Stability for Functional Differential Equations with Delay in Banach Spaces

(1)

ISSN Online: 2160-0384 ISSN Print: 2160-0368

DOI: 10.4236/apm.2019.94016 Apr. 28, 2019 337 Advances in Pure Mathematics

Stability for Functional Differential Equations

with Delay in Banach Spaces

Anna Kisiolek

1

, Ireneusz Kubiaczyk

1

, Aneta Sikorska-Nowak

2 1_{The Great Poland University of Social and Economics in Środa Wlkp, Środa Wlkp, Poland} 2_{Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland}

Abstract

In this paper, using Sadovskii’s fixed point theorem and properties of the measure of noncompactness, we obtain asymptotic stability results for solu-tions of nonlinear differential equation with variable delay. Results presented

in this paper extend some previous results due to Burton [1], Becker and

Burton [2], Ardjouni and Djoudi [3], and Jin and Luo [4].

Keywords

Fixed Point, Differential Equations, Banach Space, Stability, Measure of Noncompactness

1. Introduction

Since 1892, the Lyapunov’s direct method has been used for the study of stability properties of ordinary, functional, differential and partial differential equations. Nevertheless, the applications of this method to problem of stability in differential equations with delay have encountered serious difficulties if the delay

is unbounded or if the equation has unbounded terms (see [1][5][6]). Recently,

investigators, such as Burton and Furumochi have noticed that some of these difficulties vanish or might be overcome by means of fixed point theory (see [1]-[17]). The fixed point theory does not only solve the problem on stability but has a significant advantage over Lyapunov’s direct method. Sadovskii’s fixed

point theorem (see [18]) and techniques of the theory of the measure of

noncompactness are used to prove the existence and stability of the solution of the problem investigated in this paper.

Let E denote a Banach space. We consider the nonlinear differential equation

with variable delay

How to cite this paper: Kisiolek, A., Ku-biaczyk, I. and Sikorska-Nowak, A. (2019) Stability for Functional Differential Equa-tions with Delay in Banach Spaces. Ad-vances in Pure Mathematics, 9, 337-346.

https://doi.org/10.4236/apm.2019.94016

Received: March 14, 2019 Accepted: April 25, 2019 Published: April 28, 2019

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/apm.2019.94016 338 Advances in Pure Mathematics

( )

=

( )

(

1

( )

)

( )

(

1

( )

)

( )

(

( )

,

(

2

( )

)

x t′ −a t x t r t− +b t x t r t′ − +c t G x t x t r t− (1)

with the initial condition

( )

, for

( )

0 ,0 ,

x t =ψ t t∈ _m _

where ψ∈C m

(

_

( )

0 ,0 ,_ E

)

_, mj

( )

0 =inf

{

t r t t− j

( )

, ≥0

}

,

( )

0 min

{

j

( )

0 , 1,2

}

m = m j= _.

Here CB R E

(

+,

)

_{denotes the set of all continuous and bounded functions}

:

f R+ _→E_{with the supremum norm} _⋅ _defined

( )

{

}

sup : 0

x = x t m ≤ < ∞t _{. Throughout this paper, we assume that}

(

)

1

, , ,

a b c CB R R_∈ + _, 2

(

)

(

)

1 , , 2 ,

r C R R r C R R_∈ + _∈ + _with

_{( )}

j

t r t− → ∞ _as

t→ ∞_, j=1,2. We also assume that the function G E E: × →E_{is uniformly}

continuous. Moreover, we assume that the function G is bounded, so there exists

a constant L≥0_{, such that} G

( )

⋅ ⋅ ≤, L_and G

( )

0,0 =0_.

Special cases of the equation (1) have been investigated by many authors. For

example, Burton in [6] and Zhang in [17] have studied boundedness and

stability of the linear equation:

( )

(

1

( )

)

.

x t′ = −a t x t r t−

In [13], Burton and Furumochi using the fixed point theorem of Krasnosielski

obtained boundedness and asymptotic stability for the equation:

( )

=

( ) (

1

) ( )

(

2

( )

)

,

x t_′ ₋a t x t r₋ ₊b t x t r tγ ₋

where r1 >0 is constant; γ∈

( )

0,1 and

γ

is a quotient with odd positive

integer denominator and a C R_∈

(

+, 0,

(

_∞

)

_.

