PRECISE ASYMPTOTIC BEHAVIOR OF STRONGLY DECREASING SOLUTIONS OF FIRST-ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

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1 Elecronic Journal of Differenial Equaions, Vol ), No. 206, pp. 4. ISSN: URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu PRECISE ASYMPTOTIC BEHAVIOR OF STRONGLY DECREASING SOLUTIONS OF FIRST-ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS GEORGE E. CHATZARAKIS, KUSANO TAKAŜI, IOANNIS P. STAVROULAKIS Absrac. In his aricle, we sudy he asympoic behavior of srongly decreasing soluions of he firs-order nonlinear funcional differenial equaion x ) + p) xg)) α xg)) = 0, where α is a posiive consan such ha 0 < α <, p) is a posiive coninuous funcion on [a, ), a > 0 and g) is a posiive coninuous funcion on [a, + ) such ha lim g) =. Condiions which guaranee he exisence of srongly decreasing soluions are esablished, and heorems are saed on he asympoic behavior of such soluions, a infiniy. The problem i is sudied in he framework of regular variaion, assuming ha he coefficien p) is a regularly varying funcion, and focusing on srongly decreasing soluions ha are regularly varying. In addiion, g) is required o saisfy he condiion g) lim =. Examples illusraing he resuls are also given.. Inroducion Consider he firs-order nonlinear funcional differenial equaion x ) + p) xg)) α xg)) = 0,.) where α is a posiive consan such ha 0 < α <, p) is a posiive coninuous funcion on [a, ), a > 0 and g) is a posiive coninuous funcion on [a, + ) such ha lim g) =. By a soluion of he equaion.), we mean a funcion x) which saisfies.) for all a. A soluion x) of he equaion.) is called oscillaory, if he erms x) of he funcion are neiher evenually posiive nor evenually negaive. Oherwise, he soluion is said o be nonoscillaory. Assume ha he soluion x) of.) is nonoscillaory. Then i is eiher evenually posiive or evenually negaive. As x) is also a soluion of.), we may resric ourselves o he case where x) > 0 for all large Mahemaics Subjec Classificaion. 34C, 26A2. Key words and phrases. Funcional differenial equaion; srongly decreasing soluion; regularly varying funcion; slowly varying soluion. c 204 Texas Sae Universiy - San Marcos. Submied June 26, 204. Published Ocober 2, 204.

2 2 G. E. CHATZARAKIS, K. TAKAŜI, I. P. STAVROULAKIS EJDE-204/206 We are ineresed in he asympoic behavior of posiive soluions of.) exising in some neighborhood of infiniy and decreasing o zero as. Such soluions are ofen referred o as srongly decreasing soluions of.). An ineresing quesion hen arises wheher.) possess srongly decreasing soluions. If his is he case, is i possible o deermine he asympoic behavior a infiniy of is srongly decreasing soluions precisely? I seems o be difficul o answer he equaion in general. So, we sudy he problem in he framework of regular variaion, which means ha he coefficien p) is assumed o be a regularly varying funcion and our aenion is focused only on srongly decreasing soluions which are regularly varying. Then, he quesion is answered in he affirmaive in he case ha he deviaing argumen g) is required o saisfy he condiion g) lim =..2) For he reader s convenience we recall here he definiion of regularly varying funcions, noaions and some of basic properies including Karamaa s inegraion heorem which will play an imporan role in esablishing he main resuls of his paper. Definiion.. A measurable funcion f : [0, ) 0, ) is called regularly varying of index ρ R if i saisfies fλ) lim f) = λρ for all λ > 0. The se of all regularly varying funcions of index ρ is denoed by RVρ). The symbol SV is ofen used o denoe RV0) in which case members of SV are called slowly varying funcions. Since any funcion f) RVρ) is expressed as f) = ρ g) wih g) SV, he class SV of slowly varying funcions is of fundamenal imporance in he heory of regular variaion. Typical examples of slowly varying funcions are all funcions ending o posiive consans as, N log n ) αn, α n R, exp { N log n ) βn}, β n 0, ), n= where log n denoes he n-h ieraion of log and log denoes he naural logarihm. I is known ha he funcion 2 + sinlog log ) is regularly varying, whereas 2 + sinlog ) is no. The funcion n= L) = exp { log ) θ coslog ) θ}, θ 0, 2 ), is a slowly varying funcion which is oscillaing in he sense ha lim sup L) = and lim inf L) = 0. n One of he mos imporan properies of regularly varying funcions is he following represenaion heorem.

