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<title>Necessary and sufficient conditions for oscillation of the solutions of even order differential equations\(^{\bullet }\): Necessary and sufficient conditions for oscillation of the solutions of even order differential equations\(^{\bullet }\)</title>
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<h1>Necessary and sufficient conditions for oscillation of the solutions of even order differential equations\(^{\bullet }\)</h1>
<p class="authors">
<span class="author">Cheng Jin-Fa\(^{\S }\) Chu Yu-Ming\(^{\ast }\)</span>
</p>
<p class="date">December 15, 2010.</p>
</div>
<p>\(^\S \) School of Mathematical Sciences, Xiamen University, Siming South Str. no. 422, 361005, Siming District, Xiamen, China, e-mail: <span class="tt">jfcheng@xmu.edu.cn</span>. </p>
<p>\(^\ast \)Department of Mathematics, Huzhou Teachers College, Xueshi Str. no. 1, 313000, Huzhou, China, e-mail: <span class="tt">chuyuming2005@yahoo.com.cn</span>. </p>
<p>\(^{\bullet }\)The work of the second author has been supported by the Natural Science Foundation of China (Grant No. 11071069) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant No. T200924). </p>

<div class="abstract"><p> In this paper, we establish several necessary and sufficient conditions for oscillation of the solutions of the following even order differential equation </p>
<div class="displaymath" id="a0000000002">
  \[  x^{(n)}(t) + q(t)x^\gamma (t) = 0, \quad \mbox{$n$ is even},  \]
</div>
<p> where \( q(t) \in C([t_0 ,\infty ),{\mathbb R}^+ )\) and \(\gamma \) is the quotient of odd positive integers. </p>
<p><b class="bf">MSC.</b> 34K11. </p>
<p><b class="bf">Keywords.</b> Oscillation, nonoscillatory solution, even order differential equation. </p>
</div>
<h1 id="a0000000003">1 Introduction</h1>
<p>Considering the \(n\)-order differential equation </p>
<div class="equation" id="a0000000004">
<p>
  <div class="equation_content">
    \begin{equation}  x^{(n)}(t) + q(t)x^\gamma (t) = 0,\quad \mbox{ $n$ is even}, \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where \( q(t) \in C([t_0 ,\infty ),{\mathbb R}^+ )\) and \(\gamma \) is the quotient of odd positive integers. </p>
<p>In the recent past, the asymptotic and oscillatory properties of the solutions of \(n\)-order differential equations have been researched by many authors (see [1–3, 7–9]). </p>
<p>A solution of Eq.(1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, the solution is said to be nonoscillatory. </p>
<p>We say that Eq.(1) is strictly superlinear if \(\gamma {\gt}1\), strictly sublinear if \(0{\lt}\gamma {\lt}1\), and linear if \(\gamma =1\). </p>
<p>In particular, if \(n=2\), then Eq.(1) reduced to </p>
<div class="equation" id="a0000000005">
<p>
  <div class="equation_content">
    \begin{equation}  x''(t) + q(t)x^\gamma (t) = 0, \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> Eq.(2) is the well-known Emden-Fowler equation (see [10–12]). </p>
<p>Many remarkable results have been established for the oscillation of solutions of the second and higher order functional differential equations. For example, the following well-known Theorems A-C are presented in [4–6]. </p>
<p><i class="sc">Theorem A</i> (see [4, 6]). <i class="itshape">If \(\gamma {\gt}0\), then Eq.<span class="rm">(2)</span> has a bounded nonoscillatory solution if and only if <div class="displaymath" id="a0000000006">
