LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS

Electronic Journl of Differentil Equtions, Vol. 2017 (2017), No. 139, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS MUSTAFA FAHRI AKTAŞ, DEVRIM ÇAKMAK Dedicted to the memory of Aydın Tiryki (June 1956 - My 2016) Abstrct. In this rticle, we estblish new Lypunov-type inequlities for third-order liner differentil equtions y q(t)y = 0 under the three-point boundry conditions nd y() = y(b) = y(c) = 0 y() = y (d) = y(b) = 0 by bounding Green s functions G(t, s) corresponding to pproprite boundry conditions. Thus, we obtin the best constnts of Lypunov-type inequlities for three-point boundry vlue problems for third-order liner differentil equtions in the literture. 1. Introduction Lypunov [18] obtined the remrkble result: If q C([0, ), R) nd y is nontrivil solution of y q(t)y = 0 (1.1) under the Dirichlet boundry conditions nd y(t) 0 for t (, b), then y() = y(b) = 0 (1.2) 4 b q(s) ds. (1.3) Thus, this inequlity provides lower bound for the distnce between two consecutive zeros of y. The inequlity (1.3) is the best possible in the sense tht if the constnt 4 in the left hnd side of (1.3) is replced by ny lrger constnt, then there exists n exmple of (1.1) for which (1.3) no longer holds (see [14, p. 345], [16, p. 267]). In this pper, our im is to obtin the best constnts of Lypunov-type inequlities for third-order liner differentil equtions with three-point boundry conditions. The bove result of Lypunov hs found mny pplictions in res 2010 Mthemtics Subject Clssifiction. 34C10, 34B05, 34L15. Key words nd phrses. Lypunov-type inequlities; Green s Functions; three-point boundry conditions. c 2017 Texs Stte University. Submitted Jnury 5, 2017. Published My 24, 2017. 1

2 M. F. AKTAŞ, D. ÇAKMAK EJDE-2017/139 like eigenvlue problems, stbility, oscilltion theory, disconjugcy, etc. Since then, there hve been severl results to generlize the bove liner eqution in mny directions; see the references. There re vrious methods used to obtin Lypunov-type inequlities for different types of boundry vlue problems. One of the most useful methods is s follows: Nehri [21] strted with the Green s function of the problem (1.1) with (1.2), which is { (t )(b s) b, s t, nd he wrote G(t, s) = y(t) = (s )(b t) b, t s b, (1.4) G(t, s)q(s)y(s)ds. (1.5) Then by choosing t = t 0, where y(t) is mximized nd cnceling out y(t 0 ) on both sides, he obtined 1 mx t b G(t, s) q(s) ds. (1.6) Note tht if we tke the bsolute mximum vlue of the function G(t, s) for ll t, s [, b] in (1.6), then we obtin the inequlity (1.3). Following the ides of these ppers, this method hs been pplied in huge number of works to different second nd higher order ordinry differentil equtions with different types of boundry conditions. We see tht by bounding the Green s function G(t, s) in vrious wys, we cn obtin the best constnts in the Lypunov-type inequlities in other differentil equtions with ssocited boundry conditions s well. Thus, we obtin the best constnts of the Lypunov-type inequlities for three-point boundry vlue problems for third-order liner differentil equtions by using the bsolute mximum vlues of the Green s functions G(t, s) in the literture. In this rticle, we consider the third-order liner differentil eqution of the form y q(t)y = 0, (1.7) where q C([0, ), R) nd y(t) is rel solution of (1.7) stisfying the three-point boundry conditions y() = y(b) = y(c) = 0 (1.8) nd y() = y (d) = y(b) = 0, (1.9), b, c, d R with < b < c nd d b re three points nd y(t) 0 for t (, b) (b, c) nd t (, b), respectively. Some of the recent studies bout Lypunov-type inequlities for third nd higher order boundry vlue problems re s follows: In 1999, Prhi nd Pnigrhi [22] estblished the inequlities similr to the clssicl Lypunov inequlity (1.3) for the third-order liner differentil eqution (1.7) under the three-point boundry conditions (1.8) nd (1.9) s follows. Theorem 1.1 ([22, Theorem 2]). If y(t) is nontrivil solution of the problem (1.7) with (1.8), then 4 c (c ) 2 < q(s) ds. (1.10)

