QUALITATIVE PROPERTIES OF A THIRD-ORDER DIFFERENTIAL EQUATION WITH A PIECEWISE CONSTANT ARGUMENT

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1 Electronic Journl of Differentil Equtions, Vol (2017), No. 193, pp ISSN: URL: or QUALITATIVE PROPERTIES OF A THIRD-ORDER DIFFERENTIAL EQUATION WITH A PIECEWISE CONSTANT ARGUMENT HUSEYIN BEREKETOGLU, MEHTAP LAFCI, GIZEM S. OZTEPE Abstrct. We consider third order differentil eqution with piecewise constnt rgument nd investigte oscilltion, nonoscilltion nd periodicity properties of its solutions. 1. Introduction For mny yers, oscilltion, non-oscilltion nd periodicity of third order differentil equtions hve been investigted. Kim [15] studied oscilltion properties of the eqution y + py + qy + ry = 0, where p, q nd r re continuous on n intervl. Tryhuk [24] estblished sufficient conditions for the existence of two linerly independent oscilltory solutions of the third order differentil eqution y + p(t)y + q(t)y = 0. Cecchi [6] investigted the oscilltory behvior of the liner third-order differentil eqution of the form y + p(x)y + q(x)y = 0, where the function p(x) chnges sign on the positive x-xis. Prhi nd Ds [18] considered the eqution (r(t)y ) + q(t)y + p(t)y = F (t), nd gve necessry nd sufficient conditions for the existence of nonoscilltory or oscilltory solutions of this eqution. In [19], they lso investigted oscilltory nd symptotic properties of solutions of the eqution y + (t)y + b(t)y + c(t)y = 0, where C 2, b C 1, c C 0, (t), b(t), c(t) 0 eventully nd b(t) 0, c(t) 0 on ny intervl of positive mesure. The oscilltion of the solutions of (b(t)((t)y (t)) ) + (q 1 (t)y(t)) + q 2 (t)y (t) = Mthemtics Subject Clssifiction. 34K11. Key words nd phrses. Third order differentil eqution; piecewise constnt rgument; oscilltion; periodicity. c 2017 Texs Stte University. Submitted July 7, Published August 4,

2 2 H. BEREKETOGLU, M. LAFCI, G. S. OZTEPE EJDE-2017/193 nd (b(t)((t)y (t)) ) + q 1 (t)y(t) + q 2 (t))y(τ(t)) = 0 ws studied in [8] by Dhiy. Admets nd Lomttidze [1] nlyzed oscilltory properties of solutions of the third-order differentil eqution u + p(t)u = 0, where p is loclly integrble function on [0, ) which is eventully of one sign. Hn, Sun nd Zhng [13] deduced new sufficient conditions which gurntee tht every solution x of the delyed third order differentil eqution (x(t) (t)x(τ(t))) + p(t)x(δ(t)) = 0 is either oscilltory or tends to zero. In [9], the uthors stted necessry nd sufficient conditions for the oscilltion of the third-order nonhomogenous differentil eqution y + (t)y + b(t)y + c(t)y = f(t), under certin conditions given in terms of differentibility, continuity nd signs of the coefficient functions nd their derivtives. [10] ws dedicted to studying the nonoscilltory solutions of the eqution with mixed rguments ((t)(x (t)) γ ) = q(t)f(x(τ(t))) + p(t)g(x(σ(t))), where τ(t) < t, σ(t) > t. In 2015, Brtušek nd Došlá [2] gve conditions under which every solution of the eqution x (t) + q(t)x (t) + r(t) x λ (t) sgn x(t) = 0, t 0, is either oscilltory or tends to zero. They lso studied Kneser solutions vnishing t infinity nd the existence of oscilltory solutions. Shoukku [21] considered m y (t) (t)y (t) b(t)y (t) c i (t)y(σ i (t)) = 0 i=1 using Riccti inequlity. Ezeilo [11] studied the eqution x + x + bx + h(x) = p(t), where, b re constnts, p(t) is continuous nd periodic with lest period w. Using the Lery-Schuder technique, under certin conditions