DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT

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1 Electronic Journal of Differential Equations, Vol. 217 (217, No. 227, pp ISSN: URL: or DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT BRIAN STREET Abstract. The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations ehibit uniqueness but not eistence, while in others they ehibit eistence but not uniqueness. An eample of such a differential equation is, for a polynomial P and continuous functions f(t, : [, 1] [, 1] R, P (f(t, P (f(t, f(t, =, >. t These differential equations are related to inverse problems. 1. Introduction The purpose of this paper is to study a family of ill-posed differential equations. In some instances, these equations ehibit eistence, but not uniqueness. In other instances, they ehibit uniqueness, but not eistence. The questions studied here can be seen as a family of forward and inverse problems, which in special cases become well-known eamples from the literature. This is discussed more below and detailed in Section 3. In this introduction, we informally state the main results, and present their relationship to inverse problems. However, before we enter into the results in full generality, to help the reader understand our somewhat technical results, we give some very simple special cases, where some of the basic ideas already appear in a simple form: Eample 1.1 (Eistence without uniqueness. Fi ɛ 1, ɛ >. We consider the differential equation, defined for functions f(t, C([, ɛ 1 ] [, ɛ ] by f(t, f(t, f(t, =, >. (1.1 t We claim that (1.1 has eistence: i.e., given f ( C([, ɛ ], there eists a solution f(t, to (1.1 with f(, = f (. Indeed, given a(t C([, ɛ 1 ] with a( = f ( set { e t/ f ( + 1 t f(t, = e(s t/ a(s ds, >, (1.2 a(t, =. 21 Mathematics Subject Classification. 34K29, 34K9. Key words and phrases. Ill-posed; differential equation with difference quotient; eistence without uniqueness; uniqueness without eistence; inverse problem. c 217 Teas State University. Submitted January 7, 217. Published September 21,

2 2 B. STREET EJDE-217/227 It is immediate to verify that f(t, C([, ɛ 1 ] [, ɛ ] and satisfies (1.1. Furthermore, this is the unique solution, f(t,, to (1.1 with f(, = f ( and f(t, = a(t. 1 Thus, to uniquely determine the solution to (1.1 one needs to give both f(, and f(t,. We call this eistence without uniqueness, since there are many solutions corresponding to any initial condition f(, -one for each choice of a(t. Eample 1.2 (Uniqueness without eistence. Fi ɛ 1, ɛ >. We consider the differential equation, defined for functions f(t, C([, ɛ 1 ] [, ɛ ] by f(t, f(t, f(t, =, >. (1.3 t We claim that (1.3 has uniqueness: i.e., if f(t,, g(t, C([, ɛ 1 ] [, ɛ ] both satisfy (1.3 and f(, = g(,, for all, then f(t, = g(t,, for all t,. Indeed, suppose f(t, satisfies (1.3. Then, by reversing time, treating f(ɛ 1, as our initial condition, and treating a(t := f(t, as a given function, we may solve the differential equation (1.3, for >, to see f(, = e ɛ1/ f(ɛ 1, e u/ a(u du, >. (1.4 From (1.4 uniqueness follows. Indeed, if f(t, and g(t, are two solutions to (1.3 with f(, = g(, for all, then (1.4 shows 1 1 e u/ f(u, du = 1 1 e u/ g(u, du + O(e ɛ1/. It then follows (see Corollary 8.4 that f(t, = g(t, for all t. With f(t, = g(t, in hand, (1.3 is a standard ODE for > and it follows that f(t, = g(t, for all t,. This proves uniqueness. Furthermore, (1.4 shows that (1.3 does not have eistence: not every initial condition gives rise to a solution. In fact, every initial condition that does give rise to a solution must be of the form given by (1.4, for some continuous functions a(t and f(ɛ 1,. I.e., the initial condition must be of Laplace transform type, modulo an appropriate error. Furthermore, it is easy to see that for such an initial condition, there eists a solution. Hence, we have eactly characterized the initial conditions which give rise to a solution to (1.3. The goal of this paper is to etend the above ideas to a nonlinear setting. Consider the following simplified eample. Eample 1.3. Let P (y = D j=1 c jy j be a polynomial without constant term. Consider the differential equation, defined for functions f(t, C([, ɛ 1 ] [, ɛ ], given by P (f(t, P (f(t, f(t, =, >. (1.5 t (Uniqueness without eistence If we restrict our attention to solutions f(t, with P (f(t, > for all t and we insist that f(t, C 2 ([, ɛ 1 ], then (1.5 has uniqueness (but not eistence. I.e., if f(t,, g(t, C([, ɛ 1 ] [, ɛ ] are two solutions to (1.5 with f(, = g(, for all, P (f(t,, P (g(t, > for all t, and f(t,, g(t, C 2 ([, ɛ 1 ], then f(t, = g(t, for all t,. However, not every initial condition gives rise to 1 Uniqueness is immediate here, since for >, if f(t, is assumed to be a(t, then (1.1 is a standard ODE and standard uniqueness theorems apply.

3 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 3 a solution. See Section 2.2. This generalizes 2 Eample 1.2 where P (y = y and therefore P (y 1 >. (Eistence without uniqueness Given f ( C([, ɛ ] and a(t C 2 ([, ɛ 1 ] with a( = f ( and P (a(t < for all t, there eists δ > and a unique solution f(t, C([, ɛ 1 ] [, δ] to (1.5 satisfying f(, = f ( and f(t, = a(t. See Section 2.1. This generalizes Eample 1.1 where P (y = y and therefore P (y 1 <. In short, if one has P (f(t, > for all t, one has uniqueness but not eistence, and if one has P (f(t, < for all t, one has eistence but not uniqueness. We now turn to the more general setting of our main results. Fi m N and ɛ, ɛ 1 >. For t [, ɛ 1 ], [, ɛ ], and y, z R m, let P (t,, y, z be a polynomial in y given by m P (t,, y, z = c α,j (t,, zy α e j, j=1 α D where e j R m denotes the jth standard basis element. For f(t, C([, ɛ 1 ] [, ɛ ]; R m we consider the differential equation P (t,, f(t,, f(t, P (t,, f(t,, f(t, f(t, =, >. (1.6 t We state our assumptions more rigorously in Section 2, but we assume: c α,j (t,, z = 1 e w/ b α,j (t, w, z dw, where the b α,j (t, w, z have a certain prescribed level of smoothness. We consider only solutions f(t, C([, ɛ 1 ] [, ɛ ]; R m such that f(t, C 2 ([, ɛ 1 ]; R m. For y R m, set M y (t := d y P (t,, y, y, so that M y (t is an m m matri. We consider only solutions f(t, such that there eists an invertible matri R(t which is C 1 in t and such that R(tM f(t, (tr(t 1 is a diagonal matri. When m = 1, this is automatic. Under the above assumptions, we prove the following: (Uniqueness without eistence Under the above hypotheses, if M f(t, (t is assumed to have all strictly positive eigenvalues, then (1.6 has uniqueness, but not eistence. I.e., if f(t,, g(t, C([, ɛ 1 ] [, ɛ ]; R m are solutions to (1.6 which satisfy all of the above hypothesis and such that the eigenvalues of M f(t, (t and M g(t, (t are strictly positive, for all t, then if f(, = g(, for all, we have f(t, = g(t, for all t,. Furthermore, in this situation we prove stability estimates. Finally, in analogy to Eample 1.2, we will see that only certain initial conditions give rise to solutions. See Section 2.2. (Eistence without uniqueness Suppose f ( C([, ɛ ]; R m and A(t C 2 ([, ɛ 1 ]; R m are given such that f ( = A( and M A(t (t has all strictly negative eigenvalues. Suppose further that there eists an invertible matri R(t, which is C 1 in t such that R(tM A(t R(t 1 is a diagonal matri. Then we show that there eists δ > and a unique function 2 Since we insisted f(t, C 2, this is not strictly a generalization of Eample 1.2, however it does generalize the basic ideas of Eample 1.2. A similar remark holds for the net part where we discuss eistence without uniqueness.

