UNIQUENESS OF nth ORDER DIFFERENTIAL SYSTEMS WITH STRONG SINGULARITY

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1 UNIQUENESS OF nth ORDER DIFFERENTIAL SYSTEMS WITH STRONG SINGULARITY YIFEI PAN AND MEI WANG Abstract. By a Lipschitz type condition, we obtain the uniqueness of solutions for a system of nth order nonlinear ordinary differential equations where the coefficients are allowed to have singularities. 1. Introduction and main results Lipschitz condition was a key part in proving classical results on the existence and uniqueness of ordinary differential equations, as expensively surveyed and summarized in [1]. In this paper, we use a Lipschitz type condition to obtain the uniqueness of solutions of nth order nonlinear ordinary differential systems where the coefficients are allowed to have singularities. Our results are in the spirit of the Carathéodory theorem on the existence of ordinary differential equations ([5], Chapter 2), which gives a Lipschitz condition in first order differential equations. The conditions in this paper are in terms of absolute continuity, thus the uniqueness result is on solutions in weaker or more general sense. For first order differential equations, Nagumo s Theorem [8] and its generalizations ([2, 3]) give precise coefficients and sharp order of sigularity in the lipschitz condition. A natural generalization of the classical Carathéodory condition is to higher order linear and nonlinear differential systems (e.g. [4], [6]). Our results are on higher order differnetial equations with coefficients of sigularities. The main results are stated below. The proofs are provided in the next section. We conclude with corollaries and an example to illustrate the sharpness of the singularity order allowed in the main condition of the Lipschitz type in an n order differential equation. Let L 1 (a, b) denote the set of real Lebesgue integrable functions on the interval (a, b), the Euclidean norm in R d, and C k (a, b) the set of d-dimensional functions with kth continuously differentiable components on (a, b). The following theorem is on the uniqueness of solutions of differential systems. 2 Mathematics Subject Classification. 34A12, 65L5. Key words and phrases. unique continuation, uniqueness, Carathéodory theorem, Grönwall inequality. 1

2 2 UNIQUENESS OF DIFFERENTIAL SYSTEMS WITH SINGULARITY Theorem 1. Consider the system of differential equations of y : (a, b) R d, y (n) = f ( x, y, y,, y ()), x (a, b), a < < b, y() = a y () = a 1. y () () = a (1) where f : (a, b) R nd R d satisfies the Lipschitz type condition f(x, s,, s ) f(x, r,, r ) λ k (x) s k r k, x n k 1 a. e. x (a, b) (2) for λ k (x), λ k L 1 (a, b) and (s,, s ), (r,, r ) in a domain in R nd containing (a,, a ). If u, v C (a, b) are solutions of (1) and u (), v () are absolutely continuous with respect to the Lebesgue measure on (a, b), then k= v(x) u(x) for x (a, b). The follow theorem gives the condition for a function that satisfies a differential inequality to be identically zero, which can be considered as an nth order Grönwall [7] uniqueness theorem. Theorem 2. Let y : (a, b) R d, y C (a, b) and y () absolute continuous with respect to the Lebesgue measure on (a, b), a < < b. If and y() = y (1) () = = y () () = y (n) (x) λ k (x) y(k) (x) x n k 1 a. e. x (a, b) (3) k= with λ k L 1 (a, b), λ k, k =, 1,, n 1, then y(x) x (a, b). 2. Proofs of the main results The absolute continuity assumption in the theorems is needed in applying the Fundamental Theorem of Calculus to prove the following lemma. Lemma 3. Assume φ : (a, b) R d, φ C (a, b), φ () absolutely continuous on (a, b) for a < < b. If φ (k) () = R d, k =, 1,, n 1, then for any λ(x) L 1 (a, b), λ(x) and k =, 1,, n 1, λ(t) φ(k) (t) x dt t n k 1 λ(t)dt φ (n) (t) dt, x (a, b). (4)

3 UNIQUENESS OF DIFFERENTIAL SYSTEMS WITH SINGULARITY 3 Proof. By the absolute continuity assumption of φ (), the Fundamental Theorem of Calculus grants the form φ (k) (x) = tn k 1 for k =, 1,, n 1. Define g(x) = t t1 φ (n) (t)dt dt 1 d, x [, b) t1 φ (n) (t) dt dt 1 dt, x [, b). Then g C [, b), g (n) = φ (n) a. e. in (a, b) and for k =, 1,, n 1, g (k) (x) = g (k) () =. tn k 1 t1 φ (n) (t) By Taylor s formula, for any t (, b) and k =,, n 2, dt dt 1 d w. r. t. x (, b) g (k) (t) = g (k) () + g (k+1) (c)(t ) c (, t) g (k+1) (t) t g (k+2) (t) t 2 g () (t) By φ (k) (x) g (k) (x) and the monotonicity of g (k) on [, b), λ(t) φ(k) (t) dt = λ(t) g(k) (t) dt x [, b), k =, 1,, n 1 tn k 1 λ(t)g () (t)dt g () (x) g (n) (t)dt λ(t)dt = λ(t)dt φ (n) (t) dt λ(t)dt. Thus the result holds for x [, b). For x (a, ], consider x = x and define φ (x ) = φ( x ), g (x ) = t t1 Repeating the above derivation for x [, b) leads to λ( t) φ (k) (t) dt φ (n) (t) dt φ (n) (t) dt dt 1 dt, λ( t)dt, x [, a). Subsisting t by t in the integrands and replacing x by x = x (a, ], x [, a). x λ(t) φ(k) (t) dt t n k 1 x φ (n) (t) dt x λ(t)dt, x (a, ]. Therefore (4) holds for all x (a, b). This completes the proof of Lemma 3. The lemma is to be used in the proof of Theorem 1 below.

