generalized bibasic hypergeometric series and theiru n extensions we survey summation theorems for generalized bibasic hypergeometric series found recently by chu and perfected by macdonald these contain arbitrary sequences of parameters and generalize bibasic summation theorems found by gosper gasper and rahman they also contain a host of matrix inversions for infinite lower triangular matrices including those found by gould hsu carlitz and krattenthaler further special cases of krattenthaler s theorem are andrews matrix formulation of the bailey transform and bressoud s and gasper s inversions we extend the telescoping arguments used for bibasic summation theorems to provideu n 1 extensions of gosper s and gasper s bibasic summation theorems special cases includeu n 1 extensions of carlitz s and gasper s matrix inversions further specializations correspond to theu n 1 bailey transform andu n 1 bressoud matrix inversion found previously by milne applications includeu n 1 transformation and expansion formulas and au n 1 extension of aq analogue of abel s binomial theorem thisq analogue is different from those found by jackson and johnson

positive unbounded and monotone solutions of the singular second painleve equation on the half line abstract in this paper we study singular boundary value problems on the half line and we prove the existence of a global monotone positive and unbounded solution the latter satisfies a neumann condition at the origin and has prescribed asymptotic behavior at infinity our approach is based on a generalization of the kneser s property continuum of the cross sections of the solutions funnel i e on the properties of the so called consequent mapping and on properties of the associated vector field on the face space two applications one on the well known second painleve type equation which is related to superconductivity theory and a second in the theory of colloids clarify our results

a sparsity regularization and total variation based computational framework for the inverse medium problem in scattering abstract we present a fast computational framework for the inverse medium problem in scattering i e we look at discretization reconstruction and numerical performance the helmholtz equation in two and three dimensions is used as a physical model of scattering including point sources and plane waves as incident fields as well as near and far field measurements for the reconstruction of the medium we set up a rapid variational regularization scheme and indicate favorable choices of the various parameters the underlying paradigm is roughly speaking to minimize the discrepancy between the reconstruction and measured data while at the same time taking into account various structural a priori information via suitable penalty terms in particular the involved penalty terms are designed to promote information expected in real world environments to this end a combination of sparsity promoting terms total variation and physical bounds of the inhomogeneous medium e g positivity constraints is employed in the regularization penalty a primal dual algorithm is used to solve the minimization problem related to the variational regularization the computational feasibility performance and efficiency of the proposed approach is demonstrated for synthetic as well as experimentally measured data

multiple solutions of singular three point boundary value problems on 0 abstract in this paper we study the existence of single and multiple solutions of three point boundary value problems for the following nonlinear singular second order differential equations x t p x t q x t h t f t x t 0 t 0 a x 0 b x 0 k x c 0 lim t x t e r t d 0 where p b 0 a k 0 q 0 0 r 0 p p 2 4 q 2 h 0 0 is continuous and may be singular at t 0 f 0 0 is continuous and may take a negative value by applying the technique of lower and upper solutions and the theory of topological degree we obtain the conditions for the existence of at least one solution and at least three solutions respectively

unbounded solutions for three point boundary value problems with nonlinear boundary conditions on 0 abstract the existence and multiplicity of unbounded solutions is investigated for second order three point differential equations with nonlinear boundary conditions on 0 where the growth of the nonlinear term with respect to the third variable is allowed to be quadratic by applying the techniques of unbounded upper and lower solutions and the topological degree theory the existence of at least one solution and three solutions respectively is shown for the above problem one example is worked out to indicate the applications of our main results

inverse source problems for the helmholtz equation and the windowed fourier transform ii in this work we extend the qualitative reconstruction method for inverse source problems for time harmonic acoustic and electromagnetic waves in free space recently developed in r griesmaier m hanke and t raasch siam j sci comput 34 2012 pp a1544 a1562 to a relevant three dimensional setting the reconstruction algorithm relies on the fact that a windowed fourier transform of the far field pattern of the wave radiated by a compactly supported source approximates an exponential ray transform with a purely imaginary exponent of a mollified version of the source a filtered backprojection scheme for the standard ray transform applied to the absolute values of the windowed fourier transform of the far field pattern is used to recover information on the support of the source we provide the theoretical foundation of the method discuss a numerical implementation of the fully three dimensional algorithm and present a series of numerical examples including an inverse scattering problem to s

