Elecronic Journal of Differenial Equaions, Vol. 206 (206, No. 39, pp.. ISSN: 072-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu EXISTENCE OF NON-OSCILLATORY SOLUTIONS TO FIRST-ORDER NEUTRAL DIFFERENTIAL EQUATIONS TUNCAY CANDAN Absrac. This aricle presens sufficien condiions for he exisence of nonoscillaory soluions o firs-order differenial equaions having boh delay advance erms, known as mixed equaions. Our main ool is he Banach conracion principle.. Inroducion In his aricle, we consider a firs-order neural differenial equaion d d [x( + P (x( τ + P 2 (x( + τ 2 ] + Q (x( σ Q 2 (x( + σ 2 = 0, (. where P i C([ 0,, R, Q i C([ 0,, [0,, τ i > 0 σ i 0 for i =, 2. We give some new crieria for he exisence of non-oscillaory soluions of (.. Recenly, he exisence of non-oscillaory soluions of firs-order neural funcional differenial equaions has been invesigaed by many auhors. Yu Wang [6] showed ha he equaion d d [x( + px( c] + Q(x( σ = 0, 0 has a non-oscillaory soluion for p 0. Laer, in 993, Chen e al [9] sudied he same equaion hey exended he resuls o he case p R\{ }. Zhang e al [7] invesigaed he exisence of non-oscillaory soluions of he firs-order neural delay differenial equaion wih variable coefficiens d d [x( + P (x( τ] + Q (x( σ Q 2 (x( σ 2 = 0, 0. They obained sufficien condiions for he exisence of non-oscillaory soluions depending on he four differen ranges of P (. In [0], exisence of non-oscillaory soluions of firs-order neural differenial equaions was sudied. d [x( a(x( τ] = p(f(x( σ d 200 Mahemaics Subjec Classificaion. 34K, 34C0. Key words phrases. Neural equaions; fixed poin; non-oscillaory soluion. c 206 Texas Sae Universiy. Submied Ocober 4, 205. Published January 27, 206.
2 T. CANDAN EJDE-206/39 On he oher h, here has been research aciviies abou he oscillaory behavior of firs higher order neural differenial equaions wih advanced erms. For insance, in [] [5], n-h order neural differenial equaions wih advanced erm of he form [x( + ax( τ + bx( + τ] (n + δ (q(x( g + p(x( + h = 0 ( d [x(+λax( τ+µbx(+τ] (n +δ q(, ξx( ξdξ + c d c p(, ξx(+ξdξ = 0, were sudied, respecively. This aricle was moivaed by he above sudies. To he bes of our knowledge, his curren paper is he only paper regarding o he exisence of non-oscillaory soluions of neural differenial equaion wih advanced erm. Some oher papers for he exisence of non-oscillaory soluions of firs, second higher order neural funcional differenial difference equaions; see [3, 8, 6, 7, 8, 5] he references conained herein. We refer he reader o he books [4, 2, 4,, 2, 3] on he subjec of neural differenial equaions. Le m = max{τ, σ }. By a soluion of (. we mean a funcion x C([ m,, R, for some 0, such ha x( + P (x( τ + P 2 (x( + τ 2 is coninuously differeniable on [, (. is saisfied for. As i is cusomary, a soluion of (. 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 heorems. Theorem. (Banach s Conracion Mapping Principle. 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 ( P 2 (. Theorem 2.. Assume ha 0 P ( p <, 0 P 2 ( p 2 < p 0 Q (sds <, hen (. has a bounded non-oscillaory soluion. Proof. Because of (2., we can choose a > 0, sufficienly large such ha 0 Q 2 (sds <, (2. 0 + max{τ, σ } (2.2 Q (sds M 2 α M 2,, (2.3 Q 2 (sds α (p + p 2 M 2 M M 2,, (2.4 where M M 2 are posiive consans such ha (p + p 2 M 2 + M < M 2 α ( (p + p 2 M 2 + M, M 2.
