Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 193169, 19 pages
doi:10.1155/2009/193169
Research Article
Existence and Analytic Approximation of
Solutions of Duffing Type Nonlinear
Integro-Differential Equation with Integral
Boundary Conditions
Ahmed Alsaedi and Bashir Ahmad
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203,
Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Ahmed Alsaedi, aalsaedi@hotmail.com
Received 16 November 2008; Accepted 14 January 2009
Recommended by Yong Zhou
A generalized quasilinearization technique is developed to obtain a sequence of approximate
solutions converging monotonically and quadratically to a unique solution of a boundary value
problem involving Duffing type nonlinear integro-differential equation with integral boundary
conditions. The convergence of order k k ≥ 2 for the sequence of iterates is also established. It is
found that the work presented in this paper not only produces new results but also yields several
old results in certain limits.
Copyright q 2009 A. Alsaedi and B. Ahmad. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Vascular diseases such as atherosclerosis and aneurysms are becoming frequent disorders
in the industrialized world due to sedentary way of life and rich food. Causing more
deaths than cancer, cardiovascular diseases are the leading cause of death in the world. In
recent years, computational fluid dynamics CFD techniques have been used increasingly
by researchers seeking to understand vascular hemodynamics. CFD methods possess the
potential to augment the data obtained from in vitro methods by providing a complete
characterization of hemodynamic conditions blood velocity and pressure as a function of
space and time under precisely controlled conditions. However, specific difficulties in CFD
studies of blood flows are related to the boundary conditions. It is now recognized that the
blood flow in a given district may depend on the global dynamics of the whole circulation
and the boundary conditions e.g., the instantaneous velocity profile at the inlet section
of the computed domain for an in vitro blood flow computation need to be prescribed.
Taylor et al. 1 assumed very long circular vessel geometry upstream the inlet section to
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Journal of Inequalities and Applications
obtain the analytic solution due to Womersley 2. However, it is not always justified to
assume a circular cross-section. In order to cope with this problem, an alternative approach
prescribing integral boundary conditions is presented in 3. The validity of this approach is
verified by computing both steady and pulsated channel flows for Womersley number up to
15. In fact, the integral boundary conditions have various applications in other fields such as
chemical engineering, thermoelasticity, underground water flow, population dynamics, and
so forth, see for instance, 4–10 and references therein.
Integro-differential equations are encountered in many areas of science where it is
necessary to take into account aftereffect or delay. Especially, models possessing hereditary
properties are described by integro-differential equations in practice. Also, the governing
equations in the problems of biological sciences such as spreading of disease by the dispersal
of infectious individuals 11–13, the reaction-diffusion models in ecology to estimate the
speed of invasion 14, 15, and so forth, are integro-differential equations. For the theoretical
background of integro-differential equations, we refer the reader to the text 16.
In this paper, we study a boundary value problem for Duffing type nonlinear integrodifferential equation Duffing equation with both integral and nonintegral forcing terms of
nonlinear type with integral boundary conditions given by
−u t − γu t f t, ut u0 − μ1 u 0 1
t
K t, s, us ds,
0
q1 us ds,
u1 μ2 u 1 0
0 < t < 1,
1
q2 us ds,
1.1
0
where f : 0, 1 × R → R, K : 0, 1 × 0, 1 × R → R , qi : R → R i 1, 2 are continuous
functions, γ ∈ R − {0}, and μi i 1, 2 are nonnegative constants.
A generalized quasilinearization QSL technique due to Lakshmikantham 17, 18
is applied to obtain the analytic approximation of the solutions of the integral boundary
value problem 1.1. In recent years, the QSL technique has been extensively developed
and applied to a wide range of initial and boundary value problems for different types
of differential equations, for instance, see 19–30 and the references therein. Section 2
contains some basic results. A monotone sequence of approximate solutions converging
uniformly and quadratically to a unique solution of the problem 1.1 is obtained in
Section 3. The convergence of order k k ≥ 2 for the sequence of iterates is established in
Section 4 .
2. Preliminary Results
For any ρ, θ1 , θ2 ∈ C0, 1, the nonhomogeneous linear problem
−u t − γu t ρt, 0 < t < 1,
1
1
u0 − μ1 u 0 θ1 sds,
u1 μ2 u 1 θ2 sds,
0
0
2.1
Journal of Inequalities and Applications
3
has a unique solution
ut G1 t 1
2.2
Gt, sρsds,
0
where G1 t is the unique solution of the problem
−u t − γu t 0, 0 < t < 1,
1
1
u1 μ2 u 1 θ2 sds,
u0 − μ1 u 0 θ1 sds,
0
2.3
0
and is given by
1
1 γμ1 − 1 − γμ2 e−γ
1
1
×
− 1 γμ2 e−γ e−γt
θ1 sds 1 γμ1 − e−γt
θ2 sds ,
G1 t 0
⎧
⎨ 1 − γμ2 − eγ1−s 1 γμ1 − e−γt ,
Gt, s Λ ⎩ 1 − γμ − eγ1−t 1 γμ − e−γs ,
2
1
0
2.4
0 ≤ t ≤ s,
s ≤ t ≤ 1,
eγs
Λ .
