Theory of First Order Differential Equations 53 § 5 Tools from Functional Analysis
Some Integrable Cases § 1 Explicit First Order Equations
We consider the explicit first order differential equation y' = f(x,y) (1)
The right-hand side f(x,y) of the equation is assumed to be defined as a real- valued function on a set D in the xy-plane.
In the context of differential equations, let J represent an interval that can take various forms, including open, closed, half-open, half-line, or the entire real line, with specific restrictions outlined as needed A function y(x): J → ℝ qualifies as a solution to the differential equation (1) within the interval J if it is differentiable throughout J, its graph is a subset of the designated domain D, and it satisfies the equation (1).
The differential equation has a geometric interpretation where an integral curve, y(x), passing through a point (x₀, y₀) indicates the slope at that point as y'(x₀) = f(x₀, y₀) This concept introduces the ideas of line elements and direction fields A numerical triple (x, y, p) represents a point in the plane, with p indicating the slope of a line through (x, y) The angle of inclination of the line, defined by tan(θ) = p, is visualized geometrically The set of all line elements of the form (x, y, f(x, y)), where p = f(x, y), is referred to as a direction field.
The relationship between direction fields and differential equations can be understood geometrically: a solution y(x) aligns with its direction field, meaning the slope of the solution curve matches the slope of the corresponding line element at each point Specifically, if y(x) is a solution within an interval J, the line elements (x, y(x), y'(x)) are included in the broader set of line elements (x, y, f(x, y)) defined in the direction field D By sketching several line elements in the direction field and attempting to draw curves that conform to these elements, one can gain insight into the nature of the differential equation's solutions This approach implies that for every point (x, y) in D, there exists a unique solution curve y(x) that passes through it, leading to a formal understanding of this concept.
The Initial Value Problem involves finding a differentiable function y(x) defined on an interval J, which includes a fixed point (a, b) in the (x,y)-plane This function must satisfy the condition that its derivative y'(x) equals f(x, y(x)) within the interval J.
Equation (3) is called the initial condition Naturally, in (2) it is assumed that graphy = {(x,y(x)) :.T E J} cD (otherwise the right-hand side of (2) would not even be defined).
Differential equations can be derived from families of curves that entirely cover a set D in the plane Specifically, when dealing with a set of differentiable functions whose graphs are disjoint and collectively form the union D, there exists a corresponding differential equation with these curves as solutions To determine the right-hand side of this equation, one must identify the unique function φ that belongs to the family M for each point (xo, Yo) in D, ensuring that φ(xo) = Yo, and then set f(xo, Yo) = φ'(xo).
While this relationship may not offer significant mathematical insights, it provides valuable perspectives on the behavior of differential equations and serves as a useful tool for constructing illustrative examples.
We will now briefly discuss some mathematical shorthand that is frequently used.
In certain cases, especially in examples, the solution obtained from a differential equation may only be valid on a specific subinterval of its defined domain.
The statement "¢J(x) is a solution to the differential equation in the interval J" indicates that ¢J is defined at least within the interval J, and that its restriction ¢J|J qualifies as a solution according to the established definition.
If ¢J : J > lR is a solution to the differential equation (1) and J' is a subinterval of J, then the restriction 'lj! = ¢JIJI is also a solution of (1), but it is not considered a new solution For example, when stating that "the differential equation has exactly one solution existing in the interval J," it implies that there is a solution defined over J, and all other solutions are merely restrictions of this primary solution.
Before giving a detailed investigation of initial value problems, we will study some simple examples.
If the function f(x) is continuous over an interval J, the set D forms a strip J x ℝ The direction field remains independent of y, suggesting that all solutions can be derived by translating a specific solution along the y-axis Analysis supports this conjecture, confirming its validity For a fixed value E in J, the fundamental theorem of calculus indicates that the function φJ(x) := ∫_0^x f(t) dt serves as a solution to the differential equation with the initial condition y(0) = 0.
, , X Direction field in the case where the
" right side depends only onx
The general solution can be expressed as y = y(x;C) = ¢(x) + C, where C represents an arbitrary constant This indicates that the initial value problem has a unique solution, specifically y(x) = ¢(x) + '!].
The solution exists in all ofJ.
If the interval J is not compact, the function f may not be bounded and, consequently, may not be integrable over J However, since [~, x] is a compact subinterval of J and f is continuous on [~, x], the integral defining ¢(x) exists for each x in J, ensuring that the equation ¢' = f is valid throughout J.
Example The equation y' = x 3 +cosx has as solutions
Ifthe initial condition is y(l) = 1, then the corresponding solution is
Finding a solution to a differential equation of type (IV) essentially involves integral calculus, specifically the task of determining an antiderivative of the function f(x) This leads to the popular phrase "integrating a differential equation," which is often understood as the process of discovering its solutions.
Let g(y) be continuous over the interval J The direction field resembles that of IV, but with the roles of x and y reversed, indicating that we can interchange x and y to express the solution curves as x = x(y).
