MATHEMATICA COMMUNICATIONS 151 Mah. Commun. 22(2017), 151 164 Exisence of non-oscillaory soluions of a kind of firs-order neural differenial equaion Fanchao Kong Deparmen of Mahemaics, Hunan Normal Universiy, Changsha, Hunan 410 081, P.R.China Received April 16, 2016; acceped Ocober 13, 2016 Absrac. This paper deals wih he exisence of non-oscillaory soluions o a kind of firs-order neural equaions having boh delay advance erms. The new resuls are esablished using he Banach conracion principle. AMS subjec classificaions: 34C10, 34K11 Key words: Neural equaions, Banach conracion principle, Non-oscillaory soluion 1. Inroducion In his paper, we sudy he firs-order neural differenial equaion d x()+p1 ()x( τ 1 )+P 2 ()x(+τ 2 ) d +Q 1 ()g 1 (x( σ 1 )) Q 2 ()g 2 (x(+σ 2 )) f() = 0, (1) where P i C( 0, ),R), Q i C( 0, ),0, )), i = 1,2. τ i > 0 σ i 0, i = 1,2. f C( 0, ),R) g i C(R,R). We assume ha g i,i = 1,2 saisfy he local ipschiz condiion xg i (x) > 0, i = 1,2, for x 0. The problem of oscillaion of soluions of neural funcional differenial equaions is of boh heoreical pracical ineres. Recenly, here has been an ineres in esablishing he oscillaory he non-oscillaory behavior of firs, second, hird higher order neural funcional differenial equaions. In 2002, Zhou Zhang 31 exended he resuls of Kulenović Hadžiomerspahić in 18 o a higher order linear neural delay differenial equaion of he form d n d nx()+cx( τ)+( 1)n+1 P()x( σ) Q()x( σ) = 0. In 2005, he exisence of non-oscillaory soluions of firs-order linear neural delay differenial equaions of he form d d x()+p()x( τ)+q 1()x( σ 1 ) Q 2 ()x( σ 2 ) = 0, Corresponding auhor. Email address: fanchaokong88@sohu.com (F.Kong) hp://www.mahos.hr/mc c 2017 Deparmen of Mahemaics, Universiy of Osijek
152 F. Kong was invesigaed by Zhang e al. 30 in he same year, Yu Wang 29 sudied non-oscillaory soluions of second-order nonlinear neural delay equaions of he form r()x() +P()x( τ) +Q1 ()f(x( σ 1 )) Q 2 ()g(x( σ 2 )) = 0. For more relaed works, we refer he reader o papers 1, 4, 5, 7, 8, 9, 10, 12, 13, 16, 17, 20, 21, 22, 23, 24, 25, 26, 28, 32 books 2, 3, 6, 14, 15, 19, 27. However, o he bes of our knowledge, here are no resuls of he exisence of non-oscillaory soluions for he firs-order neural differenial equaions having boh delay advance erms. Moivaed by he works menioned above, in his paper, we sudy he exisence of non-oscillaory soluions for (1). e m = max{τ 1,σ 1,σ 2 }. By a soluion of (1) we mean a funcion x C( 1 m, ),R), for some 1 0, such ha x() + P 1 ()x( τ 1 ) + P 2 ()x( + τ 2 ) is coninuously differeniable on 1, ) (1) is saisfied for 1. As cusomary, a soluion of (1) is said o be oscillaory if i has arbirarily large zeros. Oherwise he soluion is called non-oscillaory. The following heorem will be used o prove he main resuls in he nex secion. Theorem 1 (Banach s Conracion Mapping Principle, see 11). A conracion mapping on a complee meric space has exacly one fixed poin. 2. Main resuls To show ha an operaor S saisfies he condiions for he conracion mapping principle, we consider differen cases for he ranges of he coefficiens P 1 () P 2 (). Theorem 2. Assume ha 0 P 1 () < 1, 0 P 2 () p 2 < 1 0 Q i (s)ds <,i = 1,2, hen (1) has a bounded non-oscillaory soluion. 