R E S E A R C H
Open Access
Nonlocal boundary value hyperbolic
problems involving integral conditions
Allaberen Ashyralyev
1,2and Necmettin Aggez
1**Correspondence:
1Department of Mathematics, Fatih
University, Istanbul, 34500, Turkey Full list of author information is available at the end of the article
Abstract
Stability estimates for the solution of the nonlocal boundary value problem with two integral conditions for hyperbolic equations in a Hilbert spaceHare established. In applications, stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained.
MSC: 35L10
Keywords: hyperbolic equation; stability; nonlocal boundary value problems
1 Introduction
It is well known that nonlocal boundary value problems with integral conditions are widely used for thermo-elasticity, chemical engineering, heat conduction, and plasma physics [–]. Some problems arising in dynamics of ground waters are defined as hyperbolic equations with nonlocal conditions [] and []. The authors of [] investigate nonclassi-cal problems for multidimensional hyperbolic equation with integral boundary conditions and the uniqueness of classical solution. In [] a linear second-order hyperbolic equation with forcing and integral constraints on the solution is converted to a nonlocal hyperbolic problem. Using the Riesz representation theorem and the Schauder fixed point theorem, existence and uniqueness of a generalized solution are proved. The solutions of hyperbolic equations with nonlocal integral conditions were investigated in [–]. The method of operators as a tool for investigation of the solution to hyperbolic equations in Hilbert and Banach spaces has been used extensively in [–].
In [] the nonlocal boundary value problem
⎧ ⎪ ⎨ ⎪ ⎩
du(t)
dt +Au(t) =f(t), ≤t≤, u() =nr=αru(λr) +ϕ, ut() =
n
r=βrut(λr) +ψ,
<λ≤λ≤ · · · ≤λn≤
was investigated. Stability estimates for the solution of the problem were established. First order of accuracy difference schemes for the approximate solutions of the problem were presented. Stability estimates for the solution of these difference schemes were estab-lished. Theoretical statements were supported by numerical examples.
In the present paper, we consider the nonlocal boundary value problem with integral conditions
⎧ ⎪ ⎨ ⎪ ⎩
du(t)
dt +Au(t) =f(t), ≤t≤, u() =α(ρ)u(ρ)dρ+ϕ,
ut() =
β(ρ)ut(ρ)dρ+ψ
()
in a Hilbert spaceHwith a self-adjoint positive definite operatorA. We are interested in studying the stability of solutions of problem () under the assumption
+
α(s)β(s)ds>
α(s)+β(s)ds. ()
As in [], a functionu(t) is called asolutionof problem () if the following conditions are satisfied:
(i) u(t)is twice continuously differentiable on the interval(, )and continuously differentiable on the segment[, ].
(ii) The elementu(t)belongs toD(A)for allt∈[, ], and the functionAu(t)is continuous on the segment[, ].
(iii) u(t)satisfies the equation and nonlocal boundary conditions ().
2 The main theorem
LetHbe a Hilbert space,Abe a positive definite self-adjoint operator withA≥δI, where δ>δ> . Throughout this paper,{c(t),t≥}is a strongly continuous cosine
operator-function defined by
c(t) = e
itA/+e–itA/
.
Then, from the definition of sine operator-functions(t),
s(t)u=
t
c(ρ)u dρ
it follows that
s(t) =A–/e
itA/
–e–itA/
i .
For the theory of cosine operator-function we refer to [] and [].
Lemma . The following estimates hold:
c(t)H→H≤, A/s(t)H→H≤. ()
Lemma . Suppose that assumption()holds.Then the operator T,
T=
+
α(s)β(s)ds ds
I–
has the inverse
T–=
+
α(s)β(s)ds ds
I–
β(s) +α(s)c(s)ds
–
and the following estimate is satisfied:
T–
H→H≤
| +
α(s)β(s)ds|–
(|α(s)|+|β(s)|)ds
. ()
Proof Applying the triangle inequality and estimates (), we obtain
+
α(s)β(s)ds ds
I–
β(s) +α(s)c(s)ds
H→H
≥ +
α(s)β(s)ds–
β(s) +α(s)c(s)ds
H→H
≥ +
α(s)β(s)ds–
α(s)+β(s)c(s)H→Hds
≥ +
α(s)β(s)ds–
α(s)+β(s)ds> .
