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Theorems on the Equations differentiate them ordinary Equations differentiate them of 1° degree1) Theorem of the contractions of Banach - Caccioppoli : In a complete metric space a contraction admits always an only fixed point. Existence : taken one point x0 arbitrary we will construct one succession that converges to one point x~ : F(x~) = x~ La succession is defined ricorsivamente like xn 1 = F(xn). Taking advantage of the fact that F is contraction it succeeds to establish that d(xn 1 , xn) £ rn d(x1 , x0) turned out that it comes used in order to
demonstrate that the succession is fundamental in how much of given
Cauchy that d(xn , xm) £ and . It is had in
fact : Supponiamo that the limit ofthe argument converges then to x ~ taking advantage of the continuity of the F contraction pu² to become the limit of the function and therefore to demonstrate that to F(x~) = x~ . Oneness : We suppose that 2 fixed points, the distance between of they taking advantage of the contraction definition, cannot that exist be 0, therefore the 2 points in truth are the same point.
2) Condition of Lipschitzianità for f : If f and its partial derivative respect to y is continuous in D f is locally lipschitziana respect to y, uniform in t. It is demonstrated applying the theorem of the valor
medium
3) If the succession of continuous functions fn converges uniform its limit f is a continuous function. It is demonstrated applying the theorem of the exchange of
the limits to the function
4) Lemma di Volterra: If j ? C1(the d) is solution of the problem of
Cauchy j satisfies the equation integral of
Volterra E' clearly that if j is solution of the problem of Cauchy it can integrate the first equation of the problem between t and t and replacing the condition begins them j(t) = x . ? It is obtained deriving in how much j it is continuous and it has derived continuous in virtue of the equality with 2° the member.
5) Theorem of existence and local oneness : It is f : D® "n with open D of "n 1 , continues in locally lipschitziana D and in D, respect to y and uniform in t for every point (t,x) ? D exists aroundthe d of t such thatthe d = [t-d , t d] in which one is defined solution of the problem of Cauchy. Such solution is only in the sense that every other solution coincides with j in the common interval of definition. The demonstration is articulated in the three following steps : to) the correspondent is associated
to the problem of Cauchy equation of Volterra b) a complete metric space is characterized A complete metric space is the space of the continuous functions on a compact one, therefore " (t, x) ? D characterizes a compact G : = {(t,y) ? "n 1 : ||t-t|| < a and ||y-x||< b}, to its inside we characterize the metric space Y of the continuous functions with diagram contained in G , Y : = { j ? C(Id) : ||j(t) -x||< b} it is a complete metric space to pact to adopt the metric one of the uniform convergence d(j, y) = max || j(t) -y(t) || c) is demonstrated that correspondent to the equation of Volterra works them is a contraction and therefore characterizes an only fixed point that is an only solution. To the equation of Volterra it is associated works them to) We want that it is F[y ] ? Y
b) We want that it is contraction
that is Choosing d = min (to, 1/L, b/M) the theorem is demonstrated.
6) Condition for the existence of one skillful massimale solution : It is f : D® "n with open D of "n 1 , continues in locally lipschitziana D and in D, respect to y and uniform in t is y :[t0 , b) ® "n one limited skillful massimale solution b = b To demonstrate.
7) Lemma di Gronwall : They are The ? " an interval and t ? They are moreover u,v :The ® " two continuous in I, not negative functions and c and " . If
8) Theorem of existence and total oneness : It is S : = (t1, t2) x "n Supponiamo that f it is defined in S and that in S f is continuous and locally lipschitziana respect to y and uniform in t. If moreover 2 such positive B exist and constants To that ||f(t, y)|| £ To B||y|| " (t,y) ? S " (t,x) ? S, j(t ;t,x) are defined in t1 , t2 . For the previous theorem enough to demonstrate
that the y(t) it is limited so that the interval to us he is
massimale, to such aim normando the integral equation of Volterra
9) Theorem of existence and total oneness : It is f : (t1, t2) x "n ®" a continuous and totally lipschitziana function in Y with L constant " (t,x) ? S, j(t ;t,x) are defined in t1 , t2 . To demonstrate.
10) Theorem of the dependency of the solutions from the data begins them : It is f :D® "n continues and locally lipschitziana respect to y, uniform in t. Supponiamo moreover that in around of point (t0, x0) are stung in which every solution it is defined in a common interval to tutte moreover if (t ,x) ®(t0 ,x0 ) j(t ;t,x)®j(t ; t0 , x0) Methods of resolution of equations differentiate them of 1° the order
11) resolutive Formula of the equations differentiates them linear : They are equations in the shape
12) resolutive Formula of the equations differentiates them exact : They are equations in the shape
being F(t, y) a function upgrades them
13) Equations to variable separabili : They are equations in the shape
14) Equations of Bernoulli : They are equations in the shape
15) homogenous Equations or of Manfredi : They are equations in the shape
16) Like resolving equations of the type y = F(x, y') : It must be placed y' = p therefore to derive respect to x, replacing therefore p' = dp/dx an equation to variable separabili is obtained resolving which gain x and y according to p
17) Equations of Of Alembert - Lagrange : Draft of equations in shape y = x g(y') f(y'), is resolved placing y' = p therefore to derive respect to x, filler therefore to a linear equation in function of the x resolving itself which obtains one parametric solution of x and y according to c. Equations differentiate them of order n18) Criterion base for the trattazione : Every equation differentiates them of order n can be led back to a linear system of n equations of 1° the order therefore in order to demonstrate the existence and the oneness of the solution, can be made resorted to how much already demonstrated per le equations differentiates them of 1° the valid order and also for the linear systems to pact to change the simbolismo.
19) necessary and sufficient Condition because n solutions of the equation are linearly independent : The determining one of the wronskiana matrix must be 0
20) Theorem of Liouville : The wronskiano of an equation differentiates them of order
n satisfies la To demonstrate
21) Theorem of Lagrange: They are y1 ..., yn n independent solutions of the homogenous one, a solution of not - homogenous it is supplied from the formula
22) Theorem of superimposition: If b(t) = b1(t) b2(t) and we know a particular integral of the equation with famous term b1(t) and one of the equation with famous term b2(t) the sum of the 2 integrals sarè an integral of the equation with famous term b(t) Systems of equations differentiate them of 1° the order23) Criterion base for the trattazione : The criteria of existence and local and total oneness are same ad.eccezione.della the notation.
24) necessary and sufficient Condition because n solutions of the equation are linearly independent : The determining one of the wronskiana matrix does not have
to be null
25) Theorem of Liouville : The wronskiano of a system of solutions of the equation
26) Corollario of the theorem of Liouville : The wronskiano of an equation differentiates them of order
n satisfies la
27) Method of variation of the arbitrary constants (Theorem of Lagrange): In order to obtain the general solution of the not
homogenous system |