on, b, h, p, the uthor gurnteed the existence of one solution of this eqution with lest period w. Tbuev In [22] studied the existence of periodic solution of Ezeilo [12] showed tht the eqution x + αx + βx + sin x = e(t). x + ψ(x )x + φ(x)x + θ(x) = p(t) + q(t, x, x ) hs n w-periodic solution, where ψ, φ, θ, p nd q re continuous in their respective rguments nd p, q hve given period w, w > 0, in t. In 1979, Tejumol [23] proved the existence of t lest one w periodic solution of the third order differentil eqution x + f(x )x + g(x)x + h(x) = p(t, x, x, x ), where p is w-periodic in its first rgument. In [25], the uthor gve theorem on the existence of 2π-periodic solutions of the nonliner third order differentil eqution with multiple deviting rguments 2 c(t)x (t)+ [ i (x (i) ) 2k 1 +b i (x (i) ) 2k 1 (t τ i )]+g(t, x(t τ(t)), x (t τ 3 )) = p(t), i=0

3 EJDE-2017/193 THIRD-ORDER DIFFERENTIAL EQUATIONS 3 where i, b i (i = 0, 1, 2) nd τ i (i = 0, 1, 2, 3) re constnts, k is positive integer. Chen nd Pn [7] proved sufficient conditions for the existence of periodic solutions of third order differentil equtions with deviting rguments of the type x (t) + x (t τ 2 (t)) + bx (t τ 1 (t)) + cx(t) + f(t, x(t τ(t)) = p(t). As fr s we know, there re some ppers on the third-order differentil equtions with piecewise constnt rguments. The oldest one ws published in 1994 by Ppschinopoulos nd Schins [17]. They considered the eqution (y(t) + py(t 1)) = qy(2[ t ]) nd proved existence, uniqueness nd symptotic stbility of the solutions. Here t [0, ), p, q re rel constnts nd [ ] denotes the gretest integer function. Ling nd Wng [16] stted severl sufficient conditions which insure tht ny solution of the eqution (r 2 (t)(r 1 (t)x (t)) ) + p(t)x (t) + f(t, x([t])) = 0, t 0 oscilltes or converges to zero. Sho nd Ling [20] estblished sufficient conditions for the oscilltion nd symptotic behviour of the eqution (r(t)x (t)) + f(t, x([t])) = 0. In [5], the uthors showed tht every solution x(t) of third-order nonliner differentil eqution with piecewise constnt rguments of the type (r 2 (t)(r 1 (t)x (t)) ) + p(t)x (t) + f(t, x([t 1])) + g(t, x([t])) = 0 oscilltes or converges to zero, where t 0, r 1 (t), r 2 (t) re continuous on [0, ) with r 1 (t), r 2 (t) > 0 nd r 1(t) 0, p(t) is continuously differentible on [0, ) with p(t) 0. On the other hnd, the first nd third uthors considered n impulsive first order dely differentil eqution with piecewise constnt rgument in [14]. They investigted its oscilltory nd periodic solutions. Then in [3], the sme uthors studied the oscilltion, nonoscilltion, periodicity nd globl symptotic stbility of n dvnced type impulsive first order nonhomogeneous differentil eqution with piecewise constnt rguments. Also, in 2011, oscilltion, nonoscilltion nd periodicity of second order x (t) 2 x(t) = bx([t 1]) + cx([t] + dx([t + 1] differentil eqution with mixed type piecewise constnt rguments were investigted [4]. In this pper, we extend our results on oscilltion, nonoscilltion nd periodicity of solutions to first nd second order liner differentil equtions with piecewise constnt rguments to third order liner differentil eqution with piecewise constnt rgument. For this purpose, we consider the following third order liner differentil eqution with piecewise constnt rgument with the initil conditions x (t) 2 x (t) = bx([t 1]) (1.1) x( 1) = α 1, x(0) = α 0, x (0) = α 1, x (0) = α 2, (1.2) where 0 nd, b, α 1, α 0, α 1, α 2 R.

