# Qualitative theory of differential equations in Banach spaces

A branch of functional analysis in which one studies the behaviour on the real axis $J$ or on the positive (or negative) semi-axis $J ^ {+}$( or $J ^ {-}$) of the solution of the evolution equation in a Banach space. Consider the equations

$I$. $\dot{u} = A ( t) u$;

$II$. $\dot{u} = A ( t) u + f ( t)$;

$III$. $\dot{u} = A ( t) u + f ( t , u )$, where $u ( t)$ is the required function and $f ( t)$ the given function with values in a complex Banach space $E$; $A ( t)$ is a linear operator and $f ( t , u )$ is a non-linear operator on $E$. By the derivative $\dot{u}$ one means the limit in the norm of $E$ of $\Delta u / \Delta t$ as $\Delta t \rightarrow 0$.

Uniform stability holds for the equation $I$ if there exists a constant $M$ such that for any solution $\| u ( t) \| \leq M \| u ( s) \|$, $t \geq s$, and exponential stability if for any solution $\| u ( t) \| \leq M \mathop{\rm exp} ( - \alpha ( t - s ) ) \| u ( s) \|$ for some $M$ and $\alpha > 0$.

For equation $I$ with a constant bounded operator $A ( t) \equiv A$, the solution of the Cauchy problem ( $u ( s)$ given) has the form $u ( t) = \mathop{\rm exp} ( ( t - s ) A ) u ( s)$. The estimate $\| \mathop{\rm exp} ( A t ) \| \leq M \mathop{\rm exp} ( \sigma t )$ holds, where $\sigma$ is a number greater than the real parts of all points of the spectrum of $A$. Thus, for exponential stability it is necessary and sufficient that the spectrum of $A$ lies in the interior of the left half-plane. In a Hilbert space this holds if and only if there exists a positive-definite form $( W x , y )$ for which $( d / d t) ( W u , u ) / 2 = \mathop{\rm Re} ( W u , A u ) \leq - \alpha ( u , u )$ for every solution of the equation (Lyapunov's theorem). If the spectrum of $A$ is distributed on both sides of the imaginary axis and does not intersect it, then $E$ can be decomposed into a direct sum of subspaces $E _ {+}$ and $E _ {-}$ which are invariant with respect to $A$, and all the solutions are exponentially increasing (decreasing) in $E _ {+}$( $E _ {-}$) as $t \rightarrow \infty$. In this case, exponential dichotomy holds for the equation.

If $A$ is closed, unbounded and has a dense domain in $E$, then the Cauchy problem with $u ( 0) = u _ {0}$ is not, in general, well-posed. The existence and properties of the solutions are not determined merely by the distribution of the spectrum of $A$; the behaviour of its resolvent $( A - \lambda I ) ^ {-} 1$ must also be specified. Commonly used conditions ensuring the uniform well-posedness of the Cauchy problem on $J ^ {+}$ are provided by the inequalities

$$\| ( A - \lambda I ) ^ {-} m \| \leq \ \frac{M}{( \lambda - \sigma ) ^ {m} } ,\ \ \lambda > \sigma ,\ \ m = 1 , 2 \dots$$

a sufficient condition for their fulfillment being the Hille–Yosida condition

$$\| ( A - \lambda I ) ^ {-} 1 \| \ \leq \frac{1}{\lambda - \sigma } ,\ \ \lambda > \sigma ,$$

or the inequality

$$\| ( A - \lambda I ) ^ {-} 1 \| \ \leq \frac{M}{| \lambda - \sigma | } ,\ \ \mathop{\rm Re} \lambda > \sigma .$$

When these conditions hold, the solution of the Cauchy problem has the form $u ( t) = T ( t - s ) u ( s)$, where $T ( t)$ is a strongly-continuous semi-group of operators for $t \geq 0$ and $\| T ( t) \| \leq M \mathop{\rm exp} ( \sigma t )$. For uniform (or exponential) stability it suffices that $\sigma = 0$( respectively, $\sigma < 0$). If $A$ generates a strongly-continuous semi-group on a Hilbert space, then the sufficiency part of Lyapunov's theorem holds for it, while if it generates a group, then the necessity part holds as well. In a Hilbert space exponential stability is equivalent to $L _ {2}$- stability, the property that for all solutions $\| T ( t) u _ {0} \| \in L _ {2} ( 0 , \infty )$.

