on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

use of equivalent boundary conditions for the solution of a class of strip grating structures equivalent boundary conditions for an electromagnetic strip grating are used to derive reflection and transmission coefficients for a boundary value problem consisting of three nonconducting media where the strip grating exists at one of the planar boundaries the results are used to generate data for a number of cases where data and or theories exist the theory presented compares favorably with the literature cited for the limiting condition that the period of the grating be small when compared with the free space wavelength of the radiation

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

the reciprocity and power conservation principles for boundary conditions on a semitransparent surface generalized boundary conditions of coupling on a semitransparent surface are valid only if the lorentz reciprocity theorem and the electromagnetic power conservation principle are satisfied on the surface with these boundary conditions these criteria are reformulated as relations between the tensors in the boundary conditions some known boundary conditions of coupling are considered

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent in this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power like variable reaction term one of the main features of our work is that the differential operator involves partial derivatives with different variable exponents so that the functional analytic framework relies upon anisotropic sobolev and lebesgue spaces existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations

practical stability of impulsive functional differential equations in terms of two measurements in this paper we consider practical stability of impulsive functional differential equations in terms of two measurements some sufficient conditions of uniform practical stability for functional differential equation with impulses are obtained by using piecewise continuous lyapunov functions and razumikhin techniques an example illustrates the effectiveness of the proposed result

asymptotic stability of competitive systems with delays and impulsive perturbations abstract a general impulsive nonautonomous lotka volterra system of integro differential equations with infinite delay is considered the impulses are realized at fixed moments of time sufficient conditions for uniform stability and asymptotic stability of solutions are investigated the main results are obtained by using the comparison principle and the lyapunov method

mountain pass solutions for non local elliptic operators abstract the purpose of this paper is to study the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions these equations have a variational structure and we find a non trivial solution for them using the mountain pass theorem to make the nonlinear methods work some careful analysis of the fractional spaces involved is necessary we prove this result for a general integrodifferential operator of fractional type and as a particular case we derive an existence theorem for the fractional laplacian finding non trivial solutions of the equation s u f x u in u 0 in r n as far as we know all these results are new

local and global existence and uniqueness results for impulsive functional differential equations with multiple delay abstract in this paper we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay we shall rely on a fixed point theorem of schaefer and a nonlinear alternative of leray schauder for the global existence and uniqueness we apply a recent nonlinear alternative of leray schauder type in frechet spaces due to frigon and granas m frigon a granas resultats de type leray schauder pour des contractions sur des espaces de frechet ann sci math quebec 22 2 1998 161 168

multiplicity of solutions for variable order fractional kirchhoff equations with nonstandard growth abstract the aim of this paper is to study a degenerate kirchhoff type elliptic problem driven by the fractional laplace operator with variable order derivative and variable exponents more precisely we consider u s 2 1 s u a x u p x 2 u b x u q x 2 u in u 0 in r n where u s is the garliado seminorm 1 n 1 s r n r n 0 1 is a continuous function 0 is a parameter is a bounded domain in r n with n 2 s x y for all x y s is the variable order fractional laplacian a b l are two positive weight functions and p q c with 1 p x 2 q x 2 n n 2 s x x under some suitable assumptions we obtain that the problem admits at least two distinct nonnegative solutions by applying the nehari manifold approach provided is sufficiently small moreover the existence of infinitely many solutions is also investigated by the symmetric mountain pass theorem

two analytic frequency dependent current density approximations for te scattering from a conducting strip grating two analytic frequency dependent current density approximations are presented for te mode incidence on a conducting strip grating the current densities are assumed to have both a spatial and a frequency dependence the first is based upon using the classic edge mode approximation the second is derived from applying conformal mapping methods and invoking babinet s theorem the solutions are compared to the moment method solution the resultant reflection coefficients are found to be good approximations at all frequencies for narrow strips and at moderately low frequencies for arbitrary strip widths although the solutions are not as accurate as a moment method solution at all frequencies they are computationally faster and provide analytic expressions that demonstrate the relationship between grating dimensions and performance

existence of entire solutions for a class of variable exponent elliptic equations abstract the paper deals with the existence of entire solutions for a quasilinear equation e in r n depending on a real parameter which involves a general variable exponent elliptic operator a in divergence form and two main nonlinearities the competing nonlinear terms combine each other under some conditions we prove the existence of a critical value 0 with the property that e admits nontrivial nonnegative entire solutions if and only if furthermore under the further assumption that the potential a of a is uniform convex we give the existence of a second independent nontrivial nonnegative entire solution of e when our results extend the previous work of autuori and pucci 2013 6 from the case of constant exponents p q and r to the case of variable exponents more interesting we weaken the condition max 2 p q min r p to the simple request that 1 q r furthermore we extend the previous work of alama and tarantello 1996 2 from dirichlet laplacian problems in bounded domains of r n to the case of a general variable exponent differential equation in the entire r n and also remove the assumption q 2 hence the results of this paper are new even in the canonical case p 2

topological structure of solution sets for impulsive differential inclusions in frechet spaces abstract in this paper we consider the existence of solutions as well as the topological and geometric structure of solution sets for first order impulsive differential inclusions in some frechet spaces both the initial and terminal problems are considered using ingredients from topology and homology the topological structures of solution sets closedness and compactness as well as some geometric properties contractibility acyclicity a r and r are investigated some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

second method of lyapunov and existence of integral manifolds for impulsive differential difference equations by means of piecewise continuous functions which are analogues of lyapunov s functions sufficient conditions are obtained for the existence of integral manifolds for impulsive differential difference equations with variable impulsive perturbations

higher order impedance boundary conditions for sparse wire grids higher order impedance boundary conditions designed for modeling wire grids of thin conducting wires are established the derivation is based on the exact analytical summation of the individual wire fields this allows one to write an approximate boundary condition on the grid surface which connects the averaged electric field and the averaged current or the electric field and the averaged magnetic fields on the two sides of the grid surface the condition depends on the tangential derivatives of the averaged current up to the sixth order this approach provides an extension of the averaged boundary conditions method well established for dense grids to sparse grids numerical examples demonstrate very good accuracy of the solutions for the field reflected from grids with the wire separation as large as half of the wavelength

existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay abstract in this paper we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay we shall rely on a nonlinear alternative of leray schauder for the global existence and uniqueness we apply a recent nonlinear alternative of leray schauder type in frechet spaces due to m frigon and a granas resultats de type leray schauder pour des contractions sur des espaces de frechet ann sci math quebec 22 2 1998 161 168 the goal of this paper is to extend the problems considered by a ouahab local and global existence and uniqueness results for impulsive differential equations with multiple delay j math anal appl 323 2006 456 472

existence of solutions for kirchhoff type problem involving the non local fractional p laplacian abstract the purpose of this paper is to investigate the existence of weak solutions for a kirchhoff type problem driven by a non local integro differential operator of elliptic type with homogeneous dirichlet boundary conditions as follows m r 2 n u x u y p k x y d x d y l k p u f x u in u 0 in r n where l k p is a non local operator with singular kernel k is an open bounded subset of r n with lipshcitz boundary m is a continuous function and f is a caratheodory function satisfying the ambrosetti rabinowitz type condition we discuss the above mentioned problem in two cases when f satisfies sublinear growth condition the existence of nontrivial weak solutions is obtained by applying the direct method in variational methods when f satisfies suplinear growth condition the existence of two nontrivial weak solutions is obtained by using the mountain pass theorem