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NOTE: Monotone Iterative Technique for a Nonlinear Integral Equation
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
205, 280]283 Ž1997.
AY965175
NOTE Monotone Iterative Technique for a Nonlinear Integral Equation Adrian Constantin Courant Institute of Mathematical Sciences, 251 Mercer Street, New York City, New York 10012 Submitted by William F. Ames Received February 22, 1996
A result for the existence of a positive solution to a nonlinear integral equation is proved using the monotone iterative technique and an application in the mathematical theory of water percolation phenomena is given. Q 1997 Academic Press
1 Let us consider the nonlinear integral equation uŽ t . s LŽ t . q
t
t
H0 M Ž t , s . u Ž s . ds q H0 K Ž t , s . g Ž u Ž s . . ds,
t G 0, Ž 1 .
where M, K g C Ž Rq= Rq, Rq ., L, g g C Ž Rq, Rq . with LŽ t . g Ž t . ) 0 for t ) 0 and g nondecreasing on Rq. We are looking for solutions u g C Ž Rq, Rq . of Eq. Ž1. with uŽ t . ) 0 for t ) 0. Let us introduce the class R of nondecreasing functions w g C Ž Rq, Rq . with w Ž t . ) 0 for t ) 0 and H1` Ž dsrwŽ s .. s `. THEOREM. If g g R then Eq. Ž1. has a solution u g C Ž Rq, Rq . with uŽ t . ) 0 for t ) 0 and this solution may be obtained by successi¨ e approximations starting with the identical zero function. by
Proof. We consider the operator O: C Ž Rq, Rq . ª C Ž Rq, Rq . defined
Ou Ž t . s L Ž t . q
t
t
H0 M Ž t , s . u Ž s . ds q H0 K Ž t , s . g Ž u Ž s . . ds, 280
0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
t G 0,
281
NOTE
where C Ž Rq, Rq . is endowed with the topology of uniform convergence on compact subintervals of Rq. Let u 0 Ž t . s 0, t g Rq, and define for n G 0, u nq 1 Ž t . s Ou n Ž t . ,
t g Rq .
Ž 2.
It is easy to verify by recurrence that u nŽ t .4nG 0 is an increasing sequence for any t g Rq. Let T ) 0 be fixed and choose A ) 0 such that sup tg w0, T x
L Ž t . 4 F A,
M Ž t , s . , K Ž t , s . 4 F A.
sup 0FsFtFT
Since g g R we have Žsee w2x. that H1` Ž dsrŽ s q g Ž s ... s ` so that the function G: A, ` . ª 0, ` . ,
GŽ t . s
t
ds
HA s q g Ž s . ,
is strictly increasing and onto. We define aŽ t . s Gy1 Ž At . for t g w0, T x. Let us show that for any n G 0 we have t g w 0, T x .
u n Ž t . F aŽ t . ,
Ž 3.
This is clearly true for n s 0. Assume it is true for n G 0 and let us prove it for n q 1. We have that u nq 1 Ž t . F A q A
t
u n Ž s . q g Ž u n Ž s . . ds
FAqA
t
a Ž s . q g Ž a Ž s . . ds
H0 H0
s aŽ t . ,
t g w 0, T x ,
the last equality being true by the way we defined aŽ t ., 0 F t F T. The monotonicity of u nŽ t .4nG 0 on w0, T x and relation Ž3. enable us to define uŽ t . s lim n ª` u nŽ t ., t g w0, T x. Letting n ª ` in the relation u nq 1 Ž t . s Ou n Ž t . ,
t g w 0, T x ,
we deduce by the Lebesgue dominated convergence theorem that uŽ t . s LŽ t . q
t
t
H0 M Ž t , s . u Ž s . ds q H0 K Ž t , s . g Ž u Ž s . . ds,
t g w 0, T x ,
282
NOTE
and uŽ t . g L1w0, T x, g Ž uŽ t .. g L1w0, T x so that, by the previous relation, we have that uŽ t . g C w0, T x is a solution of Ž1. on w0, T x. By Dini’s theorem Žsee w4x. we obtain that u nŽ t .4nG 0 converges uniformly to uŽ t . on w0, T x. Now, T ) 0 being arbitrary, we conclude that uŽ t . s lim nª` u nŽ t . is a solution of Ž1. on Rq. Due to the assumption LŽ t . ) 0 for t ) 0 we have that uŽ t . ) 0 for t ) 0 and the proof of the theorem is completed. Remark. The condition g g R is suggested by the following observation: if LŽ t . s L0 ) 0 for t g Rq, M Ž t, s . s 0 for Ž s, t . g Rq= Rq, K Ž t, s . s K 0 ) 0 for Ž t, s . g Rq= Rq, and g g C Ž Rq, Rq . is such that g Ž t . ) 0 for t ) 0 then Žsee w2x. the necessary and sufficient condition for Ž1. to have a solution on Rq for any choice of L0 ) 0 is that H1` Ž dsrg Ž s .. s `.