4 4 H. BEREKETOGLU, M. LAFCI, G. S. OZTEPE EJDE-2017/ Existence nd uniqueness First we give the definition of solution to (1.1). Then we use the technique in [26] to investigte the solution of this eqution. A function x(t) defined on [0, ) is sid to be solution of the initil vlue problem (1.1) (1.2) if it stisfies the following conditions: (i) x is continuous on [0, ), (ii) x exists nd continuous on [0, ), (iii) x exists on [0, ) with the possible exception of the points [t] [0, ), where one-sided derivtives exist, (iv) x stisfies (1.1) on ech intervl [n, n + 1) with n N. Theorem 2.1. Eqution (1.1) hs solution on [0, ). Proof. Let x n (t) be solution of (1.1) on the intervl [n, n + 1) with the conditions x(n) = c n, x(n 1) = c n 1, x (n) = d n, x (n) = e n. Then (1.1) reduces to x (t) 2 x (t) = bx(n 1). The solution of the bove eqution is found s x n (t) = K n + L n cosh (t n) + M n sinh (t n) b 2 tx n(n 1) (2.1) with rbitrry constnts K n, L n nd M n. Writing t = n in (2.1), we obtin c n = K n + L n b 2 nc n 1. (2.2) If we tke t = n in the first nd second derivtives of (2.1), respectively, we find From (2.2) nd (2.3), M n = d n + b 3 c n 1, L n = e n 2. (2.3) K n = c n e n 2 + b 2 nc n 1 (2.4) is obtined. Substituting (2.3) nd (2.4) in (2.1), we hve 1 + cosh (t n) sinh (t n) x n (t) = 2 e n + d n + c n + [ b 2 n b 2 t + b 3 sinh (t n)]c n 1. First nd second derivtives of (2.5) re found s x n(t) = sinh (t n) (2.5) e n + cosh (t n)d n + [ b b cosh (t n) 2 2 ]c n 1, (2.6) x n(t) = cosh (t n)e n + sinh (t n)d n + b sinh (t n)c n 1. (2.7) Writing t = n + 1 in (2.5), (2.6) nd (2.7), it follows tht c n+1 = c n + sinh d n + ( cosh 2 3 d n+1 = (cosh )d n + sinh ( b cosh e n + ( ) en + ( b sinh b 2 ) cn 1, (2.8) b 2 ) c n 1, (2.9) e n+1 = (sinh )d n + (cosh )e n + b sinh c n 1. (2.10)

5 EJDE-2017/193 THIRD-ORDER DIFFERENTIAL EQUATIONS 5 Now, let us introduce the vector v n = col(c n, d n, e n ) nd the mtrices sinh cosh 1 1 b sinh A = 2 2 b sinh 0 cosh, B = b cosh b b sinh 0 sinh cosh 0 0 so we cn rewrite the system (2.8)-(2.10) s v n+1 = Av n + Bv n 1. (2.11) Looking for nonzero solution of this difference eqution system in the form of v n = kλ n, with constnt vector k, leds us to nd chrcteristic eqution det(λ 2 I λa B) = 0, λ 4 + ( 1 2 cosh )λ 3 + (1 + b 2 b sinh + 2 cosh )λ2 3 + ( 1 2b 2b b cosh + sinh )λ + ( b sinh ) = 0. 3 Assuming tht these roots re simple, we write the generl solution of (2.12), (2.12) v n = λ n 1 k 1 + λ n 2 k 2 + λ n 3 k 3 + λ n 4 k 4, (2.13) where v n = col(c n, d n, e n ) nd k j = col(k ij ), i = 1, 2, 3, 4 which cn be found from dequte initil or boundry conditions. If some λ j is multiple zero of (2.12), then the expression for v n lso includes products of λ n j by n, n2 or n 3. Finlly, the solution x n (t) is obtined by substituting the pproprite components of the vectors v n nd v n 1 in (2.5). Remrk 2.2. From (2.8), (2.9) nd (2.10), we obtin d n = 2 sinh c (1 + cosh ) n+2 + c n+1 sinh + ( cosh b + b sinh ) 2 2 sinh b + 2b cosh 3b sinh c n 1, sinh 2 e n = 2(1 cosh ) c n cosh 1 cosh c n+1 ( cosh + b b sinh ) 2(1 cosh ) b 2b cosh + b sinh c n 1. 