On the Positive Periodic Solutions of the Nonlinear Duffing Equations with Delay and Variable Coefficients

Transcription

1 On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions wih Delay Variable Coefficiens Yuji Liu Weigao Ge Absrac We consider he exisence nonexisence of he posiive periodic soluions of he non-auonomous nonlinear Duffing equaion wih delay variable coefficiens x () p()x () + q()x() = λh()f(, x( τ())) + r(), I is shown ha he equaion has posiive periodic soluions under cerain condiions no posiive T periodic soluion under some oher condiions by using a fixed heorem in cones. 1 Inroducion Consider he exisence nonexisence of he posiive periodic soluions of he non-auonomous nonlinear delay Duffing equaion x () p()x () + q()x() = λh()f(, x( τ())) + r(), (1) where λ > is a parameer. p : R R +, q : R R +, h : R R +, r : R R, τ : R R f : R R + R are coninuous T periodic abou he variable. R = (, + ) R + = [, + ). The firs auhor was suppored by he Science Foundaion of Educaional Commiee of Hunan Province he second auhor by he Naional Naural Sciences Foundaion of P.R.China Received by he ediors November. Communicaed by J. Mawhin. Key words phrases : nonlinear Duffing equaion, posiive periodic soluion, cone, fixed poin heorem. Bull. Belg. Mah. Soc. 11 (4),

2 44 Y. Liu W. Ge Recenly, here is exensive lieraure relaed o he exisence uniqueness of he periodic soluions(almos periodic soluions, or bounded soluions) of second order differenial equaions wih or wihou delay. To idenify a few, we refer he reader o [1-13] he references cied here. Zeng in [1] sudied he exisence of almos periodic soluions of he equaion x () x() + x 3 () = f(), () where f is a almos periodic funcion. Very recenly, Wang in [] invesigaed he same problem for he following equaion x () x() ± x m ( r) = f() (3) by using he changing of variable fixed poin mehod mehod of approximaion. In [3], he auhors gave he periodic soluions of he equaion x () + Cx () + g(, x()) = e() (4) by consrucing mehod. In [14], Zima sudied he exisence of posiive soluions of he problem x () k x() + f(, x()) =, x() = lim x() =. + To he bes of our knowledge, he exisence nonexisence of he posiive periodic soluions for second order nonlinear delay differenial equaions, especially for Duffing equaions, has no been invesigaed ill now. In his paper, we sudied he exisence nonexisence of he posiive periodic soluions of he equaion (1). Our aims are o esablish he exisence non-exisence crierion of posiive T periodic soluions of (1). Since (1) is nonauonomous has variable ime lag coefficiens, he mehods used in he papers menioned above are no effecive. Our mehod in his paper is differen from hose menioned above. In his paper, we give he assumpions as follows. These assumpions will be used in he main resuls.. (H 1 ) p, q C(R, R + ) are T-periodic funcions wih p(s)ds > o q(s)ds > (H ) h C(R, R + ) is T-periodic wih h(s)ds >. (H 3 ) τ C(R, R) is T-periodic. (H 4 ) r C(R, R) is T-periodic. (H 5 ) r() = for all R. (H 6 ) There is M > such ha f(, x) M for (, x) R R +. (H 7 ) λ > is a parameer, T > a consan. (H 8 ) f : R R + R + is coninuous is T-periodic abou. (H 9 ) lim x + f(, x)/x = N (, + ] uniformly on [, T ]. (H 1 ) lim x + f(, x)/x = l [, + ] lim x f(x)/x = L [, + ] uniformly on [,T]. This paper is organized as follows: in secion, we give some useful lemmas. Our main resuls will be given in secion 3.

