Mathematical Inequalities - Chapter 5 ppsx

Chapter 5 Levin- and Lyapunov-Type Inequalities 5.1 Introduction The importance of basic comparison inequalities has been long recognized in the study of qualitative behavior of solutions of ordinary second-order differential equations. The history of these inequalities for continuous differential systems goes far back starting with the famous paper of Sturm [414] which gives inequal- ities for the zeros of solutions of linear second-order differential equations. In his fundamental work [201] Lyapunov has given one of the most basic and inspiring inequalities which provides a lower bound for the distance between consecutive zeros of the solutions of the linear second-order differential equation. Lyapunov’s inequality has become a versatile tool in the study of qualitative nature of solu- tions of ordinary second-order differential equations. Over the years there have appeared a number of generalizations, extensions, variants and applications re- lated to the basic Sturmain comparison theorem and the original Lyapunov in- equality. This chapter considers basic inequalities developed in the literature re- lated to the Sturmain comparison theorem and to the Lyapunov inequality which occupies a fundamental place in the theory of ordinary differential equations. 5.2 Inequalities of Levin and Others In this section we give results involving comparison of the solutions of linear and nonlinear second-order differential equations investigated by Levin [187], Kreith [171] and Ladas [177]. Here we consider only solutions which are defined on the whole interval of definition of the independent variable and their existence and uniqueness will be assumed without further mention. An oscillatory solution is (by definition) one which has arbitrary large zeros. 485 486 Chapter 5. Levin- and Lyapunov-Type Inequalities The basic Sturmain comparison theorem deals with functions u(x) and v(x) satisfying u  +c(x)u = 0, (5.2.1) v  +γ(x)v= 0. (5.2.2) If γ(x)  c(x), then solutions of (5.2.2) oscillate more rapidly than solutions of (5.2.1). More precisely, if u(x) is a nontrivial solution of (5.2.1) for which u(x 1 ) = u(x 2 ) = 0, x 1 <x 2 , and γ(x) c(x) for x 1  x  x 2 , then v(x) has a zero in (x 1 ,x 2 ]. In 1960, Levin [187] extended Sturm’s theorem in a direction somewhat dif- ferent from other earlier investigators. The method used by Levin involves the transformation of the differential equations (5.2.1), (5.2.2) into the Riccati equa- tions w  = w 2 +c(x), (5.2.3) z  = z 2 +γ(x), (5.2.4) by the substitutions w =−u  /u, z =−v  /v, respectively, and assuming that c(x) and γ(x)are continuous on [α, β]. In the following theorems we present the main comparison theorems estab- lished by Levin in [187]. T HEOREM 5.2.1. Let u and v be nontrivial solutions of (5.2.1) and (5.2.2), re- spectively, such that u(x) does not vanish on [α, β], v(α) = 0 and the inequality − u  (α) u(α) +  x α c(t) dt>     − v  (α) v(α) +  x α γ(t)dt     (5.2.5) holds for all x on [α, β]. Then v(x) does not vanish on [α, β] and − u  (x) u(x) >     v  (x) v(x)     ,α x  β. (5.2.6) The same theorem holds if the inequality signs in (5.2.5) and (5.2.6) are replaced by “”. P ROOF. Since u(x) does not vanish, w =−u  /u is continuous on [α, β] and sat- isfies the Riccati equation (5.2.3), which is equivalent to the integral equation w(x) =w(α) +  x α w 2 (t) dt +  x α c(t) dt. (5.2.7) 5.2. Inequalities of Levin and Others 487 By the hypothesis (5.2.5), w(x)  − u  (α) u(α) +  x α c(t) dt>0. (5.2.8) Since v(α) = 0, z =−v  /v is continuous on some interval [α, δ], α<δ β.On this interval, (5.2.4) is well defined and implies the integral equation z(x) =z(α) +  x α z 2 (t) dt +  x α γ(t)dt. (5.2.9) From (5.2.9), (5.2.5) and (5.2.8), we observe that z(x)  z(α) +  x α γ(t)dt > −w(α) −  x α c(t) dt  −w(x), and consequently, w(x) > −z(x). In order to show that   z(x)   <w(x) on α  x  δ, (5.2.10) it is sufficient to show that w(x) > z(x) on this interval. Suppose to the contrary that there exists a point x 0 on [α, δ] such that z(x 0 )  w(x 0 ). Then, since |z(α)| < w(α) from (5.2.5) (with x = α) and since w and z are continuous on [α, δ], there exists x 1 in α<x 1  x 0 such that z(x 1 ) =w(x 1 ) and z(x) < w(x) for α  x<x 1 . Since w(x) > −z(x) was established, it follows that |z(x)| <w(x) for α  x<x 1 , and consequently,  x 1 α z 2 (t) dt<  x 1 α w 2 (t) dt. Using (5.2.9), (5.2.5) and (5.2.7) yields z(x 1 ) = z(α) +  x 1 α γ(t)dt +  x 1 α z 2 (t) dt <w(α)+  x 1 α c(t) dt +  x 1 α w 2 (t) dt = w(x 1 ), contradicting z(x 1 ) =w(x 1 ). 488 Chapter 5. Levin- and Lyapunov-Type Inequalities Thus (5.2.10) holds on any interval [α, δ] of continuity of z, α<δ β, but this implies that z is continuous on the entire interval [α, β], since w(x) is bounded and z(x) has only poles at its points of discontinuity (if any). Thus (5.2.10) holds on all of the interval [α, β]. This result proves (5.2.6), and since the left member is bounded on [α, β], v(x) cannot have a zero on this interval. A slight modification of the proof shows that if “>” is replaced by “”inthe hypothesis (5.2.5), then the conclusion is still valid provided “>” is replaced by “” in (5.2.6). The proof is complete.  T HEOREM 5.2.2. Let u and v be nontrivial solutions of (5.2.1) and (5.2.2), re- spectively, such that u(x) does not vanish on [α, β], v(β) = 0, and the inequality u  (β) u(β) +  β x c(t) dt>     v  (β) v(β) +  β x γ(t)dt     (5.2.11) holds for all x on [α, β]. Then v(x) does not vanish on [α, β] and u  (x) u(x) >     v  (x) v(x)     ,α x  β. (5.2.12) The same result holds if “>” in (5.2.11) and (5.2.12) is replaced by “”. P ROOF. Let new functions u 1 ,v 1 ,c 1 ,γ 1 be defined on α  x  β by the equa- tions u 1 (x) = u(α +β −x), v 1 (x) =v(α + β −x), c 1 (x) = c(α +β −x), γ 1 (x) =γ(α+β −x). Then u 1 (x) does not vanish on [α, β], v 1 (α) =v(β) =0 and − u  1 (α) u 1 (α) +  α+β−x α c 1 (t) dt = u  (β) u(β) +  β x c(t) dt, − v  1 (α) v 1 (α) +  α+β−x α γ 1 (t) dt = v  (β) v(β) +  β x γ(t)dt. Thus the hypothesis (5.2.11) is equivalent to the hypothesis (5.2.5) of Theo- rem 5.2.1. Since x ∈[α, β] if and only if α +β − x ∈[α, β], and the conclusion (5.2.12) follows from Theorem 5.2.1.  