A general class of periodic boundary value problems for controlled nonlinear impulsive evolution equations on Banach spaces

Abstract

This paper deals with the periodic boundary value problems for nonlinear impulsive evolution equations with controls. By using the theory of semigroup and fixed point methods, we address some conditions ensuring the existence and uniqueness. Finally, two examples are provided to prove the effectiveness of the proposed results.

Melliani et al. Advances in Dierence Equations (2016) 2016:290 DOI 10.1186/s13662-016-1004-2

A general class of periodic boundary value problems for controlled nonlinear impulsive evolution equations on Banach spaces

http://crossmark.crossref.org/dialog/?doi=10.1186/s13662-016-1004-2&domain=pdf

Web End = Said Melliani*, Abdelati El Allaoui and Lalla Saadia Chadli

*Correspondence: mailto:[email protected]

Web End [email protected] Laboratoire de Mathmatiques Appliques & Calcul Scientique, Sultan Moulay Slimane University, BP 523, Beni Mellal, 23000, Morocco

1 Introduction

The theory of impulsive dierential equations has lately years been an object of increasing interest because of its vast applicability in several elds including mechanics, electrical engineering, biology, medicine, and so on. Therefore, it has drawn wide attention of the researchers in the recent years, among them we nd JinRong Wang, Michal Feckan, Yong Zhou, and others [].

For a wide bibliography and exposition on dierential equations with impulses, see for instance [], and there are many papers discussing the impulsive dierential equations and impulsive optimal controls with the classic initial condition: x() = x (see []).

In this paper, we consider the following problems for nonlinear impulsive evolution equations with periodic boundary value:

(IEE)

u (t) = Au(t) + f (t, u(t), u((t))) + B(t)c(t), t (si, ti+], i = , , , . . . , m, c Uad,

u(t) = T(t ti)gi(t, u(t)), t (ti, si], i = , , . . . , m, u() = u(a) X.

The operator A : D(A) : X X is the generator of a strongly continuous semigroup {T(t), t } on a Banach space X with a norm , and the xed points si and ti satisfying

= s < t s t < < tm sm tm+ = a

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Melliani et al. Advances in Dierence Equations (2016) 2016:290 Page 2 of 13

are pre-xed numbers, f : [, a] X X X is continuous, : [, a] [, a] is continuous, and gi : [ti, si] X X is continuous for all i = , , . . . , m.

2 Preliminaries

Next, we review some basic concepts, notations, and technical results that are necessary in our study.

Throughout this paper, I = [, a], C(I, X) is the Banach space of all continuous functions from I into X with the norm u C = suptI{ u(t) : t I} for u C(I, X), and we consider the space

PC(I, X) = u : I X : u C (ti, ti+], X , i = , , . . . , m and there exist u t

i and u t+i , i = , . . . , m with u ti = u(ti) ,endowed with the Chebyshev PC-norm u PC = suptI{ u(t) : t I} for u PC(I, X).

Denote M = suptI T(t) .

Let Y be another separable reexive Banach space where the controls c take values. Denote by Pf (Y) a class of nonempty closed and convex subsets of Y . We suppose that the multivalued map w : [, T] Pf (Y) is measurable, w() E,where E is a bounded set of

Y , and the admissible control set

Uad = c Lp(E) : c(t) w(t), a.e. , p > .

Then Uad = , which can be found in []. Some of our results are proved using the next well-known results.

Theorem (Krasnoselskiis xed point theorem) Assume that K is a closed bounded convex subset of a Banach space X. Furthermore assume that and are mappings from K into X such that:

. (u) + (v) K for all u, v K, . is a contraction,. is continuous and compact. Then + has a xed point in K.

To begin our discussion, we need to introduce the concept of a mild solution for (IEE). Assume that u : [, a] X is a solution of

u (t) = Au(t) + f t, u(t), u (t) + B(t)c(t), t a.

