Preface

This section presents some theorems regarding uniqueness of solutions to the initial value problem. It is supported by examples.

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Glossary

Existence and Uniqueness

An ordinary differential equation may have no solution, unique solution or infinitely many solutions. For instance, \( x\,y' = y, \quad y(0) =3 \) has no solution. The ODE \( y' = x^2 +y, \quad y(0) =-2 \) has the unique solution \( y = -2-2x-x^2 , \) whereas the ODE \(x\,y' = y-2 , \quad y(0) =2 \) has infinitely many solutions y = 2 + αx, where α is any real number.

A theoretical question arises as to whether a solution to the differential equation does exist. There is a group of theorems known as existence-theorems (most of them also address uniqueness as well) that provides an answer to the above question. Actually, two entirely distinct (constructive) proofs of these existence-theorems are known: the method of successive approximations, usually credited to Charles Émile Picard (1856--1941), and the Cauchy--Lipschitz method, originated from the pionering 1768 work by Leonhard Euler (1707--1783). Both approaches made a substantional impact on developing mathematics and we will demonstrate their applications in numerous cases.

Baron Augustin-Louis Cauchy (1789--1857) started working on improvements of Euler's approach at least as early as 1821 by proving the local existence of solutions to ordinary differential equations. Cauchy later in 1835 improved it by introducing the method of majorants. It was refined by the German mathematician Rudolf Otto Sigismund Lipschitz (1832--1903), which is now called the Cauchy--Lipschitz method. It is striking that the method requires only continuity and boundedness of the derivative (which was later improved by Lipschitz) and the result is local.

Giuseppe Peano.

Theorem (Peano): Suppose that f(x,y) is a continuous function defined in some rectangular region

\[ R = \left\{ (x,y)\,: \,| x_0 - x | \le \delta , \quad |y_0 - y | \le \epsilon \right\} \]
containing the point (x0,y0). Then there exists a number h (possibly smaller than δ)
\[ h = \min \left\{ \delta , \ \frac{\epsilon}{\max |f(x,y)|} \right\} , \]
so that a solution y = ϕ(x) to the initial value problem
\[ y' = f(x,y) , \qquad y(x_0 ) = y_0 \]
is defined for \( x \in (x_0 - h , x_0 + h ) . \)
   ⧫

This theorem was proved in 1886 by the Italian mathematician Giuseppe Peano (1858--1932). Giuseppe Peano was a founder of symbolic logic whose interests centered on the foundations of mathematics and on the development of a formal logical language. In 1890 Peano founded the journal Rivista di Matematica, which published its first issue in January 1891. In 1891, Peano started the Formulario Project. It was to be an "Encyclopedia of Mathematics", containing all known formulae and theorems of mathematical science using a standard notation invented by Peano. In 1888 Peano published the book Geometrical Calculus that begins with a chapter on mathematical logic.

In addition to his teaching at the University of Turin, Peano lectured at the Military Academy in Turin in 1886. The following year he discovered, and published, a method for solving systems of linear differential equations using successive approximations. However Émile Picard had independently discovered this method and had credited the German mathematician Hermann Schwarz (1843--1921) with discovering the method first.

Theorem: Let f(x,y) be continuous for all (x,y) in open rectangle \( R= \left\{ (x,y)\,:\, |x-x_0 | < a, \quad |y- y_0 | < b \,\right\} \) and Lipschitz continuous in y, with constant L independent of x, that is, \( \left\vert f\left( x, y_1 \right) - f\left( x, y_2 \right) \right\vert \le L\left\vert y_1 - y_2 \right\vert . \) Then there exists a unique solution to the initial value problem
\[ y' = f(x,y) , \qquad y(x_0 ) = y_0 \]
on the interval. Moreover, if z(x) is the solution to the same problem with the initial condition \( z(x_0 ) = z_0 , \) then
\[ \left\vert y(x) - z(x) \right\vert \le e^{L(x- x_0 )} \,|y_0 - z_0 | . \qquad ⧫ \]