  \[  \int _{t_0 }^\infty {sq(s){\rm d}s {\lt} \infty .} \]
</div></i> </p>
<p><i class="sc">Theorem B</i> (see [4, 5]). <i class="itshape">If \(\gamma {\gt}1\), then all solutions of Eq.<span class="rm">(2)</span> are oscillatory if and only if <div class="displaymath" id="a0000000007">
  \[  \int _{t_0 }^\infty {sq(s){\rm d}s = \infty .} \]
</div></i> </p>
<p><i class="sc">Theorem C</i> (see [6]). <i class="itshape">If \(0{\lt} \gamma {\lt}1\), then Eq.<span class="rm">(2)</span> is oscillatory if and only if <div class="displaymath" id="a0000000008">
  \[  \int _{t_0 }^\infty {s^{\gamma }q(s){\rm d}s = \infty .} \]
</div></i> </p>
<p>For Eq.(1) with \(\gamma =1\), the following Theorem D is presented in [7]. </p>
<p><i class="sc">Theorem D.</i> <i class="itshape">If \(\gamma =1\), then every bounded solution of Eq.<span class="rm">(2)</span> oscillates if and only if <div class="displaymath" id="a0000000009">
  \[  \int _{t_0 }^\infty {s^{n - 1}q(s){\rm d}s = \infty }. \]
</div></i> </p>
<p>Due to some obstacles of theoretical and technical character in handling with higher order nonlinear differential equation, and there are a few results which presented the necessary and sufficient conditions for the oscillatory behavior when \(\gamma \ne 1\). So there are a lot of problems worth to be considered further for the Eq.(1). </p>
<p>The main aim of this paper is to prove the following Theorem 1.1: </p>
<p><i class="sc">Theorem 1.1.</i> <i class="itshape">If \(\gamma \ne 1\) is the quotient of odd positive integers and \(n\) is even, then the following statements are true</i>: </p>
<p><i class="itshape">\((a)\) If <div class="equation" id="a0000000010">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{t_0 }^\infty {s^{n - 1} q(s){\rm d}s < \infty }, \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div> then Eq.<span class="rm">(1)</span> has a bounded nonoscillatory solution</i>; </p>
<p><i class="itshape">\((b)\) If \(\gamma {\gt}1\), then every solution of Eq.<span class="rm">(1)</span> oscillates if and only if <div class="equation" id="a0000000011">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{t_0 }^\infty {s^{n - 1} q(s){\rm d}s = \infty }; \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div></i> </p>
<p><i class="itshape">\((c)\) If \(0{\lt}\gamma {\lt}1\), then every solution of Eq.<span class="rm">(1)</span> oscillates if and only if <div class="equation" id="a0000000012">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{t_0 }^\infty {s^{(n - 1)\gamma } q(s){\rm d}s = \infty }. \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div></i> </p>
<p>We clearly see that Theorems A-C are the special case of our Theorem 1.1. </p>
<h1 id="a0000000013">2  Proof of Theorem 1.1</h1>
<p> In order to prove Theorem 1.1 we need the following Lemma 2.1. </p>
<p><i class="sc">Lemma 2.1</i> (see [1-2, 7]). <i class="it">Let \(x(t)\) be a positive and \(n\)-times differentiable function on \([t_0,\infty )\), and \(x^{(n)}(t)\) be nonpositive and not identically zero on any subinterval \([t_1,\infty )\). Then there exist \(T \ge t_0\) and integer \(k \in {\{ 0,1,...,n-1\} }\), such that \(n+k\) is odd and</i> </p>
<p><i class="itshape">\((i)\) \(x^{(i)}(t) \ge 0\) for \(t \ge T, i=0,1,...,k-1;\)</i> </p>
<p><i class="itshape">\((ii)\) \((-1)^{i+k}x^{(i)}(t){\gt}0\) for \(i=k,k+1,...,n;\)</i> </p>