Estimate () follows from this estimate. Lemma . is proved.
Now, we will obtain the formula for the solution of problem () forϕ∈D(A) andψ ∈
D(A/). It is clear that [] the initial value problem
du
dt +Au(t) =f(t), <t< , u() =u, u () =u
()
has a unique solution,
u(t) =c(t)u+s(t)u+
t
s(t–λ)f(λ)dλ, ()
where the functionf(t) is not only continuous but also continuously differentiable on [, ],
u∈D(A) andu∈D(A/).
Using () and the nonlocal boundary condition
u=
α(y)u(y)dy+ϕ,
we get
u=
α(y)
c(y)u+s(y)u+
y
s(y–λ)f(λ)dλ
dy+ϕ.
Then
I–
α(y)c(y)dy
u–
α(y)s(y)dyu=
α(y)
y
Differentiating both sides of (), we obtain
u(t) = –As(t)u+c(t)u+
t
c(t–λ)f(λ)dλ.
Using this formula and the integral condition
u=
β(y)u(y)dy+ψ,
we get
u=
β(y)
–As(y)u+c(y)u+
y
c(y–λ)f(λ)dλ
dy+ψ.
Thus,
β(y)As(y)dyu+
I–
β(y)c(y)dy
u
=
β(y)
y
c(y–λ)f(λ)dλdy+ψ. ()
Now, we have a system of equations () and () for the solution ofuandu. Solving it,
we get
u=T–
I–
β(y)c(y)dy
α(y)
y
s(y–λ)f(λ)dλdy+ϕ
+
α(y)s(y)dy
β(y)
y
c(y–λ)f(λ)dλdy+ψ
()
and
u=T–
I–
α(y)c(y)dy
β(y)
y
c(y–λ)f(λ)dλdy+ψ
–
β(y)As(y)dy
α(y)
y
s(y–λ)f(λ)dλdy+ϕ
. ()
Hence, for the solution of the nonlocal boundary value problem () we have (), (), and ().
Theorem . Suppose thatϕ∈D(A),ψ∈D(A/),and f(t)is a continuously differentiable on[, ];assumption()holds.Then there is a unique solution of problem()and the fol-lowing stability inequalities:
max
≤t≤
u(t)H≤M
ϕH+A–/ψH+
A–/f(λ)Hdλ
, ()
max
≤t≤
dudt(t)
H
+max
≤t≤A /u(t)
H
≤MA/ϕH+ψH+
f(λ)Hdλ
max
≤t≤
ddtu(t)
H
+max
≤t≤
Au(t)H
≤M
AϕH+A/ψH+f()H+
f(λ)Hdλ
()
are valid,where M does not depend on f(t),t∈[, ],ϕ,andψ.
Proof We take the estimates
max
≤t≤
u(t)H≤u()H+A–/u()H+
A–/f(t)Hdt
, ()
max
≤t≤u
(t)
H+≤maxt≤A
/u(t)
H
≤MA/u()H+u()H+
f(t)Hdt
, ()
max
≤t≤
ddtu(t)
H
+max
≤t≤
Au(t)H+max
≤t≤
A/u(t)
H
≤MAu()H+A/u()H+f()H+
f(t)Hdt
()
from [] for the solution of problem (). The proof of Theorem . is based on estimates (), (), (), and the estimates for the norms ofu,A–/u,A/u,u,Au,A/u.