Important for applications is the property of almost-periodicity of the solution $u ( t)$ or weak almost-periodicity (that is, almost periodicity of the scalar functions $\langle u ( t) , \phi \rangle$ for all $\phi \in E ^ {*}$). If all values of an almost-periodic solution lie in a compact set, then the solution is compact. Compactness and weak almost-periodicity imply almost-periodicity. For the equation $\dot{u} = Au$, the question of almost-periodicity of the solution is related to the structure of the intersection of the spectrum of $A$ with the imaginary axis. If $A$ is the generating operator of a bounded strongly-continuous semi-group and the above intersection is countable, then for the almost-periodicity of a bounded solution defined on the whole axis it is necessary and sufficient that the limit $\lim\limits _ {\epsilon \rightarrow 0 } \epsilon ( A - \lambda + \epsilon ) ^ {-} 1 u ( 0)$ exists at each limit point $\lambda$ of the spectrum on the imaginary axis. Furthermore, each uniformly-continuous solution is weakly almost-periodic; it is almost-periodic if it is weakly compact, or if $E$ contains no subspaces isomorphic to the space $c _ {0}$ of sequences converging to zero, endowed with the max norm. If $A$ is the generating operator of a strongly-continuous semi-group $T ( t)$ with the property that the functions $\| T ^ {*} ( t) \phi \|$ are bounded on $J ^ {+}$ for a set of functionals $\phi$ which are dense in $E ^ {*}$, then compactness of a solution implies almost-periodicity.

For equation $I$ with a bounded operator $A ( t)$ that is continuous in $t$, the solutions of the Cauchy problem are defined on the whole axis and, by means of the evolution operator $U ( t , s )$, can be written in the form $u( t) = U ( t , s ) u ( s)$. The property of uniform stability is equivalent to the requirement that $\| U ( t , s ) \| \leq M$; the property of exponential stability is equivalent to $\| U ( t , s ) \| \leq M \mathop{\rm exp} ( - \alpha ( t - s ) )$. If $A ( t)$ is bounded in integral norm, that is, $\sup _ {t} \int _ {t} ^ {t+} 1 \| A ( \tau ) \| d \tau < \infty$, then the general index

$$\kappa = \lim\limits _ {t , s \rightarrow \infty } \ \frac{ \mathop{\rm ln} \| U ( t + s , s ) \| }{t}$$

is finite. If $\kappa < 0$, one has exponential stability. For equations with a constant or periodic operator, the formula $\kappa = \overline{\lim\limits}\; _ {t \rightarrow \infty } ( \mathop{\rm ln} \| u ( t , 0 ) \| ) / t$ holds. It does not hold in the general case. The general index is not altered if $A ( t)$ is perturbed by adding a term $B ( t)$ for which $B ( t) \rightarrow 0$ as $t \rightarrow \infty$, or for which the integral $\int \| B ( t) \| d t$ converges at infinity. The size of the general index depends on the behaviour of $A ( t)$ at infinity. If the limit $A _ \infty = \lim\limits _ {t \rightarrow \infty } A ( t)$ exists and the spectrum of $A _ \infty$ lies in the interior of the left half-plane, then $\kappa < 0$. If the operators $A ( t)$, $\alpha \leq t < \infty$, form a compact set in the space of bounded operators, if the spectra of all limit operators belong to a half-plane $\mathop{\rm Re} \lambda \leq - \nu < 0$ and if the operator function $A ( t)$ has small oscillation, for example, it has the form $B ( \epsilon t )$ for sufficiently small $\epsilon$, or if, for sufficiently large $t$, it satisfies the Lipschitz condition $\| A ( t) - A ( \tau ) \| \leq \epsilon | t - \tau |$ with $\epsilon$ sufficiently small, then $\kappa < 0$. In a Hilbert space the condition $\kappa < 0$ is equivalent to the existence of a Hermitian form $( W ( t) x , y )$ such that $0 < \alpha _ {1} ( x , x ) \leq ( W ( t ) x , x ) \leq \alpha _ {2} ( x , x )$ and $( d / d t ) ( W ( t ) u , u ) \leq - \alpha ( u , u )$ for every solution $u ( t)$.