Next, Jin and Luo (see [4]), proved the boundedness and stability of solutions

of the equation

( )

(

( )

)

( )

1 3

(

( )

)

1 2

= ,

x t′ −a t x t r t− +b t x t r t−

and generalized the results claimed in [6][12][17]. Ardjouni and Djoudi in [3]

considered the more general neutral nonlinear differential equation

( )

(

1

( )

)

( )

(

1

( )

)

( )

(

2

( )

)

x t_′ _{= −}a t x t r t₋ ₊b t x t r t_′ ₋ ₊c t G x t r tγ ₋

and obtained the boundedness and stability results.

We mentioned here that the neutral delay differential equations appear in modelling of the networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar, as the Euler equation in some variational problems, theory of automatic control and in neuromechanical systems in which inertia plays an

important role, we refer the reader to the papers by Boe and Chang [19], Brayton

and Willoughby [20] and to the books by Driver [21], Hale [14] and Popov [22]

and reference cited therein.

The fundamental tool in this paper is the Kuratowski measure of

(3)

For any bounded subset A of E we denote by α

( )

A the Kuratowski measure

of noncompactness of A, i.e. the infimum of all >0 such that there exists a

finite covering of A by sets of diameters smaller than . The properties of the

measure of noncompactness α are:

1) if A B⊂ then α

( )

A ≤α

( )

B ;

2) α

( )

A =α

( )

A _{, where} A denotes the closure of A; 3) α

( )

A =0 if and only if A is relatively compact; 4) α

(

A B

)

=max

{

α

( ) ( )

A ,α B

}

;

5) α λ

( )

A = λ α

( )

A ,

(

λ∈R

)

; 6) α

(

A B+

)

≤α

( )

A +α

( )

B

7) α

(

convA

)

=α

( )

A , where conv

( )

A denotes the convex hull of A, 8) α

( )

A <δ

( )

A _{, where} δ

( )

A =supx y A_, ∈

{

x y−

}

.

Lemma 1. [24]. Let K C I E⊂

(

,

)

_{be a family of strongly equicontinuous}

functions. Let K t

( )

=

{

k t

( )

∈E k K, ∈

}

_, _for t I∈ and K I

( )

=



_{t I}_∈ K t

( )

.

Then

( )

sup

(

( )

)

(

( )

)

C

t I

K K t K I

α α α

∈

= =

where αC

( )

K denotes the measure of noncompactness in C I E

(

,

)

and the

function t_α

(

K t

( )

)

_{is continuous}_.

Let us denote by S∞ the set of all nonnegative real sequences. For

( )

n S

ξ = ξ ∈ ∞, η=

( )

ηn ∈S∞ , we write ξ η< if ξn≤ηn (i.e. ξn<ηn, for

1,2,

n= ) and ξ η≠ .

Let X be a closed convex subset of C I E

(

,

)

and let Φ be a function which assigns to each nonempty subset Z of X a sequence Φ

( )

Z ∈S∞ such that

{ }

(

z Z

)

( )

Z , forz X,

Φ  = Φ ∈ (2)

(

convZ

)

( )

Z ,

Φ = Φ ₍₃₎

if Φ

( )

Z = Θ_{(the zero sequence), then}Z is compact. (4)

In the proof of the main theorem, we will apply the following fixed point theorem

Theorem 1 [18] If F X: →X _{is continuous mapping satisfying}

( )

(

F Z

)

( )

Z

Φ < Φ _{for arbitrary nonempty subset Z of X with} Φ

( )

Z >0_,_then

F has a fixed point in X.

2. Main Results

A solution of the problem (1) is a continuous function x m:_

( )

0 ,∞ →

)

E_such

that x satisfies (1) on

[

0,∞

)

_and x=

ψ

on _m

( )

0 ,0__{. Stability definitions} may found in [1], for example.

In this paper, we extend stability theorem proved in [3] by giving a necessary

and sufficient conditions for asymptotic stability of the zero solution of the Equation (1). By using some conditions expressed in terms of the measure of

noncompactness which G satisfies, we define a continuous, bounded operator

(4)

point theorem of Sadovskii is used to prove the existence of a fixed point of the

operator Pφ. To construct our mapping, we begin transforming (1) to a more

tractable, but equivalent equation, which we invert to obtain an equivalent integral equation for which we derive a fixed point mapping. We need the following lemma in our proof of the main theorem.