4 4 UNIQUENESS OF DIFFERENTIAL SYSTEMS WITH SINGULARITY Proof of Theorem 1. Let φ(x) = u(x) v(x), x (a, b). Then φ satisfies the conditions in Lemma 3: φ C (a, b), φ () is absolutely continuous on (a, b), and φ (k) () = for k =, 1,, n 1. By the Lipschitz property (2), φ (n) (t) = f(t, u(t), u (t),, u () (t)) f(t, v(t), v (t),, v () (t)) (5) k= λ k (t) u(k) (t) v (k) (t) = k= λ k (t) φ(k) (t) Here we assume f is well defined so that for solutions u, v on (a, b), (u, u,, u () ), (v, v,, v () ) are in the domain for which (2) holds. For any ε (, b), define ( ε φ (k) (t) ) A(ε) = λ k (t) dt k= k= Applying inequalities (4) within the sum and then using (2), we have φ (k) (t) A(ε) = λ k (t) k= dt λ k (t)dt φ (n) (t) dt k= ( ε φ (k) (t) ) λ k (t)dt λ k (t) dt If A(ε), then k= ( ) ε = λ k (t)dt A(ε). k= However, λ k L 1 (a, b) means λ k (t)dt 1. k= λ k (t)dt as ε. k= This contradiction implies that there exists ε (, b) such that A( ε) =, ε ε. Integrating (5) on both sides, ( ε φ φ (n) (k) (t) ) (t) dt λ k (t) dt = A(ε) = Hence k= φ (n) (t) = a. e. t (, ε) = φ (n) (t) = a. e. t (, ε). Recall φ () () = u () () v () () =. Thus φ () (ε) = Repeating the argument results in φ (n) (t)dt = = φ () (t) = a. e. t (, ε). φ(t) = u(t) v(t) = Since u, v C (a, b) and u() = v(), we obtain φ(t) = u(t) v(t), a. e. t (, ε). t [, ε).

5 UNIQUENESS OF DIFFERENTIAL SYSTEMS WITH SINGULARITY 5 Let ε = max{ε : φ(t) =, t (, ε)}. If ε < b, then φ(t) = for t (, ε ] by the continuity of u and v. Then applying the above derivation to functions u(x ε ), v(x ε ) on the interval [ε, b) would yield u(t) v(t) = for t (ε, ε + ε ) for some ε >, which contradicts the definition of ε. Therefore we must have u(t) v(t) =, t [, b). To obtain the results on (a, ], replacing u(x), v(x), x (a, b) by u (x) = u( x), v (x) = v( x), x ( b, a) to obtain Combining the results we arrived at u (t) v (t) = u(t) v(t) = This concludes the proof for Theorem 1. t [, a). t (a, b). The proof of Theorem 2 is similar to the proof of Theorem 1. Proof of Theorem 2. Notice that y satisfies the assumptions in Lemma 3. Define ( ε y (k) (t) ) B(ε) = λ k (t) dt k= k= for any ε (, b). Applying (4) in Lemma 3 and then (3), y (k) (t) B(ε) = λ k (t) k= dt λ k (t)dt y (n) (t) dt k= ( ε y (k) (t) ) ( ) ε λ k (t)dt λ k (t) dt = λ k (t)dt B(ε). B(ε) would imply k= λ k (t)dt 1. k= On the other hand, λ k L 1 (a, b) implies λ k (t)dt as ε. k= Thus there must exist ε (, b) such that B( ε) =, ε ε. Integrating (3) on both sides, ( ε y y (n) (k) (t) ) (t) dt λ k (t) dt = B(ε) = which leads to Consequently, k= k= y (n) (t) = a. e. t (, ε) = y (n) (t) = a. e. t (, ε). y () (ε) = This argument leads to y (n) (t)dt = = y () (t) = a. e. t (, ε). y(t) = a. e. t (, ε).

