Existence and Uniqueness of Anti-Periodic Solutions for Nonlinear Higher-Order Differential Equations with Two Deviating Arguments

Transcription

1 Advances in Dynamical Systems and Applications ISSN , Volume 7, Number 1, pp (1) Existence and Uniqueness of Anti-Periodic Solutions for Nonlinear Higher-Order Differential Equations with wo Deviating Arguments Xiaosong ang Jinggangshan University Department of Mathematics and Physics Ji an, Jiangxi, 3439, P.R.China Abstract In this paper, we use the Leray Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with two deviating arguments of the form x (n) (t) f(t, x (n 1) (t)) g 1 (t, x(t τ 1 (t))) g (t, x(t τ (t))) = e(t). As an application, we also give an example to demonstrate our results. AMS Subject Classifications: 34C5, 34D4. Keywords: Higher-order differential equations, deviating argument, anti-periodic solution, existence and uniqueness, Leray Schauder degree. 1 Introduction During the past twenty years, anti-periodic problems of nonlinear differential equations have been extensively studied by many authors, see [1 5] and references therein. For example, anti-periodic trigonometric polynomials are important in the study of interpolation problems [6, 7], and anti-periodic wavelets are discussed in [8]. Recently, anti-periodic boundary conditions have been considered for the Schrodinger and Hill differential operator [9, 1]. Also anti-periodic boundary conditions appear in the study of difference equations [11,1]. Moreover, anti-periodic boundary conditions appear in physics in a variety of situations [13 15]. Received May 4, 1; Accepted September 4, 1 Communicated by Adina Luminiţa Sasu

2 13 Xiaosong ang In this paper, we discuss the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with two deviating arguments of the form x (n) (t) f(t, x (n 1) (t)) g 1 (t, x(t τ 1 (t))) g (t, x(t τ (t))) = e(t). (1.1) where τ i, e : R R and f, g i : R R R are continuous functions, τ i and e are -periodic, f and g i are -periodic in their first arguments, n is an integer, > and i = 1,. Clearly, when n = and f(t, x(t)) = f(x(t)), (1.1) reduces to x (t) f(x (t)) g 1 (t, x(t τ 1 (t))) g (t, x(t τ (t))) = e(t), (1.) which has been known as the delayed Rayleigh equation with two deviating arguments. herefore, we can consider (1.1) as a high-order Rayleigh equation with two deviating arguments. he dynamic behaviors of Rayleigh equation and Rayleigh system have been widely investigated [16 19] due to their application in many fields such as physics, mechanics and the engineering technique fields. In such applications, it is important to know the existence of periodic solutions of Rayleigh equation, and some results on existence of periodic solutions were obtained in [ 3]. However, to the best of our knowledge, few authors have considered the existence and uniqueness of anti-periodic solutions for (1.1). his equation can stand for analog voltage transmission, and voltage transmission process is often an anti-periodic process. hus, it is worth continuing the investigation of the existence and uniqueness of anti-periodic solutions of (1.1). he main purpose of this paper is to establish sufficient conditions for the existence and uniqueness of anti-periodic solutions of (1.1). Our results are different from those of the references listed above. In particular, an example is also given to illustrate the effectiveness of our results. It is convenient to introduce the following assumptions: (A ) Assume that there exist nonnegative constants C 1 such that f(t, x 1 ) f(t, x ) C 1 x 1 x for all t, x 1, x R. (Ã) Assume that there exist nonnegative constants C such that f(t, u) = f(u), C x 1 x (x 1 x )(f(x 1 ) f(x )) for all x 1, x, u R. (A 1 ) for all t, x R, i = 1,, f (t ), x = f(t, x), g i (t ), x = g i (t, x), ( e t ) ( = e(t), τ i t ) = τ i (t).