γ 1 − γμ2 − 1 γμ1 eγ
Here, we note that the associated homogeneous problem has only the trivial solution and
Gt, s > 0 on 0, 1 × 0, 1.
Definition 2.1. A function α ∈ C2 0, 1 is a lower solution of 1.1 if
−α t − γα t ≤ f t, αt α0 − μ1 α 0 ≤
1
0
t
K t, s, αs ds,
0
q1 αs ds,
α1 μ2 α 1 ≤
0 < t < 1,
1
2.5
q2 αs ds.
0
Similarly, β ∈ C2 0, 1 is an upper solution of 1.1 if the inequalities in the definition of lower
solution are reversed.
Now, we present some basic results which are necessary to prove the main
results.
Theorem 2.2. Let α and β be lower and upper solutions of the boundary value problem 1.1,
respectively, such that αt ≤ βt, t ∈ 0, 1. If ft, ut and Kt, s, us are strictly decreasing
in u for each t ∈ 0, 1 and for each t, s ∈ 0, 1 × 0, 1, respectively, and qi satisfy a one-sided
Lipschitz condition: qi u − qi v ≤ Li u − v, 0 ≤ Li < 1, i 1, 2, then there exists a solution u of
1.1 such that αt ≤ ut ≤ βt, t ∈ 0, 1.
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Journal of Inequalities and Applications
Proof. We define P : J × R → R by P t, ut max{αt, min{ut, βt}} and consider the
modified problem
−u t − γu t f t, ut u0 − μ1 u 0 1
t
K t, s, us ds,
0
Q1 us ds,
u1 μ2 u 1 0 < t < 1,
1
0
2.6
Q2 usds,
0
where
f t, ut f t, P t, ut P t, ut − ut,
K t, s, us K t, s, P t, us ,
⎧
⎪
qi β, if u > β,
⎪
⎪
⎨
2.7
qi u qi u, if α ≤ u ≤ β,
⎪
⎪
⎪
⎩q α, if u < α.
i
As f, K, and qi are continuous and bounded, an application of Schauder’s fixed point theorem
16 ensures the existence of a solution of the problem 2.6. We note that any solution u of the
problem 2.6 such that αt ≤ ut ≤ βt, t ∈ 0, 1, is also a solution of 1.1. Thus we need
to show that αt ≤ ut ≤ βt, t ∈ 0, 1. As P t, αt max{αt, min{αt, βt}} αt
and P t, βt max{αt, min{βt, βt}} βt, we have
f t, αt t
K t, s, αs ds f t, αt 0
α0 − μ1 α 0 ≤
1
t
K t, s, αs ds ≥ −α t − γα t,
0
q1 αs ds 0
α1 μ2 α 1 ≤
f t, βt t
1
0
q2 αs ds 0
K t, s, βs ds f t, βt 0
β0 − μ1 β 0 ≥
1
t
β1 μ2 β 1 ≤
0
1
0
q1 αs ds,
q2 αs ds,
2.8
K t, s, βs ds ≤ −β t − γβ t,
0
q1 βs ds 0
1
1
1
0
q2 βs ds 1
0
q1 βs ds,
q2 βs ds,
which imply that α, β are lower and upper solutions of 2.6, respectively, on 0, 1.
Now, we show that α ≤ u on 0, 1. If the inequality α ≤ u is not true on 0, 1,
then mt αt − ut will have a positive maximum at some t0 ∈ 0, 1. We first
suppose that t0 ∈ 0, 1. Then mt0 > 0, m t0 0, m t0 ≤ 0. On the other hand,
Journal of Inequalities and Applications
5
we have
f t0 , ut0 t0
K t0 , s, us ds
0
−u t0 − γu t0 ≤ −α t0 − γα t0 t0
≤ f t0 , αt0 K t0 , s, αs ds,
2.9
0
which leads to a contradiction. Thus, there exists t1 ∈ 0, 1 such that mt1 ≤ 0. Assuming
t0 < t1 the case t0 > t1 is similar, we can find t2 ∈ t0 , t1 such that mt2 0 and mt > 0 for
every t ∈ t0 , t2 . Then
m t γm t ≥ −f t, αt −
t
K t, s, αs ds
0
f t, P t, ut
P t, ut − ut t
K t, s, P s, us ds
2.10
0
mt > 0,
which can alternatively be written as m teγt > 0. Integrating from t0 to t, and using
m t0 0, we obtain m t > 0 for every t ∈ t0 , t2 which together with m t0 0 implies that
m t ≥ 0 for every t ∈ t0 , t2 . Thus, mt is nondecreasing on t0 , t2 which is a contradiction
as mt has a positive maximum value at t t0 . Hence t0 is an isolated maximum point. If
t0 0, then m0 > 0, m 0 0, m 0 ≤ 0. On the other hand, we find that
m0 α0 − u0 ≤ μ1 m 0 1
0
1
0
q1 αs − q1 us ds
q1 αs − q1 us ds.