2(1 cosh ) Substituting (2.14) nd (2.15) in (2.8), gives us the difference eqution c n+3 + ( 1 2 cosh )c n+2 + (1 + b 2 b 3 sinh + 2 cosh )c n+1 + ( 1 2b 2b cosh sinh )c n + ( b 2 b 3 sinh )c n 1 = 0 whose chrcteristic eqution is the sme s (2.12). c n c n (2.14) (2.15) (2.16)

6 6 H. BEREKETOGLU, M. LAFCI, G. S. OZTEPE EJDE-2017/193 Theorem 2.3. The boundry-vlue problem for (1.1) with the conditions x( 1) = c 1, x(0) = c 0, x(1) = c 1, x(n 1) = c N 1 (2.17) hs unique solution on 0 t < if N > 2 is n integer nd both of the following hypotheses re stisfied: (i) The roots of (2.12) λ j (chrcteristic roots) re nontrivil nd distinct, (ii) λ N 1 λ 2 λ 3 λ 4 ( λ 2 4λ 1 λ 2 + λ 4 λ 1 λ λ 2 4λ 1 λ 3 λ 1 λ 2 2λ 3 λ 4 λ 1 λ λ 1 λ 2 λ 2 3) + λ 1 λ 2 λ N 3 λ 4 ( λ 2 4λ 1 λ 3 + λ 4 λ 2 1λ 3 + λ 2 4λ 2 λ 3 λ 2 1λ 2 λ 3 λ 4 λ 2 2λ 3 + λ 1 λ 2 2λ 3 ) λ 1 λ 2 λ 3 λ N 4 ( λ 4 λ 2 1λ 2 + λ 4 λ 1 λ λ 4 λ 2 1λ 3 λ 4 λ 2 2λ 3 λ 4 λ 1 λ λ 4 λ 2 λ 2 3) + λ 1 λ N 2 λ 3 λ 4 ( λ 2 4λ 1 λ 2 + λ 4 λ 2 1λ 3 + λ 2 4λ 2 λ 3 λ 2 1λ 2 λ 3 λ 4 λ 2 λ λ 1 λ 2 λ 2 3). Proof. The first row of the vector eqution (2.13) gives us c n = λ n 1 k 11 + λ n 2 k 21 + λ n 3 k 31 + λ n 4 k 41. (2.18) We get following system by pplying the boundry conditions (2.17) to (2.18), respectively. λ 1 1 k 11 + λ 1 2 k 21 + λ 1 3 k 31 + λ 1 4 k 41 = c 1, (2.19) k 11 + k 21 + k 31 + k 41 = c 0, (2.20) λ 1 k 11 + λ 2 k 21 + λ 3 k 31 + λ 4 k 41 = c 1, (2.21) λ N 1 1 k 11 + λ N 1 2 k 21 + λ N 1 3 k 31 + λ N 1 4 k 41 = c N 1. (2.22) From hypothesis (ii), the determintion of the coefficients of this system is different from zero. Hence, we cn find k ij nd lso c n uniquely. Furthermore, once the vlues c n hve been found, we clculte d n nd e n from (2.14) nd (2.15), respectively. Substituting c n, d n, e n in (2.5), the unique solution x n (t) is obtined. The following four theorems depend on the chrcteristic roots. Their proofs re omitted becuse they re very similr to the proof of Theorem 2.3. Theorem 2.4. Let us ssume tht ll chrcteristic roots re nontrivil nd two of them re equl (λ 1 = λ 2 ), others re different from ech other (λ 3 λ 4 ). If [ λ N 1 (1 N)λ 1 λ 3 λ (N 2)λ 2 1λ 3 λ Nλ 3 4λ (2 N)λ 2 1λ 2 3λ 4 ] Nλ 2 4λ (N 1)λ 1 λ 3 3λ 4 λ N [ 3 λ1 λ 3 λ 3 4 2λ 2 1λ 3 λ λ 3 ] [ 1λ 3 λ 4 + λ N 4 2λ 2 1 λ 2 3λ 4 λ 1 λ 3 3λ 4 λ 3 ] 1λ 3 λ 4, then the boundry-vlue problem for (1.1) with the conditions (2.17) hs unique solution on [0, ). Theorem 2.5. If the chrcteristic roots λ j re nontrivil, λ 1 = λ 2, λ 3 = λ 4 nd [ ] λ N 1 (N 2)λ 2 1λ 2 + 2(1 N)λ 1 λ Nλ 3 2 [ ] λ N 2 (2 N)λ 1 λ (N 1)λ 2 1λ 2 + Nλ 3 1, then the boundry-vlue problem for (1.1) with the conditions (2.17) hs unique solution on [0, ).