2 Describing the infiltration of a fluid from a cylindrical reservoir into an isotropic homogeneous porous medium in the Boussinesq model Žseew1x., we are led to the nonlinear integral equation. Žsee also w5x.
¨ 2 Ž t . s LŽ t . q
t
H0 K Ž t y s . ¨ Ž s . ds,
t g Rq ,
Ž 4.
where K and L are known nonnegative smooth functions depending on physical parameters. The unknown function ¨ denotes the height of the percolating fluid above the horizontal base, multiplied by a positive factor. For this reason, from the physical point of view, nonnegative solutions of Ž1. are most interesting. PROPOSITION. Assume that L g C 1 Ž Rq, Rq . is such that L9Ž0. / 0 and LŽ t . ) 0 for t ) 0. Then Eq. Ž4. has a unique solution ¨ g C Ž Rq, Rq . such that ¨ Ž t . ) 0 for t ) 0 and this solution may be obtained by successi¨ e approximations starting with the identical zero function. Proof. The existence part is a consequence of our theorem Žletting uŽ t . s ¨ 2 Ž t ., we transform Ž4. into an equation of type Ž1. with g Ž t . s 't for t g Rq .. In order to prove the uniqueness of the solution, assume the existence of a second solution w g C Ž Rq, Rq . and let z Ž t . s < ¨ Ž t . y w Ž t . <,
t g Rq .
283
NOTE
If B s sup t g w0, 1x K Ž t .4 , the relations ¨ 2 Ž t . y w2 Ž t . s
t
H0 K Ž t y s . ¨ Ž s . y w Ž s . ds, ¨ Ž t . , w Ž t . G 'L Ž t . , 0 F t F 1,
0 F t F 1,
imply zŽ t. F
B
'
t
2 LŽ t .
H0 z Ž s . ds,
0 F t F 1.
Since L9Ž0. / 0 implies that 1r L Ž t . g L1 w0, 1x, an application of Gronwall’s lemma Žsee w6x. enables us to deduce that z Ž t . s 0 for 0 F t F 1. Clearly the argument can be repeated on any interval w0, n x with n g N and so the solution is unique.
'
Equation Ž4. was also considered in w5x under the assumption that the function t ª LŽ t .r 't is nondecreasing and convex on Ž0, `. and in w3x under the assumption that LŽ t . s L0 ) 0 for t g Rq and that K g C 1 Ž Rq, Rq . is nondecreasing. It is easy to see that our proposition improves the result in w3x and covers cases where the results from w5x are not applicable.
REFERENCES 1. J. Bear, ‘‘Dynamics of Fluids in Porous Media,’’ Amer. Elsevier, New York, 1972. 2. A. Constantin, Solutions globales d9equations differentielles perturbees, ´ ´ ´ C. R. Acad. Sci. Paris 320 Ž1995., 1319]1322. 3. A. Constantin, Topological transversality: Application to an integrodifferential equation, J. Math. Anal. Appl. 197 Ž1996., 855]863. 4. J. Dieudonne, ´ ‘‘Foundations of Modern Analysis,’’ Academic Press, New York, 1969. 5. W. Okrasinski, On a non-linear convolution equation occurring in the theory of water percolation, Ann. Polon. Math. 37 Ž1980., 223]229. 6. G. Sansone and R. Conti, ‘‘Non-linear Differential Equations,’’ Pergamon, New York, 1964.
