Non-real eigenvalues of nonlocal indefinite Sturm\u2013Liouville problems

Fu Sun,Kun Li,Baochao Liao,Jiangang Qi

doi:10.1186/s13661-019-1288-8

Fu Sun, Kun Li + Show 2 more

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https://doi.org/10.1186/s13661-019-1288-8

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Abstract

The present paper deals with non-real eigenvalues of regular nonlocal indefinite Sturm–Liouville problems. The existence of non-real eigenvalues of indefinite Sturm–Liouville differential equation with nonlocal potential K(x,t) associated with self-adjoint boundary conditions is studied. Furthermore, a priori upper bounds of non-real eigenvalues for a class of indefinite differential equation involving nonlocal point interference potential function is obtained.

Highlights

We impose the symmetry conditions on q, K and w, namely, q(x) = q(–x), K(x, t) = K(–x, t), w(–x) = –w(x) to prove the existence of non-real eigenvalues of (2.1). It follows from the hypothesis on q, K, w in (2.2) and the symmetry conditions (2.3) that if λ ∈ C is an eigenvalue of the problem (2.1) and φ is the corresponding eigenfunction, –λ is an eigenvalue of
We say that the self-adjoint operator T has k negative squares, k ∈ N0, if there exists a k-dimensional subspace X of Kin D(T) such that [Tf, f ] < 0, f ∈ X, f = 0, but no (k + 1)
If the boundary condition is given in the form y(–1) = 0, y (1) = 0 for (1.1) and let c = 1, from (3.4) we see that the nonlocal indefinite eigenvalue problem takes the form

Summary

Introduction

–1 associated to suitable boundary conditions, where λ is the spectral parameter, q ∈ L1[–1, 1]. Determining a priori bounds and determining the exact number of non-real eigenvalues are an interesting and difficult problems in Sturm–Liouville theory These open problems have been solved by Qi et al, [4, 14, 21, 28] for the regular (local) indefinite problem and by Behrndt et al, [5,6,7,8,9, 25] for the singular case, respectively. 3, we obtain the nonlocal indefinite Sturm–Liouville problem through the Dirac distribution in (1.2), the upper bounds of non-real eigenvalues in terms of q, v, w are shown (see Theorems 4.1 and 4.2) in Sect.

Existence of non-real eigenvalues

A priori bounds of non-real eigenvalues

Conclusion