7 EJDE-2017/193 THIRD-ORDER DIFFERENTIAL EQUATIONS 7 Theorem 2.6. If the chrcteristic roots λ j re nontrivil, λ 1 = λ 2 = λ 3 nd [ λ N 1 ( N 2 + 3N 2)λ 3 1λ 2 + (2N 2 5N + 2)λ 2 1λ (2 N)λ 4 1λ 2 2 ] ] + ( N 2 + 2N 1)λ 1 λ (N 1)λ 3 1λ 3 2 λ N 2 [λ 5 1λ 2 λ 3 1λ 2, then the boundry-vlue problem for (1.1) with the conditions (2.17) hs unique solution on [0, ). Theorem 2.7. If λ 1 = λ 2 = λ 3 = λ 4 = λ nd 2N 3N 2 + N 3 + ( 2N + 3N 2 N 3 )λ 2 0, then the boundry-vlue problem for (1.1) with the conditions (2.17) hs unique solution on [0, ). 3. Min results This section dels with the oscilltion, nonoscilltion nd the periodicity of the solutions of (1.1). Also, we give n exmple to illustrte our results. Theorem 3.1. If 0 < b 2 < cosh 2 cosh 2, then there exist oscilltory solutions of (1.1). Proof. Eqution (2.12) cn be written s polynomil of λ, where f(λ) = λ 4 + β 1 λ 3 + β 2 λ 2 + β 3 λ + β 4, (3.1) β 1 = 1 2 cosh, β 2 = 1 + b 2 b sinh + 2 cosh, 3 β 3 = 1 2b 2b (3.2) cosh + sinh, 2 3 β 4 = b 2 b sinh. 3 To prove the oscilltion of solutions, we need to show tht there exists unique negtive root of the chrcteristic eqution (2.12). For this reson let us tke the polynomil f( λ) = λ 4 β 1 λ 3 + β 2 λ 2 β 3 λ + β 4. Now, if hypothesis is true, then we find tht β 1 < 0, β 2 > 0, β 3 < 0, β 4 < 0. By using Descrtes rule of signs, we conclude tht there exists unique negtive root of (2.12). Let us tke λ 1 s this root. Now, consider the following boundry conditions x(0) = c 0, x( 1) = c 1 = c 0 λ 1 1, x(1) = c 1 = c 0 λ 1, x(2) = c 2 = c 0 λ 2 1. Applying these conditions to (2.18), the coefficients k i1, i = 1, 2, 3, 4 re found s k 11 = c 0, k 21 = k 31 = k 41 = 0 nd therefore, (2.18) becomes c n = x(n) = c 0 λ n 1. Since λ 1 < 0, we see tht x(n)x(n + 1) = λ 1 c 2 0λ 2n 1 < 0, c 0 0

8 8 H. BEREKETOGLU, M. LAFCI, G. S. OZTEPE EJDE-2017/193 nd so the solution x(t) of (1.1) hs zero in ech intervl (n, n +1). So there exist oscilltory solutions. Theorem 3.2. If or 0 < b < 3 (1 + 2 cosh ) sinh b < 3 2(sinh cosh ) is stisfied, then there exist nonoscilltory solutions of (1.1). (3.3) (3.4) Proof. From (3.3), we hve β 1 < 0, β 2 > 0, β 3 < 0, β 4 < 0. Also, we obtin β 1 < 0, β 2 > 0, β 3 > 0, β 4 < 0 by using (3.4) where β 1, β 2, β 3, β 4 re given by (3.2). So, from Descrtes rule of sign, if ny of the bove conditions is stisfied, then we obtin tht the chrcteristic eqution (2.12) hs t lest one positive root. Therefore, there re nonoscilltory