Lemma 2 Let g m:

( )

0 ,_{∞ →}

)

R+

 be an arbitrary continuous function and

suppose that

( )

1 1,

r t_′ _{≠ ∀ ∈}t R+₍₅₎

Then x is a solution of (1) if and only if

( )

_{( )}

(

( )

)

_{( )}

( ) ( )

( )

(

( )

)

( )

( ) ( )

( )

_{( )}

( )

( ) ( )

(

)

( )

₍

_{( )}

₎

₍

_{( )}

₎

_{( )}

₍

_{( )}

₎

( )

_{( )}

(

_{( )}

₍

_{( )}

₎

)

0 1 1 1 0 d 1 ₀ 1 1 1 d 0 d

1 1 1

0 d

2 0

0

0 0 d e

1 0

d 1

e d d

e 1 d

e , d ,

t t s t s t s

g u u r

t t r t

t g u u s

s r s

t g u u

b

x t x x r g u x u u

r b t

x t r t g u x u u

r t

g s g u x u u s

g s r s r s a s s x s r s s

c s G x s x s r s s

µ − − − − − − − ∫ ∫ ∫ ∫   =_ − − − _ ′ −   + − + ′ − −  ′  + _ − − − − _ − + −

∫

(6) where

( )

(

( ) ( ) ( )

)

(

( )

)

( ) ( )

( )

(

1

)

2 1

1

1 =

1

b s b s g s r s r s b s

s

r s

µ ′ + − ′ + ′′

′

− (7)

Now we will present our main results. We set ( )

{

0 d

}

0

sup e tg u u

t

K −

≥

∫

= (8)

Let ψ∈CB m

(

_

( )

0 ,0 ,_ E

)

be a fixed. Put

1 2 , U p U

= ₍₉₎

where

( )

_{( )}

2 0 d

1 ₀ ₀

1

0

1 d e d

1 0

t s

t g u u

r

b

U g u u K L c s s

r − − ∫   = +_ + _ + ′ − 

∫



∫

( )

_{( )}

( )

(

)

( )

{

₍

_{( )}

₎

₍

_{( )}

₎

_{( )}

}

1 1 d 2 ₀ 1 d 1 1 0

1 d e d d

1

e 1 d

t s

t t g u u s

t r t s r s

t g u u

b t

U g u u g s g u u s

r t

g s r s r s a s µ s s

− − − − ∫ ∫  = − + + ′ −   ′ + − − − − _ 

∫

and U2>0, this mean that

( )

_{( )}

( )

(

)

( )

{

₍

_{( )}

₎

₍

_{( )}

₎

_{( )}

}

1 1 d 0 1 d 1 1 0

d e d d

1

e 1 d 1.

t s

t t g u u s

t r t s r s

t g u u

b t

g u u g s g u u s

r t

g s r s r s a s µ s s

− − − − ∫ ∫  + +  ′ −   ′

+ − − − − _<



∫

(5)

Let us recall that for continuous function f R: →E_{, the first order modulus}

of smoothness for f is the function ω:R→R₊_

{ }

0 _{defined for any}

σ

≥0 by

(

f,

)

sup

{

f t

( )

f t

( )

: ,t t R t t,

}

.

ω σ = ′ − ′′ ′ ′′∈ ′ ′′− ≤σ

Define

( )

)

(

)

( )

{

( )

( ) ( )

(

)

}

0 , , , for 0 ,0 , 0

as , , .

S CB m E t t t m t

t p t z t z

ψ

ϕ

ψ

ϕ

ω

= ∈ _ ∞ = ∈_ _ →

→ ∞ ⋅ ≤ − ≤ −

Now, we use (6) to define the operator P S: ψ →Sψ by

( )( )

Pϕ t =ψ

( )

t if

( )

0 ,0

)

t_{∈ }m and for t≥0 we let

( )( )

6

( )

1 i ,

i

Pϕ t I t

=

∑

where

( )

_{( )}

(

( )

)

_{( )}

( ) ( )

₀ ( )

1

0 d

1 1 ₀

1

(0)

0 0 d e ,

1 0

t_{g u u}

r

b

I r g u u u

r

ϕ ϕ ϕ −

−

∫

 

=_ − _′ − − _

−



∫



( )

(

( )

)

2 1

1

, 1

b t

I t r t

r t ϕ

= −

′ −

( )

