Tomus 44 (2008), 295–305
APPROXIMATION OF SOLUTIONS OF THE FORCED DUFFING EQUATION WITH NONLOCAL DISCONTINUOUS
TYPE INTEGRAL BOUNDARY CONDITIONS
Ahmed Alsaedi
Abstract. A generalized quasilinearization technique is applied to obtain a sequence of approximate solutions converging monotonically and quadra- tically to the unique solution of the forced Duffing equation with nonlocal discontinuous type integral boundary conditions.
1. Introduction
Integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow and population dynamics, see for example [16, 17, 24]. In fact, boundary value problems involving integral boundary conditions have received considerable attention, see for instance, [3, 10], [12]–[15], [18, 19, 26] and the references therein. In a recent reference [2], Ahmad, et. al. discussed the existence and uniqueness of the solutions of a boundary value problem with discontinuous type integral boundary conditions.
The monotone iterative technique coupled with the method of upper and lower solutions [5, 8, 20, 23, 25] manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions. In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear [11, 21]. To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization (QSL) [9]. This method has been developed for a variety of problems [1, 4, 6, 7, 22]. In view of its diverse applications, this approach is quite an elegant and easier for application algorithms. To the best of our knowledge, the method of quasilinearization has not been developed for Duffing equation with nonlocal discontinuous type integral boundary conditions.
In this paper, we apply a quasilinearization technique to obtain the analytic approximation of the solution of the forced Duffing equation with nonlocal dis- continuous type integral boundary conditions. In fact, we obtain a sequence of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand. The concept of nonlocal discontinuous integral
2000Mathematics Subject Classification: primary 34B10; secondary 34B15.
Key words and phrases: duffing equation, integral boundary conditions, quasilinearization, quadratic convergence.
Received December 12, 2007. Editor O. Došlý.
boundary conditions corresponds to a situation when some forcing term is present at an arbitrary intermediate point of the boundary segment and thereby generates a discontinuity in the integral boundary conditions.
2. Preliminaries We consider the following boundary value problem
(2.1)
u00(t) +σu0(t) +f(t, u) = 0, t∈[0,1], σ∈R\{0}, u(0)−µ1u0(0) =g1(u(γ)) +
Z γ−
0
q1 u(s) ds+
Z 1 γ+
q1 u(s) ds , u(1) +µ2u0(1) =g2 u(γ)
+ Z γ−
0
q2 u(s) ds+
Z 1 γ+
q2 u(s) ds , 0< γ <1,
where f: [0,1]×R→ R, gi: R→ R (i = 1,2) are continuous functions, qi are continuous functions on (0, γ) and (γ,1) andµi are nonnegative constants.
The quasilinearization technique is applied to obtain a sequence of approximate solutions converging monotonically and quadratically to the unique solution of the problem (2.1).
Definition 2.1. A functionα∈C2[0,1] is a lower solution of (2.1) if α00(t) +σα0(t) +f t, α(t)
≥0, t∈[0,1], α(0)−µ1α0(0)≤g1 α(γ)
+ Z γ−
0
q1 α(s) ds+
Z 1 γ+
q1 α(s) ds , α(1) +µ2α0(1)≤g2 α(γ)
+ Z γ−
0
q2 α(s) ds+
Z 1 γ+
q2 α(s) ds .
Similarly, β ∈ C2[0,1] is an upper solution of (2.1) if the inequalities in the definition of lower solution are reversed.
Since the associated homogeneous problem of (2.1) has only the trivial solution, therefore, by Green’s function method, the solutionu(t) of (2.1) can be written as
u(t) = 1
(1 +σµ1)−(1−σµ2)e−σ
h (−1 +σµ2)e−σ+e−σt
×n
g1 u(γ) +
Z γ−
0
q1 u(s) ds+
Z 1 γ+
q1 u(s) dso + (1 +σµ1)−e−σtn
g2 u(γ) +
Z γ−
0
q2 u(s) ds+
Z 1 γ+
q2 u(s) dsoi +
Z 1 0
G(t, s)f s, u(s) ds ,
where
G(t, s) = Λ
((1−σµ2)−eσ(1−s)
(1 +σµ1)−e−σt
, 0≤t≤s , (1−σµ2)−eσ(1−t)
(1 +σµ1)−e−σs
, s≤t≤1,
Λ = eσs
σ[(1−σµ2)−(1 +σµ1)eσ].
Observe that G(t, s)>0 on (0,1)×(0,1). We state the following results which lay a foundation to establish the main result. We omit the proof as the method of proof is similar to the one employed in [2].
