Differential inequality

An inequality which interconnects the argument, the unknown function and its derivatives, e.g.

$$ \tag{1 } y ^ \prime ( x) > f ( x , y ( x) ) , $$

where $ y $ is an unknown function of the argument $ x $. 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):

$$ \tag{2 } y ( x) < z ( x) \ \textrm{ if } x _ {1} \leq x < x _ {0} , $$

$$ y ( x) > z ( x) \ \textrm{ if } x _ {0} < x \leq x _ {2} , $$

where

$$ z ^ \prime = f ( x , z) ,\ z ( x _ {0} ) = y ( x _ {0} ) , $$

on any interval $ [ x _ {1} , x _ {2} ] $ 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

$$ y ^ {(} m) + a _ {1} ( x) y ^ {(} m- 1) + \dots + a _ {m} ( x) y > f ( x) . $$

Here, estimates of the type (2) for solutions satisfying identical initial conditions at $ x = x _ {0} $ are only certainly true on some interval determined by the coefficients $ a _ {1} \dots a _ {m} $. E.g., this is the interval $ [ x _ {0} - \pi , x _ {0} + \pi ] $ for $ y ^ {\prime\prime} + y > f $.

For a system of differential inequalities

$$ y _ {i} ^ \prime ( x) > f _ {i} ( x , y _ {1} \dots y _ {n} ),\ \ i = 1 \dots n , $$

it has been shown [3] that if each function $ f _ {i} $ is non-decreasing with respect to the arguments $ y _ {j} $( for all $ j \neq i $), the estimate

$$ y _ {i} ( x) > z _ {i} ( x) \ \textrm{ if } x _ {0} < x \leq x _ {2} ; \ \ i = 1 \dots n , $$

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:

$$ \frac{d}{dx} F ( x , y _ {1} \dots y _ {n} ) \equiv \frac{\partial F }{\partial x } + \frac{\partial F }{\partial y _ {1} } y _ {1} ^ \prime + \dots + \frac{\partial F }{\partial y _ {n} } y _ {n} ^ \prime \leq 0 . $$

This requirement is used in stability theory.

A representative of another class is the differential inequality

$$ \tag{3 } \max _ {i = 1 \dots n } | y _ {i} ^ \prime - f _ {i} ( x , y _ {1} \dots y _ {n} ) | \leq \epsilon $$

( $ \epsilon > 0 $ 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 $ x \rightarrow \infty $, 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 $ \Delta u \geq 0 $, where $ \Delta $ is the Laplace operator, defines subharmonic functions; the differential inequality $ \partial u / \partial t - \Delta u \leq 0 $ 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

$$ f ( t) \leq T ( f ) ( t) , $$

where $ T $ is a mapping of some space $ X $ of functions on an interval into itself. Two useful uniqueness theorems in this setting are as follows. Let $ C ^ {+} [ 0 , a ] $ be the space of non-negative continuous functions on $ [ 0 , a ] $. Let $ K ( t) \in L ( 0 , a ) $ be continuous and non-negative. Now, if for $ 0 \leq t \leq a $,

$$ f ( t) \leq \int\limits _ { 0 } ^ { t } K ( s) f ( s) d s , $$

then $ f $ is identically zero. Now let $ f \in C ^ {+} [ 0 , a ] $ be such that $ f ( 0) = 0 $ and $ \lim\limits _ {h \downarrow 0 } h ^ {-} 1 f ( h) = 0 $. Then if

$$ f ( t) \leq \int\limits _ { 0 } ^ { t } f ( s) \frac{ds}{s} , $$

also $ f ( t) \equiv 0 $( Nagumo's lemma). let $ K \in C ^ {+} [ a , b ] \cap L ( a , b ) $, let $ f , g \in C ^ {+} [ a , b ] $ and suppose

$$ f ( t) \leq g ( t) + \int\limits _ { a } ^ { t } K ( s) f ( s) d s . $$

Then

$$ f ( t) \leq g ( t) + \int\limits _ { a } ^ { t } K ( s) \mathop{\rm exp} \left [ \int\limits _ { s } ^ { t } K ( u) d u \right ] g ( s) d s . $$

The last result is known as Gronwall's lemma (Gronwall's inequality). The case $ K = \textrm{ constant } $ is important. Another variant of Gronwall's lemma is as follows. Let $ f , K \in C ^ {+} [ a , b ] $ and for some constant $ c $,

$$ f ( t) \leq c + \int\limits _ { a } ^ { t } K ( s) f ( s) d s, $$

then

$$ f ( t) \leq c \mathop{\rm exp} \left ( \int\limits _ { a } ^ { t } K ( s) d s \right ) . $$

This last result is useful in discussing, e.g., the stability of (constantly acting) perturbations $ \dot{x} = A x + B ( t) x $( with $ A $ constant) in terms of the stability of $ \dot{x} = A x $.

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)