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In a recent paper, a series solution method based on combining the Laplace transform and Adomian polynomial expansion was proposed to find an approximate solution of nonlinear differential equations \cite{FA2016}. It uses the expansion in Adomian polynomials defined in \cite {A1,A2}. An important drawback of the Laplace transform method is the fact that it cannot be applied in the case of nonlinear differential equation in general. In order to cope with this problem, the authors of \cite{FA2016} suggested the use of Adomian polynomial expansion of the nonlinear function of the dependent variable involved in the differential equation. In this work, we propose a counterpart of this method on an arbitrary time scale and derive its general formulation for a dynamic equation of any order. We confirm that when the time scale is the set of real numbers, our method reduces to that in \cite{FA2016}. Our presentation is organized as follows. First, we recollect some preliminary information on time scales in Secton 2. In Section 3, we derive the method for an $n$-th order nonlinear dynamic equation. The next section contains the application of the method to specific examples of first order nonlinear dynamic equations. The last section is devoted to conclusion and some further directions for study.

In this paper, we consider an implicit relation to generalize iterative fixed point results in the literature in the context of metric spaces. We conclude that several existing results are immediat...

This study is devoted to the development of alternative conditions for existence and uniqueness of nonlinear fractional differential equations of higher-order with integral and multi-point boundary conditions. It uses a novel approach of employing a fixed point theorem based on contractive iterates of the integral operator for the corresponding fixed point problem. We start with developing an existence-uniqueness theorem for self-mappings with contractive iterate in a b-metric-like space. Then, we obtain the unique solvability of the problem under suitable conditions by utilizing an appropriate b-metric-like space.

In this paper we investigate the existence and uniqueness of fixed points of certain $$(\phi ,F)$$ -type contractions in the frame of metric-like spaces. As an application of the theorem we consider the existence and uniqueness of solutions of nonlinear Fredholm integral equations of the second kind on time scales. We also present a particular example which demonstrates our theoretical results.

In this paper we introduce contraction mappings of Meir-Keeler types on modular metric spaces and investigate the existence and uniqueness of their fixed points. We give an example which demonstrates our theoretical results.

In this paper fixed points of α-admissible contraction mappings of Geraghty type defined on Branciari b-metric spaces are studied. Existence and uniqueness theorems for these types of mappings of are proved. Some consequences of these theorems are given and specific examples are presented.

WOS: 000423281300008, In this paper, alpha-Meir-Keeler and generalized alpha-Meir-Keeler contractions on Branciari b-metric spaces are introduced. Existence and uniqueness of fixed points of such contractions are discussed and related theorems are proved. Various consequences of the main results are also presented.

In this paper existence and uniqueness of fixed points for a general class of contractive and nonexpansive mappings on modular metric spaces is discussed. As an application of the theoretical results, the existence of a solution of anti-periodic boundary value problems for nonlinear first order differential equations of Carath?odory?s type is considered in the framework of modular metric spaces.

WOS: 000393617200001, In this paper a class of general type alpha-admissible contraction mappings on quasi-b-metric-like spaces are defined. Existence and uniqueness of fixed points for this class of mappings is discussed and the results are applied to Ulam stability problems. Various consequences of the main results are obtained and illustrative examples are presented.

The Taylor series method for initial value problems associated with dynamic equations of first order on time scales with delta differentiable graininess function is introduced. The trapezoidal rule for the same types of problems is derived and applied to specific examples. Numerical results are presented and discussed.

WOS: 000338938300001, A class of alpha-admissible contractions defined via altering distance functions is introduced. The existence and uniqueness conditions for fixed points of such maps on complete metric spaces are investigated and related fixed point theorems are presented. The results are reconsidered in the context of partially ordered metric spaces and applied to boundary value problems for differential equations with periodic boundary conditions.

In this manuscript, the existence and uniqueness of fixed points of a class of cyclic operators defined on a closed subset of a Banach space is discussed. Fixed point theorems for some contractions from this class are introduced and illustrative examples are given.

Fixed point theorems for operators of a certain type on partial metric spaces are given. Orbitally continuous operators on partial metric spaces and orbitally complete partial metric spaces are defined, and fixed point theorems for these operators are given.