2.11
If ut < αt, then q1 us q1 αs, which, on substituting in 2.11, yields m0 ≤ 0.
This contradicts that m0 > 0. If ut > βt, then q1 us q1 βs. Hence, in view of the
fact that q1 satisfies a one-sided Lipschitz condition, we have q1 αs − q1 us q1 αs −
q1 βs ≤ L1 αs−βs which leads to m0 ≤ L1 maxt∈0,1 αt−βt L1 α0−β0 ≤ 0,
a contradiction. For αt ≤ ut ≤ βt, we also get a contradiction m0 ≤ 0. As before, there
exists t2 ∈ 0, t1 such that mt2 0 and mt > 0 for every t ∈ 0, t2 which provides a
contradiction that mt is nondecreasing on 0, t2 . Thus, t0 0 is an isolated maximum point.
In a similar manner, t0 1 yields a contradiction. Hence it follows that αt ≤ ut, t ∈ 0, 1.
On the same pattern, it can be shown that ut ≤ βt, t ∈ 0, 1. Thus, we conclude that
αt ≤ ut ≤ βt, t ∈ 0, 1. This completes the proof.
Theorem 2.3. Let α and β be, respectively, lower and upper solutions of the boundary value problem
1.1. If ft, ut and Kt, s, us are strictly decreasing in u for each t ∈ 0, 1 and for each
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Journal of Inequalities and Applications
t, s ∈ 0, 1 × 0, 1 respectively, and qi satisfies a one-sided Lipschitz condition: qi u − qi v ≤
Li u − v, 0 ≤ Li < 1, i 1, 2, then αt ≤ βt, t ∈ 0, 1.
Proof. We omit the proof as it follows the procedure employed in the proof of Theorem 2.2.
3. Analytic Approximation of the Solution with
Quadratic Convergence
Theorem 3.1. Assume that
A1 α and β ∈ C2 0, 1 are, respectively, lower and upper solutions of 1.1 such that αt ≤
βt, t ∈ 0, 1;
A2 f ∈ C2 0, 1×R is such that fu t, u < 0 for each t ∈ 0, 1 and fuu t, uφuu t, u ≥ 0,
where φuu t, u ≥ 0 for some continuous function φt, u on 0, 1 × R;
A3 K ∈ C2 0, 1 × 0, 1 × R is such that Ku t, s, u < 0 for each t, s ∈ 0, 1 × 0, 1 with
Kuu t, s, u ≥ 0.
A4 qi ∈ C2 R is such that 0 ≤ qi u < 1, and qi u ≥ 0, i 1, 2.
Then, there exists a sequence {αn } of approximate solutions converging monotonically and
quadratically to a unique solution of the problem 1.1.
Proof. Let F : 0, 1 × R → R be defined by Ft, u ft, u φt, u so that Fuu t, u ≥ 0. Using
the generalized mean value theorem together with A2 , A3 , and A4 , we obtain
ft, u ≥ ft, v Fu t, vu − v φt, v − φt, u,
Kt, s, u ≥ Kt, s, v ku t, s, vu − v,
qi u ≥ qi v q i vu − v,
3.1
u, v ∈ R.
We set
gt, u, v ft, v Fu t, vu − v φt, v − φt, u,
s, u; v Kt, s, v ku t, s, vu − v.
Kt,
3.2
Observe that gu t, u, v Fu t, v − φu t, u ≤ Fu t, u − φu t, u fu t, u < 0,
ft, u ≥ gt, u, v,
ft, u gt, u, u,
3.3
s, u; u.
Kt, s, u Kt,
3.4
u t, s, u, v < 0,
and K
s, u; v,
Kt, s, u ≥ Kt,
Let us define
Qi u, v qi v qi vu − v,
3.5
Journal of Inequalities and Applications
7
so that 0 ≤ ∂/∂uQi u, v qi < 1 and
qi u ≥ Qi u, v,
qi u Qi u, u.
Now, we fix α0 α and consider the problem
t
t, s, us; α0 s ds 0 ,
u t γu t gt, u, α0 K
0
1
u0 − μ1 u 0 Q1 us, α0 s ds,
u1 μ2 u 1 0
1
3.6
0 < t < 1,
3.7
Q2 us, α0 s ds.