First of all, let us find an estimate foru()H. By using () and estimates (), (), we
obtain
u()H≤M
A–/f(λ)Hdλ+ϕH+A–/ψH
. ()
ApplyingA–/to (), we get
A–/u=T–
I–
α(y)c(y)dy
β(y)
y
c(y–λ)A–/f(λ)dλdy+A–/ψ
–
β(y)A/s(y)dy
×
α(y)
y
A/s(y–λ)A–/f(λ)dλdy+ϕ
.
Using estimates (), (), we obtain
A–/u()H≤M
A–/f(λ)Hdλ+A–/ψH+ϕH
. ()
Thus, estimates (), (), and () yield estimate (). Second, applying operatorA/to (), we get
A/u=T–
I–
β(y)c(y)dy
α(y)
y
A/s(y–λ)f(λ)dλdy+A/ϕ
+
α(y)A/s(y)dy
β(y)
y
c(y–λ)f(λ)dλdy+ψ
Using estimates () and (), we obtain
A/u()H≤M
f(λ)Hdλ+A/ϕH+ψH
. ()
Using (), and estimates (), (), we get
u()H≤M
f(λ)Hdλ+ψH+A/ϕH
. ()
Then estimate () follows from estimates (), (), and (). Third, applyingAto () and using Abel’s formula, we have
Au=T–
I–
β(y)c(y)dy
×
α(y)
f(y) –c(y)f() –
y
c(y–λ)f(λ)dλ
dy+Aϕ
+
α(y)A/s(y)dy
×
β(y)
A/s(y)f() +
y
A/s(y–λ)f(λ)dλ
dy+A/ψ
and using estimates (), (), we get
Au()H≤M
AϕH+A/ψH+f()H+
f(λ)Hdλ
. ()
In the same manner, applyingA/to () and using Abel’s formula, and estimates (), (),
we obtain
A/u()H≤M
AϕH+A/ψH+f()H+
f(λ)Hdλ
. ()
Thus, estimate () follows from estimates () and (), and ().
3 Applications
Now, we consider the applications of Theorem .. First, a nonlocal boundary value prob-lem for a hyperbolic equation
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
utt– (a(x)ux)x+σu=f(t,x), <t< , <x< ,
u(,x) =α(ρ)u(ρ,x)dρ+ϕ(x),
ut(,x) =
β(ρ)ut(ρ,x)dρ+ψ(x), ≤x≤, u(t, ) =u(t, ), ux(t, ) =ux(t, ), ≤t≤
()
under assumption () is considered. Problem () has a unique smooth solutionu(t,x) for (), smooth functionsa(x)≥a> (x∈(, )),a() =a(),ϕ(x),ψ(x) (x∈[, ]) andf(t,x) (t,x∈[, ]),σ a positive constant and under some conditions. This allows us to reduce problem () to nonlocal boundary value problem () in the Hilbert spaceH=L[, ]
Theorem . For the solution of problem(),we have the following stability inequalities:
max
≤t≤ux(t,·)L[,]≤M
max
≤t≤f(t,·)L[,]+ϕxL[,]+ψL[,]
, ()
max
≤t≤
uxx(t,·)L[,]+max
≤t≤
utt(t,·)L[,]
≤M
max
≤t≤
ft(t,·)L[,]+f(,·)L[,]+ϕxxL[,]+ψxL[,]
, ()
where M does not depend onϕ(x),ψ(x),and f(t,x).
The proof of Theorem . is based on Theorem . and the symmetry properties of the space operator generated by problem ().
Proof Problem () can be written in the abstract form
⎧ ⎪ ⎨ ⎪ ⎩
du(t)
dt +Au(t) =f(t) (≤t≤), u() =α(ρ)u(ρ)dρ+ϕ,
ut() =
β(ρ)ut(ρ)dρ+ψ
()
in the Hilbert spaceL[, ] of all square integrable functions defined on [, ] with a
self-adjoint positive definite operatorA=Axdefined by the formula
Axu(x) = –a(x)ux
x+σu(x)
with the domain
DAx=
u(x) :u,ux,(a(x)ux)x∈L[, ], u() =u(),u() =u()
.
