OSCILLATION AND NONOSCILLATION THEOREMS FOR A CLASS OF EVEN-ORDER QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS JELENA MANOJLOVI ´ C AND TOMOYUKI TANIGAWA Received 13 November 2005; Accepted 30 January 2006 We are concerned with the o scillatory and nonoscillatory behavior of solutions of even- order quasilinear functional differential equations of the type ( |y (n) (t)| α sgn y (n) (t)) (n) + q(t) |y(g(t))| β sgn y(g(t)) = 0, where α and β are positive constants, g(t)andq(t)arepos- itive continuous functions on [0, ∞), and g(t)isacontinuouslydifferentiable function such that g  (t) > 0, lim t→∞ g(t) =∞. We first give criteria for the existence of nonoscilla- tory solutions with specific asymptotic behavior, and then derive conditions (sufficient as well as necessary and sufficient) for all solutions to be o scillatory by comparing the above equation with the related differential equation without deviating argument. Copyright © 2006 J. Manojlovi ´ c and T. Tanigawa. This is an open access article distrib- uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction We consider even-order quasilinear functional differential equations of the form    y (n) (t)   α sgn y (n) (t)  (n) + q(t)   y  g(t)    β sgn y  g(t)  = 0, (A) where (a) α and β are positive constants; (b) q :[0, ∞) → (0,∞) is a continuous function; (c) g :[0, ∞) → (0,∞)isacontinuouslydifferentiable function such that g  (t) > 0, t ≥ 0, and lim t→∞ g(t) =∞. By a solution of (A) we mean a function y :[T y ,∞) → R which is n times continu- ously differentiable together with |y (n) | α sgn y (n) and satisfies (A)atallsufficiently large t. Those solutions which vanish in a neig hborhood of infinity will be excluded from our consideration. A solution is said to be oscillatory if it has a sequence of zeros clustering around ∞, and nonoscillatory otherwise. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 42120, Pages 1–22 DOI 10.1155/JIA/2006/42120 2 Quasilinear functional differential equations The objective of this paper is to study the oscillatory and nonoscillatory behavior of solutions of (A). In Section 2 we begin with the classification of nonoscillatory solutions of (A) according to their asymptotic behavior as t →∞.Itsuffices to restr ict our consid- eration to eventually positive solutions of (A), since if y(t) is a solution of (A), then so is −y(t). Let P denote the totally of eventually positive solutions of (A). It w ill be shown that it is natural to divide P into the following two classes: P(I) = P  I 0  ∪ P  I 1  ∪···∪ P  I 2n−1  , P(II) = P  II 1  ∪ P  II 3  ∪···∪ P  II 2n−1  , (1.1) where P(I j ), j ∈{0, 1, ,2n − 1},andP(II k ), k ∈{1,3, ,2n − 1}, consist of solutions y(t) satisfying lim t→∞ y(t) ϕ j (t) = const > 0, lim t→∞ y(t) ϕ k−1 (t) =∞,lim t→∞ y(t) ϕ k (t) = 0, (1.2) respectively. Here the functions ϕ i (t), i = 0,1, ,2n − 1, are defined by ϕ i (t) = t i (i = 0,1, ,n − 1), ϕ i (t) = t n+(i−n)/α (i = n,n +1, ,2n − 1). (1.3) Moreover, we will give the integral representations for positive solutions belonging to each of these two classes. Next, In Section 3 we will give