Theorem 2.1. Let αand β be lower and upper solutions of the boundary value problem(2.1)respectively. Letf: [0,1]×R→Rbe such thatfu(t, u)<0andqiare continuous functions on (0, γ) and(γ,1) satisfying one sided Lipschitz condition:
qi(u)−qi(v) ≤Li(u−v), 0≤ Li < 1, i = 1,2, and gi:R →R are continuous functions satisfying one sided Lipschitz condition: gi(u)−gi(v) ≤ L∗i(u−v), 0≤L∗i <1,i= 1,2. Thenα(t)≤β(t).
Theorem 2.2. Assume thatαandβare lower and upper solutions of the boundary value problem (2.1) respectively such that α(t) ≤ β(t). If f: [0,1]×R → R, gi:R→Rare continuous functions andqi are continuous functions on (0, γ)and (γ,1)withgi and qi satisfying one sided Lipschitz condition, then there exists a solution u(t)of (2.1)such thatα(t)≤u(t)≤β(t),t∈[0,1].
3. Main result Theorem 3.1. Assume that
(A1) αand β∈C2[0,1] are respectively lower and upper solutions of (2.1)such that α(t)≤β(t);
(A2) f(t, u)∈C2([0,1]×R)be such thatfu(t, u)<0and fuu(t, u)+φuu(t, u)
≥ 0, whereφuu(t, u)≥0 for some continuous functionφ(t, u)on [0,1]×R; (A3) qi are continuous functions on(0, γ) and(γ,1)satisfying 0≤qi0(u)<1,
and q00i(u) +χ00i(u)
≥0 with χ00i ≥0,i= 1,2;
(A4) gi ∈ C2(R) be such that 0 ≤ gi0(u) < 1 and gi00(u) +ψ00i(u)
≤ 0 with ψi00≤0,i= 1,2.
Then, there exists a sequence {αn} of approximate solutions converging monotoni- cally and quadratically to the unique solution of the problem (2.1).
Proof. LetF: [0,1]×R→RandGi, Ki:R→Rbe defined byF(t, u) =f(t, u) + φ(t, u), Gi(u) = gi(u) +ψi(u), Ki(u) = qi(u) +χi(u) so that Fuu(t, u) ≥ 0, G00i(u)≤0,Ki00(u)≥0. Using the generalized mean value theorem together with (A2), (A3) and (A4), we obtain
f(t, u)≥f(t, v) +Fu(t, v)(u−v) +φ(t, v)−φ(t, u), (3.1)
gi(u)≤gi(v) +G0i(v)(u−v) +ψi(v)−ψi(u), u, v∈R, (3.2)
qi(u)≥qi(v) +Ki0(v)(u−v) +χi(v)−χi(u), u, v∈R. (3.3)
We set
F¯(t, u, α) =f(t, α) +Fu(tα)(u−α) +φ(t, α)−φ(t, u), Qi(u, α) =qi(α) +Ki0(α)(u−α) +χi(α)−χi(u),
¯
gi(u(γ), α, β) =gi(α(γ)) +G0i β(γ)
u(γ)−α(γ)
+ψi(α)−ψi(u), and note that
F¯u(t, u, α)<0, 0≤(∂/∂u)Qi(u, α)<1, 0≤(∂/∂u)¯gi u(γ), α, β
<1. Now, we fix α=α0and consider the problem
(3.4)
u00(t) +σu0(t) + ¯F(t, u, α0) = 0, t∈[0,1], u(0)−µ1u0(0) = ¯g1 u(γ), α0, β
+ Z γ−
0
Q1 u(s), α0(s) ds +
Z 1 γ+
Q1 u(s), α0(s) ds ,
u(1) +µ2u0(1) = ¯g2 u(γ), α0, β +
Z γ−
0
Q2 u(s), α0(s) ds +
Z 1 γ+
Q2 u(s), α0(s) ds .
