MODELLING WITH C2 FUNCTIONS ...

MODELLING WITH C 2 FUNCTIONS,CHARACTERIZING ZERO-DERIVATIVE POINTS (OLD AND NEW, IN PREPARATION) ´ MIODRAG MATELJEVIC

1. Modelling with C 2 functions,Characterizing zero-derivative points: A pure mathematician’s viewpoint This is very very rough version for temporarily use and we will correct file as soon as possible. In series of papers [6, 5, 9, 7, 8, 11] Zlobec considered some results which can be related to classical analysis; see also [1, 2, 3, 4]. It seems that the idea behind the proofs in XX is to make them as simple as possible (mainly for ”applied” audience). Note that the results from [XX] do not require Taylor’s formula in their proofs. The author has avoided it using only Lipschitz property of C 1 functions, CauchySchwarz inequality, DC decomposition of C 1 functions into convex and quadratic convex functions and definition of convex functions. It is clear that one does not need notions such as integrals, absolute continuity and bounded variation to prove these results, and that these notions are not so relevant for the results given in [XX] (except when talking about average values of functions). Here we shortly discuss some of Zlobec’s result using some novelty. We also consider these results from point of view of pure mathematics. More precisely we use notions such as integrals, absolute continuity and bounded variation, Morse theorey and Taylor formula to prove (or extend) the corresponding versions of these results. For example, around zero-derivative points, and only around such points, the functions have Morse property. Morse theorem: Suppose that f is real valued C 2 in a nbgh V of x∗ which is non-degenerate stationary point(zero-derivative point). Then f has Morse property around x∗ . I think we can prove also opposite statement. Suppose that g is change of coordinates x = g(z), x∗ = g(z ∗ ) and that function h = f ◦ g has Morse property around z ∗ . It means that h0 (z ∗ ) = 0. Hence f 0 (x∗ ) = 0. For example, we can use h0 = f 0 ◦ g 0 . Let h ∈ Tx . Then there is v ∈ Tz such that h = dg(v). Hence df (h) = df (dg(v)) = dh(v) = 0. Of course we can give more details. In [6], the author studies smooth functions in several variables with a Lipschitz derivative. ”It is shown that these functions have the envelope property: Around zero-derivative points, and only around such points, the functions are envelopes of a quadratic parabolloid. The property is used to reformulate Fermats extreme value theorem and the theorem of Lagrange under slightly more restrictive assumptions but without the derivatives.” Note that in pure mathematics absolute continuity and bounded variation have important roles. Using these notation we can extend some result concerning C 2 functions or functions whose derivatives have Lipschitz Date: September 19, 2017. 1

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property. But if we have ”applied” audience in mind we can try to make proofs as simple as possible (it was done in [6, 9, 7, 8]). Consider a smooth function f : Rn → R on a compact convex set K with interior points. We say that f has the envelope property at an interior point x0 of K if there is a constant L ≥ 0 such that f (x) − f (x0 ) ≤ L(x − x0 )2 for every x in K. It seems that we can use Taylor formula with the remainder. In particular, we have: Let f : R → R be 2 times differentiable on the open 0 interval with f continuous on the closed interval between a and x. Then there is 00 a c ∈ (a, x), such that f (x) − f (a) = f 0 (a)(x − a) + f 2!(c) (x − a)2 . Hence one can prove: Proposition 1.1. If f has extreme value at a and |f 00 | ≤ L on nbgh V(a), then |f (x) − f (a)| ≤ L(x − a)2 , x ∈ V (a). Proposition 1.2. Supp that f is absolutely continuous on [a, b]. The following cond are equivalent (i) f 0 ∈ L∞ [a, b] (ii) f is L-Lipschitz on [a, b]. x, y ∈ [a, b]. Since f is absolutely continuous R y Let R y 0 on [a, b],0 f (x) − f (y) = 0 f (t)dt. Hence by (i), we find |f (x) − f (y)| ≤ |f (t)|dt ≤ |f |∞ |x − y|. x x Using a similar way we can prove: Proposition 1.3. let D be a domain in Rn space. suppose that f is ACL function on D and ∇f ∈ L∞ (D). Then f is L-lipschitz on D, with L = |∇f |∞ . Theorem 1.1 ([6] Decomposition of Smooth Functions with a Lipschitz Derivative). Consider a smooth function f : Rn → R on a compact convex set K in its open domain. If the derivative of f has the Lipschitz property on K then f is both convexifiable and concavifiable on K. Moreover, if L is a Lipschitz constant of the derivative of f on K , then any α ≤ −L is a convexifier and any β ≥ L is a concavifier. Theorem 1.2 ([6] Characterization of Zero-Derivative Points). Consider a smooth function f : Rn → R on a compact convex set K with interior points. Assume that the derivative of f has a Lipschitz derivative with a constant L on K. Then f has the envelope property at an arbitrary interior point x0 of K if, and only if, ∇f (x0 ) = 0. In particular, one can specify Λ = 1/2L. The statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding of Lebesgue integration theory for the full generality. However, it holds also in the sense of Riemann integral provided the (k + 1)-st derivative of f is continuous on the closed interval [a, x]. Integral form of the remainder. Let f (k) be absolutely continuous on the closed interval between a and x. Then x

f (k+1) (t) (x − t)k dt. k! a Due to absolute continuity of f (k) on the closed interval between a and x its derivative f (k+1) exists as an L1 -function, and the result can be proven by a formal calculation using fundamental theorem of calculus and integration by parts. Z