5.2. Inequalities of Levin and Others 489 In 1972, Kreith [171] has given the Levin-type comparison theorems for the differential equations of the form u  −2b(x)u  +c(x)u = 0, (5.2.13) v  −2e(x)v  +γ(x)v = 0, (5.2.14) whose coefficients are assumed to be real and continuous, satisfying the initial conditions u  (x 1 ) +σu(x 1 ) = 0, (5.2.15) v  (x 1 ) +τv(x 1 ) = 0, (5.2.16) respectively, where σ and τ are finite constants. By means of the transformation w =− u  u ,z=− v  v , equations (5.2.13), (5.2.14) are transformed into Riccati equations w  = w 2 +2bw +c, (5.2.17) z  = z 2 +2ez +γ, (5.2.18) and the initial conditions − u  (x 1 ) u(x 1 ) =σ, − v  (x 1 ) v(x 1 ) =τ, (5.2.19) for (5.2.13), (5.2.14), become initial values w(x 1 ) =σ, z(x 1 ) =τ, (5.2.20) for (5.2.17) and (5.2.18). The differential equations (5.2.17) and (5.2.18) subject to (5.2.20) can be written as equivalent integral equations w(x) = σ +  x x 1 w 2 dt +  x x 1 2bw dt +  x x 1 c dt, (5.2.21) z(x) =τ +  x x 1 z 2 dt +  x x 1 2ez dt +  x x 1 γ dt. (5.2.22) It is obvious from these equations that if τ  σ  0, e(x)  b(x)  0 and  x x 1 γ(t)dt   x x 1 c(t) dt  0 490 Chapter 5. Levin- and Lyapunov-Type Inequalities on an interval [x 1 ,x 2 ], then z(x)  w(x)  0 as long as z(x) can be continued on [x 1 ,x 2 ]. Since the singularities of w(x) and z(x) correspond to the zeros of u(x) and v(x), respectively, these observations lead to the following comparison theorem for (5.2.13) and (5.2.14). T HEOREM 5.2.3. Suppose u(x) is a nontrivial solution of (5.2.13) satisfying −u  (x 1 )/u(x 1 ) =σ  0, u(x 2 ) =0. If (i) e(x)  b(x)  0 for x 1  x  x 2 , (ii)  x x 1 γ(t)dt   x x 1 c(t) dt  0 for x 1  x  x 2 , then every solution of (5.2.14) satisfying −v  (x 1 )/v(x 1 )  σ has a zero in (x 1 ,x 2 ]. In [171] Kreith has also given the variation of Theorem 5.2.3 which do not require the nonnegativity of σ, τ, b(x) and  x x 1 c(t) dt. We note that the integral equations (5.2.21) and (5.2.22) can be written as w(x) = σ +  x x 1 (w +b) 2 dt +  x x 1  c −b 2  dt, (5.2.23) z(x) =τ +  x x 1 (z +e) 2 dt +  x x 1  γ − e 2  dt. (5.2.24) This formulation shows that condition (ii) of Theorem 5.2.3 can be replaced by  x x 1  γ − e 2  dt   x x 1  c −b 2  dt  0. In order to obtain the generalization of Levin’s Theorem 5.2.1, Kreith [171] has given the following lemmas which are of independent interest. L EMMA 5.2.1. Let w(x) and z(x) be solutions of (5.2.23) and (5.2.24), respec- tively, for which σ>−∞ and (i) τ +  x x 1  γ − e 2  dt>     σ +  x x 1  c −b 2  dt     for x 1  x  x 2 , (ii) e(x)    b(x)   for x 1  x  x 2 . Then z(x)  |w(x)| as long as z(x) can be continued on [x 1 ,x 2 ]. 5.2. Inequalities of Levin and Others 491 P ROOF. From (5.2.24) we have z(x)  τ +  x x 1  γ − e 2  dt for x 1  x  x 2 . Using (i) and (5.2.23) this result implies that z(x) > −σ −  x x 1  c −b 2  dt > −σ −  x x 1  c −b 2  dt −  x x 1 (w +b) 2 dt>−w(x) for x 1  x  x 2 . It remains to show that z(x) > w(x). We assume to the contrary that there exists x 0 ∈ (x 1 ,x 2 ] such that z(x 0 )  w(x 0 ). Then there exists an ¯x ∈ (x 1 ,x 0 ] such that z( ¯x) = w( ¯x) and z(x) > |w(x)| for x 1  x<¯x. Using (ii) we have that z(x) +e(x) >   w(x)   +   b(x)      w(x) +b(x)   for x 1  x<¯x, and consequently that  ¯x x 1 (z + e) 2 dt>  ¯x x 1 (w + b) 2 dt. Using (5.2.24), (i) and (5.2.23) yields w(¯x) = σ +  ¯x x 1  c −b 2  dt +  ¯x x 1 (w +b) 2 dt <τ+  ¯x x 1  γ − e 2  dt +  ¯x x 1 (z +e) 2 dt = z( ¯x), which is a contradiction and establishes the lemma.  