necessary and sufficient condi- tions for the existence of positive solutions belonging to the class P(I) as well as sufficient conditions for the existence of positive solutions belonging to the class P(II). In Section 5 we derive criteria for all solutions of (A) to be oscil latory. Our derivations depend heavily on oscillation theory of even-order nonlinear differential equations    y (n) (t)   α sgn y (n) (t)  (n) + q(t)   y(t)   β sgn y(t) = 0(B) recently developed by Tanigawa in [7]. Comparison theorems which will be established in Section 4 enable us to deduce oscillation of an equation of the form (A)fromthatofa similar equation with a different functional argument. We note that oscillation properties of second-order functional differential equations involving nonlinear Sturm-Liouville-type differential operators have been investigated by Kusano and Lalli [2], Kusano and Wang [4], and Wang [9]. Moreover, in a recent paper by Tanigawa [6] oscillation criteria for fourth-order functional differential equations    y  (t)   α sgn y  (t)   + q(t)   y  g(t)    β sgn y  g(t)  = 0(C) have been presented. 2. Classification and integral representations of positive solutions Our purpose here is to make a detailed analysis of the structure of the set P of all possible positive solutions of (A). J. Manojlovi ´ c and T. Tanigawa 3 Classification of positive solutions. Let y(t) be an eventually positive solution of (A) on [t 0 ,∞), t 0 ≥ 0. Then, we have the following lemma which was proved by Tanigawa and Fentao in [8] and which is a natural generalization of the well-known Kiguradze lemma [1]. It will be convenient to make use of the symbols L i , i = 1,2, ,2n − 1, to denote the “quasiderivatives” generating the differential operator L 2n y = (|y (n) | α sgn y (n) ) (n) : L i y = y (i) , i = 1,2, ,n − 1, L i y =    y (n)   α sgn y (n)  (i−n) , i = n,n +1, ,2n, L i+1 y =  L i y   , i = 1,2, ,n − 2, n,n +1, ,2n − 1, L n y =    L n−1 y     α sgn  L n−1 y   , L 0 y = y. (2.1) Lemma 2.1. If y(t) is a positive solution of (A)on[t 0 ,∞),thenthereexistanoddinteger k ∈{1,3, ,2n − 1} and a t 1 >t 0 such that L i y(t) > 0, t ≥ t 1 , for i = 0,1, ,k − 1, ( −1) i−k L i y(t) > 0, t ≥ t 1 , for i = k,k +1, ,2n − 1. (2.2) We denote by P k thesubsetofP consisting of all positive solutions y(t)of(A) satisfying (2.2). The above lemma shows that P has the decomposition P = P 1 ∪ P 3 ∪···∪P 2n−1 . (2.3) Since L i y(t), i ∈{0,1, ,2n − 1}, are eventually monotone, they tend to finite or infi- nite limits as t →∞, that is, lim t→∞ L i y(t) = ω i , i ∈{0, 1, ,2n − 1}. (2.4) One can easily show that if y ∈ P k for k ∈{1, 3, ,2n − 1},thenω k is a finite nonnegative number and the set of its asymptotic values {ω i } falls into one of the following three cases: ω 0 = ω 1 =··· = ω k−1 =∞, ω k ∈ (0,∞), ω k+1 = ω k+2 =··· = ω 2n−1 = 0, ω 0 = ω 1 =··· = ω k−1 =∞, ω k = ω k+1 =··· = ω 2n−1 = 0, ω 0 = ω 1 =··· = ω k−2 =∞, ω k−1 ∈ (0,∞), ω k = ω k+1 =··· = ω 2n−1 = 0. (2.5) 4 Quasilinear functional differential equations Observing that by L’Hospital’s rule, we have, for every j ∈{1,2, ,2n − 1},that