From the theory of strongly continuous semigroups, we get

u(t) = T(t)u() +

t

T(t s)

T(a tm)gm sm, u(sm)

t

T(t s)

f s, u(s), u (s) + B(s)c(s) ds

= T(t)u(a) +

f s, u(s), u (s) + B(s)c(s) ds

= T(t)

Melliani et al. Advances in Dierence Equations (2016) 2016:290 Page 3 of 13

a

+ sm T(a s)

f s, u(s), u (s) + B(s)c(s) ds

f s, u(s), u (s) + B(s)c(s) ds for all t [, t]

and

u(t) = T(t si)u(si) +

t

si T(t s)

t

+ T(t s)

f s, u(s), u (s) + B(s)c(s) ds

= T(t si)T(si ti)gi si, u(si) +

t

si T(t s)

f s, u(s), u (s) + B(s)c(s) ds

= T(t ti)gi si, u(si) +

t

si T(t s)

f s, u(s), u (s) + B(s)c(s) ds

for all t (si, ti+], i = , , . . . , m.

This expression motivates the following denition.

Denition We say that a function u PC(I, X) is called a mild solution of the problem (IEE), if u satises

u(t) = T(t)[T(a tm)gm(sm, u(sm)) +

a

sm T(a s)(f (s, u(s), u((s))) + B(s)c(s)) ds]

+

t

T(t s)(f (s, u(s), u((s))) + B(s)c(s)) ds, t [, t], u(t) = T(t ti)gi(t, u(t)), t (ti, si], i = , , . . . , m,u(t) = T(t ti)gi(si, u(si)) +

t

si T(t s)(f (s, u(s), u((s))) + B(s)c(s)) ds,

t (si, ti+], i = , , . . . , m.

3 Existence and uniqueness of mild solutions

To establish our results, we introduce the following assumptions:

(H) . A : D(A) X X is the generator of a strongly continuous semigroup

{T(t), t } on X with a norm .. B : [, a] L(Y, X) is essentially bounded, i.e., B L([, a], L(Y, X)).

(H) We have the functions f C(I X X, X), gi C([ti, si] X, X), i = , , . . . , m, and

: I I is continuous.(H) There is a constant Cf , Lf > such that

f (t, u, v) f (t, u, v) Cf u u + Lf v v

for each t [si, ti+], u, u, v, v X and i = , , . . . , m. (H) There is a constant L > such that

f (t, u, v) L + u + v

for all t [si, ti+] and all u, v X, i = , , . . . , m, and , [, ]. (H) There is a constant Cgi > , i = , , . . . , m, such that

gi(t, u) gi(t, v) Cgi u v

for each t [ti, si], and all u, v En, i = , , . . . , m.

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(H) There is a function t i(t), i = , , . . . , m, such that

gi(t, u) i(t)

for each t [ti, si] and all u X.

We put C = maxim Cgi and Ni = supt[ti,si] i(t) < +.

Remark From the assumptions (H)-(H) and the denition of Uad, it is also easy to verify that Bc Lp([, a]; X) with p > for all c Uad.

Therefore, Bc L([, a]; X) and Bc L

< .

Now, we can establish our rst existence result.

Theorem Let assumptions (H), (H), (H), and (H) be satised. Suppose, in addition, that the following property is veried:

:= M max max

im

Cgi + (Cf + Lf )(ti+ si) ,

C, MCgm + (Cf + Lf )M(a sm) + (Cf + Lf )t < .

Then the problem (IEE) has a unique mild solution.

Proof Dene a mapping : PC(I, X) PC(I, X) by

( u)(t) =

T(t)[T(a tm)gm(sm, u(sm))

+

a

sm T(a s)(f (s, u(s), u((s))) + B(s)c(s)) ds] +

t

T(t s)(f (s, u(s), u((s))) + B(s)c(s)) ds, t [, t],

T(t ti)gi(t, u(t)), t (ti, si], i = , , . . . , m,

T(t ti)gi(si, u(si))

+

t

si T(t s)(f (s, u(s), u((s))) + B(s)c(s)) ds, t (si, ti+], i = , , . . . , m.

