Asymptotic Behavior of Nonoscillatory Solutions of Nonlinear ...

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$Â¥leftÂ¥{Â¥begin{array}{l}Â¥tilde{x}(t)=Â¥sum_{j=0}^{n(t)-1}H_{j}(t)x(Â¥tau^{j}(t))+Â¥frac{Â¥mathrm{x}(T)}{1-h(T)}H_{n(t)}(t),・・・tilde{Â¥mathrm{x}}(t)=Â¥frac{x(T)}{1-h(T)} ...
Funkcialaj Ekvacioj, 32 (1989) 251-263

Asymptotic Behavior of Nonoscillatory Solutions of Nonlinear Functional Differential Equations of Neutral Type By

Jaroslav JAROS and Takasi KUSANO (Komensky University, Czechoslovakia and Hiroshima University, Japan)

1.

Introduction We consider nonlinear neutral functional differential equations of the type

(1.1)

$¥frac{d^{2n}}{dt^{2n}}[x(t)-h(t)x(¥tau(t))]+f(t, x(g_{1}(t)), ¥ldots, x(g_{N}(t)))=0$

,

where the following conditions are assumed to hold:

is continuous, where is a constant; is a continuous, increasing function such that $¥tau(t)0$ , (c) $f:[a,$ ) $1¥leqq i¥leqq N$ and ¥ , and is non-decreasing in each , $1¥leqq i¥leqq N$ , for fixed ; each is continuous and nondecreasing, and (d) )

(1.2) (a) (b)

$h:[a, ¥infty)¥rightarrow(0, ¥lambda]$

$¥tau:[a,$ $t$

$¥geqq a$

$¥infty$

$¥lambda¥in(0,1)$

$¥rightarrow R$

$¥infty$

$¥lim_{t¥rightarrow¥infty}¥tau(t)=¥infty$

$¥times R^{N}¥rightarrow R$

$t$

$y_{1}f(t, y_{1^{ }}, ldots, y_{N}) geqq 0$

$¥geqq a$

$f(t, y_{1},

$t$

$g_{i}:[a,$

$¥infty$

ldots, y_{N})$

$y_{i}$

$¥geqq a$

$¥rightarrow R$

$¥lim_{¥mathrm{r}¥rightarrow¥infty}g_{i}(t)=¥infty$

,

$1¥leqq i¥leqq N$

.

The main objective of this paper is to investigate the asymptotic behavior as of those nonoscillatory solutions $x(t)$ of (1.1) which satisfy $x(t)[x(t)-$ $h(t)x(¥tau(t))]>0$ for all sufficiently large . We will first show that such a solution $x(t)$ satisfies either $ t¥rightarrow¥infty$

$t$

(1.1)

$0T$ ,

,

where

(2.8)

$H_{0}(t)=1$ ,

$H_{n}(t)=¥prod_{i=0}^{n-1}h(¥tau^{i}(t))$

,

$n=1,2$ ,

denotes the least positive integer such that for some , $0¥leqq k¥leqq (2.4)?(2.8) we see that if

and

$n(t)$

$x$

(2.9) and that if

(2.10)

$k$

$¥in¥ovalbox{¥tt¥small REJECT}_{k}$

$0T$ ,

(3.5)

$¥left¥{¥begin{array}{l}¥tilde{x}(t)=¥sum_{j=0}^{n(t)-1}H_{j}(t)x(¥tau^{j}(t))+¥frac{¥mathrm{x}(T)}{1-h(T)}H_{n(t)}(t),¥¥¥tilde{¥mathrm{x}}(t)=¥frac{x(T)}{1-h(T)},T_{0}¥leqq t¥leqq T,¥end{array}¥right.$

, where $n(t)$ denotes the least positive integer such that $T_{0}0$ such that $Proo/$.

(4.5)

$¥int_{T}^{¥infty}t^{2n-¥iota-1}f(t,$

Consider the set

(4.1)

$¥alpha$

$X$

$¥frac{¥gamma}{1-¥lambda}[g_{1}(t)+T+1]^{l}$

of functions

$¥frac{¥gamma t^{l-1}}{(l-1)!}¥leqq x(t)¥leqq¥frac{¥gamma t^{l-1}}{(l-1)!}+¥frac{¥gamma t^{l}}{l!}$

and define the mapping

(4.7)

$x¥in C[T_{0},$

$F:X¥rightarrow C[T_{0},$

$¥infty$

for $¥infty$

,

$t$

$¥ldots$

,

$¥frac{¥gamma}{1-¥lambda}[g_{N}(t)+T+1]^{l})dt¥leqq¥gamma$

.

) which satisfy $¥geqq T$

and

$x(t)=x(T)$

for

$T_{0}¥leqq t¥leqq T$

,

) by

$¥left¥{¥swarrow^{¥tau}¥ovalbox{¥tt¥small REJECT} ¥mathrm{x}(t)=¥frac{¥gamma t^{l-1}}{((¥gamma T^{¥mathrm{I}-1}l-1)^{|}l-1)!},+x(t)=¥frac¥int^{t}¥tau_{T_{0}¥leqq t¥leqq}¥frac{(t-s)^{¥iota-1}}{(l-1)!}¥int_{T},s¥infty¥frac{(r-s)^{2n-I-1}}{(2n-l-1)!}f(r,¥tilde{x}(g(r)))drds,t¥geqq T,¥right.$

where the function in the integrand is defined by (3.5). If $¥in X$ , then, since $x(t)¥leqq¥gamma(t+1)^{¥iota}/(l-1)!$ , $t¥geqq T$, by (4.6), we have from $¥tilde{x}$