3 On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions 443 Finally, for easy reference we sae here he fixed poin heorem [15,p.94] which is employed in his paper. LEMMA K [15]. Le X be a Banach space, K be a cone of X. Assume Ω 1, Ω are open subses of X wih Ω 1, Ω 1 Ω, le A : K (Ω Ω 1 ) K be a compleely coninuous operaor such ha or (1). Ax x for every x K Ω 1 Ax x for every x K Ω (). Ax x for every x K Ω 1 Ax x for every x K Ω Then A has a leas one fixed poin in K (Ω Ω 1 ). Lemmas To begin wih, we consider he following equaion x () a()x() = λg(x()) m(), (5) where a is a non-negaive coninuous T-periodic funcion wih a(s)ds >, m a coninuous posiive T-periodic funcion, λ > a parameer, g : R + R + a coninuous funcion. Lemma 1. Suppose g(x) lim x + x = N (, + ]. (6) Then (5) has a leas one posiive T-periodic soluion if A < B λ (A, B). A B are defined as follows. where A = [ exp( a(u)du) + 1] NσT exp( a(u)du), B = R [ exp( a(u)du) + 1], σm 1 T R = R(), σ = exp( { a(u)du), g(x), x [, + ), M 1 = max x [,R (σ+1)/σ] h(x), h(x) = g(), x (, ), R() = +T G 1 (, s)m(s)ds, G 1 (, s) = exp( s a(u)du) exp( a(u)du)+1. Proof. Suppose ha y : R [, + ) is coninuous T-periodic funcion x() is a soluion of he equaion One ges afer inegraion from o + T, we ge x () = a()x() y(). (7) (x()e a(s)ds ) = e a(s)ds y(), +T x() = λ G 1 (, s)y(s)ds,

4 444 Y. Liu W. Ge where Since we have Hence x() = λ G 1 (, s) = +T +T x() = λ exp( s a(u)du) exp( for s + T. (8) a(u)du) + 1 G 1 (, s) G 1 (, s) G 1 (, s)y(s)ds 1 exp( a(u)du) + 1 exp( a(u)du) exp( a(u)du) + 1, λ exp( a(u)du) +T exp( a(u)du) + 1 y(s)ds 1 G 1 (, s)y(s)ds λ exp( a(u)du) + 1 x() σ x, R. +T y(s)ds. Now, le X be he se of all real coninuous T periodic funcions endowed wih he usual linear srucure as well as he norm x = max [,T ] x(). Then X is a Banach space. Le K = {x X : x() σ x, [, T ]}. Then K is a cone of space X. I is easy o know ha R() saisfies Nex, consider he following equaion x () = a()x() + m(). y () = a()y() λg (y() R()). (9) We find ha (5) has a posiive T-periodic soluion x() if only if y() = x() R() is a T-periodic soluion of (9) y() + R() > for all [, T ]. Le he operaor F be defined as follows. F y() = λ +T G 1 (, s)g (y(s) + R(s)) ds for y X. I is easy o see ha F is compleely coninuous F K K by he periodic assumpions. Since λ < B, we choose α > 1 such ha λ B α. Se Ω 1 = { x X : x < R /σ }. Then one finds y() + R() y + R = R (σ + 1)/σ, y() + R() σ y R = if y K Ω 1. Thus +T F y() λm 1 G 1 (, s)ds λm 1 T exp( a(u)du) + 1 < R ασ < y.

5 On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions 445 i.e., F y < y for all y K Ω 1. Now, choosing ɛ > such ha λ(n ɛ)σt exp( a(u)du) [ exp( a(u)du) + 1] 1 by λ > A. Again, one can choose H > R /σ > R such ha h(y) y = g(y) y > N ɛ for y H. Se Ω = { x X : x < (H + R )/σ }. If y K Ω, we find Then y() + R() σ y R σ H + R σ R = H, y = H + R. σ +T F y() λ λ(n ɛ) λh(n ɛ) G 1 (, s)(n ɛ)(y(s) + R(s))ds +T G 1 (, s)(σ y R )ds exp( a(u)du) a(u)du) + 1ds exp( = λ(n ɛ)h T exp( a(u)du) exp( a(u)du) + 1 > λ(n ɛ) H + R H + R σ = y. T exp( a(u)du) exp( a(u)du) + 1 i.e., F y y for y K Ω. Hence F has a leas one fixed poin y by applying Theorem K such ha R /σ < y (H + R )/σ ha is a T-periodic soluion of (9). We claim ha y > R, for he conradicion, here is σ so ha R = y( σ ) = F y( ) < R, which is condiion. On he oher h, ασ y() + R() σ y R > σ R σ R =. So, x() = y() + R() is a posiive T-periodic soluion of equaion (5). Remark: If N = +, hen (5) has a leas one posiive T-periodic soluion if < λ < B. Lemma. Suppose ha (H 1 ) holds R 1 [ exp( p(u)du) + 1] M 1 T > 1. (1) Then here are coninuous T-periodic funcions a b such ha b() >, a(u)du > a() + b() = p(), b () + a()b() = q() for all R, (11)

