Showing papers in "Abstract and Applied Analysis in 2012"
TL;DR: In this paper, fixed point theorems for cyclic contractive mappings are established for metric spaces endowed with a partial order and for a class of contractive mapping mappings.
Abstract: We establish fixed point theorems for a new class of contractive mappings. As consequences of our main results, we obtain fixed point theorems on metric spaces endowed with a partial order and fixed point theorems for cyclic contractive mappings. Various examples are presented to illustrate our obtained results.
290 citations
TL;DR: In this paper, a review of the variational approach, the Hamiltonian approach, variational iteration method, the homotopy perturbation method, parameter expansion method, Yang-Laplace transform, and Yang-Fourier transform is presented.
Abstract: This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.
239 citations
TL;DR: A survey of some selected recent developments in the theory of Ulam's type stability can be found in this paper, where the authors provide some information on hyperstability and the fixed point methods.
Abstract: We present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability In particular we provide some information on hyperstability and the fixed point methods
187 citations
TL;DR: In this article, the uniqueness of the coupled fixed point for mixed monotone mapping satisfying nonlinear contraction in the framework of generalized metric space endowed with partial order was proved for such mappings.
Abstract: Two concepts—one of the coupled fixed point and the other of the generalized metric space—play a very active role in recent research on the fixed point theory. The definition of coupled fixed point was introduced by Bhaskar and Lakshmikantham (2006) while the generalized metric space was introduced by Mustafa and Sims (2006). In this work, we determine some coupled fixed point theorems for mixed monotone mapping satisfying nonlinear contraction in the framework of generalized metric space endowed with partial order. We also prove the uniqueness of the coupled fixed point for such mappings in this setup.
144 citations
TL;DR: In this article, the authors show that two recent definitions of discrete fractional sum operators are related, and prove power rule and commutative property of fractional difference operators with respect to the power rule.
Abstract: We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.
134 citations
TL;DR: In this article, a new closed Newton-Cotes trigonometrically fitted differential method of high algebraic order was proposed, which gives much more efficient results than the well-known ones.
Abstract: The closed Newton-Cotes differential methods of high algebraic order for small number of function evaluations are unstable. In this work, we propose a new closed Newton-Cotes trigonometrically fitted differential method of high algebraic order which gives much more efficient results than the well-know ones.
122 citations
TL;DR: In this paper, the authors present some new common fixed point theorems for a pair of nonlinear mappings defined on an ordered Banach space and extend several earlier works.
Abstract: We present some new common fixed point theorems for a pair of nonlinear mappings defined on an ordered Banach space. Our results extend several earlier works. An application is given to show the usefulness and the applicability of the obtained results.
121 citations
TL;DR: In this article, two iterative forward-backward splitting methods with relaxations and errors are introduced to find zeros of the sum of two accretive operators in the Banach spaces.
Abstract: Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem.
113 citations
TL;DR: In this article, an alternative approach to construct the homotopy equation with an auxiliary term was proposed, where the auxiliary term is used as an example to illustrate the solution procedure.
Abstract: The two most important steps in application of the homotopy perturbation method are to construct a suitable homotopy equation and to choose a suitable initial guess. The homotopy equation should be such constructed that when the homotopy parameter is zero, it can approximately describe the solution property, and the initial solution can be chosen with an unknown parameter, which is determined after one or two iterations. This paper suggests an alternative approach to construction of the homotopy equation with an auxiliary term; Dufing equation is used as an example to illustrate the solution procedure.
113 citations
TL;DR: In this paper, the Montgomery identities for Riemann-Liouville fractional integrals are extended for convex functions, and some new integral inequalities for the fractional integral are developed.
Abstract: We extend the Montgomery identities for the Riemann-Liouville fractional integrals. We also use these Montgomery identities to establish some new integral inequalities. Finally, we develop some
integral inequalities for the fractional integral using differentiable convex functions.
111 citations
TL;DR: In this paper, the continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values.
Abstract: In this paper, the continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values. The proposed technique might be considered as a variation of the finite difference method in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation. This novel approach possesses main advantages; it can be applied without any limitation on the nature of the problem, the type of singularity, and the number of mesh points. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the presented technique. The results reveal that the algorithm is very effective, straightforward, and simple.
TL;DR: In this paper, the convergence analysis of the regularized methods for the split feasibility problem (SFP) has been studied under some different control conditions, and the suggested algorithms strongly converge to the minimum norm solution of the SFP.
Abstract: Many applied problems such as image reconstructions and signal processing can
be formulated as the split feasibility problem (SFP). Some algorithms have been introduced in the literature for solving the (SFP). In this paper, we will continue to consider the convergence analysis of the regularized methods for the (SFP). Two
regularized methods are presented in the present paper. Under some different control
conditions, we prove that the suggested algorithms strongly converge to the minimum
norm solution of the (SFP).
TL;DR: In this article, the authors give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.
Abstract: Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.
TL;DR: In this paper, the numerical solution of nonlinear Fredholm-volterra integro-differential equations using reproducing kernel Hilbert space method is investigated, where the solution is represented in the form of series in the Reproducing kernel space.
Abstract: This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution 𝑢(𝑥) is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution 𝑢𝑛(𝑥) is obtained and it is proved to converge to the exact solution 𝑢(𝑥). Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.