Very recently, in a series of subsequent papers, Nan and Charoensawan introduced the notion of g-coincidence point of two mappings in different settings (metric spaces and G-metric spaces) and proved some theorems in order to guarantee the existence and uniqueness of such kind of points. Although their notion seems to be attractive, in this paper, we show how this concept can be reduced to the unidimensional notion of coincidence point, and how their main theorems can be seen as particular cases of existing results. Moreover, we prove that the proofs of their main statements have some gaps.

An expansion method for the stationary Schrödinger equation of a three-dimensional quantum billiard system whose boundary is defined by an arbitrary analytic function is introduced. The method is based on a coordinate transformation and an expansion in spherical harmonics. The effectiveness is verified and confirmed by a numerical example, which is a billiard system depending on a parameter.

An eigenfunction expansion for the Schrodinger equation for a particle moving in an arbitrary non-central potential in the cylindrical polar coordinates is introduced, which reduces the partial differential equation to a system of coupled differential equations in the radial variable r. It is proved that such an orthogonal expansion of the wavefunction into the complete set of Chebyshev polynomials is uniformly convergent on any domain of (r,θ). As a benchmark application, the bound states calculations of the quartic oscillator show that both analytical and numerical implementations of the present method are quite satisfactory.

A class of weak ψ -contractions satisfying the C -condition is defined on metric spaces. The existence and uniqueness of fixed points of such maps are discussed both on metric spaces and on partially ordered metric spaces. The results are applied to a first order periodic boundary value problem. MSC: 47H10, 54H25.

In this paper, we introduce the new notion of almost ψ-Geraghty contractive mappings and investigate the existence of a best proximity point for such mappings in complete metric spaces via the weak P-property. We provide an example to validate our best proximity point theorem. The obtained results extend, generalize, and complement some known fixed and best proximity point results from the literature. MSC:47H10, 54H25, 46J10, 46J15.

In this paper, a general class ofα-admissible contractions on partial metric spaces is introduced. Fixed point theorems for these contractions on partial metric spaces and their consequences are stated and proved. Illustrative example is presented.

WOS: 000338221900001, Existence and uniqueness of fixed points of a mapping defined on partially ordered G-metric spaces is discussed. The mapping satisfies contractive conditions based on certain classes of functions. The results are applied to the problems involving contractive conditions of integral type and to a particular type of initial value problems for the nonhomogeneous heat equation in one dimension. This work is a generalization of the results published recently in (Gordji et al. in Fixed Point Theory Appl. 2012:74, 2012, doi:10.1186/1687-1812-2012-74) to G-metric space.

WOS:000317262100043 In this paper, the concept of a quasi-partial metric space is introduced, and some general fixed point theorems in quasi-partial metric spaces are proved. (C) 2012 Elsevier Ltd. All rights reserved.

The purpose of this paper is to extend some recent coupled fixed point theorems in the context of partially ordered G-metric spaces in a virtually different and more natural way. MSC:46N40, 47H10, 54H25, 46T99.

In this manuscript, Meir-Keeler contractions on partial metric spaces are introduced. It is shown that if a self-mapping T on a complete partial metric spaces is a Meir-Keeler contraction, then T has a unique fixed point.

Existence and uniqueness of fixed points of a general class of contractive mappings on complete rectangular metric spaces are discussed. One of the theorems is a generalization of a fixed point theorem recently introduced by Lakzian and Samet. Fixed points of contractions under conditions involving rational expressions are also investigated. Several particular cases and applications as well as an illustrative example are given. MSC:46T99, 47H10, 54H25.

Conditions for existence and uniqueness of fixed points of two types of cyclic contractions defined on G -metric spaces are established and some illustrative examples are given. In addition, cyclic maps satisfying integral type contractive conditions are presented as applications. © 2012 E. Karapinar et al.

We prove coupled coincidence point and coupled fixed point results of 𝐹 ∶ 𝑋 × 𝑋 → 𝑋 and 𝑔 ∶ 𝑋 → 𝑋 involving Meir-Keeler type contractions on the class of partially ordered metric spaces. Our results generalize some recent results in the literature. Also, we give some illustrative examples and application.

