We study the existence of positive solutions for the boundary value problem of nonlinear fractional differential equationsD0+αu(t)+λf(u(t))=0,0<t<1,u(0)=u(1)=u'(0)=0, where2<α≤3is a real number,D0+αis the Riemann-Liouville fractional derivative,λis a positive parameter, andf:(0,+∞)→(0,+∞)is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results.
By means of Riccati transformation technique, we establish some new oscillation criteria for the secondorder Emden-Fowler delay dynamic equationson a time scale T; here γ is a quotient of odd positive integers with p(t) real-valued positive rd-continuous functions defined on T. To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales. Our results in this paper not only extend the results given in [R.P. Agarwal, M. Bohner, S.H. Saker, Oscillation of second-order delay dynamic equations, Can. Appl. Math. Q. 13 (1) (2005) 1-18] but also unify the oscillation of the second-order Emden-Fowler delay differential equation and the second-order Emden-Fowler delay difference equation.
In this paper, we study the following second-order Emden-Fowler neutral delay differential equationWe establish some new oscillation results which handle some cases not covered by known criteria.
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