In 1925, Mikhail Lavrentyev (also Lavrentiev, Lavrentev, Lavrentieff) (1900--1980) gave an example of a real-valued function f(x,y), continuous on a rectangle R, with the property that for every choice of initial point (x0, y0) interior to R, the initial value problem \( y' = f(x,y), \quad y(x_0 ) = y_0 , \) has more than one solution on every interval [x0, x0 + ε] and [x0 - ε, x0] for small ε > 0.
Example: Consider the initial value problem for autonomous differential equation
\[ y' = 4\left( y -3 \right)^{3/4} , \qquad y(0) = 3. \]
The slope function is continuous, but not Lipschitz. It has two solutions y = 3 (equilibrium solution) and y = 3 + x4.

Our next example deals with a separable equation

\[ \frac{{\text d}y}{{\text d}x} = \frac{2y}{x-2} , \qquad y(2) = 0 , \]
having a singular point (2,0) at which the slope function is discontinuous. This initial value problem has infinite many piecewise defined solutions
\[ y = \begin{cases} C \left( x-2\right)^2 , & \ \mbox{ if } \ x < 2, \\ c \left( x-2\right)^2 , & \ \mbox{ if } \ x \ge 2, \end{cases} \]
where C and c are arbitrary constants.

The initial value problem for autonomous differential equation

\[ y' = 1 + y^2 , \qquad y(0) = 0 , \]
has a unique solution y = tanx, which blows up at x = π/2.

We plot corresponding phase portraits.
StreamPlot[{1, 4 (y - 1)^(3/4)}, {x, 0, 1}, {y, 1, 2}]
StreamDensityPlot[{1, 2*y/(x - 2)}, {x, 0, 4}, {y, -2, 2}]
StreamDensityPlot[{1, 1 + y^2}, {x, 0, 4}, {y, -2, 2}, ColorFunction -> "RainbowReverse"]
Phase portrait for y' = 4 (y-3)(3/4)
 
Phase portrait for y' = 2y/(x-2)
 
Phase portrait for y' = 1 + y²
   ■

 

Theorem (Picard): Suppose that f(x,y) and \( \frac{\partial f}{\partial y} \) are continuous functions defined in some rectangular region
\[ R = \left\{ (x,y)\,: \,| x_0 - x | \le \delta , \quad |y_0 - y | \le \epsilon \right\} \]
containing the point \( \left( x_0 , y_0 \right) . \) If these functions are bounded in R:
\[ \left\vert f(x,y) \right\vert \le M \qquad \mbox{and}\qquad \left\vert \frac{\partial f}{\partial y} \right\vert \le K \]
for some positive constants M and K, then the initial value problem
\[ y' = f(x,y) , \qquad y(x_0 ) = y_0 \]
has a unique solution in the interval
\[ \left[ x_0 -h , x_0 +h \right] , \qquad \mbox{where} \qquad h \le \min \left\{ \delta , \frac{\epsilon}{M} \right\} . \qquad ⧫ \]

Theorem (Picard--Lindelöf): Suppose that a slope function f(x,y) is defined and continuous within an rectangular domain
\[ R = \left\{ (x,y)\,:\, \left\vert x- x_0 \right\vert \le a, \quad \left\vert y- y_0 \right\vert \le b\right\} , \qquad\mbox{for some positive constants }a, b. \]
We also assume that f(x,y) satisfies the following conditions.
  • The slope function is bounded: \( M = \max_{(x,y) \in R} \, |f(x,y)| . \)
  • The slope function is Lipschitz continuous: \( |f(x,y) - f(x,z) | \le L\,|y-z| . \) for all (x,y) from R.
Then for
\[ h = \min \left\{ a, \ \frac{b}{M}, \ \frac{1}{2L} \right\} \]
the initial value problem
\[ y' = f(x,y), \qquad y(x_0 ) = y_0 \]
has a unique solution y = φ(x) for \( |x-x_0 | < h . \)    ⧫
Emile Picard.
The above theorem is usually referred to as Picard's theorem (or sometimes Picard--Lindelöf theorem or the method of successive approximations) named after Émile Picard (1858--1941) who proved this result based on an iteration procedure. Ernst Leonard Lindelöf (1870--1946) was a Finnish mathematician and Charles Émile Picard was a French mathematician. Rudolf Otto Sigismund Lipschitz (1832--1903) was a German mathematician who gave his name to the Lipschitz continuity condition. Picard's mathematical papers, textbooks, and many popular writings exhibit an extraordinary range of interests, as well as an impressive mastery of the mathematics of his time. In addition to his theoretical work, Picard made contributions to applied mathematics, including the theories of telegraphy and elasticity. Picard's popular writings include biographies of many leading French mathematicians, including his father in law, Charles Hermite.