205, 280]283 Ž1997.
AY965175
NOTE Monotone Iterative Technique for a Nonlinear Integral Equation Adrian Constantin Courant Institute of Mathematical Sciences, 251 Mercer Street, New York City, New York 10012 Submitted by William F. Ames Received February 22, 1996
A result for the existence of a positive solution to a nonlinear integral equation is proved using the monotone iterative technique and an application in the mathematical theory of water percolation phenomena is given. Q 1997 Academic Press
1 Let us consider the nonlinear integral equation uŽ t . s LŽ t . q
t
t
H0 M Ž t , s . u Ž s . ds q H0 K Ž t , s . g Ž u Ž s . . ds,
t G 0, Ž 1 .
where M, K g C Ž Rq= Rq, Rq ., L, g g C Ž Rq, Rq . with LŽ t . g Ž t . ) 0 for t ) 0 and g nondecreasing on Rq. We are looking for solutions u g C Ž Rq, Rq . of Eq. Ž1. with uŽ t . ) 0 for t ) 0. Let us introduce the class R of nondecreasing functions w g C Ž Rq, Rq . with w Ž t . ) 0 for t ) 0 and H1` Ž dsrwŽ s .. s `. THEOREM. If g g R then Eq. Ž1. has a solution u g C Ž Rq, Rq . with uŽ t . ) 0 for t ) 0 and this solution may be obtained by successi¨ e approximations starting with the identical zero function. by
Proof. We consider the operator O: C Ž Rq, Rq . ª C Ž Rq, Rq . defined
Ou Ž t . s L Ž t . q
t
t
H0 M Ž t , s . u Ž s . ds q H0 K Ž t , s . g Ž u Ž s . . ds, 280
0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
t G 0,
281
NOTE
where C Ž Rq, Rq . is endowed with the topology of uniform convergence on compact subintervals of Rq. Let u 0 Ž t . s 0, t g Rq, and define for n G 0, u nq 1 Ž t . s Ou n Ž t . ,
t g Rq .
Ž 2.
It is easy to verify by recurrence that u nŽ t .4nG 0 is an increasing sequence for any t g Rq. Let T ) 0 be fixed and choose A ) 0 such that sup tg w0, T x
L Ž t . 4 F A,
M Ž t , s . , K Ž t , s . 4 F A.
sup 0FsFtFT
Since g g R we have Žsee w2x. that H1` Ž dsrŽ s q g Ž s ... s ` so that the function G: A, ` . ª 0, ` . ,
GŽ t . s
t
ds
HA s q g Ž s . ,
is strictly increasing and onto. We define aŽ t . s Gy1 Ž At . for t g w0, T x. Let us show that for any n G 0 we have t g w 0, T x .
u n Ž t . F aŽ t . ,
Ž 3.
This is clearly true for n s 0. Assume it is true for n G 0 and let us prove it for n q 1. We have that u nq 1 Ž t . F A q A
t
u n Ž s . q g Ž u n Ž s . . ds
FAqA
t
a Ž s . q g Ž a Ž s . . ds
H0 H0
s aŽ t . ,
t g w 0, T x ,
the last equality being true by the way we defined aŽ t ., 0 F t F T. The monotonicity of u nŽ t .4nG 0 on w0, T x and relation Ž3. enable us to define uŽ t . s lim n ª` u nŽ t ., t g w0, T x. Letting n ª ` in the relation u nq 1 Ž t . s Ou n Ž t . ,
t g w 0, T x ,
we deduce by the Lebesgue dominated convergence theorem that uŽ t . s LŽ t . q
t
t
H0 M Ž t , s . u Ž s . ds q H0 K Ž t , s . g Ž u Ž s . . ds,
t g w 0, T x ,
282
NOTE
and uŽ t . g L1w0, T x, g Ž uŽ t .. g L1w0, T x so that, by the previous relation, we have that uŽ t . g C w0, T x is a solution of Ž1. on w0, T x. By Dini’s theorem Žsee w4x. we obtain that u nŽ t .4nG 0 converges uniformly to uŽ t . on w0, T x. Now, T ) 0 being arbitrary, we conclude that uŽ t . s lim nª` u nŽ t . is a solution of Ž1. on Rq. Due to the assumption LŽ t . ) 0 for t ) 0 we have that uŽ t . ) 0 for t ) 0 and the proof of the theorem is completed. Remark. The condition g g R is suggested by the following observation: if LŽ t . s L0 ) 0 for t g Rq, M Ž t, s . s 0 for Ž s, t . g Rq= Rq, K Ž t, s . s K 0 ) 0 for Ž t, s . g Rq= Rq, and g g C Ž Rq, Rq . is such that g Ž t . ) 0 for t ) 0 then Žsee w2x. the necessary and sufficient condition for Ž1. to have a solution on Rq for any choice of L0 ) 0 is that H1` Ž dsrg Ž s .. s `.
