Search

Abstract We consider the Banach function space consisting of real functions defined on the half-axis and having increments tempered by a given modulus of continuity. Next, we formulate and prove a sufficient condition for relative compactness of a bounded subset of the mentioned Banach space. On the basis of that condition we construct a measure of noncompactness in the function space in question. Using that measure we prove a theorem on the existence of solutions of a quadratic integral equation in the space of functions satisfying the Hölder condition on the real half-axis. [ABSTRACT FROM AUTHOR]

Abstract The article suggests a new approach using the so-called consistency method for the inversion of the spherical mean Radon transform in 2D with detectors on a line. We present a new iterative formula which gives an algorithm to recover an unknown function supported completely on one side of a line L from its spherical means over circles centered on the line L. Our reconstruction formula has the benefit of being local. Such an inversion is required in problems of thermo- and photo-acoustic tomography, ultrasound reflection tomography, radar imaging, and a few other. [ABSTRACT FROM AUTHOR]

For refracted spectrally negative Lévy processes, we identify expressions for several quantities related to the Laplace transforms of their weighted occupation times until first exit times. Such quantities are expressed in terms of the unique solutions to integral equations involving weight functions and scale functions for the associated spectrally negative Lévy processes. Previous results on refracted Lévy processes are recovered. [ABSTRACT FROM AUTHOR]

Let N be an integral operator of the form ( N u ) ( x ) = ∫ R c n ( x , x − y ) u ( y ) d y acting in L p ( R c ) with a measurable kernel n satisfying the estimate | n ( x , y ) | ≤ β ( y ) , where β ∈ L 1 . It is proved that if the function t ↦ n ( t , ⋅ ) is continuous in the norm of L 1 and the operator 1 + N has an inverse, then ( 1 + N ) − 1 = 1 + M , where M is an integral operator possessing the same properties. [ABSTRACT FROM AUTHOR]

Abstract: This article deals with boundary integral equation preconditioning for the multiple scattering problem. The focus is put on the single scattering preconditioner, corresponding to the diagonal part of the integral operator, for which two results are proved. Indeed, after applying this geometric preconditioner, it appears that, firstly, every direct integral equations become identical to each other, and secondly, that the indirect integral equation of Brakhage–Werner becomes equal to the direct integral equations, up to a change of basis. These properties imply in particular that the convergence rate of a Krylov subspaces solver will be exactly the same for every preconditioned integral equations. To illustrate this, some numerical simulations are provided at the end of the paper. [Copyright &y& Elsevier]

Abstract: In this paper we study the quadratic integral equation of the form Several existence theorems for a.e. monotonic solutions in Orlicz spaces are proved for strongly nonlinear functions f. The presented method of the proof can be easily extended to different classes of solutions. [Copyright &y& Elsevier]

Abstract: In this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing Henstock–Kurzweil integrable functions. [Copyright &y& Elsevier]

Abstract: In this paper, we generalize some integral inequalities to more general situations, and the inequalities of Pachpatte type are corollaries of our''s. We establish bounds on the solutions, and we show the usefulness of our results in investigating the asymptotic behavior and the stability on the solutions of integral equations, differential equations and integro-differential equations with time delay. [Copyright &y& Elsevier]

Abstract: In this paper we prove monotone convergence theorems for Henstock–Kurzweil integrable functions from a compact real interval to an ordered Banach space. These theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing Henstock–Kurzweil integrable functions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we generalize some integral inequalities to more general situations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the global existence of the solutions of integral equations and differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the numerical solution of the integral equation of the second kind reduced by acoustic scattering in shallow oceans with Dirichlet condition. Based on analyzing the singularity of the truncating kernel with a sum of infinite series, using our trigonometric interpolatory wavelets and collocation method, we obtain the numerical solution which possesses a fast convergence rate like . Moreover, the entries of the stiffness matrix can be obtained by FFT, which lead the computational complexity to decrease obviously. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to present some fixed point results for self-generalized contractions in ordered metric spaces. Our results generalize and extend some recent results of A.C.M. Ran, M.C. Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435–1443], J.J. Nieto, R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205–2212], J.J. Nieto, R.L. Pouso, R. Rodríguez-López [J.J. Nieto, R.L. Pouso, R. Rodríguez-López, Fixed point theorem theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505–2517], A. Petruşel, I.A. Rus [A. Petruşel, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411–418] and R.P. Agarwal, M.A. El-Gebeily, D. O''Regan [R.P. Agarwal, M.A. El-Gebeily, D. O''Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., in press]. As applications, existence and uniqueness results for Fredholm and Volterra type integral equations are given. [Copyright &y& Elsevier]

Abstract: The plan of this paper is to obtain one-point iterative methods with any R-order of convergence, when they are applied to approximate solutions of quadratic equations in Banach spaces. To do this, we consider real Cauchy''s method and, under certain natural modifications, it is extended to Banach spaces. Some applications are also provided. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the exponential dichotomy of linear skew-product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on Banach bundles. We characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy. [Copyright &y& Elsevier]

Abstract: We study the existence of periodic solutions of the nonlinear neutral system of differential equations of the form In the process we use the fundamental matrix solution of and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii''s fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution of the equation. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family the subspace . Supposing that is closed and complemented, we prove that the admissibility of the pair implies the uniform dichotomy of Φ. Under the same hypothesis on , we obtain that the admissibility of the pair with is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations. [Copyright &y& Elsevier]

Abstract: In this study, transformation operator, some properties of kernel, properties of spectral characteristics and also uniquiness theorems for the inverse problems are studied for a class of Sturm–Liouville differential operators which have discontinuity conditions inside a finite interval. [Copyright &y& Elsevier]

Abstract: The concern of this paper is to study the optimization of adaptive direct algorithms for approximate solution of the second kind Fredholm integral equations. We derive the exact order of the problem with integral kernels belonging to the Besov classes. Furthermore, we also present an almost optimal adaptive direct algorithm for above equation class. [Copyright &y& Elsevier]

Abstract: In this paper we present a two-norms version of Krasnoselskii''s fixed point theorem in cones. The abstract result is then applied to prove the existence of positive solutions of Hammerstein integral equations with better integrability properties on the kernels. [Copyright &y& Elsevier]

In this paper, we derive some properties of a Ramanujan type entire function. A mild generalization of the Garret-Ismail-Stanton m -version of the Rogers-Ramanujan identities is obtained. Moreover, we investigate the zeros of the Ramanujan type entire function, and our results generalize those for the zeros of the Ramanujan function. Finally, an integral equation related to the Ramanujan type entire function is also derived. [ABSTRACT FROM AUTHOR]

In this paper, our purpose is to study the nonlinear integral equation of Hénon type (0.1) (u (x)) q − 1 = ∫ Ω | x | α | y | β u (y) | x − y | n − λ d y , where Ω is the unit ball in R n centred at the origin, α > 0 , β > 0 , and 1 < λ < n. We prove that there exists a positive radial solution of problem (0.1) provided that q ∈ (2 n n + λ + α + β , 2) and analyse the symmetry breaking of the maximizer of supremum Q α , β , λ on L q (Ω) for α + β large enough. [ABSTRACT FROM AUTHOR]