A continuity argument can be used to establish the following lemma. L EMMA 5.2.2. Let w(x) and z(x) be solutions of (5.2.23) and (5.2.24), respec- tively, for which σ>−∞ and (i) τ +  x x 1  γ − e 2  dt      σ +  x x 1  c −b 2  dt     for x 1  x  x 2 , (ii) e(x)    b(x)   for x 1  x  x 2 . Then z(x)  w(x) as long as z(x) can be continued on [x 1 ,x 2 ]. As an immediate consequence of Lemma 5.2.2 we have the following general- ization of Theorem 5.2.1. 492 Chapter 5. Levin- and Lyapunov-Type Inequalities THEOREM 5.2.4. Suppose u(x) and v(x) are nontrivial solutions of (5.2.13) and (5.2.14), respectively, and that u(x) =0 for x 1  x<x 2 , u(x 2 ) =0. If (i) − v  (x 1 ) v(x 1 ) +  x x 1  γ − e 2  dt      − u  (x 1 ) u(x 1 ) +  x x 1  c −b 2  dt     for x 1  x  x 2 , (ii) e(x)    b(x)   for x 1  x  x 2 , then v(x) has a zero in (x 1 ,x 2 ]. In 1969, Ladas [177] has established the following generalizations of Levin’s comparison theorems for the pair of nonlinear differential equations x  +p(t)g(x) = 0, (5.2.25) y  +q(t)g(y) = 0, (5.2.26) for t ∈[a,b], under some suitable conditions on the functions involved in (5.2.25) and (5.2.26). T HEOREM 5.2.5. Let the following conditions be satisfied: (i) p(t) and q(t) are real-valued continuous functions for t ∈[a,b]; (ii) g(s) is a real-valued continuous function for s ∈R such that g  (s)  0 for all s ∈[a,b], and g(0) =0; (iii) x(t) and y(t) are solutions of (5.2.25) and (5.2.26), respectively, such that x(t) = 0 for t ∈[a,b], y(a) = 0 and, for all t ∈[a,b], − x  (a) g(x(a)) +  t a p(s)ds>     − y  (a) g(y(a)) +  t a q(s)ds     (5.2.27) and g  (s)   s=x(t)  g  (s)   s=y(t) . (5.2.28) Then, for all t ∈[a,b], y(t) = 0 and − x  (t) g(x(t)) >     y  (t) g(y(t))     . P ROOF. Since x(t) = 0, it follows that g(x(t)) =0, t ∈[a,b]. Setting w(t) =− x  (t) g(x(t)) ,t∈[a,b], (5.2.29) 5.2. Inequalities of Levin and Others 493 it is easily verified that w satisfies the differential equation w  =p(t) +  d dx g  x(t)   w 2 ,t∈[a,b]. (5.2.30) The differential equation (5.2.30) will play the role of the Riccati equation to which a linear second-order differential equation is transformed by (5.2.29) in case g(s) ≡s. The rest of the proof, which we present for completeness, is an adaptation of Levin’s proof with minor modifications (see [177]). We set for simplicity d dx g(x(t)) = g  (x(t)). Integrating (5.2.30) over [a,t], t  b, and using (ii) and (iii) we obtain w(t) =w(a) +  t a p(s)ds +  t a g   x(s)  w 2 (s) ds  w(a) +  t a p(s)ds>0. (5.2.31) Since y(a) = 0, it follows from the continuity of y(t) that there is a closed interval [a,c], a<c b such that y(t) = 0, t ∈[a, c]. Then g(y(t)) = 0, t ∈ [a,b], and z(t) =−y  (t)/g(y(t)) satisfies the equation z  =q(t) +  d dy g  y(t)   z 2 ,t∈[a,c]. (5.2.32) Integrating (5.2.32) over [a,t], t  c, and using (ii), (iii) and (5.2.31) we obtain z(t) = z(a) +  t a q(s)ds +  t a g   y(s)  z 2 (s) ds  z(a) +  t a q(s)ds>−w(a) −  t a q(s)ds  −w(t). (5.2.33) We claim now that also