EJDE-2017/139 LYAPUNOV-TYPE INEQUALITIES 3 Theorem 1.2 ([22, Theorem 1]). If y(t) is nontrivil solution of the problem (1.7) with (1.9), then 4 b (b ) 2 < q(s) ds. (1.11) In 2010, Yng et l. [25] extended the inequlity (1.11) for the third-order liner differentil eqution (r 2 (t)(r 1 (t)y )) ) q(t)y = 0. (1.12) Theorem 1.3 ([25, Theorem 1]). If y(t) is nontrivil solution of (1.12) stisfying the conditions then min t 0 b < 2 [( t 0 q(s) ds, y() = y(b) = 0, (1.13) (r 1 (t)y (t)) hs zero d [, b], (1.14) t0 ) 1 ( ) 1 ] r 1 (s)ds r 2 (s)ds r 1 (s)ds r 2 (s)ds t 0 t 0 where y(t 0 ) = mx{ y(t) : t b}. (1.15) In 2013, Kiselk [17] extended the Lypunov-type inequlities from liner differentil eqution to the third-order hlf-liner differentil eqution ( 1 ( 1 ) ( y (t) p1 1 y p (t) 2 1 1 ) ) r 2 (t) r 1 (t) r 1 (t) y (t) p1 1 y (t) (1.16) q(t) y(t) p3 1 y(t) = 0, where 0 < p 1, p 2 nd p 3 = p 1 p 2. Theorem 1.4 ([17, Theorem 2.1]). If y(t) is nontrivil solution of (1.16) stisfying the conditions (1.13), nd ( 1 ) r 1 (t) y (t) p1 1 y (t) hs zero d [, b], (1.17) then [( t 0 ( t 0 min r 1/p1 1 (s)ds t 0 b ( ( r 1/p b 1 1 (s)ds t 0 where y(t 0 ) = mx{ y(t) : t b}. min t 0 [,c] ) 1/p1) r 1/p2 1 2 (s)ds ) r 1/p 1/p1 ) 1 ] 2 2 (s)ds t 0 ( < 2 ) (1.18) 1/p3, q(s) ds Theorem 1.5 ([17, Theorem 2.2]). If y(t) is nontrivil solution of the problem (1.16) with (1.8), then [( t 0 t0 ) 1 r 1/p1 1 (s)ds( r 1/p2 2 (s)ds) 1/p1 ( r 1/p 1 1 (s)ds t 0 ( < 2 q(s) ds ( ) 1/p1 ) 1 ] r 1/p 2 2 (s)ds (1.19) t 0 ) 1/p3,

4 M. F. AKTAŞ, D. ÇAKMAK EJDE-2017/139 where y(t 0 ) = mx{ y(t) : t c}. In 2014, Dhr nd Kong [13] obtined the following Lypunov-type inequlities for third-order hlf-liner differentil eqution (1.16). Theorem 1.6 ([13, Theorem 2.5]). If y(t) is nontrivil solution of (1.16) with (1.13) nd (1.17), then ( ) p3 ( ) p2 2 p2p3 r 1/p1 1 (s)ds r 1/p2 2 (s)ds (1.20) where < d q (s)ds d q (s)ds, q (t) = mx{ q(t), 0}, (1.21) q (t) = mx{q(t), 0}. (1.22) From (1.21) nd (1.22), it is esy to see tht q (t) q(t) q (t), q (t) q(t), q (t) q(t), nd q(t) = q (t) q (t) for t [, b]. Theorem 1.7 ([13, Theorem 2.6]). If y(t) is nontrivil solution of (1.16) with (1.8), then ( ) p3 ( ) p2 2 p2p3 r 1/p1 1 (s)ds r 1/p2 2 (s)ds (1.23) < mx d b [ d q (s)ds where d [, b] is given in (1.17), or 2 p2p3 ( b [ d < mx b d c r 1/p1 1 (s)ds b q (s)ds d ) p3 ( d ] q (s)ds, b ] q (s)ds, ) p2 r 1/p2 2 (s)ds where d [b, c] is given in (1.17). As result, ( ) p3 ( ) p2 2 p2p3 r 1/p1 1 (s)ds r 1/p2 2 (s)ds [ d < mx d c q (s)ds d ] q (s)ds, (1.24) (1.25) where d [, c], q (t), nd q (t) re given in (1.17), (1.21) nd (1.22), respectively. In 2016, Dhr nd Kong [12] obtined the following result for third-order liner differentil eqution (1.7). Theorem 1.8 ([12, Theorem 2.1]). If y(t) is nontrivil solution of (1.7) with (1.8), then one of the following holds: () 2 < (s )(c s)q (s)ds (b) 2 < (s )(c s)q (s)ds (c) 2 < (s )(b s)q (s)ds b (s b)(c s)q (s)ds, where q (t) nd q (t) re given in (1.21) nd (1.22), respectively.

