R E S E A R C H
Open Access
Application of measures of
noncompactness to the infinite system of
second-order differential equations in
p
spaces
Syed Abdul Mohiuddine
1*, Hari M Srivastava
2,3and Abdullah Alotaibi
1*Correspondence: [email protected] 1Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Kingdom of Saudi Arabia
Full list of author information is available at the end of the article
Abstract
In this article, we use the technique based upon measures of noncompactness in conjunction with a Darbo-type fixed point theorem with a view to studying the existence of solutions of infinite systems of second-order differential equations in the Banach sequence space
p. An illustrative example is also given in support of our
existence result.
MSC: Primary 47H10; secondary 34A34
Keywords: measure of noncompactness; Darbo-type fixed point theorem; infinite system of second-order differential equations; sequence spaces
1 Introduction
Measures of noncompactness endow helpful information, which is extensively used in the theory of integral and integro-differential equations. Besides, it is very helpful in the study of optimization, differential equations, functional equations, fixed point theory,etc. Some of the well-known measures of noncompactness are the Kuratowski measure (α), the Hausdorff measure (χ), and the Istrăţescu measure (β), which were introduced by Kuratowski [], Goldenšteinet al.[] (also studied by Goldenštein and Markus []), and Istrăţescu [], respectively. Darbo [] was the first who presented a fixed point theorem by using the idea of Kuratowski measures of noncompactness, the functionα, which is popu-larly called the Darbo fixed point theorem. This fixed point theorem generalized two very important and famous fixed point theorems, namely, (i) the classical Schauder fixed point theorem and (ii) special variant of the Banach fixed point theorem. The Darbo fixed point theorem has been generalized in many different directions. In fact, there is a vast amount of literature dealing with extensions and/or generalizations of this remarkable theorem. Recently, Aghajaniet al.[] presented a generalization of the Darbo fixed point theorem and used it to investigate the existence result concerning a general system of nonlinear integral equations. For some other recent works related to these concepts, we refer the interested reader to (for example) [–], and []. We also refer to the recent work by
Srivastavaet al.[] for some applications of fixed point theorems tofractional differen-tial equations (for details, see []).
Mursaleen and Mohiuddine [] earlier reported the existence theorems in the classical sequence spacepfor an infinite system of differential equations. On the other hand,
exis-tence theorems for infinite systems of linear equations inandpwere given by Alotaibi
et al.[]. Our main object in this sequel is to determine sufficient conditions for the solv-ability of an infinite system of second-order differential equations. We use the Dardo-type fixed point theorem given by Aghajani and Pourhadi [] for a new type of condensing operator and the method based upon the measures of noncompactness to establish the existence theorem for the above-mentioned infinite systems in the Banach sequence space
pwith p<∞. Our existence theorem is an extension of those obtained by Aghajani
and Pourhadi [] in the sequence space.
2 Preliminaries and notation
Letωdenote the space of all complex sequencesx= (xj)∞j=or, simply,x= (xj). Any vector
subspace ofωis called a sequence space. We use the standard notation∞,c, andcto
de-note the set of all bounded, convergent, and null sequences of real numbers, respectively. ByN,R, andCwe denote the sets of natural, real, and complex numbers, respectively. We recall that the notion of littleois used for comparison of growth of two arbitrary sequences xjandyjand is defined by
xj=o(yj) ⇐⇒ lim j→∞
xj
yj
= (yj= ).
We introduce the spacepof all absolutelyp-summable series as follows:
p=
x∈ω:
∞
j=
|xj|p<∞
(p<∞).
Clearly,pis a Banach space with norm
xp= ∞
j= |xj|p
/p
(p<∞).
Bye(j)we denote the sequence withjth term and all other terms zero (j∈N); we also
denotee= (, , , . . .). For any sequencex= (xj), let itsn-section be given by
x[n]=
n
j=
xje(j).