6 446 Y. Liu W. Ge where Proof. From (11), one ges R 1 = R 1 (), R 1 () = +T G(, s)q(s)ds, M 1 = R 1 (σ+1), σ = exp( σ p(s)ds). b () p()b() = b () q(). (1) Le g(x) = x, a() = p(), m() = q() λ = 1. Then N = lim x + g(x)/x = +. By Lemma 1, if < λ = 1 < R 1[ exp( p(u)du) + 1], σm 1 T hen (1) has a leas one posiive coninuous T-periodic soluion b(). Now, since q(s)ds > b ()/b() + a() = q()/b(), we ge a(u)du = q(u) du >. b(u) Remark: Condiion (1) in Lemma is a sharp condiion. If p, q, a b are real numbers, (11) becomes he following a + b = p ab = q. I is well known ha he condiion ha guaranee a b is p 4q. On he oher h, i is easy o know ha R 1 = q p, M 1 = [R 1 (σ + 1)/σ] σ = e pt. Thus lim T R 1 [ exp( p(u)du) + 1] M 1 T = lim q T p e pt + 1 T e pt ( e pt + 1 ) = p 4q. This is p 4q. Nex, assume (H 1 ) y a T-periodic funcion, we consider he equaion x () p()x () + q()x() = y(). (13) When (1) holds, y(), by Lemma, here are non-negaive coninuous T- periodic funcions a b such ha (11) holds. We ransform (13) ino he following equaion Then x () a()x () b ()x() b()x () + a()b()x() = y(). (14) (x ()e a(u)du ) (b()x()e a(u)du ) = e a(u)du y(). Inegraing i from o + T, we ge Similarly, we ge x() = +T +T x () b()x() = exp( s b(u)du) [ s+t exp( b(u)du) + 1 s exp( s a(u)du) exp( a(u)du) + 1y(s)ds. exp( v s a(u)du) ] exp( ds. a(u)du) + 1y(v)dv

7 On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions 447 By exchanging he inegraing order, one ges x() = where G(, s) is defined as follows. G(, s) = s exp[ u +T a(v)dv s u b(v)dv]du + +T s [ exp( a(v)dv) + 1][ exp( for s [, + T ]. On he oher h, From (11), we ge a(u)du + b(u)du = n + i=1 G(, s)y(s)ds, (15) p(u)du, exp[ u a(v)dv s+t u b(v)dv]du b(v)dv) + 1] a(u)du = q(u) du. (16) b(u) Since b q are coninuous b() >, choosing < T < T n n < < nt n = T ξ i ( (i 1)T, it ), one ges n n n b(u)du = lim b(ξ i ) T n, q(u) n b(u) du = lim q(ξ i ) T n + b(ξ i ) n. So a(u)du b(u)du = = lim T n q(u) b(u)du b(u) du n b(ξ i ) T n lim n + n + i=1 lim n + n n = T lim n + = T exp( 1 T Togeher wih (15), if ( T 1 [ p(u)du] 4T exp T i follows ha min{ max{ a(u)du, a(u)du, b(u)du} n n n n b(ξ i ) n i=1 i=1 q(ξ i ) ln q(u)du). ln q(u)du i=1 i=1 n [ p(u)du p(u)du] 4T exp( 1 T b(u)du} [ p(u)du + p(u)du] 4T exp( 1 T ) q(ξ i ) T b(ξ i ) n i=1 q(ξ i ) b(ξ i ), (17) ln q(u)du) ln q(u)du) := n := m.

