the existence of asymptotically stable solutions for a nonlinear functional integral equation with values in a general banach space abstract motivated by the recent known results about the solvability and existence of asymptotically stable solutions for nonlinear functional integral equations in spaces of functions defined on unbounded intervals with values in the n dimensional real space we establish asymptotically stable solutions for a nonlinear functional integral equation in the space of all continuous functions on r with values in a general banach space via a fixed point theorem of krasnosel skii type in order to illustrate the result obtained here an example is given

sharp error bounds for the derivatives of lidstone spline interpolation ii abstract in this paper we shall derive explicit error estimates in l norm between a given function f x pc n a b 4 n 6 and its quintic lidstone spline interpolate the results obtained are then used to establish precise error bounds for the approximated and biquintic lidstone spline interpolates we also include applications to integral equations and boundary value problems as well as sufficient numerical examples which dwell upon the sharpness of the obtained results

on existence of positive solutions of sturm liouville boundary value problems for a nonlinear singular differential system abstract by employing the fixed point theorem of cone expansion and compression of norm type we investigate the existence of positive solutions of sturm liouville boundary value problems for a nonlinear singular differential system some well known results in the literature are generalized and improved an example is presented to illustrate the application of our main result

monotone convergence of iterative methods for right focal point boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for the right focal point boundary value problems converges monotonically an example which dwells upon the usefulness of this technique is also included

existence of positive solutions for a system of generalized lidstone problems in this paper we study the existence of positive solutions for the system of generalized lidstone problems for ordinary differential equations 1 mu 2 m f 1 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 1 nv 2 n f 2 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 a 0u 2 i 0 b 0u 2 i 1 0 a 1u 2 i 1 b 1u 2 i 1 1 0 i 0 1 m 1 a 0v 2 j 0 b 0v 2 j 1 0 a 1v 2 j 1 b 1v 2 j 1 1 0 j 0 1 n 1 here f 1 f 2 c 0 1 xr m n r r 0 our hypotheses imposed on the nonlinearities f 1 and f 2 are formulated in terms of nonnegative matrices we use fixed point index theory to establish our main results based on a priori estimates of positive solutions

a singular boundary value problem for odd order differential equations abstract the odd order differential equation 1 nx 2n 1 f t x x 2n together with the lidstone boundary conditions x 2j 0 x 2j t 0 0 j n 1 and the next condition x 2n 0 0 is discussed here f satisfying the local caratheodory conditions can have singularities at the value zero of all its phase variables existence result for the above problem is proved by the general existence principle for singular boundary value problems

kneser s property for a parabolic partial differential equation au at au 1 1 for t 0 and x e r 1 with the initial condition u gx l 0 for x e r 2 the problem has a smooth solution u t x t 4 other than the trivial solution the purpose of this paper is to investigate properties of the set of solutions for problem 1 with condition 2 in section 2 we shall present the main result the cross section of the solutions is a continuum i e a compact and connected set in a given function space whose fact is known as kneser s property l in the theory of ordinary differential equations with nonuniqueness of solutions in ordinary differential equations hukuhara 2 31 made more detailed investigations on kneser s property and found hukuhara s phenomenon for partial differential equations of parabolic type we have tried to discuss the same problem stated above 4 regarding which goldstein 5 pointed out that our assertion is valid only for equations which generate a compact c semigroup on the other hand ballotti 6 has shown aronszajn s theorem 7 and hence kneser s property for a partial differential equation generating a compact c semigroup in ballotti s paper 6 such an example has been treated

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

higher order lidstone boundary value problems for elliptic partial differential equations the aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth order elliptic lidstone boundary value problems where the nonlinear functions depend on the higher order derivatives sufficient conditions are given for the existence and uniqueness of a solution it is also shown that there exist two sequences which converge monotonically from above and below respectively to the unique solution the approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions all the results are directly applicable to 2nth order two point lidstone boundary value problems

piecewise complementary lidstone interpolation and error inequalities the purpose of this paper is to develop piecewise complementary lidstone interpolation in one and two variables and establish explicit error bounds for the derivatives in l and l 2 norms

an atomic energy lower bound that agrees with scott s correction abstract in this paper we prove that the energy of an atom with nuclear charge z is greater than c tf z 7 3 q 8 z 2 o z 17 9 log z atomic units as z q is the number of spin states q 2 for electrons and c tf is the thomas fermi constant for such systems