EJDE-2017/139 LYAPUNOV-TYPE INEQUALITIES 5 In 2003, Yng [26] obtined the Lypunov-type inequlities for the following (2n 1)-th order differentil equtions for n N nd n-th order differentil equtions for n 2, n N, s follows. y (2n1) q(t)y = 0 (1.26) y (n) q(t)y = 0 (1.) Theorem 1.9 ([26, Theorem 1]). If y(t) is nontrivil solution of (1.26) stisfying the conditions y (i) () = y (i) (b) = 0 (1.28) for i = 0, 1,..., n 1 nd then y (2n) (t) hs zero d (, b), (1.29) n!2 n1 (b ) 2n < q(s) ds. (1.30) Theorem 1.10 ([26, Theorem 2]). If y(t) is solution of (1.) stisfying the conditions y() = y(t 2 ) = = y(t n 1 ) = y(b) = 0, (1.31) where = t 1 < t 2 < < t n 1 < t n = b y(t) 0 for t (t k, t k1 ), k = 1, 2,..., n 1, then (n 2)!n n 1 (b ) n 1 (n 1) n 2 < q(s) ds. (1.32) In 2010, Çkmk [8] obtined the following Lypunov-type inequlity for problem (1.) with (1.31) by fixing the fult in Theorem 1.10 given by Yng [26]. Theorem 1.11 ([8, Theorem 1]). If y(t) is nontrivil solution of (1.) with (1.31), then (n 2)!n n (b ) n 1 (n 1) n 1 < q(s) ds. (1.33) Recently, Dhr nd Kong [11] obtined Lypunov-type inequlities for odd-order liner differentil equtions for n N. y (2n1) ( 1) n 1 q(t)y = 0 (1.34) Theorem 1.12 ([11, Theorem 2.1]). If y(t) is nontrivil solution of (1.34) stisfying the conditions y (i1) () = y (i1) (c) = 0 (1.35) for i = 0, 1,..., n 1 nd then y(b) = 0 for b [, c], (1.36) (2n 1)!2 2n (c ) 2n S n < q(s) ds, (1.37)

6 M. F. AKTAŞ, D. ÇAKMAK EJDE-2017/139 where S n = B(α, β) = n 1 j j=0 k=0 1 0 2 2k 2j ( n 1 j j )( ) j B(n 1, n k j), (1.38) k z α 1 (1 z) β 1 dz the Bet function for α, β > 0. (1.39) Theorem 1.13 ([11, Theorem 2.2]). Assume tht y(t) is nontrivil solution of (1.34) with (1.35). () If y(b) = 0 for b (, c) nd y(t) 0 for t [, b) (b, c], then one of the following holds: (i) (2n 1)!22n (c ) 2n S n (ii) (2n 1)!22n (c ) 2n S n (iii) (2n 1)!22n (c ) 2n S n < q (s)ds < q (s)ds < q (s)ds b q (s)ds. (b) If y() = 0 nd y(t) 0 for t (, c], then (2n 1)!2 2n (c ) 2n S n < (c) If y(c) = 0 nd y(t) 0 for t [, c), then (2n 1)!2 2n (c ) 2n S n < q (s)ds. (1.40) q (s)ds, (1.41) where q (t), q (t), nd S n re given in (1.21), (1.22), nd (1.38), respectively. In this pper, we use Green s functions to obtin the best constnts of Lypunovtype inequlities for the problems (1.7) with (1.8) or (1.9) in the literture. In ddition, we obtin lower bounds for the distnce between two points of solution of the problems (1.7) with (1.8) or (1.9). 