8 448 Y. Liu W. Ge So. i is easy o know ha G(, s) = s du + +T s du [ exp( a(v)dv) + 1][ exp( b(v)dv) + 1] T [ exp( a(v)dv) + 1][ exp( b(v)dv) + 1] T [ e n + 1] G(, s) s exp [ +T a(v)dv +T b(v)dv ] du [ ( exp T a(v)dv) + 1 ] [ exp ( b(v)dv) + 1 ] + +T s exp [ +T a(v)dv +T b(v)dv ] du [ ( exp T a(v)dv) + 1 ] [ exp ( b(v)dv) + 1 ] T exp( p(s)ds) [ e m + 1]. Then we ge he following. Lemma 3. Suppose ha y is a T-periodic funcion wih y() for all, (H 1 ), (1) (17) hold. Then he soluion of (13) saisfies x() µ x for all R, where ( e n + 1) µ = exp(. p(v)dv)[ e m + 1] Proof. Le x() be a T-periodic soluion of (13). We ge x() = +T G(, s)y(s)ds T +T y(s)ds ( e n + 1) x() T exp( a(u)du) +T y(s)ds. ( e m + 1) So x() µ x for every R. The proof is complee. Remark: If p q are real numbers, (17) becomes p 4q. 3 Main resuls proofs Theorem 1. Suppose ha (H 1 ) (H 4 ), (H 7 ) (H 9 ), (1) (17) hold. Then equaion (1) has a leas one posiive T periodic soluion if a < B λ (A, B). A B are defined as follows. A = [ e n + 1] NµT exp( p(s)ds) h(s)ds

9 On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions 449 B = R ( e n + 1) µm 1 T h(s)ds, where G(, s) µ are defined in Lemma 3 m n in secion, R = R(), M 1 = max g(, x), g(, x) = (,x) [,T ] [,R (µ+1)/µ] Proof. I is easy o know ha R() saisfies Now, consider he following R() = +T G(, s)r(s)ds, x () p()x () + q()x() = r(). { f(, x), (, x) [, T ] [, + ), f(, ), (, x) [, T ] (, ). y () p()y () + q()y() = λh()f(, y( τ()) + R( τ())). (18) We find ha (1) has a posiive T-periodic soluion x() if only if y() = x() R() is a T-periodic soluion of (17) y() + R() > for all [, T ]. Le he operaor F be defined as follows. +T F y() = λ G(, s)h(s)g(s, y(s τ(s)) + R(s τ(s)))ds for y X. I is easy o see ha F is compleely coninuous F K K by he periodic assumpions. Se Ω 1 = { x X : x < R /µ }. Then one finds y() + R() y + R = R (µ + 1)/µ, if y K Ω 1, y() + R() σ y R =. Choose α > 1 so ha λ < B. Thus µ +T F y() λm 1 G(, s)h(s)ds λm 1 T h(s)ds ( e n + 1) < R αµ < y. i.e., F y < y for all y K Ω 1. Now, choosing ɛ > such ha λ(n ɛ)µt exp( p(s)ds) h(s)ds [ e m + 1] 1 by λ > A. Again, by (H 9 ), one can choose H > R /µ > R such ha g(, y) y = f(, y) y > N ɛ for [, T ] y H. Se Ω = { x X : x < (H + R )/µ }. We find y() + R() µ y R µ H + R µ R = H.

10 45 Y. Liu W. Ge Then +T F y() λ λ(n ɛ) λh(n ɛ) G(, s)h(s)(n ɛ) (y(s τ(s)) + R(s τ(s))) ds +T G(, s)h(s)(µ y R )ds T exp( p(s)ds) h(s)ds ( e m + 1) = λ(n ɛ)h T exp( p(s)ds) h(s)ds ( e m + 1) > λ(n ɛ) H + R H + R µ = y. T exp( p(s)ds) h(s)ds ( e m + 1) i.e., F y y for y K Ω. Hence F has a leas one fixed poin y such ha R /σ < y (H + R )/µ ha is a T-periodic soluion of (17). We claim ha y > R. For he conradicion, here is µ so ha R = y( µ ) = Ay( ) < R, αµ which is conradicion. On he oher h, y() + R() σ y R > µ R µ R =. So, x() = y() + R() is a posiive T-periodic soluion of sysem (1). Remark: When N = + in (9), hen (1) has posiive T-periodic soluions if (H 1 ) (H 4 ), (H 7 ), (H 8 ), (1) (16) hold λ (, B). Theorem. Assume ha (H 1 ) (H 7 ), (H 9 ), (1) (16) hold. Then (1) has a leas one posiive T-periodic soluion if A < B λ (A, B). A B are defined as follows. ( e m + 1) A = µnt exp( p(s)ds) h(s)ds B = min{ ( e n + 1) M 1 T µ( e n + 1) T h(s)ds, MT }, h(s)ds where G(, s) µ are defined in Lemma 3 m n in secion, M 1 = max g(, x), g(, x) = (,x) [,T ] [,1] Proof. Se w() = +T { f(, x) + M, (, x) [, T ] [, + ), f(, ) + M, (, x) [, T ] (, ). G(, s)h(s)ds R, z() = λmw(). Then (1) has posiive T-periodic soluion x() if only if x+z := y is a T-periodic soluion of he following equaion y () p()y () + q()y() = λh()g(, y( τ()) z( τ())), (19)