Since the initial value problem \( y' = f(x,y) , \qquad y(x_0 ) = y_0 \) is equivalent to the Volterra integral equation (subject to the condition that the slope function f(x,y) and the solution y(x) are continuous functions)
\[ y(x) = y_0 + \int_{x_0}^x f(s,y(s))\, {\text d}s , \]
this problem has a unique solution that is the limit of the sequence of function \( \left\{ \phi_n (x) \right\}_{n\ge 0} \) that satisfies the recurrence:
\[ \phi_{n+1} (x) = y_0 + \int_{x_0}^x f(s,\phi_n (s))\, {\text d}s , \qquad n=0,1,2,\ldots ; \quad \phi_0 = y_0 . \]
Proof: Starting with the initial constant function \( \phi_0 = y_0 , \) other term are defined recursively:
\[ \phi_{n+1} (x) = y_0 + \int_{x_0}^x f(s,\phi_n (s))\, {\text d}s , \qquad n=0,1,2,\ldots ; \quad n=0,1,\ldots . \]
We represent the limit functions as a telescopic sum:
\[ \phi (x) = \phi_0 + \sum_{k\ge 1} \left( \phi_k (x) - \phi_{k-1} (x) \right) . \]
The general term of the above series can be estimated as
\[ \left\vert \phi_k (x) - \phi_{k-1} (x) \right\vert = \left\vert \int_{x_0}^x \left[ f \left( t, \phi_{k-1} (t) \right) - f \left( t, \phi_{k-2} (t) \right) \right] {\text d}t \right\vert \le \int_{x_0}^x \left\vert f \left( t, \phi_{k-1} (t) \right) - f \left( t, \phi_{k-2} (t) \right) \right\vert {\text d}t \]
By the Lipschitz condition,
\[ \left\vert f \left( t, \phi_{k-1} (t) \right) - f \left( t, \phi_{k-2} (t) \right) \right\vert \le L \left\vert \phi_{k-1} (t) - \phi_{k-2} (t) \right\vert . \]
We are going to prove by induction that
\[ \left\vert \phi_{k} (t) - \phi_{k-1} (t) \right\vert \le \frac{M\,L^{k-1}}{k!} \left\vert x- x_0 \right\vert^k , \qquad k= 1,2,\ldots . \]
For k = 1, it is true because
\[ \left\vert \phi_{1} (t) - \phi_{0} (t) \right\vert = \left\vert y_0 + \int_{x_0}^x y_0 \,{\text d}s - y_0 \right\vert = \left\vert y_0 \int_{x_0}^x {\text d}s \right\vert \le \left\vert y_0 \left( x - x_0 \right) \right\vert \le M \left\vert x- x_0 \right\vert , \]
where M is some positive number.

Let us make the induction hypothesis that this holds for n−1 in place of n. Then we have

\[ \left\vert \phi_n (x) - \phi_{n-1} (x) \right\vert = \left\vert \int_{x_0}^x {\text d}s \left[ f\left( s, \phi_{n-1} (s) \right) - f\left( s, \phi_{n-2} (s) \right) \right] \right\vert \le L \int_{x_0}^x {\text d}s \left\vert \phi_{n-1} (s) - \phi_{n-2} (s) \right\vert \]
According to the induction hypothesis, it follows that:
\[ \left\vert \phi_{n-1} (s) - \phi_{n-2} (s) \right\vert \le \frac{M\,L^{n-2}}{(n-1)!} \left\vert s - x_0 \right\vert^{n-1} . \]
Substituting this inequality into the above integral, we get
\[ L \int_{x_0}^x {\text d}s \left\vert \phi_{n-1} (s) - \phi_{n-2} (s) \right\vert \le L \, \frac{M\,L^{n-2}}{(n-1)!} \int_{x_0}^x {\text d}s \left( s - x_0 \right)^{n-1} = \frac{M\,L^{n-1}}{n!} \left\vert x- x_0 \right\vert^n . \]