lim t→∞ y(t) ϕ j (t) = const ≥ 0or∞⇐⇒lim t→∞ L j y(t) = const ≥ 0or∞, (2.6) equivalent expressions for these classes of positive solutions of (A) are the following: (i) lim t→∞ y(t) ϕ k (t) = const > 0, (ii) lim t→∞ y(t) ϕ k (t) = 0, lim t→∞ y(t) ϕ k−1 (t) =∞, (iii) lim t→∞ y(t) ϕ k−1 (t) = const > 0, (2.7) where ϕ 0 (t), ,ϕ 2n−1 (t)aredefinedby(1.3). Note that these functions are particular solutions of the unperturbed equation L 2n y(t) = 0. Observing that cases (i) and (iii) are of the same category, it is natural to classify P broadly into the two classes P(I) = P(I 0 ) ∪ P(I 1 ) ∪ ··· ∪P(I 2n−1 )andP(II) = P(II 1 ) ∪ P(II 3 ) ∪ ··· ∪P(II 2n−1 ) consisting, respectively , of P  I j  =  y ∈ P :lim t→∞ y(t) ϕ j (t) = const > 0  , P  II k  =  y ∈ P :lim t→∞ y(t) ϕ k−1 (t) =∞,lim t→∞ y(t) ϕ k (t) = 0  . (2.8) Integral representations for positive solutions. We will establish the existence of eventually positive solutions for each of the above classes P(I) and P(II). For this purpose a crucial role will be played by integral representations for P(I j )andP(II k ) types of solutions of (A) established below. Let y(t) be a positive solution of (A)suchthaty(t) > 0, y(g(t)) > 0on[t 0 ,∞). Let us first derive an integral representation of the solution y(t)fromtheclassP(I j ), j ∈{0, 1, ,2n − 1}. If j ∈{n,n +1, ,2n − 1},thenweintegrate(A)2n − j times from t to ∞ and then integrate the resulting equation j times from t 0 to t to obtain (i) for j ∈{n +1,n +2, ,2n − 1}, y(t) = ζ(t)+  t t 0 (t−s) n−1 (n−1)!  ξ j (s)+(−1) 2n−j−1  s t 0 (s−r) j−n−1 ( j −n−1)! ×  ∞ r (σ −r) 2n−j−1 (2n−j− 1)! q(σ)y  g(σ)  β dσ dr  1/α ds; (2.9) J. Manojlovi ´ c and T. Tanigawa 5 (ii) for j = n, y(t) = ζ(t)+  t t 0 (t − s) n−1 (n − 1)!  ω n +(−1) n−1  ∞ s (r − s) n−1 (n − 1)! q(r)  y  g(r)  β dr  1/α ds, (2.10) where ξ j (t) = j−1  i=n L i y  t 0   t − t 0  i−n (i − n)! + ω j  t − t 0  j−n ( j − n)! (n +1 ≤ j ≤ 2n − 1), ζ(t) = n−1  i=0 L i y  t 0   t − t 0  i i! . (2.11) If j ∈{0,1, ,n − 1}, then first integrating (A)2n − j(= n +(n − j)) times from t to ∞ and then integrating j times from t 0 to t,wehave (i) for j ∈{1,2, ,n − 1}, y(t) = ζ ∗ j (t) +( −1) 2n−j−1  t t 0 (t−s) j−1 ( j −1)!  ∞ s (r−s) n−j−1 (n− j−1)!   ∞ r (σ − r) n−1 (n − 1)! q(σ)  y  g(σ)  β dσ  1/α dr ds; (2.12) (ii) for j = 0, y(t) = ω 0 +(−1) 2n−1  ∞ t (s − t) n−1 (n − 1)!   ∞ s (r − s) n−1 (n − 1)! q(r)  y  g(r)  β dr  1/α ds, (2.13) where ζ ∗ j (t) = j−1  i=0 L i y  t 0   t − t 0  i i! + ω j  t − t 0  j j! (1 ≤ j ≤ n− 1). (2.14) As regards y ∈ P(II k ), k ∈{1,3, ,2n − 1}, an integral representation is expressed by (2.9)–(2.13)withω j = 0for j = k. 3. Nonoscillat ion criteria It will be shown that necessary and sufficient conditions can be established for the exis- tence of positive solutions from class P(I). Theorem 3.1. Let j ∈{0,1, ,2n − 1}. There exists a positive solutions of (A)belongingto P(I j ) if and only if  ∞ 0 t n− j−1   ∞ t s n−1 q(s)  ϕ j  g(s)  β ds  1/α dt < ∞, j = 0,1, ,n − 1, (3.1)  ∞ 0 t 2n− j−1 q(t)  ϕ j  g(t)  β dt < ∞, j = n,n +1, ,2n − 1. (3.2) 6 Quasilinear functional differential equations Proof (the “only if” part). Suppose that (A) has a positive solution y(t)ofclassP(I j ). Notice that since y(t) satisfies asymptotic relations (2.7)(i) and (iii), there exist positive constants c j , C j such that c j ϕ j (t) ≤ y(t) ≤ C j ϕ j (t), t ≥ t 0 . (3.3) In deriving (2.9)–(2.13) we found the convergence of the integrals  ∞ t 0 t 2n− j−1 q(t)  y  g(t)  β dt < ∞,forj = n,n +1, ,2n − 1,  ∞ t 0 t n− j−1   ∞ t s n−1 q(s)  y  g(s)  β ds  1/α dt < ∞,forj = 0, 1, ,n − 1. (3.4) These together with (3.3), show that the conditions (3.1)and(3.2) are satisfied. (The “if” part.) We will distinguish two cases for j ∈{0,1, ,n − 1} and for j ∈{n, n +1, ,2n − 1}. Case 1. Let j ∈{n, n +1, ,2n − 1} and suppose that (3.2)issatisfied.Letc>0bean arbitrarity fixed constant and choose t 0 > 0suchthat  ∞ t 0 t 2n− j−1 (2n − j − 1)! q(t)  ϕ j  g(t)  β dt ≤ A  ( j − n)!  β/α  1+ j − n α  ···  n+ j − n α  β c 1−β/α , (3.5) where A = 2 −β/α if 2n − j − 1 is even, A = 2 −1 if 2n − j − 1isodd. (3.6) Define the constants k 1 and k 2 by k i = c i  ( j − n)!  1/α  1+(j − n)/α  ···  n +(j − n)/α  , i = 1,2, , (3.7) where c 1 = c 1/α , c 2 = (2c) 1/α if 2n − j − 1iseven, c 1 =  c 2  1/α , c 2 = c 1/α if 2n − j − 1isodd. (3.8) Put t ∗ = min{t 0 ,inf t≥t 0 g(t)},anddefine ϕ j (t) = ⎧ ⎨ ⎩ ϕ j  t − t 0  , t ≥ t 0 0, t ≤ t 0 . (3.9) J. Manojlovi ´ c and T. Tanigawa 7 Let Y denote the set Y =  y ∈ C  t ∗ ,∞  : k 1 ϕ j (t) ≤ y(t) ≤ k 2 ϕ j (t), t ≥ t ∗  , (3.10) and define the mapping Ᏺ j : Y → C[t ∗ ,∞)asfollows:forj ∈{n +1,n +2, ,2n − 1}, Ᏺ j y(t) =  t t 0 (t − s) n−1 (n − 1)! ×  c  s − t 0  j−n ( j − n)! +( −1) 2n− j−1  s t 0 (s − r) j−n−1 ( j −n−1)! ×  ∞ r (σ − r) 2n−j−1 (2n− j−1)! q(σ)  y  g(σ)  β dσ dr  1/α ds, t ≥ t 0 , Ᏺ j y(t) = 0, t ∗ ≤ t ≤ t 0 , (3.11) and for j = n, Ᏺ n y(t) =  t t 0 (t − s) n−1 (n − 1)!  c +(−1) n−1  ∞ s (r − s) n−1 (n − 1)! q(r)  y  g(r)  β dr  1/α ds, t ≥ t 0 , Ᏺ n y(t) = 0, t ∗ ≤ t ≤ t 0 . (3.12) It can be verified that Ᏺ j maps Y continuously into a relatively compact subset of Y. First, we can show that Ᏺ j (Y) ⊂ Y by using the expression  t t 0 (t − s) n−1 (n − 1)!  s − t 0  (j−n)/α ds = ϕ j  t − t 0   1+(j − n)/α  ···  n +(j − n)/α  . (3.13) Next, let {y m (t)} be a sequence of functions in Y converging to y 0 (t)onanycompact subinterval of [t ∗ ,∞). Then, by virtue of the Lebesgue convergence theorem it follows that the sequence {Ᏺ j y m (t)} con verges t o Ᏺ j y 0 (t) on compact subintervals of [t ∗ ,∞), which implies the continuity of the mapping Ᏺ j . Finally, since the sets Ᏺ j (Y)andᏲ  j (Y) ={(Ᏺ j y)  : y ∈ Y} are locally bounded on [t ∗ ,∞), the Arzel ´ a theorem implies that Ᏺ j (Y) is relatively compact in C[t ∗ ,∞). Thus, all the hypotheses of the Schauder-Tychonoff fixed point theorem are satisfied, and so there exists a y ∈ Y such that y = Ᏺ j y.Inviewof (3.11)and(3.12) the fixed element y = y(t) is a solution of the integral equation which is a special case of (2.9)withζ(t) = 0, ξ j (t) = (c/( j − n)!)(t − t 0 ) j−n as well as it is a special case as of (2.10)withζ(t) = 0, ω n = c.Bydifferentiation of these integ ral equations 2n times, we see that y(t) is a solution of the differential equation (A)on[t ∗ ,∞) satisfying L j y(∞) = c, that is, y ∈ P(I j ). 