2. Some preliminry lemms We stte importnt lemms which we will use in the proofs of our min results. In the following lemm, we construct Green s function for the third order nonhomogeneous differentil eqution y = g(t) (2.1) with the three-point boundry conditions (1.8) inspired by Murty nd Sivsundrm [20] s follows. Lemm 2.1. If y(t) is solution of (2.1) stisfying y() = y(b) = y(c) = 0 with < b < c nd y(t) 0 for t (, b) (b, c), then y(t) = G c (t, s)g(s)ds, (2.2)

EJDE-2017/139 LYAPUNOV-TYPE INEQUALITIES 7 where, for t [, b], G c1 (t, s) = (s )2 (b t)(c t) 2(b )(c ), s < t b < c G c2 (t, s) = (t )2 (b s)(c s) 2(b )(c ) G c (t, s) = (s t)(t )[(b s)(c )(c s)(b )] 2(b )(c ), t s < b < c G c3 (t, s) = (c s)2 (b t)(t ) 2(c )(c b), < t < b < s < c nd for t [b, c], G c4 (t, s) = (s )2 (t b)(c t) 2(b )(c ), < s < b < t < c G G c (t, s) = c5 (t, s) = (c t)2 (s b)(s ) 2(c )(c b) (t s)(c t)[(s b)(c )(s )(c b)] 2(c )(c b), < b < s t c G c6 (t, s) = (c s)2 (t b)(t ) 2(c )(c b), < b t < s c. Proof. Integrting (1.7) from to t to find y, we obtin y (t) = d 2 t y (t) = d 1 d 2 (t ) t y(t) = d 0 d 1 (t ) d 2 (t ) 2 2 (2.3) (2.4) g(s) ds, (2.5) (t s)g(s) ds, (2.6) t (t s) 2 g(s) ds. (2.7) 2 Thus, the generl solution of (1.7) is (2.7). Now, by using the boundry conditions (1.8), we find the constnts d 0, d 1, nd d 2. Thus, y() = 0 implies d 0 = 0 nd y(b) = y(c) = 0 imply d 1 = d 2 = (c s) 2 (b ) g(s) ds 2(c )(c b) (b s) 2 g(s) ds (c b)(b ) (b s) 2 (c ) g(s) ds, (2.8) 2(c b)(b ) (c s) 2 g(s) ds. (2.9) (c )(c b) Substituting the constnts d 0, d 1, nd d 2 in the generl solution (2.7), we obtin y(t) = t ( (c s) 2 (t )(b t) 2(c )(c b) t b for t [, b] nd y(t) = ( (c s)2 (b t)(t ) 2(c )(c b) (c s) 2 (b t)(t ) g(s)ds 2(c )(c b) ( (c s) 2 (b t)(t ) 2(c )(c b) t b ( (c s)2 (b t)(t ) 2(c )(c b) (b s)2 (t )(c t) 2(b )(c b) (b s)2 (t )(c t) )g(s)ds 2(b )(c b) (b s)2 (t )(c t) 2(b )(c b) (t s)2 )g(s)ds 2 (t ) s)2 g(s)ds 2 (t ) s)2 g(s)ds 2 (2.10)