11 On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions 451 y() > z() for all R. Define an operaor F as follows. +T F y() = λ G(, s)h(s)g(s, y(s τ(s)) z(s τ(s)))ds for y X. We know ha F is compleely coninuous F K K. Le λ be fixed A < λ < B. Choosing Ω 1 = { x X : x < 1}. One has y( τ()) z( τ())) y( τ()) y = 1 for y K Ω 1. Then +T F y() λm 1 G(, s)h(s)ds T λm h(s)ds 1 ( e n + 1) 1 = y (using λ < B), i.e., F y y for all y K Ω 1. Now, choosing ɛ > such ha λ(n ɛ)µt exp( p(s)ds) h(s)ds ( e m + 1) 1. Le R > 1 such ha g(, x) x Se Ω = { x X : µ λm R T h(s)ds ( e n + 1) µ f(, x) = N ɛ for (, x) [, T ] [ µr x, + ). x < R }. If y K Ω, we find () Hence +T F y() λ(n ɛ) y() z() µ y λmw() Rµ µr λm T h(u)du ( e n + 1) (using ()). G(, s)h(s)(y(s τ(s)) z(s τ(s)))ds +T λµr(n ɛ) G(, s)h(s)ds λµr(n ɛ) T exp( p(s)ds) h(s)ds ( e m + 1) R = y. i.e., F y y for y K Ω. Hence F has a leas one fixed poin y such ha 1 y R. On he oher h, y() µ y µ > λmt h(s)ds ( e n + 1)

12 45 Y. Liu W. Ge implies w() T h(s)ds ( e n + 1) (using λ < B) y() > λmw() = z() for all R. So x() = y() z() is a posiive T-periodic soluion of (1). Remark: When N = + in (H 9 ), (1) has a leas one posiive T-periodic soluion if (H 1 ) (H 7 ), (1) (16) hold λ (, B). Theorem 3. Assume ha (H 1 ) (H 5 ), (H 7 ), (H 8 ), (H 1 ), (1) (16) hold. Then (1) has a leas one posiive T periodic soluion if A < B λ (A, B). A B are defined as follows A = ( e m + 1) µlt exp( p(s)ds) h(s)ds B = ( e n + 1) LT h(s)ds. Proof. Define an operaor F as follows. +T F y() = λ G(, s)h(s)f(s, y(s τ(s)))ds for y X. We see ha A is compleely coninuous F K K. Choosing ɛ > R > such ha λ(l + ɛ) T h(s)ds ( e 1 n + 1) f(, x)/x L + ɛ for (, x) [, T ] [R, + ). Se Ω 1 = { x X : x < R /µ }. If y K Ω 1, hen y() µ y = R +T F y() = λ λ(l + ɛ) G(, s)h(s)f(s, y(s τ(s)))ds +T λ(l + ɛ) y g(, s)h(s)y(s τ(s))ds +T G(, s)h(s)ds λ(l + ɛ) y T h(s)ds ( e n + 1) y. i.e., F y y for all y K Ω. Now, choosing ɛ > such ha λµ(l ɛ)t exp( p(s)ds) h(s)ds ( e m + 1) 1.