If $A ( t)$ is periodic with period $\omega$, i.e. if $A ( t + \omega ) = A ( t)$, then $U ( t + \omega , 0 ) = U ( t , 0 ) U ( \omega , 0 )$. The operator $U ( \omega ) = U ( \omega , 0 )$ is called the monodromy operator of equation $I$. Its spectral radius $r _ {U ( \omega ) }$ is related to the general index by the formula $\kappa = ( 1 / \omega ) \mathop{\rm ln} r _ {U ( \omega ) }$. The equation has a periodic solution if and only if 1 is an eigen value of $U ( \omega )$. If the operator $U ( \omega )$ has a logarithm, then the Floquet representation $U ( t , 0 ) = \Omega ( t) \mathop{\rm exp} ( t \Gamma )$ holds, where $\Omega ( t)$ is periodic with period $\omega$ and $\Gamma = ( 1 / \omega ) \mathop{\rm ln} U ( \omega )$. In particular, the Floquet representation holds if the spectrum of $U ( \omega )$ does not surround the origin; and for this it suffices that $( 1 / \omega ) \int _ {0} ^ \omega \| A ( t) \| d t$ be less than a certain constant, which depends on the geometry of the sphere in $E$ and which is not less than $\mathop{\rm ln} 4$. For Hilbert space this constant is $\pi$. The Floquet representation reduces the question of the behaviour of the solution of the equation to the same question for the equation $\dot{v} = \Gamma v$ with a constant operator $\Gamma$.

Exponential dichotomy holds for equation $I$ if for some $s$ the space decomposes into a direct sum of subspaces $E _ {+} ( s)$ and $E _ {-} ( s)$ such that

$$\| U ( t , s ) x \| \leq N _ {1} e ^ {- \nu _ {1} ( t - s ) } \| x \| ,\ \nu _ {1} > 0 ,$$

when $x \in E _ {-} ( s)$, $t \geq s$, and

$$\| U ( t , s ) x \| \leq N _ {2} e ^ {- \nu _ {2} ( t - s ) } \| x \| ,\ \nu _ {2} > 0 ,$$

when $x \in E _ {+} ( s)$, $t \leq s$. It is supposed here that the subsequences $U ( t , s ) E _ {+} ( s)$ and $U ( t , s ) E _ {-} ( s)$ are, in a certain sense, not close to each other. If the general index is finite, this latter requirement holds automatically. For an equation with a periodic $A ( t)$, a necessary and sufficient condition for exponential dichotomy to hold is that the spectrum of the monodromy operator be distributed outside and inside the unit disc without intersecting the unit circle.

For equation $I$ with an unbounded operator $A ( t)$ whose domain does not depend on $t$ and which satisfies the well-posedness conditions for the Cauchy problem (see above) for every $t$, the existence of an evolution operator $U ( t , s )$, defined and strongly continuous in $t$ and $s$ for $t \geq s$, has been proved, under supplementary smoothness conditions. This enables one to carry over to this case a number of the ideas and results described above. However, some difficulties are encountered. E.g., the Floquet representation can be obtained only in a Hilbert space when $A ( t) = A + B ( t)$, where $A$ is a negative-definite self-adjoint operator and $B ( t)$ is a bounded periodic operator satisfying certain extra conditions.

Suppose that $A ( t)$ is periodic and that the Cauchy problem for equation $I$ is uniformly well-posed. If the intersection of the spectrum of the monodromy operator $U ( \omega )$ with the unit circle is countable, then each bounded uniformly-continuous solution on $J$ is weakly almost-periodic. It is almost-periodic in the case of weak compactness or if $E$ does not contain $c _ {0}$. A reflexive space $E$ admits a direct sum decomposition $E _ {1} + E _ {2}$ such that $E _ {1}$ and $E _ {2}$ are invariant with respect to $U ( \omega )$ and all the solutions starting in $E _ {1}$ are almost-periodic, while those starting in $E _ {2}$ are in a certain sense decreasing: For $u _ {2} ( 0) \in E _ {2}$, $\phi \in E ^ {*}$,

$$\lim\limits _ {n \rightarrow \infty } \ \frac{1}{n} \sum _ { k= } 0 ^ { n } | \langle u _ {2} ( k \omega ) , \phi \rangle | ^ {2} = 0 .$$