0 f(s) ds < ; (2) Proof. e Λ be he se of all coninuous bounded funcions on 0, ) wih he supremum norm. Se Ω = {x Λ : M 1 x() M 2, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. e i, i = 1,2 denoe he ipschiz consans of funcions g i, i = 1,2 on he se Ω, = max{ 1, 2 }, β i = max g i(x),i = 1,2,... Because of (2), we can choose 1 > 0, 1 0 +max{τ 1,σ 1,σ 2 } (3) sufficienly large such ha Q 1 (s)β 1 + f(s) ds M 2 α, 1, (4)
Exisence of non-oscillaory soluions 153 Q 2 (s)β 2 + f(s) ds α ( +p 2 )M 2 M 1, 1, (5) Q 1 (s)+q 2 (s) ds 1 ( +p 2 ), 1, (6) where M 1 M 2 are posiive consans such ha ( +p 2 )M 2 +M 1 < M 2 α ( ( +p 2 )M 2 +M 1,M 2 ). Consider he operaor S : Ω Λ defined by α P 1 ()x( τ 1 ) P 2 ()x(+τ 2 ) (Sx)() = + Q 1 (s)g 1 (x(s σ 1 )) Q 2 (s)g 2 (x(s+σ 2 )) f(s) ds, 1, (Sx)( 1 ), 0 1. Clearly, Sx is coninuous. For 1 x Ω, i follows from (4) (5) ha (Sx)() α+ Q 1 (s)g 1 (x(s σ 1 )) f(s) ds α+ Q 1 (s)β 1 + f(s) ds M 2, (Sx)() α P 1 ()x( τ 1 ) P 2 ()x(+τ 2 ) Q 2 (s)g 2 (x(s+σ 2 ))+f(s) ds α M 2 p 2 M 2 M 1 M 1. This means ha SΩ Ω. Since Ω is a bounded, closed, convex subse of Λ, in order o apply he Banach conracion principle we have o show ha S is a conracion mapping on Ω. For x 1,x 2 Ω 1, (Sx 1 )() (Sx 2 )() P 1 () x 1 ( τ 1 ) x 2 ( τ 1 ) +P 2 () x 1 (+τ 2 ) x 2 (+τ 2 ) + + Q 1 (s) g 1 (x 1 (s σ 1 )) g 1 (x 2 (s σ 1 )) ds Q 2 (s) g 2 (x 1 (s+σ 2 )) g 2 (x 2 (s+σ 2 )) ds
154 F. Kong or by (6) (Sx1 )() (Sx 2 )() ( x1 x 2 +p 2 + which implies ha x 1 x 2 = x 1 x 2, +p 2 + 1 ( +p 2 ) ) Q 1 (s)+q 2 (s)ds which shows ha S is a conracion mapping on Ω. Thus, S has a unique fixed poin which is a posiive bounded soluion of (1). This complees he proof. Theorem 3. Assume ha 0 P 1 () < 1, 1 < p 2 P 2 () 0 (2) hold; hen (1) has a bounded non-oscillaory soluion. Proof. e Λ be he se of all coninuous bounded funcions on 0, ) wih he supremum norm. Se Ω = {x Λ : N 1 x() N 2, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. e i, i = 1,2 denoe he ipschiz consans of funcions g i, i = 1,2 on he se Ω, = max{ 1, 2 }, β i = max g i(x),i = 1,2,... Because of (2), we can choose a 1 > 0, 1 0 +max{τ 1,σ 1,σ 2 } sufficienly large such ha Q 1 (s)β 1 + f(s) ds (1+p 2 )N 2 α, 1, (7) Q 2 (s)β 2 + f(s) ds α N 2 N 1, 1, (8) Q 1 (s)+q 2 (s) ds 1 ( +p 2 ), 1, (9) where N 1 N 2 are posiive consans such ha N 1 + N 2 < (1+p 2 )N 2 α (N 1 + N 2,(1+p 2 )N 2 ). Consider he operaor S : Ω Λ defined by α P 1 ()x( τ 1 ) P 2 ()x(+τ 2 ) (Sx)() = + Q 1 (s)g 1 (x(s σ 1 )) Q 2 (s)g 2 (x(s+σ 2 )) f(s) ds, 1, (Sx)( 1 ), 0 1.