4 4 B. STREET EJDE-217/227 f(t, C([, ɛ 1 ] [, δ]; R m such that f(, = f (, f(t, = A(t, and f(t, solves (1.6. See Section 2.1. The main idea is the following. If f(t, were assumed to be of Laplace transform type, f(t, = 1 e w/ A(t, w dw, then (1.6 can be restated as a partial differential equation on A(t, w and this partial differential equation is much easier to study. As eemplified in Eamples 1.1 and 1.2, not every solution is of Laplace transform type. However, we will show (under the above discussed hypotheses that every solution is of Laplace transform type modulo an error which can be controlled. Once this is done, the above results follow Motivation and relation to inverse problems. It is likely that the methods of this paper are the most interesting aspect. The differential equations in this paper seem to not fall under any current methods (the equations are too unstable, and the methods in this paper are largely new. Moreover, as we will see, special cases of the above appear in some inverse problems. Furthermore, there are other (harder inverse problems where differential equations similar to (but more complicated than the ones studied in this paper appear. For eample, we will see in Section 9.2 that the anisotropic version of the famous Calderón problem involves a non-commutative version of some of these differential equations. We hope that the ideas in this paper might shed light on such questions and, indeed, one of our motivation for these results is as a simpler model case for full anisotropic version of the Calderón problem. We briefly outline the relationship between these results and inverse problems; these ideas are discussed in greater detail in Section 3. We begin by eplaining that the results in this paper can be thought of as a class of forward and inverse problems. For simplicity, consider the setting in Eample 1.3, with ɛ = ɛ 1 = 1. Thus, we are given a polynomial without constant term, P (y = D j=1 c jy j. We consider the differential equation, for functions f(t,, given by P (f(t, P (f(t, f(t, =, >. (1.7 t Forward Problem: Given a function f ( C([, 1] and a(t C 2 ([, 1] with P (a(t <, for all t and f ( = a(, the results below imply that there eists δ > and a unique solution f(t, C([, 1] [, δ] to (1.7 with f(, = f ( and f(t, = a(t. The forward problem is the map (f (, a( f(1,. Inverse Problem: The inverse problem is given f(1,, as above, to find f ( and a(. To see how the inverse problem relates to the main results of the paper, let f(t, be the solution as above. Set g(t, = f(1 t,. If Q(y = P (y, then g(t, = a(1 t and g(t, satisfies Q(g(t, Q(g(t, g(t, =, >. (1.8 t Also, Q (g(t, >, for all t. The main results of this paper imply (1.8 has uniqueness in this setting: g(, C([, δ] uniquely determines g(t, C([, 1] [, δ]. Since g(t, = f(1 t,, f(1, C([, δ] uniquely determines both f [,δ] and a(t. Thus, the inverse problem has uniqueness. In short, the map

5 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 5 (f [,δ] (, a( f(1, is injective (though it is far from surjective as we eplain below. We go further than just proving eistence and uniqueness, though. We have: In the forward problem, we do the following (see Section 2.1: Beyond just proving eistence, we show that every solution f(t, must be of Laplace transform type, modulo an appropriate error, for every t >. This is despite the fact that the initial condition, f(, = f (, can be any continuous function with P (f ( <. We reduce the problem to a more stable PDE, so that solutions can be more easily studied. In the inverse problem, we do the following (see Section 2.2.1: We characterize the initial conditions g(, which give rise to solutions to (1.8. In other words, we characterize the image of the map (f (, a( f(1,. We see that all such functions are of Laplace transform type, modulo an appropriate error. We give a procedure to reconstruct a(t and f [,δ] from f(1,. This is necessarily unstable, but we reduce the instability to the instability of the Laplace transform, which is well understood. We prove a kind of stability for the inverse problem. Namely if one has two solutions g 1 (t, and g 2 (t, to (1.8 such that g 1 (, g 2 (, vanishes sufficiently quickly as, then g 1 (t, = g 2 (t, on a neighborhood of (the size of the neighborhood depends on how quickly g 1 (, g 2 (, vanishes in a way which is made precise. In other words if one only knows f(1, modulo functions which vanish sufficiently quickly at, one can still reconstruct a(t on a neighborhood of t = 1, in a way which we make quantitative. Some special cases of the main results in this paper can be interpreted as some standard inverse problems in the following way: When P (y = y, we saw in Eamples 1.1 and 1.2 that the forward problem is essentially taking the Laplace transform, and the inverse problem is essentially taking the inverse Laplace transform. See Section 3.1 for more details on this. As a consequence, the results in this paper can be interpreted as nonlinear analogs of the Laplace transform. In our main results, we allow the coefficients of the polynomial to be functions of. We will see in Section 3.2 that the special case of P (, y = y 2 y 2 is closely related to Simon s approach [22] to the theorem of Borg [4] and Marčenko [18] that the principal m-function for a finite interval or half-line Schrödinger operator determines the potential. In our main results, we allow f to be vector valued, and also allow the coefficients to depend on f(t,. By doing this, we see in Section 9.1 that the translation invariant version of the anisotropic version of Calderón s inverse problem can be seen in this framework. Thus, the results in this paper can be viewed as a family of inverse problems which generalize and unify the above eamples, and for which we have good results on uniqueness, characterization of solutions, a reconstruction procedure, and stability estimates.

6 6 B. STREET EJDE-217/227 Furthermore, as argued in Section 9.2, a non-commutative analog 3 of these equations arise in the full anisotropic version of the Calderón problem. Thus, a special case of results in this paper can be seen as a simplified model case for the full Calderón problem. Moreover, by replacing functions in our results with pseudodifferential operators, one gives rise to an entire family of conjectures which generalize the Calderón problem Selected notation. All functions take their values in real vector spaces or spaces of real matrices. Other than in Section 8, there are no comple numbers in this paper. Let ɛ 1, ɛ 2 >. For n 1, n 2 N, we write b(t, w C n1,n2 ([, ɛ 1 ] [, ɛ 2 ] if for j n 1, k n 2, j k t b(t, w C([, ɛ j w k 1 ] [, ɛ 2 ]. If U R m is open, and n 3 N, we write c(t, w, z C n1,n2,n3 ([, ɛ 1 ] [, ɛ 2 ] U if for j n 1, k n 2, and α n 3, we have j t j C([, ɛ 1 ] [, ɛ 2 ] U. We define the norms n 1 n 2 b C n 1,n 2 := j k, t j w k b(t, w n 1 n 2 c C n 1,n 2,n 3 := j= k= sup t,w sup j t,w,z t j j= k= α n 3 k w k k w k α z α c(t, w, z. α z α c(t, w, z If V R n is open, and U R m, we write C j (V ; U to be the usual space of C j functions on V taking values in U. We use the norm g C j (V ;U := sup α z α g(z. z V α j We write M m n to be the space of m n real matrices. We write GL m to be the space of m m real, invertible matrices. For a(w, b(w C([, ɛ 2 ] we write (a b(w := w a(w rb(r dr C([, ɛ 2 ]. (1.9 Note that is commutative and associative. If A(w C([, ɛ 2 ]; R m and α = (α 1,..., α m N m is a multi-inde, we write α A = A 1 A }{{} 1 A j A j A m A m. }{{}}{{} α 1 terms α j terms α m terms and with a slight abuse of notation, if α = and b(w is another function, we write b ( α A = b. If A(t, w is a function of t and w, we write A = t A and A = w A. For λ 1,..., λ m R, we write diag(λ 1,..., λ m to denote the m m diagonal matri with diagonal entries λ 1,..., λ m. We write A B to mean A CB, where C depends only on certain parameters. It will always be clear from contet what C depends on. We write a b to mean min{a, b}. 3 Achieved by replacing functions with pseudodifferential operators: here the frequency plays the role that 1 plays in our main results.