A numerical method for solving the time-independent Schrödinger equation of a particle moving freely in a three-dimensional axisymmetric region is developed. The boundary of the region is defined by an arbitrary analytic function. The method uses a coordinate transformation and an expansion in eigenfunctions. The effectiveness is checked and confirmed by applying the method to a particular example, which is a prolate spheroid., {"references":["M. Sieber and F. Steiner, Classical and quantum mechanics of a strongly\nchaotic billiard system, Physica D, vol.44, pp. 248-266, August 1990.","M. V. Berry, Quantizing a classically ergodic system: Sinai-s billiard and\nthe KKR method, Ann.Phys., vol. 131, pp. 163-216, 1981.","D. A. McGrew and W. Bauer, Constraint operator solution to quantum\nbilliard problem, Phys. Rev. E, vol. 54, pp. 5809-5818, November 1996.","M. Sieber and F. Steiner, Quantum chaos in the hyperbola billiard, Phys.\nLett. A, vol. 148, pp. 415-420, September 1990.","M. Robnik, Quantizing a generic family of billiards with analytic boundaries,\nJ. Phys. A: Math. Gen., vol. 17, pp. 1049-1074, 1984.","T. Papenbrock, Numerical study of a three-dimensional generalized\nstadium billiard, Phys. Rev. E, vol. 61, pp. 4626-4628, April 2000.","H. Primak and U. Smilansky, The quantum three-dimensional Sinai\nbilliard - A semiclassical analysis, Phys. Rep., vol. 327, pp. 1-107,\nApril 2000.","W. A. Strauss, Partial Differential Equations, New York:\nJohn-Wiley, 1992.","H. Tas┬©eli, İInci M. Erhan and O┬¿ . Ug╦ÿur, An eigenfunction expansion for\nthe Schr┬¿odinger equation with arbitrary non-central potentials, J. Math.\nChem., vol. 32, pp. 323-338, November 2002.\n[10] M. Abramovitz and I. A. Stegun , Handbook of Mathematical Functions,\nNew York:Dover, 1970.\n[11] S. A. Moszkowski, Particle states in spheroidal nuclei, Phys Rev. vol.\n99, pp. 803-809, 1955."]}

In this paper, we introduce the concept of $(\alpha,\psi)$ -rational type contractive mappings and provide sufficient conditions for the existence and uniqueness of a fixed point for such class of generalized nonlinear contractive mappings in the setting of generalized metric spaces. We also deduce several interesting corollaries.

In this paper, some multifunctions on partial metric space are defined and common fixed points of such multifunctions are discussed. The results presented in the paper generalize some of the existing results in the literature. Several conclusions of the main results are given.

WOS: 000350677900001, We introduce alpha-admissible Meir-Keller and generalized alpha-admissible Meir-Keller contractions on quasi-metric spaces and discuss the existence of fixed points of such contractions. We apply our results to G-metric spaces and express some fixed point theorems in G-metric spaces as consequences of the results in quasi-metric spaces.

In this paper, following the idea of Samet et al. (J. Nonlinear. Sci. Appl. 6:162-169, 2013), we establish a new fixed point theorem for a Meir-Keeler type contraction via Gupta-Saxena rational expression which enables us to extend and generalize their main result (Gupta and Saxena in Math. Stud. 52:156-158, 1984). As an application we derive some fixed points of mappings of integral type.

Very recently, Hussain et al. (Fixed Point Theory Appl. 2014:62, 2014) announced the existence and uniqueness of some coupled coincidence point. In this short note we remark that the announced results can be derived from the coincidence point results in the literature. MSC: 47H10, 54H25.

Recently, some (common) coupled fixed theorems in various abstract spaces have appeared as a generalization of existing (usual) fixed point results. Unexpectedly, we noticed that most of such (common) coupled fixed theorems are either weaker or equivalent to existing fixed point results in the literature. In particular, we prove that the very recent paper of Turkoglu and Sangurlu ‘Coupled fixed point theorems for mixed g-monotone mappings in partially ordered metric spaces [Fixed Point Theory and Applications 2013, 2013:348]’ can be considered as a consequence of the existing fixed point theorems on the topic in the literature. Furthermore, we give an example to illustrate that the main results of Turkoglu and Sangurlu (Fixed Point Theory Appl. 2013:348, 2013) has limited applicability compared to the mentioned existing fixed point result. MSC:47H10, 54H25.