Exisence of non-oscillaory soluions 155 Obviously, Sx is coninuous. For 1 x Ω, i follows from (7) (8) ha (Sx)() α p 2 N 2 + N 2, Q 1 (s)β 1 + f(s) ds (Sx)() α N 2 N 1. Q 2 (s)β 2 + f(s) ds This proves ha SΩ Ω. Since Ω is a bounded, closed, convex subse of Λ, in order o apply he Banach conracion principle we have o show ha S is a conracion mapping on Ω. For x 1,x 2 Ω 1, by using (9), we can obain (Sx1 )() (Sx 2 )() ( ) x1 x 2 +p 2 + Q 1 (s)+q 2 (s)ds x 1 x 2 +p 2 + 1 (p 1 +p 2 ) This implies = x 1 x 2. which shows ha S is a conracion mapping on Ω. Thus, S has a unique fixed poin which is a posiive bounded soluion of (1). This complees he proof. Theorem 4. Assume ha 1 < P 1 () 0 <, 0 P 2 () p 2 < 1 (2) hold; hen (1) has a bounded non-oscillaory soluion. Proof. e Λ be he se of all coninuous bounded funcions on 0, ) wih he supremum norm. Se Ω = {x Λ : M 3 x() M 4, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. e i, i = 1,2 denoe he ipschiz consans of funcions g i, i = 1,2 on he se Ω, = max{ 1, 2 }, β i = max g i(x), i = 1,2,... In view of (2), we can choose 1 > 0, 1 +τ 1 0 +max{σ 1,σ 1 } (10) sufficienly large such ha Q 1 (s)β 1 + f(s) ds M 4 α, 1, (11) Q 2 (s)β 2 + f(s) ds α 0 M 3 (1+p 2 )M 4, 1, (12)
156 F. Kong Q 1 (s)+q 2 (s) ds ( +p 2 ) 1, 1, (13) where M 3 M 4 are posiive consans such ha 0 M 3 +(1+p 2 )M 4 < M 4 α ( 0 M 3 +(1+p 2 )M 4, M 4 ). Define a mapping S : Ω Λ as follows: 1 P 1 (+τ 1 ) (Sx)() = + +τ 1 (Sx)( 1 ), 0 1. { α x(+τ 1 ) P 2 (+τ 1 )x(+τ 1 +τ 2 ) Q 1 (s)g 1 (x(s σ 1 )) Q 2 (s)g 2 (x(s+σ 2 )) f(s) } ds, 1, Clearly, Sx is coninuous. For 1 x Ω, from (11) (12) i follows ha ( 1 ) (Sx)() α+ Q 2 (s)β 2 + f(s) ds P 1 (+τ 1 ) 1 ) (α+ Q 2 (s)β 2 + f(s) ds M 4, 1 (Sx)() P 1 (+τ 1 ) 1 0 M 3. ( α (1+p 2 )M 4 ( α (1+p 2 )M 4 ) Q 2 (s)β 2 + f(s) ds ) ds Q 2 (s)β 2 + f(s) This means ha SΩ Ω. Since Ω is a bounded, closed, convex subse of Λ, in order o apply he Banach conracion principle we have o show ha S is a conracion mapping on Ω. For x 1,x 2 Ω 1, by using (13), we can obain (Sx1 )() (Sx 2 )() ( 1 ) x 1 x 2 1+p 2 + (Q 1 (s)+q 2 (s))ds This implies x 1 x 2. which shows ha S is a conracion mapping on Ω. Thus, S has a unique fixed poin which is a posiive bounded soluion of (1). This complees he proof. Theorem 5. Assume ha 1 < P 1 () 0 <, 1 < p 2 P 2 () 0 (2) hold; hen (1) has a bounded non-oscillaory soluion.