Let h > be very small and u PC(I, X), we have the following.Case : For t [, t], we have

( u)(t + h) ( u)(t)

= T(t + h)

T(a tm)gm sm, u(sm) +

a

sm T(a s)

f s, u(s), u (s) + B(s)c(s) ds

t+h

+ T(t + h s)

f s, u(s), u (s) + B(s)c(s) ds

T(t)

T(a tm)gm sm, u(sm) +

a

sm T(a s)

f s, u(s), u (s) + B(s)c(s) ds

t

T(t s)

f s, u(s), u (s) + B(s)c(s) ds

M

T(h)

T(a tm)gm sm, u(sm) +

a

sm T(a s)

f s, u(s), u (s) + B(s)c(s) ds

T(a tm)gm sm, u(sm) +

a

sm T(a s)

f s, u(s), u (s) + B(s)c(s) ds

Melliani et al. Advances in Dierence Equations (2016) 2016:290 Page 5 of 13

+ M

h

f s, u(s), u (s) + B(s)c(s) ds + M

t

B(s + h)c(s + h) B(s)c(s) ds

+ M

f s + h, u(s + h), u (s + h) f s, u(s), u (s) ds as h .

Case : For t (ti, si], i = , . . . , m, we have

( u)(t + h) ( u)(t) = T(t + h ti)gi t + h, u(t + h) T(t ti)gi t, u(t)

M T(h)gi t + h, u(t + h) gi t, u(t) as h .

Case : For t (si, ti+], i = , . . . , m, we have

( u)(t + h) ( u)(t)

= T(t + h ti)gi

si, u(si) +

t

t+hsi T(t + h s)

f s, u(s), u (s) + B(s)c(s) ds

T(t ti)gi si, u(si)

t

si T(t s)

f s, u(s), u (s) + B(s)c(s) ds

M T(h)gi si, u(si) gi si, u(si) + M

si+h si

f s, u(s), u (s) + B(s)c(s) ds

+ M

t

si

B(s + h)c(s + h) B(s)c(s) ds

+ M

f s + h, u(s + h), u (s + h) f s, u(s), u (s) ds as h .

Then is well dened and u PC(I, X) for all u PC(I, X). Now we only need to show that is a contraction mapping.

Case : For u, v PC(I, X) and t [, t], we have

( u)(t) ( v)(t)

=

T(t)

t

si

T(a tm)gm sm, u(sm) +

a

sm T(a s)

f s, u(s), u (s) + B(s)c(s) ds

t

+ T(t s)

f s, u(s), u (s) + B(s)c(s) ds

T(t)

T(a tm)gm sm, v(sm) +

a

sm T(a s)

f s, v(s), v (s) + B(s)c(s) ds

t

T(t s)

f s, v(s), v (s) + B(s)c(s) ds

M

MCgm u(sm) v(sm) + M

a

sm

Cf u(s) v(s) + Lf u (s) v (s) ds

Cf u(s) v(s) + Lf u (s) v (s) ds

M MCgm + (Cf + Lf )M(a sm) + (Cf + Lf )t u v PC

u v PC.

+ M

t

Melliani et al. Advances in Dierence Equations (2016) 2016:290 Page 6 of 13

Case : For u, v PC(I, X) and t (ti, si], i = , . . . , m, we have

( u)(t) ( v)(t) = T(t ti)gi t, u(t) T(t ti)gi t, v(t)

MCgi u v PC MC u v PC u v PC.

Case : For u, v PC(I, X) and t (si, ti+], i = , . . . , m, we have

( u)(t) ( v)(t)

= T(t ti)gi

si, u(si) +

t

si T(t s)

f s, u(s), u (s) + B(s)c(s) ds

T(t ti)gi si, v(si)

t

si T(t s)

f s, v(s), v (s) + B(s)c(s) ds

M Cgi + (Cf + Lf )(ti+ si) u v PC

M max

im

Cgi + (Cf + Lf )(ti+ si) u v PC

u v PC.