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

explicit error bounds for the derivatives of piecewise lidstone interpolation abstract the purpose of this paper is to provide explicit error bounds for the derivatives of piecewise lidstone interpolation in l and l2 norms the results obtained are best possible sharp and supplement those established in earlier papers

polynomial approximation of cm functions by means of boundary values and applications a survey we collect classical and more recent results on polynomial approximation of sufficiently regular real functions defined in bounded closed intervals by means of boundary values only the problem is considered from the point of view of the existence of explicit formulas interpolation to boundary data bounds for the remainder and convergence of the polynomial series applications to some problems of numerical analysis are pointed out such as nonlinear equations numerical differentiation and integration formulas special associated differential boundary value problems some polynomial expansions for smooth enough functions defined in rectangles or in triangles of r 2 as well as in cuboids or in tetrahedrons in r 3 and their applications are also discussed

explicit polynomial expansions of regular real functions by means of even order bernoulli polynomials and boundary values for a function f c2n 1 a b an explicit polynomial interpolant in a and in the even derivatives up to the order 2n 1 at the end points of the interval is derived explicit cauchy and peano representations and bounds for the error are given and the analysis of the remainder term allows to find sufficient conditions on f so that the polynomial approximant converges to f these results are applied to derive a new summation formula with application to rectangular quadrature rule the polynomial interpolant is related to a fairly interesting boundary value problem for odes we will exhibit solutions for this problem in some special situations

lidstone polynomials and boundary value problems abstract we define lidstone polynomials provide their explicit representations give their relations with bernoulli polynomials obtain best possible error estimates in lidstone interpolation and establish several best possible sharp inequalities which compare favourably with the known results next we use these results to study even order differential equations together with lidstone boundary conditions

upper and lower solution methods for fully nonlinear boundary value problems abstract sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions the conditions assume the existence of a strong upper solution lower solution pair a concept that is defined in the paper the differential equation has nonlinear dependence on all lower order derivatives of the unknown in particular appropriate nagumo conditions are obtained and employed

eigenvalues and the one dimensional p laplacian we consider the boundary value problem p u f t u 0 with p 1 t 0 1 u 0 u 1 0 and with 0 the value of is chosen so that the boundary value problem has a positive solution in addition we derive an explicit interval for such that for any in this interval the existence of a positive solution to the boundary value problem is guaranteed in addition the existence of two positive solutions for in an appropriate interval is also discussed

on the connectedness and asymptotic behaviour of solutions of reaction diffusion systems abstract in this paper we consider reaction diffusion systems in which the conditions imposed on the nonlinearity provide global existence of solutions of the cauchy problem but not uniqueness we prove first that for the set of all weak solutions the kneser property holds that is that the set of values attained by the solutions at every moment of time is compact and connected further we prove the existence and connectedness of a global attractor in both the autonomous and nonautonomous cases the obtained results are applied to several models of physical or chemical interest a model of fractional order chemical autocatalysis with decay the fitz hugh nagumo equation and the ginzburg landau equation

hukuhara kneser property for a nonlinear dirichlet problem abstract the purpose of this paper is to show that the set of solutions of a nonlinear dirichlet problem satisfies the usual hukahara kneser property for ordinary differential equations

monotone iterative technique and positive solutions of lidstone boundary value problems the purpose of this paper is to investigate the existence and iteration of n symmetric positive solutions for lidstone boundary value problem 1 nw 2 n t f t w t 0

sufficiency of condition ψ for local solvability in two dimensions on etablit l existence de solutions locales de l equation pu f ou p est un operateur pseudodifferentiel classique a 2 dimensions de type principal d ordre m qui satisfait la condition la partie imaginaire p 2 du symbole principal de p ne change pas de signe de a le long d une bicaracteristique orientee de la partie reelle p 1 du symbole principal

on local solvability of a class of abstract underdetermined systems abstract in this work we present a necessary and sufficient condition for a class of abstract underdetermined systems to be solvable we develop j f treves ideas presenting the so called condition and its connection with the study of the solvability in consideration we also prove the existence of finite order regularity solutions

positive solutions for higher order lidstone boundary value problems a new approach via sperner s lemma abstract consider the higher order nonlinear scalar differential equation where associated to the lidstone boundary conditions existence of a solution of boundary value problems bvp 1 2 such that are given under superlinear or sublinear growth in f similarly existence for the bvp 1 3 under the same assumptions is proved such that we further prove analogous results for the case when i e derivatives of the obtaining solution satisfy inverse inequalities the approach is based on an analysis of the corresponding vector field on the face plane and the well known from combinatorial analysis knaster kuratowski mazurkiewicz s principle or as it is known sperner s lemma

