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This paper investigates the nonlinear functional-integral equation comprising of Hadamard fractional operator. Using the concept of measure of non compactness, existence of solutions in Banach algebra has been studied under certain relevant assumptions in conjunction with fixed point theory. Finally an example has been considered to substantiate the validity of the result. [ABSTRACT FROM AUTHOR]

In this paper we discuss and present a least square and a QR point of view to reproducing kernel methods to approximate solutions to some linear and nonlinear functional equations. The procedure we discuss here may includes ordinary, partial differential, and integral equations. We also give new proofs to some known results on this subject. The most interesting contribution is that the proposed algorithm may work even when we know a reproducing kernel but nothing more about the associated reproducing kernel Hilbert space, including the inner product structure. [ABSTRACT FROM AUTHOR]

Abstract Two-phase composites with n equal non-overlapping inclusions randomly embedded in the matrix are investigated. It is considered the case when the inclusions are bounded by some Lyapunov's boundary curve. The problem is reduced to a vector-matrix problem of dimension n for one inclusion. The generalized alternating method of Schwarz applied to the vector-matrix problem is decomposed onto n scalar problems for one inclusion which are solved numerically by the method of integral equations for any smooth shape of the inclusions. A symbolic computation method is developed to solve the same problem by means of conformal mapping and functional equations. As a purpose, the effective conductivity of such models is exactly expressed through all geometrical and mechanical properties of its components. [ABSTRACT FROM AUTHOR]

Abstract We consider existence of positive solutions for a difference equation with continuous time, variable coefficients and delays x (t + 1) − x (t) + ∑ k = 1 m a k (t) x (h k (t)) = 0 , a k (t) ≥ 0 , h k (t) ≤ t , t ≥ 0 , k = 1 , ... , m. We prove that for a fixed h (t) ≢ t , a positive solution may exist for a k exceeding any prescribed M > 0, as well as for constant positive a k with h k (t) ≤ t − n , where n ∈ N is arbitrary and fixed. The point is that for equations with continuous time, non-existence of positive solutions with inf x (t) > 0 on any bounded interval should be considered rather than oscillation. Sufficient conditions when such solutions exist or do not exist are obtained. We also present an analogue of the Grönwall–Bellman inequality for equations with continuous time, and examine the question when the equation has no positive non-increasing solutions. Counterexamples illustrate the role of variable delays. [ABSTRACT FROM AUTHOR]

Abstract Recently, Njeru and Guo presented an accelerated SOR-like (ASOR) method for solving the large and sparse augmented systems. In this paper, we establish an accelerated symmetric SOR-like (ASSOR) method, which is an extension of the ASOR method. Furthermore, the convergence properties of the ASSOR method for augmented systems are studied under suitable restrictions, and the functional equation between the iteration parameters and the eigenvalues of the relevant iteration matrix is established in detail. Finally, numerical examples show that the ASSOR is an efficient iteration method. [ABSTRACT FROM AUTHOR]

In this work, we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein–Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart. [ABSTRACT FROM AUTHOR]

Classical solutions of nonlinear second-order partial differential functional equations of parabolic type with the Robin condition are approximated in the paper by solutions of associated boundedness-preserving implicit difference functional equations. It is proved that the discrete solutions uniquely exist, they are uniformly bounded with respect to meshes and the numerical method is convergent and stable. We also find the error estimate and its asymptotic behavior. The properties of some auxiliary nonlinear discrete recurrent equations are showed. The proofs are based on the comparison technique and the Banach fixed-point theorem. [ABSTRACT FROM AUTHOR]

In this work we introduce and analyse a stochastic functional equation, which contains both delayed and advanced arguments. This equation results from adding a stochastic term to the discrete FitzHugh–Nagumo equation which arises in mathematical models of nerve conduction. A numerical method is introduced to compute approximate solutions and some numerical experiments are carried out to investigate their dynamical behaviour and compare them with the solutions of the corresponding deterministic equation. [ABSTRACT FROM AUTHOR]

In this work, we prove the existence of integrable solutions for the following generalized mixed-type nonlinear functional integral equation x ( t ) = g ( t , ( T x ) ( t ) ) + f ( t , ∫ 0 t k ( t , s ) u ( t , s , ( Q x ) ( s ) ) d s ) , t ∈ [ 0 , ∞ ) . Our result is established by means of a Krasnosel’skii type fixed point theorem proved by Taoudi (2009). In the last section we give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