As a first step, it will be shown that α0, β are respectively lower and upper solutions of (3.4). Using (A1) together with the fact that ¯F(t, α0, α0) =f(t, α0),
¯
g1 α(γ), α0, β
=gi α0(γ)
andQi(α0, α0) =qi(α0), we have
α000(t) +σα00(t) + ¯F(t, α0, α0) =α000(t) +σα00(t) +f(t, α0)≥0, t∈[0,1], α0(0)−µ1α00(0)≤g1 α0(γ)
+ Z γ−
0
q1 α0(s) ds+
Z 1 γ+
q1 α0(s) ds
= ¯g1(α0(γ), α0, β) + Z γ−
0
Q1 α0(s), α0(s) ds +
Z 1 γ+
Q1 α0(s), α0(s) ds ,
α0(1) +µ2α00(1)≤g2 α0(γ) +
Z γ−
0
q2 α0(s) ds+
Z 1 γ+
q2 α0(s) ds
= ¯g2 α0(γ), α0, β +
Z γ−
0
Q2 α0(s), α0(s) ds +
Z 1 γ+
Q2 α0(s), α0(s) ds and
β00(t) +σβ0(t) + ¯F(t, β, α0)≤β00(t) +σβ0(t) +f(t, β)≤0, t∈[0,1].
Moreover, there exists c0,c1∈ α0(γ), β(γ)
andc2,c3∈(α0, β) so that g1 β(γ)
−¯g1 β(γ), α0, β
=g1 β(γ)
−g1 α0(γ)
−G01 β(γ)
β(γ)−α0(γ)
−ψ1 α0(γ)
+ψ1 β(γ)
=
g01(c0)−g01 β(γ)
β(γ)−α0(γ) +
ψ10(c1)−ψ10 β(γ)
β(γ)−α0(γ)
≥0, q1 β(s)
−Q1 β(s), α0(s)
=q1 β(s)
−q1 α0(s)
−K10 α0(s)
β(s)−α0(s)
−χ1 α0(s)
+χ1 β(s)
=
q10(c2)−q01 α0(s)
β(s)−α0(s) +
χ01(c3)−χ01 α0(s)
β(s)−α0(s)
≥0 and consequently, we obtain
β(0)−µ1β0(0)≥g1 β(γ) +
Z γ−
0
q1 β(s) ds+
Z 1 γ+
q1 β(s) ds
≥g¯1 β(γ), α0, β +
Z γ−
0
Q1 β(s), α0(s) ds+
Z 1 γ+
Q1 β(s), α0(s) ds.
Similarly, it can be shown that β(1)+µ2β0(1)≥g¯2 β(γ), α0, β
+ Z γ−
0
Q2 β(s), α0(s) ds+
Z 1 γ+
Q2 β(s), α0(s) ds . Thus we conclude thatα0andβ are respectively lower and upper solutions of (3.4).
Hence, by Theorems 2.1 and 2.2, there exists the unique solutionα1of (3.4) such that
α0(t)≤α1(t)≤β(t), t∈[0,1]. Next, we consider
(3.5)
u00(t) +σu0(t) + ¯F(t, u, α1) = 0, t∈[0,1], u(0)−µ1u0(0) = ¯g1 u(γ), α1, β
+ Z γ−
0
Q1 u(s), α1(s) ds +
Z 1 γ+
Q1 u(s), α1(s) ds ,
u(1) +µ2u0(1) = ¯g2 u(γ), α1, β +
Z γ−
0
Q2 u(s), α1(s) ds +
Z 1 γ+
Q2 u(s), α1(s) ds .
Using the earlier arguments, it can be shown thatα1 andβ are lower and upper solutions of (3.5) respectively and by Theorems 2.1 and 2.2, there exists the unique solutionα2 of (3.5) such that
α1(t)≤α2(t)≤β(t), t∈[0,1].
Continuing this process successively yields a sequence{αn} of solutions satisfying α0(t)≤α1(t)≤α2(t)≤ · · · ≤αn(t)≤β(t), t∈[0,1],
where the elementαn of the sequence{αn} is a solution of the problem u00(t) +σu0(t) + ¯F(t, u, αn−1) = 0, t∈[0,1],
u(0)−µ1u0(0) = ¯g1 u(γ), αn−1, β +
Z γ−
0
Q1 u(s), αn−1(s) ds +
Z 1 γ+
Q1 u(s), αn−1(s) ds ,
u(1) +µ2u0(1) = ¯g2 u(γ), αn−1, β +
Z γ−
0
Q2 u(s), αn−1(s) ds +
Z 1 γ+
Q2 u(s), αn−1(s) ds
and is given by
αn(t) = −(1−σµ2)e−σ+e−σt (1 +σµ1)−(1−σµ2)e−σ
h
¯
g1 αn(γ), αn−1, β +
Z γ−
0
Q1 αn(s), αn−1(s) ds+
Z 1 γ+
Q1 αn(s), αn−1(s) dsi + (1 +σµ1)−e−σt
(1 +σµ1)−(1−σµ2)e−σ h
¯
g2 αn(γ), αn−1, β +
Z γ−
0
Q2 αn(s), αn−1(s) ds+
Z 1 γ+
Q2 αn(s), αn−1(s) dsi +
Z 1 0
G(t, s) ¯F s, αn(s), αn−1(s) ds . (3.6)
Using the fact that [0,1] is compact and the monotone convergence of the sequence {αn} is pointwise, it follows that the convergence of the sequence is uniform. If u(t) is the limit point of the sequence, taking the limitn→ ∞in (3.6), we obtain
u(t) = −(1−σµ2)e−σ+e−σt (1 +σµ1)−(1−σµ2)e−σ
hg1 u(γ) +
Z γ−
0
q1 u(s) ds+
Z 1 γ+
q1 u(s) dsi + (1 +σµ1)−e−σt
(1 +σµ1)−(1−σµ2)e−σ
hg2 u(γ) +
Z γ−
0
q2 u(s) ds+
Z 1 γ+
q2 u(s) dsi +
Z 1 0
G(t, s)f s, u(s) ds .