EJDE-206/39 EXISTENCE OF NON-OSCILLATORY SOLUTIONS 3 Le Λ be he se of all coninuous bounded funcions on [ 0, wih he supremum norm. Se Ω = {x Λ : M x( M 2, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. Define an operaor α P (x( τ P 2 (x( + τ 2 (Sx( = + [Q (sx(s σ Q 2 (sx(s + σ 2 ]ds,, (Sx(, 0. Obviously, Sx is coninuous. For x Ω, from (2.3 (2.4, respecively, i follows ha (Sx( α + Q (sx(s σ ds α + M 2 Q (sds M 2 (Sx( α P (x( τ P 2 (x( + τ 2 α p M 2 p 2 M 2 M 2 Q 2 (sds M. Q 2 (sx(s + σ 2 ds This means ha SΩ Ω. To apply he conracion mapping principle, he remaining is o show ha S is a conracion mapping on Ω. Thus, if x, x 2 Ω, or (Sx ( (Sx 2 ( P ( x ( τ x 2 ( τ + P 2 ( x ( + τ 2 x 2 ( + τ 2 + (Q (s x (s σ x 2 (s σ + Q 2 (s x (s + σ 2 x 2 (s + σ 2 ds (Sx ( (Sx 2 ( ( x x 2 p + p 2 + (Q (s + Q 2 (s ds ( p + p 2 + M 2 α + α (p + p 2 M 2 M x x 2 M 2 M 2 = λ x x 2, where λ = ( M M 2. This implies ha Sx Sx 2 λ x x 2, where he supremum norm is used. Since λ <, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (.. This complees he proof. Theorem 2.2. Assume ha 0 P ( p <, p < p 2 P 2 ( 0 (2. hold, hen (. has a bounded non-oscillaory soluion.
4 T. CANDAN EJDE-206/39 Proof. Because of (2., we can choose a > 0 sufficienly large saisfying (2.2 such ha Q (sds ( + p 2N 2 α,, (2.5 N 2 where N N 2 are posiive consans such ha Q 2 (sds α p N 2 N N 2,, (2.6 N + p N 2 < ( + p 2 N 2 α (N + p N 2, ( + p 2 N 2. Le Λ be he se of all coninuous bounded funcions on [ 0, wih he supremum norm. Se Ω = {x Λ : N x( N 2, 0 }. I is clear ha Ω is a bounded, closed convex subse of Λ. Define an operaor α P (x( τ P 2 (x( + τ 2 (Sx( = + [Q (sx(s σ Q 2 (sx(s + σ 2 ] ds,, (Sx(, 0. Obviously, Sx is coninuous. For x Ω, from (2.5 (2.6, respecively, i follows ha (Sx( α p 2 N 2 + N 2 Q (sds N 2, (Sx( α p N 2 N 2 Q 2 (sds N. This proves ha SΩ Ω. To apply he conracion mapping principle, i remains o show ha S is a conracion mapping on Ω. Thus, if x, x 2 Ω, ( (Sx ( (Sx 2 ( x x 2 p p 2 + (Q (s + Q 2 (s ds where λ 2 = ( N N 2. This implies λ 2 x x 2, Sx Sx 2 λ 2 x x 2, where he supremum norm is used. Since λ 2 <, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (.. This complees he proof. Theorem 2.3. Assume ha < p P ( p 0 <, 0 P 2 ( p 2 < p (2. hold, hen (. has a bounded non-oscillaory soluion. Proof. In view of (2., we can choose a > 0, sufficienly large such ha + τ 0 + σ, (2.7 Q (sds p M 4 α M 4,, (2.8 Q 2 (sds α p 0 M 3 ( + p 2 M 4 M 4,, (2.9
EJDE-206/39 EXISTENCE OF NON-OSCILLATORY SOLUTIONS 5 where M 3 M 4 are posiive consans such ha p 0 M 3 + ( + p 2 M 4 < p M 4 α ( p 0 M 3 + ( + p 2 M 4, p M 4. Le 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. Define a mapping P {α x( + τ (+τ P 2 ( + τ x( + τ + τ 2 (Sx( = + +τ [Q (sx(s σ Q 2 (sx(s + σ 2 ] ds},, (Sx(, 0. Clearly, Sx is coninuous. For x Ω, from (2.8 (2.9, respecively, i follows ha (Sx( (α + M 4 Q (sds (α + M 4 Q (sds M 4 P ( + τ p (Sx( P ( + τ ( α ( + p 2 M 4 M 4 p 0 (α ( + p 2 M 4 M 4 Q 2 (sds Q 2 (sds M 3. This means ha SΩ Ω. To apply he conracion mapping principle i remains o show ha S is a conracion mapping on Ω. Thus, if x, x 2 Ω, (Sx ( (Sx 2 ( ( x x 2 + p 2 + (Q (s + Q 2 (s ds p λ 3 x x 2, where λ 3 = ( p 0 M3 p M 4. This implies Sx Sx 2 λ 3 x x 2, where he supremum norm is used. Since λ 3 <, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (.. This complees he proof. Theorem 2.4. Assume ha < p P ( p 0 <, p < p 2 P 2 ( 0 (2. hold, hen (. has a bounded non-oscillaory soluion. Proof. In view of (2., we can choose a > 0 sufficienly large saisfying (2.7 such ha Q (sds (p + p 2 N 4 α,, (2.0 N 4 Q 2 (sds α p 0 N 3 N 4 N 4,, (2. where N 3 N 4 are posiive consans such ha p 0 N 3 + N 4 < (p + p 2 N 4 α ( p 0 N 3 + N 4, (p + p 2 N 4.