8 Quasilinear functional differential equations Case 2. Let j ∈{0,1, ,n − 1} and suppose that (3.1)issatisfied.Letc>0beanygiven constant and choose t 0 > 0sothat  ∞ t 0 t n− j−1 (n − j − 1)!   ∞ t (s − t) n−1 (n − 1)! q(s)  ϕ j (s)  β ds  1/α dt ≤ B(j!) β/α c 1−β/α , (3.14) where B = 2 −β/α if 2n − j − 1iseven, B = 2 −1 if 2n − j − 1isodd. (3.15) Define the constants k 1 and k 2 as follows: k 1 = c j! , k 2 = 2c j! if 2n − j − 1iseven, k 1 = c 2 j! , k 2 = c j! if 2n − j − 1 is odd, (3.16) and define the set Y by (3.10) with these k 1 , k 2 . We define the mapping Ᏺ j : Y → C[t ∗ ,∞) in the following manner: for j ∈{1,2, ,n − 1}, Ᏺ j y(t) = c(t − t 0 ) j j! +( −1) 2n− j−1  t t 0 (t − s) j−1 ( j − 1)!  ∞ s (r − s) n− j−1 (n − j − 1)! ×   ∞ r (σ − r) n−1 (n − 1)! q(σ)  y  g(σ)  β dσ  1/α dr ds, t ≥ t 0 , Ᏺ j y(t) = 0, t ∗ ≤ t ≤ t 0 (3.17) and for j = 0, Ᏺ 0 y(t) = c +(−1) 2n−1  ∞ t (s − t) n−1 (n − 1)!   ∞ s (r − s) n−1 (n − 1)! q(r)  y  g(r)  β dr  1/α ds, t ≥ t 0 , Ᏺ 0 y(t) = 0, t ∗ ≤ t ≤ t 0 . (3.18) Then it is routinely verified that Ᏺ j (Y) ⊂ Y,thatᏲ j is continuous, and that Ᏺ j (Y)is relatively compact in C[t ∗ ,∞). Consequently, there exists a fixed element y ∈ Y such that y = Ᏺ j y, which is the integral equation (2.13)withω 0 = c for j = 0aswellasitisthe integral equation (2.12)withζ ∗ j (t) = (c/ j!)(t − t 0 ) j for j ∈{1,2, ,n − 1}. It is clear that the fixed element y = y(t) is a solution of (A)belongingtoP(I j ). This completes the proof.  Unlike the solutions of class P(I) it seems to be very difficult (or impossible) to char- acterize the existence of solutions of class P(II), and we will be content to give sufficient conditions under which (A) possesses such solutions. J. Manojlovi ´ c and T. Tanigawa 9 Theorem 3.2. (i) Let k be an odd integer less than n.Equation(A)hasasolutionofclass P(II k ) if  ∞ 0 t n−k−1   ∞ t s n−1 q(s)  ϕ k  g(s)  β ds  1/α dt < ∞, (3.19)  ∞ 0 t n−k   ∞ t s n−1 q(s)  ϕ k−1  g(s)  β ds  1/α dt =∞. (3.20) (ii) Let n be odd and let k = n.Equation(A)hasasolutionofclassP(II k ) if  ∞ 0 t n−1 q(t)  ϕ n  g(t)  β dt < ∞,  ∞ 0   ∞ t s n−1 q(s)  ϕ n−1  g(s)  β ds  1/α dt =∞. (3.21) (iii) Let k be an odd integer greater than n and less than 2n.Equation(A)hasasolution of class P(II k ) if  ∞ 0 t 2n−k−1 q(t)  ϕ k  g(t)  β dt < ∞,  ∞ 0 t 2n−k q(t)  ϕ k−1  g(t)  β dt =∞. (3.22) Proof. (i) Let k be an odd integer less than n. The desired solution y(t)willbeobtained as a solution of the integral equation y(t) = cϕ k−1 (t) +  t t 0 (t−s) k−1 (k−1)!  ∞ s (r−s) n−1−k (n−1−k)!   ∞ r (σ −r) n−1 (n−1)! q(σ)  y  g(σ)  β dσ  1/α dr ds, t ≥ t 0 , (3.23) where c>0isfixedandt 0 > 0 is chosen so large that t ∗ = min{t 0 ,inf t≥t 0 g(t)}≥1and  ∞ t 0 t n−1−k (n − 1 − k)!   ∞ t s n−1 (n − 1)! q(s)  ϕ k  g(s)  β ds  1/α dt ≤ 2 −β/α c 1−β/α . (3.24) In order to show the existence of solution y(t) of the integral equation (3.23) we will show that mapping Ᏻ k y(t) defined on the set Y =  y ∈ C  t ∗ ,∞  : cϕ k−1 (t) ≤ y(t) ≤ 2cϕ k (t), t ≥ t ∗  (3.25) 10 Quasilinear functional differential equations by Ᏻ k y(t) = cϕ k−1 (t) +  t t 0 (t−s) k−1 (k−1)!  ∞ s (r−s) n−1−k (n−1−k)!   ∞ r (σ −r) n−1 (n−1)! q(σ)  y(g(σ)  β dσ  1/α dr ds, t ≥ t 0 , Ᏻ k y(t) = 0, t ∗ ≤ t ≤ t 0 (3.26) has a fixed element in Y.Ify ∈ Y, then, using (3.24), we have cϕ k−1 (t) ≤ Ᏻ k y(t) ≤ cϕ k−1 (t)+c  t t 0 (t − s) k−1 (k − 1)! ds = cϕ k−1 (t)+cϕ k (t) ≤ 2cϕ k (t), t ≥ t ∗ , (3.27) which implies that Ᏻ k maps Y into itself. Since it could b e shown without difficulty that Ᏻ k is continuous in the topology of C[t ∗ ,∞) and that Ᏻ k (Y) is relatively compact in C[t ∗ ,∞), there exists a fixed element y of Ᏻ k in Y.Repeateddifferentiation of (3.26) shows that L k−1 y(t) = c(k − 1)! +  t t 0  ∞ s (r − s) n−k−1 (n − k − 1)!   ∞ r (σ − r) n−1 (n − 1)! q(σ)  y  g(σ)  β dσ  1/α dr ds, (3.28) L k y(t) =  ∞ t (s − t) n−k−1 (n − k − 1)!   ∞ s (r − s) n−1 (n − 1)! q(r)  y  g(r)  β dr  1/α ds, (3.29) for t ≥ t 0 . It is obvious that L k y(∞) = 0. Evaluating the right-hand side of (3.28), we see that it is bounded from below by  t t 0  s − t 0  n−k (n − k)!   ∞ s (r − s) n−1 (n − 1)! q(r)  y  g(r)  β dr  1/α ds ≥ c β/α  t t 0  s − t 0  n−k (n − k)!   ∞ s (r − s) n−1 (n − 1)! q(r)  ϕ k−1  g(r)  β dr  1/α ds, (3.30) from which, in view of (3.20), it follows that L k−1 y(∞) =∞. This shows that y(t)belongs to P(II k ). (ii) Let n be odd and let k = n.Chooset 0 > 0 large enough so that t ∗ = min{t 0 , inf t≥t 0 g(t)}≥1and  ∞ t 0 t n−1 q(t)  ϕ n  g(t)  β dt ≤ 2 −β c α−β (n − 1)!, (3.31) [...]... Tanigawa and W Fentao, On the existence of positive solutions for a class of even order quasilinear differential equations, Advances in Mathematical Sciences and Applications 14 (2004), no 1, 75–85 [9] J Wang, Oscillation and nonoscillation theorems for a class of second order quasilinear functionaldifferential equations, Hiroshima Mathematical Journal 27 (1997), no 3, 449–466 Jelena Manojlovi´ : Department... differential equations, Pacific Journal of Mathematics 83 (1979), no 1, 187–197 [6] T Tanigawa, Oscillation and nonoscillation theorems for a class of fourth order quasilinear functional differential equations, Hiroshima Mathematical Journal 33 (2003), no 3, 297–316 , Oscillation criteria for a class of higher order nonlinear differential equations, Memoirs [7] on Differential Equations and Mathematical Physics 37... M Naito, Comparison theorems for functional- differential equations with deviating arguments, Journal of the Mathematical Society of Japan 33 (1981), no 3, 509–532 [4] T Kusano and J Wang, Oscillation properties of half-linear functional- differential equations of the second order, Hiroshima Mathematical Journal 25 (1995), no 2, 371–385 [5] W E Mahfoud, Comparison theorems for delay differential equations,... Jelena Manojlovi´ : Department of Mathematics and Computer Science, Faculty of Science and c Mathematics, University of Niˇ, Viˇegradska 33, 18000 Niˇ, Serbia and Montenegro s s s E-mail address: jelenam@pmf.ni.ac.yu Tomoyuki Tanigawa: Department of Mathematics, Faculty of Science Education, Joetsu University of Education, Niigata 943-8512, Japan