This paper describes a numerical verification of solutions for infinite-dimensional functional equations based on residual forms and Newton-like iteration. The method is based upon a verification method previously developed by the authors. Several computer-assisted proofs for differential equations, including nonlinear partial differential equations, are presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate comparison criteria for third order nonlinear dynamic equations with mixed nonlinearities on time scales. Our results are essentially new. Some applications illustrating the importance of our results are included and these applications solve a problem posed in [2, Remark 3.3]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the definition of generalized coupled fixed point in the space of the bounded functions on a set S and we prove a result about the existence and uniqueness of such points. As an application of our result, we study the problem of existence and uniqueness of solutions for a class of systems of functional equations which appears in dynamic programming. [ABSTRACT FROM AUTHOR]

The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified integral equation method to regularize the nonlinear problem with globally and locally Lipschitz source terms. Convergence estimates are established under priori assumptions on exact solution. A numerical test is provided to illustrate that the proposed method is feasible and effective. These results extend some earlier works on a Cauchy problem for elliptic equations [ABSTRACT FROM AUTHOR]

In this article we approximate the solutions of the nonlinear Fredholm integral equations of the second kind, by the method based on using the properties of RH wavelets and matrix operator. Also, the Banach fixed point theorem guarantees the convergence of the method. Also we get an upper bound for the error. Furthermore, the order of convergence is analyzed. The algorithm to compute the solutions and some numerical examples are also illustrated. The numerical results obtained by our method have been compared with other methods. [ABSTRACT FROM AUTHOR]

We consider Fredholm–Hammerstein integral equations of the second kind over a rectangular region in plane. As in Kumar and Sloan (1987) [5], we reformulate it into an equivalent integral equation. For the alternative equation, we triangulate the rectangular domain and on each triangle use a collocation method based on constant spline approximation. We discuss the convergence of the approximate solutions and conclude the paper with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we unify the system of functional equations defining a multi-additive-quadratic mapping to obtain a single equation. We also prove, using the fixed point method, the generalized Hyers–Ulam stability of this equation thus generalizing some known results. [ABSTRACT FROM AUTHOR]

We construct generating functions for beta-type rational functions and the beta polynomials. By using these generating functions, we derive a collection of functional equations and PDEs. By using these functional equations and PDEs, we give derivative formulas, a recurrence relation and a variety of identities related to these polynomials. We also give a relation between the beta-type rational functions and the Bernstein basis functions. Integrating these identities and relations, we derive various combinatorial sums involving binomial coefficients, some old and some new, for the beta-type rational functions and the Bernstein basis functions. Finally, by applying the Laplace transform to these generating functions, we obtain two series representations for the beta-type rational functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the functional equation, f ( x + y ) - f ( x ) f ( y ) = d sin x sin y . Some generalizations of the above functional equation are also considered. [ABSTRACT FROM AUTHOR]

Abstract: We prove some results on stability and non-stability of the linear functional equation of the first order, in single variable. Moreover we suggest some possible definitions of stability and non-stability. [Copyright &y& Elsevier]

Abstract: The concept of invariance factors to obtain first integrals for difference equations was presented by Van Horssen (2002) [6]. It was shown that all invariance factors have to satisfy a functional equation. One of the main difficulties in finding first integrals for a system of first order difference equations is solving the aforementioned functional equation. In this paper, we consider a functional equation which is related to a system of two first order, linear difference equations. Linear transformations and an adapted version of the method of separation of variables will be used to construct solutions of this functional equation. The constructed solutions of this functional equation do not seem to be available in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the generalized Hyers–Ulam stability of the functional equation for integer values of n such that , where f is a function from a normed space X to a Banach space Y. The solutions of the equation are called additive–quadratic mappings. [Copyright &y& Elsevier]

Abstract: In this paper, we prove some common fixed point theorems for two pairs of non-self weakly compatible mappings enjoying common limit range property, besides satisfying a generalized φ-weak contractive condition in symmetric spaces. We furnish some illustrative examples to highlight the realized improvements in our results over the corresponding relevant results of the existing literature. We extend our main result to four finite families of mappings in symmetric spaces using the notion of pairwise commuting mappings. Finally, we utilize our results to discuss the existence and uniqueness of solutions of certain system of functional equations arising in dynamic programming. [Copyright &y& Elsevier]