z(t) < w(t), t ∈[a,c], (5.2.34) so that (5.2.33) and (5.2.34) would imply   z(t)    w(t), t ∈[a,c]. (5.2.35) 494 Chapter 5. Levin- and Lyapunov-Type Inequalities Indeed if (5.2.35) were false, there should exist a point t 1 ∈[a,c] such that z(t 1 ) = w(t 1 ) and w(t)  |z(t)| for t ∈[a,t 1 ]. (We used the fact that (5.2.27) for t = a gives w(a) > |z(a)|.) Then using (5.2.33), (iii) and (5.2.31) we obtain a contradiction z(t 1 ) = z(a) +  t 1 a q(s)ds +  t 1 a g   y(s)  z 2 (s) ds <w(a)+  t 1 a p(s)ds +  t 1 a g   x(s)  w 2 (s) ds = w(t 1 ). Therefore, inequality (5.2.35) is established for every interval [a,c] of conti- nuity of z(t).Butw(t) is bounded on [a,b] and z(t) can have only pole disconti- nuities on [a,b] so (5.2.35) holds throughout [a,b], that is, − x  (t) g(x(t)) >     y  (t) g(y(t))     ,t∈[a,b], and g(y(t)) = 0, that is, y(t) = 0. (Here we also used the uniqueness of solutions of (5.2.26).)  T HEOREM 5.2.6. Let in addition to the hypotheses (i) and (ii) of Theorem 5.2.5, the following condition be satisfied. (iii  ) x(t) and y(t) are solutions of (5.2.25) and (5.2.26), respectively, such that x(t) = 0, t ∈[a,b], y(b) =0 and, for all t ∈[a,b], x  (b) g(x(b)) +  b t p(s)ds>     y  (b) g(y(b)) +  b t q(s)ds     (5.2.36) and g  (s)   s=x(t)  g  (s)   s=y(t) . (5.2.37) Then, for all t ∈[a,b], y(t) = 0 and x  (t) g(x(t)) >     y  (t) g(y(t))     . P ROOF. It follows from Theorem 5.2.5 by setting −t + a +b in place of t.  In 1970, Bobisud [34] has established Levin-type results involving comparison of the solutions of nonlinear second-order differential equations and inequalities [...]... t ∈ [α, c] (5. 3.33) 50 3 5. 3 Levin-Type Inequalities Differentiating (5. 3.33) with respect to t and using (5. 3.26), it follows that z (t) = q(t) + v (t) f (v(t)) z2 (t), b(t) g(v (t)) Integrating (5. 3.34) over [α, t], t t z(t) = z(α) + (5. 3.34) c, we obtain t q(s) ds + α t ∈ [α, c] α v (s) f (v(s)) z2 (s) ds b(s) g(v (s)) (5. 3. 35) Using (ii), (iii), (iv), (5. 3.27) and (5. 3.32) in (5. 3. 35) , we observe... γ , and use monotonicity of f t ∈ [α, γ ] (5. 3.20) 50 0 Chapter 5 Levin- and Lyapunov-Type Inequalities From (5. 3.20), (5. 3. 15) and (5. 3.19), for t ∈ [α, γ ], we have t z(α) + K1 z(t) q(s) ds α t z(α) + q(s) ds α t > −w(α) − K1 q(s) ds α −w(t), that is, z(t) > −w(t) for t ∈ [α, γ ] (5. 3.21) z(t) < w(t) for t ∈ [α, γ ] (5. 3.22) Now we shall show that Suppose (5. 3.22) fails to hold for all t ∈ [α, γ ]...4 95 5.3 Levin-Type Inequalities of the forms y + a(t)f (y) 0 (5. 2.38) and y + p(t, y)y + g(t, y)y 0 (5. 2.39) under some suitable conditions on the functions involved in (5. 2.38) and (5. 2.39) Here we do not discuss the details 5. 3 Levin-Type Inequalities In this section we are concerned with Levin-type inequalities established by Lalli and Jahagirdar [180,181] and Pachpatte [327] for certain second-order... and Q(d) = 0 51 8 Chapter 5 Levin- and Lyapunov-Type Inequalities If d is the largest extreme point of y in [d, b), then y (t) 0 and thus r(t) 0 t for t ∈ [d, b) Set Q∗ = supd t b d q(s) ds By Lemma 5. 5.1, Q∗ > 0 and from (5. 5.7), 0 r(t) Q∗ + w(t) The proof of the second part of theorem now follows in a way similar to that of the first T HEOREM 5. 