u n very well posed 10 p 9 transformations abstract in this paper we derive multivariable generalizations of bailey s classical terminating very well poised 10 o 9 transformation we work in the setting of multiple basic hypergeometric series very well poised on unitary groups u n 1 we obtain these u n 1 10 o 9 transformations from an alternate form of our general u n 1 classical bailey transform and the u n 1 generalizations of jackson s q analog of dougall s terminating 7 f 6 1 summation one of these 10 o 9 transformations contains the u n 1 10 o 9 transformation of denis and gustafson 1992 as a speci we also deduce a u n 1 extension of one of sears 4 o 3 transformations the classical case of all this work corresponds to u 2 or equivalently a 1

balanced3ϕ2summation theorems foru n basic hypergeometric series abstract in this paper we begin the theory and application of the u n 1 generalization of the classical bailey transform and bailey lemma we work in the setting of multiple basic hypergeometric series very well poised on unitary groups u n 1 the u n 1 bailey transform is obtained from a suitably modified u n 1 terminating very well poised 4 3 summation theorem and termwise transformations it is then interpreted as a matrix inversion result for two infinite lower triangular matrices this provides a higher dimensional generalization of andrews matrix inversion formulation of the bailey transform as in the classical case the concept of a u n 1 bailey pair is introduced and then inverted this u n 1 inversion enables us to derive u n 1 terminating balanced 3 2 summations directly from suitable u n 1 terminating very well poised 6 5 summations and vice versa these pairs of u n 1 dual identities extend the classical one variable case of andrews special limiting cases of the u n 1 terminating balanced 3 2 summation yield u n 1 q gauss summations several u n 1 q chu vandermonde summations u n 1 q binomial theorems and a u n 1 cauchy identity many other consequences of the u n 1 6 2 and 3 2 summations and the u n 1 bailey transform are further developed in our subsequent papers these include a derivation of the u n 1 bailey lemma several terminating u n 1 q whipple transformations u n 1 10 9 transformations and nonterminating u n 1 q whipple transformations the classical case of all this work corresponding to a 1 or equivalently u 2 contains a substantial amount of the theory and application of one variable basic hypergeometric series

the range test and the no response test for oseen problems the reconstruction of unknown shapes and inclusions is an important task for many applied sciences in electromagnetics and acoustics waves are used to probe some unknown region however often different types of measurements are available in industrial or natural applications such as measurements of fluid flow at some boundary of a region in space then the task is to detect the location and shape of an unknown inclusion from these remote flow measurements the goal of this work is to develop the theory to establish two new methods for finding unknown inclusions embedded in some fluid which is governed by the oseen equation we will introduce and analyze a range test based on ideas of kusiak potthast and sylvester 2003 and the no response test as suggested by luke and potthast 2003 for acoustic and electromagnetic problems in this work we will provide a new approach to show convergence of the no response test by showing its equivalence to a regularized version of the range test so in addition to two novel methods in inverse fluid dynamics this also provides further basic steps of methodological development on sampling and probing methods

boundary value problems on half line for functional differential equations with infinite delay in a banach space in this paper the monch fixed point theorem is used to investigate the existence of unbounded solutions of boundary value problem for functional differential equations with infinite delay in a banach space moreover an uniqueness theorem is obtained also

singular boundary value problems on a half line singular boundary value problems of nonlinear second order differential equations on a half line are considered the main methods used are modified upper and lower solution techniques necessary and sufficient conditions for the existence of solutions to a class of sublinear singular boundary value problems are obtained

positive solutions of even higher order singular superlinear boundary value problems abstract this paper investigates 2 m th order m 1 superlinear singular two point boundary value problems and obtains some necessary and sufficient conditions for existence of c 2 m 1 or c 2 m 1 positive solutions on the closed interval

existence and iteration of positive solutions for multi point boundary value problems on a half line in this study using fixed point index theorems and monotone iterative techniques we present iterative schemes as well as the existence of positive solutions to the m point boundary value problem on a half line

boundary value problems on infinite intervals and semiconductor devices the nonlinear differential equation y f x y y 0 x f satisfying bernstein type growth conditions we also examine an important application which occurs in the theory of semiconductor devices

existence theory for nonlinear fredholm and volterra integral equations on half open intervals in this chapter we present existence theory for the nonlinear fredholm integral equation y t h t smallint 0 t k t s g s y s ds 4 1 1 and the nonlinear volterra integral equation y t h t smallint 0 t k t s g s y s ds 4 1 2 when both are defined on the compact interval 0 t naturally we first concern ourselves with existence principles for both equations

