Eigenvalue Problems for Nonlinear Differential Equations on a Measure Chain

Solvability of a nonlinear second order conjugate eigenvalue problem on a time scale

“…This paper constitutes an extension of the recent work by Erbe and Peterson [6] and Chyan and Henderson [3] in which they obtained positive solutions of (1.8) and (1.9) for all 0 < λ < ∞ assuming that f is either superlinear or sublinear. The solutions obtained in [3,6] were found to belong to the intersection of a cone with an annular type region. Now we state a Green's function inequality which is fundamental in the proof of our main result.…”

“…Assumption (A1) is important because it admits a larger class of functions than those allowed in [3]. This paper constitutes an extension of the recent work by Erbe and Peterson [6] and Chyan and Henderson [3] in which they obtained positive solutions of (1.8) and (1.9) for all 0 < λ < ∞ assuming that f is either superlinear or sublinear. The solutions obtained in [3,6] were found to belong to the intersection of a cone with an annular type region.…”

Second-order n-point eigenvalue problems on time scales

“…To establish eigenvalue intervals we will employ the following fixed point theorem due to Krasnosel'skiȋ [18]; for more on the establishment of eigenvalue intervals for time-scale boundary value problems, see, for example, Chyan and Henderson [11] and Davis et al [13].…”

Impulsive Diffusion Equation on Time Scales

“…The first studies on these type problems for linear ∆− differential equations on T were fulfilled by Chyan, Davis, Henderson and Yin [8] in 1998 and Agarwal, Bohner and Wong [9] in 1999. In [8], the theory of positive operators according to a cone in a Banach space is applied to eigenvalue problems related to the second order linear ∆−differential equations on T to prove existence of a smallest positive eigenvalue and then a theorem proved to compare the smallest positive eigenvalue for two problems of that type. In [9], an oscillation theorem is given for Sturm-Liouville (SL) eigenvalue problem on T with separated boundary conditions and Rayleigh's principle is studied.…”