Ordinary differential equations in locally convex spaces.

*(English. Russian original)*Zbl 0834.34076
Russ. Math. Surv. 49, No. 3, 97-175 (1994); translation from Usp. Mat. Nauk 49, No. 3(297), 93-168 (1994).

This paper, mostly expository, is on the ordinary differential equation
\[
x'(t)= f(t, x(t))\tag{1}
\]
in a locally convex space \(E\). Extending the theory of (1) from finite-dimensional spaces to Banach spaces already presents some surprises, among them the failure of Peano’s theorem; there exist equations \(x'(t)= f(x(t))\) with \(f\) continuous that have no solution for any initial condition. However results for equations with \(f(t, x)\) Lipschitz continuous in \(x\) survive, for instance the construction of local solutions by Picard iteration; in particular, the solution of the linear equation
\[
x'(t)= Ax(t),\quad x(0)= x_0\tag{2}
\]
with \(A\) continuous is given by the exponential series
\[
\sum^\infty_{n= 0} {t^n\over n!} A^n x_0.\tag{3}
\]
If the space \(E\) is not normable, even Picard approximation collapses: among many seemingly pathological examples, there exist linear equations (2) with \(A\) continuous, where the series (3) is divergent for every \(t\neq 0\), \(x_0\neq 0\); yet, (2) is uniquely solvable for all \(t\). In other examples, where (3) is divergent, solutions exist but are not unique or fail to exist at all.

The main motivation for the study of (1) is the modelling of partial differential equations, where \(f(t, x)\) is a differential operator in a set of space variables. Here, one can go two ways. The first is to stick to the Banach space setting, in which case \(f(t, x)\) will be discontinuous. If \(f(t, x)= f(x)\) is fully nonlinear but time independent one can apply the theory of nonlinear semigroups, with extensions to the time dependent case; other theories exist for particular cases such as semilinear equations, where \(f(t, x)= Ax+ g(t, x)\), \(A\) the generator of a linear semigroup and the nonlinear part \(g(t, x)\) “less discontinuous” than \(A\) in various senses.

The second way is to model the partial differential equation in locally convex spaces. Differential operators are continuous (even smooth) in a variety of locally convex function spaces, thus we may use (1) with smooth \(f(t, x)\). This approach is not necessarily easier or sharper than the first, but is in some senses preferable; for instance it is more faithful to Hadamard’s original formulation of well posed Cauchy problems, where convergence of sequences of initial data and of solutions means uniform convergence of functions and of a number of their derivatives (depending of the sequence); this can be modelled in a FrĂ©chet space but not in a Banach space. This approach to (1) also allows some amazing sleights of hand such as the reduction of nonlinear equations to linear (§4 in the paper) although these feats attest more to the complexity of locally convex spaces than to anything else.

This paper deals mainly with the second way, although there is some material on the first. It studies existence, uniqueness, continuation of solutions and their continuous dependence on the initial data. It contains some proofs, and numerous examples and counterexamples. Applications are mentioned but not discussed in detail. The paper contains extensive references (125 items) and it will probably be useful to specialists, occasional users and beginners alike.

The main motivation for the study of (1) is the modelling of partial differential equations, where \(f(t, x)\) is a differential operator in a set of space variables. Here, one can go two ways. The first is to stick to the Banach space setting, in which case \(f(t, x)\) will be discontinuous. If \(f(t, x)= f(x)\) is fully nonlinear but time independent one can apply the theory of nonlinear semigroups, with extensions to the time dependent case; other theories exist for particular cases such as semilinear equations, where \(f(t, x)= Ax+ g(t, x)\), \(A\) the generator of a linear semigroup and the nonlinear part \(g(t, x)\) “less discontinuous” than \(A\) in various senses.

The second way is to model the partial differential equation in locally convex spaces. Differential operators are continuous (even smooth) in a variety of locally convex function spaces, thus we may use (1) with smooth \(f(t, x)\). This approach is not necessarily easier or sharper than the first, but is in some senses preferable; for instance it is more faithful to Hadamard’s original formulation of well posed Cauchy problems, where convergence of sequences of initial data and of solutions means uniform convergence of functions and of a number of their derivatives (depending of the sequence); this can be modelled in a FrĂ©chet space but not in a Banach space. This approach to (1) also allows some amazing sleights of hand such as the reduction of nonlinear equations to linear (§4 in the paper) although these feats attest more to the complexity of locally convex spaces than to anything else.

This paper deals mainly with the second way, although there is some material on the first. It studies existence, uniqueness, continuation of solutions and their continuous dependence on the initial data. It contains some proofs, and numerous examples and counterexamples. Applications are mentioned but not discussed in detail. The paper contains extensive references (125 items) and it will probably be useful to specialists, occasional users and beginners alike.

Reviewer: H.O.Fattorini (Los Angeles)

##### MSC:

34G20 | Nonlinear differential equations in abstract spaces |