3 EJDE-204/206 FIRST-ORDER SUBLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 3 Proposiion.2. A funcion f) RVρ) if and only if f) is represened in he form f) = c) exp { δs) 0 s ds}, 0,.3) for some 0 > 0 and for some measurable funcions c) and δ) such ha lim c) = c 0 0, ) and lim δ) = ρ. If c) c 0, hen f) is called a normalized regularly varying funcion of index ρ. The following resul illusraes operaions which preserve slow variaion. Proposiion.3. Le L), L ), L 2 ) be slowly varying. Then, L) α for any α R, L ) + L 2 ), L )L 2 ) and L L 2 )) if L 2 ) ) are slowly varying. A slowly varying funcion may grow o infiniy or decay o zero as. Bu is order of growh or decay is several limied as in shown in he following. Proposiion.4. Le f) SV. Then, for any ε > 0, lim ε f) = and lim ε f) = 0. A simple crierion for deermining he regularly of differeniable posiive funcions follows see [6]). Proposiion.5. A differeniable posiive funcion f) is a normalized regularly varying funcion of index ρ if and only if lim f ) f) = ρ. The following proposiion, known as Karamaa s inegraion heorem [9, 20], is of highes imporance in handling slowly and regularly varying funcions analyically. Here and hroughou he symbol is used o denoe he asympoic equivalence of wo posiive funcions, ha is f) g), Proposiion.6. Le L) SV. Then i) if α >, a ii) if α <, iii) if α =, l) = m) = g) lim f) =. s α Ls)ds α + α+ L), ; s α Ls)ds α + α+ L), ; a Ls) s L) ds SV, and lim Ls) ds SV, and lim s l) = 0, L) m) = 0. Here in defining m) i is assumed ha L)/ is inegrable near he infiniy.

4 4 G. E. CHATZARAKIS, K. TAKAŜI, I. P. STAVROULAKIS EJDE-204/206 We now define he class of nearly regularly varying funcions. To his end i is convenien o inroduce he following noaion. Noaion. Le f) and g) be wo posiive coninuous funcions defined in a neighborhood of infiniy, say for T. We use he noaion f) g),, o denoe ha here exis posiive consans k and K such ha kg) f) Kg) for T. Clearly, f) g),, implies f) g),, bu no conversely. I is easy o see ha if f) g), and if lim g) = 0, hen lim f) = 0. Definiion.7. If f) saisfies f) g),, for some g) which is regularly varying of index ρ, hen f) is called a nearly regularly varying funcion of index ρ. For example, he funcion 2 + sinlog ) is nearly slowly varying because 2 + sinlog ) 2+sinlog log ),. I follows ha, for any ρ R, ρ 2+sinlog )) is nearly regularly varying, bu no regularly varying, of index ρ. For a complee exposiion of heory of regular variaion and is applicaions we refer he reader o he book by Bingham, Goldie and Teugels []. See also Senea [2], Geluk and Haan [4]. A comprehensive survey of resuls up o 2000 on he asympoic analysis of second order ordinary differenial equaions by means