Exisence of non-oscillaory soluions 157 Proof. e Λ be he se of all coninuous bounded funcions on 0, ) wih he supremum norm. Se Ω = {x Λ : N 3 x() N 4, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. e i, i = 1,2 denoe he ipschiz consans of funcions g i, i = 1,2 on he se Ω, = max{ 1, 2 }, β i = max g i(x),i = 1,2,... In view of (2), we can choose 1 > 0, 1 +τ 1 0 +max{σ 1,σ 1 } sufficienly large such ha Q 1 (s)β 1 + f(s) ds ( +p 2 )N 4 α, 1, (14) Q 2 (s)β 2 + f(s) ds α 0 N 3 N 4, 1, (15) Q 1 (s)+q 2 (s) ds ( +p 2 ) 1, 1, (16) where N 3 N 4 are posiive consans such ha 0 N 3 +N 4 < ( +p 2 )N 4 α ( 0 N 3 +N 4,( +p 2 )N 4 ). Define a mapping S : Ω Λ as follows: 1 P 1 (+τ 1 ) (Sx)() = + +τ 1 (Sx)( 1 ), 0 1. { α x(+τ 1 ) P 2 (+τ 1 )x(+τ 1 +τ 2 ) Q 1 (s)g 1 (x(s σ 1 )) Q 2 (s)g 2 (x(s+σ 2 )) f(s) } ds, 1, Clearly, Sx is coninuous. For 1 x Ω, from (14) (15) i follows ha ( 1 ) (Sx)() α p 2 N 4 + Q 1 (s)β 1 + f(s) ds P 1 (+τ 1 ) 1 ) (α p 2 N 4 + Q 1 (s)β 1 + f(s) ds N 4, 1 (Sx)() P 1 (+τ 1 ) 1 0 N 3. ( α N 4 ( α N 4 ) Q 2 (s)β 2 + f(s) ds ) ds Q 2 (s)β 2 + f(s)
158 F. Kong This proves ha SΩ Ω. To apply he Banach conracion principle i remains o show ha S is a conracion mapping on Ω. For x 1,x 2 Ω 1, by using (16), we can have (Sx1 )() (Sx 2 )() ( 1 x 1 x 2 1 p 2 + This implies x 1 x 2. ) Q 1 (s)+q 2 (s) ds which shows ha S is a conracion mapping on Ω. Thus, S has a unique fixed poin which is a posiive bounded soluion of (1). This complees he proof. Theorem 6. Assume ha 1 < P 1 () 0, 0 P 2 () p 2 < 1+ (2) hold; hen (1) has a bounded non-oscillaory soluion. Proof. e Λ be he se of all coninuous bounded funcions on 0, ) wih he supremum norm. Se Ω = {x Λ : M 5 x() M 6, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. e i, i = 1,2 denoe he ipschiz consans of funcions g i, i = 1,2 on he se Ω, = max{ 1, 2 }, β i = max g i(x),i = 1,2,... Because of (2), we can choose 1 > 0, 1 0 +max{τ 1,σ 1,σ 2 } sufficienly large such ha Q 1 (s)β 1 + f(s) ds (1+ )M 6 α, 1, (17) Q 2 (s)β 2 + f(s) ds α p 2 M 6 M 5, 1, (18) Q 1 (s)+q 2 (s) ds 1+ p 2, 1, (19) where M 5 M 6 are posiive consans such ha M 5 +p 2 M 6 < (1+ )M 6 α (M 5 +p 2 M 6,(1+ )M 6 ). Define an operaor S : Ω Λ as follows: α P 1 ()x( τ 1 ) P 2 ()x(+τ 2 ) (Sx)() = + Q 1 (s)g 1 (x(s σ 1 )) Q 2 (s)g 2 (x(s+σ 2 )) f(s) ds, 1, (Sx)( 1 ), 0 1.
