Solvability of second order linear differential equations in the sequence space \(n(\phi)\)

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R E S E A R C H

Open Access

Solvability of second order linear

differential equations in the sequence

space

n(

φ

)

Abdullah Alotaibi

1

_{, M. Mursaleen}

1,2*

_{and Badriah A.S. Alamri}

1

*_{Correspondence:} [email protected] 1_{Operator Theory and Applications} Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

2_{Department of Mathematics,} Aligarh Muslim University, Aligarh, India

Abstract

We apply the concept of measure of noncompactness to study the existence of solution of second order diﬀerential equations with initial conditions in the sequence spacen(

φ

).

MSC: Primary 47H09; 47H10; secondary 34A34

Keywords: Measure of noncompactness; Darbo’s fixed point theorem; Sequence spaces; Infinite system of second order differential equations

1 Introduction and preliminaries

In recent years, the notion of measure of noncompactness has been effectively utilized in sequence spaces for different classes of differential equations (see [4,5,8,11–15]). By applying this notion, Aghajani and Pourhadi [2] investigated the infinite system of second-order differential equations in an1-space. Since then Mohiuddine et al. [10] and Banaś et al. [6] focused on this system in the sequence spacep.

A measure of noncompactness is a nonnegative real-valued map deﬁned on a collection of bounded subsets of a normed (metric) space which maps the class of relatively compact sets (known as kernel) to zero, while other sets are mapped to a positive value. There are several ways to deﬁne this notion on a given space. The widely used approach is the axiomatic one, introduced in [3], which is given below.

LetMEdenote the family of all nonempty bounded subsets of a Banach spaceEandNE

be its subfamily consisting of all relatively compact sets. LetB(x,r) denote the closed ball centered atxwith radiusrandBr=B(θ,r).

We recall the following deﬁnition given in [3].

Deﬁnition 1.1([3, Deﬁnition 3.1.3]) A mappingμ: ME−→R+ is called ameasure of

noncompactness(MNC for short) if

(i) kerμis nonempty and a subset ofNE.

(ii) μ(X)≤μ(Y)forX⊂Y.

(iii) μ(X) =μ(X).

(iv) μ(ConvX) =μ(X).

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(v) For allλ∈[0, 1],

μλX+ (1 –λ)Y≤λμ(X) + (1 –λ)μ(Y).

(vi) If(Xn)n∈Nis a sequence of closed sets fromMEsatisfying

Xn+1⊂Xn for alln∈N and μ(Xn)→0 asn→ ∞,

then

X∞= ∞

n=1

Xn=∅.

Deﬁnition 1.2([5, Deﬁnition 3.1.3]) For a measure of noncompactnessμinE, the map-pingG:B⊆E−→Eis said to be aμE-contractionif there exists a constant 0 <k< 1 such

that

μG(Y)≤kμ(Y) (1.1)

for any bounded closed subsetY⊆B.

Darbo [7] used the idea of measure of noncompactness to obtain a new ﬁxed point the-orem which generalizes the Banach contraction principle and assures the existence of a ﬁxed point concerning the so-called condensing operators.

Theorem 1.1([7]) Letbe a nonempty,closed,bounded,and convex subset of a Banach space E,and letG: →be a continuous mapping such that there exists a constantθ ∈ [0, 1)with the propertyμ(G())≤θ μ().ThenGhas a ﬁxed point in.

The following deﬁnition was given in [1] which is a generalization of Meir–Keeler con-traction (MKC) given in [9].

Deﬁnition 1.3([1]) For an arbitrary measure of noncompactnessμon a Banach spaceX, we say that an operatorT:B →Bis a Meir–Keeler condensing operator if for any> 0 there existsδ> 0 such that

≤μ(E) <+δ ⇒ μT(E)< (1.2)

for any bounded subsetEofB; whereBis a nonempty subset ofX.

Now we state the following theorem for Meir–Keeler condensing operators which will be applied in our main results.

Theorem 1.2([1]) Letμbe an arbitrary measure of noncompactness on a Banach space X.

IfT:B →B is a continuous and Meir–Keeler condensing operator,thenThas at least one ﬁxed point and the set of all ﬁxed points ofTin B is compact,where B is a nonempty,

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2 The sequence spacen(

φ

)

We denote by Cthe space of ﬁnite sets of distinct positive integers. For any σ∈C, we deﬁneα(σ) ={αn(σ)}such thatαn(σ) is 1 ifnis inσ; and 0 elsewhere. Write

Cr=

σ∈C: ∞

n=1

αn(σ)≤r

,

and deﬁne

=φ= (φk) : 0 <φ1≤φn≤φn+1and (n+ 1)φn≥nφn+1 .

