental Science

Kaohsiung Medical University Kaohsiung 807, Taiwan

and

Department of Mathematics King Abdulaziz University

P.O. Box 80203, Jeddah 21589, Saudi Arabia e-mail: yaojc@kmu.edu.tw

Abstract

We define a non-smooth guiding function for a functional differential inclusion and apply it to the study the asymptotic behavior of its solutions.

Keywords: asymptotic behavior, functional differential inclusion, integral guiding function, non-smooth guiding function.

2010 Mathematics Subject Classification: Primary: 34E05; Secondary:

34C37, 34K09.

1

The corresponding author.

1. Introduction

The method of guiding functions, whose foundations were laid by M.A. Kras- noselskii and A.I. Perov (see, for example, [15, 16]) was extended to differential inclusions and demonstrated its effectiveness to the study of periodic problems (see, e.g., [7, 9, 13, 14, 18]). It should be mentioned that the role of differen- tial inclusions in the description of problems in mathematical control theory and optimization is well known (see, e.g., [1] – [3], [7, 10, 11, 12, 17, 18]). In the present paper we define a non-smooth guiding function for a functional differen- tial inclusion and apply it to the study the asymptotic behavior of its solutions.

This type of behavior of solutions, closely related to the existence of heteroclinic and homoclinic solutions, for the case of differential equations was described, for example, in [4] – [6].

2. Preliminaries

Recall some notions from the theory of multivalued maps (see, e.g., [7, 9, 10]).

Let (X, d

) and (Y, d

) be metric spaces. By the symbol K(Y ) we will denote the collection of all nonempty compact subsets of the space Y. If Y is a normed space, the symbol Kv(Y ) will denote the collection of all nonempty convex compact subsets of Y.

Definition 1. A multivalued map (multimap) F : X → K(Y ) is called upper semicontinuous (u.s.c.) at a point x

∈ X if for every ε > 0 there exists δ > 0 such that d

(x

, x) < δ implies F (x) ⊂ U

(F (x

)), where the symbol U

denotes ε-neighborhood of a set.

Definition 2. A multimap F : X → K(Y ) is called upper semicontinuous (u.s.c.) provided it is u.s.c. at each point x ∈ X.

A multimap will be called multifunction if it is defined on a subset of R.

Definition 3. Let I be a closed subset of R, endowed with the Lebesgue measure;

Y a Banach space. A multifunction F : I → K(Y ) is called measurable if for each open set W ⊂ Y, its pre-image

F

(W ) = {t ∈ I : F (t) ⊂ Y } is a measurable subset of I.

Remark 1. A u.s.c. multifunction is measurable. Each measurable multifunc-

tion F : I → K(Y ) admits a measurable selection, i.e., there exists a measurable

function f : I → Y, such that f (t) ∈ F (t) for a.e. t ∈ I.

For h > 0, denote by the symbol C the space C([−h, 0]; R

) of continuous func- tions x : [−h, 0] → R

with the norm kxk = sup

kx(t)k. For a given function ψ ∈ C, the symbol D

will denote the set of all continuous functions x : [−h, +∞) → R

such that x(t) = ψ(t), t ∈ [−h, 0] and the restriction of x to R

= [0, +∞) is absolutely continuous.

For a function x ∈ D

and t ≥ 0 we use notation x

∈ C for the function given as x

(θ) = x(t + θ), θ ∈ [−h, 0].

Considering the Cauchy problem for a functional differential inclusion of the following form:

(1) x

(t) ∈ F (t, x

) a.e. t ∈ R

,

(2) x(t) = ψ(t), t ∈ [−h, 0],

we will study the problem of existence of solutions satisfying the estimate of the type

(3) kx(t)k ≤ k

g(t) , t ∈ R

,

where k > 0 and g is a given function, under assumption that the right-hand part F : R

× C → Kv(R

) satisfies the following conditions:

(F

) for each ϕ ∈ C the multifunction F (·, ϕ) : R

→ Kv(R

) admits a mea- surable selection;

(F

) for a.e. t ∈ R

the multimap F (t, ·) : C → Kv(R

) is u.s.c.;

(F

) there exists a positive and summable on each compact interval function α(·) ∈ L

(R

) such that for every ϕ ∈ C we have

kF (t, ϕ)k := max

y∈F (t,ϕ)

kyk ≤ α(t) (1 + kϕk) for a.e. t ∈ R

.