<p><i class="itshape">\((iii)\) \((t-T)|x^{(k-i)}(t)| \le (1+i)|x^{(k-i-1)}(t)|\) for \(t\ge T,i=0,1,...,k-1,k=1,...,n-1.\)</i> </p>
<p><i class="itshape">Proof of the Theorem 1.1.</i> </p>
<p>\((a)\) Assume that (3) holds, we first prove that Eq.(1) has a nonoscillatory solution. </p>
<p>Observing that if \(x(t)\) satisfies the equation </p>
<div class="equation" id="a0000000014">
<p>
  <div class="equation_content">
    \begin{equation}  x(t) = 1 - \tfrac {1}{{(n - 1)!}}\int _t^\infty {(s - t)^{n - 1} q(s)x^\gamma (s){\rm d}s}, \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> then \(x(t)\) is a solution of Eq.(1). Therefore it suffices to show that Eq.(6) has bounded nonoscillatory solution. To this end, choose sufficient large \(t \ge T\) such that </p>
<div class="equation" id="a0000000015">
<p>
  <div class="equation_content">
    \begin{equation}  \max \left\{  \int \limits _t^\infty s^{n - 1} q(s){\rm d}s,2\gamma \int \limits _t^\infty s^{n - 1} q(s){\rm d}s \right\}  < \tfrac {1}{2} (n - 1)! .\end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> Next, we consider the functional set </p>
<div class="displaymath" id="a0000000016">
  \[  M = \{  x \in C([T,\infty ),{\mathbb R}):\tfrac {1}{2} \le x(t) \le 1\}   \]
</div>
<p> and define the operator \(S:M \to C([T,\infty ),{\mathbb R})\) as follows: </p>
<div class="equation" id="a0000000017">
<p>
  <div class="equation_content">
    \begin{equation}  Sx(t) = 1 - \tfrac {1}{{(n - 1)!}}\int _t^\infty {(s - t)^{n - 1} q(s)x^\gamma (s){\rm d}s}.\end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> We clearly see that \( x(t) ^\gamma \le 1\) and </p>
<div class="displaymath" id="a0000000018">
  \[  (Sx)(t) \ge 1 - \tfrac {1}{{(n - 1)!}}\int \limits _{t }^\infty {(s - t)^{n - 1} q(s){\rm d}s \ge \tfrac {1}{2} \quad for \  t \ge T.}  \]
</div>
<p> Therefore, \((Sx)(t)\le 1\) and \( S:M \to M\). Now, we claim that \(S\) is a contraction on \(M\). In fact, let \(f(x)=x^\gamma \), then for \(x_1,x_2 \in (\tfrac {1}{2},1)\) one has </p>
<div class="displaymath" id="a0000000019">
  \[  |x_1^\gamma - x_2^\gamma | = |f'(\xi )||x_1^{} - x_2^{} |, \  \text{ where} \  \xi \in (\min \{  x_1 ,x_2 \}  ,\max \{  x_1 ,x_2 \}  ),  \]
</div>
<p> where </p>
<div class="displaymath" id="a0000000020">
  \begin{eqnarray*} |f’(\xi )|=|\gamma \xi ^{\gamma -1}| & \le &  \left\{  \begin{array}{lll} \gamma , &  \  if \  \gamma \ge 1 \,  , & \\[ 4pt] 2\gamma , &  \  if \  0{\lt}\gamma {\lt}1 \, . & \end{array} \right. \end{eqnarray*}
</div>
<p>Therefore </p>
<div class="displaymath" id="a0000000021">
  \[  |x_1^\gamma - x_2^\gamma | \le 2\gamma |x_1^{} - x_2^{} |, \quad for \  x_1 ,x_2 \in (\tfrac {1}{2},1).  \]
</div>
<p> Let \(x,w \in M\), then for \(n\ge N\) one has </p>
<div class="displaymath" id="a0000000022">
  \begin{align*}  |(Sx)(t) - (Sw)(t)| & \le \tfrac {1}{{(n - 1)!}}\int \limits _t^\infty {(s - t)^{n - 1} } q(s)|x^\gamma (s) - w^\gamma (s)|{\rm d}s\\ &  \le \tfrac {2\gamma }{(n-1)!} \int \limits _t^\infty {(s - t)^{n - 1} } q(s)|x(s) - w(s)|{\rm d}s\\ &  \le \tfrac {2\gamma }{(n-1)!} ||x(s) - w(s)||\int \limits _t^\infty {(s - t)^{n - 1} } q(s){\rm d}s \le \tfrac {1}{2}||x - w||. \end{align*}
</div>
<p> Hence</p>
<div class="equation" id="a0000000023">
<p>
  <div class="equation_content">