13 On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions 453 Choosing < R 1 < R such ha f(, x)/x > l ɛ for (, x) [, T ] (, R 1 ]. Se Ω 1 = { x X : x < R 1 }. For y K Ω 1, we find +T F y() λ λ(l ɛ) G(, s)h(s)y(s τ(s))(l ɛ)ds +T = λ(l ɛ)µ y G(, s)h(s)µ y ds +T G(, s)h(s)ds λ(l ɛ)µ y T exp( p(s)ds) h(s)ds ( e m + 1) y. i.e., F y y for y K Ω 1. Hence F has a leas one fixed poin y such ha R 1 y R ha is a posiive periodic soluion of (1). Theorem 4. Assume ha (H 1 ) (H 5 ), (H 7 ), (H 8 ), (H 1 ), (1) (16) hold. Then (1) has a leas one posiive T periodic soluion if A < B λ (A, B). A B are defined as follows. A = ( e m + 1) µt exp( p(s)ds)l h(s)ds B = ( e n + 1) lt h(s)ds. The proof is similar o ha of Theorem 3 hen omied. Remark: For he case where f is sub-linear(i.e., l = + L = ) or superlinear(i.e., l = L = + ), Theorem 3 or 4 is effecive respecively. Theorem 5. If (H 1 ) (H 3 ), (H 5 ), (H 7 ) (H 8 ), (1) (16) hold λ T h(s)ds f(, x) < x (1) ( e n + 1) for (, x) [, T ] (, + ), hen (1) has no posiive T-periodic soluion. Proof. Assume o he conrary ha y() is a posiive T-periodic soluion of (1). There is [, T ] such ha y = y( ). Thus we have +T y = y( ) = λ G(, s)h(s)f(s, y(s τ(s)))ds [ T < h(s)ds ] 1 +T G( ( e n + 1), s)h(s)y(s τ(s))ds [ T h(s)ds ] 1 +T G( ( e n + 1), s)h(s)ds y [ T h(s)ds ] 1 [ T h(s)ds ] y ( e n + 1) ( e n + 1) = y

14 454 Y. Liu W. Ge which is a conradicion. Theorem 6. If (H 1 ) (H 3 ) (H 5 ), (H 7 ), (H 8 ), (1) (16) hold λ µt exp( p(s)ds) h(s)ds ( e m + 1) f(, x) > x () for (, x) [, T ] (, + ), hen (1) has no posiive T-periodic soluion. The proof is similar o ha of Theorem 5. References [1] Zeng W. Y., Almos periodic soluions for nonlinear Duffing equaions. Aca Mahemaica Sinica, New Series, 1997, 13(3): [] Wang Q. Y., The exisence uniqueness of almos periodic soluions for nonlinear differenial equaions wih ime lag. Aca Mahemaica Sinica, 1999, 4(3): [3] Li W. G., Shen Z. H., An consrucive proof of he exisence Theorem for periodic soluions of Duffing equaions. Chinese Science Bullein, 1997, 4(15): [4] Mawhin J., Ward J. R., Nonuniform nonresonance condiions a he wo firs eigenvalues for periodic soluion of forced linear Duffing equaions. Rocky Mounain J. Mah., 198, 1(4): [5] Ding T. R., Nonlinear oscillaion a he uniform non-resonance poin. Science in China, 198, 1:1-13. [6] Li W. G., A necessary sufficien condiion on he exisence uniqueness of π soluion of Duffing. Chinese Ann. of Mah., B, 199, 11: [7] Shen Z. H., On he periodic soluions o he Newonian equaion of moion. Nonlinear Analysis, 1989, 13(): [8] Li Y., Lin Zh. H., A consrucive proof of he Poincare-Birkhoff heorem. Transacion of American Mah. Soc., 1995, 1(6):5-17. [9] K. Wang, Posiive periodic soluions of neural delay equaions(in Chinese). Aca Mah. Sinica, 1996, 6: [1] Mawhin J., Periodic soluion of some vecor rearded funcional differenial equaions. J.of Mah. Anal. Appl., 1974, 45: [11] Guo D. J., Sun J. X., Zh.L.Liu, Funcional mehods for nonlinear ordinary differenial equaions. Jinan:Shong Science Technology Press, 1995(In Chinese). [1] Gyori I., Ladas G., Oscillaion Theory of Delay Differenial Equaions wih Applicaions. Oxford, Clarendon Press, 1991.

15 On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions 455 [13] Ding W. Y., Periodic soluions of ordinary differenial equaions A fixed poin Theorem. Aca Mahemaica Sinica, 198, 5():7-35. [14] Zima M., On he posiive soluions of boundary value problems on half-line, J. of Mah. Anal. Appl., 1, 59: [15] Guo D., Lakshmikanhan V., Nonlinear Problems in Absrac Cones, Academic Press, Son Diego, CA Deparmen of Mahemaics, Hunan Insiue of Technology, Yueyang,414, P.R.China Deparmen of Applied Mahemaics, Beijing Insiue of Technology, Beijing 181, P.R.China