of regular variaion can be found in he monograph of Marić [6]. Since he publicaion of [6] here has been an increasing ineres in he analysis of ordinary differenial equaions by means of regularly varying funcions, and hus heory of regular variaion has proved o be a powerful ool of deermining he accurae asympoic behavior of posiive soluions for a variey of nonlinear differenial equaions of Emden-Fowler and Thomas-Fermi ypes. See, for example, he papers [3, 6, 7, 9, 0,, 2, 3, 4, 5, 7, 8]. 2. Main resuls In his secion we esablish condiions under which.) possess srongly decreasing soluions, and sudy he asympoic behavior of such soluions. To his end, he following lemmas provide useful ools. Lemma 2.. Assume ha.2) is saisfied. Then, for any regularly varying funcion f), i holds ha fg)) f) as. Proof. Suppose ha f RVσ). By Proposiion.2, f) is represened as f) = c) exp { 0 δs) s ds}, 0, for some 0 > 0 and for some measurable funcions c) and δ) such ha lim c) = c 0 0, ) and lim δ) = σ. We may assume ha cg))/c) 2 and δ) 2 σ for 0. Then, fg)) f) = cg)) c) exp { g) δs) s ds} 2 exp { 2 σ = 2 exp { 2 σ log g) ) },, g) ds s }

5 EJDE-204/206 FIRST-ORDER SUBLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 5 which means ha fg)) f),. The proof is complee. Nex we have a generalized L Hospial s rule. Lemma 2.2 [5]). Le f), g) C [T, ) and suppose ha or Then lim f) = lim g) = and g ) > 0 for all large, lim f) = lim g) = 0 and g ) < 0 for all large. lim inf f ) f) g lim inf ) g), f) f ) lim sup lim sup g) g ). The main resuls of his aricle are described in he following wo heorems. Theorem 2.3. Suppose ha p) is regularly varying and g) saisfies.2). Then,.) possesses srongly decreasing slowly varying soluions if and only if p RV ) and a p)d <, 2.) in which case any such soluion x) obeys he unique decay law ) x) α) ps)ds,. 2.2) Theorem 2.4. Suppose ha p) is regularly varying and g) saisfies.2). Then,.) possesses srongly decreasing regularly varying soluions of negaive index ρ if and only if p RVλ) wih λ <, 2.3) in which case ρ is given by ρ = λ + α, 2.4) and any such soluion x) obeys he unique decay law x) p) ρ ),. 2.5) Proof. We give simulaneous proof of boh Theorems 2.3 and 2.4. We assume ha p RVλ) is represened in he form p) = λ l), l SV, 2.6) and seek srongly decreasing regularly varying soluions x) of.) expressed as Our aim is o solve he inegral equaion x) = ρ ξ), ξ SV, ρ ) x) = ps)xgs)) α ds, 2.8) in some neighborhood of infiniy in he class of regularly varying funcions. Suppose ha.) has a srongly decreasing soluion x RVρ). Using 2.6), 2.7),.2) and Lemma 2., we have ps)xgs)) α ds = s λ gs) αρ ls)ξgs)) α ds s λ+αρ ls)ξs) α ds, 2.9) as. The convergence of he las inegral means ha λ + αρ.