8 M. F. AKTAŞ, D. ÇAKMAK EJDE-2017/139 (c s) 2 (b t)(t ) g(s)ds t 2(c )(c b) for t [b, c]. This completes the proof. Consider the function G c (t, s) for t [, b]. It is esy to see tht 0 G c1 (t, s) G c1 (s) = (s )2 (b s)(c s) 2(b )(c ) G c1(s) = (s )2 (c s) 2 2(b )(c ) (2.11) for s < t b < c. Since the function G c1(s) tkes the mximum vlue t c 2, i.e. G c1(s) mx s b G c1(s) = G c1( c (c )3 ) = 2 32(b ). (2.12) Thus, 0 G c1 (t, s) G c1(s) (c )3 32(b ) for s < t b. Now, we consider 0 G c2 (t, s) for t s < b < c. Let g c1 (t, s) = (t )2 (b s)(c s) 2(b )(c ) nd g c2 (t, s) = (s t)(t )[(b s)(c )(c s)(b )] 2(b )(c ). We know tht mx G c2(t, s) t s<b mx g c1(t, s) mx g c2(t, s). (2.13) t s<b t s<b Thus, we find the mximum vlue of the functions g c1 (t, s) nd g c2 (t, s). It is esy to see tht from (2.12), 0 g c1 (t, s) G c1(s) (c )3 32(b ) for t s < b. Now, we find the bsolute mximum of g c2 (t, s). g c2 (t, s) tkes its mximum vlue t the point ( 4 2 b c 2bc (t 0, s 0 ) =, b c 22 4bc ), 3(2 b c) 3(2 b c) nd its mximum vlue is g c2 ( 42 b c 2bc 3(2 b c) Thus, we hve G c (t, s) min { (c ) 3 32(b ), (c ) 3 32(b ) 4 for t, s < b [1, 26]. Similrly, we obtin, b c 22 4bc ) = 4 3(2 b c) ( (c )(b ) ) 2 } = 2 b c 0 G c3 (t, s) G c3 (s) = (c s)2 (s ) 2 2(c )(c b) for < t < b < s < c. Therefore, we hve G c (t, s) { (c ) 3 )(b ) ((c ) 2. 2 b c (c )3 32(c b) (c )3 32(b ) (2.14) (2.15) 32(b ), t, s b, (c ) 3 32(c b), t < b < s < c (2.16) for t [, b]. Similrly, it is esy to see tht we hve { (c ) 3 32(b ) G c (t, s), < s < b < t, (c ) 3 32(c b), b t, s c, (2.17)

EJDE-2017/139 LYAPUNOV-TYPE INEQUALITIES 9 for t [b, c]. Now, we give nother importnt lemm. In the following lemm, we construct Green s function for the third order nonhomogeneous differentil eqution (2.1) with the three-point boundry conditions (1.9) inspired by Moorti nd Grner [19] s follows. Lemm 2.2 ([19, Tble 1]). If y(t) is solution of (2.1) stisfying y() = y (d) = y(b) = 0 with d b nd y(t) 0 for t (, b), then y(t) = G d (t, s)g(s) ds, (2.18) holds, where for d < s, G d1 (t, s) = (t )(b s)2 2(b ), t s b, G d (t, s) = G d2 (t, s) = (s )(b t)2 2(b ) (t s)(b t)(bs 2) 2(b ), s < t b, nd for s d, G d3 (t, s) = (t )2 (b s) 2(b ) G d (t, s) = (s t)(t )(2b s) 2(b ), t s b, G d4 (t, s) = (s )2 (b t) 2(b ), s < t b. Consider the function G d (t, s) for d < s. It is esy to see tht 0 G d1 (t, s) G d1 (s) = (s )(b s)2 2(b ) (2.19) (2.20) (2.21) for t s b. Since the function G d1 (s) tkes the mximum vlue t 2b 3, i.e. G d1 (s) mx G d1(s) = G d1 ( 2 b ) = s