    \begin{equation} ||Sx - Sw|| \le \tfrac {1}{2}||x - w|| \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> and \(S\) is a contraction on \(M\). The (unique) fixed point of \(T\) is the desired bounded, nonoscillatory solution of Eq.(1). </p>
<p>\((b)\) Sufficiency. Assume that \(\gamma {\gt}1\) and \( \int _{t_0 }^\infty {s^{n - 1} q(s){\rm d}s = \infty }\), we prove that every solution of Eq.(1) oscillates. Otherwise, Eq.(1) has a nonoscillatory solution \(x(t)\). Without loss of generality, we assume that \(x(t){\gt}0\) for \(t\ge t_0\). Then Lemma 2.1 implies that there exist odd integer \(k \in {\{ 1,...,n-1\} }\) and \(T_k \ge t_0\) such that </p>
<div class="equation" id="a0000000024">
<p>
  <div class="equation_content">
    \begin{equation} x^{(i)}(t)>0, \quad for \quad t \ge T_k,0 \le i \le k,\end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<div class="equation" id="a0000000025">
<p>
  <div class="equation_content">
    \begin{equation} (-1)^{i+k}x^{(i)}(t)>0, \quad for \quad t \ge T_k, k \le i \le n.\end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p> The proof is divided into two cases. </p>
<p>Case 1 \(k=1\). That is </p>
<div class="equation" id="a0000000026">
<p>
  <div class="equation_content">
    \begin{equation} x'(t)>0,x''(t)<0,x^{(3)}(t)>0,...,x^{n}(t)<0.\end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> From (10) and (11) together with the Taylor expansion we get </p>
<div class="equation" id="a0000000027">
<p>
  <div class="equation_content">
    \begin{equation}  x'(t) = \sum \limits _{j = 0}^{n - 2} {\tfrac {{( - 1)^j }}{{j!}}} x^{(1 + j)} (\tau )(\tau - t)^j + \tfrac {{( - 1)^{n - 1} }}{{(n - 2)!}}\int \limits _t^\tau {(s - t)^{n - 2} x^{(n)} (s){\rm d}s.} \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p> Using (12) we have </p>
<div class="equation" id="a0000000028">
<p>
  <div class="equation_content">
    \begin{equation}  x'(t) > \int \limits _t^\tau {\tfrac {{(s - t)^{n - 2} }}{{(n - 2)!}}} q(s)x^\gamma (s){\rm d}s, \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p> which implies </p>
<div class="equation" id="a0000000029">
<p>
  <div class="equation_content">
    \begin{equation}  x'(t) > \int \limits _t^\infty {\tfrac {{(s - t)^{n - 2} }}{{(n - 2)!}}} q(s)x^\gamma (s){\rm d}s > \int \limits _t^\infty {\tfrac {{(s - t)^{n - 2} }}{{(n - 2)!}}} q(s){\rm d}sx^\gamma (t). \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> From inequality </p>
<div class="displaymath" id="a0000000030">
  \begin{align*}  \int _T^t (u - s)^{n - k - 1} {\rm d}s & = - \tfrac {{(u - s)^{u - k} }}{{n - k}}|_T^t = \tfrac {1}{{n - k}}[(u - T)^{n - k} - (u - t)^{n - k} ]\\ &  \ge \tfrac {1}{{n - k}}(t - T)(u - T)^{n - k - 1} \end{align*}
</div>
<p> we obtain </p>
<div class="displaymath" id="a0000000031">
  \begin{align*}  \int \limits _{T_{} }^t {\tfrac {{x’(s)}}{{x^\gamma (s)}}} {\rm d}s&  {\gt} \int \limits _{T_{} }^t {{\rm d}s\int \limits _s^\infty {} \tfrac {{(u - s)^{n - 2} }}{{(n - 2)!}}q(u)} {\rm d}u \\ &  = \int \limits _{T_{} }^t {q(u){\rm d}u\int \limits _T^u {\tfrac {{(u - s)^{n - 2} }}{{(n - 2)!}}} } {\rm d}s + \int \limits _t^\infty {q(u){\rm d}u} \int \limits _T^t {\tfrac {{(u - s)^{n - 2} }}{{(n - 2)!}}} {\rm d}s\\ &  \ge \int \limits _{T_{} }^t {\tfrac {{(u - T)^{n - 1} }}{{(n - 1)!}}q(u)} {\rm d}u + (t - T)\int \limits _{t_{} }^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}q(u)} {\rm d}u. \end{align*}
</div>
<p> Therefore </p>
<div class="equation" id="a0000000032">