6 6 UNIQUENESS OF DIFFERENTIAL SYSTEMS WITH SINGULARITY where a. e. can be removed by the continuity of y. The interval (, ε) on which y can be extended to (, b) and (a, ) using arguments analogous to the ones used in the proof of Theorem 1. This concludes the proof of Theorem Corollaries and an example When f in Theorem 1 is linear, the results can be stated as the corollary below. Corollary 4. Let a < < b and y : (a, b) R d be a solution of y (n) + a (x, y)y () + + a o (x, y)y =, x (a, b), where y () is absolutely continuous on (a, b), and the coefficient functions If then a k (x, y) λ k(x) x n k 1, λ k L 1 (a, b), k =, 1,, n 1. y() = y (1) () = = y () () =, y on (a, b). Theorem 1 is on uniqueness of nth order differential systems where the nth derivative of the solution only needs to exist almost everywhere, which is in the spirit of the classical Carathéodory theorem on the existence of ordinary differential equations. Theorem 1 for n = 1 can be stated as the following. Consider the differential equation If f : (a, b) R d R d satisfies y = f(x, y(x)), x (a, b), y(x o ) = y o. f(x, y 1 ) f(x, y 2 ) λ(x) y 1 y 2, a. e. x (a, b) where λ L 1 (a, b), λ, then the solution of the differential equation is unique. We conclude by an example to show the sharpness of the orders in (2) and (3). An example. For p (, 1/2), let y = e 1/ x p, x ( 1, 1), y (k) () =, k =,, n 1. Then y and its derivatives are even functions. for x (, 1), we have y = p y x p+1 y = p y y p(p + 1) xp+1 x p+2 y = p y y y 2p(p + 1) + p(p + 1)(p + 2) xp+1 xp+2 The general form can be written as y (m) (x) = m 1 k= C n k x p+3 y (k), x ( 1, 1) \ {}, m = 1,, n. (6) x m k+p

7 UNIQUENESS OF DIFFERENTIAL SYSTEMS WITH SINGULARITY 7 where Ck m are constants depending on n, k and p. Formula (6) can be verified by induction. Taking derivative of (6) for x (, 1), m 1 y (m+1) (x) = Ck m y (k+1) m 1 x m k+p + Ck m y (k) ( m + k p) x m k+p+1 (k = k + 1) = = k= m k= C m y (k ) m 1 k 1 x m+1 k +p + k =1 k= m k= C m+1 k y (k) x m+1 k+p Ck m ( m + k p) y (k) x m+1 k+p where C m+1 k are constants depending on n, k and p. Therefore (6) holds for all m = 1,, n. We may write the case of m = n as ( ) C y (n) n (x) k y (k) (x) x 1 p x n k 1+2p = y (k) (x) λ k (x), x ( 1, 1) \ {} x n k 1+2p k= k= where λ k L 1 ( 1, 1) for p (, 1/2). In (2) and (3), the order of singularity corresponding to y (k) is n k 1. In this example, the corresponding order is n k 1 + 2p, where p (, 1/2) can be arbitrarily small. However it is enough to make y on ( 1, 1). Alternatively, we may write ) y (k) (x) y (n) (x) k= ( C n k x 1+p x n k 1 = k= λ k(x) y(k) (x), x ( 1, 1) \ {} x n k 1 Now the order of singularity corresponding to y (k) is n k 1 as in (2) and (3), however λ k(x) = Cn k x 1+p L q ( 1, 1), q < 1, p (, 1/2). 1 + p Thus we do not have y as in the conclusions of the theorems. We are interested in the uniqueness of solutions ordinary differential systems when the coefficients are allowed to have singularities [9, 1]. In this paper, We give a Lipschitz type condition for the uniqueness of weak solutions in the style of the Carathéodory theorem for nonlinear nth order nonlinear ordinary differential systems. We also give a unique continuation condition for functions satisfying an nth order Grönwall differential inequality. We use an example to show the sharpness of the order of singularity required in the conditions. References [1] R. P. Agarwal and V. Lakshmikantham (1993). Uniqueness and nonuniqueness criteria for ordinary differential equations. World Scientific. Singapore. [2] Z. S. Athanassov(199). Uniqueness and convergence of successive approximations for ordinary dfferential equations, Math. Japon 35 2, [3] Z. S. Athanassov(21). On Nagumo s theorem, Proc. Japan Acad. 86 Ser. A, [4] (1998) On existence of oscillatory solutions of nth order differential equations with quasiderivatives, Archivum Mathematicum (Brno), Vol. 34: [5] E. A. Coddington and N. Levinson (1955). Theory of Ordinary Differential Equations, McGraw- Hill, New York. [6] B. C. Dhage, T. L. Holambe and S. K. Ntouyas (24). The method of upper and lower solutions for caratheodory n-th order differential inclusions. Electron. J. Diff. Eqns. 24(8): 1-9.

8 8 UNIQUENESS OF DIFFERENTIAL SYSTEMS WITH SINGULARITY [7] T. H. Grönwall (1919). Note on the derivative with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. 2 (4): [8] M. Nagumo(1926). Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung, Japan J. Math. 3, [9] Y. Pan and M. Wang (28). When is a function not flat? Journal of Mathematical Analysis and Applications 34 (1): [1] Y. Pan and M. Wang (29) A uniqueness result on ordinary differential equations with singular coefficients. Electron. J. Diff. Eqns. 29(56): 1-6. Yifei Pan, Department of Mathematical Sciences, Indiana University - Purdue University Fort Wayne, Fort Wayne, IN School of Mathematics and Informatics, Jiangxi Normal University, Nanchang, China address: pan@ipfw.edu Mei Wang, Department of Statistics, University of Chicago, Chicago, IL address: meiwang@galton.uchicago.edu