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Optical fiber distributed acoustic sensing (DAS) based on phase sensitive optical time domain reflectometry (ϕ-OTDR) has a great demand in many application fields with large signals, such as railway safety monitoring, cable vibration detection, etc. However, due to the inherent limits of phase unwrapping algorithm, the large signal whose phase change exceeds π cannot be recovered correctly by DAS. In this work, the differential-unwrapping-integral (DUI) algorithm is proposed and demonstrated to replace the traditional unwrapping algorithm for large signal recovery. By utilizing differential operation to compress the signal amplitude, the large phase signal can be converted into small signal which can be recovered through unwrapping. Then, multiple integral operations and polynomial fitting compensation algorithm are applied onto the differential signal to recover the original phase signal without any signal-to-noise ratio(SNR) sacrificed. In theory, the maximum measurable phase of DUI can be improved 1390 times compared with traditional DUI algorithm. Further, the experimental results demonstrate that the dynamic range of DAS with DUI can reach up to 131.7 dB. Moreover, DUI algorithm has been tested and verified in the field test for recording the digging signals with large amplitude. This scheme can be applicable for all phase-demodulation-based sensing techniques without increasing the system complexity. [ABSTRACT FROM AUTHOR]

In this paper we discuss the existence of a solution between well-ordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1. [ABSTRACT FROM AUTHOR]

In this paper, using fixed point index and the mixed monotone technique, we present some multiplicity and uniqueness results for the singular nonlocal boundary value problems involving nonlinear integral conditions. Our nonlinearity may be singular in its dependent variable and it is allowed to change sign. [ABSTRACT FROM AUTHOR]

In this paper, using the theory of fixed point index on a cone and the Leray-Schauder fixed point theorem, we present the multiplicity of positive solutions for the singular nonlocal boundary-value problems involving nonlinear integral conditions and the existence of at least one positive solution for the singular nonlocal boundary-value problems with sign-changed nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, using fixed-point index theory and approximation techniques, we consider the existence and multiplicity of fixed points of some nonlinear operators with singular perturbation. As an application we consider the existence and multiplicity of positive solutions of singular systems of multi-point boundary value problems, which improve the results in the literature. [ABSTRACT FROM AUTHOR]

This paper considers the existence of positive solutions of the boundary value problems Vn +λ(V +Vp -Vq) = 0 and V(-1) = V(1) = 0, where p > r > q > -1 and λ is a positive parameter. Using a time-map approach, we obtain the exact number of positive solutions in different cases. [ABSTRACT FROM AUTHOR]

Abstract  The existence of multiple positive solutions for the singular Dirichlet boundary-value problem $$\begin{array}{*{20}c} {{x^{{\prime \prime }} + \Phi {\left( t \right)}f{\left( {t,x{\left( t \right)},x^{\prime } {\left( t \right)}} \right)} = 0,\,\,\,\,0 is presented by using the fixed point index; here f may be singular at x = 0. [ABSTRACT FROM AUTHOR]

Abstract  The existence of at least one positive solution and the existence of multiple positive solutions are established for the singular second-order boundary value problem $$\left\{\begin{array}{l}{\frac{1}{p(t)}(p(t)x'(t))'}+\Phi(t)f(t,x,px')=0,\quad 0 using the fixed point index, where f may be singular at x=0 and px′=0. [ABSTRACT FROM AUTHOR]

Positive solutions to the singular initial-boundary value problems x″ = - f(t, xt), 0 < t < 1, x0 = 0, x(1) = 0, are obtained by applying the Schauder fixed-point theorem, where xt(u) = x(t+u) (0 ≤ t ≤ 1) on [-r, 0] and f(.,.) : (0, 1) × (C+\{0})→R+(C+ = {x ϵ C([-r, 0], R), x(t) ≥ 0, ∀t ϵ [-r, 0]}) may be singular at φ(u) = 0 (-r ≤ u ≤ 0) and t = 0. As an application, an example is given to demonstrate our result. [ABSTRACT FROM AUTHOR]

Abstract  The existence of multiple positive solutions is presented for the singular second-order boundary value problems $$\left\{\begin{array}{ll} x^{\prime\prime}\Phi(t)f(t,x,x') =0, 0 using the fixed point index, where f may be singular at x  =  0 and x′  = 0. [ABSTRACT FROM AUTHOR]

Abstract??This paper presents a lower and upper solution approach for singular second order boundary value problems on the half line and establishes the existence of positive, unbounded and monotone solutions. [ABSTRACT FROM AUTHOR]

Using the theory of fixed point index, this article discusses the existence of positive solutions for singular second-order boundary value problems, where f may be singular at x = 0 and x'=0, and f may change sign. [ABSTRACT FROM AUTHOR]