5.2 Let y denote a nontrivial solution of (5. 5.1) satisfying... sup a t c q(s) ds > 1 (5. 5.9) t Moreover, if there are no extreme values of y in (a, c), then c (c − a) sup q(s) ds > 1 (5. 5.10) a t c t The proof is similar to the proof of Theorem 5. 5.1 and is omitted C OROLLARY 5. 5.1 If t (b − d) sup d t b q(s) ds 1, q(s) ds 1, d then (5. 5.1) is right disfocal on [d, b) If c (c − a) sup a t c t then (5. 5.1) is left disfocal on (a, c] T HEOREM 5. 5.3 Let a and b denote... and I2 , of [α, β], we have (β − α) I1 ∪I2 q(s) ds 4 (5. 5.13) Then (5. 5.1) is disconjugate on [α, β] P ROOF Suppose the contrary, then there exists a nontrivial solution y of (5. 5.1) with y(a) = y(b) = 0 for α a < b β Without loss of generality we assume 52 0 Chapter 5 Levin- and Lyapunov-Type Inequalities that y(t) = 0 for t ∈ (a, b) By Theorem 5. 5.3, there exist two disjoint intervals, I1 and I2 ,... appeared in the literature In this section 50 6 Chapter 5 Levin- and Lyapunov-Type Inequalities we are concerned with inequalities related to Lyapunov’s inequality established by Hartman [1 45] , Patula [361], Kwong [176] and Harris [143] for second-order differential equations In [1 45, p 3 45] Hartman has given the following theorem T HEOREM 5. 4.1 Let q(t) be real-valued and continuous for a m(t) 0 be a... c 0 and x γ (s) ds Q(t) dt , exp 2 t 0 x x γ (s) ds dt exp 2 t 51 2 Chapter 5 Levin- and Lyapunov-Type Inequalities If 4A(c)B(c) < 1, then (5. 4.1) is right disfocal on [0, c) C OROLLARY 5. 4.4 If 4c sup0 cal on [0, c) x x c| 0 q(t) dt| < 1, then (5. 4.1) is right disfo- P ROOF We set γ (t) = 0 for t ∈ [0, c) in Theorem 4.4 .5 C OROLLARY 5. 4 .5 If B(c) = sup c x s q(r) dr ds dt exp 2 t 0 x c 0 0 and x A(c)... [0, c], then y has no zeros in [0, c] 51 6 Chapter 5 Levin- and Lyapunov-Type Inequalities 5. 5 Extensions of Lyapunov’s Inequality In this section we deal with inequalities similar to Lyapunov’s inequality established by Harris and Kong [144] and Brown and Hinton [46] Consider the linear second-order differential equation y + q(t)y = 0, (5. 5.1) where q is a real-valued function belonging to L1 In [144]... p(s) ds α w(t1 ), which is a contradiction to (5. 3.38) Thus, from (5. 3.36) and (5. 3.37), we have w(t) > z(t) , t ∈ [α, c] (5. 3.39) 50 4 Chapter 5 Levin- and Lyapunov-Type Inequalities Therefore, (5. 3.39) is true for every interval [α, c] of continuity of z(t) Since w(t) is bounded on I and z(t) can have only poles discontinuities on I , it follows that (5. 3.39) holds throughout I Thus f (v(t)) = 0, . (5. 2.33) We claim now that also z(t) < w(t), t ∈[a,c], (5. 2.34) so that (5. 2.33) and (5. 2.34) would imply   z(t)    w(t), t ∈[a,c]. (5. 2. 35) 494 Chapter 5. Levin- and Lyapunov-Type Inequalities Indeed. obtain z(t)  z(α) +K 1  t α q(s)ds, t ∈[α, γ ]. (5. 3.20) 50 0 Chapter 5. Levin- and Lyapunov-Type Inequalities From (5. 3.20), (5. 3. 15) and (5. 3.19), for t ∈[α, γ ],wehave z(t)  z(α) +K 1  t α q(s)ds . contradiction to (5. 3.38). Thus, from (5. 3.36) and (5. 3.37), we have w(t) >   z(t)   ,t∈[α, c]. (5. 3.39) 50 4 Chapter 5. Levin- and Lyapunov-Type Inequalities Therefore, (5. 3.39) is true