<p>
  <div class="equation_content">
    \begin{equation}  \int \limits _{T_{} }^t {\tfrac {{(u - T)^{n - 1} }}{{(n - 1)!}}q(u)}{\rm d}u< \int \limits _{T_{} }^t {\tfrac {{x’(s)}}{{x^\gamma (s)}}} {\rm d}s \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p> or </p>
<div class="equation" id="a0000000033">
<p>
  <div class="equation_content">
    \begin{equation}  \int \limits _{T_{} }^t {\tfrac {{(u - T)^{n - 1} }}{{(n - 1)!}}q(u)} {\rm d}u < \tfrac {{x^{1 - \gamma } (t)}}{{\gamma - 1}} < \infty , \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p> which contradicts with </p>
<div class="displaymath" id="a0000000034">
  \[  \int \limits _{T_{} }^\infty {u^{n - 1} q(u)} {\rm d}u = \infty . \]
</div>
<p>Case 2 \(k{\gt}1\). It follows from \((iii)\) of Lemma 2.1 that for \(t \ge T_k\),</p>
<div class="equation" id="a0000000035">
<p>
  <div class="equation_content">
    \begin{equation}  x(t) \ge \tfrac {{(t - T_k )^{k - 1} }}{{k!}}x^{(k - 1)} (t).\end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> For sufficient large \(t\), we have </p>
<div class="displaymath" id="a0000000036">
  \[  x^\gamma (t) \ge \tfrac {{(t - T_k )^{(k - 1)\gamma } }}{{(k!)^\gamma }}(x^{(k - 1)} (t))^\gamma {\gt} \tfrac {{(t - T_k )^{k - 1} }}{{(k!)^\gamma }}(x^{(k - 1)} (t))^\gamma ,\gamma {\gt} 1.  \]
</div>
<p> Let \(z(t)=x^{(k-1)}(t)\), then </p>
<div class="displaymath" id="a0000000037">
  \[ z(t){\gt}0,z'(t){\gt}0,z''(t){\lt}0,... \]
</div>
<p> and so </p>
<div class="equation" id="a0000000038">
<p>
  <div class="equation_content">
    \begin{equation}  z^{(n - k + 1)} (t) + q(t)\tfrac {{(t - T_k )^{k - 1} }}{{(k!)^\gamma }}z^\gamma (t) < 0. \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>Making use of the same method as in the proof of case 1, we get </p>
<div class="displaymath" id="a0000000039">
  \[  \int _{t_0 }^\infty {s^{n - k}q(s)\tfrac {(s - T_k )^{k - 1}}{(k!)^{\gamma }}{\rm d}s {\lt} \infty }  \]
</div>
<p> or </p>
<div class="equation" id="a0000000040">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{t_0 }^\infty {s^{n - 1} q(s)ds < \infty } , \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p> which also contradicts with </p>
<div class="displaymath" id="a0000000041">
  \[  \int \limits _{t_0{} }^\infty {s^{n - 1} q(s)}{\rm d}s = \infty .  \]
</div>
<p>Conversely, we prove that (4) holds if every solution of Eq.(1) oscillates and \(\gamma {\gt}1\). Otherwise (3) holds, then from Theorem 1.1\((a)\) we get the contradiction that Eq.(1) has a nonoscillatory solution. </p>
<p>\((c)\) Sufficiency. For \(0{\lt}\gamma {\lt}1\), there are two cases as follows. </p>
<p>Case 1 \(k=1\). That is </p>
<div class="displaymath" id="a0000000042">
  \[ x(t){\gt}0,x'(t){\gt}0,x''(t){\lt}0,...,x^{n}(t){\lt}0. \]
</div>
<p> Making use of the same method as Case 1 in Theorem 1.1\((b)\), we have </p>
<div class="equation" id="a0000000043">
<p>
  <div class="equation_content">
    \begin{equation}  x'(t) > \int \limits _t^\infty {\tfrac {{(s - t)^{n - 2} }}{{(n - 2)!}}} q(s)x^\gamma (s){\rm d}s. \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p> Integrating (21) from \(T\) to \(t\) yields </p>
<div class="displaymath" id="a0000000044">
  \begin{align*}  x(t)&  {\gt} x(t) - x(T)\\ &  {\gt} \int \limits _T^t {\tfrac {{(u - T)^{n - 1} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u + (t - T)\int \limits _t^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u\\ &  {\gt} (t - T)\int \limits _t^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u \end{align*}