a new representation for the error function for the hermite interpolation and sharper pointwise and uniform error bounds for the derivatives let a a a r 2 2 be the given points and x e c a a n 2 2 be a given function satisfying the conditions a a j l j i 0 1 k j 1 2 r 1 1 where j kj r n let p r be the hermite interpolation polynomial of degree n 1 satisfying 1 1 for the error function e r x f p t everyone knows the integral representation in terms of peano s kernel green s function however the explicit representation of the green s function is unwieldy l 21 and hence except in some simple cases it cannot be used to obtain the bounds for the derivatives of e t the situation further worsens because the derivatives of the green s function change sign in the interval at a in fact even for the case n 6 r 2 k k 2 the computation is quite lengthy 3 the purpose of this paper is to give a new representation of the error function e t and use it to obtain pointwise as well as uniform error bounds for je k t 0 5 k 5 n 1 in terms of max ix t in

a proof of scott s correction for matter this paper contains a strengthening of stability of matter which in particular shows that thomas fermi theory which is already known to give the leading order contribution to the energy of matter if supplemented with the so called scott correction is correct uniformly in the number of nuclei new more precise estimates of the volume of matter also follow

on the kneser property for the complex ginzburg landau equation and the lotka volterra system with diffusion as we have seen in the previous chapters when we consider the cauchy problem of a differential equation and uniqueness fails to hold or it is not known to hold then we have to consider a set of solutions corresponding to a given initial data

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

on the local solvability and the local integrability of systems of vector fields on etudie des systemes surdetermines de champs vectoriels dans un sous ensemble ouvert de r n 1 de type l j t j j t x x j 1 m ayant des coefficients analytiques on suppose que les champs de vecteurs l j satisfont la condition de frobenius

numerical methods for unilateral problems abstract the theory of variational inequalities besides being elegant and synthetic also provides a unified general framework for applying the numerical methods for solving many unrelated free boundary value problems in this paper we describe numerical experience on the use of variational inequalities and pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity like those describing the equilibrium configuration of an elastic beam stretched over an elastic obstacle in the case of a known obstacle these problems can be alternatively formulated as nonlinear boundary value problems without constraints for which the technique of pade approximants can be successfully employed the variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution

eigenvalues of a general class of boundary value problem with derivative dependent nonlinearity we consider a general class of boundary value problem bvp comprising the differential equation y m t f t y t y t y t y q t t 0 1 where 1 q m 1 and 0 together with multi point boundary conditions y 0 y 0 y 0 y q 1 0 0 a i y q t j y q 1 t j y m 1 t j 0 j r 0 1 i m q where 0 t 0 t 1 t r 1 particular cases of this general bvp include the well known abel gontscharoff focal m p sturm liouville and complementary lidstone bvps it is noted that bvps with derivative dependent nonlinear terms are less investigated in the literature due to technical difficulty in this paper a new technique is developed to characterize the eigenvalues so that the bvp has a positive solution explicit eigenvalue intervals are also established we include several well known examples in the literature to illustrate the results obtained

the hukuhara kneser property for quasilinear parabolic equations on etudie le probleme de dirichlet quasilineaire u t a x t u lu f x t u x t q t u x t t t ou a est une matrice diagonale et l un operateur elliptique on demontre un resultat d invariance pour un ensemble convexe borne d la limitation de l ensemble d donne une borne a priori que l on utilise avec une application de la theorie du degre de leray schauder et un theoreme de structure pour montrer l existence de solutions et la propriete de hukuhara kneser

singular lidstone boundary value problem with given maximal values for solutions abstract the singular problem 1 nx 2n f t x x 2n 2 x 2j 0 x 2j t 0 0 j n 1 max x t 0 t t a depending on the parameter is considered here the positive caratheodory function f may be singular at the zero value of all its phase variables the paper presents conditions which guarantee that for any a 0 there exists a 0 such that the above problem with a has a solution x ac2n 1 0 t which is positive on 0 t the proofs are based on the regularization and sequential techniques and use the leray schauder degree and vitali s convergence theorem

eigenvalues of lidstone boundary value problems we consider the following boundary value problem 1 ny 2 n lf t y n 1 t 0 1 y 2 i 0 y 2 i 1 0 0 0 the values of l are characterized so that the boundary value problem has a positive solution in addition we derive explicit intervals of l such that for any l in the interval existence of a positive solution of the boundary value problem is guaranteed several examples are also included to dwell upon the importance of the results obtained

monotone convergence of iterative methods for n p and p n boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for n p and p n boundary value problems converges monotonically an example which illustrates the usefulness of this technique is also included

polynomial approximation to a non local boundary value problem in this paper we construct a special class of polynomials which converge uniformly to the solution of a non local boundary value problem nbvp the use of this special class is justified by the physics of the model which is described by this nbvp this nbvp has been studied by palamides et al 2009 in 2 where the existence of solutions is established