Next we show that our sequence converges uniformly to a limit for | x - x0 | \le; h. By the Weierstrass M-Test, we have

\[ \left\vert \phi (x) \right\vert \le \left\vert \phi_0 (x) \right\vert + \sum_{k\ge 1} \left\vert \phi_{k} (x) - \phi_{k-1} (x) \right\vert \le M + \sum_{k\ge 1} \frac{M\,L^{k-1}}{k!} = \frac{M}{L}\, e^{L |x - x_0 |} . \]
Therefore, the sequence of Picard's iteration converges uniformly for all x from some finite interval [x0 - h, x0 + h].

The last step is to prove that the limit function φ(x) satisfies the differential equation \( y' = f(x,y) \) and the initial condition y(0) = y0. This follows immediately from equivalence of the initial value problem and the corresponding Volterra integral equation. Note that the latter statement is valid only for continuous functions, so requirement that the slope function f(x,y) is continuous is essential.   ▣

Corollary: The continuous dependence of the solutions on the initial conditions holds whenever slope function f satisfies a global Lipschitz condition.    ⧫
Corollary: If the solution y(x) of the initial value problem \( y' = f(x,y), \ y(x_0 )= y_0 \) has an a priori bound M, i.e., \( |y(x)| \le M \) whenever y(x) exists, then the solution exists for all \( x \in \mathbb{R} . \)    ⧫
Not all assumptions in Picard's theorem are neseccary and some of them can be replaced with the less restrictive conditions. For example, the assumption of continuity of the slope function f(x, y) can be replaced with a condition of integrability and finite number of finite discontinuities (usually referred to as the Dirichlet conditions). Also, the Lipschitz condition can be replaced by the less restrictive condition
\[ \left\vert f(x,y_1 ) - f(x, y_2 ) \right\vert < K |y_1 - y_2 |\,\ln \frac{1}{|y_1 - y_2 |} . \]
A substantional approvement of Picard's theorem was proposed by several researchers, including the Finnish mathematician by the name of Ernst Leonard Lindelöf (1870--1946).

The ideal method for proving the existence theorem would be one that leads to a solution that converges uniformly throughout the largest possible interval, called the validity interval. Such interval can be obtained, in principle, using the Cauchy--Lipschitz method that originated from the Euler method, discussed in Part III of this tutorial. Picard's method provides only local existence of solutions, but not the validity interval. The following theorem provides the best known interval of convergence for Picard--Lindelöf theorem.

Sergey Lozinsky.

Theorem (Lozinsky): Let f(x,y) be a continuous function in some domain Ω and M(x) be a nonnegative continuous function on some finite interval \( I \ (x_o \le x \le x_1 \) inside Ω. Let \( |f(x,y)| \le M(x) \) for x ∈ I and (x,y) ∈ Ω. Suppose that the closed domain Q, defined by inequalities

\[ x_0 \le x \le x_1 , \qquad \left\vert y - y_0 \right\vert \le \int_{x_0}^x M(u)\,{\text d}u , \]
is a subset of Ω and there exists a nonnegative integrable function k(x), k ∈ I.such that
\[ \left\vert f(x, y_1 ) - f(x, y_2 ) \right\vert \le k(x) \left\vert y_1 - y_2 \right\vert , \qquad x_0 \le x \le x_1 , \quad (x,y_1)\ (x,y_2 ) \in Q . \]
Then the formula
\[ \phi_{n+1} (x) = y_0 + \int_{x_0}^x f \left( s, \phi_n (s) \right) {\text d}s \]
defines the sequence of functions { φn(x) } that converges to a unique solution of the initial value problem \( y' = f(x,y), \ y(x_0 ) = y_0 \) provided that all points (x, φn(x)) are included in Q when x0 ≤ x ≤ x1. Moreover,
\[ \left\vert y(x) - \phi_{n} (x) \right\vert \le \frac{1}{n!} \int_{x_0}^x M(u)\,{\text d}u \left[ \int_{x_0}^x k(u)\,{\text d}u \right]^n . \qquad ⧫ \]

