Positive solutions of periodic boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative

Abstract

In this article, we study the existence of positive solutions of periodic boundary value problems for the second-order differential equation with the nonlinearity depends on the derivative

$$\begin{aligned} \left\{ {\begin{array}{l}\left( { Lu} \right) \left( t \right) = h\left( t \right) f\left( {u\left( t \right) ,{u}'\left( t \right) } \right) ,\quad 0 \le t \le \omega ,\\ R_1 \left( \! u \right) \equiv u\left( 0 \right) - u\left( \omega \right) = 0,\\ R_2 \left( \! u \right) \equiv p\left( 0 \right) {u}'\left( 0 \right) - p\left( \omega \right) {u}'\left( \omega \right) = 0, \end{array}} \right. \end{aligned}$$

where $\left( {Lu} \right) \left( t \right) = - \left( {p\left( t \right) {u}'} \right) ^\prime + q\left( t \right) u$. By applying coincidence degree theorem, some conditions guaranteeing the existence of at least one positive solution are given in terms of the relative behaviors of the quotient $\displaystyle \frac{f\left( {u,v} \right) }{\left| u \right| + \left| v \right| }$ for $|u|+ |v|$ near 0 and $+ \infty$. The result discussed in the paper is a generalization of recent one and the difference is that a nonlinear term depends on the derivative of unknown function.

Appendix

Like the well-known Arzela–Ascoli theorem in [6], it is easy to prove the lemma below.

Lemma 4.1

$F \subset C^1\left[ {0,1} \right]$ ( the norm is defined by (2.1)) is relatively compact if and only if both $F$ and ${F}{^\prime } = \left\{ {\left. {u}{^\prime } \right| u \in F} \right\}$ are uniformly bounded and equicontinuous.

Thus we can prove the following lemma which is applied in Sect. 2.

Lemma 4.2

$L^{ - 1}:Z \rightarrow X$ is a completely continuous linear operator.

Proof

Firstly, we shall show the continuity of $L^{ - 1}$.

Let $u_n ,u_0 \in Z$ with $\left\| {u_n - u_0 } \right\| _0 \rightarrow 0;$ then $\forall \varepsilon > 0,$ there exists a positive integer $N$ such that whenever $n > N,\left\| {u_n - u_0 } \right\| _0 < K^{ - 1}\varepsilon ,$ where

$$\begin{aligned} K&= \omega \max \{a\left\| \phi \right\| _0^2 + b\left\| \psi \right\| _0^2 + \left( { A + B } \right) \left\| \phi \right\| _0 \left\| \psi \right\| _0,\\&\quad a {\left\| {{\phi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 + b\left\| {{\psi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 + C\left( {\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 + \left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 } \right) } \}, \end{aligned}$$

and $a,b,A,B,\phi \left( t \right) ,\psi \left( t \right) ,C$ given by Lemma 1.1. So for $n > N$ we have

$$\begin{aligned}&\left| {\left( {L^{ - 1}u_n } \right) \left( t \right) - \left( {L^{ - 1}u_0 } \right) \left( t \right) } \right| \le \int _0^\omega {G\left( {t,s} \right) } \left| {u_n \left( s \right) - u_0 \left( s \right) } \right| ds \\&\quad \le \displaystyle \frac{K}{\omega }\int _{0}^{\omega }{\left| {u_n ( s ) - u_0 \left( s \right) } \right| } ds < \varepsilon , \left| {\left( {L^{ - 1}u_n } \right) ^\prime \left( t \right) - \left( {L^{ - 1}u_0 } \right) ^\prime \left( t \right) } \right| \\&\quad \le \left| {\left( {\int _0^\omega {G\left( {t,s} \right) } u_n \left( s \right) ds} \right) ^\prime - \left( {\int _0^\omega {G\left( {t,s} \right) } u_0 \left( s \right) ds} \right) ^\prime } \right| \\&\quad =\left| {\left( {\int _0^\omega {\left( {a\phi \left( t \right) \phi \left( s \right) - b\psi \left( t \right) \psi \left( s \right) } \right) (u_n \left( s \right) -u_0 \left( s \right) )} ds} \right) ^\prime } \right. \\&\qquad +\left( {\int _0^t {\left( {A\phi \left( t \right) \psi \left( s \right) - B\phi \left( s \right) \psi \left( t \right) } \right) (u_n \left( s \right) -u_0 \left( s \right) )} ds} \right) ^\prime \\&\qquad +\left. \left( {\int _t^\omega {\left( {A\phi \left( s \right) \psi \left( t \right) - B\phi \left( t \right) \psi \left( s \right) } \right) (u_n \left( s \right) -u_0 \left( s \right) )} ds} \right) ^\prime \right| \\&\quad =\left| {\left( {\int _0^\omega {a\phi \left( t \right) \phi \left( s \right) (u_n \left( s \right) \!-\!u_0 \left( s \right) )} ds} \right) ^\prime } \right. \\&\qquad -\left( {\int _0^\omega {b\psi \left( t \right) \psi \left( s \right) (u_n \left( s \right) \!-\!u_0 \left( s \right) )} ds} \right) ^\prime \\&\qquad + \left( {\int _0^t {A\phi \left( t \right) \psi \left( s \right) (u_n \left( s \right) \!-\!u_0 \left( s \right) )} ds} \right) ^\prime \end{aligned}$$