TL;DR: In this paper, the authors introduce the concept of s-geometrically convex functions and establish some integral inequalities of Hermite-Hadamard type related to the s-Geometric Convex Functions.
Abstract: The authors introduce the concept of the s-geometrically convex functions. By the well-known Holder inequality, they establish some integral inequalities of Hermite-Hadamard type related to the s-geometrically convex functions and apply these inequalities to special means.
TL;DR: In this paper, a generalized fractional integral with Lagrangians depending on classical derivatives and derivatives was used to obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems.
Abstract: We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.
TL;DR: In this article, the wave equation in fractal vibrating string was introduced in the framework of local fractional calculus and the technique of the LFTF series was applied to derive the solution of the Local fractional wave equation.
Abstract: We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the MittagLeffler function.
TL;DR: In this article, it was shown that the system of three difference equations, where all elements of the sequences were real numbers, can be solved, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three were deduced.
Abstract: We show that the system of three difference equations , , and , , where all elements of the sequences , , , , , and initial values , , , , are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.
TL;DR: In this paper, a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term was studied and an exponent for which there does not exist any global solution for the problem was obtained.
Abstract: This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.
TL;DR: In this paper, the authors define and study statistical convergence of double sequences in a locally solid Riesz space and define statistical -convergence, statistical -Cauchy, and statistical -Convergence of Cauchy double sequences.
Abstract: Recently, the notion of statistical convergence is studied in a locally solid Riesz space by Albayrak and Pehlivan (2012). In this paper, we define and study statistical -convergence, statistical -Cauchy and -convergence of double sequences in a locally solid Riesz space.
TL;DR: In this article, the best possible lower and upper bounds for the Neuman-Sandor mean in terms of convex combinations of either the harmonic and quadratic means or the geometric and quadralatic means were presented.
Abstract: We present the best possible lower and upper bounds for the Neuman-Sandor mean in terms of the convex combinations of either the harmonic and quadratic means or the geometric and quadratic means or the harmonic and contraharmonic means
TL;DR: In this paper, a computer virus model with time delay based on an SEIR model is proposed, where the authors regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf Bifurcation.
Abstract: By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.
TL;DR: In this paper, a fixed-point theorem for a class of operators with suitable properties, in very general conditions, was proved and an affirmative answer to the open problem of Brzdek and Cieplinski (2011) is given.
Abstract: In this paper we prove a fixed-point theorem for a class of operators with suitable properties, in very general conditions. Also, we show that some recent fixed-points results in Brzdek et al., (2011) and Brzdek and Cieplinski (2011) can be obtained directly from our theorem. Moreover, an affirmative answer to the open problem of Brzdek and Cieplinski (2011) is given. Several corollaries, obtained directly from our main result, show that this is a useful tool for proving properties of generalized Hyers-Ulam stability for some functional equations in a single variable.
TL;DR: In this article, the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method was obtained.
Abstract: We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.
TL;DR: In this paper, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference was studied using the Lyapunov direct method, and conditions for uniform stability, uniform asymmetric stability, and uniform global stability were discussed.
Abstract: Using the Lyapunov direct method, the stability of
discrete nonautonomous systems within the frame of the Caputo fractional
difference is studied. The conditions for uniform stability, uniform
asymptotic stability, and uniform global stability are discussed.
TL;DR: In this article, the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary value conditions,,, where, and is the Caputo fractional derivative and is a continuous function, were investigated.
Abstract: We investigate the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary value conditions, , , where , and is the Caputo fractional derivative and is a continuous function. Our analysis relies on a fixed point theorem in partially ordered sets. Moreover, we compare our results with others that appear in the literature.
TL;DR: In this article, it was shown that a self-mapping on a complete partial metric space has a fixed point if it satisfies the cyclic weak ǫ-contraction principle.
Abstract: A new fixed point theorem is obtained for the class of cyclic weak 𝜙-contractions on partially metric spaces. It is proved that a self-mapping 𝑇 on a complete partial metric space 𝑋 has a fixed point if it satisfies the cyclic weak 𝜙-contraction principle.
TL;DR: In this article, the greatest values, and least values, such that the double inequalities and hold for all with and present some new bounds for the complete elliptic integrals were found.
Abstract: We find the greatest values , and least values , such that the double inequalities and hold for all with and present some new bounds for the complete elliptic integrals. Here , , and are the arithmetic-geometric, Toader, and th Gini means of two positive numbers and , respectively.
TL;DR: In this article, the authors examined oscillatory properties of the third-order neutral delay differential equation and presented oscillatory and asymptotic criteria to improve and complement those results in the literature.
Abstract: The purpose of this paper is to examine oscillatory properties of the third-order neutral delay differential equation . Some oscillatory and asymptotic criteria are presented. These criteria improve and complement those results in the literature. Moreover, some examples are given to illustrate the main results.
TL;DR: In this article, the equivalence between variational inequalities and the Wiener-Hopf equations was established using essentially the projection technique, which was used to suggest and analyse some iterative methods for solving the general multivalued variational inequality in conjunction with nonexpansive mappings.
Abstract: We introduce and study some new classes of variational inequalities and the Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and
analyze some iterative methods for solving the general multivalued variational in
equalities in conjunction with nonexpansive mappings. We prove a strong convergence
result for finding the common element of the set of fixed points of a nonexpansive
mapping and the set of solutions of the general multivalued variational inequalities
under some mild conditions. Several special cases are also discussed.