Improvements on the Leighton oscillation theorem for second-order dynamic equations

Abstract The time scales version of the Leighton oscillation theorem states that if (*) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:munderover> <m:mo>∫</m:mo> <m:mrow> <m:msub> <m:mi>t</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:munderover> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>a</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfrac> <m:mspace width="thinmathspace"/> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mrow> <m:munderover> <m:mo>∫</m:mo> <m:mrow> <m:msub> <m:mi>t</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:munderover> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mspace width="thinmathspace"/> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo>,</m:mo> </m:math> $$ \int\limits_{t_0}^\infty \frac{1}{a(t)}\,{\Delta}t=\int\limits_{t_0}^\infty {p(t)}\,{\Delta}t=\infty,$$ where a and p are rd-continuous with a ( t ) &gt; 0 for t ≥ t 0 , then every solution of the second-order self-adjoint dynamic equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo stretchy="false">(</m:mo> <m:mi>a</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi mathvariant="normal">Δ</m:mi> </m:msup> <m:mrow> <m:msup> <m:mo stretchy="false">)</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>σ</m:mi> </m:msup> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> $$ (a(t)x^\Delta)^\Delta+p(t)x^\sigma=0$$ is oscillatory. The theorem turns into the famous Leighton oscillation theorem when the time scale is taken as the set of real numbers, and its discrete version when the time scale is taken as the set of integers. The divergence of the first improper integral in ( ∗ ) means that the dynamic equation is in canonical form. The equation is called noncanonical when the integral is convergent. In this study, we establish an improved version of the Leighton oscillation theorem on time scales that can be applied to both canonical and noncanonical types of dynamic equations. Furthermore, we allow the second improper integral in ( ∗ ) to be convergent. In the special case, we derive completely new Leighton-type oscillation theorems for second-order self-adjoint difference equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>a</m:mi> <m:mi>k</m:mi> </m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:msub> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>p</m:mi> <m:mi>k</m:mi> </m:msub> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:math> $$ {\Delta}(a_k\Delta x_k)+p_kx_{k+1}=0,$$ where Δ is the forward difference operator, defined by Δ x k = x k +1 − x k (the derivative). Examples are given to illustrate the significance of these theorems.