b 3 2(b )2. (2.22) Thus, 0 G d1 (t, s) G d1 (s) 2(b )2 for t s b. Now, we consider 0 G d2 (t, s) for s < t b. Let g d1 (t, s) = (s )(b t)2 2(b ) nd g d2 (t, s) = (t s)(b t)(bs 2) 2(b ). We know tht mx G d2(t, s) s<t b mx g d1(t, s) mx g d2(t, s). (2.23) s<t b s<t b Thus, we find the mximum vlue of the functions g d1 (t, s) nd g d2 (t, s). It is esy to see tht from (2.22), 0 g d1 (t, s) G d1 (s) 2(b )2 for s < t b. Now, we find the bsolute mximum of g d2 (t, s). g d2 (t, s) tkes its mximum vlue t the point (t 0, s 0 ) = ( 2 b, 4 b ), 3 3 nd its mximum vlue is Thus, g d2 ( 2 b 3 G d (t, s) min { 2(b ) 2,, 4 b ) = 3 2(b )2 4(b )2. 4(b )2 } 2(b ) 2 = (2.24)

10 M. F. AKTAŞ, D. ÇAKMAK EJDE-2017/139 for d < s. Similrly, it is esy to see tht we hve for s d. G d (t, s) 2(b )2 (2.25) Remrk 2.3. It is esy to see tht if we tke d = or d = b in Lemm 2.2, the problems (1.7) with (1.9) become two-point boundry vlue problems. 3. Min results Now, we give one of min results of this pper. Theorem 3.1. If y(t) is nontrivil solution of the problem (1.7) with (1.8), then where C q(s) ds, (3.1) C = min { 32(c b) 32(b ) }, (c ) 3. (c ) 3 Proof. Let y() = y(b) = y(c) = 0 where, b, c R with < b < c re three points, nd y is not identiclly zero on (, b) (b, c). From (2.2), (2.16), nd (2.17), we obtin y(t) G c (t, s) y (s) ds (3.2) (c ) 3 32(b ) y (s) ds b From (1.7) nd inequlity (3.3), we obtin y (t) = q(t) y(t) q(t) C (c ) 3 32(c b) y (s) ds 1 C Integrting from to c both sides of (3.4), we obtin Next, we prove tht y (s) ds 1 C 0 < If (3.6) is not true, then we hve y (s) ds y (s) ds. (3.3) y (s) ds. (3.4) q(s) ds. (3.5) y (s) ds. (3.6) y (s) ds = 0. (3.7) It follows from (3.3) nd (3.7) tht y(t) 0 for t (, c), which contrdicts with (1.8) since y(t) 0 for ll t (, c). Thus, by using (3.6) in (3.5), we obtin inequlity (3.1). Remrk 3.2. It is esy to see tht in the specil cses, inequlity (3.1) is shrper thn (1.10), (1.19), (1.25), (1.32), (1.33), nd (1.37) in the sense tht they follow from (3.1), but not conversely. Therefore, our result improves Theorems 1.1, 1.5, 1.7, 1.8, nd 1.10 1.13, in the specil cses. In fct, the Lypunov-type inequlity (3.1) is the best possibility for problem (1.7) with (1.8) in the sense tht the constnt 32 in the left hnd side of (3.1) cnnot be replced by ny lrger constnt.