positive solution of singular dirichlet boundary value problems for second order differential equation system this paper investigates the existence of positive solutions of singular dirichlet boundary value problems for second order differential system a necessary and sufficient condition for the existence of c 0 1 c 0 1 positive solutions as well as c1 0 1 c1 0 1 positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems our nonlinearity fi t x1 x2 may be singular at x1 0 x2 0 t 0 and or t 1 i 1 2

multiple positive solutions of the singular boundary value problems for second order differential equations on the half line abstract in this paper the existence of multiple positive solutions for singular sturm liouville boundary value problems on the half line is investigated by applying the fixed point index theorem of cone map some existence and multiplicity results of positive solutions are derived our results improve and generalize many well known results two examples are presented to demonstrate the application of our main results

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

the convex back scattering support a monochromatic i e fixed frequency back scattering kernel measured at all angles does not uniquely determine the index of refraction in an inhomogeneous medium nor can it guarantee any upper bound on the support of the inhomogeneity we show that it is possible to associate with any such kernel its convex back scattering support a convex set which must be a subset of the convex hull of the support of any inhomogeneity with that back scattering kernel for the born approximation we further demonstrate that there is an inhomogeneity supported in any neighborhood of the convex back scattering support which has exactly that back scattering kernel last we discuss a practical implementation of these results and include a numerical example

solutions in weighted spaces of singular boundary value problems on the half line in this paper we establish the existence of an unbounded solution to a second order boundary value problem on the half line also the existence of multiple unbounded positive solutions is discussed using the theory of fixed point index

existence of positive solutions for second order semipositone differential equations on the half line in this paper by means of constructing a special cone we obtain a sufficient condition for the existence of positive solution for semipositone singular sturm liouville boundary value problem on the half line the results obtained generalize and improve the corresponding results of zhang and liu in papers x g zhang l s liu positive solutions of superlinear semipositone singular dirichlet boundary value problems j math anal appl 316 2006 525 537 y s liu existence and unboundedness of positive solutions for singular boundary value problems on half line appl math comput 144 2003 543 556

existence of positive solutions for sturm liouville boundary value problems on the half line in this paper we establish sufficient conditions to guarantee the existence of at least one positive solution a unique positive solution and multiple positive solutions for the sturm liouville boundary value problem on the half line by using an effective operator the fixed point theorems in cone especially krasnoselskii fixed point theorem can be applied to such systems and then existence criteria are established the interesting point of the results is that the nonlinear term f can be sign changing

calculus of variations for a boundary value problem of differential system on the half line boundary value problem of second order differential equations on the half line is investigated in this paper the method mainly depends on the calculus of variations to the classical functionals sufficient conditions are obtained for the existence of the solutions

positive solutions of singular sub linear boundary value problems for fourth order and second order differential equation systems abstract this paper investigates the existence of positive solutions of singular sub linear boundary value problems for fourth order and second order differential equation systems first of all we establish some important lemmas then we define a partial ordering in c 2 a b c 4 a b c a b c 2 a b and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c 2 0 1 c 0 1 positive solutions as well as c 3 0 1 c 1 0 1 positive solutions our nonlinearity f t x 1 x 2 x 3 g t x 1 x 2 may be singular at x 1 0 x 2 0 x 3 0 t 0 and or t 1

locating an obstacle in a 3d finite depth ocean using the convex scattering support we consider an inverse scattering problem in a 3d homogeneous shallow ocean specifically we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle this reconstruction is obtained from the far field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency the technique is based on an implementation of the theory of the convex scattering support s kusiak j sylvester the scattering support commun pure appl math 2003 1525 1548 a few examples are presented to show the feasibility of the method

structure of a class of singular boundary value problem with superlinear effect abstract this paper investigates the structure of solutions of singular boundary value problem with superlinear effect it is proved that the closure of positive solution set possesses a maximal subcontinuum c i e a maximal closed connected subset of solutions which comes from 0 and tends to 0 finally as a corollary the existence of multiple positive solutions and the behavior of solutions according to parameter are obtained