0
Using A1 , 3.3, 3.4, and 3.6, we obtain
t
t, s, α0 s; α0 s ds
α0 t γα0 t g t, α0 , α0 K
0
t
α0 t γα0 t f t, α0 K t, s, α0 s ds ≥ 0, 0 < t < 1,
0
1
1
α0 0 μ1 α0 0 ≤ q1 α0 s ds Q1 α0 s, α0 sds,
0
0
1
1
q2 α0 s ds Q2 α0 s, α0 s ds,
0
0
t
t, s, βs; α0 s ds
β t γβ t g t, β, α0 K
0
t
≤ β t γβ t ft, β K t, s, βs ds ≤ 0, 0 < t < 1,
α0 1 μ2 α1 0 ≤
β0 − μ1 β 0 ≥
1
0
q1 βs ds ≥
0
β1 μ2 β 1 ≥
1
3.8
1
Q1 βs, α0 s ds,
0
q2 βs ds ≥
1
0
Q2 βs, α0 s ds,
0
which imply that α0 and β are, respectively, lower and upper solutions of 3.7. It follows by
Theorem 2.2 and 2.3 that there exists a unique solution α1 of 3.7 such that
α0 t ≤ α1 t ≤ βt,
t ∈ 0, 1.
3.9
Next, we consider
t
t, s, us; α1 s ds 0,
K
0
1
u0 − μ1 u 0 Q1 us, α1 s ds,
u t γu t gt, u, α1 u1 μ2 u 1 0
1
0
Q2 us, α1 s ds.
0 < t < 1,
3.10
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Journal of Inequalities and Applications
Using the earlier arguments, it can be shown that α1 and β are lower and upper solutions of
3.10, respectively, and hence by Theorem 2.2 and 2.3, there exists a unique solution α2 of
3.10 such that
α1 t ≤ α2 t ≤ βt,
t ∈ 0, 1.
3.11
Continuing this process successively yields a sequence {αn } of solutions satisfying
α0 t ≤ α1 t ≤ α2 t ≤ · · · ≤ αn ≤ βt,
t ∈ 0, 1,
3.12
where the element αn of the sequence {αn } is a solution of the problem
u t γu t gt, u, αn−1 t
0
u0 − μ1 u 0 s, us; αn−1 sds 0,
Kt,
1
0 < t < 1,
Q1 us, αn−1 s ds,
3.13
0
u1 μ2 u 1 1
Q2 us, αn−1 s ds,
0
and is given by
1
− 1 − γμ2 e−γ e−γt
αn t −γ Q1 αn s, αn−1 s ds
1 γμ1 − 1 − γμ2 e
0
1
1 γμ1 − e−γt
−γ Q2 αn s, αn−1 s ds
1 γμ1 − 1 − γμ2 e
0
1
t
Gt, s g s, αn s, αn−1 s K s, l, ul; αn−1 l dl ds.
0
3.14
0
Using the fact that 0, 1 is compact and the monotone convergence of the sequence {αn } is
pointwise, it follows by the standard arguments Arzela Ascoli convergence criterion, Dini’s
theorem 21 that the convergence of the sequence is uniform. If u is the limit point of the
sequence, taking the limit n → ∞ in 3.14, we obtain
1
− 1 − γμ2 e−γ e−γt
q1 usds
ut 1 γμ1 − 1 − γμ2 e−γ 0
1
1 γμ1 − e−γt
−γ q2 us ds
1 γμ1 − 1 − γμ2 e
0
1
t
Gt, s f s, us K s, l, ul dl ds,
0
0
3.15
Journal of Inequalities and Applications
9
thus, u is a solution of 1.1. Now, we show that the convergence of the sequence is quadratic.