7 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 7 2. Statement of results Fi m N, ɛ, ɛ 1, ɛ 2 (,, U R m open, and D N. For j {1,..., m}, α N m a multi-inde with α D, let b α,j (t, w, z C,3, ([, ɛ 1 ] [, ɛ 2 ] U, with b α,j (t,, z, ( w b α,j (t,, z C 1 ([, ɛ 1 ] U. Define c α,j (t,, z C([, ɛ 1 ] [, ɛ ] U by c α,j (t,, z := 1 We assume there is a C < with 2 e w/ b α,j (t, w, z dw. b α,j C,3, ([,ɛ 1] [,ɛ 2] U, b α,j C 1 ([,ɛ 1] U, b α,j C 1 ([,ɛ 1] U C. 2 Eample 2.1. Because 1 wl e w/ l! dw = l + e ɛ2/ G(, with G C([,, any polynomial in can be written in the form covered by the c α,j, modulo error terms of the form e ɛ2/ G(, G C([,. The results below are invariant under such error terms, so polynomials in can be considered as a special case of the c α,j. Define P (t,, y, z := (P 1 (t,, y, z,..., P m (t,, y, z, where for y R m, P j (t,, y, z = α D c α,j (t,, zy α. Let V R m be an open set with U V. Let G(t,, y, z C([, ɛ 1 ] [, ɛ ] V U; R m be such that for every γ (, ɛ 2, G(t,, y, z = e γ/ G γ (t,, y, z, where G γ (t,, y, z C([, ɛ 1 ] [, ɛ ] V U; R m satisfies for any compact sets K 1 U, K 2 V, G γ (t,, y 1, z G γ (t,, y 2, z sup <. t [,ɛ 1], [,ɛ ],z K 1 y 1 y 2 y 1,y 2 K 2,y 1 y 2 We will be considering the differential equation, defined for f(t, C([, ɛ 1 ] [, ɛ ]; V with f(t, C([, ɛ 1 ]; U, P (t,, f(t,, f(t, P (t,, f(t,, f(t, f(t, = t + G(t,, f(t,, f(t,, >. (2.1 Corresponding to P (t,, y, z, for δ (, ɛ 2 ] and A C 1 ([, δ]; R m, we define ( P (t, A(, z(w = P1 (t, A(, z(w,..., P m (t, A(, z(w by P j (t, A(, z(w = α D α +1 w α +1 (b α,j(t,, z ( α A (w.

8 8 B. STREET EJDE-217/ Eistence without uniqueness. Theorem 2.2. Suppose f ( C([, ɛ ]; V and A (t C 2 ([, ɛ 1 ]; U are given, with f ( = A (. Set M(t := d y P (t,, A (t, A (t. 4 We suppose that there eists R(t C 1 ([, ɛ 1 ]; GL m with R(tM(tR(t 1 = diag(λ 1 (t,..., λ m (t, where λ j (t > for all j, t. Then, there eists δ > and a unique solution f(t, C([, ɛ 1 ] [, δ ]; R m to (2.1, satisfying f(, = f ( and f(t, = A (t. Remark 2.3. As in the introduction, we call this eistence without uniqueness because one has to specify both f(, and f(t, (as opposed to just f(,. Beyond proving eistence, we can show that the solution given in Theorem 2.2 is of Laplace transform type, modulo an appropriate error, as shown in the net theorem. Theorem 2.4. Assume the same assumptions as in Theorem 2.2, and let f(t, be the unique solution guaranteed by Theorem 2.2. Take c, C 1, C 2, C 3, C 4 > such that min t,j λ j (t c >, R C 1 C 1, R 1 C 1 C 2, M 1 C 1 C 3, A C 2 C 4. Then, there eists δ = δ(m, D, c, C, C 1, C 2, C 3, C 4 > and A(t, w C,2 ([, ɛ 1 ] [, δ ɛ 2 ]; R m such that t A(t, w = P (t, A(t,, A(t, (w, and such that if λ (t = min j λ j (t, then for all γ [, 1, f(t, = 1 δ ɛ2 A(t, = A (t, e w/ A(t, w dw + O ( e γ(δ ɛ2/ + e (γ/ R t λ(s ds, (2.2 for (, δ ], where the implicit constant in the O in (2.2 does not depend on (t, [, ɛ 1 ] (, δ ]. Furthermore, the representation (2.2 is unique in the following sense. Fi t [, ɛ 1 ]. Suppose there eists < δ < δ ɛ 2 ( t λ (s ds and B C([, δ ]; R m with f(t, = 1 δ Then, A(t, w = B(w, for all w [, δ ]. e w/ B(w dw + O ( e δ /, as Uniqueness without Eistence. In addition to the above assumptions, for the net result we assume for every compact set K U, b α,j (t, w, z 1 b α,j (t, w, z 2 sup <, t [,ɛ 1],w [,ɛ 2] z 1 z 2 z 1,z 2 K,z 1 z 2 w sup b α,j(t, w, z 1 w b α,j(t, w, z 2 <. t [,ɛ 1],w [,ɛ 2] z 1 z 2 z 1,z 2 K,z 1 z 2 (2.3 4 Notice the minus sign in the definition of M(t. This is in contrast to the notation in the introduction, which lacked the minus sign.

9 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 9 Theorem 2.5. Suppose f 1 (t,, f 2 (t, C([, ɛ 1 ] [, ɛ ]; V satisfy f j (t, C 2 ([, ɛ 1 ]; U, both satisfy (2.1, and f 1 (, = f 2 (,, for all [, ɛ ]. Set M k (t := d y P (t,, f k (t,, f k (t,. We suppose that there eists R k (t C 1 ([, ɛ 1 ]; GL m with R k (tm k (tr k (t 1 = diag(λ k 1(t,..., λ k m(t, where λ k j (t >, for all j {1,..., m}, t [, ɛ 1]. Then f 1 (t, = f 2 (t,, for all t [, ɛ 1 ], [, ɛ ]. Theorem 2.5 shows uniqueness, but we will show more. We will further investigate the following questions: Stability: If f 1 (, f 2 (, vanishes sufficiently quickly at, and under the hypotheses of Theorem 2.5, we will prove that f 1 (t, and f 2 (t, agree for small t, and we will make this quantitative. See Theorem 2.1. Reconstruction: Given the initial condition f(, for (2.1, and under the hypotheses of Theorem 2.5, we will show how to reconstruct the solution f(t,, for all t. This is an unstable process, but we will reduce the instability to that of inverting the Laplace transform, which is well understood. See Remark 2.9. Characterization: We will show that if f(t, is a solution to (2.1, and under the hypotheses of Theorem 2.5, then f(t, must be of Laplace transform type, modulo an appropriate error term. In particular, only initial conditions f(, which are of Laplace transform type modulo an appropriate error give rise to solutions. See Theorem 2.6 and Remark 2.7. We now turn to making these ideas more precise Stability, Reconstruction, and characterization. For our first result, we take P as in the start of this section, but we drop the assumption (2.3. Theorem 2.6 (Charaterization. Suppose f(t, C([, ɛ 1 ] [, ɛ ]; R m is such that for all γ [, ɛ 2, P (t,, f(t,, f(t, P (t,, f(t,, f(t, f(t, = +O(e γ/, [, ɛ, t where the implicit constant in O is independent of t,. We suppose f(t, C 2 ([, ɛ 1 ]; U. Set M(t := d y P (t,, f(t,, f(t,. We suppose there eists R(t C 1 ([, ɛ 1 ]; GL m with R(tM(tR(t 1 = diag(λ 1 (t,..., λ m (t, where λ j (t >, for all j, t. Take c, C 1, C 2, C 3, C 4 > such that min t,j λ j (t c >, R C 1 C 1, R 1 C 1 C 2, M 1 C 1 C 3, f(, C 2 C 4. Then, there eist δ = δ(m, D, c, C, C 1, C 2, C 3, C 4 > and A(t, w C,2 ([, ɛ 1 ] [, δ ɛ 2 ]; R m such that t A(t, w = P (t, A(t,, A(t,, A(t, = f(t,, (2.4 and such that if λ (t = min j λ j (t, then for all γ (, 1, f(t, = 1 δ ɛ2 e w/ A(t, w dw + O (e γ(δ ɛ2/ + e (γ/ R ɛ 1 t λ (s ds, (2.5