For the solution of the non-homogeneous equation $II$, the following formula holds:

$$u ( t) = U ( t , s ) u ( s) + \int\limits _ { s } ^ { t } U ( t , \tau ) f ( \tau ) d \tau .$$

For an equation with a bounded operator, this equation is equivalent to the differential equation. In the case of an unbounded operator, this is generally not so, but then this equation is taken as the definition of the (generalized) solution. The fundamental problem for equation $II$ is to investigate the properties of the solutions under prescribed properties of the right-hand side. These properties are usually described in terms of the function $f$ belonging to some Banach space of functions on $J$ or $J ^ {+}$ with values in $E$. If corresponding to each bounded continuous function $f \in C ( E)$ there is at least one bounded solution, then the operator $L = d / d t - A ( t)$ is called weakly regular. If corresponding to each $f \in C ( E)$ there is a unique solution $u \in C ( E)$, then $L$ is called regular. For bounded constant $A$, weak regularity implies regularity. This assertion is no longer true for unbounded $A$ or for bounded periodic $A ( t)$, even in a Hilbert space. If the general index of the equation $\dot{u} = A ( t) u$ is finite, then exponential dichotomy for this equation is equivalent to regularity of $L$ on $J$. For exponential dichotomy to hold on $J ^ {+}$ it is necessary and sufficient that $L$ be weakly regular on $J ^ {+}$ and that the set of those initial values $u ( 0)$ to which the bounded solutions of equation $I$ correspond be a complemented subspace of $E$. If for all solutions of $II$ the inequality

$$\| u ( t) \| \leq k \sup _ { t } \ \left \{ \int\limits _ { 0 } ^ { 1 } \| f ( t + s ) \| ^ {2} d s \right \} ^ {1/2}$$

holds, and if for the solution of the formal adjoint equation $- d v / d t = A ^ {*} ( t) v + g ( t)$ the inequality $\| v ( t) \| \leq k \sup \| g ( t) \|$ holds, then the operators $L$ and $L ^ {*} = - d / d t - A ^ {*} ( t)$ are regular. It is not known (1990) whether regularity is preserved on replacing the right-hand side of the first inequality by $k \sup _ {t} \| f ( t) \|$. For the regularity of $L$ and $L ^ {*}$ both, a priori estimates are necessary.

If $A ( t)$ is periodic, then a necessary and sufficient condition for the existence of periodic solutions for each periodic $f$ is that the mapping $I - U ( \omega )$ be surjective, while for such a solution to be unique, it is necessary and sufficient that the operator $I - U ( \omega )$ be invertible.

The verification of regularity under known conditions can be reduced to the verification of the regularity of operators with constant coefficients. In the case when $A ( t)$ is strongly oscillating (for example $A ( t) = B ( \omega t)$ with $\omega$ large), under the assumption that the mean

$$\overline{A}\; = \ \lim\limits _ {T \rightarrow \infty } \ \frac{1}{2T} \int\limits _ {- T + \alpha } ^ { {T } + \alpha } A ( t) d t$$

exists uniformly with respect to $\alpha \in J$, the operator $A ( t)$ is regular if and only if the operator $d / d t - \overline{A}\;$ is regular.

For almost-periodic solutions the specifics of an infinite-dimensional space are already encountered in a generalization of the well-known Bohl–Bohr theorem on the almost-periodicity of a bounded integral of an almost-periodic function, that is, on the almost-periodicity of a solution of the simplest differential equation $\dot{u} = f ( t)$. If the indefinite integral of an almost-periodic function with values in $E$ is bounded in $E$ and $E$ does not contain $c _ {0}$, then the function is almost-periodic. The Bohl–Bohr theorem fails in $c _ {0}$. If $A ( t)$ is periodic, $f ( t)$ is almost-periodic and the intersection of the spectrum of the monodromy operator $U ( \omega )$ with the unit circle is countable, then the same conclusions on the almost-periodicity of solutions stated earlier for the homogeneous equation hold for this case. If $A ( t)$ is almost-periodic (as a function with values in the space of bounded operators on $E$), then in order that for each almost-periodic function $f$ there exists a unique almost-periodic solution, it is necessary and sufficient that $L$ be regular. If equation $II$ has a (weakly) compact solution for $t > 0$, and the non-trivial (weakly) compact solutions of the homogeneous equation $\dot{u} = A ( t) u$ have the property that $\lim\limits _ {t \rightarrow \infty } \| u ( t) \| > 0$, then equation $II$ has a (weakly) almost-periodic solution.