Remark 2. To satisfy condition (F

) it is sufficient to assume that the multi- function F (·, ϕ) is measurable for each ϕ ∈ C (see Remark 1).

Remark 3. From conditions (F

) − (F

) it follows that for each function

x ∈ D

, the multifunction t ⊸ F (t, x

), t ∈ R

admits a locally summable

selection (see, e.g., [7, 9, 10]).

For a given initial function ψ ∈ C, by a solution of problem (1), (2) we mean a function x ∈ D

satisfying inclusion (1) for a.e. t ∈ R

.

Let us recall some notions of non-smooth analysis (see [8]).

Let V : R

→ R be a locally Lipschitz function. For x ∈ R

and ν ∈ R

, the generalized derivative V

(x; ν) of a function V (·) at a point x in the direction ν is defined as

V

(x; ν) = lim z → x t → 0+

V (z + tν) − V (z)

t ,

where z ∈ R

. Then the generalized gradient ∂V (x) of a function V (·) at a point x is defined in the following way:

∂V (x) = y ∈ R

: hy, νi ≤ V

(x; ν) for all ν ∈ R

.

It is known that the multimap ∂V : R

⊸ R

has convex compact values and it is u.s.c. It means, in particular, that for each continuous function x : [a, b] → R

the set of summable selections of the multifunction t → ∂V (x(t)) is non-empty (see Remark 1).

Recall also that locally Lipschitz function V : R

→ R is called regular if for each x ∈ R

and ν ∈ R

there exists the directional derivative V

(x, ν) and it coincides with V

(x, ν). It is known, in particular, that convex functions are regular.

3. Main result

Denote by V the collection of all regular functions V : R

→ R satisfying the coercivity condition

(4) lim

kxk→+∞

V (x) = −∞.

Notice that, given a function V ∈ V, for each r > 0 there exists k(r) > r such that if

(5) α

:= inf{V (x), kxk ≤ r}.

then

(6) V (x) < α

, kxk ≥ k(r).

Now, let g : R

→ R

be a given C

-function such that

inf{g(t), t ∈ R} ≥ 1.

Definition 4. A function V ∈ V is called an integral guiding function for inclu- sion (1) along the function g if there exists

(7) r

> g(0)kψ(0)k

such that for every function x ∈ D

satisfying conditions (i) there exists a largest finite number τ

> 0 such that

g(t)kx(t)k ≤ r

for all t ∈ [0, τ

);

(ii) there exists a (least) finite number τ

> τ

such that g(τ

)kx(τ

)k = k

:= k(r

).

(iii) kx

(t)k ≤ kF (t, x

)k for a.e. t ∈ R

we have

(8)

Z

τ_♯^x

v(s), g

(s)x(s) + g(s)f (s)ds ≥ 0

for each summable selections v(s) ∈ ∂V (g(s)x(s)) and f (s) ∈ F (s, x

), where τ

:= sup{τ ∈ [τ

, τ

), kg(τ )x(τ )k = r

}.

Now we are in position to formulate the main result of the paper.

Theorem 1. If V ∈ V is an integral guiding function for inclusion (1) along the function g then each solution of Cauchy problem (1), (2) satisfies the estimate

(9) kx(t)k ≤ k

· 1

g(t) , t ∈ R

. To prove this theorem we need the following assertion.

Lemma 1. Let V : R

→ R be a regular function, x : [a, b] → R

an absolutely continuous function. Then the the function V (x(t)) is also absolutely continuous and the following relation holds true:

(10) V (x(t)) − V (x(a)) = Z

a

V

(x(s), x

(s))ds, t ∈ [a, b].