existence and uniqueness for an m point boundary value problem at resonance on infinite intervals this paper investigates an m point boundary value problem with nonlinear boundary conditions at resonance on infinite intervals by establishing a general continuation theorem the existence and uniqueness of unbounded solutions is obtained on the basis of a suitable homotopy leray schauder degree properties and a priori estimate this paper provides a new method to solve boundary value problems with nonlinear boundary conditions at resonance on infinite intervals

positive solutions for three point boundary value problems for second order impulsive differential equations on infinite intervals this paper discusses the existence of positive solutions for three point boundary value problems for second order nonlinear impulsive differential equations on an infinite interval by using the diagonalization process fixed point theory and inequality techniques sufficient conditions for guaranteeing the existence of positive solutions are obtained

approximate controllability results for non densely defined fractional neutral differential inclusions with hille yosida operators abstractin this paper we mainly focused the approximate controllability results for a class of non densely defined fractional neutral differential control systems first we establish a set of suf

boundary value problems for nonlinear impulsive integrodifferential equations we consider boundary value problems for nonlinear impulsive integrodifferential equations in which impulses occur at fixed times by suitably modifying the notions of upper and lower solutions we establish existence results in a sector then develop the monotone method and prove the existence of extremal solutions of the impulsive integrodifferential equations as an application we derive some existence results for a boundary value problem for a third order impulsive differential equation

unbounded solutions for singular boundary value problems on the semi infinite interval upper and lower solutions and multiplicity this paper presents a lower and upper solution technique for singular second order boundary value problems on the half line and discusses sufficient and necessary conditions for the existence of positive solutions also multiplicity is discussed using the fixed point index

unbounded upper and lower solutions method for sturm liouville boundary value problem on infinite intervals abstract unbounded upper and lower solutions theories are established for the sturm liouville boundary value problem of a second order ordinary differential equation on infinite intervals by using such techniques and the schauder fixed point theorem the existence of solutions as well as the positive ones is obtained nagumo conditions play an important role in the nonlinear term involved in the first order derivatives

exponential stability of linear delay impulsive differential equations for an impulsive delay differential equation with bounded delay and bounded coefficients the following result is established if each solution is bounded on 0 together with its derivative for each bounded right hand side then the equation is exponentially stable an explicit stability theorem is presented

numerical implementations of dynamical probe method for non stationary heat equation the dynamical probe method for non stationary heat equation is developed recently which aims to detect an unknown inclusion of conductive material from the boundary measurement data the runge approximation is used to define some indicator function for this method which is a mathematical testing machine to detect the inclusion a numerical realization of the runge approximation is the key to this method by using a regularizing method a realization scheme is given for the runge approximation and numerical examples are given to show the validity of the dynamical probe method

unbounded solutions of the singular boundary value problems for second order differential equations on the half line this paper presents some results on the existence of multiple unbounded solutions for the boundary value problems for second order differential equations on half line and on the uniqueness of unbounded solution

the existence of solutions to nonlinear second order periodic boundary value problems abstract we consider the existence of periodic boundary value problems of nonlinear second order ordinary differential equations under certain half lipschitzian type conditions several existence results are obtained as applications positive periodic solutions of some laplacian type equations and duffing type equations are investigated

existence results for impulsive neutral functional integrodifferential equations with infinite delay abstract this paper is concerned with the existence of mild solutions for impulsive neutral functional integrodifferential equations with infinite delay in banach spaces the results are obtained by using the krasnoselski schaefer type fixed point theorem an example is provided to illustrate the theory

the scattering support we discuss inverse problems for the helmholtz equation at fixed energy specifically the inverse source problem and the inverse scattering problem from a medium or an obstacle we introduce the convex scattering support of a far field a set which will be a subset of the convex hull of the support of any source which can produce it we give several theorems which explain how to compute the convex scattering support and how to relate it to the actual support of a source medium or obstacle 2003 wiley periodicals inc

a q analog of the balanced 3f2 summation theorem for hypergeometric series in u n abstract in this paper we derive a q analog of holman s u n generalization of the pfaff saalschutz or balanced 3 f 2 summation theorem this provides a multivariable generalization of jackson s classical q analog of the balanced 3 f 2 summation theorem we obtain our main result from our new general q difference equations for balanced basic hypergeometric series in u n and both q analog of holman s u n generalization of the vandermonde summation theorem even the classical case of our proof appears to be new the q analog of u n pfaff saalschutz is invariant under inversion of the base q and also multivariable reversal of series corresponding theorems for ordinary series are deduced by taking the limit as q 1 of all these results