Sargent [16] deﬁned the following sequence spaces which were further studied in [11]. WriteS(x) for the set of all sequences that are rearrangements ofx. Forφ∈,

m(φ) =

x= (xk) :xm(φ)=sup

r≥1

sup

σ∈Cr

1 φr

k∈σ

|xk|

<∞

,

n(φ) =

x= (xk) :xn(φ)= sup

u∈S(x)

_∞

k=1

|uk|φk

<∞

,

whereφk=φk–φk–1. Note that, for alln∈N={1, 2, 3, . . .},m(φ) =1,n(φ) =∞ifφn= 1;

andm(φ) =∞,n(φ) =1ifφn=n.

We have the following important result.

Theorem 2.1([12]) For any bounded subsetQof n(φ),we have

χ(Q) = lim

k→∞sup_x_∈_Q

sup

u∈S(x)

_∞

n=k

|un|φn

,

whereχ(Q) denotes theHausdorﬀ measure of noncompactnessof the setQwhich is de-ﬁned by

χ(Q) :=inf

> 0 :Q⊂

n

i=1

B(xi,ri),xi∈X,ri<(i= 1, 2, . . .)

.

3 Inﬁnite system of second order differential equations inn(

φ

)

We study the following inﬁnite system:

d2_u

i

dt2 = –fi

t,u1(t),u2(t),u3(t), . . .

; ui(0) =ui(T) = 0,t∈[0,T],i= 1, 2, 3 . . . (3.1)

LetC(I,R) be the space of all continuous real functions on the intervalI= [a,b] and

C2₍_I_,_R_{) be the class of functions with the second continuous derivative on}_I_{. A function}

u= (ui)∈C2(I,R) is a solution of (3.1) if and only ifu∈C(I,R) is a solution of the system

of integral equations

ui(t) =

T

0

G(t,s)fi

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wherefi(t,u)∈C(I×R∞,R),i= 1, 2, 3, . . . ; and the Green’s function associated with (3.1)

is given by

G(t,s) =

⎧ ⎨ ⎩

t

T(T–s), 0≤t≤s≤T, s

T(T–t), 0≤s≤t≤T.

(3.3)

From (3.2) and (3.3)

ui(t) =

t

0

s

T(T–t)fi

s,u(s)ds+

T

t

T(T–s)fi

s,u(s)ds.

Now compute

d

dtui(t) = –

1

T t

0

sfi

s,u(s)ds+ 1

T

t

(T–s)fi

s,u(s)ds.

Again diﬀerentiating we get

d2_u

i(t)

dt2 = – 1

T

tfi

t,u(t)+ 1

T(t–T)fi

t,u(t)) = –fi

t,u(t)).

The solution of the infinite system (3.1) in the sequence space1was discussed by Agha-jani and Pourhadi [2] by establishing a generalization of Darbo type fixed point theorem using the concept ofα-admissibility function and Schauder’s fixed point theorem. Here, we determine the solvability of system (3.1) in Banach sequence spacesn(φ). Our result is more general than that of [2].

Assume that

(i) The functionsfiare deﬁned on the setI×R∞and take real values. The operatorf

deﬁned on the spaceI×n(φ)inton(φ)as

(t,u)→(fu)(t) =f1

t,u(t),f2

t,u(t),f3

t,u(t), . . .

is such that the class of all functions((fu)(t))t∈Iis equicontinuous at every point of

the spacen(φ).

(ii) The following inequality holds:

fn

t,u1(t),u2(t),u3(t), . . .≤gn(t) +hn(t)un(t),

wheregn(t)andhn(t)are real functions deﬁned and continuous onIsuch that

∞

k=1gk(t)φkconverges uniformly onIand the sequence(hn(t))is equibounded

onI.

Write

G=sup

t∈I

∞

k=1

gk(t)φk

and

H= sup

n∈N,t∈I

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Theorem 3.1 Let conditions(i)–(ii)hold.Then system(3.1)has at least one solution u(t) = course a nonempty, bounded, closed, and convex subset ofn(φ). Consider the operator

F= (Fi) onC(I,B¯) deﬁned as follows. Fort∈I,

Also since (Fiu)(t) satisﬁes the boundary conditions, we have

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and

Pourhadi [2] but our proof is quite diﬀerent.

4 Example

In order to illustrate the above result, we provide the following example.

Example4.1 Let us consider the system of second order diﬀerential equations

–d considered system (3.1). Clearly j

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Notice that, for anyt∈I= [0,T], (fk(t,u(t)))∈n(φ) if (uk(t))∈n(φ). Moreover, we have

which implies continuity as in assumption (i). Now, we show that assumption (ii) is satis-ﬁed.

j4 is continuous and

Thus, from Theorem3.1, for a suitable value ofr1(as discussed in Theorem3.1) the operatorFas deﬁned in Theorem3.1onB¯(u0_,_r

1) has a ﬁxed pointu(t) = ((ui(t))∈n(φ),

which is a solution of system (4.1).

Funding

This project was funded by the Deanship of Scientiﬁc Research (DSR) at King Abdulaziz University, Jeddah, under grant No. RG-18-130-37. The authors, therefore, acknowledge with thanks DSR for technical and ﬁnancial support.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Publisher’s Note

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Received: 13 June 2018 Accepted: 18 September 2018

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