2 Describing the infiltration of a fluid from a cylindrical reservoir into an isotropic homogeneous porous medium in the Boussinesq model Žseew1x., we are led to the nonlinear integral equation. Žsee also w5x.
¨ 2 Ž t . s LŽ t . q
t
H0 K Ž t y s . ¨ Ž s . ds,
t g Rq ,
Ž 4.
where K and L are known nonnegative smooth functions depending on physical parameters. The unknown function ¨ denotes the height of the percolating fluid above the horizontal base, multiplied by a positive factor. For this reason, from the physical point of view, nonnegative solutions of Ž1. are most interesting. PROPOSITION. Assume that L g C 1 Ž Rq, Rq . is such that L9Ž0. / 0 and LŽ t . ) 0 for t ) 0. Then Eq. Ž4. has a unique solution ¨ g C Ž Rq, Rq . such that ¨ Ž t . ) 0 for t ) 0 and this solution may be obtained by successi¨ e approximations starting with the identical zero function. Proof. The existence part is a consequence of our theorem Žletting uŽ t . s ¨ 2 Ž t ., we transform Ž4. into an equation of type Ž1. with g Ž t . s 't for t g Rq .. In order to prove the uniqueness of the solution, assume the existence of a second solution w g C Ž Rq, Rq . and let z Ž t . s < ¨ Ž t . y w Ž t . <,
t g Rq .
283
NOTE
If B s sup t g w0, 1x K Ž t .4 , the relations ¨ 2 Ž t . y w2 Ž t . s
t
H0 K Ž t y s . ¨ Ž s . y w Ž s . ds, ¨ Ž t . , w Ž t . G 'L Ž t . , 0 F t F 1,
0 F t F 1,
imply zŽ t. F
B
'
t
2 LŽ t .
H0 z Ž s . ds,
0 F t F 1.
Since L9Ž0. / 0 implies that 1r L Ž t . g L1 w0, 1x, an application of Gronwall’s lemma Žsee w6x. enables us to deduce that z Ž t . s 0 for 0 F t F 1. Clearly the argument can be repeated on any interval w0, n x with n g N and so the solution is unique.
'
Equation Ž4. was also considered in w5x under the assumption that the function t ª LŽ t .r 't is nondecreasing and convex on Ž0, `. and in w3x under the assumption that LŽ t . s L0 ) 0 for t g Rq and that K g C 1 Ž Rq, Rq . is nondecreasing. It is easy to see that our proposition improves the result in w3x and covers cases where the results from w5x are not applicable.
REFERENCES 1. J. Bear, ‘‘Dynamics of Fluids in Porous Media,’’ Amer. Elsevier, New York, 1972. 2. A. Constantin, Solutions globales d9equations differentielles perturbees, ´ ´ ´ C. R. Acad. Sci. Paris 320 Ž1995., 1319]1322. 3. A. Constantin, Topological transversality: Application to an integrodifferential equation, J. Math. Anal. Appl. 197 Ž1996., 855]863. 4. J. Dieudonne, ´ ‘‘Foundations of Modern Analysis,’’ Academic Press, New York, 1969. 5. W. Okrasinski, On a non-linear convolution equation occurring in the theory of water percolation, Ann. Polon. Math. 37 Ž1980., 223]229. 6. G. Sansone and R. Conti, ‘‘Non-linear Differential Equations,’’ Pergamon, New York, 1964.