3 Anti-Periodic Solutions for Nonlinear Differential Equations 131 Preliminary Results For convenience, we introduce a continuation theorem [4] as follows. Lemma.1. Let Ω be open bounded in a linear normal space X. Suppose that f is a complete continuous field on Ω. Moreover, assume that the Leray Schauder degree deg{ f, Ω, p}, for p X \ f( Ω). hen equation f(x) = p has at least one solution in Ω. Let u(t) : R R be continuous in t. u(t) is said to be anti-periodic on R if, u(t ) = u(t), u(t ) = u(t) for all t R. For ease of exposition, throughout this paper we will adopt the following notations: C k := {x C k (R, R), x is -periodic}, k {, 1,,...}, ( x q = ) 1 x(t) q q dt, x = max x(t), t [, ] x(k) = max t [, ] x(k) (t). { ( C k, 1 = x C k, x t ) } = x(t) for all t R which is a linear normal space endowed with the norm defined by x = max { x, x,..., x (k) } for all x C k, 1 t [, ] he following lemmas will be useful for proving our main results in Section 3. Lemma.. If x C (R, R) with x(t ) = x(t), then x (t). ( ) x (t). (.1) π Proof. Lemma. is known as Wirtinger inequality, and see [5] for the proof of it. Lemma.3. Suppose one of the following conditions is satisfied: (A ) (A ) holds and there exist nonnegative constants b 1, b such that C 1 π (b n 1 b ) < 1 and (π) n 1

4 13 Xiaosong ang g i (t, x 1 ) g i (t, x ) b i x 1 x for all t, x i R, i = 1, ; (A 3 ) (Ã) holds and there exist nonnegative constants b 1, b such that (b 1 b ) < C (π) n n 1 and g i (t, x 1 ) g i (t, x ) b i x 1 x for all t, x i R, i = 1,. hen (1.1) has at most one anti-periodic solution. Proof. Suppose that x 1 and x are two anti-periodic solutions of (1.1). hen, we have (x 1 (t) x (t)) (n) (f(t, x (n 1) 1 (t)) f(t, x (n 1) (t))) (g 1 (t, x 1 (t τ 1 (t))) g 1 (t, x (t τ 1 (t)))) (g (t, x 1 (t τ (t))) g (t, x (t τ (t)))) =. (.) Set Z(t) = x 1 (t) x (t). hen, from (.), we obtain Z (n) (t) (f(t, x (n 1) 1 (t)) f(t, x (n 1) (t))) (g 1 (t, x 1 (t τ 1 (t))) g 1 (t, x (t τ 1 (t)))) (g (t, x 1 (t τ (t))) g (t, x (t τ (t)))) =. (.3) Since Z(t) = x 1 (t) x (t) is an anti-periodic function on R, we have ( Z(t)dt = Z(t)dt Z(t)dt = Z t ) dt It follows that there exists a constant γ (, ) such that Z(t)dt =. (.4) Z( γ) =. (.5) hen, we have t Z(t) = Z( γ) t Z (s)ds Z (s) ds, t [ γ, γ ], (.6) and γ γ Z(t) = Z(t ) = Z( γ) γ t γ t Z (s) ds, t [ γ, γ ]. Combining the above two inequalities, we obtain Z = max Z(t) = max Z(t) t [, ] t [ γ, γ ] { ( 1 t Z (s) ds max t [ γ, γ ] 1 γ Z (s) ds 1 Z. γ t Z (s)ds )} Z (s) ds (.7) (.8)

5 Anti-Periodic Solutions for Nonlinear Differential Equations 133 Now suppose that (A ) (or(a 3 )) holds. We shall consider two cases as follows. Case (i) If (A ) holds, multiplying both sides of (.3) by Z (n) (t) and then integrating them from to, we have Z (n) = = C 1 Z (n) (t) dt (f(t, x (n 1) 1 (t)) f(t, x (n 1) (t)))z (n) (t)dt (g 1 (t, x 1 (t τ 1 (t))) g 1 (t, x (t τ 1 (t))))z (n) (t)dt (g (t, x 1 (t τ (t))) g (t, x (t τ (t))))z (n) (t)dt x (n 1) 1 (t) x (n 1) (t) Z (n) (t) dt b 1 x 1 (t τ 1 (t)) x (t τ 1 (t)) Z (n) (t) dt b x 1 (t τ (t)) x (t τ (t)) Z (n) (t) dt. From this, (.1), (.8) and the Schwarz inequality, we have ( ) 1 ( Z (n) C 1 x (n 1) 1 (t) x (n 1) (t) dt Z (n) (t) dt (b 1 b ) Z 1 Z (n) (t) dt C 1 Z (n 1) Z (n) (b 1 b ) Z ( C 1 Z (n 1) Z (n) (b 1 b ) Z Z (n) C 1 π Z(n) (b 1 b ) Z Z (n) ( C 1 π (b ) 1 b ) n Z (n) (π). n 1 It follows from C 1 π (b 1 b ) n < 1 that (π) n 1 ) 1 ) 1 ( ) 1 1 dt Z (n) (t) dt Z (n) (t) for all t R. (.9) Since Z (n ) () = Z (n ) ( ), there exists a constant ξ n 1 [, ] with Z (n 1) (ξ n 1 ) =, then, in view of (.9), we get Z (n 1) (t) for all t R. (.1)