Exisence of non-oscillaory soluions 159 Obviously, Sx is coninuous. For 1 x Ω, from (17) (18) i follows ha (Sx)() α M 6 + Q 1 (s)β 1 + f(s) ds M 6, (Sx)() α p 2 M 6 M 5. Q 2 (s)β 2 + f(s) ds This proves ha SΩ Ω. To apply he Banach conracion principle i remains o show ha S is a conracion mapping on Ω. For x 1,x 2 Ω 1, by using (19), we can ge (Sx1 )() (Sx 2 )() ( x1 x 2 +p 2 + which implies ha x 1 x 2, ) Q 1 (s)+q 2 (s) ds where he supremum norm is used. Thus, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (1). This complees he proof. Theorem 7. Assume ha 1 < P 1 () 0, 1 < p 2 P 2 () 0 (2) hold; hen (1) has a bounded non-oscillaory soluion. Proof. e Λ be he se of coninuous bounded funcions on 0, ) wih he supremum norm. Se Ω = {x Λ : N 5 x() N 6, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. e i, i = 1,2 denoe he ipschiz consans of funcions g i, i = 1,2 on he se Ω, = max{ 1, 2 }, β i = max g i(x),i = 1,2,... Because of (2), we can choose 1 > 0, 1 0 +max{τ 1,σ 1,σ 2 } sufficienly large such ha Q 1 (s)β 1 + f(s) ds (1+ +p 2 )N 6 α, 1, (20) Q 2 (s)β 2 + f(s) ds α N 5, 1, (21)
160 F. Kong Q 1 (s)+q 2 (s) ds 1+ +p 2, 1, (22) where N 5 N 6 are posiive consans such ha N 5 < (1+ +p 2 )N 6 α (N 5,(1+ +p 2 )N 6 ). Define an operaor S : Ω Λ as follows: α P 1 ()x( τ 1 ) P 2 ()x(+τ 2 ) (Sx)() = + Q 1 (s)g 1 (x(s σ 1 )) Q 2 (s)g 2 (x(s+σ 2 )) f(s) ds, 1, (Sx)( 1 ), 0 1. Obviously, Sx is coninuous. For 1 x Ω, from (20) (21) i follows ha (Sx)() α N 6 p 2 N 6 + Q 1 (s)β 1 + f(s) ds N 6, (Sx)() α N 5. Q 2 (s)β 2 + f(s) ds This proves ha SΩ Ω. To apply he Banach conracion principle i remains o show ha S is a conracion mapping on Ω. Thus, for x 1,x 2 Ω 1, by using (22), we can obain (Sx 1 )() (Sx 2 )() x 1 x 2 x 1 x 2, ( p 2 + ) Q 1 (s)+q 2 (s) ds which implies ha where he supremum norm is used. Thus, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (1). This complees he proof. Theorem 8. Assume ha < 0 P 1 () < 1, 0 P 2 () p 2 < 1 (2) hold; hen (1) has a bounded non-oscillaory soluion. Proof. e Λ be he se of all coninuous bounded funcions on 0, ) wih he supremum norm. Se Ω = {x Λ : M 7 x() M 8, 0 }.
Exisence of non-oscillaory soluions 161 I is clear ha Ω is a bounded, closed convex subse of Λ. e i, i = 1,2 denoe he ipschiz consans of funcions g i, i = 1,2 on he se Ω, = max{ 1, 2 }, β i = max g i(x),i = 1,2,... In view of (2), we can choose 1 > 0, 1 +τ 1 0 +max{σ 1,σ 1 } sufficienly large such ha Q 1 (s)β 1 + f(s) ds 0 M 7 +α, 1, (23) Q 2 (s)β 2 + f(s) (s)ds ( 1 p 2 )M 8 α, 1, (24) Q 1 (s)+q 2 (s) ds 1 p 2, 1, (25) where M 7 M 8 are posiive consans such ha 0 M 7 < ( 1 p 2 )M 8 α ( 0 M 7,( 1 p 2 )M 8 ). Define a mapping S : Ω Λ as follows: 1 P 1 (+τ 1 ) (Sx)() = +τ 1 (Sx)( 1 ), 0 1. { α+x(+τ 1 )+P 2 (+τ 1 )x(+τ 1 +τ 2 ) Q 1 (s)g 1 (x(s σ 1 )) Q 2 (s)g 2 (x(s+σ 2 )) f(s) } ds, 1 Clearly, Sx is coninuous. For 1 x Ω, i follows from (23) (24) ha (Sx)() 1 ( ) α+m 8 +p 2 M 8 + Q 1 (s)β 1 + f(s) ds M 8, (Sx)() 1 ( α 0 M 7. ) Q 2 (s)β 2 + f(s) ds This implies ha SΩ Ω. To apply he Banach conracion principle i remains o show ha S is a conracion mapping on Ω. Thus, for x 1,x 2 Ω 1, by using (25), we ge (Sx 1 )() (Sx 2 )() 1 ( x 1 x 2 1+p 2 + x 1 x 2. ) Q 1 (s)+q 2 (s) ds
