nhóm 2 khi đi trên xe đò có một cụ già phải đứng vì hết ghế ngồi em sẽ ứng xử như thế nào khi gặp tình huống trên vì sao em ứng xử như vậy em có ý kiến gì khi có người không nhường ghế
... T, ≤ t ≤ s ≤ T 2. 4 Remark 2. 4 Notice that it is not necessary to assume that v / 2. 5 In fact, if a ∈ Λ , then from 13, Remark in Page 3 328 , we have π T λ1 a ≥ 1− a p K 2p∗ > 0, 2. 6 where λ1 a ... theorem, see 15, Theorem 19.3 , we may get the desired results Remark 3.4 Theorem 3 .2 is a partial generalization of Lemma 3.1 It is enough to prove that the condition i on f in Theorem 3 .2 holds ... 122 9– 123 6, 20 08 D Jiang, J Chu, and M Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations,” Journal of Differential Equations, vol 21 1, no 2, pp 28 2–3 02, ...
... contradicts to Lemma 2. 3 Proof of main results Proof of Theorems 1.1 and 1 .2 We only prove Theorem 1.1 since the proof of Theorem 1 .2 is similar ν It is clear that any solution of (2. 4) of the form ... Lemma 2.2 Spr(λk L) < 1, which is impossible since Spr(L) = λk f0 ν Lemma 2. 5 If (µ, u) ∈ E is a non-trivial solution of (2. 4), then u ∈ Sk for ν and some k ∈ N ν Proof Taking into account Lemma ... form u = λAu (2. 1) Equations of the form (2. 1) are usually called nonlinear eigenvalue problems L´pez-G´mez [7] studied a o o nonlinear eigenvalue problem of the form u = G(r, u), (2. 2) where r...
... problem 2. 4 - 2. 5 Lemma 2. 1 see 12 The boundary value problem 2. 4 - 2. 5 is equivalent to integral equation 2 ut G t, s F s, J u s − ω ds, 2. 6 where G t, s ⎧ ⎪ 2e ρ /2 ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ 2e ρ /2 ⎪ ⎪ ... t0 2 dt n 3.9 for some t0 ∈ 0, 2 Then t un max un t 0≤t 2 max 0≤t 2 t0 t max 0≤t 2 ≤ Fn s, J un s Fn s, J un s 2 un s ds −ω t0 222 un s ds −ω 2 n 2 ds n ρ un t − un t0 ρun s − 2 un ... ⎪ ⎩ t−s sin 2 t−s √ e−ρπ sin 3 /2 ρ t − s , ≤ s ≤ t ≤ 2 , √ √ 3ρ eρπ e−ρπ − cos 3ρπ √ √ sin 3 /2 ρ s − t e−ρπ sin 3 /2 ρ 2 − s t , ≤ t ≤ s ≤ 2 √ √ 3ρ eρπ e−ρπ − cos 3ρπ 2. 7 √ 3 /2 ρ 2 − t s Moreover,...
... (2. 3) Thus ≤ u(k) ≤ M n0 for k ∈ N + (2. 20) Let β(k) ≡ u(k) for k ∈ N + Now (2. 7) and (2. 20) guarantee α(k) ≤ β(k) for k ∈ N + (2. 21) Now (2. 2) and (2. 20) imply f (k,β(k)) ≤ h(β(k)) ≤ h(M) ... k + 1) (2. 27) j =k Thus (2. 3) holds Theorem 2. 1 guarantees that (2. 23) has a solution u ∈ C(N + , R) with u(k) > for k ∈ N Next we present a result for initial value problems Theorem 2. 3 Let ... ϕ p β(k − 1) (2. 22) Now Theorem 1 .2 guarantees that (1.1) has a solution u ∈ C(N + , R) with u(k) ≥ α(k) > for k ∈ N 21 0 Discrete initial and boundary value problems Example 2.2 Consider the...
