By making use of Merle's general shooting method we investigate Dirac equations of the formHere it is possible that F(0) = −∞ and that F(s) defined on (0,+∞) is not monotonously nondecreasing. Our results cover some known ones as a special case.
We investigate solutions for nonlinear operator equations and obtain some abstract existence results by linking methods. Some well-known theorems about periodic solutions for second-order Hamiltonian systems by M. Schechter are special cases of these results.
In this paper, we first investigate the classification of positively homogeneous equations (φp(u )) + q(t)φp(u) = 0, u(0) = 0 = u(1), where p > 1 is fixed, φp(u) = |u| p−2 u and q ∈ L ∞ (0, 1), and then discuss the existence of solutions for non-homogeneous equations. The main method of classification is by using a generalized Prufer equationwhere sinp : R → [−1, 1] is a periodic function and cosp t = d sinp t/dt for t ∈ R.
In this paper we study the solvability of Sturm-Liouville BVPs for Duffing equations by means of homotopy continuation methods. We propose a new kind of solvable conditions on the nonlinear function in the equation. This kind of conditions can be seen as some limiting cases of the well-known asymptotically positive linear conditions. The obtained results generalize and unify some previous results by S. Villegas, T. Ma and L. Sanchez, and Y. Dong, respectively.
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