A formal calculation gives dy dy
- = g(y) ¢? - = dx dx g(y) and hence the solution
If9 f= 0, then (5) gives a function x = x(y) whose inverse function y(x) is a solution to the differential equation, as we will show in VII Finding a solution §1 Explicit First Order Equations 13 y x
In scenarios where the right side of the direction field is solely dependent on y, and given the initial condition y(~) = TJ, it is essential to select the integration constant to ensure that x(TJ) equals ~ This leads to the expression x(y) = ~ + iT! g(z).
First Order Systems Equations of Higher Order 105 § 10 The Initial Value Problem for a System of First Order 105
Differential Equations § 5 Tools from Functional Analysis
Functional analysis provides elegant solutions to many questions in differential equations theory In this chapter and those to follow, we will utilize functional-analytic methods to establish theorems regarding the existence, uniqueness, and parameter dependence of solutions We will start by introducing the concept of a Banach space.
A vector space, also known as a linear space, is defined as a set L = {a, b, c, } where addition and scalar multiplication are established, allowing for each pair of elements a and b in L to yield a unique element a + b in L Additionally, for each element a in L and each scalar λ, there exists an element λa in L These operations must adhere to specific laws, ensuring that the set L forms an abelian group under addition, meaning that for any elements a, b, and c in L, the following properties hold: closure, associativity, commutativity, the existence of an additive identity, and the existence of additive inverses.
(a+b)+c a+b a+(b+c) b+ a, and there is a unique zero element, denoted bye, and to each a E L a unique inverse, denoted by -a,such that a + e a a+(-a) e
For a,bEL and arbitrary numbers A,/-L,scalar multiplication satisfies the rules
The space is called a real or a complex vector space, depending on whether the scalars A, /-L come from the field of real or complex numbers.
A nonempty subset ofL that (with the previously described operations) is again a linear space is called a (linear) sllbspace ofL.
II Normed Space Let L be a real or complex linear space A real- valued function Iiall, defined for a E L, is called a norm if it has the properties
Iia + bll ::; Iiall + Ilbll triangle ineq1lality.
The space L is often referred to as being "normed" by the norm II II To prevent confusion with the numeral 0, we denote the zero element of L with a unique symbol B Moving forward, we will use the standard symbol 0 to represent the zero element in any vector space, ensuring clarity in equations such as 0 a = 0, as discussed in Section I.
For future reference we mention two simple consequences of the triangle inequality:
Note that a norm defines a distance jllnction (or metric) p(a, b) = Iia - bll that satisfies the axioms of a metric space: p(a, b) >0 for a =I-b, p(a, a) = 0 positivity, p(a, b) = p(b, a) symmetry, p(a, b) ::; p(a,c)+ p(c, b) triangle ineqllality.
A normed space is fundamentally a metric space, allowing us to apply a standard distance function to extend the definitions of various mathematical concepts from Euclidean space \( \mathbb{R}^n \) to any normed space \( L \) This includes the notions of balls, neighborhoods, interior points, boundary points, as well as open and closed sets, and importantly, the concept of convergence.
III Examples (a) The n-dimensional Euclidean space IR n This space is the set of all n-tuples of real numbers a = (al, , an) = (ai)
Addition and scalar multiplication (A real) are defined by a+b = (ai + bi ), Aa = (Aai).
The space IR n can be normed in many ways, for example, by anyone of the following: lal e Ja 2 1 + + a 2 n Euclidean norm, lal lall + + lanl, lal maxi lail maximum norm.
Inthis text, elements from IR n are denoted by boldface italic type and norms inIR n by the ordinary absolute value symbol.
The n-dimensional complex or unitary space, denoted as C n, is defined similarly to the previous example, with the distinction that both ai and A are complex numbers When defining the Euclidean norm in this context, it is essential to incorporate absolute value bars.
Let \( K \) be a compact set and \( C(K) \) represent the collection of all continuous real-valued functions \( f(x) \) defined on \( K \) In this context, the operations of addition and scalar multiplication are naturally defined for functions \( f \) and \( g \) within \( C(K) \) Specifically, the sum of two functions is given by \( h(x) = f(x) + g(x) \), while scalar multiplication is expressed as \( k(x) = Af(x) \), where \( A \) is a real number and \( x \) belongs to \( K \).
As a norm one can choose, for instance,
Ilfllo =max{If(x)1 : x E K} maximum norm, or, more generally, a weighted maximum norm
Ilflh = sup {If(x)lp(x) : x E K}, wherep(x) is a given, fixed function with 0< 0 :::; p(x) :::;(3< 00
In the study of complex differential equations, it is essential to consider a domain \( G \subset \mathbb{C} \) and the set \( H_0(G) \) comprising holomorphic and bounded functions \( u(z) : G \to \mathbb{C} \) Additionally, if \( p(z) \) is a real-valued function defined in \( G \) that satisfies \( 0 < \alpha \leq p(z) \leq \beta \) for appropriate positive constants \( \alpha \) and \( \beta \), this framework is crucial for the investigation.