Thus,u(t) is a solution of (2.1). Now, we show that the convergence of the sequence is quadratic. For that we set ωn(t) = u(t)−αn(t)
≥0,t∈[0,1]. In view of (A2),
it follows by Taylor’s theorem that
ω00n(t) +σω0n(t) =u00+σu0−(α00n+σα0n) =−f(t, u) + ¯F(t, αn, αn−1)
=−f(t, u) +f(t, αn−1) +Fu(t, αn−1)(αn−αn−1) +φ(t, αn−1)−φ(t, αn)
=−fu(t, c4)(u−αn−1)−Fu(t, αn−1)(u−αn) +Fu(t, αn−1)(u−αn−1)−φu(t, c5)(αn−αn−1)
=
−fu(t, c4) +Fu(t, αn−1)−φu(t, c5) ωn−1 +
−Fu(t, αn−1) +φu(t, c5) ωn
=
−Fu(t, c4) +Fu(t, αn−1) +φu(t, c4)−φu(t, c5) ωn−1 +
−Fu(t, αn−1) +φu(t, c5) ωn
≥
−Fu(t, u) +Fu(t, αn−1) +φu(t, αn−1)−φu(t, αn) ωn−1 +
−Fu(t, αn−1) +φu(t, αn−1) ωn
≥
−Fuu(t, c6)−φuu(t, c7)
ωn−12 −fu(t, αn−1)ωn≥ −A1kωn−1k2, (3.7)
where αn−1 ≤c4, c6 ≤ u, αn−1 ≤ c5, c7 ≤αn, A is a bound onkFuuk, B is a bound onkφuuk andA1=A+B. Further, we have
ωn(0)−µ1ωn0(0) =g1 u(γ)
−g¯1 αn(γ), αn−1, β +
Z γ−
0
q1 u(s)
−Q1 αn(s), αn−1(s) ds +
Z 1 γ+
q1 u(s)
−Q1 αn(s), αn−1(s) ds
=g1 u(γ)
−g1 αn−1(γ)
−G01 β(γ)
(αn−αn−1) +
Z γ−
0
q1 u(s)
−q1= αn−1(s)
−K10 αn−1(s)
(αn−αn−1)−χ1(αn−1) +χ1(αn)
ds+ Z 1
γ+
q1 u(s)
−q1 αn−1(s)
−K10 αn−1(s)
(αn−αn−1)
−χ1(αn−1) +χ1(αn) ds
≤h1
2g100(ξ1) +ψ100(η2)i
ωn−12 (γ) +
G01 β(γ)
−ψ01(η1) ωn(γ) +
Z γ−
0
h
K10 αn−1(s)
−χ01(η3)
ωn(s) +1
2q001(ξ2) +χ001(η4)
ω2n−1(s)i ds +
Z 1 γ+
h
K10 αn−1(s)
−χ01(η3)
ωn(s) +1
2q001(ξ2) +χ001(η4)
ω2n−1(s)i ds
and
ωn(1) +µ2ωn0(1) =g2(u(γ))−¯g2(αn(γ), αn−1, β) +
Z γ−
0
q2 u(s)
−Q2 αn(s), αn−1(s) ds +
Z 1 γ+
q2 u(s)
−Q2 αn(s), αn−1(s) ds
≤h1
2g200(ξ3) +ψ200(η5)i
ω2n−1(γ) +
G02 β(γ)
−ψ02(η6) ωn(γ) +
Z γ−
0
h
K20 αn−1(s)
−χ02(η7)
ωn(s) +1
2q002(ξ4) +χ002(η8)
ωn−12 (s)i ds +
Z 1 γ+
h
(K20(αn−1(s))−χ02(η7))ωn(s) +1
2q200(ξ4) +χ002(η8)
ω2n−1(s)i ds , whereαn−1 ≤ξj ≤u, j = 1, . . . ,4, αn−1 ≤ην ≤ αn ≤u, ν = 1, . . . ,8. In view of (A3) and (A4), there exists λi < 1, λ∗i <1, Mi ≥0 and Mi∗ ≥0 such that
|G0i−ψi0| ≤λ∗i,|Ki0−χ0i| ≤λi,|12qi00+χ00i| ≤Miand