( ) ( )

1

3 d ,

t t r t

I g u ϕ u u

−

=

_∫

( )

_{( )}

( )

( ) ( )

(

1

)

d

4 ₀e d d ,

t s

t g u u s

s r s

I − g s g u ϕ u u s

−

∫

=

_∫

( )d

₍

_{( )}

₎

₍

_{( )}

₎

_{( )}

₍

_{( )}

₎

5 ₀e 1 1 1 1 d ,

t s

t g u u

I − g s r s r s a s µ s ϕ s r s s

′

∫ _ _

=

_∫

_ − − − − _ −

( )d

_{( )}

(

_{( )}

₍

_{( )}

₎

)

6 ₀e , 2 d , for 0,

t s

t g u u

I ₌ −∫ c s G x s x s r s₋ s t_≥

∫

where

( )

(

( ) ( ) ( )

)

(

( )

)

( ) ( )

( )

(

1

)

2 1

1

. 1

s

r s

µ = ′ + − ′ + ′′

′

− (10)

Theorem 2 Assume that the function G CB E E E∈

(

× ,

)

_{is uniformly}

continuous, satisfies the condition G

( )

0,0 =0 _{and there exists a constant}

1

0<k <1 _{such that}

( ) ( )

(

)

(

G V J V J1 , 2

)

k1max

{

(

V J1

( )

)

,

(

V J2

( )

)

}

,

α ≤ α α (11)

for each V V1, 2⊂Sψ and for each J =m

( )

0 , ,T T ∈m

( )

0 ,∞

)

, where α

denotes the Kuratowski measure of noncompactness. Moreover, we assume that there exists a constant p1∈

( )

0,1 satisfies, for t≥0:

( )

_{( )}

( )

(

)

( )

{

₍

_{( )}

₎

₍

_{( ) ( )}

_{( )}

₎

}

1 1

d

1 ₀

1

d

1 1 1

0

d e d d

1

e 1 d ,

t s

t t g u u s

t r t s r s

t g u u

b t

k g u u g s g u u s

r t

g s r s r s a s µ s s p

−

− −

−

∫

∫ 

+ _′ + +

− 

 ′

+ − − − − _≤



∫

(6)

DOI: 10.4236/apm.2019.94016 342 Advances in Pure Mathematics where

( )

(

( ) ( ) ( )

)

(

( )

)

( ) ( )

( )

(

)

1 1 2 1 1 1

s

r s

µ = ′ + − ′ + ′′

′ −

and ( )d

( )

0e d 0

t s

t −∫g u u _{c s s}_→

∫

, as t→ ∞.

Then there exists at least one solution of the problem (1), which tends to zero if

( )

0 d , as .

t

g s s→ ∞ t→ ∞

∫

(13)

Proof. We divide our proof into parts. In the first part, we show that

( )

Pϕ :Sψ →Sψ complete with is well defined. Next, using Sadovskii’s fixed

point theorem and techniques of the theory of the measure of noncompactness,

we prove, that there exists a fixed point of the operator

( )

Pϕ , which is a

solution of the problem (1). In the last part of our proof, we will show, that the solution of (1) tends to zero as t→ ∞ if

_∫

₀tg s s

( )

d → ∞, as t→ ∞.

Part I. It is clear that

( )

Pϕ :_m

( )

0 ,∞ →

)

E_{is continuous. We will show the}

boundedness of this operator. Notice that G CB E E E∈

(

× ,

)

_{, so there exists a}

constant L>0 such that G x t x t r t

(

( )

,

(

− 2

( )

)

≤L, x S t∈ ψ, ∈m

( )

0 ,∞.

For t∈ _m

( )

0 ,0__{, the boundedness of the operator}

( )

Pϕ _{is clear. For}

0

t> we have

( )

_{( )}

(

( )

)

_{( )}

( ) ( )

( )

(

( )

)

( )

( ) ( )

( )

_{( )}

( )

( ) ( )

(

)

( )

₍

_{( )}

₎

₍

_{( ) ( )}

_{( )}

₎

₍

_{( )}

₎

( )

_{( )}

(

_{( )}

₍

_{( )}

₎

)