6 T. CANDAN EJDE-206/39 Le 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. Define a mapping P {α x( + τ (+τ P 2 ( + τ x( + τ + τ 2 (Sx( = + +τ [Q (sx(s σ Q 2 (sx(s + σ 2 ] ds},, (Sx(, 0. Clearly, Sx is coninuous. For x Ω, from (2.0 (2., respecively, i follows ha (Sx( (α p 2 N 4 + N 4 Q (sds P ( + τ ( α p 2 N 4 + N 4 Q (sds N 4 p (Sx( P ( + τ ( α N 4 N 4 p 0 (α N 4 N 4 Q 2 (sds Q 2 (sds N 3. This proves ha SΩ Ω. To apply he conracion mapping principle i remains o show ha S is a conracion mapping on Ω. Thus, if x, x 2 Ω, (Sx ( (Sx 2 ( ( x x 2 p 2 + (Q (s + Q 2 (s ds p λ 4 x x 2, where λ 4 = ( p 0 N3 p N 4. This implies Sx Sx 2 λ 4 x x 2, where he supremum norm is used. Since λ 4 <, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (.. This complees he proof. Theorem 2.5. Assume ha < p P ( 0, 0 P 2 ( p 2 < + p (2. hold, hen (. has a bounded non-oscillaory soluion. Proof. Because of (2., we can choose a > 0 sufficienly large saisfying (2.2 such ha Q (sds ( + p M 6 α, M 6, (2.2 Q 2 (sds α p 2M 6 M 5, M 6, (2.3 where M 5 M 6 are posiive consans such ha M 5 + p 2 M 6 < ( + p M 6 α (M 5 + p 2 M 6, ( + p M 6.
EJDE-206/39 EXISTENCE OF NON-OSCILLATORY SOLUTIONS 7 Le 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. Define an operaor α P (x( τ P 2 (x( + τ 2 (Sx( = + [Q (sx(s σ Q 2 (sx(s + σ 2 ] ds,, (Sx(, 0. Obviously, Sx is coninuous. For x Ω, from (2.2 (2.3, respecively, i follows ha (Sx( α p M 6 + M 6 Q (sds M 6, (Sx( α p 2 M 6 M 6 Q 2 (sds M 5. This proves ha SΩ Ω. To apply he conracion mapping principle i remains o show ha S is a conracion mapping on Ω. Thus, if x, x 2 Ω,, ( (Sx ( (Sx 2 ( x x 2 p + p 2 + (Q (s + Q 2 (s ds λ 5 x x 2, where λ 5 = ( M5 M 6. This implies Sx Sx 2 λ 5 x x 2, where he supremum norm is used. Since λ 5 <, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (.. This complees he proof. Theorem 2.6. Assume ha < p P ( 0, p < p 2 P 2 ( 0 (2. hold, hen (. has a bounded non-oscillaory soluion. Proof. Because of (2., we can choose a > 0 sufficienly large saisfying (2.2 such ha Q (sds ( + p + p 2 N 6 α,, (2.4 N 6 where N 5 N 6 are posiive consans such ha Q 2 (sds α N 5 N 6,, (2.5 N 5 < ( + p + p 2 N 6 α (N 5, ( + p + p 2 N 6. Le Λ be he se of coninuous bounded funcions on [ 0, wih he supremum norm. Se Ω = {x Λ : N 5 x( N 6, 0 }.