In the qualitative study of a non-linear equation $\dot{z} = F ( t , z )$, it is usually supposed in advance that conditions hold which ensure that solutions exist on $J$ or on $J ^ {+}$; this imposes an essential constraint on the form of non-linearity of $F ( t , z )$. A solution $z _ {0} ( t)$ on $J ^ {+}$ is said to be uniformly stable if there exists for each $\epsilon > 0$ a $\delta$ such that for every other solution $z ( t)$ the inequality $\| z ( s) - z _ {0} ( s) \| \leq \delta$, $s \geq 0$, implies that $z ( t)$ is defined for $t \geq s$ and that $\| z ( t) - z _ {0} ( t) \| \leq \epsilon$. A solution is called asymptotically stable if it is uniformly stable and if for some $\delta$ the inequality $\| z ( s) - z _ {0} ( s) \| \leq \delta$ implies that $\lim\limits _ {t \rightarrow \infty } \| z ( t) - z _ {0} ( t) \| = 0$. If one makes the substitution $u = z - z _ {0}$ in the equation, then after linearization (if this is possible) it takes the form $III$, where $A ( t) = F _ {z} ^ { \prime } ( t , z _ {0} ( t) )$, and the non-linearity has the property that $f ( t , 0 ) \equiv 0$. Thus the problem of uniform (asymptotic) stability of the solution $z _ {0} ( t)$ reduces to that of the uniform (or asymptotic) stability of the zero solution of $III$. In this connection, the equation $\widetilde{u} dot = A ( t) \widetilde{u}$ is called the linearization equation (or variational equation) of the original equation with respect to the solution $z _ {0} ( t)$.

If the non-linear part $f ( t , u )$ is sufficiently small, then the properties of the solutions are determined by those of the linearized equation. If the general index of the linearized equation is negative and the inequality $\| f ( t , u ) \| \leq q \| u \|$, $t \geq 0$, $\| u \| \leq \rho$, holds, then for sufficiently small $q$ the zero solution of $III$ is asymptotically stable. If $A ( t) \equiv A$ and the spectrum of $A$ does not intersect the imaginary axis and has a point in the right half-plane, then for sufficiently small $q$ the zero solution is unstable. If the inequality

$$\| f ( t , u ) \| \leq c \| u \| ^ {1+} p ,\ \ p > 0 ,\ t \geq 0 ,\ \ \| u \| \leq \rho ,$$

holds, then the requirement that there be no points of the spectrum on the imaginary axis can be deleted.

If exponential dichotomy holds on $J$ for the linearized equation, then for every $\rho > 0$ there exist $M$ and $c$ depending on $A ( t)$ and $\rho$ such that the inequalities

$$\| f ( t , u ) \| \leq M$$

and

$$\| f ( t , u ) - f ( t , v ) \| \leq c \| u - v \| ,\ \ t \in J ,\ \| u \| , \| v \| \leq \rho ,$$

imply the existence, in some neighbourhood of the origin, of two manifolds $M _ {+}$ and $M _ {-}$, intersecting at the single point $u _ {0}$, such that the solution $u _ {0} ( t)$ starting at $u _ {0}$ is bounded on the entire axis and $\| u _ {0} ( t) \| \leq \rho$; solutions starting on $M _ {-}$( $M _ {+}$) approach $u _ {0} ( t)$ exponentially as $t \rightarrow \infty$( $t \rightarrow - \infty$) and move away from it as $t$ goes to infinity in the opposite direction. Furthermore, $M _ {+}$( $M _ {-}$) is homeomorphic to the neighbourhood of the space $E _ {+} ( 0)$( $E _ {-} ( 0)$) involved in the definition of exponential dichotomy. If under the above conditions $A ( t) \equiv A$ and $g ( t , u )$ is almost-periodic in $t$ for each $u$ with $\| u \| \leq \rho$, then the solution $u _ {0} ( t)$ is almost-periodic.