$x$

(3. 5)

Jaroslav JAROS and Takasi KUSANO

260

$¥tilde{x}(g_{i}(t))¥leqq¥frac{¥gamma}{1-¥lambda}¥{[g_{i}(t)+1]^{l}+T^{l}¥}¥leqq¥frac{¥gamma}{1-¥lambda}[g_{i}(t)+T+1]^{¥mathrm{t}}$

for

,

$t¥geqq T$

$1¥leqq i¥leqq N$

, and using this in (4.7), we obtain

$¥frac{¥gamma t^{l-1}}{(l-1)!}¥leqq¥swarrow^{¥Gamma}x(t)$

$¥leqq¥frac{¥gamma t^{l-1}}{(l-1)!}+¥int_{T}^{t}¥frac{(t-s)^{¥iota-1}}{(l-1)!}ds$

$¥times¥int_{T}^{¥infty}r^{2n-¥iota-1}f$

$¥leqq¥frac{¥gamma t^{l-1}}{(l-1)!}+¥frac{¥gamma t^{¥iota}}{l!}$

(

$r$

,

$¥frac{¥gamma}{1-¥lambda}[g_{1}(r)+T+1]^{l}$

,

$t¥geqq T$

,. , $¥cdot$

$¥frac{¥gamma}{1-¥lambda}[g_{N}(r)+T+1]^{l}$

)

$dr$

,

maps $X$ into itself. The continuity of . Thus which implies that can be proved without difficulty. Thereand the relative compactness of associated with in $X$ . That the function has a fixed element fore, gives a solution of equation (1.1) follows, exactly as in the proof of Theorem 1, . To show that from differentiation of the integral equation satisfied by behavior, we note that asymptotic has the desired $J^{¥varpi}x¥in X$

$¥swarrow¥Gamma$

$¥swarrow¥varpi$

$¥ovalbox{¥tt¥small REJECT}(X)$

$x^{*}$

$¥swarrow¥Gamma$

$¥tilde{x}^{*}$

$¥tilde{x}^{*}$

(4.8)

$¥frac{d^{l-1}}{dt^{l-1}}[¥tilde{¥mathrm{x}}^{*}(t)-h(t)¥tilde{¥mathrm{x}}^{*}(¥tau(t))]=¥gamma+¥mathrm{H}_{Ts}^{t¥infty}¥frac{(r-s)^{2n-l-1}}{(2n-l-1)!}f(r,¥tilde{x}^{*}(g(r)))drds$

and

(4.9) for

$t$

$¥frac{d^{¥iota}}{dt^{l}}[¥tilde{x}^{*}(t)-h(t)¥tilde{¥chi}^{*}(¥tau(t))]=¥int_{t}^{¥infty}¥frac{(s-t)^{2n-l-1}}{(2n-l-1)!}f(s,¥tilde{¥chi}^{*}(g(s)))ds$

$¥geqq T$

From (3.6) (with

(4. 10)

$x=x^{*}$

) and (4.6) it follows that

$¥tilde{x}^{*}(t)=x^{*}(t)+h(t)¥tilde{x}^{*}(¥tau(t))¥geqq x^{*}(t)¥geqq¥frac{¥gamma t^{l-1}}{(l-1)!}$

for all large . $t$

Combining (4.10) with the inequality

$¥frac{d^{¥iota-1}}{dt^{1-1}}[¥tilde{x}^{*}(t)-h(t)¥tilde{x}^{*}(¥tau(t))]¥geqq¥gamma+¥int_{T}^{t}¥frac{(r-T)^{2n-l}}{(2n-l)!}f(r,¥tilde{x}^{*}(g(r)))dr$

which follows from (4.8), we see that

(4. 11)

$¥lim_{t¥rightarrow¥infty}¥frac{d^{l-1}}{dt^{l-1}}[¥tilde{¥mathrm{x}}^{*}(t)-h(t)¥tilde{x}^{*}(¥tau(t))]=¥infty$

On the other hand, from (4.9) we have

.

$x^{*}$

$¥tilde{x}^{*}$

Nonlinear FDE

(4. 12)

of Neutral

261

Type

$¥lim_{t¥rightarrow¥infty}¥frac{d^{¥iota}}{dt^{l}}[¥tilde{¥mathrm{x}}^{*}(t)-h(t)¥tilde{¥chi}^{*}(¥tau(t))]=0$

.

In view of the equivalence of (4. 11) and (4. 12) to $¥lim_{t¥rightarrow¥infty}¥frac{¥tilde{¥mathrm{x}}^{*}(t)-h(t)¥tilde{¥mathrm{x}}^{*}(¥tau(t))}{t^{l}}=0$

we conclude that the solution Example 2.

(4. 13)

$¥tilde{x}^{*}(t)$

,

$¥lim_{t¥rightarrow¥infty}¥frac{¥tilde{¥mathrm{x}}^{*}(t)-h(t)¥tilde{¥mathrm{x}}^{*}(¥tau(t))}{t^{l-1}}=¥infty$

satisfies (2.5).

,

This completes the proof.

Consider the equation $¥frac{d^{2n}}{dt^{2n}}[x(t)-¥lambda x(¥mu t)]+p(t)|x(t^{¥theta})|^{¥rho}¥mathrm{s}¥mathrm{g}¥mathrm{n}x(t^{¥theta})=0$

,

where $01$ , and strongly sublinear if $¥rho