doi:10.1155/2010/810453
Research Article
Singular Cauchy Initial Value Problem for Certain
Classes of Integro-Differential Equations
Zden ˇek ˇSmarda
Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic
Correspondence should be addressed to Zdenˇek ˇSmarda,[email protected]
Received 30 December 2009; Accepted 10 March 2010
Academic Editor: A ˘gacik Zafer
Copyrightq2010 Zdenˇek ˇSmarda. 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.
The existence and uniqueness of solutions and asymptotic estimate of solution formulas are stud-ied for the following initial value problem:gtyt ayt1ft, yt,t0Kt, s, yt, ysds, y0 0,t∈0, t0, wherea >0 is a constant andt0>0. An approach which combines topological method of T. Wa ˙zewski and Schauder’s fixed point theorem is used.
1. Introduction and Preliminaries
The singular Cauchy problem for first-order differential and integro-differential equations resolvedor unresolvedwith respect to the derivatives of unknowns is fairly well studied
see, e.g.,1–16, but the asymptotic properties of the solutions of such equations are only partially understood. Although the singular Cauchy problems were widely considered by using various methodssee, e.g.,1–13,16–18, the method used here is based on a different approach. In particular, we use a combination of the topological method of T. Wa ˙zewski
see, e.g., 19, 20 and Schauder’s fixed point theorem 21. Our technique leads to the existence and uniqueness of solutions with asymptotic estimates in the right neighbourhood of a singular point.
Consider the following problem:
gtyt ayt
1f
t, yt, t
0K
t, s, yt, ysds ,
y0 0,
wheref∈C0J×R×R,R, K∈C0J×J×R×R,R, J 0, t
0, t0>0.Denote
ft ogtast → 0if there is valid limt→0ft/gt 0,
ft∼gtast → 0if there is valid limt→0ft/gt 1.
The functionsg, f, Kwill be assumed to satisfy the following.
ia > 0 is a constant,gt ∈ C1J, gt > 0, g0 0, gt ∼ ψtgλtast →
0, λ >0, ψtgτt o1ast → 0for eachτ >0, ψ∈CJ,R.
ii|ft, u, v| ≤ |u| |v|, |t0Kt, s, yt, ysds| ≤ rt|y|,0 < rt ∈ CJ, rt φt, Co1ast → 0,whereφt, C Cexptt0a/gsdsis the general solution of the equationgtyt ayt.
In the text we will apply the topological method of Wa ˙zewski and Schauder’s theorem. Therefore, we give a short summary of them.
Letft, ybe a continuous function defined on an opent, y-setΩ ⊂ R×Rn,Ω0 an
open set ofΩ, ∂Ω0 the boundary of Ω0 with respect to Ω,and Ω0 the closure of Ω0 with
respect toΩ. Consider the system of ordinary differential equations
yft, y. 1.2
Definition 1.1see19. The pointt0, y0∈Ω∩∂Ω0is called an egressor an ingress point
ofΩ0with respect to system1.2if for every fixed solution of system1.2,yt
0 y0, there
exists an >0 such thatt, yt∈Ω0fort
0−≤t < t0 t0 < t≤t0. An egress pointingress
point t0, y0ofΩ0is called a strict egress pointstrict ingress pointofΩ0ift, yt/∈Ω 0
on intervalt0< t≤t01t0−1≤t < t0for an1.
Definition 1.2see19. An open subsetΩ0 of the set Ωis called au, v-subset ofΩwith
respect to system1.2if the following conditions are satisfied.
1There exist functionsuit, y∈ C1Ω,R, i 1, . . . , m,andvjt, y ∈CΩ,R, j
1, . . . , n, mn >0 such that
Ω0
t, y∈Ω:ui
t, y<0, vj
t, y<0 ∀i, j. 1.3
2u˙αt, y < 0 holds for the derivatives of the functionsuαt, y,α 1, . . . , m,along
trajectories of system1.2on the set
Uα
t, y∈Ω:uα
t, y0, ui
t, y≤0, vj
t, y≤0,∀j, i /α. 1.4
3v˙βt, y > 0 holds for the derivatives of the functions vβt, y,β 1, . . . , n,along
trajectories of system1.2on the set
Vβ
t, y∈Ω:vβ
t, y0, ui
t, y≤0, vj
t, y≤0, ∀i, j /β. 1.5
The set of all points of egressstrict egressis denoted byΩ0
Lemma 1.3see19. Let the setΩ0 be au, v-subset of the setΩwith respect to system1.2.