The theorem above was proved by the famous Russian mathematician Sergey Mikhailovich Lozinskii/Lozinsky (1914--1985), student of V.I. Smirnov and S.N. Bernstein. Sergey Lozinsky made many important contributions to the error estimation methods for various types of approximation solutions of ordinary differential equations. During two years (1941-1942) he was in active military duties defending Leningrad (now St. Petersburg).

Example. Consider the initial value problem for the Riccati equation

\[ y' = x^2 + y^2, \quad y(0)=0 , \]
which has a unique solution \( y = \phi (x) \) expressed via Bessel functions
\[ \phi (x) = x \, \dfrac{Y_{-3/4} \left( \frac{x^2}{2} \right) - J_{-3/4} \left( \frac{x^2}{2} \right)}{J_{1/4} \left( \frac{x^2}{2} \right) - Y_{1/4} \left( \frac{x^2}{2} \right)} . \]
This function blows up at \( x \approx 2.003147359 \) --- the first positive root of the transcendent equation \( J_{1/4} \left( \frac{x^2}{2} \right) = Y_{1/4} \left( \frac{x^2}{2} \right) . \)
Plot[{BesselJ[1/4, x^2 /2], BesselY[1/4, x^2 /2]}, {x, 0.5, 8}, PlotStyle -> Thick]
Actually, Picard's theorem guarantees a unique solution within the interval \( \left[ 0, 2^{-1/2} \right] \approx [0,0.707107 ] . \) To find its solution, we use Picard's iteration procedure:
\begin{align*} \phi_0 (x) &= 0 , \\ \phi_1 (x) &= \int_0^x \left( x^2 + 0^2 \right) {\text d}x = \frac{x^3}{3} , \\ \phi_2 (x) &= \int_0^x \left( x^2 + \left( \frac{x^3}{3} \right)^2 \right) {\text d}x = \frac{x^3}{3} + \frac{x^7}{63} , \\ \phi_3 (x) &= \int_0^x \left( x^2 + \left( \phi_2 (x) \right)^2 \right) {\text d}x = \frac{x^3}{3} + \frac{x^7}{63} + \frac{2\,x^{11}}{2079} + \frac{x^{15}}{59535} , \\ \phi_4 (x) &= \int_0^x \left( x^2 + \left( \phi_3 (x) \right)^2 \right) {\text d}x \\ &= \frac{x^3}{3} + \frac{x^7}{63} + \frac{2\,x^{11}}{2079} + \frac{13\,x^{15}}{218,295} + \frac{82\,x^{19}}{37,328,445} + \frac{662\,x^{23}}{10,438,212,015} + \cdots , \end{align*}
and so on.
  1. Hsu, S.-B., Ordinary Differential Equations with Applications, Second edition, World Scientific, 2013.
  2. Lavrentieff, M., Sur une équation différentielle du premier ordre, Mathematische Zeitschrift, 1925, Vol. 23, Issue 1, pp. 197--209.
  3. Murray, F.J. and Miller, K.S. (1954), Existence Theorems for Ordinary Differential Equations, New York University Press, New York, NY.
  4. Petrovski, I.G., Ordinary Differential Equations, Dover, NY, 1973.
  5. Tisdell, C.C., On Picard's iteration method to solve differential equations and a pedagogical space for otherness, International Journal of Mathematical Education in Science and Technology, 2019, Vol. 50, Issue 5, pp. 788--799. https://doi.org/10.1080/0020739X.2018.1507051

 

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