10 1 B. STREET EJDE-217/227 where the implicit constant in O is independent of t,. Furthermore, the representation in (2.5 of f(t, is unique in the following sense. Fi t [, ɛ ]. Suppose there eists < δ < δ ɛ 2 1 t λ(s ds and B C([, δ ]; R m with f(t, = 1 δ Then, A(t, w = B(w, for all w [, δ ]. e w/ B(w dw + O ( e δ /, as. Remark 2.7. By taking t = in (2.5, we see that f(, is of Laplace transform type, modulo an error: for all γ (, 1, f(, = 1 δ ɛ2 ( e w/ A(, w dw + O e γ(δ ɛ2/ + e (γ/ R ɛ 1 λ (s ds. Thus, under the hypotheses of Theorem 2.6, the only initial conditions that give rise to a solution are of Laplace transform type, modulo an appropriate error. Furthermore, by taking t = in the last conclusion of Theorem 2.6, we see that f(, uniquely determines A(, w. For the remainder of the results in this section, we assume (2.3. Proposition 2.8. The differential equation (2.4 has uniqueness in the following sense. Let δ > and A(t, w, B(t, w C,2 ([, ɛ 1 ] [, δ ]; R m satisfy t A(t, w = P (t, A(t,, A(t, (w, t B(t, w = P (t, B(t,, B(t, (w, (2.6 and A(, w = B(, w for w [, δ ]. Set A (t = A(t,, and suppose A (t C 2 ([, ɛ 2 ]; R m and set M(t = d y P (t,, A (t, A (t. Suppose there eists R(t C 1 ([, ɛ 1 ]; GL m with R(tM(tR(t 1 = diag(λ 1 (t,..., λ m (t, t where λ j (t > for all j, t. Set γ (t := ma j λ j(s ds, and { γ 1 δ := (δ, if γ (ɛ 1 δ, ɛ 1, else. Then, A(t, = B(t, for t [, δ ]. Remark 2.9 (Reconstruction. Proposition 2.8 leads us to the reconstruction procedure, which is as follows: (i Given a solution f(t, to (2.1, satisfying the assumptions of Theorem 2.5, we use Theorem 2.6 to see that f(t, can be written in the form (2.5. In particular, as discussed in Remark 2.7, f(, uniquely determines A(, w. Etracting A(, w from f(, involves taking an inverse Laplace transform, and this step therefore inherits any instability inherent in the inverse Laplace transform. (ii With A(, w in hand, and with the knowledge that A(t, w satisfies (2.4, Proposition 2.8 shows that A(, w uniquely determines A(t, = f(t, for t δ, for some δ. (iii With f(t, in hand, for > (2.1 is a standard ODE, and so uniquely determines f(t, for t δ. (iv Iterating his procedure gives f(t,, for all t.

11 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 11 The above procedure reduces the reconstruction of f(t, from f(, to the reconstruction of A(t, w from A(, w. As we will see in the proof of Proposition 2.8, the differential equation satisfied by A is much more stable than that satisfied by f. In particular, we will be able to prove Proposition 2.8 by a straightforward application of Grönwall s inequality. Theorem 2.1 (Stability. Suppose f 1 (t,, f 2 (t, C([, ɛ 1 ] [, ɛ ]; R m satisfy, for k = 1, 2, for all γ (, ɛ 2, t f k(t, = P (t,, f k(t,, f k (t, P (t,, f k (t,, f k (t, + O ( e γ/, for (, ɛ ], where the implicit constant in O may depend on γ, but not on t or. Suppose, further, for some r > and all s [, r, We assume the following for k = 1, 2: f 1 (, = f 2 (, + O ( e s/. (2.7 f k (t, C 2 ([, ɛ 1 ]; U. Set M k (t := d y P (t,, f k (t,, f k (t,. We suppose that there eists R k (t in C 1 ([, ɛ 1 ]; GL m with R k (tm k (tr k (t 1 = diag(λ k 1(t,..., λ k m(t, where λ k j (t > for all j, t. Take c, C 1, C 2, C 3, C 4 > such that for k = 1, 2, min t,j λ k j (t c >, R k C 1 C 1, R 1 k C 1 C 2, M 1 k C 1 C 3, f k (, C 2 C 4. Set γ (t := ma j t λ 1 j(s ds, λ k (t = min λ k j (t. j Then there eists δ = δ(m, D, c, C, C 1, C 2, C 3, C 4 > such that the following holds. Define 1 1 δ = δ ɛ 2 λ 1 (s ds λ 2 (s ds >, and set { γ 1 δ := (r δ, if γ (ɛ 1 r δ, ɛ 1, otherwise. Then, f 1 (t, = f 2 (t, for t [, δ ]. 3. Forward problems, inverse problems, and past work The results in this paper can be seen as studying a class of nonlinear forward and inverse problems. Indeed, suppose we have the same setup as described at the start of Section 2. Forward Problem. Given f ( C([, ɛ 1 ]; V and A (t C 2 ([, ɛ 1 ]; U with f ( = A (. Let M(t be as in Theorem 2.2. Suppose there eists R(t C 1 ([, ɛ 1 ]; GL m with R(tM(tR(t 1 = diag(λ 1 (t,..., λ m (t, and λ j (t >, for all t. Let f(t, be the solution to (2.1 described in Theorem 2.2, with f(, = f (, f(t, = A (t. The forward problem is the map: (f, A f(ɛ 1,.

12 12 B. STREET EJDE-217/227 Inverse Problem: The inverse problem is, given f(ɛ 1, as described above, find f and A. Note that if f(t, is the function described above, f(t, = f(ɛ 1 t, satisfies all the hypotheses of Theorem 2.5 (here we assume (2.3. We have the following: The map (f, A f(ɛ 1, is injective Theorem 2.5. The map (f, A f(ɛ 1, is not surjective. In fact, the only functions in the image of are Laplace transform type, modulo an appropriate error term Theorem 2.4. The inverse map f(ɛ 1, (f, A is unstable, but we do have some stability results. Indeed, if one only knows f(ɛ 1, up to error terms of the form O(e r/, then f(ɛ 1, determines A (t for t [δ ɛ 1, ɛ 1 ], where δ is described in Theorem 2.1. We have a procedure to reconstruct A (t and f ( from f(ɛ 1, Remark 2.9. The above class of inverse problems has, as special cases, some already well understood inverse problems. We net describe two of these. For these problems, we reverse time in the above discussion since we are focusing on the inverse problem. In addition, the results in this paper are related to the famous Calderón problem, and we describe this connection in Section Laplace Transform. As see in Eamples 1.1 and 1.2 the Laplace transform is closely related to the case P (t,, y, z = y studied in this paper. In fact, the following proposition makes this even more eplicit. For a L ([, define the Laplace transform: L(a( = 1 e w/ a(w dw. Proposition 3.1. Let a C([, L ([,. unique solution to the differential equation For each > there is a f(t, a(t f(t, =, (3.1 t such that sup t f(t, <. For t, t define a t (t = a(t + t. This solution f(t, is given by f(t, = L(a t (. Furthermore, f(t, etends to a continuous function f C([, [, by setting f(t, = a(t. Proof. If we set f(t, = L(a t ( = 1 e s/ a(t + s ds = 1 t e (t s/ a(s ds, then it is clear that f satisfies (3.1, sup t f(t, <, and that f etends to a continuous function f C([, [, by setting f(t, = a(t. Suppose g(t, is another solution to (3.1 such that sup t g(t, <. Let h = f g. Then h(t, satisfies t h(t, = h(t, /, sup t h(t, <. This implies that h(t, = e t/ h(, and we conclude h(, = = h(t,, for all t. Thus f(t, = g(t,, proving uniqueness. In light of Proposition 3.1 one may define L(a (at least for a C([, L ([, in another way: there is a unique f(t, C([, [, with