Then
Ω0
se Ω0e m
α1
Uα\ n
β1
Vβ. 1.6
Definition 1.4see19. LetXbe a topological space andB⊂X.
LetA⊂B. A functionr∈CB, Asuch thatra afor alla∈Ais a retraction from BtoAinX.
The setA⊂Bis a retract ofBinXif there exists a retraction fromBtoAinX.
Theorem 1.5Wa ˙zewski’s theorem19. LetΩ0be someu, v-subset ofΩwith respect to system
1.2. LetSbe a nonempty compact subset ofΩ0∪Ω0
esuch that the setS∩Ω0eis not a retract ofSbut is a retractΩ0
e. Then there is at least one pointt0, y0∈S∩Ω0such that the graph of a solutionyt
of the Cauchy problemyt0 y0for1.2lies inΩ0on its right-hand maximal interval of existence.
Theorem 1.6 Schauder’s theorem21. Let E be a Banach space and S its nonempty convex and closed subset. If P is a continuous mapping of S into itself and PS is relatively compact then the mapping P has at least one fixed point.
2. Main Results
Theorem 2.1. Let assumptions (i) and (ii) hold, then for eachC /0,there exists one solutionyt, C of initial problem1.1such that
yit, C−φit, C≤δφ2t, Ci, i0,1, 2.1
fort∈0, tΔ,where0< tΔ≤t0, δ >1is a constant, andtΔdepends onδ, C.
Proof. 1DenoteEthe Banach space of continuous functionshton the interval0, t0with
the norm
ht max
t∈0,t0|ht|. 2.2
The subsetS of Banach space E will be the set of all functions ht from Esatisfying the inequality
ht−φt, C≤δφ2t, C. 2.3
The setSis nonempty, convex and closed.
2 Now we will construct the mapping P. Let h0t ∈ S be an arbitrary function.
Substitutingh0t, h0sinstead ofyt, ysinto1.1, we obtain the differential equation
gtyt ayt
1f
t, yt, t
Set
yt φt, C Cexp
t
t0
1−αa gs ds
Y0t, 2.5
yt φt, C 1
gtCexp
t
t0
1−αa gs ds
Y1t, 2.6
where 0 < α < 1 is a constant and new functions Y0tandY1t satisfy the differential
equation
gtY0t α−1aY0t Y1t. 2.7
From2.3, it follows that
h0t φt, C H0t, |H0t| ≤δφ2t, C. 2.8
Substituting2.5,2.6and2.8into2.4we get
Y1t aY0t
aCexp
t
t0 αa gsds
aY0t
×f
t, φt, C Cexp
t
t0
1−αa gs ds
Y0t,
t
0
Kt, s, φt, C H0t, φs, C H0s
ds
.
2.9
Substituting2.9into2.7we get
gtY0t αaY0t
aCexp
t
t0 αa gsds
aY0t
×f
t, φt, C Cexp
t
t0
1−αa gs ds
Y0t,
t
0K
t, s, φt, C H0t, φs, C H0s
ds
.
2.10
Now we will use Wa ˙zewski’s topological method. Consider an open setΩ⊂ R×R. Investigate the behaviour of integral curves of2.10with respect to the boundary of the set
Ω0⊂Ω, Ω0{t, Y0: 0< t < t0, u0t, Y0<0}, 2.11
where
u0t, Y0 Y02−
δCexp
t
t0
1αa gs ds
2
. 2.12
Calculating the derivative ˙u0t, Y0along the trajectories of2.10on the set
∂Ω0{t, Y0: 0< t < t0, u0t, Y0 0}, 2.13
we obtain
˙ u0t, Y0
2a gt
αY02t
Y0tCexp
t
t0 αa gsds
Y02t
×f
t, φt, C Cexp
t
t0
1−αa gs ds
Y0t,
t
0
Kt, s, φt, C H0t, φs, C H0s
ds
−δ21αC2exp
t
t0
21αa
gs ds .