0 2 1 1 0 d 1 ₀ 1 1 1 d 0 d

1 1 1

0 d

2 0

0

0 0 d e

1 0

d 1

e d d

e 1 d

e , d

t t s t s t s

g u u r

t t r t

t g u u s

s r s

t g u u

b

P t r g u u u

r

b t

t r t g u u u

r t

g s g u u u s

g s r s r s a s s s r s s

c s G s s r s s

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ µ ϕ ϕ ϕ − − − − − − − ∫ ∫ ∫ ∫   ≤ _ − − − _ ′ −     + − + ′ − −  ′  + _ − − − − _ − + −

∫

( )

_{( )}

( )

(

)

( )

{

₍

_{( )}

₎

₍

_{( ) ( )}

_{( )}

₎

}

( )

_{( )}

0 2 1 1 0 d 0 1 1 d 0 d 1 1 0 d 0 0

1 d e

1 0

d 1

e d d

e 1 d

e d t t s t s t s

g u u r

t t r t

t g u u s

s r s

t g u u

b

g u u r

b t

g u u r t

g s g u u s

g s r s r s a s s s

L c s s

(7)

( )

_{( )}

( )

(

)

( )

{

₍

_{( )}

₎

₍

_{( ) ( )}

_{( )}

₎

}

( )

_{( )}

2 1 1 0 0 1 d 0 1 d 1 1 0 d 0 0 1 d 1 0

d e d d

1

e 1 d

e d ,

t s t s t s r

t t g u u s

t r t s r s

t g u u

b

g u u K r

b t

g u u g s g u u s

r t

g s r s r s a s s s

L c s s

ϕ µ − − − − − − ∫ ∫ ∫   ≤ +_ + _ ′ −    +  _′ + + −   ′ + − − − −  +

∫

where

( )

(

( ) ( ) ( )

)

(

( )

)

( ) ( )

( )

(

1

)

2 1

1

. 1

s

r s

µ = ′ + − ′ + ′′

′ −

Therefore, because ϕ∈Sψ satisfies ϕ ⋅ ≤

( )

p , we have, by (9), that

( )

Pϕ ⋅ ≤ p_.

Now we will show that for each ϕ∈Sψ we obtain

( )( ) ( )( )

0 δ 0t z δ Pϕ t Pϕ z .

∀ > ∃ > − < ⇒ − <

Let’s note that

( )( ) ( )( )

6

( )

1 i i ,

i

Pϕ t Pϕ z A t A z

=

− ≤

∑

−

where

( )

_{( )}

(

( )

)

_{( )}

( ) ( )

(

₀ ( ) ₀ ( )

)

1

0 d d

1 1 ₀

1

0

0 0 d e e ,

1 0

t_{g u u} z_{g u u}

r

b

A r g u u u

r

ϕ ϕ ϕ − −

− ∫ ∫   =_ − _′ − − _ − − 

∫



( )

(

( )

)

( )

(

( )

)

2 1 1

1 1

,

1 1

b t b z

A t r t z r z

r t ϕ r z ϕ

= − − −

′ ′

− −

( )

( ) ( )

( )

( ) ( )

1 1

3 d d ,

t z

t r t z r z

A g u ϕ u u g u ϕ u u

− − =

_∫

−

_∫

( ) ( )

(

)

( )

(

_{( )}

( ) ( )

)

1 d d

4 ₀ e e d d ,

t z

s s

t g u u g u u s

s r s

A − − g s g u ϕ u u s

−

∫ ∫

=

_∫

−

_∫

( ) ( )

(

d d

)

(

( )

)

(

( )

)

( )

(

( )

)

5 ₀ e e 1 1 1 1 d ,

t z

s s

t g u u g u u

A −_∫ −_∫ _g s r s r s_′ a s µ s _ϕ s r s s

=

_∫

− _ − − − − _ −

( ) ( )

(

d d

)

( )

(

( )

(

( )

)

6 ₀ e e , 2 d .

t z

s s

t g u u g u u

A −_∫ −_∫ c s G ϕ s ϕ s r s s

=

_∫

− −

Because the exponential function is continuous and using assumptions about functions ϕ, ,g b_{and properties of Bochner integral we have}

( )

1 1 1 1 1

0 0 .

6

t z A t A z

δ δ

∀ > ∃ > − < ⇒ − < 

By equicontinuity of function G and equicontinuity of family Sψ we have

that

( )

(

( )

)

(

)

(

( )

(

( )

)

2 2 1

0 0 , , .