162 F. Kong This implies where he supremum norm is used. Thus, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (1). This complees he proof. Theorem 9. Assume ha < 0 P 1 () < 1, +1 < p 2 P 2 () 0 (2) hold; hen (1) has a bounded non-oscillaory soluion. Proof. e Λ be he se of coninuous bounded funcions on 0, ) wih he supremum norm. Se Ω = {x Λ : N 7 x() N 8, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. e i, i = 1,2 denoe he ipschiz consans of funcions g i, i = 1,2 on he se Ω, = max{ 1, 2 }, β i = max g i(x),i = 1,2,... In view of (2), we can choose 1 > 0, 1 +τ 1 0 +max{σ 1,σ 1 } sufficienly large such ha Q 1 (s)β 1 + f(s) ds 0 N 7 +p 2 N 8 +α, 1, (26) Q 2 (s)β 2 + f(s) ds ( 1)N 8 α, 1, (27) Q 1 (s)+q 2 (s) ds p 2 1, 1, (28) where N 7 N 8 are posiive consans such ha 0 N 7 p 2 N 8 < ( 1)N 8 α ( 0 N 7 p 2 N 8,( 1)N 8 ). Define a mapping S : Ω Λ as follows: 1 P 1 (+τ 1 ) (Sx)() = +τ 1 (Sx)( 1 ), 0 1. { α+x(+τ 1 )+P 2 (+τ 1 )x(+τ 1 +τ 2 ) Q 1 (s)g 1 (x(s σ 1 )) Q 2 (s)g 2 (x(s+σ 2 )) f(s) } ds, 1, Clearly, Sx is coninuous. For 1 x Ω, from (26) (27) i follows ha (Sx)() 1 ( ) α+n 8 + Q 1 (s)β 1 + f(s) ds N 8
Exisence of non-oscillaory soluions 163 (Sx)() 1 ( α+p 2 N 8 0 N 7. ) Q 2 (s)β 2 + f(s) ds These prove ha SΩ Ω. To apply he Banach conracion principle i remains o show ha S is a conracion mapping on Ω. Thus, for x 1,x 2 Ω, 1, by using (28), we can obain (Sx 1 )() (Sx 2 )() 1 ( x 1 x 2 1 p 2 + This implies x 1 x 2. ) Q 1 (s)+q 2 (s) ds where he supremum norm is used. Hence, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (1). This complees he proof. Acknowledgemen The auhors would like o express heir grea hanks o he reviewers who carefully reviewed he manuscrip. References 1 R. Agarwal, S. Grace, Oscillaion heorems for cerain neural funcional differenial equaions, Compu. Mah. Appl. 38(1999), 1 11. 2 R.Agarwal, S.Grace, D.O Regan, Oscillaion Theory for Difference Funcional Differenial Equaions, Kluwer Academic Publishers, 2000. 3 R. Agarwal, M. Bohner, W. i, Nonoscillaion Oscillaion: Theorey for Funcional Differenial Equaions, Marcel Dekker Inc., New York, 2004. 4 R. Agarwal, M. Bohner, T. i, Oscillaion of second-order nonlinear neural dynamic equaions, Dynam. Sysems Appl. 22(2013), 535 542. 5 R. Agarwal, C. Zhang, T. i, Some remarks on oscillaion of second order neural differenial equaions, Appl. Mah. Compu. 274(2016), 178 181. 6 D. Bainov, D. Mishev, Oscillaion Theory for Neural Differenial Equaions wih Delay, Adam Hilger, 1991. 7 B. Baculikova, T. i, J. Dzurina, Oscillaion heorems for second order neural differenial equaions, Elecron. J. Qual. Theory Differ. Equ. 74(2011), 1 13. 8 T. Can, R. Dahiya, Oscillaion heorems for nh-order neural funcional differenial equaions, Mah. Compu. Modelling. 43(2006), 357 367. 9 T. Can, R. Dahiya, Exisence of nonoscillaory soluions of firs second order neural differenial equaions wih disribued deviaing argumens, J. Franklin Ins. 347(2010), 1309 1316. 10 T. Can, Exisence of nonoscillaory soluions of firs-order nonlinear neural differenial equaions, Appl. Mah. e. 26(2013), 1182 1186.