Suppose that the autonomous equation $\dot{z} = F ( z)$ has an $\omega$- periodic solution $z _ {0} ( t)$; then the linearized equation ${\widetilde{u} } dot = F _ {z} ^ { \prime } ( z _ {0} ( t) ) \widetilde{u}$ has a single periodic solution $\dot{z} _ {0} ( t)$ and, consequently, its monodromy operator $U ( \omega )$ has 1 as eigen value. If this eigen value is simple and the remaining spectrum of $U ( \omega )$ lies in the interior of the unit disc and does not surround the origin, then there exist numbers $\mu$, $\tau _ {0}$ and $N$ such that $\| z ( t - z _ {0} ) - z ( t) \| \leq N \mathop{\rm exp} ( - \mu t )$ for all other solutions of the original equation. This property is called asymptotic orbital stability of the periodic solution. Studies have also been made of other stable invariant manifolds for non-linear equations.

Equations of type $III$ with an unbounded operator $A ( t)$ correspond in applications to quasi-linear equations of parabolic or hyperbolic type. A theory of "properly" non-linear equations of a specific type has been developed. Thus, if $B$ is a continuous everywhere-defined dissipative operator, then the Cauchy problem $\dot{u} = B u$, $u ( 0) = u _ {0}$, is uniquely solvable on the semi-axis $J ^ {+}$ for any $u _ {0} \in E$. The definition of a dissipative operator carries over in a natural fashion to the case of a many-valued operator; for such operators one considers a differential inclusion $\dot{u} \in B u$ rather than a differential equation. If on a reflexive space $E$, the operator $B$ is closed and dissipative and if the values of $I - B$ cover the entire space $E$, then the Cauchy problem is uniquely solvable on $J ^ {+}$ for each $u _ {0}$ in the domain of $B$. There are many variants of the latter statement. The solution is given by the formula $u ( t) = S ( t) u _ {0}$, where $S ( t)$ is a semi-group of non-linear operators, $S ( t + \tau ) x = S ( t) ( S ( \tau ) x )$, $t , \tau \geq 0$. For equations with dissipative operators there are also existence theorems for periodic and almost-periodic solutions.

#### References

 [1] Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian) MR0352638 [2] H.H. Schaefer, "Linear differential equations and function spaces" , Acad. Press (1966) [3] K. Yosida, "Functional analysis" , Springer (1968) MR0239384 Zbl 0830.46001 Zbl 0435.46002 Zbl 0365.46001 Zbl 0286.46002 Zbl 0217.16001 Zbl 0152.32102 Zbl 0126.11504 [4] S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) MR0342804 Zbl 0179.20701 [5] V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici (1976) (Translated from Rumanian) MR0390843 Zbl 0328.47035 [6] A.G. Baskakov, "On the spectral analysis of isometric representations of locally compact Abelian groups" Trudy Mat. Fak. Vorozhen. Inst. , 10 (1973) pp. 96–101 (In Russian) MR0466403 [7] V.V. Zhikov, "Some admissibility and dichotomy questions. The averaging principle" Math. USSR Izv. , 10 (1976) pp. 1307–1332 Izv. Akad. Nauk SSSR Ser. Mat. , 40 : 6 (1976) pp. 1380–1408 [8] V.V. Zhikov, B.M. Levitan, "Favard theory" Russian Math. Surveys , 32 : 2 (1977) pp. 129–180 Uspekhi Mat. Nauk , 32 : 2 (1977) pp. 123–171 MR470405 Zbl 0235.34095 Zbl 0224.34036 Zbl 0198.48103 [9] A.I. Miloslavskii, "Floquet theory for parabolic equations" Funct. Anal. Appl. , 10 : 2 (1976) pp. 151–153 Funktsional. Anal. i Prilozhen. , 10 : 2 (1976) pp. 80–81 MR0473390 [10] V.M. Tyurin, "Functional analysis" , 1 , Ul'yanovsk (1973) (In Russian) [11] B.M. Levitan, V.V. Zhikov, "Almost-periodic functions and differential equations" , Cambridge Univ. Press (1982) (Translated from Russian) MR0690064 Zbl 0499.43005