Therefore, we obtain

u v PC u v PC, u, v PC(I, X).

Finally, we nd that is a contraction mapping on PC(I, X), and there exists a unique u PC(I, X) such that u = u.

So we conclude that u is the unique mild solution of (IEE).

By using Krasnoselskiis xed point theorem, we also obtain the existence of a mild solution.

Theorem Let assumptions (H), (H), (H), and (H) be satised. Suppose, in addition, that the semigroup {T(t), t } is compact and

:= max LM M(a sm) + t , LM(ti+ si) <

, i = , . . . , m,

:= max MCgm , MCgi < .

Then the problem (IEE) has at least one mild solution.

Proof Let N = max(N, N, . . . , Nm) and Br = {u PC(I, X) : u PC < r} the ball with radius r > .

Here

r max{, , MN},

Melliani et al. Advances in Dierence Equations (2016) 2016:290 Page 7 of 13

with

= MNm + (M + M) Bc L

+ LM(M(a sm) + t)

and

= MNi + M Bc L

+ ML(ti+ si) .

We introduce the decomposition = + , where

( u)(t) =

T(t)T(a tm)gm(sm, u(sm)), t [, t],T(t ti)gi(t, u(t)), t (ti, si], i = , , . . . , m,

T(t ti)gi(si, u(si)), t (si, ti+], i = , , . . . , m,

and

T(t)

a

sm T(a s)(f (s, u(s), u((s))) + B(s)c(s)) ds +

t

T(t s)(f (s, u(s), u((s))) + B(s)c(s)) ds, t [, t],

, t (ti, si], i = , , . . . , m,

t

si T(t s)(f (s, u(s), u((s))) + B(s)c(s)) ds, t (si, ti+], i = , , . . . , m. We distinguish in the proof several steps.

Step . We prove that u = u + u Br for all u Br. Indeed: Case . For t [, t], we have

( u + u)(t)

T(t) T(a tm)gm sm, u(sm)

+ T(t)

a

sm

( u)(t) =

T(a s) f s, u(s), u (s) + B(s)c(s) ds

t

+ T(t s) f s, u(s), u (s) + B(s)c(s) ds

MNm + LM

a

sm

+ u(s) + u (s) ds + M + M Bc L

+ u(s) + u (s) ds

MNm + M + M Bc L

+ LM( + r)(a sm) + LM( + r)t

= MNm + M + M Bc L

+ LM

t

+ LM M(a sm) + t + rLM M(a sm) + t

r( ) + r = r.

Case . For t (ti, si], i = , . . . , m, we have

( u + u)(t) T(t ti) gi t, u(t)

MNi MN r.

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Case . For t (si, ti+], i = , . . . , m, we have

( u + u)(t) T(t ti) gi si, u(si)

t

si

+ T(t s) f s, u(s), u (s) + B(s)c(s) ds

MNi + LM( + r)(ti+ si) + M Bc L

+ LM(ti+ si) + rLM(ti+ si)

r( ) + r = r.

Then we deduce that u + u Br.

Step . is contraction on Br. Let u, v Br. Case . For t [, t], we have

( u)(t) ( v)(t) T(t) T(a tm) gm sm, u(sm) gm sm, v(sm)

MCgm u(sm) v(sm)

MCgm u v PC u v PC.

Case . For t (ti, si], i = , . . . , m, we have

( u)(t) ( v)(t) T(t ti) gi t, u(t) gi t, v(t)

MCgm u v PC u v PC.

Case . For t (si, ti+], i = , . . . , m, we have

( u)(t) ( v)(t) T(t ti) gi si, u(si) gi si, v(si)

MCgm u v PC u v PC.

This implies that is a contraction.

Step . is continuous.

Let (un)n be a sequence such that limn+ un u PC = .