Ilull =sup{lu(z)lp(z) : z E G} defines a norm in Ho(G).
The norm properties in the given examples are straightforward to verify, as they are homogeneous, nonnegative, finite, and only vanish for the zero function The triangle inequality is well-established for the Euclidean norm and can be easily confirmed for the other norms In function spaces, the triangle inequality holds under general conditions, where G can be any set, and the functions f, g can be complex-valued, while p is real-valued and nonnegative.
Ilfll =sup{If(x)jp(x) : :£ E G}, then
IfCr) + g(l;)lp(x) :s; If(x)lp(x) + Ig(x)lp(x) :s; Ilfll + Ilgl/ for x E G
Therefore, the triangle inequality Ilf + gil :s; Ilfll + Ilgll holds if the norms of f and g are finite.
IV Convergence and Completeness The notion of the convergence of a sequence of numbers can be extended in a natural way to a normed space
L A sequence Xl, X2, of elements ofL converges "strongly" or "in the norm" to an element X E L if
In this case, we also write x n -> x (n ->(0) or lim r n = X n -,>oo
Convergence for infinite series is defined similarly: fXk =.r ¢=} Ilt.rk-xll -> 0 as n -> 00. k=l k=l
A sequenceXl, X2, is called a Cauchy sequence or a fundamental sequence if it satisfies the Cauchy convergence criterion: For everyc>0, there exists an
No(c) such that Ilx n - xmll < c faT' n, m ;::: No(c); or more briefly, lim Ilxn - X m II = O. n,nl~OO
Every Cauchy sequence of real or complex numbers converges to a limit, which is the fundamental principle of the Cauchy convergence criterion However, this property does not universally apply to all normed spaces; rather, it is a distinct characteristic known as the completeness property, found in specific linear spaces.
A normed lineal' space L is called complete if every Cauchy sequence of elements ofL has a limit in L (in the sense of convergence in the norm). §5 Tools from Functional Analysis 57
V Banach Spaces A Banach space is a complete normed linear space, that is, a set with the properties given in sections I II, and IV.
Examples III.(a) and III.(c) represent real Banach spaces, while III.(b) and III.(d) are classified as complex Banach spaces The completeness of the first two examples is directly derived from the inherent completeness of the spaces lR and C In the third example, we emphasize a crucial observation that underpins its completeness.
Convergence in the maxim'um norm 'is equivalent to uniform convergence.
Indeed if(fn) is a Cauchy sequence, then the statement Ilfn- Imllo < E for m,n ~ no is precisely the Cauchy criterion for uniform convergence:
Ifn(:r) - fm(x)1 < E for m,n ~ no and all x E K.
The completeness of the space follows from the established theorem that states the limit of a uniformly convergent sequence of continuous functions is also continuous Consequently, there exists a function f in C(K) such that the limit of fn(x) as n approaches infinity equals f(x) uniformly.
12 -00 formly in K Ifx and n in (*) are fixed and m > 00, then it follows that
Ifn(:r) - f(ãr)1 S: E for n ~ no and x E!{; i.e., Ilfn - fllo S: E for n ~ noã Thus fn > f in the sense of convergence in the norm, and hence C(K) is complete.
This argument is also valid for the weighted maximum norm Ilflll in III.(c). There we assumed that 0< a S: p(x) S: (3 in K, so that allfllo S: Ilflll S: (3llfllo.
It follows that these two norms are equivalent:
Two norms \( ||\cdot|| \) and \( ||\cdot||' \) on a vector space \( L \) are considered equivalent if there exist constants \( \alpha \) and \( \beta \) such that \( ||x|| \leq \alpha ||x||' \) and \( ||x||' \leq \beta ||x|| \) for all \( x \in L \) This equivalence indicates that convergence in one norm guarantees convergence in the other For instance, the norms \( ||f||_1 \) and \( ||f||_2 \) are equivalent, and it will be demonstrated in section 10.III that all norms in \( \mathbb{R}^n \) are equivalent Further details on the equivalence of norms can be found in section D.II.
In Example (d), the completeness of the space is achieved similarly; however, it is essential to apply the theorem stating that the limit of a uniformly convergent sequence of holomorphic functions is also holomorphic.
In the context of normed spaces E and F, a function T: D → F is classified as an operator or functional, particularly when F is either ℝ or ℂ An operator T is deemed linear if its domain D is a linear subspace of E, satisfying the condition T(αx + βy) = αT(x) + βT(y) for all x, y in D and scalars α, β in ℝ or ℂ Typically, the value of T at a point x is denoted as Tx rather than T(x).
An operatorT :D > F is said to becontinuous at a point Xo E Dif X n ED,
:r n > :];0 implies that TX n > T,TO' The equivalent 0, E-formulation reads: For every E > 0, there e.Tists ° > 0 such that from xED 11:1' - J:oll < 0, itfollo'ws that IITx - Txoll