solutions of (1.1). Theorem 3.3. If b > 3 (1 + 2 cosh ), (3.5) sinh then there exist both oscilltory nd nonoscilltory solutions of (1.1). Proof. Condition (3.5) implies tht β 1 < 0, β 2 < 0, β 3 < 0, β 4 < 0, where β 1, β 2, β 3, β 4 re given by (3.2). Hence, from Descrtes rule of sign, we conclude tht there exists single positive root of (2.12). So, the other roots re negtive or complex. Positive root genertes nonoscilltory solutions, nd others give us the oscilltory solutions of (1.1). Theorem 3.4. A necessry nd sufficient condition for the solution of problem (1.1)-(1.2) to be k periodic, k N {0}, is c(k) = c(0), c(k 1) = c( 1), d(k) = d(0), e(k) = e(0). (3.6) Here {c(n)} n 1 is the solution of (2.16) with the initil conditions c( 1) = α 1, c(0) = α 0, d(0) = α 1, e(0) = α 2. Proof. In this proof, we use technique in [14]. If x(t) is periodic with period k, then x(t + k) = x(t) for t [0, ). This implies tht the equlities (3.6) is true. For the proof of sufficiency cse, suppose tht (3.6) is stisfied. From (2.5), 1 + cosh (t k) sinh (t k) x k (t) = 2 e k + d k + c k ( b + 2 k b 2 t + b ) 3 sinh (t k) c k 1, k t < k + 1, 1 + cosh t sinh t x 0 (t) = 2 e 0 + d 0 + c 0 ( + b 2 t + b ) 3 sinh t c 1, 0 t < 1. (3.7) (3.8)

9 EJDE-2017/193 THIRD-ORDER DIFFERENTIAL EQUATIONS 9 So x k (t) = x 0 (t k), k t < k + 1. Moreover x k+1 (t) 1 + cosh (t (k + 1)) = 2 e k+1 + ( b b + (k + 1) 2 2 t + b To show tht x 1 (t) = we need to show tht 1 + cosh (t 1) 2 e 1 + sinh (t (k + 1)) d k+1 + c k+1 ) 3 sinh (t (k + 1)) c k, k + 1 t < k + 2, sinh (t 1) d 1 + c 1 + ( b 2 b 2 t + b 3 sinh (t 1))c 0, 1 t < 2. (3.9) (3.10) x k+1 (t) = x 1 (t k), (3.11) c(k + 1) = c(1), d(k + 1) = d(1), e(k + 1) = e(1). (3.12) For this purpose, by using the continuity t t = k + 1, we obtin x k (k + 1) = x k+1 (k + 1), k t < k + 1. Here, x k (k + 1) nd x k+1 (k + 1) re obtined by tking t = k + 1 in (3.7) nd (3.9), respectively. Hence 1 + cosh x(k + 1) = 2 e k + sinh ( b d k + c k b ) 3 sinh c k 1 (3.13) where x(k + 1) = x k+1 (k + 1). By using the sme procedure t t = 1, we find x(1) = 1 + cosh 2 e 0 + sinh ( b d 0 + c b ) 3 sinh c 1. (3.14) Considering (3.6) in (3.13) nd (3.14), we obtin x(k + 1) = x(1), tht is c(k + 1) = c(k). Tking the derivtives of (3.7) nd (3.8) gives us x sinh (t k) k(t) = e k + cosh (t k)d k ( b b ) (3.15) + cosh (t k) 2 2 c k 1, k t < k + 1, x sinh t ( b b ) 0(t) = e 0 + (cosh t)d 0 + cosh t 2 2 c 1, 0 < t < 1. (3.16) Using continuity t t = k + 1 nd t = 1 in (3.15) nd (3.16), we find x (k + 1) = sinh e k + (cosh )d k + ( b 2 cosh b ) ck 1 2, k t < k + 1, (3.17) x (1) = sinh e 0 + (cosh )d 0 + ( b 2 cosh b ) c 1 2, 0 < t < 1. (3.18) Considering (3.6) in (3.17) nd (3.18), we hve x (k + 1) = x (1) i.e. d(k + 1) = d(1).