Case . For t [, t], we have

( un)(t) ( u)(t)

T(t)

= MNi + M Bc L

a

sm

+ T(t s) f s, un(s), un (s) f s, u(s), u (s) ds

M(a sm) f , un(), un () f , u(), u () PC

T(a s) f s, un(s), un (s) f s, u(s), u (s) ds

t

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+ Mt f , un(), un () f , u(), u () PC

= M M(a sm) + t f , un(), un () f , u(), u () PC.

Case . For t (ti, si], i = , . . . , m, we have

( un)(t) ( u)(t) = .

Case . For t (si, ti+], i = , . . . , m, we have

( un)(t) ( u)(t)

t

si

T(t s) f s, un(s), un (s) f s, u(s), u (s) ds

= M(ti+ si) f , un(), un () f , u(), u () PC.

This implies that limn+ un u PC = , then we deduce that is continuous.Step . is compact.. We have Br Br, then is uniformly bounded on Br.

. For u Br, we have the following.

Case . For l < l t, we have

( u)(l) ( u)(l)

T(l) T(l)

a

sm

T(a s) f s, u(s), u (s) + B(s)c(s) ds

l

+ T(l s) T(l s) f s, u(s), u (s) + B(s)c(s) ds

l l

+ T(l s) f s, u(s), u (s) + B(s)c(s) ds

M L( + r)(a sm) + Bc L

T(l l) I

+ M L( + r)t + Bc L

T(l l) I

+ LM( + r)(l l) + M

B(s)c(s) ds

= LM( + r) M(a sm) + t + M + M Bc L

l l

T(l l) I

B(s)c(s) ds as l l.

Since {T(t), t } is compact, T(l l) I as l l.

Case . For ti l < l si, i = , . . . , m, we have

( u)(l) ( u)(l) = .

Case . For si l < l ti+, i = , . . . , m, we have

( u)(l) ( u)(l)

=

lsi T(l s)

+ LM( + r)(l l) + M

l l

f s, u(s), u (s) + B(s)c(s) ds

Melliani et al. Advances in Dierence Equations (2016) 2016:290 Page 10 of 13

lsi T(l s)

f s, u(s), u (s) + B(s)c(s) ds

l l

T(l s) f s, u(s), u (s) + B(s)c(s) ds

l si

+ T(l s) T(l l) I f s, u(s), u (s) + B(s)c(s) ds

LM( + r)(l l) + M

l l

B(s)c(s) ds

T(l l) I as l l.

This permits us to conclude that is equicontinuous.We have Br Br, let := Br, (t) := Br(t) = {( u)(t) : u Br} for t [, a]. . (t) is relatively compact. Indeed:

T(t) is compact, hence

() =

a

sm T(a s)

+ M L( + r)(ti+ si) + Bc L

f s, u(s), u (s) + B(s)c(s) ds

,

is relatively compact. For < < t a, dene

(t) := Br(t) = T()( u)(t ) : u Br .

Clearly, (t) is relatively compact for t (, a], since T(t) is compact.Case . For t (, t], we have

(t) := u (t) = T()( u)(t )

=

T()T(t )

a

sm T(a s)

f s, u(s), u (s) + B(s)c(s) ds

+ T()

t

T(t

s) f s, u(s), u (s) + B(s)c(s) ds : u Br

=

T(t)

a

sm T(a s)

f s, u(s), u (s) + B(s)c(s) ds

f s, u(s), u (s) + B(s)c(s) ds : u Br

,

t

+ T(t s)

and we get

( u)(t) u (t)

t

= T(t s)

f s, u(s), u (s) + B(s)c(s) ds

t

T(t s)

f s, u(s), u (s) + B(s)c(s) ds

T(t s) f s, u(s), u (s) + B(s)c(s) ds

LM( + r) + M

t

t

t

t

B(s)c(s) ds as .