EJDE-2017/139 LYAPUNOV-TYPE INEQUALITIES 11 Note tht we cn rewrite Green s functions G c (t, s) given in (2.3) nd (2.4) s follows: for t [, b], G c1 (s) = (s )2 (b s)(c s) 2(b )(c ), s < t b G c2 (s) = (s )2 (b s)(c s) 2(b )(c ) 0 G c (t, s) (3.8) (s )2 [(b s)(c )(c s)(b )] 2(b )(c ), t s < b G c3 (s) = (c s)2 (s ) 2 b < s < c 2(c )(c b), nd for t [b, c], G c4 (s) = (s )2 (c s) 2 2(b )(c ), < s < b G c5 (s) = (c s)2 (s b)(s ) 2(c )(c b) G c (t, s) (c s)2 [(s b)(c )(s )(c b)] 2(c )(c b), b < s t c G c6 (s) = (c s)2 (s b)(s ) 2(c )(c b), b t < s c. (3.9) Thus, for t [, b], we hve { 0 G c (t, s) mx Gc1 (s), G c2 (s), G c3 (s) } { = mx Gc2 (s), G c3 (s) } (3.10) s c s c nd for t [b, c], { G c (t, s) mx Gc4 (s), G c5 (s), G c6 (s) } { = mx Gc4 (s), G c5 (s) }. (3.11) s c s c Thus, from (3.10) nd (3.11), we obtin { G c (t, s) G c (s) = mx Gc2 (s), G c3 (s) } { or mx Gc4 (s), G c5 (s) } (3.12) s c s c for s [, c] [1, 26]. The proof of the following result proceeds s in Theorem 3.1 by using (3.12) insted of (3.2) nd hence it is omitted. Theorem 3.3. If y(t) is nontrivil solution of the problem (1.7) with (1.8), then 1 where G c (s) is given in (3.12). G c (s) q(s) ds, (3.13) Theorem 3.4. If y(t) is nontrivil solution of the problem (1.7) with (1.8), then 1 where y(t 0 ) = mx{ y(t) : t c}. G c (t 0, s) q(s) ds, (3.14) Proof. Let y() = y(b) = y(c) = 0 where, b, c R with < b < c re three points, nd y is not identiclly zero on (, b) (b, c). Pick t 0 (, c) so tht y(t 0 ) = mx{ y(t) : t c}. From (1.7) nd (2.2), we obtin y(t 0 ) = G c (t 0, s)[ q(s)y(s)]ds G c (t 0, s) q(s) y(s) ds (3.15)

12 M. F. AKTAŞ, D. ÇAKMAK EJDE-2017/139 nd hence y(t 0 ) y(t 0 ) Dividing both sides by y(t 0 ), we obtin inequlity (3.14). G c (t 0, s) q(s) ds. (3.16) Now, we give other min results of this pper under three-point boundry conditions (1.9). The proofs of following results re similr to tht of Theorems 3.1 3.4 nd hence they re omitted. Theorem 3.5. If y(t) is nontrivil solution of problem (1.7) with (1.9), then 2(b ) 2 q(s) ds. (3.17) Remrk 3.6. It is esy to see in the specil cses tht the inequlity (3.17) is shrper thn (1.11), (1.15), (1.18), (1.20), nd (1.30) in the sense tht they follow from (3.17), but not conversely. Therefore, our result improves Theorems 1.2 1.4, 1.6, nd 1.9 in the specil cses. In fct, the Lypunov-type inequlity (3.17) is the best possibility for the problem (1.7) with (1.9) in the sense tht the constnt 2 in the left hnd side of (3.17) cnnot be replced by ny lrger constnt. From (2.19) nd (2.20), it is esy to see tht for d < s, G d1 (s) = (s )(b s)2 2(b ), t s G d (t, s) G d2 (s) = (s )(b s)2 2(b ) (b s)2 (bs 2) 2(b ), s t nd for s d, G d (t, s) Therefore, we hve { Gd3 (s) = (s )2 (b s) 2(b ) (s )2 (2b s) 2(b ), t s b G d4 (s) = (s )2 (b s) 2(b ), s < t b. (3.18) (3.19) G d (t, s) G d (s) = mx s b {G d2(s), G d3 (s)}. (3.20) Theorem 3.7. If y(t) is nontrivil solution of the problem (1.7) with (1.9), then 1 where G d (t) is given in (3.20). G d (s) q(s) ds, (3.21) Theorem 3.8. If y(t) is nontrivil solution of the problem (1.7) with (1.9), then 1 where y(t 0 ) = mx{ y(t) : t b}. G d (t 0, s) q(s) ds, (3.22) We my dopt different point of view nd use (3.1) or (3.17) to obtin n extension of the following oscilltion criterion due originlly to Lipounoff (cf. [4]): y (t) nd y (t)y 1 (t) re continuous for t b, with y() = y(b) = 0, then 4 b < y (s)y 1 (s) ds. (3.23)