13 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 13 sup t f(t, < and satisfying f(t, f(t, f(t, =, f(t, = a(t. t L(a( is then defined to be L(a( = f(,. Thus, the well known fact that a L(a is injective follows from uniqueness for the differential equation f(t, f(t, f(t, =. t Eample 3.2. The above discussion leads naturally to the following nonlinear inverse Laplace transform. Indeed, let P (y be a polynomial in y R. Let f 1 (t,, f 2 (t, C([, ɛ 1 ] [, ɛ ] satisfy, for j = 1, 2, t f j(t, = P (f j(t, P (f j (t,, (, ɛ ]. Suppose: f 1 (, = f 2 (,, for all [, ɛ ]. f j (t, C 2 ([, ɛ 1 ], j = 1, 2. P (f j (t, >, for t [, ɛ 1 ], j = 1, 2. Then, by Theorem 2.5, f 1 (t, = f 2 (t, for (t, [, ɛ 1 ] [, ɛ ]. When P (y = y, this amounts to the inverse Laplace transform as discussed above Inverse spectral theory. In this section, we describe the results due to Simon in the influential work [22], where he gave a new approach to the theorem of Borg-Marčenko that the principal m-function for a finite interval or half-line Schrödinger operator determines the potential. As we will show, this is closely related to the special case P (t,, y, z = 2 y 2 + y of the results studied in this y+1 y paper. We will contrast our theorems and methods with those of Simon. Let q L 1 loc ([, with sup y> q(t dt <, and consider the Schrödinger operator d2 dt 2 + q(t. For each z C \ [β, (with β sufficiently large, there is a unique solution (up to multiplication by a constant u(, z L 2 ([, of ü + qu = zu. The principal m-function is defined by m(t, z = u(t, z u(t, z. It is a theorem of Borg [4] and Marčenko [18] that m(, z uniquely determines q Simon [22] saw this as an instance of uniqueness for a generalized differential equation, which we now eplain in the framework of this paper. Indeed, it is easy to see that m satisfies the Riccati equation ṁ(t, z = q(t z m(t, z 2, (3.2 and well-known that m has the asymptotics m(t, κ 2 = κ q(t κ + o(κ 1, as κ. Thus, q(t can be obtained from m(t, and (3.2 is a differential equation involving only m. Thus, if the equation (3.2 has uniqueness, then m(, z uniquely determines q(t. However, one does not need to full power of uniqueness for (3.2. In fact, one needs only know uniqueness under the additional assumption that m(t, z is a principal m-function: i.e., if m 1 (t, z and m 2 (t, z both satisfy (3.2 with m 1 (, = m 2 (, and are both principal m-functions, then m 1 (t, z = m 2 (t, z, for all t, z. Simon proceeds via this weaker statement.

14 14 B. STREET EJDE-217/227 At this point, we rephrase these ideas into the language used in this paper. For, y R define P (, y = 2 y 2 + y. Note that P is of the form covered in this paper (Eample 2.1 and d y P (, y = 1. Given a principal m-function as above, define for small, { ( 1 f(t, := m(t, (2 2 + (2 1, if >, (3.3 q(t, if =. It is easy to see from the above discussion that f satisfies P (, f(t, P (, f(t, f(t, =, >. (3.4 t Furthermore, if q is continuous then f is continuous as well. Thus to show m(t, z uniquely determines q(t it suffices to show that (3.4 has uniqueness. In this contet, our results and the results of [22] are closely related but have a few differences: As discussed above, [22] only considers solutions to (3.2 which are principal m-functions. This forces f(t, in (3.3 to be eactly of Laplace transform type, for all t. As we have seen, not all solutions to (3.4 are eactly of Laplace transform type. In this way, our results are stronger than [22] in that we prove uniqueness when the initial condition is not necessarily of Laplace transform type we do not even require any sort of analyticity. 5 We require q C 2, while [22] requires no additional regularity on q. The constant δ in Theorems 2.6 and 2.1 is taken to be in [22]. Our results work for much more general polynomials than P. The reason for the differences above is that, once m is assumed to be a principal m-function, one is able to use many theorems regarding Schrödinger equations to deduce the stronger results, which we did not obtain in our more general setting. That we assumed q C 2 is likely not essential. For the specific case discussed in this section, our methods do yield results for q with lower regularity than C 2, though we chose to not pursue this. Moreover, even for the more general setting of our main results, it seems likely that a more detailed study of the partial differential equations which arise in this paper would lead to lower regularity requirements, though this would require some new ideas. That δ is assumed small in Theorems 2.6 and 2.1 seems much more essential this has to do with the fact that the equations studied in this paper are non-linear in nature, unlike the results in [22] which rested on the underlying linear theory of the Schrödinger equation. Remark 3.3. Many works followed [22], some of which dealt with m taking values in square matrices; e.g., [8]. All of the above discussion applies to these cases as well. 4. Convolution In this section, we record several results on the commutative and associative operation defined in (1.9. In Section 4.2 we distill the consequences of these results into the form in which they will be used in the rest of the paper and the reader may wish to skip straight to those results on a first reading. For this section, fi some ɛ >. 5 We learn a posteriori, in Theorem 2.6, that the initial condition must be of Laplace transform type modulo an error, but this is not assumed.

15 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 15 Lemma 4.1. Let a C([, ɛ], b C 1 ([, ɛ]. Then w (a b(w = a(wb( + (a b (w. In particular, if b( =, then w (a b(w = (a b (w. The proof of the above lemma is immediate from the definitions. Lemma 4.2. Let l 1 and let a C([, ɛ], b C l+1 ([, ɛ]. Suppose for j l 1, j w b( =. Then a b C l+1 ([, ɛ] and for j l, j j w (a b( =. j Furthermore, if a C 1 ([, ɛ], then a b C l+2 ([, ɛ]. Proof. By repeated applications of Lemma 4.1, for j l, w (a b = a j j w b, j and this epression clearly vanishes at. Applying Lemma 4.1 again, we see l+1 (a b = l w l+1 w (a b = a(w l b ( + (a l+1 b. This epression is continuous, w l w l w l+1 so a b C l+1. Furthermore, if( a C 1, it follows from one more application of Lemma 4.1 that l+2 (a b = w l+2 w a(w l b ( + (a l+1 b w l w l+1 is continuous, and therefore a b C l+2. For the net few results, suppose a 1,..., a L C 1 ([, ɛ] are given. For J = {j 1,..., j k } {1,..., L}, we define j J a = a j1 a jk. With an abuse of notation, for b C([, ɛ], we define b ( j a = b. Lemma 4.3. For each n {1,..., L}, a 1 a n C n ([, ɛ] and if j n 2, j w j (a 1 a n ( =. Proof. For n = 1, the result is trivial. We prove the result by induction on n, the base case being n = 2 which follows from Lemma 4.1. We assume the result for n 1 and prove it for n. By the inductive hypothesis, a 1 a n 1 C n 1 and vanishes to order n 3 at. From here, the result follows from Lemma 4.2 with l = n 2. Define and let I =. Lemma 4.4. I L (a 1,..., a L := L 1 w L 1 (a 1 a L = ( L 1 j=1 J {1,...,L} ( j J j ( a j ( k J ca k, a j ( a L + I L 1 (a 1,..., a L 1 a L, (4.1 L w L (a 1 a L = I L (a 1,..., a L. (4.2 Proof. We prove the result by induction on by induction on L. The base case, L = 1, is trivial. We assume (4.1 and (4.2 for L 1 and prove them for L. We have, using repeated applications of Lemmas 4.1 and 4.3, L 1 w L 1 (a 1 a L = (( L 2 w w L 2 (a 1 a L 1 a L ( L 2 ( L 1 = w L 2 (a 1 a L 1 (a L + w L 1 (a 1 a L 1 a L

