One of the interesting results was given by Geraghty [6] in the setting of complete metric spaces by considering an auxiliary function.
By employing an auxiliary function on the contraction condition, the results of this article extend and improve previously known results.
In 1973, Geraghty [27] generalized the Banach contraction principle by considering an auxiliary function in the following way.
By introducing an auxiliary function, we can change the coupled governing equations with variable coefficients for the deflection and rotation to a single governing equation.
In particular, Geraghty [6] obtained a generalization of the Banach contraction principle in the setting of complete metric spaces by considering an auxiliary function.
By introducing an auxiliary function of two variables and using a Riccati transformation, we establish some Kamenev type (cf. Philos [24]) interval oscillation criteria.
By using the Laplace transform, the mixed initial-boundary-value problem is reduced to triple series equations, then to a singular integral equation of the first kind by introducing an auxiliary function.
By introducing an auxiliary function of two variables and using Riccati transformation, several Kamenev type interval oscillation criteria are established, which generalize and extend some known ones, such as those of Huang and Feng.
Recently, Li et al. [22] investigated equation (1.2) with impulses and obtained several interval oscillation theorems by introducing an auxiliary function of one variable and generalized and extended some known results in [9, 17, 23].
So, by employing an auxiliary function p t) on the contraction condition, we get new criteria for asymptotic stability of the zero solution by using the fixed point method and the differential inequality techniques which not only includes the results on sufficient part in [17 19], but also includes several equations that previously known related results can not be applied to.