on the kneser property for reaction diffusion equations in some unbounded domains with an h 1 valued non autonomous forcing term abstract in this paper we prove the kneser property for a reaction diffusion equation on an unbounded domain satisfying the poincare inequality with an external force taking values in the space h 1 using this property of solutions we check also the connectedness of the associated global pullback attractor we study also similar properties for systems of reaction diffusion equations in which the domain is the whole r n finally the results are applied to a generalized logistic equation

on 2nth order lidstone boundary value problems this paper is concerned with the solutions of a class of 2nth order lidstone boundary value problems sufficient conditions for the existence and uniqueness of a solution are given a monotone iteration is developed so that the iteration sequence converges monotonically to a maximal or a minimal solution the approach to the problem is by the method of upper and lower solutions with a new maximum principle

a fourth order compact finite difference method for higher order lidstone boundary value problems a compact finite difference method is proposed for a general class of 2nth order lidstone boundary value problems the existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term a monotone iteration process is provided for solving the resulting discrete system efficiently and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations the convergence of the finite difference solution and the fourth order accuracy of the proposed method are proved numerical results demonstrate the high efficiency and advantages of this new approach

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

existence and uniqueness of positive solutions for a higher order boundary value problem the goal of this paper is to study the existence and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 n 1u 2 n 2 a 0u 2 i 0 email protected 0u 2 i 1 0 0 i 0 1 n 1 a 1u 2 i 1 email protected 1u 2 i 1 1 0 i 0 1 n 1 the krein rutman theorem and the krasnoselskii zabreiko fixed point theorem are the main tools that have been used to develop our work

wavelet like bases for thin wire integral equations in electromagnetics in this paper wavelets are used in solving by the method of moments a modified version of the thin wire electric field integral equation in frequency domain the time domain electromagnetic quantities are obtained by using the inverse discrete fast fourier transform the retarded scalar electric and vector magnetic potentials are employed in order to obtain the integral formulation the discretized model generated by applying the direct method of moments via point matching procedure results in a linear system with a dense matrix which have to be solved for each frequency of the fourier spectrum of the time domain impressed source therefore orthogonal wavelet like basis transform is used to sparsify the moment matrix in particular dyadic and m band wavelet transforms have been adopted so generating different sparse matrix structures this leads to an efficient solution in solving the resulting sparse matrix equation moreover a wavelet preconditioner is used to accelerate the convergence rate of the iterative solver employed these numerical features are used in analyzing the transient behavior of a lightning protection system in particular the transient performance of the earth termination system of a lightning protection system or of the earth electrode of an electric power substation during its operation is focused the numerical results obtained by running a complex structure are discussed and the features of the used method are underlined

the hukuhara kneser property for a nonlinear integral equation abstract applying a structure theorem of krasnosel skii and perov we show that the solution set of a nonlinear integral equation satisfies the classical hukuhara kneser property

upper and lower estimates for positive solutions of the higher order lidstone boundary value problem we consider the higher order lidstone boundary value problem new upper and lower estimates for positive solutions of the problem are obtained a discussion of the sharpness of the estimates is included

positive solutions of a 2nth order boundary value problem involving all derivatives via the order reduction method this paper is mainly concerned with the existence multiplicity and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 i 2 u i 1 n 1u 2 n 1 u 2 i 0 u 2 i 1 1 0 i 0 1 n 1 where n 2 and f c 0 1 xr 2 n r r 0 we first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first order integro differential equation and then use the fixed point index theory to prove the existence multiplicity and uniqueness of positive solutions for the resulting problem based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators finally as a byproduct our main results are applied for establishing the existence multiplicity and uniqueness of symmetric positive solutions for the lidstone problem involving all derivatives

the hukuhara kneser property for parabolic systems with nonlinear boundary conditions the domain of the problem is a cylinder qt r x 0 7 1 the parabolic operator l is the same for each component of u and the nonlinearity f may be strongly coupled the purpose of this paper is to develop some of the fundamental theory for this problem we allow conditions general enough for nonunique solutions and start by giving a set of conditions that requires all the solutions to remain in an invariant region using these conditions we show the existence of at least one such solution and that the problem satisfies the hukuhara kneser property these results are thus extensions of berbernes and schmitt 5 to nonlinear boundary conditions a set d c rm is said to be invariant for the above problem if w g c d implies u q c d let n u denote an outer normal at a point u in the boundary of a nonempty open convex set d in theorem 3 1 the maximum principle is used to show that if the strong tangent inequality