16 16 B. STREET EJDE-217/227 Using our inductive hypothesis for (4.1 and the fact that (b c( = for any b, c, ( L 2 w L 2 (a 1 a L 1 (a L = [ L 1 j=1 ] a j ( a L, and using our inductive hypothesis for (4.2, ( L 1 w L 1 (a 1 a L 1 a L = I L 1 (a 1,..., a L 1 a L. Combining the above equations yields (4.1. Taking Lemma 4.1, (4.2 follows, completing the proof. Corollary 4.5. Let A C 1 ([, ɛ]; R m, b C 1 ([, ɛ]. α N m, α +1 w α +1 (b ( α A ( α β = β α β < α b(a( β( α β A + β α The above corollary follows immediately from Lemma 4.4. w of (4.1 and applying Then, for a multi-inde ( α ( A( β b ( α β A. β Lemma 4.6. Let b 1,..., b L, c 1,..., c L C([, ɛ]. Then, b 1 b L c 1 c L = ( 1 J +1 ( l J (b l c l ( l J b l =J {1,...,L} The proof of the above lemma is standard, uses only the multilinearity of, and can be proved using a simple induction. Lemma 4.7. Suppose a 1,..., a L C 2 ([, ɛ]. Then, L w L (a 1 a L ( L 1 = a l ( a L + l=1 ( L 1 + l=1 J {1,...,L 1} l=min J c ( L 1 ( l=1 ( j J 1 k L 1 k l ( ( a j ( a l( Proof. Using Lemmas 4.1 and 4.4, we have L w L (a 1 a L = w = ( L 1 j=1 (( L 1 j=1 a k ( a l( a L k J c k l a j ( a L + I L 1 (a 1,..., a L 1 a L a k + a l ( k J c a k k l a L ( a j ( a L + I L 1 (a 1,..., a L 1 (a L + w I L 1(a 1,..., a L 1 a L

17 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 17 Since (b c( = for any b, c, Using Lemma 4.1, L 1 ( I L 1 (a 1,..., a L ( = w I L 1(a 1,..., a L 1 L 1 ( = l=1 J {1,...,L 1} l=min J c j J l=1 ( ( a j ( a l( Combining the above equations yields the result. Corollary 4.8. Let a 1,..., a L C 2 ([, ɛ]. where F 2 (w L w L (a 1 a L (w = ( L 1 l=1 1 k L 1 k l k J c k l a k a k ( a l(. + a l ( k J c a k k l. a l ( a L(w + F 1 (w, (4.3 L w L (a 1 a L (w = F 2 (w, (4.4 F 1 (w sup r w ( a L 1 ( + sup r w a L (r a L (r, ( a L(w + sup a L (r, r w where the implicit constants may depend on L, and upper bounds for ɛ and a j C 2, 1 j L. Proof. The bound for F 1 follows immediately from Lemma 4.7. The bound for F 2 follows from (4.3 and the bound for F 1. Lemma 4.9. Let a, b C 1 ([, ɛ]. Let f( = 1 ɛ e w/ a(w dw and g( = e w/ b(w dw. Then, there eists G C([, such that 1 Also, f(g( = 1 f( f( e w/ = 1 w (a b(w dw + 1 e ɛ/ G(. (4.5 w/ a e w (w dw 1 e ɛ/ a(ɛ. (4.6 Proof. A straightforward computation shows f(g( = 1 u 2 e u/ a(w 1 b(u w 1 dw 1 du We have 1 2 2ɛ ɛ e u/ u ɛ + 1 2ɛ 2 e u/ ɛ u ɛ a(w 1 b(u w 1 dw 1 du a(w 1 b(u w 1 dw 1 du.

18 18 B. STREET EJDE-217/227 = 1 2 e ɛ/ e u/ a(w 1 b(u + ɛ w 1 dw 1 du =: 1 e ɛ/ G 1 (, where G 1 C([, ɛ]. Also, using that (a b( =, 1 2 = 1 u u e u/ a(w 1 b(u w 1 dw 1 du ( u e u/ (a b(u du = 1 e ɛ/ (a b(ɛ + 1 Combining the above equations yields (4.5. We have 1 which proves (4.6. e u/ (a b(u du. u w/ a e w (w dw = 1 e ɛ/ a(ɛ 1 a( = f( f( + 1 e ɛ/ a(ɛ, e w/ a(w dw Lemma 4.1. Let a 1,..., a n C 1 ([, ɛ]. Define f j ( = 1 ɛ e w/ a j (w dw. Then, there are continuous functions G 1, G 2 C([, such that n f j ( = 1 e w/ n 1 w n 1 (a 1 a n (w dw + 1 e ɛ/ G 1 (, (4.7 j=1 n j=1 f j( n j=1 f j( = 1 e w/ n w n (a 1 a n (w dw e ɛ/ G 2 (. (4.8 Proof. We prove (4.7 by induction on n. n = 1 is trivial and n = 2 is contained in Lemma 4.9. We assume the result for n 1 and prove it for n. Thus, we assume n 1 j=1 f j ( = 1 n 2 w n 2 (a 1 a n 1 (w dw + 1 e ɛ/ G1 (, (4.9 where G 1 C([,. By Lemma 4.3, a 1 a n 1 C n 1 and vanishes to order n 3 at. Using this, and repeated applications Lemma 4.1, we have ( n 2 w n 2 (a 1 a n 1 a n = n 2 w n 2 (a 1 a n. Using this and Lemma 4.9 we have, for some G 2 C([,, ( 1 ɛ n 2 w n 2 (a 1 a n 1 (w dw f n ( = 1 e w/ ( n 2 w w n 2 (a 1 a n 1 a n (w dw + 1 e ɛ/ G2 ( = 1 n 1 w n 1 (a 1 a n (w + 1 e ɛ/ G2 (. (4.1

19 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 19 Combining (4.9 with (4.1, we have n f j ( = 1 n 1 w n 1 (a 1 a n (w + 1 e ɛ/ G2 ( + 1 e ɛ/ G1 (f n (, j=1 which proves (4.7. We turn to (4.8. Using (4.6 and (4.7, we have n j=1 f j( n j=1 f j( = 1 e w/ n w n (a 1 a n (w dw e ɛ/ G 1 ( n 1 1 w=ɛ e ɛ/ w n 1 (a 1 a n (w. Since a 1 a n C n (by Lemma 4.3, this completes the proof Smoothing. The operation has smoothing properties, and this section is devoted to discussing the instances of these smoothing properties which are used in this paper. Fi m N, ɛ 1, ɛ 2 >. Definition For L, n 1, and increasing functions G 1, G 2, G 3 : (, (,, we say G : C,L ([, ɛ 1 ] [, ɛ 2 ]; R m C,L ([, ɛ 1 ] [, ɛ 2 ]; R n is an (L, G 1, G 2, G 3 operation if: G(A(t, w depends only on the values of A(t, r for r [, w]. As a result, for δ (, ɛ 2 ], G defines a map G : C,L ([, ɛ 1 ] [, δ]; R m C,L ([, ɛ 1 ] [, δ]; R n. For k L 1, there are functions G k : C([, ɛ 1 ]; R m k+1 C([, ɛ 1 ]; R m such that G k : C L ([, ɛ 1 ]; R m C L 1 ([, ɛ 1 ]; R m C L 1 ([, ɛ 1 ]; R m C L k 1 ([, ɛ 1 ]; R n, and k w k G(A(t, w w= = G k( A(,, A w (,,..., k A w k (, (t. The following holds for all M (,, δ (, ɛ 2 ]. for all A C,L ([, ɛ 1 ] [, δ]; R m with A C,L M, G(A C,L G 1 (M. for all A, B C,L ([, ɛ 1 ] [, δ]; R m with A C,L, B C,L M, G(A G(B C,L G 2 (M A B C,L. For k L 1, and g 1,..., g k with g j C L j ([, ɛ 1 ]; R m and g j C L j M, G k (g 1,..., g k C L k 1 G 3 (M. Below we use to construct several eamples, in the case n = 1, of (2, G 1, G 2, G 3 operations.