164 F. Kong 11 T.Can, Exisence of non-oscillaory soluions o firs-order neural differenial equaions, Elecron. J. Diff. Equ. 39(2016), 1 11. 12 M. Chen, J. Yu, Z. Wang, Nonoscillaory soluions of neural delay differenial equaions, Bull. Ausral. Mah. Soc. 48(1993), 475 483. 13 B. Dorociaková, A. Najmanová, R. Olach, Exisence of nonoscillaory soluions of firs-order neural differenial equaions, Absr. Appl. Anal. 2011(2011), Aricle ID 346745, 9pp. 14. Erbe, Q. Kong, B. Zhang, Oscillaion Theory for Funcional Differenial Equaions, Marcel Dekker Inc., New York, 1995. 15 I. Györi, G. adas, Oscillaion Theory of Delay Differenial Equaions Wih Applicaions, Clarendon Press, Oxford, 1991. 16 Z.Han, T.i, C.Zhang, S.Sun, Oscillaory behavior of soluions of cerain hirdorder mixed neural funcional differenial equaions, Bull. Malays. Mah. Sci. Soc. 35(2012), 611 620. 17 Y. Jiang, Y. Fu, H. Wang, T. i, Oscillaion of forced second-order neural delay differenial equaions, Adv. Difference Equ. 2015(2015), 1. 18 M. Kulenović, S. Hadžiomerspahić, Exisence of nonoscillaory soluion of secondorder linear neural delay equaion, J. Mah. Anal. Appl. 228(1998), 436 448. 19 G. adde, V. akshmikanham, B. Zhang, Oscillaion Theory of Differenial Equaions wih Deviaing Argumens, Marcel Dekker Inc., New York, 1987. 20 T. i, C. Zhang, G. Xing, Oscillaion of hird-order neural delay differenial equaions, Absr. Appl. Anal. 2012(2012), Aricle ID 569201, 11 pp. 21 T. i, E. Thapani, Oscillaion of soluions o second-order neural differenial equaions, Elecron. J. Diff. Equ. 67(2014), 1 7. 22 T. i, Y. Rogovchenko, Oscillaion heorems for second-order nonlinear neural delay differenial equaions, Absr. Appl. Anal. 2014(2014), Aricel ID 594190, 5 pp. 23 T. i, Y. Rogovchenko, Oscillaion of second-order neural differenial equaions, Mah. Nachr. 288(2015), 1150 1162. 24 T. i, Yu. V. Rogovchenko, Oscillaion crieria for even-order neural differenial equaions, Appl. Mah. e. 61(2016), 35 41. 25 R. Olah, Noe on he oscillaion of differenial equaion wih advanced argumen, Mah. Slav. 33(1983), 241 248. 26 Y. Tian, Y. Cai, T. i, Exisence of nonoscillaory soluions o second-order nonlinear neural difference equaions, J. Nonlinear Sci. Appl. 8(2015), 884 892. 27 A. Tanawy, R. Dahiya, Oscillaion of even order neural delay differenial equaions, in: Differenial Equaions Sabiliy Conrol, (S. Elaydi, Ed.), ecure Noes in Pure Applied Mahemaics, Marcel Dekker Inc., New York, 1991, 503 509. 28 J. Yu, Y. Wang, Nonoscillaion of a neural delay differenial equaion, Rad. Ma. 8(1992-1996), 127 133. 29 Y. Yu, H. Wang, Nonoscillaory soluions of second-order nonlinear neural delay equaions, J. Mah. Anal. Appl. 311(2005), 445 456. 30 W. Zhang, W. Feng, J. Yan, J. Song, Exisence of nonoscillaory soluions of firsorder linear neural delay differenial equaions, Compu. Mah. Appl. 49(2005), 1021 1027. 31 Y. Zhou, B. Zhang, Exisence of nonoscillaory soluions of higher-order neural differenial equaions wih posiive negaive coefficiens, Appl. Mah. e. 15(2002), 867 874. 32 C. Zhang, R. Agarwal, M. Bohner, T. i, Oscillaion of second-order nonlinear neural dynamic equaions wih noncanonical operaors, Bull. Malays. Mah. Sci. Soc. 38(2015), 761 778.