6 6 G. E. CHATZARAKIS, K. TAKAŜI, I. P. STAVROULAKIS EJDE-204/206 Firs consider he case ha λ + αρ =. Then, from 2.8) and 2.9) i follows ha x) s ls)ξs) α ds SV,, 2.0) which implies ha x SV; ha is, ρ = 0 x) = ξ)), and so we see ha λ =. Now le η) denoe he righ-hand side of 2.0). Then, which can be rewrien as η l)ξ) α l)η) α, η) α η l) = p), or η) α ) p), 2.) as. Since η) 0,, 2.) implies ha p) is inegrable on [a, ), and inegraing 2.) from, we easily find ha ) x) α) ps)ds,. Consider nex he case ha λ+αρ <. Then, applying Par ii) of Proposiion.6 o 2.9), we obain This shows ha ρ = λ + αρ +, or x) λ+αρ+ l)ξ) α,, 2.2) λ + αρ + ) ρ = λ + α. Since ρ < 0 in his case, we mus have λ <. Noing ha 2.2) is rewrien as x) p)x)α,, ρ we immediaely obain he asympoic relaion for x): x) p) ρ ),. The above observaions can be summarized as follows. A srongly decreasing regularly varying soluion x) of.) is eiher slowly varying ρ = 0) in which case p) saisfies 2.), or regularly varying of negaive index ρ in which case p) saisfies 2.3) and ρ is given by 2.4). Furhermore, he asympoic behavior of x) is governed by he unique formula 2.2) and 2.5) according as ρ = 0 and ρ < 0. This complees he proof of he only if pars of Theorems 2.3 and 2.4. The proof of he if pars of he heorems proceeds as follows. Assume ha p) saisfies eiher 2.) or 2.3). Le α) ps)ds) if λ =, X) = p) ) 2.3) ρ if λ <, where ρ = λ+. I can be shown ha X) saisfies he inegral asympoic relaion ps)xgs)) α ds X),. 2.4)

7 EJDE-204/206 FIRST-ORDER SUBLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 7 In fac, if λ = and p) is inegrable on [a, )), hen ps)xgs)) α ds ps)xs) α ds = ps) α) = α) ps)ds ) = X),, s ) α pr)dr ds and if λ <, hen using he expression for X) = ρ l)/ ρ)) /), we find ha ps)xgs)) α ds ps)xs) α ds = s λ+αρ ls) ls) ρ ) α ds = s ρ ls) ls) ρ ) α ds ρ l) ρ l) ρ ) α = ρ l) ρ ) = X), as. In view of 2.4) here exiss T > a such ha T 0 := inf T g) a and 2 X) ps)xgs)) α ds 2X), T. 2.5) Noice ha X) is decreasing in case λ =. We may assume ha X) is also decreasing on [T 0, ) in case λ <. This follows from [, Theorem.5.3] which assers ha any funcion f RVρ) wih nonzero index ρ is asympoic o a monoone funcion. Choose posiive consans k and K such ha and define he se X of coninuous funcions by k 2, K 2, 2.6) X = { x C[T 0, ) : kx) x) KX), T 0 }. 2.7) Finally consider he mapping F : X C[T 0, ) defined by { ps)xgs)) α ds for T, F x) = F xt ) XT ) X) for T 0 T. 2.8) We will show ha he Schauder-Tychonoff fixed poin heorem see e.g., [2, Chaper I]) is applicable o F acing on X. Le x X. Using 2.5) 2.8), we see ha F x) F x) ps)kxgs))) α ds 2K α X) KX), ps)kxgs))) α ds 2 kα X) kx), so ha kx) F x) KX) for T. The las inequaliy clearly holds for T 0 T since k F xt )/XT ) K. Thus, F maps X ino iself. Since kx) F x) KX) for T 0, he se F X ) is uniformly bounded on [T 0, ). Since 0 F x) α p)xg)) α, T, for all x X. F X ) is equiconinuous on [T, ), and hence on [T 0, ). Then, he relaive compacness of F X ) follows from he Ascoli s heorem.