For that we set en t ut − αn t ≥ 0, t ∈ 0, 1. In view of A2 , A3 , and 3.2, it follows
by Taylor’s theorem that
en t γen t
u γu − αn γαn
−ft, u −
t
K t, s, us ds g t, αn , αn−1 0
t
t, s, αn s; αn−1 s ds
K
0
−ft, u f t, αn−1 Fu t, αn−1 αn − αn−1 φ t, αn−1 − φ t, αn
t
− K t, s, us K t, s, αn−1 s Ku t, s, αn−1 s αn s − αn−1 s ds
0
−fu t, c1 u − αn−1 − Fu t, αn−1 u − αn Fu t, αn−1 u − αn−1 −φ t, c2 αn − αn−1
−
t
Ku t, s, d1 u − αn−1 − Ku t, s, αn−1 s αn s − αn−1 s ds
0
− fu t, c1 Fu t, αn−1 − φu t, c2 en−1 − Fu t, αn−1 φu t, c2 en
−
t
Ku t, s, d1 − Ku t, s, αn−1 s en−1 sds −
t
0
Ku t, s, αn−1 s en sds
0
− Fu t, c1 Fu t, αn−1 φu t, c1 − φu t, c2 en−1 − Fu t, αn−1 φu t, c2 en
−
t
Ku t, s, d1 − Ku t, s, αn−1 s en−1 sds −
t
0
Ku t, s, αn−1 s en sds
0
≥ − Fu t, u Fu t, αn−1 φu t, αn−1 − φu t, αn en−1 − Fu t, αn−1 φu t, αn−1 en
−
t
Ku t, s, us − Ku t, s, αn−1 s en−1 sds −
0
t
Ku t, s, αn−1 s en sds
0
2
− Fuu t, c3 − φuu t, c4 en−1
− fu t, αn−1 en
−
t
0
2
Kuu t, s, d2 en−1
sds
−
t
Ku t, s, αn−1 s en sds
0
2
≥ −Men−1 ,
3.16
where αn−1 ≤ c1 , c3 ≤ u, αn−1 ≤ c2 , c4 ≤ αn , αn−1 ≤ d1 , d2 ≤ u, M χ1 χ2 χ3 , χ1 is a bound
t
on Fuu , χ2 is a bound on φuu for t ∈ 0, 1, χ3 provides a bound on 0 Kuu ds. Further, in
10
Journal of Inequalities and Applications
view of 3.5, we have
en 0 − μ1 e n 0 1
q1 us − Q1 αn s, αn−1 s ds
0
1
0
en 1 μ2 e n 1 q1 us − q1 αn−1 s − q1 αn−1 s αn − αn−1 ds
1
1
2
s ds,
q1 αn−1 s en s q2 ζ1 en−1
2
0
1
3.17
q2 us − Q2 αn s, αn−1 s ds
0
1
1
2
s ds,
q2 αn−1 s en s q2 ζ2 en−1
2
0
where αn−1 ≤ ζ1 , ζ2 ≤ u. In view of A4 , there exist λi < 1 and Mi ≥ 0 such that qi αn−1 s ≤ λi
and 1/2qi ζi ≤ Mi i 1, 2. Let λ max{λ1 , λ2 } and M3 max{M1 , M2 }, then
1
en 0 − μ1 e n 0 ≤ λ
en sds M3
1
0
0
1
1
en 1 μ2 e n 1 ≤ λ
en sds M3
0
0
2
en−1
sds,
3.18
2
en−1
sds.
Using the estimates 3.16 and 3.18, we obtain
1
− 1 − γμ2 e−γ e−γt
en t q1 us − Q1 αn s, αn−1 s ds
−γ
1 γμ1 − 1 − γμ2 e
0
1
1 γμ1 − e−γt
q2 us − Q2 αn s, αn−1 s ds
−γ
1 γμ1 − 1 − γμ2 e
0
1
t
Gt, s f s, us K s, l, ul dl
0
0
− g t, αn , αn−1 t
0
s, l, αn l; αn−1 l dl
K
ds
Journal of Inequalities and Applications
1
1
− 1 − γμ2 e−γ e−γt
2
≤
e
sds
M
e
sds
λ
n
3
n−1
1 γμ1 − 1 − γμ2 e−γ
0
0
11
1
1
1 γμ1 − e−γt
2
sds
−γ λ en sds M3 en−1
1 γμ1 − 1 − γμ2 e
0
0
−
1
0
Gt, s en s γe n s ds
1
1
2 1
2
sds Men−1 Gt, sds
≤ λ en sds M3 en−1
0
0
0
2
2
≤ λen M3 en−1 M4 en−1 2
λen M5 en−1 ,
3.19
1
where M4 provides a bound on M 0 Gt, sds and M5 M4 M3 . Taking the maximum over
0, 1, we get
en ≤ M5 en−1 2 ,
1−λ
3.20
where u {|ut| : t ∈ 0, 1}. This establishes the quadratic convergence of the sequence of
iterates.
4. Higher Order Convergence
Theorem 4.1. Assume that
B1 α and β ∈ C2 0, 1 are, respectively, lower and upper solutions of 1.1 such that αt ≤
βt, t ∈ 0, 1;
B2 f ∈ Ck 0, 1 × R is such that ∂p /∂up ft, u < 0 p 1, 2, 3, . . . , k − 1 and
∂k /∂uk ft, u φt, u ≥ 0 with ∂k /∂uk φt, u ≥ 0 for some continuous function
φt, u on Ck 0, 1 × R;
B3 K ∈ C2 0, 1 × 0, 1 × R satisfies ∂p /∂up Kt, s, u < 0 p 1, 2, 3, . . . , k − 1 and
∂k /∂uk Kt, s, u ≥ 0.