20 2 B. STREET EJDE-217/227 Lemma Let α N m be a multi-inde with α 2, and let b(t, w C([, ɛ 1 ] [, ɛ 2 ]. For A C,2 ([, ɛ 1 ] [, ɛ 2 ]; R m set G(A(t, w := (b(t, ( α A (t, (w. Then, G is a (2, G 1, G 2, G 3 operation, where the functions G 1, G 2, and G 3 can be chosen to depend only on α, m, and upper bounds for ɛ 2 and b C. Proof. Let k 1 = min{l : α l } and k 2 = min{l : (α e k1 l }. Using Lemma 4.1, we have and w G(A(t, w = A k 1 (t, ( b(t, ( α e k A (t, (w + ( b(t, A k 1 (t, ( α e k 1 A (t, (w, 2 w 2 G(A(t, w = A k 1 (t, A k 2 (t, ( b(t, ( α e k 1 e k2 A (t, (w + A k 1 (t, ( b(t, A k 2 (t, ( α e k 1 e k2 A (t, (w + A k 2 (t, ( b(t, A k 1 (t, ( α e k 1 e k2 A (t, (w + ( b(t, A k 1 (t, A k 2 (t, ( α e k 1 e k2 A (t, (w. For any c 1, c 2, we have (c 1 c 2 ( =, we may therefore take G = and G 1 =. Using the above formulas, combined with Lemma 4.6, the result follows. Lemma Suppose α = 1 and b(t, w C,1 ([, ɛ 1 ] [, ɛ 2 ]. C,2 ([, ɛ 1 ] [, ɛ 2 ]; R m set G(A(t, w := (b(t, ( α A (t, (w. For A Then G is a (2, G 1, G 2, G 3 operation, where the functions G 1, G 2, and G 3 can be chosen to depend only on m and upper bounds for ɛ 2 and b C,1. Proof. Without loss of generality we take α = e 1, so that G(A(t, w = (b(t, A 1(t, (w. Using Lemma 4.1 we have w G(A(t, w = A 1(t, b(t, w + (b(t, A 1(t, (w, 2 w 2 G(A(t, w = A 1(t, wb (t, w + b(t, A 1(t, w + (b (t, A 1(t, (w. In particular, G(A(t, =, G(A(t, w = A w w= 1(t, b(t,. Using the above formulas, the result follows easily. Lemma Suppose α 2 and b(t C([, ɛ 1 ]. For A C,2 ([, ɛ 1 ] [, ɛ 2 ]; R m set G(A(t, w := b(t ( α A (t, (w. Then, G is a (2, G 1, G 2, G 3 operation, where the functions G 1, G 2, and G 3 can be chosen to depend only on m and upper bounds for ɛ 2 and b C.

21 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 21 Proof. Let k 1 = min{l : α l } and k 2 = min{l : (α e k1 l }. Using Lemma 4.1 we have w G(A(t, w = b(ta k 1 (t, ( α e k 1 A (t, (w and In particular, + b(t ( A k 1 (t, ( α e k 1 A (t, (w, 2 w 2 G(A(t, w = b(ta k 1 (t, A k 2 (t, ( α e k 1 e k2 A (t, (w G(A(t, =, + b(ta k 1 (t, ( A k 2 (t, ( α e k 1 e k2 A (t, (w + b(ta k 2 (t, ( A k 1 (t, ( α e k 1 e k2 A (t, (w + b(t ( A k 1 (t, A k 2 (t, ( α e k 1 e k2 A (t, (w. G(A(t, w = w w= {, if α > 2, b(ta k 1 (t, A k 2 (t,, if α = 2. Using the above formulas, combined with Lemma 4.6, the result follows easily. Lemma Suppose d C,2 ([, ɛ 1 ] [, ɛ 2 ] is such that d(t, C 1 ([, ɛ 1 ]. For A C,2 ([, ɛ 1 ] [, ɛ 2 ]; R m set G(A(t, w := d(t, w. Then, G is a (2, G 1, G 2, G 3 operation, where the functions G 1, G 2, and G 3 can be chosen to depend only on upper bounds for d C,2 and d(, C 1. The above lemma follows immediately from the definitions. Lemma Suppose G : C,L ([, ɛ 1 ] [, ɛ 2 ]; R m C,L ([, ɛ 1 ] [, ɛ 2 ] is an (L, G 1, G 2, G 3 operation. Let β N m be a multi-inde, and define G(A(t, w := A(t, β G(A(t, w. Then, G is an (L, G 1, G 2, G 3 operation, where G 1, G 2, and G 3 can be chosen to depend only on G 1, G 2, G 3, L, and β. The above lemma follows immediately from the definitions Polynomials. For this section, we take all the same notation and assumptions as in the beginning of Section 2. Thus, we have b α,j, c α,j, P (t,, y, z, and P (t, A(, z(w as described in that section. Lemma Let δ (, ɛ 2 ] and A(t, w C,1 ([, ɛ 1 ] [, δ]; R m with A(t, C([, ɛ 1 ]; U. Define f(t, C([, ɛ 1 ] [, ɛ ]; R m by Then f(t, = 1 δ e w/ A(t, w dw. P (t,, f(t,, f(t, P (t,, f(t,, f(t, = 1 δ where G(t, C([, ɛ 1 ] [, ɛ ]; R m. e w/ P (t, A(t,, A(t, (w dw e δ/ G(t,,

22 22 B. STREET EJDE-217/227 The above lemma follows from Lemma 4.1, using the fact that f(t, = A(t,. Proposition Let δ (, ɛ 2 ]. For A C,2 ([, ɛ 1 ] [, δ]; R m and A (t C 1 ([, ɛ 1 ]; R m, P (t, A(t,, A (t(w = d y P (t,, A(t,, A (ta (t, w + G A (A(t, w, where G A : C,2 ([, ɛ 1 ] [, ɛ 2 ]; R m C,2 ([, ɛ 1 ] [, ɛ 2 ]; R m is a (2, G 1, G 2, G 3 operation 6. The functions G 1, G 2, and G 3 can be chosen to depend only on C, m, D, 7 and upper bounds for ɛ 2 and A C 1. Proof. By linearity, it suffices to prove the result for P a monomial in y. I.e., P (t,, y, z = c α,j (t,, zy α e j, for some j {1,..., m}, α N m with α D. In this case, P (t, A(t,, z(w = α +1 w (b α,j(t,, z ( α A(t, (we α +1 j. (4.11 Using Corollary 4.5 and the fact that b α,j (t,, z = c α,j (t,, z, P (t, A(t,, A (t(w m = α l b α,j (t,, A (ta(t, α e l A l(t, we j l=1 + β α β < α 1 + β α ( α β ( α ( b α,j (t,, A (ta(t, β α β A (t, (we j β A(t, β ( b α,j(t,, A (t ( α β A (t, (we j Note that m α l b α,j (t,, A (ta(t, α e l A l(t, we j = d y P (t,, A(t,, A (ta (t, w. l=1 (4.12 Thus, it remains to show the final two terms on the right hand side of (4.12 are a (2, G 1, G 2, G 3 operation. This follows from Lemmas 4.12, 4.13, 4.14, 4.15, and 4.16, completing the proof. Proposition In addition to the other assumptions of this section, we assume (2.3. Let δ (, ɛ 2 ] and let A, B C,2 ([, ɛ 1 ] [, δ]; R m. Set g(t, w = A(t, w B(t, w. Then P (t, A(t,, A(t, (w P (t, B(t,, B(t, (w = d y P (t,, A(t,, A(t, g (t, w + F (t, w, where there eists a constant C with F (t, w C sup g(t, r, t, w. r w Here, C is allowed to depend on any of the ingredients in the proposition, including A and B. 6 See Definition 4.11 for the definition of a (2, G1, G 2, G 3 operation. 7 See Section 2 for the definitions of these various constants.