$$\begin{aligned}&\qquad - \left( {\int _0^t {B\phi \left( s \right) \psi \left( t\right) (u_n \left( s \right) \!-\!u_0 \left( s \right) )} ds} \right) ^\prime \\&\qquad + \left( {\int _t^\omega \!\! {A\phi \left( s \right) \psi \left( t \right) (u_n \left( s \right) \!-\!u_0 \left( s \right) )} ds} \right) ^\prime \\&\qquad \left. { - \left( {\int _t^\omega \!\! {B\phi \left( t \right) \psi \left( s \right) (u_n \left( s \right) \!-\!u_0 \left( s \right) )} ds} \right) ^\prime } \right| \\&\quad = \left| {a{\phi }{^\prime }\left( t \right) \int _0^\omega {\phi \left( s \right) \left( {u_n \left( s \right) - u_0 \left( s \right) } \right) } ds - b{\psi }{^\prime }\left( t \right) \int _0^\omega {\psi \left( s \right) \left( {u_n \left( s \right) - u_0 \left( s \right) } \right) } ds} \right. \\&\qquad +\, A{\phi }{^\prime }\left( t \right) \int _0^t {\psi \left( s \right) \left( {u_n \left( s \right) - u_0 \left( s \right) } \right) } ds - B{\psi }{^\prime }\left( t \right) \int _0^t {\phi \left( s \right) \left( {u_n \left( s \right) - u_0 \left( s \right) } \right) } ds \\&\qquad + \left. A{\psi }'\left( t \right) \int _t^\omega {\phi \left( s \right) \left( {u_n \left( s \right) - u_0 \left( s \right) } \right) } ds - B{\phi }{^\prime }\left( t \right) \int _t^\omega {\psi \left( s \right) \left( {u_n \left( s \right) - u_0 \left( s \right) } \right) } ds \right| \\&\quad \le \left( a {\left\| {{\phi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 + b\left\| {{\psi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 } \right) \int _0^\omega {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds \\&\qquad +\, A \left\| {{\phi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 \int _0^t {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds + B\left\| {{\phi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 \int _t^\omega {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds \\&\qquad +\,B \left\| {{\psi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 \int _0^t {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds + A\left\| {{\psi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 \int _t^\omega {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds. \\&\quad \le \left( a {\left\| {{\phi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 + b\left\| {{\psi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 } \right) \int _0^\omega {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds \\&\qquad +\, C\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 \int _0^t {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds + C\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 \int _t^\omega {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds \end{aligned}$$

$$\begin{aligned}&\qquad +\, C\left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 \int _0^t {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds + C\left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 \int _t^\omega {\left| {u_n \left( s \right) - u_0 \left( s \right) } \right| } ds \\&\quad \le \omega \left( a{\left\| {{\phi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 + b\left\| {{\psi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 } \right) \left\| {u_n \left( s \right) - u_0 \left( s \right) } \right\| _0 \\&\qquad +\, C\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 \left\| {u_n \left( s \right) - u_0 \left( s \right) } \right\| _0 t + C\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 \left\| {u_n \left( s \right) - u_0 \left( s \right) } \right\| _0 \left( {\omega - t} \right) \\&\qquad +\, C\left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 \left\| {u_n \left( s \right) - u_0 \left( s \right) } \right\| _0 t + C\left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 \left\| {u_n \left( s \right) - u_0 \left( s \right) } \right\| _0 \left( {\omega - t} \right) \\&\quad \le \omega \left( a {\left\| {{\phi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 + b\left\| {{\psi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 + C\left( {\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 + \left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 } \right) } \right) \\&\qquad \left\| {u_n \left( s \right) - u_0 \left( s \right) } \right\| _0\le \varepsilon . \end{aligned}$$

And hence $\left\| {L^{ - 1}u_n - L^{ - 1}u_0 } \right\| _1 < \varepsilon ,$ i.e. $L^{ - 1}$ is continuous.