B4 qj ∈ Ck R is such that di /dui qj u ≤ M/β − αi−1 i 1, 2, . . . , k − 1, j 1, 2
and dk /duk qj u ≥ 0, where M < 1/3.
Then, there exists a monotone sequence {αn } of approximate solutions converging uniformly and
rapidly to the unique solution of the problem 1.1 with the order of convergence k k ≥ 2.
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Journal of Inequalities and Applications
Proof. Using Taylor’s theorem and the assumptions B2 –B4 , we obtain
ft, u ≥
k−1 i
∂k
∂
u − vi
u − vk
−
,
ft,
v
φt,
ξ
i
i!
k!
∂uk
i0 ∂u
i
k−1 i
u−v
∂
,
Kt, s, u ≥
Kt, s, v
i
i!
i0 ∂u
qj u ≥
4.1
k−1
di
u − vi
,
q
v
j
i
i!
i0 du
where α ≤ v ≤ ξ ≤ u ≤ β. We set
ht, u, v k−1 i
∂k
∂
u − vi
u − vk
−
,
ft,
v
φt,
ξ
i
i!
k!
∂uk
i0 ∂u
K ∗ t, s, u, v Qj∗ u, v k−1 i
∂
u − vi
,
Kt, s, v
i
i!
i0 ∂u
4.2
k−1
di
u − vi
,
q
v
i j
i!
i0 du
and note that
ft, u ≥ ht, u, v,
Kt, s, u ≥ K ∗ t, s, u, v,
qj u ≥ Qj∗ u, v,
hu t, u, v Ku∗ t, s, u, v ∂ ∗
Q u, v ∂u j
ft, u ht, u, u,
4.3
Kt, s, u K ∗ t, s, u, u,
4.4
qj u Qj∗ u, u,
4.5
k−1 i
∂
∂k
u − vi−1
u − vk−1
− k φt, ξ
≤ 0,
ft, v
i
i − 1!
k − 1!
∂u
∂u
i1
k−1 i
∂
u − vi−1
≤ 0,
Kt,
s,
v
i − 1!
∂ui
i1
k−1
4.6
k−1
β − α
d
M
u − v
≤
qj v
i
i−1
i1
i − 1!
i − 1!
du
β − α
i1
i
1
≤ M 3 − k−2 < 1.
2
i−1
i−1
Journal of Inequalities and Applications
13
Letting α0 α, we consider the problem
u t γu t h t, u, α0 t
K ∗ t, s, u; α0 ds 0,
0 < t < 1,
0
u0 − μ1 u 0 1
0
u1 μ2 u 1 1
0
Q1∗ us, α0 s ds,
4.7
Q2∗ us, α0 s ds.
Using B1 , 4.3, 4.4, and 4.5, we obtain
α0 t γα0 t h t, α0 , α0 t
K ∗ t, s, α0 ; α0 ds
0
α0 t γα0 t f t, α0 α0 0 − μ1 α0 0 ≤
α0 1 μ2 α1 0 ≤
1
0
0
1
q2 α0 s ds 0
1
∗
0 < t < 1,
Q1∗ α0 s, α0 s ds,
Q2∗ α0 s, α0 s ds,
4.8
K t, s, β; α0 ds
t
Kt, s, βds ≤ 0,
0
q1 βs ds ≥
0
1
1
0
t
≤ β t γβ t ft, β β1 μ2 β 1 ≥
1
0
0
β0 − μ1 β 0 ≥
K t, s, α0 ds ≥ 0,
q1 α0 s ds β t γβ t h t, β, α0 t
1
0
q2 βs ds ≥
0
1
0
0 < t < 1,
Q1∗ βs, α0 s ds,
Q2∗ βs, α0 s ds.
Thus, it follows by definition that α0 and β are, respectively, lower and upper solutions of
4.7. As before, by Theorem 2.2 and 2.3, there exists a unique solution α1 of 4.7 such that
α0 t ≤ α1 t ≤ βt,
t ∈ 0, 1.
4.9
Continuing this process successively, we obtain a monotone sequence {αn } of solutions
satisfying
α0 t ≤ α1 t ≤ α2 t ≤ · · · ≤ αn ≤ βt,
t ∈ 0, 1,
4.10
14
Journal of Inequalities and Applications
where the element αn of the sequence {αn } is a solution of the problem
u t γu t h t, u, αn−1 t
K ∗ t, s, us; αn−1 s ds 0,
0 < t < 1,
0
u0 − μ1 u 0 1
0
u1 μ2 u 1 1
0
Q1∗ us, αn−1 s ds,
4.11
Q2∗ us, αn−1 s ds,
and is given by
1
− 1 − γμ2 e−γ e−γt
αn t −γ Q1∗ αn s, αn−1 s ds
1 γμ1 − 1 − γμ2 e
0
1
1 γμ1 − e−γt
−γ Q2∗ αn s, αn−1 s ds
1 γμ1 − 1 − γμ2 e
0
1
4.12
t
∗
Gt, s h s, αn s, αn−1 s K s, l, αn l; αn−1 l dl ds.