A sequence spaceXis called aBK spaceif it is a Banach space with continuous coordi-natespk:X→Candpk(x) =xkfor allx= (xj)∈Xandk∈N. ABK spaceX⊃ψ (that is,
the set of all finite sequences that terminate in zeros) is said to haveAKif every sequence x= (xj)∈Xhas a unique representation
x=
∞
j=
We denote byMX, (X,d), andB(x,r), respectively, the class of all bounded subsets ofX, the metric space, and the open ball with center atxand radiusr, that is,
B(x,r) =y∈X:d(x,y) <r.
LetF∈MX. Then theHausdorff measure of noncompactnessofFis defined by
χ(F) =inf
> :F⊂
n
j=
B(xj,rj),xj∈X,rj<(jn;n∈N)
.
The functionχ:MX→[,∞) is called theHausdorff measure of noncompactness. We now recall some basic properties of the Hausdorff measure of noncompactness. Let F,F, andFbe bounded subsets of the metric space (X,d). Then
(i) χ(F) = if and only ifFis totally bounded; (ii) χ(F) =χ(F¯), whereF¯ denotes the closure ofF; (iii) F⊂Fimplies thatχ(F)χ(F);
(iv) χ(F∪F) =max{χ(F),χ(F)};
(v) χ(F∩F) =min{χ(F),χ(F)}.
In the case of a normed space (X, · ), the functionχhas some additional properties connected with the linear structure. For example, we have
χ(F+F)χ(F) +χ(F),
χ(F+x) =χ(F) for allx∈X,
χ(αF) =|α|χ(F) for allα∈C.
Theorem (see []) Let X be a BK space with a Schauder basis(bj)∞j=and F∈MX.Also,
let Pj:X→X(j∈N)be the projector onto the linear span of{e(),e(), . . . ,e(j)}.Then
alim supj→∞
sup
x∈F
(I–Pj)(x)χ(F)lim sup j→∞
sup
x∈F
(I–Pj)(x), ()
where I is the identity operator on X,and
a=lim sup
j→∞
I–Pj.
It is known thatp(p<∞) is aBK space withAK with respect to its usual norm · p. Additionally,{e(),e(), . . .}as depicted from a Schauder basis forp, in view of (),
the following result is derivable by using Theorem (see [] and []).
Theorem Let F be a bounded subset of X=p.Then
χ(F) = lim
k→∞supx∈F
jk |xj|p
/p
. ()
Theorem LetCbe a nonempty,bounded,closed,and convex subset of a Banach space X,and let T:C→Cbe a continuous function satisfying the inequality
μT(F)ϕμ(F) ()
for each F ⊂C, whereμ is an arbitrary measure of noncompactness,andϕ: [,∞)→ [,∞)is an increasing(not necessarily continuous)function with
lim
n→∞ϕ
n(t) = .
Then T has at least one fixed point in the setC.
The notion of (α,φ,ϕ)-μ-condensing operators andα-admissible operators were re-cently demonstrated by Aghajani and Pourhadi [] by consideringϕ andφ as follows. We use the notationto denote the functionsϕ: [, +∞)→[,∞) like
lim inf
n→∞ ϕ(an) = ,
conferred that
lim
n→∞an= ,
where (an)n∈N is a nonnegative sequence. For ϕ ∈, let us consider a function φ :
[, +∞)→[, +∞) that satisfies the following conditions:
(i) φis a lower semi-continuous function withφ(t) = if and only ift= ; (ii) lim infn→∞ϕ(an) <φ(a), provided thatlimn→∞{an}=a.
We use the notationϕto denote the class of all such functions. Throughout this paper, byConvFwe denote theconvex hullofF⊂X.
LetT :W ⊆X→X is an arbitrary mapping. Further, we state thatT is (α,ϕ,φ)-μ -condensing if the functionsα:MX→[, +∞),ϕ∈, andφ=ϕare such that
α(F)φμ(TF)=ϕμ(F) (F∈W),
where bothFand its imageTFbelong toMX.