10 10 H. BEREKETOGLU, M. LAFCI, G. S. OZTEPE EJDE-2017/193 Similrly, from the continuity t t = k + 1 nd t = 1 in the following derivtives x k(t) = (cosh (t k))e k + (sinh (t k))d k + b (sinh (t k))c k 1, k t < k + 1, x 0(t) = (cosh t)e 0 + ( sinh t)d 0 + b (sinh t)c 1, 0 < t < 1, of (3.15) nd (3.16), we obtin e(k + 1) = e(1). Therefore, we find (3.12). By induction, x k+n (t) = x n (t k). As n exmple we consider the differentil eqution x (t) x (t) = (0.1)x([t 1]), (3.19) which is specil cse of (1.1) with = 1, b = 0.1. It is esily checked tht (3.19) stisfies the condition of Theorem 3.1. Thus there re oscilltory solutions of (3.19). The solution x n (t) of (3.19) with the initil conditions x( 1) = 64.91, x(0) = 1, x (0) = , x (0) = for n = 0, 1,..., 13 is shown in Figure 1 Figure 1. Solution of of (3.19) References [1] Admets, L.; Lomttidze, A.; Oscilltion conditions for third-order liner eqution. (Russin) Differ. Urvn. 37(6), , 2001; trnsltion in Differ. Equ. 37(6), , [2] Brtušek, M.; Došlá, Z.; Oscilltion of third order differentil eqution with dmping term. Czechoslovk Mth. J., 65 (140), 2, , [3] Bereketoglu, H.; Seyhn, G.; Ogun, A.; Advnced impulsive differentil equtions with piecewise constnt rguments. Mth. Model. Anl. 15(2), , [4] Bereketoglu, H.; Seyhn, G.; Krkoc, F.; On second order differentil eqution with piecewise constnt mixed rguments. Crpthin J. Mth. 27(1), 1 12, [5] Bereketoglu, H.; Lfci, M.; S. Oztepe, G.; On the oscilltion of third order nonliner differentil eqution with piecewise constnt rguments. Mediterr. J. Mth., 14 (3), Art. 123, 19 pp., 2017.