0
0
Employing the arguments used in the proof of Theorem 3.1, we conclude that the sequence
{αn } converges uniformly to a unique solution u of 1.1.
In order to prove that the convergence of the sequence is of order k k ≥ 2, we set
en t ut − αn t and an t αn1 t − αn t, t ∈ 0, 1 and note that
en t ≥ 0,
an t ≥ 0,
en1 t en t − an t,
enk ≥ akn .
Using Taylor’s theorem, we find that
en t γen t
u t γu t − αn t γαn t
−f t, ut −
t
0
K t, s, us ds h t, αn , αn−1 t
K ∗ t, s, αn s; αn−1 s ds
0
k−1 i
akn−1
ain−1
∂k
∂ −
−f t, ut f
t,
α
φt,
ξ
n−1
i
i!
k!
∂uk
i0 ∂u
t 0
k−1 i
ain−1
∂
ds
K
t,
s,
α
s
− K t, s, us n−1
i
i!
i0 ∂u
4.13
Journal of Inequalities and Applications
15
i
k−1 i
en−1
∂ f
t,
α
−f t, ut f t, αn−1 t n−1
i!
∂ui
i1
i
k−1 i
akn−1
− ain−1
en−1
∂k
∂ −
−
f
t,
α
φt,
ξ
n−1
i!
k!
∂ui
∂uk
i1
i
t k
k−1 i
en−1
en−1 − ain−1
∂
∂k
k Kt, s, η
ds
−
Kt, s, αn−1 s
i
i!
k!
∂u
0 i1 ∂u
k−1
k
k−1 i
en−1
akn−1
en−1 − an−1 ∂
∂k
∂k
i−1−l l
−
− k ft, ξ
ft,
α
e
a
−
φt,
ξ
n−1
n−1
n−1
k!
i!
k!
∂ui
∂u
∂uk
i0
i1
k−1
t k
k−1 i
en−1
en−1 − an−1 ∂
∂k
i−1−l l
ds
−
K t, s, αn−1 s
en−1 an−1 k Kt, s, η
i
i!
k!
∂u
0 i1 ∂u
i0
k
k−1 i
i−1
en−1 en ∂ ∂k
∂k
−
≥−
ft,
ξ
φt,
ξ
f
t,
α
ei−1−l al
n−1
k!
i! l0 n−1 n−1
∂ui
∂uk
∂uk
i1
t k
k−1 i
i−1
en−1
en ∂
∂k
i−1−l l
ds
−
K t, s, αn−1 s
e
a Kt, s, η
i
i! l0 n−1 n−1 ∂uk
k!
0 i1 ∂u
k
k−1 i
i−1
en−1
en ∂ ∂k
−
Ft,
ξ
f
t,
α
ei−1−l al
n−1
k!
i! l0 n−1 n−1
∂ui
∂uk
i1
t k
k−1 i
i−1
en−1
en ∂
∂k
i−1−l l
ds
−
K t, s, αn−1 s
e
a Kt, s, η
i
i! l0 n−1 n−1 ∂uk
k!
0 i1 ∂u
−
en−1 k
≥ −N
,
k!
4.14
k
k
where
t k Nk N1 N2 , N1 is a bound for ∂ /∂u Ft, ξ and N2 is a bound on
∂ /∂u Kt, s, ηds, αn−1 ≤ η ≤ u. Again, by Taylor’s theorem and using 4.2, we obtain
0
qj us −
Qj∗
i
k
k−1
u − αn−1
u − αn−1
dk
di q αn−1
qj c
αn s, αn−1 s i j
i!
k!
duk
i0 du
i
k−1
αn − αn−1
di −
q αn−1
i j
i!
du
i0
k
k−1
i−1
en−1
1
di dk
i−1−l l
en qj αn−1
e
a
qj c
i! l0 n−1 n−1
k!
dui
duk
i1
k
M en−1
γ k−1 k!
k
M en−1 ,
≤ χj ten t k−1
k!
γ
≤ χj ten t 4.15
16
Journal of Inequalities and Applications
where
χj t k−1
i−1
1
di i−1−l l
q
en−1
an−1 ,
α
j
n−1
i
i!
du
i1
l0
γ max βt − min αt.
t∈0,1
t∈0,1
4.16
Making use of B4 , we find that
χj t ≤
k−1
i1
i−1
k−1
i−1
1
M
1 i−1−l l
< 3M < 1.
β−α
i−1 i! en−1 an−1 ≤
i−1 i
−
1
!