LetTandαbe given mappings as before. ThenTisα-admissibleif
α(F) ⇒ α(ConvTF) (F∈W;F,TF∈MX).
Remark IfT follows the Darbo condition with regard to a measureμand a constant
k∈[, ), that is, if
μ(TF) =kμ(F) (F∈W;F,TF∈MX),
thenTis an (α,ϕ,φ)-μ-condensing operator, whereα(F) = for any setF∈W such that F∈MX,φis the identity mapping, and the functionϕ(t) =kt,t. In this regard,Tis a
Aghajani and Pourhadi [] also established the following fixed point theorem by using
α-admissible and (α,φ,ϕ)-μ-condensing operators.
Theorem LetC∈MXbe a closed convex subset of a Banach space X,and let T:C→C
be a continuous(α,ϕ,φ)-μ-condensing operator,whereμis an arbitrary measure of non-compactness.Moreover,T isα-admissible,andα(C).Then T has at least one fixed point that pertains tokerμ.
3 Infinite systems of second-order differential equations
Let us consider the following infinite system of second-order differential equations:
–d
x i
dt =fi(t,x,x,x, . . .) ()
with the initial conditions given by
xi() =xi(T) =
i∈N:=N∪ {}={, , , . . .};t∈I= [,T]
.
The space of all continuous real functions onI with values inRand the space of all functions with two continuous derivatives on the intervalI are shown by the standard notationsC(I,R) andC(I,R), respectively. It is evident thatx∈C(I,R) is a solution of
() if and only ifx∈C(I,R) is a solution of the following system of integral equations:
xi(t) =
T
G(t,s)fi
s,x(s),x(s),x(s), . . .
ds (t∈I), ()
where
fi(t,x,x,x, . . .)∈C(I,R) (i∈N),
and the Green functionG(t,s) associated with () is given by
G(t,s) =
t
T(T–s) (tsT), s
T(T–t) (stT).
()
For more details of green functions, we refer to []. We can rewrite () with the help of () as follows:
xi(t) =
t
s
T(T–t)fi
s,x(s),x(s),x(s), . . .
ds
+ T
t
t
T(T–s)fi
s,x(s),x(s),x(s), . . .
ds. ()
Upon differentiating both sides of () with respect tot, we get
d dt
xi(t)
= –
T t
sfi
s,x(s),x(s),x(s), . . .
ds
+ T
T
t
(T–s)fi
s,x(s),x(s),x(s), . . .
Again, by differentiating both sides of () with respect totwe obtain
d
dt
xi(t)
= –
Ttfi
t,x(t),x(t),x(t), . . .
+
T(t–T)fi
t,x(t),x(t),x(t), . . .
= –fi
s,x(s),x(s),x(s), . . .
.
We now investigate the existence result concerning the second-order differential equa-tions for the infinite system given by () in the Banach sequence spacep(p<∞) with
the help of measures of noncompactness. For this investigation, we consider the following hypotheses:
(i) The functionsfi(i∈N) are defined onI×R∞and take real values. Furthermore,
the operatorf is shown on the spaceI×pas
(t,x)→(fx)(t) =f(t,x),f(t,x),f(t,x), . . .
,
which represent the space of maps fromI×pintop; it is found that the class of
all functions{(fx)(t)}t∈I is equicontinuous at every point ofp.
(ii) There are a nonnegative mappingg:I→R+, a functionh:I×p→R, and a
super-additive mappingϕ:R+→R+, that is,
ϕ(s+t)ϕ(s) +ϕ(t)
for alls,t∈R+, such that
h(t,x) ⇒ fi(t,x,x,x, . . .) p
gi(t)ϕ
|xi|p, ()
wherex= (xi)∈p,t∈I, andikfor somek∈N.