8 8 G. E. CHATZARAKIS, K. TAKAŜI, I. P. STAVROULAKIS EJDE-204/206 Finally, le {x n )} be any sequence in X converging, as n, o x) in X uniformly on compac subinervals of [T 0, ). Then, by 2.8) we have F x n ) F x) and, for T 0 T, ps) x n gs)) α xgs)) α ds, T, 2.9) F x n ) F x) = F x nt ) F xt ) X) F x n T ) F xt ). 2.20) XT ) Applicaion of he Lebesgue dominaed convergence heorem o he righ-hand side of 2.9) ensures ha, as n, F x n ) converges o F x) uniformly on [T, ), and using his fac in 2.20) we conclude ha he convergence F x n ) F x) is uniform on he enire inerval [T 0, ). Thus all he hypoheses of he Schauder-Tychonoff fixed poin heorem are fulfilled, and hence here exiss x X such ha x = F x, which implies in paricular ha x) saisfies he inegral equaion 2.8) on [T, ), ha is, x) is a srongly decreasing soluion of equaion.). The membership x X implies ha x) is a nearly regularly varying of he same index as X). I remains o verify ha x) is cerainly a regularly varying wih he help of he generalized L Hospial s rule Lemma 2.2). Le x) be he srongly decreasing soluion of.) obained above as a soluion of he inegral equaion 2.8). Define he funcion u) by and pu u) = m = lim inf x) u), ps)xgs)) α ds, 2.2) x) M = lim sup u). 2.22) Since x) X),, i is clear ha 0 < m M <. We now apply Lemma 2.2 o M, obaining x ) p)xg)) α M lim sup u = lim sup ) p)xg)) α xg)) ) α x) = lim sup = lim sup Xg)) X) ) α = = M α, lim sup x) u) where he relaion u) X),, cf. 2.4)) has been used in he las sep. Thus, M M α, which implies M because of α <. Likewise, applicaion of Lemma 2.2 o m leads o m. I follows herefore ha m = M = ; ha is, i.e., x) lim u) = ; x) u) X),, which shows ha x) is a regularly varying funcion of index ρ = 0 or of index ρ = λ + )/ α) < 0 according as λ = or λ <. This complees he proof of Theorems 2.3 and 2.4. ) α

9 EJDE-204/206 FIRST-ORDER SUBLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 9 3. Perurbaions of equaion.) Consider he following perurbaion of equaion.), x ) + p) xg)) α xg)) + q) xh)) β xh)) = 0, 3.) where α is a posiive consan such ha 0 < α <, p) is a posiive coninuous funcion on [a, ), a > 0 and g) is a posiive coninuous funcion on [a, + ) such ha lim g) =, β is a posiive consan, q) is a posiive coninuous funcion on [a, ) and h) is a coninuous deviaing argumen on [a, ) such ha lim h) =. Our purpose here is o show ha he srucure of srongly decreasing soluions of equaion.) remains essenially unchanged provided he perurbaion is sufficienly small in a definie sense. The main resul is described in he following heorem. Theorem 3.. Assume ha p RVλ) saisfies 2.) or 2.3). Le X) denoe he funcion defined by 2.3). Suppose moreover ha lim q)xh)) β = ) p)xg)) α i) Le 2.) hold. Then, equaion 3.) possesses srongly decreasing slowly varying soluions all of which enjoy one and he same asympoic behavior ) x) α) ps)ds,. 3.3) ii) Le 2.3) hold. Then, equaion 3.) possesses srongly decreasing regularly varying of he unique negaive index ρ = λ+ all of which enjoy one and he same asympoic behavior p) ) x),. 3.4) ρ Proof. Choose posiive consans k and K saisfying k 2, K 4, 3.5) Since X) saisfies 2.4), here exiss T > a such ha T 0 := inf T g) a and 2 X) In view of 3.2) we may assume ha T is chosen so ha Le ps)xgs)) α ds 2X), T. 3.6) q)xh)) β p)xg)) α kα K β for T. 3.7) X = { x C[T 0, ) : kx) x) KX), T 0 } and consider he mapping G : X C[T 0, ) defined by ps)xgs)) α + qs)xhs)) β) ds for T, Gx) = GxT ) XT ) X) for T 0 T. 3.8) 3.9) One can prove ha i) G maps X ino iself, ii) GX ) is relaively compac in C[T 0, ) and iii) G is a coninuous mapping.