Secondly, we shall show that both $L^{ - 1}F$ and $\left( {L^{ - 1}F} \right) ^\prime$ are uniformly bounded.

Suppose $F \subset Z$ is bounded; then there exists a constant $M > 0$ such that $\left\| u \right\| _0 \le M,\forall u \in F$. Consequently for $\forall u \in F,$

$$\begin{aligned} \left| {\left( {L^{ - 1}u} \right) \left( t \right) } \right|&= \left| {\int _0^\omega {G\left( {t,s} \right) } u\left( s \right) ds} \right| \le \int _0^\omega {G\left( {t,s} \right) } \left| {u\left( s \right) } \right| ds \le KM.\\ \left| {\left( {L^{ - 1}u} \right) ^\prime \left( t \right) } \right|&= \left| {\left( {\int _0^\omega {\left( {a\phi \left( t \right) \phi \left( s \right) - b\psi \left( t \right) \psi \left( s \right) } \right) } u\left( s \right) ds} \right) ^\prime } \right. \\&\,+ \left( {\int _0^t {\left( {A\phi \left( t \right) \psi \left( s \right) - B\phi \left( s \right) \psi \left( t \right) } \right) } u\left( s \right) ds} \right) ^\prime \\&\,+\left. \left( {\int _t^\omega {\left( {A\phi \left( s \right) \psi \left( t \right) - B\phi \left( t \right) \psi \left( s \right) } \right) } u\left( s \right) ds} \right) ^{\prime } \right| \\&= \left| {a{\phi }{^\prime }\left( t \right) \int _0^\omega {\phi \left( s \right) } u\left( s \right) ds - } \right. b{\psi }{^\prime }\left( t \right) \int _0^\omega {\psi \left( s \right) } u\left( s \right) ds\\&\,+ \,A{\phi }{^\prime }\left( t \right) \int _0^t {\psi \left( s \right) } u\left( s \right) ds - B{\psi }{^\prime }\left( t \right) \int _0^t {\phi \left( s \right) } u\left( s \right) ds\\&\,+\,\left. { A{\psi }{^\prime }\left( t \right) \int _t^\omega {\phi \left( s \right) } u\left( s \right) ds - B{\phi }{^\prime }\left( t \right) \int _t^\omega {\psi \left( s \right) } u\left( s \right) ds} \right| \\&\le \omega \left( a{\left\| {{\phi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 + b\left\| {{\psi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 } \right) \left\| u \right\| _0\\&\,+\,\quad C\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 \left\| u \right\| _0 t + C\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 \left\| u \right\| _0 \left( {\omega - t} \right) \\&\,+ \,C\left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 \left\| u \right\| _0 t + C\left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 \left\| u \right\| _0 \left( {\omega - t} \right) \\&\le \omega \left( a{\left\| {{\phi }{^\prime }} \right\| _0 \left\| \phi \right\| _0 + b\left\| {{\psi }{^\prime }} \right\| _0 \left\| \psi \right\| _0 } \right) \left\| u \right\| _0\\&\,+\,\,\omega C\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 \left\| u \right\| _0 + \omega C\left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 \left\| u \right\| _0\\&= \!\omega (a \left\| {{\phi }{^\prime }} \right\| _0 \left\| \phi \right\| _0\!+\!b\left\| {{\psi }{^\prime }} \right\| _0 \left\| \psi \right\| _0\\&+\,C\left( {\left\| {\phi }{^\prime } \right\| _0 \left\| \psi \right\| _0 \!+\! \left\| {\psi }{^\prime } \right\| _0 \left\| \phi \right\| _0 } \right) )\left\| u \right\| _0 \\&\le KM. \end{aligned}$$

These imply that both $L^{ - 1}F$ and $\left( {L^{ - 1}F} \right) ^\prime$ are uniformly bounded.

Thirdly, we shall show that $L^{ - 1}F$ and $\left( {L^{ - 1}F} \right) ^\prime$ are equicontinuous.

For $\forall u \in F,t_1 ,t_2 \in [0,\omega ],$ we have

$$\begin{aligned} \left| {\left( {L^{ - 1}u} \right) \left( {t_1 } \right) - \left( {L^{ - 1}u} \right) \left( {t_2 } \right) } \right|&= \left| {\int _0^\omega {\left( {G\left( {t_1 ,s} \right) - G\left( {t_2 ,s} \right) } \right) } u\left( s \right) ds} \right| \\&\le \int _0^\omega {\left| {G\left( {t_1 ,s} \right) - G\left( {t_2 ,s} \right) } \right| } \left| {u\left( s \right) } \right| ds\\&\le M\int _0^\omega {\left| {G\left( {t_1 ,s} \right) - G\left( {t_2 ,s} \right) } \right| } ds. \end{aligned}$$

It is easy to see that $L^{ - 1}F$ is equicontinuous.

