Differential inequality
An inequality which interconnects the argument, the unknown function and its derivatives, e.g.
![]() | (1) |
where is an unknown function of the argument
. The principal problem in the theory of differential inequalities is to describe, starting from a known differential inequality and additional (initial or boundary) conditions, all its solutions.
Differential inequalities obtained from differential equations by replacing the equality sign by the inequality sign — which is equivalent to adding some non-specified function of definite sign to one of the sides of the equation — form a large class. A comparison of the solutions of such inequalities with the solutions of the corresponding differential equations is of interest. Thus, the following estimates [1] are valid for any solution of (1):
![]() | (2) |
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where
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on any interval of existence of both solutions. This simple statement is extensively employed in estimating the solutions of differential equations (by passing to the respective differential inequality with a particular solution which is readily found), the domain of extendability of solutions, the difference between two solutions, in deriving conditions for the uniqueness of a solution, etc. A similar theorem [2] is also valid for a differential inequality (Chaplygin's inequality) of the type
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Here, estimates of the type (2) for solutions satisfying identical initial conditions at are only certainly true on some interval determined by the coefficients
. E.g., this is the interval
for
.
For a system of differential inequalities
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it has been shown [3] that if each function is non-decreasing with respect to the arguments
(for all
), the estimate
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resembling (2), is valid. The development of these considerations leads to the theory of differential inequalities in spaces with a cone.
A variant of differential inequalities is the requirement that the total derivative of a given function is of constant sign:
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This requirement is used in stability theory.
A representative of another class is the differential inequality
![]() | (3) |
( is given), which was first studied in the context of the general idea of an approximate description of a real problem by differential equations [4]. Here the description of the integral funnel, i.e. the set of all points of all solutions which satisfy the given initial conditions, in particular, the behaviour of the funnel as
, is of interest. A natural generalization of the differential inequality (3) is a differential equation in contingencies, specified by a field of cones, which generalizes the concept of a field of directions.
The theory of boundary value problems was also studied for differential inequalities. The inequality , where
is the Laplace operator, defines subharmonic functions; the differential inequality
defines subparabolic functions. Studies were also made of differential inequalities of a more general type (in both the above classes) with partial derivatives for differential operators of various types.
References
[1] | M. Petrovitsch, "Sur une manière d'étendre le théorème de la moyence aux équations différentielles du premier ordre" Math. Ann. , 54 : 3 (1901) pp. 417–436 |
[2] | S.A. Chaplygin, "Fundamentals of a new method of approximate integration of differential equations" , Moscow (1919) (In Russian) |
[3] | T. Wazewski, "Systèmes des équations et des inégualités différentielles ordinaires aux deuxièmes membres monotones et leurs applications" Ann. Soc. Polon. Math. , 23 (1950) pp. 112–166 |
[4] | P. Bohl, "Ueber Differentialungleichungen" J. Reine Angew. Math. , 144 (1914) pp. 284–313 |
[5] | A. Haar, "Ueber Eindeutigkeit und analyzität der Lösungen partieller Differentialgleichungen" , Atti congress. internaz. mathematici (Bologna, 1928) , 3 , Zanichelli (1930) pp. 5–10 |
[6] | W. Walter, "Differential- und Intergralgleichungen und ihre Anwendung bei Abschätzungs- und Eindeutigkeitsproblemen" , Springer (1964) |
[7] | J. Szarski, "Differential inequalities" , PWN (1965) |
[8] | V. Lakshmikantham, S. Leela, "Differential and integral inequalities" , 1–2 , Acad. Press (1969) |
Comments
More generally one considers functional inequalities and integral inequalities of the form
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where is a mapping of some space
of functions on an interval into itself. Two useful uniqueness theorems in this setting are as follows. Let
be the space of non-negative continuous functions on
. Let
be continuous and non-negative. Now, if for
,
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then is identically zero. Now let
be such that
and
. Then if
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also (Nagumo's lemma). let
, let
and suppose
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Then
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The last result is known as Gronwall's lemma (Gronwall's inequality). The case is important. Another variant of Gronwall's lemma is as follows. Let
and for some constant
,
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then
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This last result is useful in discussing, e.g., the stability of (constantly acting) perturbations (with
constant) in terms of the stability of
.
References
[a1] | D.W. Jordan, P. Smith, "Nonlinear ordinary differential equations" , Clarendon Press (1977) |
[a2] | E. Hille, "Ordinary differential equations in the complex plane" , Wiley (Interscience) (1976) |
Differential inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_inequality&oldid=30823