6

t z G t t t G z z z

δ δ ϕ ϕ τ ϕ ϕ τ

(8)

So we obtain that A t2

( )

−A z2

( )

<1₆.

In the same way, we estimate the remaining ingredients and we get

( )( ) ( )( )

0 δ 0t z δ Pϕ t Pϕ z .

∀ > ∃ > − < ⇒ − <

Now we will show, that

( )( )

Pϕ t →0_as t→ ∞_{. It is obvious that the first}

term I t1

( )

tends to zero as t→ ∞ , by the condition (13). Because

(

t r1

)

0

ϕ − → _and t r t− ₁

( )

→ ∞_as t→ ∞_and_b_{is a bounded function so}

the second term I t2

( )

tends to zero as t→ ∞. Because functions

ϕ

,g are

bounded functions, so I3→0, t→ ∞. Analogously I4 and I5 tends to zero

if t→ ∞. Moreover,

( )

_{( )}

(

_{( )}

₍

_{( )}

₎

)

( )

_{( )}

d

6 ₀ 2

d 0

e , d

e d 0, .

t s

t g u u

A c s G x s x s r s s

L c s s t

− −

∫

≤ −

≤ → → ∞

∫

In conclusion

( )( )

Pϕ t →0_as t→ ∞. Hence Pϕ maps Sψ into Sψ .

Part II. Suppose that V ⊂Sψ and α denotes the Kuratowski measure of

noncompactness. Let J= _m

( )

0 ,T__, T∈_m

( )

0 ,∞

)

_{. Then,}

( )( )

(

)

( )

(

)

( )

_{( )}

( )

_{( )}

( )

(

)

( )

{

₍

_{( )}

₎

₍

_{( )}

₎

_{( )}

}

( ) ( )

{

}

1

1 1 d 0

d

1 1

0

1 1 2

d 1

e d d

e 1 d

max ,

t s

t t r t

t g u u s

s r s

t g u u

P V t b t

V J g u u

r t

g s g u u s

g s r s r s a s s s

k V V

α

µ

α α

− −

∫

∫ 

≤ ⋅ ₋ _′ +

 +

 ′

+ − − − − _

 + ⋅

∫

( )

(

)

( )

_{( )}

( )

_{( )}

( )

(

)

( )

{

₍

_{( )}

₎

₍

_{( )}

₎

_{( )}

}

( )

(

)

1

1 1

d 0

d

1 1 1

0

1

d 1

e d d

e 1 d

. t s

t s

t t r t

t g u u s

s r s

t g u u

b t

V J g u u

r t

g s g u u s

g s r s r s a s s s k

p V J

α

µ

α

− −

∫

∫ 

≤ ⋅ ₋ _′ +

 +

 ′

+ − − − − _{+ }



≤ ⋅

∫

So, using (12) and properties of α we have

( )

(

)

sup

(

( )

)

sup

(

( )( )

)

1

(

( )

)

.

t J t J

V J V t P V t p V J

α α α α

∈ ∈

= = ≤ ⋅

Because p1<1, we get α

(

V t

( )

)

=0 for each t J∈ .

Define Φ

( )

V =

(

sup_{t J}_∈₁α

(

V t

( )

)

,sup_{t J}_∈₂α

(

V t

( )

)

,_

)

_{for any nonempty} subset V of Sψ , where Jk = m

( )

0 ,k, k=1,2, , n. Evidently Φ

( )

V ∈S∞.

By the properties of α, the function Φ_{satisfies conditions (2)-(4). From (10)}

( )

(

P V

)

( )

V

Φ < Φ _{, whenever}Φ

( )

V >0 _{. If}Φ

( )

V =0 _{then for each}

( )

0 ,

)

t∈_m ∞ _{we have}α

(

V t

( )

)

=0_{. Hence Arzela-Ascoli’s theorem proves}

(9)

( )

Pϕ _{has a fixed point which is a solution of the problem (1) with} x t

_{( )}

= Φ

_{( )}

t

on _m

( )

0 ,0__and x t

( )

→0_as t→ ∞, if

_∫

₀tg s s

( )

d → ∞_{, for}t→ ∞. The proof is complete.

Remark 1. It is clear that we can assume about the function G other types of continuity and other conditions on measures of noncompactness. When we investigate the existence of solutions of (1) with non-continuous right-hand side, it is natural to consider the so-called Carathéodory-type solutions.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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