Since

$$\begin{aligned}&\big |{\left( {L^{ - 1}u} \right) ^\prime \left( {t_1 } \right) \!-\! \left( {L^{ - 1}u} \right) ^\prime \left( {t_2 } \right) }|=\left| {\left( {\int _0^\omega {G\left( {t_1 ,s} \right) } u\left( s \right) ds} \right) ^\prime \!-\! \left( {\int _0^\omega {G\left( {t_2 ,s} \right) } u\left( s \right) ds} \right) ^\prime }\right| \\&\quad = \left| {a\left( {{\phi }'\left( {t_1 } \right) - {\phi }'\left( {t_2 } \right) } \right) \int _0^\omega {\phi \left( s \right) } u\left( s \right) ds - } \right. b\left( {{\psi }{^\prime }\left( {t_1 } \right) - {\psi }{^\prime }\left( {t_2 } \right) } \right) \int _0^\omega {\psi \left( s \right) } u\left( s \right) ds\\&\qquad +\, A{\phi }{^\prime }\left( {t_1 } \right) \int _0^{t_1 } {\psi \left( s \right) } u\left( s \right) ds - A{\phi }{^\prime }\left( {t_2 } \right) \int _0^{t_2 } {\psi \left( s \right) } u\left( s \right) ds \\&\qquad - \, B{\psi }{^\prime }\left( {t_1 } \right) \int _0^{t_1 } {\phi \left( s \right) } u\left( s \right) ds + B{\psi }{^\prime }\left( {t_2 } \right) \int _0^{t_2 } {\phi \left( s \right) } u\left( s \right) ds\\&\qquad + \, A{\psi }{^\prime }\left( {t_1 } \right) \int _{t_1 }^\omega {\phi \left( s \right) } u\left( s \right) ds - A{\psi }{^\prime }\left( {t_2 } \right) \int _{t_2 }^\omega {\phi \left( s \right) } u\left( s \right) ds \\&\qquad - \, B{\phi }{^\prime }(t_1)\int _{t_1 }^\omega \psi (s)u(s)ds+B{\phi }{^\prime }(t_2)\int _{t_2 }^\omega \psi (s)u(s)ds|\\&\quad = \left| a({\phi }{^\prime }(t_1)-{\phi }{^\prime }(t_2))\int _0^\omega \phi (s)u(s)ds-b({\psi }'(t_1)-{\psi }{^\prime }(t_2))\int _0^\omega \psi (s)u(s)ds\right. \\&\qquad + \, A{\phi }{^\prime }\left( {t_1 } \right) \int _0^{t_1 }\! {\psi \left( s \right) } u\left( s \right) ds \!-\! A{\phi }{^\prime }\left( {t_2 } \right) \int _{t_1 }^{t_2 }\! {\psi \left( s \right) } u\left( s \right) ds \\&\qquad - \,A{\phi }{^\prime }\left( {t_2 } \right) \int _0^{t_1 }\! {\psi \left( s \right) } u\left( s \right) ds \\&\qquad - \, B{\psi }{^\prime }\left( {t_1 } \right) \int _0^{t_1 }\! {\phi \left( s \right) } u\left( s \right) ds \!+\! B{\psi }{^\prime }\left( {t_2 } \right) \int _{t_1 }^{t_2 }\! {\phi \left( s \right) } u\left( s \right) ds \\&\qquad +\, B{\psi }{^\prime }\left( {t_2 } \right) \int _0^{t_1 }\! {\phi \left( s \right) } u\left( s \right) ds \\&\qquad +\, A{\psi }{^\prime }\left( {t_1 } \right) \int _{t_1 }^\omega \! {\phi \left( s \right) } u\left( s \right) ds \!-\! A{\psi }{^\prime }\left( {t_2 } \right) \int _{t_1 }^\omega \! {\phi \left( s \right) } u\left( s \right) ds \\&\qquad - \,A{\psi }{^\prime }\left( {t_2 } \right) \int _{t_2 }^{t_1 }\! {\phi \left( s \right) } u\left( s \right) ds \\&\qquad \left. -\,B{\phi }{^\prime }(t_1)\int _{t_1 }^\omega \! \psi (s)u(s)ds\!+\!B{\phi }'(t_2)\int _{t_1 }^\omega \! \psi (s)u(s)ds+B{\phi }{^\prime }(t_2)\int _{t_2 }^{t_1 }\!\psi (s)u(s)ds \right| \end{aligned}$$