8 T. CANDAN EJDE-206/39 I is clear ha Ω is a bounded, closed convex subse of Λ. Define an operaor α P (x( τ P 2 (x( + τ 2 (Sx( = + [Q (sx(s σ Q 2 (sx(s + σ 2 ] ds,, (Sx(, 0. Obviously, Sx is coninuous. For x Ω, from (2.4 (2.5, respecively, i follows ha (Sx( α p N 6 p 2 N 6 + N 6 Q (sds N 6, (Sx( α N 6 Q 2 (sds N 5. This proves ha SΩ Ω. To apply he conracion mapping principle i remains o show ha S is a conracion mapping on Ω. Thus, if x, x 2 Ω, ( (Sx ( (Sx 2 ( x x 2 p p 2 + (Q (s + Q 2 (s ds λ 6 x x 2, where λ 6 = ( N5 N 6. This implies Sx Sx 2 λ 6 x x 2, where he supremum norm is used. Since λ 6 <, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (.. This complees he proof. Theorem 2.7. Assume ha < p 0 P ( p <, 0 P 2 ( p 2 < p (2. hold, hen (. has a bounded non-oscillaory soluion. Proof. In view of (2., we can choose a > 0 sufficienly large saisfying (2.7 such ha Q (sds p 0 M 7 + α,, (2.6 M 8 Q 2 (sds ( p p 2 M 8 α, M 8, (2.7 where M 7 M 8 are posiive consans such ha p 0 M 7 < ( p p 2 M 8 α ( p 0 M 7, ( p p 2 M 8. Le 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 }. I is clear ha Ω is a bounded, closed convex subse of Λ. Define a mapping P {α + x( + τ (+τ + P 2 ( + τ x( + τ + τ 2 (Sx( = +τ [Q (sx(s σ Q 2 (sx(s + σ 2 ] ds}, (Sx(, 0.
EJDE-206/39 EXISTENCE OF NON-OSCILLATORY SOLUTIONS 9 Clearly, Sx is coninuous. For x Ω, from (2.7 (2.6, respecively, i follows ha (Sx( (α + M 8 + p 2 M 8 + M 8 Q 2 (sds M 8 p (Sx( (α M 8 p 0 Q (sds M 7. This implies ha SΩ Ω. To apply he conracion mapping principle i remains o show ha S is a conracion mapping on Ω. Thus, if x, x 2 Ω, (Sx ( (Sx 2 ( ( x x 2 + p 2 + (Q (s + Q 2 (s ds p λ 7 x x 2, where λ 7 = ( p 0 M7 p M 8. This implies Sx Sx 2 λ 7 x x 2, where he supremum norm is used. Since λ 7 <, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (.. This complees he proof. Theorem 2.8. Assume ha < p 0 P ( p <, p + < p 2 P 2 ( 0 (2. hold, hen (. has a bounded non-oscillaory soluion. Proof. In view of (2., we can choose a > 0 sufficienly large saisfying (2.7 such ha Q (sds p 0 N 7 + p 2 N 8 + α,, (2.8 N 8 Q 2 (sds ( p N 8 α N 8,, (2.9 where N 7 N 8 are posiive consans such ha p 0 N 7 p 2 N 8 < ( p N 8 α ( p 0 N 7 p 2 N 8, ( p N 8. Le 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. Define a mapping P {α + x( + τ (+τ + P 2 ( + τ x( + τ + τ 2 (Sx( = +τ [Q (sx(s σ Q 2 (sx(s + σ 2 ] ds},, (Sx(, 0. Clearly, Sx is coninuous. For x Ω, from (2.9 (2.8, respecively, i follows ha (Sx( ( α + N 8 + N 8 Q 2 (sds N 8 p (Sx( p 0 ( α + p 2 N 8 N 8 Q (sds N 7.