10 0 G. E. CHATZARAKIS, K. TAKAŜI, I. P. STAVROULAKIS EJDE-204/206 i) GX ) X. Le x X. Then, since 3.7) implies p)xg)) α + q)xh)) β = p)xg)) α ) + q)xh))β p)xg)) α for T, using 3.6) and 3.5), we see ha Gx) 2 for T. Since Gx) p)xg)) α + Kβ q)xh)) β k α p)xg)) α ) 2p)xg)) α, ps)xgs)) α ds 2K α ps)xgs)) α ds 4K α X) KX), ps)xgs)) α ds k α ps)xgs)) α ds 2 kα X) kx), for T, we see ha kx) Gx) KX) for T. I is clear ha his inequaliy holds also for T 0 T. This shows ha G is a self-map on X. ii) GX ) is relaively compac. I is clear ha GX ) is uniformly bounded on [T 0, ). GX ) is equiconinuous on [T, ) since i holds ha 0 Gx) α p)xg)) α + K β q)xh)) β ), T, for all x X. The equiconinuiy on [T 0, T ] is eviden. iii) G is coninuous. Le {x n )} be a sequence in X converging, as n, o x) in X uniformly on any compac subinerval of [T 0, ). We hen have ) Gx n ) Gx) ps) x n gs)) α xgs)) α +qs) x n hs)) β xhs)) β ds for T, and Gx n ) Gx) Gx n T ) GxT ) for T 0 T, 3.0) from which he uniform convergence of Gx n ) Gx) on [T 0, ) follows as a consequence of applicaion of he Lebesgue dominaed convergence heorem o he righ-hand side of 3.0). Therefore, by he Schauder-Tychonoff fixed poin heorem here exiss a fixed poin x X of G, which saisfies he inegral equaion x) = ps)xgs)) α + qs)xhs)) β) ds 3.) for T. Hence x) is a srongly decreasing soluion of 3.) on [T, ) which is nearly regularly varying. Tha x) is cerainly regularly varying can be proved wih he help of Lemma 2.2. Le u) = ps)xgs)) α + qs)xhs)) β) ds 3.2) and consider he inferior and superior limis of x)/u): m = lim inf x) u), x) M = lim sup u), 3.3)

11 EJDE-204/206 FIRST-ORDER SUBLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS The fac ha x) X),, guaranees ha 0 < m M <. We noice ha p)xg)) α + q)xh)) β p)xg)) α,, 3.4) cf. 3.2)) which implies ah p)xg)) α + q)xh)) β p)xg)) α,. 3.5) We now apply Lemma 2.2 o m and M. Using 3.4), 3.5) and he relaion u) X),, which follows from 3.4), we obain M lim sup = lim sup = lim sup x ) u = lim sup ) p)xg)) α p)xg)) α = lim sup x) ) α x) = lim sup X) u) p)xg)) α + q)xh)) β p)xg)) α + q)xh)) β ) α xg)) Xg)) ) α = M α. Thus, we have M M α, which implies M because α <. Similarly, Lemma 2.2 applied o m leads o m m α which gives m. I follows ha m = M = ; ha is, x) lim = = x) u) X),. u) We conclude herefore ha x) is slowly varying if λ = and regularly varying of negaive index ρ = λ+ if λ <. The proof is complee. Remark 3.2. I is worh noicing ha in Theorem 3. he exponen β may be any consan larger or smaller han ), he coefficien q) may no be regularly varying, and he only requiremen for he deviaing argumen h) is ha lim h) =. Remark 3.3. In equaion 3.) suppose ha Then, p)xg)) α RVλ + αρ), and so condiion 3.2) is saisfied if q RVµ) and h RVν), ν ) q)xh)) β RVµ + βρν), µ + βρν < λ + αρ, 3.7) which gives, via Theorem 3., a pracical crierion for he exisence of srongly decreasing soluions for equaion 3.) wih regularly varying q) and h). Noe ha if ρ = 0 λ = ), hen 3.7) reduces o µ <. Corollary 3.4. Assume ha p) saisfies 2.) or 2.3), and ha g) saisfies.2). Suppose moreover ha q) and h) saisfy 3.6). i) Le λ =. If µ <, hen 3.) possesses srongly decreasing slowly varying soluions x) all of which enjoy he unique asympoic behavior ) x) α) ps)ds,. ii) Le λ <. If 3.7) holds, hen 3.) possesses srongly decreasing regularly varying soluions x) of negaive index ρ all of which enjoy he unique asympoic behavior p) ) x),. ρ