11 EJDE-2017/193 THIRD-ORDER DIFFERENTIAL EQUATIONS 11 [6] Cecchi, M.; Oscilltion criteri for clss of third order liner differentil equtions. Boll. Un. Mt. Itl. C (6) 2(1), , [7] Chen, X. R.; Pn, L. J.; Existence of periodic solutions for third order differentil equtions with deviting rgument. Fr Est J. Mth. Sci. (FJMS), 32(3), , [8] Dhiy, R. S.; Oscilltion of the third order differentil equtions with nd without dely. Differentil equtions nd nonliner mechnics (Orlndo, FL, 1999), 75-87, Mth. Appl. 528, Kluwer Acd. Publ., Dordrecht, [9] Ds, P.; Pti, J. K.; Necessry nd sufficient conditions for the oscilltion third-order differentil eqution. Electron. J. Differentil Equtions, no. 174, 10 pp., [10] Džurin, J.; Bculíková, B.; Property (B) nd oscilltion of third-order differentil equtions with mixed rguments. J. Appl. Anl. 19(1), 55-68, [11] Ezeilo, J. O. C.; On the existence of periodic solutions of certin third-order differentil eqution. Proc. Cmbridge Philos. Soc. 56, , [12] Ezeilo, J. O. C.; Periodic solutions of certin third order differentil equtions. Atti Accd. Nz. Lincei Rend. Cl. Sci. Fis. Mt. Ntur., (8) 57 (1974), no. 1-2, (1975). [13] Hn, Z.; Li, T.; Sun, S.; Zhng, C.; An oscilltion criteri for third order neutrl dely differentil equtions. J. Appl. Anl. 16(2), , [14] Krkoc, F.; Bereketoglu, H.; Seyhn, G.; Oscilltory nd periodic solutions of impulsive differentil equtions with piecewise constnt rgument. Act Appl. Mth. 110(1), , [15] Kim, W. J.; Oscilltory properties of liner third-order differentil equtions. Proc. Amer. Mth. Soc. 26, , [16] Ling, H.; Wng, G.; Oscilltion criteri of certin third-order differentil eqution with piecewise constnt rgument. J. Appl. Mth. Art. ID , 18 pp., [17] Ppschinopoulos, G.; Schins J. ; Some results concerning second nd third order neutrl dely differentil equtions with piecewise constnt rgument. Czechoslovk Mth. J. 44(3), , [18] Prhi, N.; Ds, P.; Oscilltion nd nonoscilltion of nonhomogeneous third order differentil equtions. Czechoslovk Mth. J. 44, 119(3), , [19] Prhi, N.; Ds, P.; On the oscilltion of clss of liner homogeneous third order differentil equtions. Arch. Mth. (Brno) 34(4), , [20] Sho, Y.; Ling, H.; Oscilltory nd symptotic behvior for third order differentil equtions with piecewise constnt rgument. Fr Est J. Dyn. Syst. 21(1), 45, [21] Shoukku, Y.; Oscilltion criteri for third order differentil equtions with functionl rguments. Mth. Slovc 65 (2015), no. 5, [22] Tbuev, V. A.; Conditions for the existence of periodic solution of third-order differentil eqution. Prikl. Mt. Meh (Russin); trnslted s J. Appl. Mth. Mech [23] Tejumol, H. O.; Periodic solutions of certin third order differentil equtions. Atti Accd. Nz. Lincei Rend. Cl. Sci. Fis. Mt. Ntur. (8) 6, no. 4, , [24] Tryhuk, V.; An oscilltion criterion for third order liner differentil equtions. Arch. Mth. (Brno) 11(1975) no:2, (1976). [25] Wng, N.; Sufficient conditions for the existence of periodic solutions to some third order differentil equtions with deviting rgument. Mth. Appl. (Wuhn) 21(4), , [26] Wiener, J.; Lkshmiknthm, V.; Excitbility of second-order dely differentil eqution. Nonliner Anl. Ser. B: Rel World Appl. 38(1), 1 11, Huseyin Bereketoglu Deprtment of Mthemtics, Fculty of Sciences, Ankr University, 06100, Ankr, Turkey E-mil ddress: bereket@science.nkr.edu.tr Mehtp Lfci Deprtment of Mthemtics, Fculty of Sciences, Ankr University, 06100, Ankr, Turkey E-mil ddress: mlfci@nkr.edu.tr

12 12 H. BEREKETOGLU, M. LAFCI, G. S. OZTEPE EJDE-2017/193 Gizem S. Oztepe (corresponding uthor) Deprtment of Mthemtics, Fculty of Sciences, Ankr University, 06100, Ankr, Turkey E-mil ddress: