# 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)

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)
How to Cite This Entry:
Differential inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_inequality&oldid=46691
This article was adapted from an original article by A.D. Myshkis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article