The Ljusternik-Schnirelman theory for indefinite and not necessarily odd nonlinear operators and its applications

In this paper, we establish a new framework to deal with the eigenvalue problems. As an application, we show that there exist positive and negative solutions to the semilinear elliptic eigenvalue problem with supercritical exponent. This is a generalization of previous results on the subcritical and critical exponents.

Consider Robin eigenvalue problem involving the p(x)-Laplacian on a smooth bounded domain Ω as follows:

We prove the existence of infinitely many eigenvalue sequences if p(x) ≢ constant and also present some sufficient conditions for which there is no principal eigenvalue and the set of all eigenvalues is not closed.

A new approach using the over-relaxed proximal point algorithm in the context of solving a class of inclusion problems based on the notion of maximal (η)-monotonicity is developed and examined. Convergence analysis seems to be reasonable, and finally, some specializations are included. Furthermore, the model developed in this communication seems to be appropriate to the Yosida approximation in the sense that it can be applied to first-order evolution equations/inclusions as well.

In this paper, we prove the existence of sign changing solutions and multiple solutions of a semilinear elliptic eigenvalue problem. One of the key points here is to construct nonempty invariant sets of the gradient flow which contain the positive and the negative solution of the problem.

In this paper, a new system of nonlinear (set-valued) variational inclusions involving $(A\text{,}\eta )$-maximal relaxed monotone and relative $(A\text{,}\eta )$-maximal monotone mappings in Hilbert spaces is introduced and its approximation solvability is examined. The notion of $(A\text{,}\eta )$-maximal relaxed monotonicity generalizes the notion of general $\eta$-maximal monotonicity, including $(H\text{,}\eta )$-maximal monotonicity (also referred to as $(H\text{,}\eta )$-monotonicity in literature). Using the general $(A\text{,}\eta )$-resolvent operator method, approximation solvability of this system based on a generalized hybrid iterative algorithm is investigated. Furthermore, for the nonlinear variational inclusion system on hand, corresponding nonlinear Yosida regularization inclusion system and nonlinear Yosida approximations are introduced, and as a result, it turns out that the solution set for the nonlinear variational inclusion system coincides with that of the corresponding Yosida regularization inclusion system. Approximation solvability of the Yosida regularization inclusion system is based on an existence theorem and related Yosida approximations. The obtained results are general in nature.

A general framework for the over-relaxed proximal point algorithm based on the notion of A-maximal relaxed monotonicity is developed, and then the convergence analysis in the context of solving a general class of nonlinear implicit inclusion problems is explored. Furthermore some auxiliary results of interest involving A-maximal relaxed monotone mappings in a Hilbert space setting are included.