2.14
Since
lim
t→0ψtg
τt 0 for anyτ >0,
gt∼ψtgλt fort → 0, λ >0,
2.15
then there exists a positive constantMsuch that
Consequently,
t
t0 ds gs <
1 M
t
t0 gsdt
gs
1 Mln
gt
gt0 −→ −∞
if t−→0. 2.17
From here limt→0φt, C 0 and by L’Hospital’s ruleφτt, Cgσt o1fort → 0, σ
is an arbitrary real number. These both identities imply that the powers ofφt, Caffect the convergence to zero of the terms in2.14, in decisive way.
Using the assumptions ofTheorem 2.1and the definition ofY0t, φt, C,we get that
the first termαY2
0tin2.14has the form
αY02t αδ2C2exp
t
t0
21αa gs ds
, 2.18
and the second term
Y0tCexp
t
t0 αa gsds
Y02t
×f
t, φt, C Cexp
t
t0
1−αa gs ds
Y0t,
t
0K
t, s, φt, C H0t, φs, C H0s
ds
2.19
is bounded by terms with exponents which are greater than
t
t0
21αa
gs ds. 2.20
From here, we obtain
sgn ˙u0t, Y0 sgn
−δ2C21αexp
t
t0
21αa gs ds
−1 2.21
for sufficiently smallt∗,depending onC, δ, 0< t∗≤t0.
The relation2.21implies that each point of the set∂Ω0 is a strict ingress point with
respect to2.10. Change the orientation of the axistinto opposite. Now each point of the set∂Ω0is a strict egress point with respect to the new system of coordinates. By Wa ˙zewski’s
topological method, we state that there exists at least one integral curve of2.10lying inΩ0
Now we will prove the uniqueness of a solution of2.10. LetY0tbe also the solution
of2.10. PuttingZ0 Y0−Y0and substituting into2.10, we obtain
gtZ0αaZ0
aCexp
t
t0 αa gsds
atZ0t
×
f
t, φt, C Cexp
t
t0
1−αa gs ds
Z0t Y0t
,
t
0K
t, s, φt, C H0t, φs, C H0s
ds
−f
t, φt, C Cexp
t
t0
1−αa gs ds
Y0t,
t
0K
t, s, φt, C H0t, φs, C H0s
ds .
2.22
Let
Ω1δ {t, Z0: 0< t < t∗, u1t, Z0<0}, 2.23
where
u1t, Z0 Z02−
δCexp
t
t0
1α−μa
gs ds
2
, 0< μ < α. 2.24
Using the same method as above, we have
sgn ˙u1t, Z0 −1 2.25
fort ∈0, t∗. It is obvious thatΩ0 ⊂Ω1δfort ∈0, t∗.LetZ0tbe any nonzero solution
of 2.14such thatt1, Z0t1 ∈ Ω1 for 0 < t1 < t∗.Letδ ∈ 0, δbe such a constant that
t1, Z0t1 ∈ ∂Ω1δ.If the curveZ0tlays inΩ1δfor 0 < t < t1, thent1, Z0t1would
have to be a strict egress point of∂Ω1δwith respect to the original system of coordinates.
This contradicts the relation2.25. Therefore, there exists only the trivial solutionZ0t≡0
of2.22, soY0Y0tis the unique solution of2.10.
From2.5, we obtain
wherey0t, Cis the solution of2.4fort∈0, t∗.Similarly, from2.6,2.9we have
3We will prove thatP Sis relatively compact andPis a continuous mapping. It is easy to see, by2.26 and 2.27, that P Sis the set of uniformly bounded and equicontinuous functions fort∈0, t∗.By Ascoli’s theorem,P Sis relatively compact.
where
Investigate the behaviour of integral curves of2.29with respect to the boundary∂Ω0k, t∈
0, tΔ.Using the same method as above, we obtain for trajectory derivatives
sgn ˙u0kt, Y0 −1 2.32
wherem >0 is a constant depending ontΔ, C, M.This estimate implies thatPis continuous. We have thus proved that the mappingPsatisfies the assumptions of Schauder’s fixed point theorem and hence there exists a functionht ∈ Swithht Pht.The proof of existence of a solution of1.1is complete.
Let
By the same method as in the case of the existence of a solution of1.1, we obtain that inΩ00there is only the trivial solution of2.41. The proof is complete.
Example 2.2. Consider the following initial value problem:
In our case a general solution of the equation
According toTheorem 2.1, there exists for every constantC /0 the unique solutionyt, Cof
2.42such that
The author was supported by the Council of Czech Government Grants MSM 00216 30503 and MSM 00216 30529.
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