A uniqueness result for a singular nonlinear eigenvalue problem

2013 ◽  
Vol 143 (4) ◽  
pp. 739-744 ◽  
Author(s):  
Alfonso Castro ◽  
Eunkyung Ko ◽  
R. Shivaji
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Uniqueness Result ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Image Position

We consider the positive solutions to singular boundary-value problems of the form where λ > 0, β ∈ (0,1) and Ω is a bounded domain in ℝN, N ≥ 1, with smooth boundary ∂Ω. Here, we assume that f: [0, ∞) → (0, ∞) is a C1 non-decreasing function and f(s)/sβ is decreasing for s large. We establish the uniqueness of the positive solution when λ is large.

A Note on the Uniqueness of Positive Solutions for Singular Boundary Value Problems

2011 ◽  
Vol 2 (3) ◽  
pp. 43-50
Author(s):  
Fu-Hsiang Wong ◽  
Sheng-Ping Wang ◽  
Hsiang-Feng Hong
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Uniqueness Theorems ◽  
Singular Boundary ◽  
Sufficient Condition ◽  
Singular Boundary Value Problems

In this paper, the authors examine sufficient condition for the uniqueness of positive solutions of singular Strum-Liouville boundary value problems. The authors use the uniqueness theorems of (E) with respect to the boundary conditions to show that the boundary value problems have one positive solution.

scholarly journals Conditions for the solvability of singular boundary value problems

1996 ◽  
Vol 53 (3) ◽  
pp. 485-497
Author(s):  
Xiyu Liu
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Uniqueness Result ◽  
Necessary And Sufficient Conditions ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Singular Boundary Value Problems ◽  
Necessary And Sufficient

Consider the singular boundary value problem (r(x′))′ + f(t, x) = 0, 0 < t < 1. We give necessary and sufficient conditions for this problem to have solutions. In addition, a uniqueness result is obtained.

Differential Operators of infinite order

1980 ◽  
Vol 86 (1-2) ◽  
pp. 85-101
Author(s):  
Einar Hille
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Boundary Value ◽  
Natural Generalization ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Linear Differential ◽  
Image Position ◽  
Special Case

SynopsisThe differential operators in question are of the form G(DZ) where G(w)is an entire function of order at most 1/n and minimal type while Dz is a linear differential operator of order n with coefficients which are entire ( = integral) functions of z, usually polynomials. This class of operators form a natural generalization of the class G(d/dz) studied during the first half of the century Muggli, Polya, Ritt and others. The class G(DZ) was introduced by the present author and his pupils in the 1940s. In fact, the present paper is partly based on a MS from that period, mostly devoted to the special casebut also containing generalizations, some of which were later worked out by Klimczak. A basic tool in this paper is the characteristic seriesExamples are given showing that the domain of absolute convergence of such a series need neither be convex nor of finite connectivity, a question which has puzzled the author for forty odd years. Characteristic series arising from regular or singular boundary value problems for the operator Dz are used to study the inversion problemfor given F(z). In particular it is shown that exp (Dx)[W(z)] = 0 has the unique solution W(z) ≡ 0. Some singular boundary value problems are considered briefly.

scholarly journals The difference of consecutive eigenvalues

1994 ◽  
Vol 57 (3) ◽  
pp. 305-315
Author(s):  
Hsu-Tung Ku ◽  
Mei-Chin Ku
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Upper Bound ◽  
Smooth Boundary ◽  
Uniform Estimate ◽  
Smooth Bounded Domain ◽  
The Difference ◽  
Image Position

AbstractLet M be a smooth bounded domain in Rn with smooth boundary, n ≥ 2, and . We prove an inequality involving the first k + 1 eigenvalues of the eigenvalue problem: where am−1 ≥ 0 are constants and at−1 = 1. We also obtain a uniform estimate of the upper bound of the ratios of consecutive eigenvalues.

scholarly journals UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

2013 ◽  
Vol 55 (2) ◽  
pp. 399-409 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problems ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problems \[ \{\begin{array}{ll} -\Delta u=\lambda f(u)\ \ &\text{in}\Omega, \ \ \ \ \ u=0 &\text{on \partial \Omega, \] where Ω is a bounded domain in ℝn with smooth boundary ∂Ω, λ is a positive parameter and f:(0,∞) → (0,∞) is sublinear at ∞ and is allowed to be singular at 0.

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