6 134 Xiaosong ang By using a similar argument as in the proof of (.1), in view of (.5), we can show Z(t) Z (t)... Z (n ) (t) for all t R. hus, x 1 (t) x (t) for all t R. herefore, (1.1) has at most one anti-periodic solution. Case (ii) If (A 3 ) holds, multiplying both sides of (.3) by Z (n 1) (t) and then integrating them from to, together with (.8), we have C Z (n 1) = = = C x n 1 1 (t) x n 1 (t) dt (f(x n 1 1 (t)) f(x n 1 (t)))(x n 1 1 (t) x n 1 Z (n) (t)z (n 1) (t)dt g 1 (t, x (t τ 1 (t))))z (n 1) (t)dt g (t, x (t τ (t))))z (n 1) (t)dt (t))dt (g 1 (t, x 1 (t τ 1 (t))) (g (t, x 1 (t τ (t))) (g 1 (t, x 1 (t τ 1 (t))) g 1 (t, x (t τ 1 (t))))z (n 1) (t)dt (g (t, x 1 (t τ (t))) g (t, x (t τ (t))))z (n 1) (t)dt b 1 x 1 (t τ 1 (t)) x (t τ 1 (t)) Z (n 1) (t) dt b x 1 (t τ (t)) x (t τ (t)) Z (n 1) (t) dt (.11) (b 1 b ) Z Z (n 1) b 1 b n 1 (π) n Z(n 1). By using a similar argument as in the proof of Case (i), in view of (.5), (A 3 ) and (.11), we obtain Z(t) Z (t)... Z (n ) (t) for all t R. hus, x 1 (t) x (t) for all t R. herefore, (1.1) has at most one anti-periodic solution. he proof of Lemma.3, is now complete. Remark.4. If f (x) > C for all t R, one can see that f(x) satisfies the assumption (Ã).

7 Anti-Periodic Solutions for Nonlinear Differential Equations Main Results In this section, we establish some sufficient conditions for the existence and uniqueness of anti-periodic solutions for (1.1). heorem 3.1. Let condition (A 1 ) hold. Assume that condition (A ) or condition (A 3 ) is satisfied. hen (1.1) has a unique anti-periodic solution. Proof. Consider the auxiliary equation of (1.1) as the following: x (n) (t) = λf(t, x (n 1) (t)) λg 1 (t, x(t τ 1 (t))) λg (t, x(t τ (t))) λe(t) = λq(t, x(t), x (n 1) (t)), λ (, 1]. (3.1) By Lemma.3, together with (A ) and (A 3 ), it is easy to see that (1.1) has at most one anti-periodic solution. hus, to prove heorem 3.1, it suffices to show that (1.1) has at least one anti-periodic solution. o do this, we shall apply Lemma.1. Firstly, we will claim that the set of all possible anti-periodic solutions of (3.1) is bounded. Let x C n 1, 1 be an arbitrary anti-periodic solution of (3.1). hen, by using a similar argument as that in the proof of (.8), we have x 1 x. (3.) In view of (A ) and (A 3 ), we consider two cases as follows. Case (i) If (A ) holds, multiplying both sides of (3.1) by x (n) (t) and then integrating them from to, in view of (.1), (3.), (A ) and the inequality of Schwarz, we obtain x (n) = x (n) dt = λ f(t, x (n 1) (t))x (n) (t)dt λ g 1 (t, x(t τ 1 (t)))x (n) (t)dt λ g (t, x(t τ (t)))x (n) (t)dt λ f(t, x (n 1) (t)) f(t, ) f(t, ) x (n) (t) dt e(t)x (n) (t)dt [ g 1 (t, x(t τ 1 (t))) g 1 (t, ) g 1 (t, ) ] x (n) (t) dt [ g (t, x(t τ (t))) g (t, ) g (t, ) ] x (n) (t) dt e(t)x (n) (t)dt C 1 x (n 1) x (n) b 1 x(t τ 1 (t)) x (n) (t) dt