• We discuss the existence of solution of stochastic functional equation with the help of condensing operators and measure of noncompactness. • This functional integral equations contains several equations in this area. • This type of equations help us to represent many real life nonlinear problems. • We provide three examples to illustrate our main results. In this paper, the existence of the solution of a class of nonlinear stochastic functional integral equations is investigated. Such that the concept of measures of noncompactness, and Petryshyn's fixed point theorem in Banach space has been used. In addition to proving the theorems of the existence of the solution, by solving some examples, it is shown that the method is efficient. We also give an example that satisfies the conditions of our main theorem, while the conditions described in some other papers do not. [ABSTRACT FROM AUTHOR]

Abstract: The dynamical behavior of the traveling wave front solutions of the delayed Fisher-KPP equation is studied by employing the theory of functional differential equation. It is found that, when the delay term τ comes through a threshold, one equilibrium point is always unstable and there exists a periodic solution near the other equilibrium point in the traveling wave equation. By using the theory of center manifold and normal form, the stability of the bifurcating periodic solution, which corresponds to the periodic traveling wave solution of the original equation, is obtained. Finally, some numerical simulations are also given to illustrate our results of theoretical analysis. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper, is to present a spectral collocation method to approximate the solution of Fredholm integral equations of the second kind on the half-line. In this approach, interpolation operator based on scaled Laguerre functions is proposed. Also, convergence theorems are established under suitable assumptions. Finally, two numerical examples are presented and commented. [Copyright &y& Elsevier]

Abstract: This paper derives the functional forms of multiattribute utility functions that lead to a maximum of one-switch change in preferences between any two uncertain and multi-period cash flows as the decision maker’s wealth increases through constant annuity payments. We derive the general and continuous non-constant solutions of the corresponding functional equations. [Copyright &y& Elsevier]

Abstract: In this paper we present a method that allows to study the Hyers–Ulam stability of some systems of functional equations connected with the Cauchy, Jensen and quadratic equations. In particular we generalize and extend some already known results. [Copyright &y& Elsevier]

Abstract: In this paper, a new projection neural network with two time delays is proposed for solving quadratic programming problems subject to linear constraints. And we give the existence of the continuous solution. By the theory of functional differential equation, the proposed neural network is proved to be globally exponentially stable under some conditions. Simulation results with some applications show the performance and characteristic of the proposed neural network. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a new class of abstract integral equations which enables us to study in a unified manner several different types of differential equations. [Copyright &y& Elsevier]

Abstract: Some sufficient conditions which guarantee the existence of a bounded solution together with its first derivative on the whole real line of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function, are given. [Copyright &y& Elsevier]

Abstract: In this paper, we obtained Suzuki type fixed point results for a generalized multi-valued mapping on a set equipped with two b-metrics. As a consequence, existence and uniqueness of solution of functional equation arising in dynamical programming is also derived. [Copyright &y& Elsevier]

Abstract: Some conditions which guarantee the local existence of Lipschitz-continuous solutions of some systems of nonlinear functional equations with complicated deviations dependent of unknown functions are presented. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to establish fixed point theorems for operators in the Fréchet space of continuous functions on the real half-axis. In our considerations we apply the technique of measures of noncompactness in conjunction with the Tikchonov fixed point theorem. The obtained results are applied in the proof of the solvability of a nonlinear functional integral equation with the initial value. Moreover, we show that solutions of that equation are uniformly globally asymptotically attractive. The results presented in the paper allow to improve existence theorems for integral and functional equations obtained earlier in many research papers. [Copyright &y& Elsevier]

Abstract: Initial boundary value problems of Dirichlet type for first order partial functional differential equations are considered. Explicit difference schemes and implicit difference methods are investigated. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are constructed. Numerical examples are presented. [Copyright &y& Elsevier]

Abstract: A new method for solving the Abel equation of the second kind using functional transformations of the form is presented. It is important to say that the method depends on the possibility of finding . The method is illustrated for the cases and . Finally, the case , where the necessity and difficulty of finding becomes explicit, is analyzed. [Copyright &y& Elsevier]