Here,f(t) =f(t,x) andu(t) =u(t,x) are known and unknown abstract functions defined on [, ] with the values in H=L[, ]. Therefore, estimates () and () follow from
estimates (), (), and (). Thus, Theorem . is proved.
Second, letΩ be the unit open cube in them-dimensional Euclidean spaceRm:{x=
(x, . . . ,xm) : <xj< , ≤j≤m}with boundaryS,Ω=Ω∪S. In [, ]×Ω, let us consider
a boundary value problem for the multidimensional hyperbolic equation
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂u(t,x)
∂t –
m
r=(ar(x)uxr)xr+σu(x) =f(t,x),
x= (x, . . . ,xm)∈Ω, <t< ,
u(,x) =α(ρ)u(ρ,x)dρ+ϕ(x), x∈Ω,
ut(,x) =
β(ρ)ut(ρ,x)dρ+ψ(x), x∈Ω, u(t,x) = , x∈S,
()
under assumption (). Here,ar(x) (x∈Ω),ϕ(x),ψ(x) (x∈Ω) andf(t,x),t∈(, ),x∈Ω
Let us introduce the Hilbert spaceL(Ω) of all square integrable functions defined onΩ,
equipped with the norm
fL(Ω)= · · ·
x∈Ω
f(x)dx· · ·dxm
.
Theorem . For the solution of problem(),the following stability inequalities hold:
max
≤t≤
m
r=
uxr(t,·)L(Ω)
≤M
max
≤t≤f(t,·)L(Ω)+
m
r=
ϕxrL(Ω)+ψL(Ω)
, ()
max
≤t≤
m
r=
uxrxr(t,·)L(Ω)+≤maxt≤utt(t,·)L(Ω)
≤M
max
≤t≤ft(t,·)L(Ω)+f(,·)L(Ω)
+
m
r=
ϕxrxrL(Ω)+
m
r=
ψxrL(Ω)
, ()
where M does not depend onϕ(x),ψ(x),and f(t,x) (t∈(, ),x∈Ω).
Proof Problem () can be written in the abstract form () in the Hilbert spaceL(Ω)
with a self-adjoint positive definite operatorA=Axdefined by the formula
Axu(x) = –
m
r=
ar(x)uxr
xr+σu(x) with domain
DAx=u(x) :u,uxr,
ar(x)uxr
xr∈L(Ω), ≤r≤m,u(x) = ,x∈S
.
Here,f(t) =f(t,x) andu(t) =u(t,x) are known and unknown abstract functions defined onΩwith the values inH=L(Ω). So, estimates () and () follow from estimates (),
(), (), and the following theorem.
Theorem .[] For the solution of the elliptic differential problem
Axu(x) =ω(x), x∈Ω,
u(x) = , x∈S,
the following coercivity inequality holds:
m
r=
uxrxrL(Ω)≤MωL(Ω),
4 Conclusion
This work is devoted to the study of the stability of the nonlocal boundary value problem with integral conditions for hyperbolic equations. For the solution of nonlocal boundary problem () in a Hilbert spaceH with a self-adjoint positive definite operatorA, Theo-rem . is established. Two applications of TheoTheo-rem . are given. Of course, stable two-step difference schemes for approximate solution of problem () can be presented. The methods given above permit us to establish the stability of these difference schemes. Ap-plying [], we can give a numerical support of the theoretical results.
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 final manuscript.
Author details
1Department of Mathematics, Fatih University, Istanbul, 34500, Turkey.2Department of Mathematics, ITTU, Gerogly
Street, Ashgabat, Turkmenistan.
Acknowledgements
This work is supported by the Scientific Research Fund of Fatih University (Project No: P50041203-B).
Received: 15 June 2014 Accepted: 19 August 2014 References
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