(iii) The functionG(t,s)g(s)is integrable onIand such that
g(s) =lim sup
i→∞
gi(s)
(i∈N)
for any fixed elementt∈I. Additionally, if a nonnegative sequence(yn)n∈N
converges to some number, then
lim inf
n→∞ ϕ(yn) <
C ()
such that
sup
t∈I
T
G(t,s)pg(s)ds
C
for some positive constantC. (iv) There is a functionxsuch that
In addition, fort∈I, we have
We are now prepared to formulate our main result.
Theorem Under assumptions(i)to(iv),the infinite system of second-order differential
equations()has at least one solution x(t) = (xi(t))such that x(t)∈pfor each t∈I.
clearly seen thatFis continuous onC(I,p). Obviously, the functionFxis also continuous,
and (Fx)(t)∈p ifx(t) = (xi(t))∈p. In view of the fact thatϕ is superadditive, together
with Eq. () and hypothesis (iii), it follows that
(Fx)(t)pp= nonempty bounded setQ∈Mp(whereMpdenotes the family of all nonempty bounded
lim
k→∞supx∈Q
Tp/p
jk
Cϕ|xj|p
/p
C lim
k→∞supx∈Q
ϕ
jk |xj|p
/p
=C lim
k→∞
ϕ
sup
x∈Q
jk |xj|p
/p
Cϕχ(Q).
This shows that
α(Q)φχ(FQ)ϕχ(Q),
whereα:Mp→[,∞) is the mapping defined by
α(Q) =
(h(t,x(t));x∈Q;t∈I), (otherwise),
and
φ(b) = b
C (b∈R+).
Obviously,φ∈ϕ, and it satisfies (). Interestingly, by hypothesis (iv) we conclude that the operatorFisα-admissible and satisfies all of the conditions of Theorem . Therefore,
Fhas at least one fixed pointx=x(t) such thatx(t)∈pfor allt∈I. Hereof, the function
x=x(t) is a solution of the infinite system ().
Remark Our existence theorem (Theorem ) is more general than that proved earlier
by Aghajani and Pourhadi []. Indeed, if we setp= in the sequence spacep, then it
reduces to the sequence space, and so Theorem . of Aghajani and Pourhadi [] is a
particular case of our Theorem .
We now present an interesting illustrative example in support of our result.
Example Consider the following second-order differential equations:
–d
dt{xq}=
t(T–t)e–qt
(q+ ) +
∞
r=q
xr(t) √
t
( +q)(r+ ), ()
whereq∈Nandt∈I= [,T] ( <T< √
). Obviously, the functionsaqr(t) given by
aqr(t) =
√
t ( +q)(r+ )
are continuous, and the series
∞
r=q
wherep> and /p+ /p= , which yields the continuity, as desired. Hence hypothesis (i) is satisfied. In order to verify hypotheses from (ii) to (iv), we reckon a functionh:I×p→
Rthat occurs on nonnegative values if and only if
x(t) =xq(t)
∈p,
where (xq(t)) is a nonincreasing sequence inR+with
x() = =x(T).
We thus find that
t(T–t)e–qt
uniformly with regard tot∈(,T). It is convenient to observe that
x∈p:ht,x(t) (t∈I)=∅.
Let
ht,x(t).
Now, from the data
x() =x(T) =
and () it follows that
which yields
By consideringϕ(t) as a kind of identity mapping, we conclude that conditions (ii), (iii), and () are satisfied. It is now left to show that () holds. Indeed, if we assume that
uniformly with respect tot∈(,T). In order to verify (), we have to show that
uniformly in (,T). By straightforward calculation we obtain
(q+ )
which converges uniformly to zero asq→ ∞. This evidently proves (), so that assump-tion (iv) is satisfied. Hence, in light of Theorem , Eq. () has a soluassump-tion in the spacep.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Kingdom of Saudi Arabia.2Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada. 3China Medical University, Taichung, 40402, Taiwan, Republic of China.
Acknowledgements
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
Received: 8 June 2016 Accepted: 31 October 2016 References
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