EJDE-2017/139 LYAPUNOV-TYPE INEQUALITIES 13 Thus, (3.1) or (3.17) leds to the following extension: If y (t) nd y (t)y 1 (t) re continuous for t c or t b, y(t) hs three points including, b, c, nd d, then C y (s)y 1 (s) ds or 2(b ) 2 y (s)y 1 (s) ds. (3.24) Now, we give nother ppliction of the obtined Lypunov-type inequlities for the eigenvlue problem y λh(t)y = 0 (3.25) under three points boundry conditions (1.8) or (1.9). Thus, if there exists nontrivil solution y(t) of liner homogeneous problem (3.25), then we hve C h(s) ds λ or 2(b ) λ. (3.26) 2 b h(s) ds References [1] R. P. Agrwl, A. Özbekler; Lypunov-type inequlities for n-th order forced differentil equtions with mixed nonlinerities, Commun. Pure Appl. Anl., 15 (2016), 2281-2300. [2] M. F. Aktş, D. Çkmk, A. Tiryki; Lypunov-type inequlity for qusiliner systems with nti-periodic boundry conditions, J. Mth. Inequl. 8 (2014), 313-320. [3] M. F. Aktş; Lypunov-type inequlities for n -dimensionl qusiliner systems, Elect. J. of Diff. Eq., 2013, No. 67 (2013), 1-8. [4] G. Borg; On Lipounoff criterion of stbility, Amer. J. of Mth. 71 (1949), 67-70. [5] A. Cbd, J. A. Cid, B. Mquez-Villmrin; Computtion of Green s functions for boundry vlue problems with Mthemtic, Appl. Mth. Comput., 219 (2012), 1919-1936. [6] D. Çkmk; On Lypunov-type inequlity for clss of nonliner systems, Mth. Inequl. Appl., 16 (2013), 101-108. [7] D. Çkmk, M. F. Aktş, A. Tiryki; Lypunov-type inequlities for nonliner systems involving the (p 1, p 2,..., p n)-lplcin, Elect. J. of Diff. Eq., 128 (2013), 1-10. [8] D. Çkmk; Lypunov-type integrl inequlities for certin higher order differentil equtions, Appl. Mth. Comput., 216 (2010), 368-373. [9] D. Çkmk, A. Tiryki; On Lypunov-type inequlity for qusiliner systems, Appl. Mth. Comput., 216 (2010), 3584-3591. [10] D. Çkmk, A. Tiryki; Lypunov-type inequlity for clss of Dirichlet qusiliner systems involving the (p 1, p 2,..., p n)-lplcin, J. Mth. Anl. Appl. 369 (2010), 76-81. [11] S. Dhr, Q. Kong; Lypunov-type inequlities for odd order liner differentil equtions, Elect. J. of Diff. Eq., 243 (2016), 1-10. [12] S. Dhr, Q. Kong; Lypunov-type inequlities for third-order liner differentil equtions, Mth. Inequl. Appl., 19 (2016), 297-312. [13] S. Dhr, Q. Kong; Lipunov-type inequlities for third-order hlf-liner equtions nd pplictions to boundry vlue problems, Nonliner Anl. 110 (2014), 170-181. [14] P. Hrtmn; Ordinry Differentil Equtions, Wiley, New York, 1964 n Birkhäuser, Boston 1982. [15] E. L. Ince; Ordinry Differentil Equtions, Dover Publictions, New York, 1926. [16] W. G. Kelley, A. C. Peterson; The Theory of Differentil Equtions, Clssicl nd Qulittive, Springer, New York 2010. [17] J. Kiselk; Lypunov-type inequlity for third-order hlf-liner differentil equtions, Tmkng J. Mth. 44 (2013), 351-357. [18] A. M. Lipunov; Probleme generl de l stbilite du mouvement, Ann. Fc. Sci. Univ. Toulouse,2 (1907), 203-407. [19] V. R. G. Moorti, J. B. Grner; Existence nd uniqueness theorems for three-point boundry vlue problems for third order differentil equtions, J. Mth. Anl. Appl., 70 (1979), 370-385. [20] K. N. Murty, S. Sivsundurm; Existence nd uniqueness of solutions to three-point boundry vlue problems ssocited with nonliner first order systems of differentil equtions, J. Mth. Anl. Appl., 173 (1993), 158-164.

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