8 136 Xiaosong ang b x(t τ (t)) x (n) (t) dt g (t, ) ] x (n) (t) dt e(t) x (n) (t) dt [ f(t, ) g 1 (t, ) C 1 π x(n) (b 1 b ) x x (n) [max{ f(t, ) g 1 (t, ) g (t, ) : t } e ] x (n) C 1 π x(n) b 1 b x x (n) [max{ f(t, ) g 1 (t, ) g (t, ) : t } e ] x (n) C 1 π x(n) b 1 b n (π) n 1 x(n) [max{ f(t, ) g 1 (t, ) g (t, ) : t } e ] x (n), (3.3) which, together with (A ), implies that there exists a positive constant D 1 x (j) ( ) n j x n < D 1, j = 1,,..., n. (3.4) π Since x (j) () = x (j) ( )(j = 1,,..., n 1), it follows that there exists a constant η j [, ] such that and x (j1) (η j ) =, x (j1) (t) = x(j1) (η j ) x (j) (s)ds η j t x (j) (t) dt x (j), where j = 1,,..., n, t [, ]. ogether with (3.) and (3.4), (3.5) implies that there exists a positive constant D (3.5) x (j) x (j1) D, j = 1,,..., n 1, (3.6) which implies that, for all possible anti-periodic solutions x(t) of (3.1), there exists a constant M 1 such that max 1jn 1 x(j) < M 1. (3.7)

9 Anti-Periodic Solutions for Nonlinear Differential Equations 137 Case (ii) If (A 3 ) holds, multiplying both sides of (3.1) by x (n 1) (t) and then integrating them from to, in view of (.1), (3.), (A 3 ) and the inequality of Schwarz, we obtain C x (n 1) = C x (n 1) (t)x (n 1) (t)dt (f(x (n 1) (t)) f())x (n 1) (t)dt g 1 (t, x(t τ 1 (t)))x (n 1) (t)dt g (t, x(t τ (t)))x (n 1) (t)dt e(t)x (n 1) (t)dt f()x (n 1) (t)dt g 1 (t, x(t τ 1 (t))) g 1 (t, ) x (n 1) (t) dt g (t, x(t τ (t))) g (t, ) x (n 1) (t) dt e(t) x (n 1) (t) dt [ f() g 1 (t, ) g (t, ) ] x (n 1) (t) dt b 1 x(t τ 1 (t)) x (n 1) (t) dt b x(t τ (t)) x (n 1) (t) dt e(t) x (n 1) (t) dt [ f() g 1 (t, ) g (t, ) ] x (n 1) (t) dt (b 1 b ) x x (n 1) (t) [max{ f() g 1 (t, ) g (t, ) : t } e ] x (n 1) (t) b 1 b x x (n 1) (t) [max{ f() g 1 (t, ) g (t, ) : t } e ] x (n 1) (t) b 1 b n (π) n x(n 1) (t) [max{ f() g 1 (t, ) g (t, ) : t } e ] x (n 1) (t). (3.8)