Abstract: The existence and uniqueness of bounded continuous solutions on the real line of a general class of nonlinear functional equations with quite complicated deviations dependent of unknown functions are established in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we study solvability of two functional equations arising in dynamic programming of multistage decision processes. By using Boyd and Wong fixed point theorem, some existence and uniqueness theorems of solutions and iterative approximation for solving these class of functional equations are established. The results presented here extend, improve and unify the corresponding results due to Bellman, Bhakta and Mitra, Bhakta and Choudhary, Liu and Kang, Liu et al., Jiang et al. and others. We also discuss some illustrative examples to highlight the realized improvements. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence of solutions for first order nonlinear impulsive functional integro-differential equations with integral boundary value conditions. We firstly build a new comparison theorem. Then by utilizing the monotone iterative technique and the method of lower and upper solutions, we obtain the existence of extremal solutions or quasi-solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we employ fixed point theorem and functional equation theory to study the existence of positive periodic solutions of the delay differential equationMoreover, new explicit conditions for the related population growth models are also given. [Copyright &y& Elsevier]

Abstract: The Barnes’G-function G(x)=1/Γ 2, satisfies the functional equation logG(x +1)−logG(x)=logΓ(x). We complement W. Krull’s work in Bemerkungen zur Differenzengleichung g(x +1)− g(x)= φ(x), Math. Nachrichten 1 (1948), 365–376 with additional results that yield a different characterization of the function G, new expansions and sharp bounds for G on x >0 in terms of Gamma and Digamma functions, a new expansion for the Gamma function and summation formulae with Polygamma functions. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of positive solutions for the nonlinear Captuo fractional order functional differential equationwhere is the Captuo fractional order derivative, subject to the boundary conditionswe obtain the existence results of positive solutions by using some fixed point theorems. [Copyright &y& Elsevier]

Abstract: It is shown that the three nonlinear dynamic Euler ordinary differential equations (ODEs), concerning the motion of a rigid body free to rotate about a fixed point, are reduced, by means of a subsidiary function which is to be determined, to three Abel equations of the second kind of the normal form. Based on a recently developed mathematical construction concerning exact analytic solutions of the Abel nonlinear ODEs of the second kind, we perform a new mathematical solution for the classical dynamic Euler nonlinear ODEs. [Copyright &y& Elsevier]

Abstract: This paper is concern analytic solutions of an iterative functional equation of the formByconstructing a convergent power series solution of an auxiliary equationanalytic solutions of the original equation are obtained. We discuss not only these α appeared in the auxiliary equation at the hyperbolic case 0<∣α∣≠1 and resonance, i.e., at a root of the unity, but also those α near resonance (i.e., near a root of the unity) under Brjuno condition. [Copyright &y& Elsevier]

Abstract: We look for nonlinear systems of difference equations which have power functions as solutions and we find a connection to the theory of invariant curves and to self-reversed polynomials. In particular, we construct algebraic solutions of a special equation of invariant curves. [Copyright &y& Elsevier]

Abstract: In this paper, a Volterra model with mutual interference and Beddington–DeAngelis functional response is investigated, some sufficient conditions which guarantee the existence and global attractivity of positive periodic solution for the system are obtained. Furthermore, our results improve the main results of paper . [Copyright &y& Elsevier]

Abstract: A method for the study of systems with renewable resources is proposed. The individual and the group parameters are separated and a discretization of time is carried out. We obtain equilibrium proportions which are functional equations with shift. A cyclic model and an open model are considered. Conditions for the existence and uniqueness of the solution are formulated for the cyclic model. For the open model, the system’s evolution is analyzed. [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence of integrable solutions for a generalized mixed type operator equation, which contains many key integral and functional equations appearing frequently in Mathematical literature. Our main tool is a Krasnosel’skii type fixed point theorem recently proved by Latrach and Taoudi, the first author. An existence theory for a class of nonlinear transport equations is also developed. [Copyright &y& Elsevier]

Abstract: In this paper solutions of conjugacy equation for a strictly decreasing continuous given function f and a continuous given function g (maybe non-monotonic) are constructed by piecewise defining. We determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g. Finally, the recursive algorithm is implemented in MATLAB software and two examples are respectively presented for a non-monotonic solution and a continuously differentiable one. [Copyright &y& Elsevier]

Abstract: In this paper, necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of a class of nonlinear higher-order neutral dynamic equations are stated related to various ranges of the coefficient associated with the neutral part, in one of which the coefficient is allowed to oscillate and not tend to zero asymptotically. The new results improve and generalize the known results stated for difference equations and differential equations, and answer some problems left open for the case for arbitrary time scales. Some certain examples, which the results in the literature fail to deliver an answer to, are given to illustrate the applicability and significance of the new results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a Volterra model with mutual interference and Beddington–DeAngelis functional response. By applying the comparison theorem of the differential equation and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained. [Copyright &y& Elsevier]