12 2 G. E. CHATZARAKIS, K. TAKAŜI, I. P. STAVROULAKIS EJDE-204/ Examples In his secion we give four examples illusraing he main resuls of his aricle. Example 4.. Consider he equaion.) wih p) saisfying p) exp log ),, 4.) log and call i equaion E). Obviously p RV ) and p) is inegrable near he infiniy, and so by Theorem 2.3 equaion E), for any g) saisfying.2), has srongly decreasing slowly varying soluions x) all of which enjoy he unique asympoic behavior x) α) If in paricular ) ) ps)ds 2 α)) exp log,. 4.2) p) = exp log ) log α exp α log g) ) log ), hen p) saisfies 4.) and equaion E) possesses an exac slowly varying soluion x 0 ) = 2 α)) exp log ). Example 4.2. Consider he equaion.) wih p) saisfying p) α L),, 4.3) where L) is any coninuous slowly varying funcion, and call i equaion E2). Since λ = α <, from Theorem 2.4 i follows ha equaion E2) possesses srongly decreasing soluions belonging o he class RV α ) and ha any such soluion x) enjoys he asympoic behavior If in paricular x) α) α g) p) = α L) α L) ) α 2 L) Lg)),. 4.4) ) α L ) ), αl) where L) is a coninuously differeniable slowly varying funcion, hen p) saisfies 4.3) use Lemma 2. and Proposiion.5) and equaion E2) has an exac srongly decreasing regularly varying soluion α ) x 0 ) = α L), α for any deviaing argumen g) saisfying.2). Example 4.3. Consider equaion 3.) in which α <, p) saisfies 4.), β > 0 is a consan and q) and h) saisfy 3.6). This equaion is referred o as equaion EP). By i) of Corollary 3.4 one concludes ha EP) possesses srongly decreasing slowly varying soluions all of which enjoy he asympoic behavior 4.2). I is o be noed ha he perurbed erm may be superlinear β > ) or sublinear β < ), and ha any deviaing argumen, rearded, advanced or oherwise, is

13 EJDE-204/206 FIRST-ORDER SUBLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 3 admied as h) as long as i is regularly varying of nonnegaive index. For insance, h) may be any one of he following: ± τ, ±, ± log, c, θ, log, where τ, c and θ are posiive consans. For example, if α <, µ < and g),, hen he equaion EP) x ) + exp log ) xg)) α xg)) + µ L) xh)) β xh)) = 0, 4.5) log always possesses srongly decreasing slowly varying soluions x) all of which behave like ) x) 2 α)) exp log,, for any consan β > 0, any L SV and any h RVν), ν 0, such ha lim h) =. Example 4.4. Consider equaion 3.) in which α <, p) saisfies 4.3), and β, q) and h) are as in he above equaion EP). We call his equaion EP2). Since λ = α and ρ = α, condiion 3.7) is reduced o µ < αβν α, 4.6) which ensures he exisence of srongly decreasing regularly varying soluions x) of negaive index for equaion EP2) having he asympoic behavior x) α) α L), α. 4.7) In paricular, if µ <, hen he equaion x α L) xg)) α x + sin ) + µ M) xlog ) β xlog ) = 0 has srongly decreasing soluions x) saisfying 4.7) for any posiive consan β and for any coninuous slowly varying funcions L) and M). Acknowledgemens. The auhors would like o hank he anonymous referee for he useful remarks. References [] N. H. Bingham, C. M. Goldie, J. L. Teugels; Regular Variaion, Encycl. Mah. Appl., Vol. 27, Cambridge Univ. Press, Cambridge, 987. [2] W. A. Coppel; Sabiliy and Asympoic Behavior of Differenial