</div>
<p> or </p>
<div class="equation" id="a0000000045">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {{x(t)}}{{t - T}} > \int \limits _t^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u. \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p> Let </p>
<div class="equation" id="a0000000046">
<p>
  <div class="equation_content">
    \begin{equation}  z(t) = \int \limits _t^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u, \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p> then \(z'(t){\lt}0,0{\lt}z(t){\lt} \tfrac {x(t)}{t-T}\) and </p>
<div class="equation" id="a0000000047">
<p>
  <div class="equation_content">
    \begin{equation}  z'(t) = - \tfrac {{(t - T)^{n - 2} }}{{(n - 1)!}}q(t)x^\gamma (t) \le -\tfrac {{(t - T)^{n - 2 + \gamma } }}{{(n - 1)!}}q(t)z^\gamma (t), \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<div class="equation" id="a0000000048">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {{z’(t)}}{{z^\gamma (t)}} \le - \tfrac {{(t - T)^{n - 2 + \gamma } }}{{(n - 1)!}}q(t) \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p> for \(T_2{\gt}T\). Then we get </p>
<div class="equation" id="a0000000049">
<p>
  <div class="equation_content">
    \begin{equation}  \int \limits _{T_2 }^t {\tfrac {{z’(u)}}{{z^\gamma (u)}}}{\rm d}u \le - \int \limits _{T_2 }^t {\tfrac {{(u - T)^{n - 2 + \gamma } }}{{(n - 1)!}}q(u)}{\rm d}u, \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<div class="equation" id="a0000000050">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {1}{{1 - \gamma }}[z^{1 - \gamma } (t) - z^{1 - \gamma } (T_2 )] \le - \tfrac {1}{{(n - 1)!}}\int \limits _{T_2 }^t {(u - T)^{n - 2 + \gamma } q(u){\rm d}u}. \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p> Therefore </p>
<div class="equation" id="a0000000051">
<p>
  <div class="equation_content">
    \begin{equation}  \int \limits _{T_2 }^t {(u - T)^{n - 2 + \gamma } q(u){\rm d}u} < + \infty . \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p> Inequality (28) and \((n-1)\gamma {\lt}n-2+\gamma \) leads to </p>
<div class="equation" id="a0000000052">
<p>
  <div class="equation_content">
    \begin{equation}  \int \limits _{T_2 }^t {(u - T)^{(n - 1)\gamma } q(u){\rm d}u} < + \infty , \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<p> which contradicts with the assumption. </p>
<p>Case 2 \(k{\gt}1\). That is </p>
<div class="displaymath" id="a0000000053">
  \[ x(t){\gt}0, x'(t){\gt}0,..., x^{(k-1)}(t){\gt}0, x^{(k)}(t){\gt}0, x^{(k+1)}(t){\lt}0,...,x^{(n)}(t){\lt}0. \]
</div>
<p> Lemma 2.1 implies </p>
<div class="displaymath" id="a0000000054">
  \[  x(t) \ge \tfrac {{(t - T_k )^{k - 1} }}{{k!}}x^{(k - 1)} (t)  \]
</div>
<p> or </p>
<div class="equation" id="a0000000055">
<p>
  <div class="equation_content">
    \begin{equation}  x^\gamma (t) \ge \tfrac {{(t - T_k )^{(k - 1)\gamma } }}{{(k!)^\gamma }}[x^{(k - 1)} (t)]^\gamma . \end{equation}
  </div>
  <span class="equation_label">30</span>
</p>
</div>
<p> Let \(z(t)=x^{(k-1)}(t)\), then \(z(t){\gt}0,z'(t){\gt}0,z''(t){\lt}0,...,z^{(n - k + 1)}{\lt}0\) and </p>
<div class="equation" id="a0000000056">
<p>
  <div class="equation_content">
