Lyapunov-type inequalities for one-dimensional Minkowski-curvature problems

π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems

“…It is worth mentioning that our results in this paper explain why de nitions of solutions in [12] are only for non π -tangential solutions. Furthermore, applying non π -tangentiality of solutions for problem (P ) as well as systems (CS ) and (SCS ) (see Section 4 and 6), we get Lyapunov-type inequalities for weight functions in Bq class corresponding to nonlinear terms associated with q, which extend the results in [12].…”

“…where r ∈ Bq and l ≥ . As mentioned in Introduction, the authors estimated Lyapunov-type inequality to give necessary condition on B -class weight functions for the existence of non π -tangential positive solution for problem (P ) when l = in [12]. In this section, we extend such a result to Bq-class weight function by providing a condition on Bq-class weight functions for the existence of non π -tangential solution of problem (P ) for any l ≥ q.…”

“…Recently, using a bifurcation technique, Yang-Lee [10] and Yang-Lee-Sim [11] investigated the existence and multiplicity of positive radial solutions as well as nodal radial solutions for a prescribed mean curvature problem on an exterior domain which is transformed into one dimensional problems such as problem (P) with a singular weight in Bq . Besides, when q = and f (u) = u, Yang-Sim-Lee [12] estimated Lyapunov-type inequalities to give necessary conditions on weight functions in B for the existence of a positive solution under the assumption of non π -tangentiality. It is worth mentioning that our results in this paper explain why de nitions of solutions in [12] are only for non π -tangential solutions.…”

“…Besides, when q = and f (u) = u, Yang-Sim-Lee [12] estimated Lyapunov-type inequalities to give necessary conditions on weight functions in B for the existence of a positive solution under the assumption of non π -tangentiality. It is worth mentioning that our results in this paper explain why de nitions of solutions in [12] are only for non π -tangential solutions. Furthermore, applying non π -tangentiality of solutions for problem (P ) as well as systems (CS ) and (SCS ) (see Section 4 and 6), we get Lyapunov-type inequalities for weight functions in Bq class corresponding to nonlinear terms associated with q, which extend the results in [12].…”

A Lyapunov-Type Inequality for Partial Differential Equation Involving the Mixed Caputo Derivative

“…The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem and an example is given to illustrate it. Moreover, we present some applications to demonstrate the effectiveness of the new results.Mathematics 2020, 8, 47 2 of 11 see for instance [6][7][8][9][10][11] and the references therein. Since fractional calculus (see for example [12][13][14]) is more effective and powerful in describing practical phenomena than integer-order calculus, more and more researchers pay more attention to this subject.…”