Proof. Let us cover the range x([a, b]) ⊂ R

of a function x(·) by a finite number of open sets on which the function V is Lipschitz. Denote by L > 0 the greatest of Lipschitz constants of the function V on these sets. Let ξ > 0 be the Lebesgue number of this covering, i.e., each open ball centered at any point of the set x([a, b]) of the radius ξ is contained in one of these open sets. Now fix ε, 0 < ε ≤ ξ.

We obtain

kV (x

) − V (x

)k ≤ Lkx

− x

k for each x

, x

∈ x([a, b]), kx

− x

k ≤ ε.

Since the function x : [a, b] → R

is absolutely continuous, for a given ε > 0 there exists δ > 0 such that

m

X

k=1

kx(t

) − x(t

)k < ε, provided

m

X

k=1

kt

− t

k < δ,

where a ≤ t

< t

≤ t

< t

≤ · · · ≤ t

< t

≤ t ≤ t

< t

≤ b, m ∈ N.

Then

m

X

k=1

kV (x(t

)) − V (x(t

))k ≤ L

m

X

k=1

kx(t

) − x(t

)k < Lε, implying the absolute continuity of the function V (x(t)).

By using now the integral representation of the absolutely continuous func- tion V (x(t)) and the regularity of the function V, we obtain

V (x(t)) − V (x(a)) = Z

a

V

(x(s), x

(s))ds = Z

a

V

(x(s), x

(s))ds.

Proof of the theorem. From conditions (F

) − (F

) and basic properties of differential inclusions (see, e.g., [3, 7, 9, 10, 11, 17]) it follows that all solutions of problem (1)–(2) are extendable to R

and the set of these solutions is non-empty.

Let x(·) be any solution of problem (1)–(2) on R

.

From (7) it follows that there exists a largest τ

> 0 such that (11) g(t)kx(t)k ≤ r

< k

, t ∈ [0, τ

).

If τ

= +∞, then

(12) kx(t)k < k

· 1

g(t) , t ∈ R

and the assertion is true.

If t

< +∞, then estimate (11) is valid only on a bounded interval. Let us show that

(13) g(t)kx(t)k ≤ k

, t ∈ R

.

Supposing the contrary, we will find τ

> τ

such that (14) g(τ

)kx(τ

)k > k

.

From (11) and (14) it follows that there exists a least τ

< τ

< τ

for which (15) g(τ

)kx(τ

)k = k

.

Taking

τ

= sup{τ ∈ [τ

, τ

), g(τ )kx(τ )k = r

}, we have

g(τ

)kx(τ

)k = r

. From (5) we obtain the following estimate

(16) V (g(τ

)x(τ

)) ≥ α

. By applying Definition 4 and the above Lemma we have

V (g(τ

)x(τ

)) − V (g(τ

)x(τ

)) = Z

τ_♯^x

V

(g(s)x(s), g

(s)x(s) + g(s)x

(s))ds

≥ Z

τ_♯^x

v(s), g

(s)x(s) + g(s)x

(s)ds ≥ 0, v(s) ∈ ∂V (g(s)x(s)).

Taking into account relations (6), (15) and (16), we have α

> V (g(τ

)x(τ

)) ≥ V (g(τ

)x(τ

)) ≥ α

. The obtained contradiction shows the validity of estimate (13).

Acknowledgements

This research is supported by the joint Taiwan NSC – Russia RFBR grant 14-01-

92004, the RFBR grant 14-01-00468 and the Russian Scientific Foundation NF

14-21-00066 as well as NSC 103-2923-E-037-001-MY3. The work on the paper

was carried out during Prof. V. Obukhovskii’s visit to the Center for Fundamental Science, Kaohsiung Medical University and the Department of Applied Mathe- matics, National Sun Yat-Sen University, Kaohsiung, Taiwan in 2014. He would like to express his gratitude to the members of the Center and the Department for their kind hospitality.

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Received 24 August 2014