Equaions, D. C. Heah and Company, Boson, 965. [3] V. M. Evukhov, A. M. Samoĭlenko; Asympoic represenaions of soluions of nonauonomous ordinary differenial equaions wih regularly varying nonlineariies Russian) Differ. Uravn ), ; English ransl.: Differ. Equ ), [4] J. L. Geluk, L. de Haan; Regular Variaion Exensions and Tauberian Theorems, CWI Trac 40, Amserdam, 987. [5] O. Haup, G. Aumann; Differnial- und Inegralrechnung, Waler de Gruyer, Berlin, 938. [6] J. Jaroš, T. Kusano; Exisence and precise asympoic behavior of srongly monoone soluions of sysems of nonlinear differenial equaions, Differ. Equ. Appl., 5 203), [7] J. Jaroš, T. Kusano; Asympoic behavior of posiive soluions of a class of sysems of second order nonlinear differenial equaions, E. J. Qualiaive Theory of Diff. Equ., ), 23. [8] J. Jaroš, T. Kusano, T. Tanigawa; Asympoic analysis of posiive soluions of a class of hird order nonlinear differenial equaions in he framework of regular variaion, Mah. Nachr., ),

14 4 G. E. CHATZARAKIS, K. TAKAŜI, I. P. STAVROULAKIS EJDE-204/206 [9] T. Kusano, J. Manojlović; Asympoic behavior of posiive soluions of sublinear differenial equaions of Emden-Fowler ype, Compu. Mah. Appl., 62 20), [0] T. Kusano, J. Manojlović; Asympoic analysis of Emden-Fowler differenial equaions in he framework of regular variaion, Annali di Maemaica, 90 20), [] T. Kusano, J. Manojlović; Posiive soluions of fourh order Thomas-Fermi ype differenial equaions in he framework of regular variaion, Aca Appl. Mah.., 2 202), [2] T. Kusano, J. Manojlovć; Complee asympoic analysis of posiive soluions of odd-order nonlinear differenial equaions, Lih. Mah. J., ), [3] T. Kusano, J. Manojlovć, T. Tanigawa; Exisence and asympoic behavior of posiive soluions of fourh order guasilinear differenial equaions, Taiwanese J. Mah., 7 203), DOI: 0.650/jm , 32 pages. [4] T. Kusano, V. Marić, T. Tanigawa; An asympoic analysis of posiive soluions of generalized Thomas-Fermi differenial equaions - The sub-half-linear case, Nonlinear Anal., ), [5] J. Manojlovć, V. Marić; An asympoic analysis of posiive soluions of Thomas-Fermi ype sublinear differenial equaions, Mem. Differenial Equaions Mah. Phys., ), [6] V. Marić; Regular Variaion and Differenial Equaions, Lec. Noes Mah., Vol. 726, Springer Verlag, Berlin-Heidelberg, [7] V. Marić, Z. Radašin; Regularly Varying Funcions in Asympoic Analysis, Obzornik ma. fiz., ), [8] V. Marić, M. Tomić; Regular variaion and asympoic properies of soluions of nonlinear differenial equaions, Publ. Ins. Mah. Beograd) N. S.) 235) 977), [9] A. Nikolić; Abou wo famous resuls of Jovan Karamaa, archives Inernaionales d Hisoires des Sciences, 4 998), [20] A. Nikolić; Jovan Karamaa ), Novi Sad J. Mah., ), 5. [2] E. Senea; Regularly Varying Funcions, Lec. Noes Mah., Vol. 508, Springer Verlag, Berlin- Heidelberg, 976. George E. Chazarakis Deparmen of Elecrical and Elecronic Engineering Educaors, School of Pedagogical and Technological Educaion ASPETE), 42, N. Heraklio, Ahens, Greece address: geaxaz@oene.gr, gea.xaz@aspee.gr Kusano Takaŝi Professor Emerius a: Deparmen of Mahemaics, Faculy of Science, Hiroshima Universiy, Higashi-Hiroshima , Japan address: kusano@zj8.so-ne.ne.jp Ioannis P. Savroulakis Deparmen of Mahemaics, Universiy of Ioannina, 45 0 Ioannina, Greece address: ipsav@cc.uoi.gr