10 138 Xiaosong ang his implies that there exists a constant D 1 > such that x (j) x (j1) D 1, j = 1,,..., n. (3.9) Multiplying x (n) (t) and (3.1) and then integrating it from to, by (A 3 ), (3.), (3.3), (3.9) and the inequality of Schwarz, we obtain x (n) = x (n) (t) dt g 1 (t, x(t τ 1 (t))) g 1 (t, ) x (n) (t) dt g (t, ) x (n) (t) dt e(t) x (n) (t) dt [ g 1 (t, ) g (t, ) ] x (n) (t) dt g (t, x(t τ (t))) b 1 x(t τ 1 (t)) x (n) (t) dt b x(t τ (t)) x (n) (t) dt e(t) x (n) (t) dt (b 1 b ) x x (n) (t) [max{ g 1 (t, ) [ g 1 (t, ) g (t, ) ] x (n) (t) dt g (t, ) : t } e ] x (n) (t) b 1 b x x (n) (t) [max{ g 1 (t, ) g (t, ) : t } e ] x (n) (t) b 1 b D 1 x (n) (t) [max{ g 1 (t, ) g (t, ) : t } e ] x (n) (t), it follows from (3.5) that there exists a positive constant D such that x (n 1) (t) x (n) D. (3.1) herefore, in view of (3.9) and (3.1), for all possible anti-periodic solutions x of (3.1), there exists a constant M 1 such that which, together with (3.7), implies that max 1jn 1 x(j) M 1, (3.11) max 1jn 1 x(j) M 1 M 1 1 := M. (3.1)

11 Anti-Periodic Solutions for Nonlinear Differential Equations 139 Set Ω = { } x C n 1, 1 = X : max 1jn 1 x(j) < M, then we know that (3.1) has no anti-periodic solution on Ω as λ (, 1]. Now, we consider the Fourier series expansion of a function x C n 1, 1, we have x(t) = i= Define an operator L : C k, 1 C k1, 1 [ π(i 1)t a i1 cos b i1 sin by setting ] π(i 1)t. (Lx)(t) = t = π x(s)ds π i= i= b i1 i 1 [ ai1 π(i 1)t sin (i 1) b i1 π(i 1)t cos (i 1) ]. (3.13) hen d(lx)(t) dt = x(t), and In view of (Lx)(t) x(s) ds π x π ( i= ( i= b i1 (i 1) i= ) 1 ( ) 1 1. (i 1) b i1 ) 1 1 (i 1) i= = π, (3.14) and the Parseval equality x(s) ds = (a i1 b i1), i=

12 14 Xiaosong ang we obtain ( (Lx)(t) x 4 (a i1 b i1) i= x ( ) 1 4 x(s) ds ( 4 ) x, l t [, ]. ) 1 (3.15) hus, (Lx)(t) ( ) x, and the operator L is continuous. 4 For all x C n 1, 1 Q 1 ( t, x ( t, from (A 1 ), we get ), x (n 1) ( t )) = Q 1 (t, x(t), x (n 1) (t)). herefore, Q 1 (t, x(t), x (n 1) (t)) C, 1. Define a operator F µ : Ω C n, 1 setting X by F µ (x) = L(... L(L(Q 1 (x)))) = µl n (Q 1 (x)), µ [, 1]. It is easy to see from the Arzela Ascoli lemma that F µ is a compact homotopy, and the fixed point of F 1 on Ω is the antiperiodic solution of (1.1). Define the homotopic continuous field as follows ogether with (3.1), we have H µ (x) : Ω [, 1] C n 1, 1, H µ (x) = x F µ (x). H µ ( Ω), µ [, 1]. Hence, using the homotopy invariance theorem, we obtain deg{x F 1 x, Ω, } = deg{x, Ω, }. By now we know that Ω satisfies all the requirement in Lemma.1, and then x F 1 x = has at least one solution in the Ω, i.e., F 1 has a fixed point on Ω. So, we have proved that (1.1) has a unique anti-periodic solution. his completes the proof. 4 Example and Remark In this section, we give an example to demonstrate the results obtained in previous sections.

13 Anti-Periodic Solutions for Nonlinear Differential Equations 141 Example 4.1. Let g 1 (t, x) = g (t, x) = (1 sin 4 (t)) 1 sin x. hen the Rayleigh 1π equation x (t) 1 8 x (t) 1 8 e sin t sin x (t) g 1 (t, x(t sin t)) g (t, x(t cos t)) = 1 (4.1) 6π cos t, has a unique anti-periodic solution with period π. Proof. From (4.1), we have f(t, x) = 1 8 x 1 8 e sin t sin x, then f(t, x 1 ) f(t, x ) 1 4 x 1 x for all t, x 1, x R. hus, b 1 = b = 1 6π, C 1 = 1 4, τ 1(t) = sin t, τ (t) = cos t and e(t) = 1 cost. It is 6π obvious that the assumptions (A 1 ) and (A ) hold. herefore, in view of heorem 3.1, (4.1) has a unique anti-periodic solution with period π. Remark 4.. Since there exist no results for the uniqueness of anti-periodic solutions of the nth-order differential equations with two deviating arguments. One can easily see that all the results in [1 3, 6] and the references therein can not be applicable to (4.1) to obtain the existence and uniqueness of anti-periodic solutions with periodic π. his implies that the results of this paper are essentially new. Acknowledgement he author would like to thank the referee for his or her careful reading and some comments on improving the presentation of this paper. References [1] H. Okochi, On the existence of periodic solutions to nonlinear abstract parabolic equations, J. Math. Soc. Japan 4(3), , [] A.R. Aftabizadeh, S. Aizicovici, N.H. Pavel, On a class of second-order antiperiodic boundary value problems, J. Math. Anal. Appl. 171, 31 3, 199. [3] S. Aizicovici, M. McKibben, S. Reich, Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities, Nonlinear Anal. 43, 33 51, 1. [4] Y. Chen, J.J. Nieto, D. ORegan, Anti-periodic solutions for fully nonlinear firstorder differential equations, Math. Comput. Modelling 46, , 7.

14 14 Xiaosong ang [5]. Chen, W. Liu, J. Zhang, M. Zhang, he existence of anti-periodic solutions for Lienard equations, J. Math. Study 4, , (in Chinese), 7. [6] F.J. Delvos, L. Knoche, Lacunary interpolation by antiperiodic trigonometric polynomials, BI 39, , [7] J.Y. Du, H.L. Han, G.X. Jin, On trigonometric and paratrigonometric Hermite interpolation, J. Approx. heory 131, 74 99, 4. [8] H.L. Chen, Antiperiodic wavelets, J. Comput. Math. 14, 3 39, [9] P. Djiakov, B. Mityagin, Spectral gaps of the periodic Schrodinger operator when its potential is an entire function, Adv. Appl. Math. 31, , 3. [1] P. Djiakov, B. Mityagin, Simple and double eigenvalues of the Hill operator with a two-term potential, J. Approx. heory 135, 7 14, 5. [11] A. Cabada, D.R. Vivero, Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations, Adv. Difference Equ. 4, 91 31, 4. [1] Y. Wang, Y.M. Shi, Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions, J. Math. Anal. Appl. 39, 56 69, 5. [13] A. Abdurahman, F. Anton, J. Bordes, Half-string oscillator approach to string field theory (Ghost sector: I), Nuclear Phys. B 397, 6 8, [14] C. Ahn, C. Rim, Boundary flows in general coset theories, J. Phys. A 3, 59 55, [15] H. Kleinert, A. Chervyakov, Functional determinants from Wronski Green function, J. Math. Phys. 4, , [16].A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orland, FL, [17] J.K. Hale, heory of Functional Differential Equations, Springer-Verlag, New York, [18] H.A. Antosiewicz, On nonlinear differential equations of the second order with integrable forcing term, J. Lond. Math. Soc. 3, 64 67, [19] G. Sansone, R. Conti, Nonlinear Differential Equations, MacMillan, New York, [] S. Lu, Z. Gui, On the existence of periodic solutions to p-laplacian Rayleigh differential equation with a delay, J. Math. Anal. Appl. 35, 685 7, 7.

15 Anti-Periodic Solutions for Nonlinear Differential Equations 143 [1] M. Zong, H. Liang, Periodic solutions for Rayleigh type p-laplacian equation with deviating arguments, Appl. Math. Lett. 6, 43 47, 7. [] F. Zhang, Y. Li, Existence and uniqueness of periodic solutions for a kind of duffing type p-laplacian equation, Nonlinear Anal.: RWA 9(3), , 8. [3] H. Chen, K. Li, D. Li, On the Existence of exactly one and two periodic solutions of Lienard equation, Acta Math. Sinica 47(3), , (in Chinese), 4. [4] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, [5] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge Univ. Press, London, [6] Qiyi Fan, Wentao Wang and Xuejun Yi. Anti-periodic solutions for a class of nonlinear nth-order differential equations with delays. Journal of Computational and Applied Mathematics 3, , 9.