Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations
The paper studies oscillation of the nonlinear differential equation \[ (r(t)\Psi (y(t))| y^{\prime }(t)| ^{\alpha -2}y^{\prime }(t))^{\prime }+q(t)f(y(t))g(y^{\prime }(t))=0,\;\alpha>1. \] by averaging of certain functional quantities on a sequence of intervals in the nonnegative half-axis. It is assumed that \(\Psi (y)\), \(yf(y)>0\) for all nonzero \(y\), and \(g(y)\geq K>0\) for a certain constant \(K\). By introducing the Philos-like kernel \(H(t,s)\), a thorough analysis of the functional quantities \(A(h;s,t,\rho )=\int_s^t\rho (v)H(t,v)h(v)\,dv\), \(t\geq s\), and \(B(h;t,s,\rho )=\int_t^s\rho (v)H(t,v)h(v)\,dv\), \(t\leq s\), is performed in a series of technical lemmas. For instance, if a solution \(y(t)\) is positive on \([c,b)\) then the quotient \(u(t)=[r(t)\Psi (y(t))| y^{\prime }(t)| ^{\alpha -2}y^{\prime }(t)]/f(y(t))\) satisfies the inequality \[ A(Kq;c,b,\rho )\leq \rho (c)H(b,c)u(c)+A((r/M^{\alpha -1})(\alpha ^{-1}| \lambda _2H^{-1/\alpha }-(\rho ^{\prime }/\rho )| )^{(\alpha )};c,b,\rho ). \] Here, \(f\) is continuously differentiable, \(f^{\prime }(y)[\Psi (y)| f(y)| ^{\alpha -2}]^{1/(1-\alpha )}\geq M>0\) and \(\rho \) is positive and continuously differentiable. The proofs are based on an integration by parts technique and the maximization of certain functions. Theorem 2.1 reads as: Assume that the above conditions hold and that for some \(c\in (a,b)\), \(H\) in the class of Philos-like kernels and \(\rho >0\), one has \[ \begin{multlined} H(c,a)^{-1}B(Kq;a,c,\rho )+H(b,c)^{-1}A(Kq;c,b,\rho )>\\ M^{1-\alpha }\alpha ^{-\alpha }[H(c,a)^{-1}B(r| \lambda _1H^{-1/\alpha }+(\rho ^{\prime }/\rho )| )^{(\alpha )};a,c,\rho )\\+H(b,c)^{-1}A((r| \lambda _2H^{-1/\alpha }-(\rho ^{\prime }/\rho )| )^{(\alpha )};c,b,\rho )].\end{multlined} \] Then, every solution of the differential equation has a zero in \((a,b)\). This result is further used in connection with a condition for the existence of an infinity of such intervals \((a,b)\) thus ensuring the oscillation of the equation. Corollaries are obtained by taking \(H(t,s)=(t-s)^{(\theta )}\), \(\theta >\alpha -1\), and respectively \(H(t,s)=[R(t)-R(s)]^{(\theta )}\), where \(R(t)=\int_l^tr(s)^{1/(1-\alpha )}\,ds\). In particular, Theorem 2.5 dealing with the latter type of kernel is an interesting and powerful extension of a known result due to \textit{Q. Kong} [J. Math. Anal. Appl. 229, 258--270 (1999; Zbl 0924.34026)]. The case of nonmonotonous \(f\) is treated in Section 3 with \(\alpha =2\). It is required that \(f(y)/y\geq M_0>0\) and \(0<\Psi (y)\leq d\) for all nonzero \(y\). Theorem 3.1 reads as: Assume that the above conditions hold and that for some \(c\in (a,b)\), \(H\) in the class of Philos-like kernels and \(\rho >0\), one has \[ \begin{multlined} H(c,a)^{-1}B(KM_0q;a,c,\rho )+H(b,c)^{-1}A(KM_0q;c,b,\rho )>\\ [4H(c,a)]^{-1}B((dr| \lambda _1H^{-1/2}+(\rho ^{\prime }/\rho )| )^2;a,c,\rho )\\+[4H(b,c)]^{-1}A((dr| \lambda _2H^{-1/2}-(\rho ^{\prime }/\rho )| )^2;c,b,\rho ).\end{multlined} \] Then, every solution of the differential equation has a zero in \((a,b)\). An investigation similar to that sketched before is performed in the remaining part of the section. Two examples conclude the paper, one dealing with the problematic case of \(\int_0^{+\infty }q(s)\,ds=-\infty \) and the other one with a nonmonotonous function \(f\). The latter exhibits the exact solution \(y(t)=\sin t\). The paper is of considerable interest for those working in the field of nonlinear oscillations.
scientific article; zbMATH DE number 2126374
scientific article