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An integral inequality
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
An Integral
(1981)
Inequality* D. ROGERS
THOMAS
Department of Mathemarics. Edmorlton
82, 470-472
APPLICATIONS
Unicersif~ of Alberla.
76G 2G1,
Canada
Submitted bv C: Lakshmikantham
In this article we obtain improved pointwise bounds for solutions integral inequalities of the form 0<
h(f)
< x(r)
+ 1-l [H(r,
s) -
G(f,
s)]
x(s)
x(f)
ds
of
(1)
-0
in the situation where H(t, s), G(f, s) > 0 and H(f, s) is nondecreasing in f, but no assumption is made on the sign of H(t, s) - G(t, s). Results of a similar nature for the case G(t, s) = 0 are found in [l] and [2]. THEOREM. Suppose f (t) is nonnegafiue and nondecreasing on the interoal Z: 0 < f < T and that H(f, s) and G(f, s) are nonnegafitre on I x I with H(f, s) nondecreasing in t for fixed s. Define. for n = 2. 3,...,
.I,(!; h(t)) = 1’ G(f, s) h(s) ds, -0
J,(f;h(f))=!;-(;‘...
j:“-‘H(f.f,)
x G(f,m,.
...
H(f,-,,f,-,)
h(f,) df, ... df,.
fn)
Then solutions x(t) of (1) also satisy the explicit pointwise bound x(f)
6 f(f)
exp [’ H(t, s) ds - f J,(t; h(f)) m-t -0
fortEIandn=1,2,....
* This work
was supported
by a grant
from
the National
470 0022.247X:8 Copyright All rights
I /080470-03$02.00/O
% 1981 by Academic Press. Inc. of reproduction in any form reserved.
Research
Council
of Canada.
471
AN INTEGRAL INEQUALITY
Prooj
From the upper bound on x(t) in (l), x(f) + if G(t, t,) x(t,) dt, "0
+ (-IWC f,) -0
[
x(f,) + if’ GO,, fz)-Q,)
df,
-0
1
df,
- if 1” H(f, f,) G(f,, f2)x(fZ) df, df,. -0-o Inductively
we obtain
x(f) + $ J,(t;x(t)) m=l G./-(f) + pqf, -0
f,)
[
X(f,) + t J,(f,;x(f,)) m=,
1
df, -J,+,(t;x(f)).
Since f(t) and H(t, s) are nondecreasing in t, it follows, from an easy modification of the Bellman-Gronwall inequality (cf. [I]), that x(f)
exp [‘H(t, S) ds -0
From the lower bound in (1) 0 < h(t)
k J,,,(f; x(f)) df. tTl=l the result now follows.
COROLLARY. In addition fo the hypothesis of fhe theorem, suppose h(f) is bounded on I, and H(f, s) and G(f, s) are bounded on I X I. Then solutions x(f) of (1) safisj.?
x(f)
exp fr H(f, s) ds -0
for O
g J,(f; h(f)) m=l
T.
Proof. In this case there exists a constant M such that J,(f; h(f)) < (Mf)“/n! and so the series converges uniformly on I. By way of comparison, consider the solution x(f) = cash t of the integral equation x(f) = 1 + Jb (f - 5:) x(s) ds. From [ 11, x(f) < exp(f2/2), whereas from the corollary x(f) ,< exp(f*/2 )$+A+
409/82,2
I2
,.;.,
+...I.
412
THOMAS D.ROGERS REFERENCE
I. G. BUTLER AND T. ROGERS,A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations, J. Math. Anal. Appl. 33 (1971), 77-81. 2. T. ROGERS.A functional integral inequality and bounds for nonlinear differential equations, Bull. Acad. Polon. Sci. 22 (1974). 269-272.
OF MATHEMATICAL
ANALYSIS
AND
An Integral
(1981)
Inequality* D. ROGERS
THOMAS
Department of Mathemarics. Edmorlton
82, 470-472
APPLICATIONS
Unicersif~ of Alberla.
76G 2G1,
Canada
Submitted bv C: Lakshmikantham
In this article we obtain improved pointwise bounds for solutions integral inequalities of the form 0<
h(f)
< x(r)
+ 1-l [H(r,
s) -
G(f,
s)]
x(s)
x(f)
ds
of
(1)
-0
in the situation where H(t, s), G(f, s) > 0 and H(f, s) is nondecreasing in f, but no assumption is made on the sign of H(t, s) - G(t, s). Results of a similar nature for the case G(t, s) = 0 are found in [l] and [2]. THEOREM. Suppose f (t) is nonnegafiue and nondecreasing on the interoal Z: 0 < f < T and that H(f, s) and G(f, s) are nonnegafitre on I x I with H(f, s) nondecreasing in t for fixed s. Define. for n = 2. 3,...,
.I,(!; h(t)) = 1’ G(f, s) h(s) ds, -0
J,(f;h(f))=!;-(;‘...
j:“-‘H(f.f,)
x G(f,m,.
...
H(f,-,,f,-,)
h(f,) df, ... df,.
fn)
Then solutions x(t) of (1) also satisy the explicit pointwise bound x(f)
6 f(f)
exp [’ H(t, s) ds - f J,(t; h(f)) m-t -0
fortEIandn=1,2,....
* This work
was supported
by a grant
from
the National
470 0022.247X:8 Copyright All rights
I /080470-03$02.00/O
% 1981 by Academic Press. Inc. of reproduction in any form reserved.
Research
Council
of Canada.
471
AN INTEGRAL INEQUALITY
Prooj
From the upper bound on x(t) in (l), x(f) + if G(t, t,) x(t,) dt, "0
+ (-IWC f,) -0
[
x(f,) + if’ GO,, fz)-Q,)
df,
-0
1
df,
- if 1” H(f, f,) G(f,, f2)x(fZ) df, df,. -0-o Inductively
we obtain
x(f) + $ J,(t;x(t)) m=l G./-(f) + pqf, -0
f,)
[
X(f,) + t J,(f,;x(f,)) m=,
1
df, -J,+,(t;x(f)).
Since f(t) and H(t, s) are nondecreasing in t, it follows, from an easy modification of the Bellman-Gronwall inequality (cf. [I]), that x(f)
exp [‘H(t, S) ds -0
From the lower bound in (1) 0 < h(t)
k J,,,(f; x(f)) df. tTl=l the result now follows.
COROLLARY. In addition fo the hypothesis of fhe theorem, suppose h(f) is bounded on I, and H(f, s) and G(f, s) are bounded on I X I. Then solutions x(f) of (1) safisj.?
x(f)
exp fr H(f, s) ds -0
for O
g J,(f; h(f)) m=l
T.
Proof. In this case there exists a constant M such that J,(f; h(f)) < (Mf)“/n! and so the series converges uniformly on I. By way of comparison, consider the solution x(f) = cash t of the integral equation x(f) = 1 + Jb (f - 5:) x(s) ds. From [ 11, x(f) < exp(f2/2), whereas from the corollary x(f) ,< exp(f*/2 )$+A+
409/82,2
I2
,.;.,
+...I.
412
THOMAS D.ROGERS REFERENCE
I. G. BUTLER AND T. ROGERS,A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations, J. Math. Anal. Appl. 33 (1971), 77-81. 2. T. ROGERS.A functional integral inequality and bounds for nonlinear differential equations, Bull. Acad. Polon. Sci. 22 (1974). 269-272.