$$\begin{aligned}&\quad \le a\left| {{{\phi }{^\prime }\left( {t_1 } \right) - {\phi }{^\prime }\left( {t_2 } \right) } } \right| \left\| \phi \right\| _0 \left\| u \right\| _0 \omega + b\left| {{{\psi }{^\prime }\left( {t_1 } \right) - {\psi }{^\prime }\left( {t_2 } \right) }} \right| \left\| \psi \right\| _0 \left\| u \right\| _0 \omega \\&\qquad + \, A\left| {{{\phi }{^\prime }\left( {t_1 } \right) - {\phi }{^\prime }\left( {t_2 } \right) }} \right| \left\| \psi \right\| _0 \left\| u \right\| _0 \omega +\, A\left| {{\phi }'\left( {t_2 } \right) } \right| \left\| \psi \right\| _0 \left\| u \right\| _0 \left| {t_2 - t_1 } \right| \\&\qquad + \, B\left| {{{\psi }{^\prime }\left( {t_1 } \right) - {\psi }{^\prime }\left( {t_2} \right) } } \right| \left\| \phi \right\| _0 \left\| u \right\| _0 \omega + \, B\left| {{\psi }{^\prime }\left( {t_2 } \right) } \right| \left\| \phi \right\| _0 \left\| u \right\| _0 \left| {t_2 - t_1 } \right| \\&\qquad + \, A\left| { {{\psi }{^\prime }\left( {t_1 } \right) - {\psi }{^\prime }\left( {t_2 } \right) } } \right| \left\| \phi \right\| _0 \left\| u \right\| _0 \omega + \, A\left| {{\psi }{^\prime }\left( {t_2 } \right) } \right| \left\| \phi \right\| _0 \left\| u \right\| _0 \left| {t_2 - t_1 } \right| \\&\qquad + \, B\left| { {{\phi }{^\prime }\left( {t_1 } \right) - {\phi }{^\prime }\left( {t_2 } \right) } } \right| \left\| \psi \right\| _0 \left\| u \right\| _0 \omega + \, B\left| {{\phi }{^\prime }\left( {t_2 } \right) } \right| \left\| \psi \right\| _0 \left\| u \right\| _0 \left| {t_2 - t_1 } \right| \\&\quad \le a\left| { {{\phi }{^\prime }\left( {t_1 } \right) - {\phi }'\left( {t_2 } \right) } } \right| \left\| \phi \right\| _0 M\omega + b\left| { {{\psi }{^\prime }\left( {t_1 } \right) - {\psi }{^\prime }\left( {t_2 } \right) } } \right| \left\| \psi \right\| _0 M\omega \\&\qquad + \, A\left| { {{\phi }{^\prime }\left( {t_1 } \right) - {\phi }'\left( {t_2 } \right) } } \right| \left\| \psi \right\| _0 M\omega + A\left| {{\phi }{^\prime }\left( {t_2 } \right) } \right| \left\| \psi \right\| _0 M\left| {t_2 - t_1 } \right| \\&\qquad + \, B\left| {{{\psi }{^\prime }\left( {t_1 } \right) - {\psi }{^\prime }\left( {t_2 } \right) } } \right| \left\| \phi \right\| _0 M\omega + B\left| {{\psi }{^\prime }\left( {t_2 } \right) } \right| \left\| \phi \right\| _0 M\left| {t_2 - t_1 } \right| \\&\qquad + \, A\left| { {{\psi }{^\prime }\left( {t_1 } \right) - {\psi }{^\prime }\left( {t_2 } \right) } } \right| \left\| \phi \right\| _0 M\omega + A\left| {{\psi }'\left( {t_2 } \right) } \right| \left\| \phi \right\| _0 M\left| {t_2 - t_1 } \right| \\&\qquad + \, B\left| { {{\phi }{^\prime }\left( {t_1 } \right) - {\phi }{^\prime }\left( {t_2 } \right) }} \right| \left\| \psi \right\| _0 M\omega + B\left| {{\phi }'\left( {t_2 } \right) } \right| \left\| \psi \right\| _0 M\left| {t_2 - t_1 } \right| , \end{aligned}$$

$\left( {L^{ - 1}F} \right) ^{\prime }$ is equicontinuous.

To sum up, it can be inferred that $L^{ - 1}F$ is relatively compact in $X$ by Lemma 4.1. According to the definition of completely continuous linear operator in [6], we can know that $L^{ - 1}$ is completely continuous.