the no response test a sampling method for inverse scattering problems we describe a novel technique which we call the no response test to locate the support of a scatterer from knowledge of a far field pattern of a scattered acoustic wave the method uses a set of sampling surfaces and a special test response to detect the support of a scatterer without a priori knowledge of the physical properties of the scatterer specifically the method does not depend on information about whether the scatterer is penetrable or impenetrable nor does it depend on any knowledge of the nature of the scatterer absorbing reflecting etc in contrast to previous sampling algorithms the techniques described here enable one to locate obstacles or inhomogeneities from the far field pattern of only one incident field the no response test is a one wave method we investigate the theoretical basis for the no response test and derive a one wave uniqueness proof for a region containing the scatterer we show how to find the object within this region we demonstrate the applicability of the method by reconstructing sound soft sound hard impedance and inhomogeneous medium scatterers in two dimensions from one wave with full and limited aperture far field data

on improper integrals and differential equations in ordered banach spaces in this paper we study the existence of improper integrals of vector valued mappings the so obtained results combined with fixed point results in partially ordered functions spaces are then applied to derive existence and comparison results for least and greatest solutions of initial and boundary value problems in ordered banach spaces the considered problems can be singular functional nonlocal implicit and discontinuous concrete examples are also solved

solutions of impulsive boundary value problems on the half line abstract this paper presents some existence results for a class of impulsive boundary value problems with nonhomogeneous boundary data on the half line by applying fixed point theorems on cones we obtain the existence of positive bounded and unbounded solutions our methods are integral equations approximate solutions a priori bounds and modified arzela techniques

far field splitting for the helmholtz equation given far field data of a time harmonic wave radiated by an ensemble of well separated acoustic or electromagnetic sources as well as a priori information on the locations of these sources we discuss an algorithm to approximate the far field data radiated by each of these sources separately the method is based on a galerkin procedure considering subspaces spanned by the singular vectors of restricted far field operators that map local source distributions to the corresponding radiated far field patterns we provide an error analysis for this algorithm and consider its stability furthermore we exemplify a means to extract the required a priori knowledge directly from the far field data and we show how to utilize the split far fields to recover information on the supports of the individual sources beyond that a priori information numerical results for the most important example of inverse obstacle scattering illustrate our theoretical findings

inverse source problems for the helmholtz equation and the windowed fourier transform we consider the inverse source problem for time harmonic acoustic or electromagnetic wave propagation in the two dimensional free space given the radiated far field pattern of the solution to the helmholtz equation for a certain source term we find that the windowed fourier transform of this far field is related to an exponential radon transform with purely imaginary exponent of a smoothed approximation of the source based on this observation we set up a filtered backprojection algorithm to recover information on the unknown source term from a single far field measurement we analyze this algorithm and provide extensive numerical results that illustrate our theoretical findings as one outcome the method is shown to work better the larger the wave number possible extensions of the reconstruction method to limited aperture data and to inverse obstacle scattering problems are briefly sketched

a necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems abstract we study the nonlinear boundary value problem u f t u u u t 0 1 u 2 i 0 u 2 i 1 0 i 0 1 and obtain a necessary and sufficient condition for the existence of symmetric positive solutions we also discuss the application of our result to the special case where f is a power function of u and its derivatives moreover similar conclusions for a more general higher order boundary value problem are established our analysis mainly relies on the lower and upper solution method

successive iteration and positive solutions for a second order multi point boundary value problem on a half line this paper deals with the existence of positive solutions for some second order multi point boundary value problem on the half line our approach is based on the fixed point theorem and the monotone iterative technique without the assumption of the existence of lower and upper solutions we obtain not only the existence of positive solutions for the problems but also establish iterative schemes for approximating the solutions

inverse source problems for maxwell s equations and the windowed fourier transform the inverse source problem for time harmonic maxwell s equations in three dimensional free space is considered given far field observations of a wave radiated by a compactly supported volume current density at a single frequency we generalize a qualitative reconstruction algorithm recently developed in r griesmaier m hanke and t raasch siam j sci comput 34 2012 a1544 a1562 to this three dimensional vector valued setting the method is based on the fact that a windowed fourier transform of the far field data approximates an exponential transverse ray transform with a purely imaginary exponent of a smoothed version of the unknown source applying a filtered backprojection for the standard transverse ray transform to a suitably generalized squared absolute value of the windowed fourier transform of the far field data we recover information on the support of the volume current density we analyze this algorithm discuss its numerical implementation and supplement our theoretical findings