    \begin{equation}  z^{(n - k + 1)} (t) + q(t)\tfrac {{(t - T_k )^{(k - 1)\gamma } }}{{(k!)^\gamma }}z^\gamma (t) < 0, \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p> where \(n-k+1\) is also even. Making use of the same method as in Case 1, we conclude that </p>
<div class="equation" id="a0000000057">
<p>
  <div class="equation_content">
    \begin{equation}  \int \limits _{t_0 }^\infty {s^{(n - k)\gamma } q(s)}\tfrac {(s - T_k )^{(k - 1)\gamma }}{(k!)^{\gamma }} {\rm d}s < + \infty \end{equation}
  </div>
  <span class="equation_label">32</span>
</p>
</div>
<p> or </p>
<div class="equation" id="a0000000058">
<p>
  <div class="equation_content">
    \begin{equation}  \int \limits _{t_0 }^\infty {s^{(n - 1)\gamma } q(s)} {\rm d}s < + \infty , \end{equation}
  </div>
  <span class="equation_label">33</span>
</p>
</div>
<p> which also contradicts with the assumption. </p>
<p>Necessity. For \(0{\lt}\gamma {\lt}1\) and (5) holds, we prove that Eq.(1) has a nonoscillatory solution. Otherwise, from (33) we know that there exists \(t \ge T\) such that </p>
<div class="equation" id="a0000000059">
<p>
  <div class="equation_content">
    \begin{equation}  \int \limits _t^\infty {s^{(n - 1)\gamma } } q(s){\rm d}s \le \tfrac {1}{2}. \end{equation}
  </div>
  <span class="equation_label">34</span>
</p>
</div>
<p>Let \(M\) be a set defined by </p>
<div class="displaymath" id="a0000000060">
  \[ M=\{ x \in C([T, \infty ),R):\tfrac {1}{2(n-1)!}(t-T)^{n-1} \le x(t) \le \tfrac {1}{(n-1)!}(t-T)^{n-1},t \ge T\}  \]
</div>
<p> and the mapping \(T\) on \(M\) defined by</p>
<div class="equation" id="a0000000061">
<p>
  <div class="equation_content">
    \begin{equation}  Sx(t)=\int \limits _T^t {{\rm d}s_1 \int \limits _T^{s_1 } {{\rm d}s_2 \cdots } } \int \limits _T^{s_{n - 2} } {[\tfrac {1}{2}} + \int \limits _{s_{n - 1} }^\infty {q(u)x^\gamma (u){\rm d}u]{\rm d}s_{n - 1} }. \end{equation}
  </div>
  <span class="equation_label">35</span>
</p>
</div>
<p>Then \((Sx)(t) \ge \tfrac {1}{2(n-1)!}(t-T)^{n-1}\) for \(x(t) \in M\) and \(t \ge T\). Moreover, from the definition of the operator \(S\) we get \((Sx)(t) \le \tfrac {1}{(n-1)!}(t-T)^{n-1}.\) Therefore, \(TM \subseteq M\). </p>
<p>Next, we define the function \(u_n:[T,\infty ) \to {\mathbb R}\) as follows </p>
<div class="equation" id="a0000000062">
<p>
  <div class="equation_content">
    \begin{equation} u_n=(Su_{n-1})(t), \quad n \in {\mathbb N}\end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p> and </p>
<div class="displaymath" id="a0000000063">
  \[ u_0(t)=\tfrac {1}{2(n-1)!}(t-T)^{n-1}, \quad t \ge T. \]
</div>
<p>A straightforward verification leads to </p>
<div class="displaymath" id="a0000000064">
  \[ \tfrac {1}{2(n-1)!}(t-T)^{n-1} \le u_{n-1}(t) \le u_n(t) \le \tfrac {1}{(n-1)!}(t-T)^{n-1}, \quad t \ge T. \]
</div>
<p> Therefore, there exists the limit \( \mathop{\lim }\limits _{n \to \infty } u_n (t) = u(t)\) for \(t \ge T.\) It follows from the Lebesgue convergence theorem that \(u \in M\) and \(u(t)=(Su)(t).\) It is easy to see that \(u(t)\) is the solution of the Eq.(1). </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000065">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgement
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
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</dd>
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</dd>
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</dd>
</dl>


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