0 T. CANDAN EJDE-206/39 These prove ha SΩ Ω. To apply he conracion mapping principle i remains o show ha S is a conracion mapping on Ω. Thus, if x, x 2 Ω,, (Sx ( (Sx 2 ( ( x x 2 p 2 + (Q (s + Q 2 (s ds p λ 8 x x 2, where λ 8 = ( p 0 N7 p N 8. This implies Sx Sx 2 λ 8 x x 2, where he supremum norm is used. Since λ 8 <, S is a conracion mapping on Ω. Thus S has a unique fixed poin which is a posiive bounded soluion of (.. This complees he proof. Example 2.9. Consider he equaion [ x( 2 x( 2π + [ 2 exp( ] 2 ] x( + 5π + 2 exp( 2 x( 4π exp( 2 x( + 5π 2 = 0, > 2 ln(/2 (2.20 noe ha P ( = 2, P 2( = 2 exp( 2, Q ( = 2 exp( 2, Q 2( = exp( 2. A sraighforward verificaion yields ha he condiions of Theorem 2.5 are valid. We noe ha x( = 2 + sin is a non-oscillaory soluion of (2.20. Example 2.0. Consider he equaion [ x( [3 exp( 4 exp( ] x( exp(/4 [ 4 + exp( ] x( + ] 4 (2.2 + exp( x( exp( + 4 x( + 4 = 0, 3 2 noe ha P ( = [3 exp( 4 exp( ], P 2 ( = exp( 4 [ 4 + exp( ], Q ( = exp(, Q 2 ( = exp( + 4. I is easy o verify ha he condiions of Theorem 2.6 are valid. x( = + exp( is a non-oscillaory soluion of (2.2. We noe ha References [] R. P. Agarwal, S. R. Grace; Oscillaion Theorems for Cerain Neural Funcional Differenial Equaions, Compu. Mah. Appl., 38 (999, -. [2] R. P. Agarwal, S. R. Grace, D. O Regan; Oscillaion Theory for Difference Funcional Differenial Equaions, Kluwer Academic, (2000. [3] R. P. Agarwal, M. Bohner, W. T. Li; Nonoscillaion Oscillaion: Theorey for Funcional Differenial Equaions, Marcel Dekker, Inc., New York, 2004. [4] D. D. Bainov, D. P. Mishev; Oscillaion Theory for Neural Differenial Equaions wih Delay, Adam Hilger, (99. [5] T. Can, R. S. Dahiya; Oscillaion heorems for nh-order neural funcional differenial equaions, Mah. Compu. Modelling, 43 (2006,357-367.
EJDE-206/39 EXISTENCE OF NON-OSCILLATORY SOLUTIONS [6] T. Can R. S. Dahiya; Exisence of nonoscillaory soluions of firs second order neural differenial equaions wih disribued deviaing argumens, J. Franklin Ins., 347 (200, 309-36. [7] T. Can; The exisence of nonoscillaory soluions of higher order nonlinear neural equaions, Appl. Mah. Le., 25(3 (202, 42-46. [8] T. Can; Exisence of nonoscillaory soluions of firs -order nonlinear neural differenial equaions, Appl. Mah. Le., 26 (203, 82-86. [9] M. P. Chen, J. S. Yu, Z. C. Wang; Nonoscillaory soluions of neural delay differenial equaions, Bull. Ausral. Mah. Soc., 48(3 (993,475-483. [0] B. Dorociaková, A. Najmanová, R. Olach; Exisence of nonoscillaory soluions of firs-order neural differenial equaions, Absr. Appl. Anal., 20, Ar. ID 346745, 9 pp. [] L. H. Erbe, Q. K. Kong, B. G. Zhang; Oscillaion Theory for Funcional Differenial Equaions, Marcel Dekker, Inc., New York, (995. [2] I. Györi, G. Ladas; Oscillaion Theory of Delay Differenial Equaions Wih Applicaions, Clarendon Press, Oxford, (99. [3] M. R. S. Kulenović, S. Hadžiomerspahić; Exisence of Nonoscillaory Soluion of Second- Order Linear Neural Delay Equaion, J. Mah.Anal. Appl., 228 (998, 436-448. [4] G. S. Ladde, V. Lakshmikanham, B. G. Zhang; Oscillaion Theory of Differenial Equaions wih Deviaing Argumens, Marcel Dekker, Inc., New York, (987. [5] Y. Tian, Y. Cai, T. Li; Exisence of nonoscillaory soluions o second-order nonlinear neural difference equaions, J. Nonlinear Sci. Appl., 8 (205, 884-892. [6] J. Yu, Y. Wang; Nonoscillaion of a neural delay differenial equaion, Rad. Ma., 8( (992/996, 27-33. [7] W. Zhang, W. Feng, J. Yan, J. Song; Exisence of Nonoscillaory Soluions of Firs-Order Linear Neural Delay Differenial Equaions, Compu. Mah. Appl., 49 (2005, 02-027. [8] Y. Zhou, B. G. Zhang; Exisence of Nonoscillaory Soluions of Higher-Order Neural Differenial Equaions wih Posiive Negaive Coefficiens, Appl. Mah. Le., 5 (2002, 867-874. Tuncay Can Deparmen of Mahemaics, Faculy of Ars Sciences, Niğde Universiy, Niğde 5200, Turkey E-mail address: can@nigde.edu.r