far field splitting by iteratively reweighted ell 1 minimization the aim of far field splitting for time harmonic acoustic or electromagnetic waves is to decompose the far field of a wave radiated by an ensemble of several compactly supported sources into the individual far field components radiated by each of these sources separately without further assumptions this is an ill posed inverse problem observing that far fields radiated by compactly supported sources have nearly sparse representations with respect to certain suitably transformed fourier bases that depend on the approximate source locations we develop an ell 1 characterization of these far fields and use it to reformulate the far field splitting problem as a weighted ell 1 minimization problem in the spirit of basis pursuit to this end we assume that some a priori information on the locations of the individual source components is available we prove that the unique solution to the weighted ell 1 minimization problem coincides with the solution to the far field splitting problem and we discuss its numerical approximation furthermore we propose an iterative strategy to successively improve the required a priori information by solving a sequence of these weighted ell 1 minimization problems where estimates of the approximate locations of the individual source components that are used as a priori information for the next iteration are computed from the value of the current solution this also gradually decreases the ill posedness of the splitting problem and it significantly improves the quality of the reconstructions we present a series of numerical examples to demonstrate the performance of this algorithm

multiple solutions of second order three point singular boundary value problems on the half line in this paper the existence of multiple solutions of second order three point singular boundary value problems on the half line is discussed by using the technique of upper and lower solutions and the theory of fixed point index in a cone moreover it generalizes and includes some known results

existence and unboundedness of positive solutions for singular boundary value problems on half line by constructing a special cone and using cone compression and expansion fixed point theorem this paper presents some existence results of positive solutions of singular boundary value problem bvp for short on half line for a class of second order differential equations at the same time the boundedness and unboundedness of positive solutions for bvp are considered also

boundary value problems on the half line with impulses and infinite delay this paper presents some results on the existence and uniqueness of solutions for the boundary value problems on the half line with impulses and infinite delay moreover an existence theorem of multiple solutions is obtained also the problems may be singular at the boundary

uncertainty principles for three dimensional inverse source problems novel uncertainty principles for far fields of time harmonic acoustic or electromagnetic waves radiated by collections of well separated localized sources in two dimensional free space have recently been established in r griesmaier and j sylvester siam j appl math 77 2017 pp 154 180 these uncertainty principles have been utilized to develop stability estimates and reconstruction algorithms for the recovery of the far field radiated by each of the individual sources and the simultaneous restoration of missing data segments in this paper we present generalizations of these uncertainty principles for a relevant three dimensional setting we consider extensions of the reconstruction schemes for far field splitting and data completion including their stability analysis and we discuss the sharpness of our results in the three dimensional case we briefly comment on the numerical implementation of the reconstruction algorithms and include a numerical example

approximate controllability results for impulsive neutral differential inclusions of sobolev type with infinite delay abstractthe paper is mainly concerned with the approximate controllability results for a class of impulsive neutral differential control systems first we establish a set of sufficient conditions

convex backscattering support in electric impedance tomography this paper reinvestigates a recently introduced notion of backscattering for the inverse obstacle problem in impedance tomography under mild restrictions on the topological properties of the obstacles it is shown that the corresponding backscatter data are the boundary values of a function that is holomorphic in the exterior of the obstacle s which allows to reformulate the obstacle problem as an inverse source problem for the laplace equation for general obstacles the convex backscattering support is then defined to be the smallest convex set that carries an admissible source i e a source that yields the given backscatter data as the trace of the associated potential the convex backscattering support can be computed numerically numerical reconstructions are included to illustrate the viability of the method

global structure of solutions for a class of two point boundary value problems involving singular and convex or concave nonlinearities abstract this paper investigates the following two point singular boundary value problems bvp x t x q x p t 0 1 x t 0 t 0 1 x 0 x 1 0 where q 0 and p 0 are fixed given numbers r 0 is a parameter the results obtained are the global structure of solutions and exact number of solutions when p 1 or p 1 and q is sufficiently small

positive solutions for singular semipositone boundary value problems on infinite intervals by using the fixed point theory on a cone with a special norm and space we discuss the existence of positive solutions for a class of semipositone boundary value problems on infinite intervals the work improves many known results including singular and non singular cases