β−α
i1 β − α
l0
M
4.17
Thus, we can find λ < 1 such that χj t ≤ λ, t ∈ 0, 1 and consequently, we have
en 0 − μ1 en 0 1
0
q1 us − Q1∗ αn s, αn−1 s ds
1
≤λ
en 1 μ2 en 1
1
0
en sds 0
M en−1 k ,
γ k−1 k!
q2 us − Q2∗ αn s, αn−1 s ds
1
≤λ
en sds 0
4.18
M en−1 k .
γ k−1 k!
By virtue of 4.14 and 4.18, we have
1
− 1 − γμ2 e−γ e−γt
en t q1 us − Q1∗ αn s, αn−1 s ds
−γ
1 γμ1 − 1 − γμ2 e
0
1
−γt
1 γμ1 − e
q2 us − Q2∗ αn s, αn−1 s ds
−γ
1 γμ1 − 1 − γμ2 e
0
1
t
Gt, s f s, us K s, l, ul dl − h t, αn s, αn−1 s
0
−
t
0
K s, l, αn l ; αn−1 l dl ds
∗
0
1
1
− 1 − γμ2 e−γ e−γt
M
k
≤
en−1
sds
−γ λ en sds k−1
γ k! 0
1 γμ1 − 1 − γμ2 e
0
1
1
−γt
1 γμ1 − e
M
k
λ en sds k−1
e sds
γ k! 0 n−1
1 γμ1 − 1 − γμ2 e−γ
0
1
− Gt, s en s γen s ds
0
1
1
M
N
k
en−1 k Gt, sds
e
sds
k!
γ k−1 k! 0 n−1
0
0
k
M
N3 k
k
≤ λen k−1 en−1 en−1 λen N4 en−1 ,
k!
γ k!
1
≤ λ en sds 4.19
Journal of Inequalities and Applications
17
1
where N4 M γ k−1 N3 /γ k−1 k! and N3 is a bound on N 0 Gt, sds. Taking the maximum
over 0, 1 and solving the above expression algebraically, we obtain
en ≤ N4 en−1 k .
1−λ
4.20
This completes the proof.
5. Discussion
An algorithm for approximating the analytic solution of Duffing type nonlinear integrodifferential equation with integral boundary conditions is developed by applying the method
of generalized quasilinearization. A monotone sequence of iterates converging uniformly and
quadratically rapidly to a unique solution of an integral boundary value problem 1.1 is
presented. For the illustration of the results, let us consider the following integral boundary
value problem:
−u t 1.5u t −teut1 − 2ut 1
t
2 − te−us−1 ds, 0 < t < 1,
0
1 1
1
us − 1 ds,
u0 − u 0 4
2 0 2
1
1
1
u1 u 1 us 1 ds.
2
0 2
5.1
Let αt −1 and βt t be, respectively, lower and upper solutions of 5.1. Clearly αt
and βt are not the solutions of 5.1 and αt < βt, t ∈ 0, 1. Moreover, the assumptions
A1 , A2 , A3 , and A4 of Theorem 3.1 are satisfied by choosing φt, u Mu2 , M > 0.
Thus, the conclusion of Theorem 3.1 applies to the problem 5.1.
The results established in this paper provide a diagnostic tool to predict the possible
onset of diseases such as cardiac disorder and chaos in speech by varying the nonlinear
forcing functions ft, u, Kt, s, u, and qi u appropriately in 1.1. If the nonlinearity ft, u
in 1.1 is of convex type, then the assumption A2 in Theorem 3.1 reduces to fuu t, u ≥ 0
and B2 in Theorem 4.1 becomes ∂k /∂uk ft, u ≥ 0 that is, φt, u 0 in this case. The
existence results for Duffing type nonlinear integro-differential equations with Dirichlet
boundary conditions can be recorded by taking q1 · 0 q2 · and μ1 0 μ2
in 1.1. Further, for q1 · a, q2 · b a and b are constants and μ1 0 μ2 in
1.1, our results become the existence results for Duffing type nonlinear integro-differential
equations with nonhomogeneous Dirichlet boundary conditions. If we take μ1 0 μ2
in 1.1, our problem reduces to the Dirichlet boundary value problem involving Duffing
type nonlinear integro-differential equations with integral boundary conditions. In case,
we fix q1 · a, q2 · b in 1.1, the existence results for Duffing type nonlinear
integro-differential equations with separated boundary conditions appear as a special case
of our results. By taking Kt, s, u ≡ 0 in 1.1, the results of 30 appear as a special
case of our work. The results for forced Duffing equation involving a purely integral type
18
Journal of Inequalities and Applications
of nonlinearity subject to integral boundary conditions follow by taking ft, u ≡ 0 in
1.1.
Acknowledgment
This work was supported by Deanship of Scientific Research, King Abdulaziz University
through Project no. 428/155.
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