23 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 23 Proof. By linearity, it suffices to prove the result for P a monomial in y. I.e., P (t,, y, z = c α,j (t,, zy α e j, for some j {1,..., m} and α N m with α D. In this case P is given by (4.11. Using Lemma 4.6, P (t, A(t,, A(t, (w P (t, B(t,, B(t, m α +1 ( = α l bα,j (t,, A(t, g w α +1 l (t, ( α e l A(t, (we j l=1 + ( α ( 1 β +1 β β α β 2 α +1 ( ( ( b(t,, A(t, β g(t, α β A(t, (we w α +1 j + α +1 w α +1 ((b α,j(t,, A(t, b α,j (t,, B(t, ( α B(t, (we j =: (I + (II + (III. We study the three terms on the right hand side of the above equation separately. Applying (4.3 to each term of the sum in (I, with g l playing the role of a L, and using the fact that b α,j (t,, z = c α,j (t,, z, (I = m α l b α,j (t,, A(t, A(t, α e l g l(t, e j + F 1 (t, w l=1 = d y P (t,, A(t,, A(t, g (t, + F 1 (t, w, where F 1 (t, w sup r w g(t, r. Turning to (II, we note that in each term in the sum defining (II, β 2, and so there are at least two coordinates (counting repetitions of g(t, in the convolution. Applying (4.4 to each term of the sum, with these two coordinates of g(t, playing the roles of a L and a L 1, we see (II = F 2 (t, w where F 2 (t, w sup r w g(t, r. Finally, for (III, we use that as t varies over [, ɛ 1 ], A(t, and B(t, range over a compact subset of U. Applying (4.4 with b α,j (t,, A(t, b α,j (t,, B(t, playing the role of a L, and using (2.3, we see (III = F 3 (t, w, where F 3 (t, w sup b α,j (t, r, A(t, b α,j (t, r, B(t, r w + b α,j(t, w, A(t, b α,j(t, w, B(t, g(t, sup g(t, r. r w Summing the above three estimates completes the proof. 5. Ordinary differential equations In this section, we prove some auiliary results concerning ODEs which are needed in the remainder of the paper Chronological Calculus. Let m N and let J = [a, b] for some a < b. Let M(t : J M m m be locally bounded and measurable.

24 24 B. STREET EJDE-217/227 Definition 5.1. For t J, we define ( t ep A(s ds = E(t, to be the unique solution E : J M m m to the differential equation a Ė(t = A(tE(t, E(a = I, where I denotes the m m identity matri. For the rest of this section, fi ɛ >. Proposition 5.2. Let M(t, C(J [, ɛ ]; M m m be such that there eists R(t C 1 (J; GL m with R(tM(t, R(t 1 = diag(λ 1 (t,..., λ m (t, with λ j (t > for all t. Set λ (t = min 1 j m λ j (t. Then, for all δ [, 1, (, ɛ ], for all (, ], for all t J, ( ep 1 t a M(s, ds R(t 1 R(a ep ( δ To prove Proposition 5.2, we introduce the following lemma. t a λ (s ds. Lemma 5.3. Let M(t, C(J [, ɛ ]; M m m and set 2λ (t to be the least eigenvalue of M(t, + M(t,. We assume λ (t >, for all t J. Then, for all δ [, 1, there eists (, ɛ ], such that for all (, ], ( ep 1 t a M(s, ds ep ( δ t a λ (s ds. Proof. Let N (t, = M(t, M(t, so that N (t, C(J [, ɛ ]; M m m and N (t, =. Fi δ [, 1 and take (, ɛ ] so small for all (t, J [, ], N (t, inf s J (1 δλ (s. Let θ R m and set θ(t, := ep ( 1 t a M(s, ds θ. Then t θ(t, 2 = θ(t,, ( 1 M(t, + 1 M(t, θ(t, By Grönwall s inequality, we have = 1 θ(t,, ( M(t, + M(t, θ(t, 1 θ(t,, ( N (t, + N (t, θ(t, 2 λ (t θ(t, N (t, θ(t, 2 δλ (t θ(t, 2. θ(t, 2 θ 2 ep Taking square roots yields the result. ( 2δ t a λ (s ds. Proof of Proposition 5.2. Let Λ(t = diag(λ 1 (t,..., λ m (t = R(tM(t, R(t 1. For θ R m, set θ(t, = ( t ep a M(s, ds θ. Let γ(t, = R(tθ(t,. γ satisfies 1 t γ(t, = 1 R(tM(t, R(t 1 γ(t, + Ṙ(tR(t 1 γ(t,

25 EJDE-217/227 DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT 25 = 1 M(t, γ(t,, where M(t, = Λ(t + R(t (M(t, M(t, R(t 1 Ṙ(tR(t 1. In particular, note M(t, = Λ(t. It follows that γ(t, = ( ep 1 M(s, ds γ(a,. Fi δ [, 1. By Lemma 5.3, there eists (, ɛ ] (independent of θ such that for (, ], ( ep 1 t ( M(s, ds ep δ t λ (s ds. Hence, for (, ], The result follows. a θ(t, R(t 1 γ(t, ( R(t 1 ep δ R(t 1 R(a ep t a a λ (s ds γ(a, ( δ t a λ (s ds θ A basic eistence result. Fi ɛ 1, ɛ > and let W R m be an open neighborhood of R m. Suppose M(t,, y C([, ɛ 1 ] [, ɛ ] W ; M m m be such that for every compact set K W, M(t,, y 1 M(t,, y 2 sup <. t [,ɛ 1], [,ɛ ] y 1 y 2 y 1,y 2 K,y 1 y 2 Let G(t,, y C([, ɛ 1 ] [, ɛ ] W ; R m be such that for every compact set K W, G(t,, y 1 G(t,, y 2 sup <. t [,ɛ 1], [,ɛ ] y 1 y 2 y 1,y 2 K,y 1 y 2 Let g C([, ɛ ]; R m have g ( =. The goal of this section is to study the differential equation t g(t, w = 1 M(t,, g(t, g(t, + G(t,, g(t,, > (5.1 with the initial condition g(, = g (. The main result is the following. Proposition 5.4. Set M (t = M(t,,. We suppose that there eists R(t C 1 ([, ɛ 1 ]; GL m such that R(tM (tr(t 1 = diag(λ 1 (t,..., λ m (t and λ j (t >, for all j, t. Then, there eists δ (, ɛ ] and a function g(t, C([, ɛ 1 ] [, δ ]; W such that g(, = g ( for all [, δ ], g(t, = for all t [, ɛ 1 ], and g satisfies (5.1. To prove Proposition 5.4, we need two lemmas. As in Proposition 5.4, set M (t = M(t,,. For these lemmas, instead of assuming the eistence of R(t as in Proposition 5.4, we let 2λ (t be the least eigenvalue of M (t + M (t and we assume λ (t >, for all t [, ɛ 1 ].