12gi00+ψ00i
≤Mi∗. Lettingλ= max{λ1, λ2}, λ∗ = max{λ∗1, λ∗2}, M∗ = max{M1∗, M2∗}, andM = max{M1, M2}, we get
(3.8)
ωn(0)−µ1ω0n(0)≤M∗ωn−12 (γ) +λ∗ωn(γ) +λhZ γ−
0
ωn(s)ds+ Z 1
γ+
ωn(s)dsi +MhZ γ−
0
ω2n−1(s)ds+ Z 1
γ+
ωn−12 (s)dsi , ωn(1) +µ2ω0n(1)≤M∗ωn−12 (γ) +λ∗ωn(γ)
+λhZ γ−
0
ωn(s)ds+ Z 1
γ+
ωn(s)dsi +MhZ γ−
0
ω2n−1(s)ds+ Z 1
γ+
ωn−12 (s)dsi . Using the estimates (3.7) and (3.8), we obtain
ωn(t) = −(1−σµ2)e−σ+e−σt (1 +σµ1)−(1−σµ2)e−σ
g1 u(γ)
−¯g1 αn(γ), αn−1, β +
Z γ−
0
q1 u(s)
−Q1 αn(s), αn−1(s) ds +
Z 1 γ+
q1 u(s)
−Q1 αn(s), αn−1(s) ds + (1 +σµ1)−e−σt
(1 +σµ1)−(1−σµ2)e−σ
g2 u(γ)
−g¯2 αn(γ), αn−1, β
+ Z γ−
0
q2 u(s)
−Q2 αn(s), αn−1(s) ds +
Z 1 γ+
q2 u(s)
−Q2 αn(s), αn−1(s) ds +
Z 1 0
G(t, s)
f s, u(s)
−F¯(t, αn, αn−1) ds
≤ −(1−σµ2)e−σ+e−σt (1 +σµ1)−(1−σµ2)e−σ
h
M∗ωn−12 (γ) +λ∗ωn(γ) +λZ γ−
0
ωn(s)ds+ Z 1
γ+
ωn(s)ds
+MZ γ−
0
ω2n−1(s)ds+ Z 1
γ+
ωn−12 (s)dsi + (1 +σµ1)−e−σt
(1 +σµ1)−(1−σµ2)e−σ h
M∗ωn−12 (γ) +λ∗ωn(γ) +λZ γ−
0
ωn(s)ds+ Z 1
γ+
ωn(s)ds
+MZ γ−
0
ω2n−1(s)ds+ Z 1
γ+
ωn−12 (s)dsi
− Z 1
0
G(t, s)
ωn00(s) +σωn0(s) ds
≤M∗ω2n−1(γ) +λ∗ωn(γ) +λZ γ−
0
ωn(s)ds+ Z 1
γ+
ωn(s)ds +MZ γ−
0
ωn−12 (s)ds+ Z 1
γ+
ω2n−1(s)ds
+A1kωn−1k2 Z 1
0
G(t, s)ds
≤M∗kωn−1k2+λ∗kωnk+λkωnk+Mkωn−1k2+A2kωn−1k2
=λ∗∗kωnk+M∗∗kωn−1k2 whereA2 provides a bound onA1
R1
0 G(t, s). We chooseλ∗ andλso thatλ∗∗ = λ∗+λ <1 andM∗∗ =M∗+M+A2. Taking the maximum over [0,1], we get
kωnk ≤ M∗∗
1−λ∗∗kωn−1k2, wherekuk= max
|u(t)|:t∈[0,1] . This establishes the quadratic convergence of
the sequence of iterates.
Remark. The results obtained in [2] appear as a special case of our results if we takeγ= 1/2 in (2.1) andψi≡0≡χi,i= 1,2 in the assumptions (A3) and (A4) of Theorem 3.1.
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Department of Mathematics, Faculty of Science King Abdulaziz University
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