Rk (x) =

CHARACTERIZING ZERO-DERIVATIVE POINTS

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Proposition 1.4. Suppose that (i) f 0 is absolutely continuous function on [a, b] and f 00 ∈ L∞ [a, b] . Then f has the envelope property at an arbitrary interior point x0 of K if f 0 (x0 ) = 0. R x 00 Outline: Since R1 (x) = a f 2!(t) (x − t) dt, we find |R1 (x)| ≤ c(x − a)2 . Note that f has a Lipschitz derivative with a constant L on [a, b], then (i) holds. Theorem 1.3 (Theorem 2 [9] (Separation property of C 2 functions)). Consider a C 2 function f (x) in n variables defined on an open set V of Rn containing a compact convex set K. Denote by ρ the global spectral radius of the Hessian matrix of f on K. Take an arbitrary interior point x0 of K and denote by G = ∇f (x0 ) the gradient of f at x0 . Then for every number Λ such thatΛ ≥ ρ, we have for every x ∈ K, 1 1 f (x0 )+∇f (x0 )·(x−x0 )− Λ(x−x0 )2 ≤ f (x) ≤ f (x0 )+∇f (x0 )·(x−x0 )+ Λ(x−x0 )2 . 2 2 It seems that one can use a version of Taylor formula to extend this result to D2 f ∈ L∞ (V ). If f is of bounded variation in [a, b], it is the difference of two positive, monotonic increasing functions; and the difference of two bounded monotonic increasing functions is a function of bounded variation. (A) If f 0 is bounded variation on [a, b], then f 0 = g − h, where g and h are monotonic increasing functions. It seems that (A) implies Proposition 1.5. If f is absolutely continuous function and f 0 is of bounded variation on [a, b], then f is the difference of two convex function. We need to compare Proposition 1.5 with Theorem 2 (Separation property of C 2 functions)[9]. It seems that we can use also a version of Taylor formula. 1.1. Characterizing fixed points. Sanjo Zlobec, Characterizing fixed points,Croatian Operational Research Review 353 CRORR 8(2017), 353-358 see [9] Theorem 1 (Quadratic envelope characterization of zero-derivative points [7]). Consider a continuously differentiable function of the single variable with Lipschitz derivative on an interval I = [a, b]. If a < x0 < b, then f 0 (x0 ) = 0 if, and only if there is a constant L ≥ 0 such that |f (x) − f (x0 )| = L(x − x0 )2 , for every x in I. Sanjo Zlobec in [10] wrote ”Note that this theorem talks about zero-derivative points without using differentiation. It was proved for functions in n variables in [7]. One can find it depicted in this authors data Formula on Researchgate. Its simplified proof for n = 2 and for C 2 functions is given in the textbook [6]. Various discussions regarding this result can be found in the Q&A section on Researchgate under the question Is there a book in English . . . We have not yet seen an affirmative answer to this question. The interested reader can find more on fixed points in, e.g., [1, 2, 5] and Journal on Fixed Point Theory and Applications. Depictions of fixed points in one, two and three dimensions using string, disc and a cup of coffee, respectively, can be found in the literature typically related to the Brouwer fixed point theorem. In this section we characterize fixed points using a particular integral. Since the results appear to be new, and possibly non-intuitive, we illustrate them by elementary examples”.

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We can check whether the following is true. Set z = (x, y), z0 = (x0 , y0 ), t = (t1 , t2 ), dt = dt1 dt2 , g(t) = f (t) − t, Q(z) = Q(x, y) = [x0 , x] × [y0 , y] and Q∗ (z) = Q(x∗ , y ∗ ), where x∗ = x0 + |x − x0 | and y ∗ = y0 + |y − y0 |. Theorem 1.4. (Primal characterization of fixed points). Consider a continuous Lipschitz function f ofR two variables x, y on a convex planar set D. For z0 and z in D, denote W (z) = Q (x, y)(f (t) − t)dt. Then (i) f (z0 ) = z0 if, and only if (1) |W (z)| = L|z − z0 |3 , z in D, for some L > 0. We give a very simple example to illustrate the result. Example 1. (a) For (sin t1 , sin t2 ), and z0 = (0, 0), we R f (t) = t + g(t), R y g(t)R = x compute W1 (z) = Q(x,y) sin t1 dt1 = 0 dt2 ( 0 sin t1 dt1 ) = 2y sin2 (x/2). (b) = (t1 + t2 , t21 , and z0 = (0, 0), we compute W1 (z) = R For f (t) = t + g(t), g(t) 2 (t + t2 )dt1 dt2 = xy /2 + x2 y/2. Q(x,y) 1 Outline. Let f be L1 -Lip. Suppose (i). By (i) g(z0 ) = 0 and R ∗ therefore there is L2 > 0 such that |g(t)| ≤ L3 |t|, t ∈ Q(z). Note that A(z) = Q (z)dt1 dt2 ≤ |z − z0 |2 . Hence R∗ |W (z)| ≤ L3 Q (z)|t|dt1 dt2 ≤ L3 |z − z0 |A(z) ≤ L3 |z − z0 |3 . Suppose (1). There is w in Q(z) such that |W (z)| = |g(w)|A(z) and therefore |g(w)| ≤ L3 |z − z0 |. Hence, |g(w)| tends 0 if z tends z0 and |g(z0 )| = 0. Theorem 1.5. (Dual characterization of fixed point). Consider a continuous Lipschitz function f of the single variable x on I = [a, b] and a point x0 such that a < x0 < b. Recall W(x) and denote R(x) = |W (x)|(x − x0 )−2 on I \ {x0 }. Now f (x0 ) = x0 is a fixed point if, and only if R(x) is bounded on I \ {x0 } by some L > 0. RG communication: Dear prof Joahim, As I understand you take care whether regularity assumption about f is enough to provide the proof. Joahim: ” A ”primal” necessary condition for fixed point proved in the paper says that abs(W (x))