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Case . For t (ti, si], i = , . . . , m, we have

(t) := {, u Br},

in this case ( u)(t) ( u)(t) = .Case . For t (si, ti+], i = , . . . , m, we have

(t) := u (t)

=

T()

tsi T(t

s) f s, u(s), u (s) + B(s)c(s) ds : u Br

=

tsi T(t s)

f s, u(s), u (s) + B(s)c(s) ds : u Br

,

and we get

( u)(t) u (t)

t

= si T(t s)

f s, u(s), u (s) + B(s)c(s) ds

tsi T(t s)

f s, u(s), u (s) + B(s)c(s) ds

T(t s) f s, u(s), u (s) + B(s)c(s) ds

LM( + r) + M

t

t

t

t

B(s)c(s) ds as .

Now, from the Arzela-Ascoli theorem we can conclude that : Br Br is completely continuous. The existence of a mild solution for (IEE) is now a consequence of Krasnoselskiis xed point theorem.

4 Examples

In this section, we give examples to illustrate our abstract results in the previous section.

Let X = L(, ), I = [, ], = t = s, t = , s = , and a = . Dene Av = xv for

v D(A) =

v X :

v x,

v

x X, v() = v() =

.

Then A is the innitesimal generator of a strongly continuous semigroup {T(t), t } on X. In addition T(t) is compact and T(t) , for all t .

Example Consider

t u(t, x) = xu(t, x) + cos(u(t, x) + u(t, x)) + c(t, x), x (, ), t [, ) (, ],

x u(t, ) = xu(t, ) = , t [, ) (, ],u(, x) = u(, x), x (, ),u(t, x) = T(t ) sin(u(t, x)), x (, ), t (, ].

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Denote v(t)(x) = u(t, x) and B(t)c(t)(x) = c(t, x), this problem can be abstracted into

v (t) = Av(t) + f (t, v(t), v((t))) + B(t)c(t), t [s, t) (s, a], v(t) = T(t ti)g(t, v(t)), t (t, s],v() = v(a) X,

()

where (t) = t, f (t, v(t), v((t)))(x) =

cos(v(t)(x) + v(t)(x)) and

g t, v(t) (x) =

sin v(t)(x) .

In this case, we have M = , Cf = Lf =

, Cg = , and = M MCg + (Cf + Lf )(a s) + (Cf + Lf )t =

< .

This implies that all assumptions in Theorem are satised. Then there exists a unique mild solution for this problem.

Example Consider

t u(t, x) = xu(t, x) +

u(t,x)+u(t,x)|

+|u(t,x)+u(t

,x)| + c(t, x), x (, ), t [, ) (, ],

x u(t, ) = xu(t, ) = , t [, ) (, ], u(, x) = u(, x), x (, ),u(t, x) = T(t )

et |

u(t,x)|

+|u(t,x)| , x (, ), t (, ].

This problem can be abstracted into (), with (t) = t,

f t, v(t), v (t) (x) =

et

|v(t)(x) + v(t)(x)| + |v(t)(x) + v(t)(x)|

,

g t, v(t) (x) =

et

|v(t)(x)| + |v(t)(x)|

,

and B(t)c(t)(x) = c(t, x).

In this case, we have L = , Cg = , = < , and = < .

This implies that all assumptions in Theorem are satised. Then this problem has at least one mild solution.

5 Conclusion

In order to describe the evolution of the temperature using a control, we consider periodic boundary value problems for controlled nonlinear impulsive evolution equations. By using operator semigroup theory, impulsive conditions, and xed point methods, we overcome some diculties from the proof of equicontinuity and compactness and obtain new existence results. In addition, future work includes expanding the idea signalized in this work and introducing observability. This is a fertile eld with vast research projects, which can lead to numerous theories and applications. We plan to devote signicant attention to this eld of research.

Melliani et al. Advances in Dierence Equations (2016) 2016:290 Page 13 of 13

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Acknowledgements

The authors express their sincere thanks to the anonymous referees for numerous helpful and constructive suggestions which have improved the manuscript.

Received: 8 February 2016 Accepted: 18 October 2016

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