E-mail address: tanigawa@juen.ac.jp ... that all solutions of (5.4) are oscillatory Application of Theorem 4.3 then shows that all solutions of (5.3) are oscillatory, and the conclusion of the theorem follows from comparison of (A) with (5.3) by means of Theorem 4.2 It will be shown below that there is a class of sublinear equations of the type (A) for which the oscillation situation can be completely characterized Theorem 5.2 Let α > β and. .. six-month stay as a Visiting Scholar at the Department of Applied Mathematics of the Fukuoka University in Japan, supported by the Matsumae International Foundation She wishes to express her sincere gratitude for warm hospitality of the host scientist, Professor Naoki Yamada The second author’s research was supported in part by Grant-in-Aid for Young Scientist (B) (no 16740084) by the Ministry of Education,... that an essential part of the proof of Lemma 4.4 has been proving the existence of the solution for each of the integral equations (4.20), (4.21), and (4.22) That has been done by the application of Schauder-Tychonoff fixed point theorem on the basis of the corresponding integral inequalities (4.12), (4.15), and (4.18) Proceeding here in a similar way, on the basis that u(t) satisfies (4.28), (4.29), and. .. Science, and Technology, Japan References [1] I T Kiguradze, On the oscillatory character of solutions of the equation d m u/dt m + a( t)|u|n signu = 0, Matematicheskii Sbornik New Series 65 (107) (1964), 172–187 (Russian) [2] T Kusano and B S Lalli, On oscillation of half-linear functional- differential equations with deviating arguments, Hiroshima Mathematical Journal 24 (1994), no 3, 549–563 [3] T Kusano and. .. − ∞, → as t − ∞ → 12 Quasilinear functional differential equations 4 Comparison theorems In order to establish criteria (preferably sharp) for all solutions of (A) to be oscillatory, we are essentially based on the following oscillation result of Tanigawa [7] for the even-order nonlinear differential equation (B) Theorem 4.1 (i) Let α > β All solutions of (B) are oscillatory if and only if ∞ ∞ β ϕ2n−1... Oscillation criteria The aim of this section is to establish criteria (preferably sharp) for all solutions of (A) to be oscillatory Oscillation theorems will be established first in the sublinear case of (A) for α > β as well as in the superlinear case for α < β We first give the sufficient condition for all of solutions of sublinear equation (A) to be oscillatory Theorem 5.1 Let α > β Suppose that there . OSCILLATION AND NONOSCILLATION THEOREMS FOR A CLASS OF EVEN-ORDER QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS JELENA MANOJLOVI ´ C AND TOMOYUKI TANIGAWA Received 13 November 2005; Accepted 30 January. enable us to deduce oscillation of an equation of the form (A) fromthatofa similar equation with a different functional argument. We note that oscillation properties of second-order functional. differential equations involving nonlinear Sturm-Liouville-type differential operators have been investigated by Kusano and Lalli [2], Kusano and Wang [4], and Wang [9]. Moreover, in a recent paper by