... in Theorem 3 .2 are satisfied Therefore by Theorem 3 .2 BVP 3.70 , 3.71 has a unique solution x x t satisfying 2 ≤ x t ≤ 2, 2t ≤ x t ≤ −2t, t ∈ −1, , −t2 ≤ x t ≤ t2 , t ∈ −1, , −2t ≤ x t ≤ 2t, t ... no 2, pp 22 7 23 2, 20 02 22 W L Zhao, “Existence and uniqueness of solutions for third order nonlinear boundary value problems,” The Tohoku Mathematical Journal, vol 44, no 4, pp 545–555, 19 92 ... λf t, w0 t, x0 , , wn 2 t, xn 2 , xn−1 xn 2 − λwn 2 t, xn 2 φ |xn−1 | ≤ 3 .25 2Mn 2 φ |xn−1 | : φE |xn−1 | Furthermore, we obtain ∞ s φE s ∞ ds s ds 2Mn 2 φ s ∞ 3 .26 Thus, Fλ satisfies the...
... sequence K i 1/2iα , i ≥ 1, if and only if either ∞ A> i 1 2iα ςα 4 .29 or A < 2 − ∞ i −1 i 2iα 2 − 2 −1 −1 ∞ i 1 2iα where ς is the well-known Riemann function Proofs of the main theorems 5.1 Proof ... 1/n n c i 4 .21 q c i is the extended binomial i∈Z 4 .22 and by using the well-known properties of the μ− i 1 c 2 μ c−1 , if μ ≥ 4 .23 c2c−1 > 1, therefore by statement α of Theorem 3.3 we get ... of Theorem 3.1 To prove Theorem 3.1 we need the next result from 20 2 − 2 −1 −1 ς α , 4.30 I Gy˝ ri and L Horv´ th o a 11 Theorem A Let us consider the initial value problem 3.1 , 3 .2 Suppose...
... (η) (2. 21) here we have used the fact pη,τ (1) = q1,τ (η) By (2. 20) and (2. 21), we have Lτ (η) ≥ q1,τ (η) p0,τ (η) , q1,τ (0) + q1,τ (η) p0,τ (1) + p0,τ (η) Lτ x (2. 22) By (2. 17) and (2. 22) , ... Theorem 2. 1 cannot be obtained by Theorems 1.1–1.4 and the abstract results in [ 12] Remark 2. 4 The nonlinear term f was assumed to be nondecreasing in Theorems 1 .2 and 1.4, but in Theorem 2.2 in ... any t1 ,t2 ∈ [0,η], |t1 − t2 | < δ, η −δ R0 δ1 ε G(τ) t1 ,s − G(τ) t2 ,s a(s)ds < , [0,η] [0,η] R0 p0,τ t2 − p0,τ t1 p0,τ (η) ε < (2. 25) By (2. 24)– (2. 25), we have for any x ∈ B and t1 ,t2 ∈ [0,η],...
... j) em k −m ω( j) = B j =1 B(k−2m− j) em ω( j) + f (k) (3 .26 ) j =1 Since eB(−m) ≡ I and m k B(k−2m− j) em j =1 k −m ω( j) = B(k−2m− j) em j =1 k ω( j) + B(k−2m− j) em j =k−m+1 ω( j) , (3 .27 ) 12 ... Δ em k −m ω( j) = B B(k−2m− j) em j =1 ω( j) + f (k) j =1 (3 .24 ) Using formula (2. 8) we get B(k−m− j) em B(k−2m− j) = Bem , (3 .25 ) and the last relation becomes eB(−m) ω(k + 1) + B m k B(k−2m− ... · j (2. 24) Finally due to (2. 21), ΔeBk = B I + m Bj · j =1 k − m − ( j − 1)m j B(k = Bem −m) (2. 25) and formula (2. 16) is proved Remark 2.2 Analyzing the formula (2. 8) we conclude that the discrete...
... no 2, pp 22 7 23 7, 20 03 [14] A O Ignatyev and O A Ignatyev, “On the stability in periodic and almost periodic difference systems,” Journal of Mathematical Analysis and Applications, vol 313, no 2, ... periodic system,” Funkcialaj Ekvacioj, vol 12, pp 23 –40, 1969 [2] R P Agarwal, Difference Equations and Inequalities, vol 22 8 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel ... York, NY, USA, 2nd edition, 20 00 [3] C T H Baker and Y Song, “Periodic solutions of discrete Volterra equations,” Mathematics and Computers in Simulation, vol 64, no 5, pp 521 –5 42, 20 04 [4] C Cuevas...
... 2F(s) f (s) f (s) 2 = + O(1)s−β (2. 8) Multiplying by (2F(s))−1 /2 we find − 2F(s) −1 /2 − + 2F(s) 2F(s) 1 /2 1 /2 f (s) f (s) −1 f (s) 2 = 2F(s) = 2F(s) −1 /2 −1 /2 + O(1) 2F(s) + O(1) 2F(s) −1 /2 ... s ∞ s −1 /2 2F(t) 2F(t) dt −1 /2 −β t dt −1 /2 −β s 2F(s) = lim s→∞ + βs−β−1 = + lim β s→∞ = + lim s→∞ −1 /2 2F(t) s s 2F(s) −β 2F(t) −1 /2 dt −1 /2 −β s 2F(s) ∞ ∞ s dt (2. 11) −1 /2 −1 − s 2F(s) f (s) ... f (s) 2 = + O(1)e−s (3.6) Claudia Anedda et al Multiplying by (2F(s))−1 /2 we find − 2F(s) −1 /2 − + 2F(s) 1 /2 2F(s) 1 /2 2 f (s) f (s) −1 f (s) = 2F(s) = 2F(s) −1 /2 −1 /2 −1 /2 −s + O(1) 2F(s) +...
... Vietnam J Math 29 (20 01), no 3, 28 1 28 6 M S Berger, Nonlinearity and Functional Analysis, Lectures on Nonlinear Problems in Mathematical Analysis Pure and Applied Mathematics, Academic Press, New ... systems of difference equations, J Difference Equ Appl (20 02) , no 12, 1085–1105 R M¨ rz, On linear differential-algebraic equations and linearizations, Appl Numer Math 18 a (1995), no 1–3, 26 7 29 2 ... following lemma Lemma 2. 1 The matrix Gn := An + Bn Qn−1,n is nonsingular if and only if Sn ∩ KerAn−1 = {0}, (2. 3) where, as in the DAE case, Sn := {ξ ∈ Rm : Bn ξ ∈ ImAn } The proof of Lemma 2. 1 repeats...
... solution of (1.1), then (2. 2) holds and un ∆un < for large n Proof Without loss of generality, assume u eventually positive If condition (2. 2) does not hold, from Lemma 2.2 we obtain lim un = n ... contradiction with (3 .2) The second statement follows from Lemma 2. 2(i) The following uniqueness result will play an important role in our later consideration Theorem 3.4 Assume (1.10) For any fixed c ∈ R ... where Φ(u) = u2 sgnu and an = n(n + 1)(n + 2) 2 , bn = 8(n + 1)(n + 2) n (n + 1)(n + 2) − (4.10) We have Φ∗ 1 < , = ∗ n(n + 1) an Φ n(n + 1)(n + 2) (4.11) Mariella Cecchi et al 20 3 so ∞ 1/Φ∗ (an...
... (2. 0 .2) Here the raised dot denotes matrix multiplication, A is the matrix of coefficients, and b is the right-hand side written as a column vector, a11 a21 A= aM a 12 a 22 ··· aM a1N a2N ... symmetric positive-definite ( 2. 9), tridiagonal ( 2. 4), band diagonal ( 2. 4), Toeplitz ( 2. 8), Vandermonde ( 2. 8), sparse ( 2. 7) • Strassen’s “fast matrix inversion” ( 2. 11) 36 Chapter Solution ... call 1-800-8 72- 7 423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) Some other topics in this chapter include • Iterative improvement of a solution ( 2. 5) • Various...
... + a 22 ) × (b11 + b 22 ) Q2 ≡ (a21 + a 22 ) × b11 Q3 ≡ a11 × (b 12 − b 22 ) Q4 ≡ a 22 × (−b11 + b21 ) Q5 ≡ (a11 + a 12 ) × b 22 Q6 ≡ (−a11 + a21 ) × (b11 + b 12 ) Q7 ≡ (a 12 − a 22 ) × (b21 + b 22 ) (2. 11.3) ... they are written explicitly: c11 = a11 × b11 + a 12 × b21 c 12 = a11 × b 12 + a 12 × b 22 c21 = a21 × b11 + a 22 × b21 (2. 11 .2) c 22 = a21 × b 12 + a 22 × b 22 Do you think that one can write formulas for ... a seemingly simple question: How many individual multiplications does it take to perform the matrix multiplication of two × matrices, a11 a21 a 12 a 22 · b11 b21 b 12 b 22 = c11 c21 c 12 c 22 (2. 11.1)...