Sum-difference Equations Research Articles

Asymptotic behavior of solutions of sum-difference equations

In this study, we present an investigation of the asymptotic behavior of solutions of sum-difference equations. Based on some mathematical inequalities, we have obtained our results. The obtained results can apply to some fractional type difference equations as well.

Existence results of sequential fractional Caputo sum-difference boundary value problem

<abstract><p>In this article, we study the existence and uniqueness results for a sequential nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and Schaefer's fixed point theorem. Furthermore, we also show the existence of a positive solution. Our problem contains different orders and four fractional difference operators. Finally, we present an example to display the importance of these results.</p></abstract>

Sum-difference equation for analytic functions generated by triangle and its applications

Sum-difference equation for analytic functions generated by triangle and its applications

Computational approach of dynamic integral inequalities with applications to timescale calculus

Based on some known results and simple technique, we emphasize in this article, certain nonlinear dynamic integral inequalities in one variable on timescales. Part of the novelty herein not only unifies and extends some integral inequalities related to different cases of positive constants but also explores the explicit bounds for discontinuous functions on timescales. We contribute to the ongoing research by providing mathematical results that can be used as necessary tools in the theory of certain classes of differential, integral, finite difference and sum–difference equations on timescales. The consequences of the computational experiments show that the proposed strategy can produce adequate and reliable results. Examples are also discussed to demonstrate the importance of the tests.

On a coupled system of fractional sum-difference equations with p-Laplacian operator

In this paper, we propose a nonlocal fractional sum-difference boundary value problem for a coupled system of fractional sum-difference equations with p-Laplacian operator. The problem contains both Riemann–Liouville and Caputo fractional difference with five fractional differences and four fractional sums. The existence and uniqueness result of the problem is studied by using the Banach fixed point theorem.

Sum-Difference Equation for Analytic Functions, Generated by a Hexagon, and Its Applications

Let D be a hexagon having a pair of parallel sides, equal in length, and L be a half of its boundary. We study solvability of a seven-element sum-difference equation equation in the class of functions holomorphic outside L and vanishing at infinity. Their boundary values satisfy the Holder condition on every compact not containing the nodes. At the nodes they have at most logarithmic singularities. To regularize the equation, on the boundary of the hexagon we introduce a Carleman shift having jump discontinuity at the vertices. We seek a solution in the form of Cauchy type integral over L with an unknown density. We find conditions providing equivalence of such regularization. We also consider a particular case for which the corresponding Fredholm equation is solvable. We give some applications to the moment problem for entire functions of exponential type (EFETs). In particular, we construct a system of EFETs biorthogonal, with a piece-wise quasipolynomial weight, to a system of three degree functions on three rays. For such EFETs, the conjugate diagram is an octagon. We note that various generalizations of our investigations are possible due to the large arbitrariness in the choice of set L.

Existence and Stability Analysis for Fractional Impulsive Caputo Difference-Sum Equations with Periodic Boundary Condition

In this paper, by using the Banach contraction principle and the Schauder’s fixed point theorem, we investigate existence results for a fractional impulsive sum-difference equations with periodic boundary conditions. Moreover, we also establish different kinds of Ulam stability for this problem. An example is also constructed to demonstrate the importance of these results.

On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations

In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example.

On Separate Fractional Sum-Difference Equations with n-Point Fractional Sum-Difference Boundary Conditions via Arbitrary Different Fractional Orders

In this article, we study the existence and uniqueness results for a separate nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and the Schauder’s fixed point theorem. Our problem contains two nonlinear functions involving fractional difference and fractional sum. Moreover, our problem contains different orders in n + 1 fractional differences and m + 1 fractional sums. Finally, we present an illustrative example.

On generalization of Pachpatte type discrete inequalities and applications

In this paper, we generalize some existing discrete Pachpatte type inequalities to more general situations. These inequalities in turn can be applied to study various qualitative as well as quantitative properties of solutions of various finite difference and sum-difference equations and its variants.

On nonlocal boundary value problems for hybrid fractional sum-difference equations involving different orders

On nonlocal boundary value problems for hybrid fractional sum-difference equations involving different orders

Existence Results of Initial Value Problems for Hybrid Fractional Sum-Difference Equations

We consider a hybrid fractional sum-difference initial value problem and a hybrid fractional sequential sum-difference initial value problem. The existence results of these two problems are proved by using the hybrid fixed point theorem for three operators in a Banach algebra and the generalized Krasnoselskii’s fixed point theorem, respectively.

Some new generalized Volterra-Fredholm type discrete fractional sum inequalities and their applications

In this paper, we present some new Volterra-Fredholm-type discrete fractional sum inequalities. These inequalities can be used as handy and powerful tools in the study of certain fractional sum-difference equations. Some applications are also presented to illustrate the usefulness of our results.

Solvability for a discrete fractional mixed type sum-difference equation boundary value problem in a Banach space

In this paper, by means of Darbo’s fixed point theorem, we establish the existence of solutions to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Additionally, as an application, we give an example to demonstrate the main result.

Sum-difference equation for analytic functions generated by pentagon and its application

For analytic functions, we study poly-element linear functional equation generated by a pentagon. Such equations are connected with the moment problem for entire functions of exponential type.

Subexponential Solutions of Linear Volterra Difference Equations

AbstractWe study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.

Existence and Uniqueness of Solutions for a Discrete Fractional Mixed Type Sum-Difference Equation Boundary Value Problem

By means of Schauder’s fixed point theorem and contraction mapping principle, we establish the existence and uniqueness of solutions to a boundary value problem for a discrete fractional mixed type sum-difference equation with the nonlinear term dependent on a fractional difference of lower order. Moreover, a suitable choice of a Banach space allows the solutions to be unbounded and two representative examples are presented to illustrate the effectiveness of the main results.

On Nonlinear Fractional Sum-Difference Equations via Fractional Sum Boundary Conditions Involving Different Orders

We study existence and uniqueness results for Caputo fractional sum-difference equations with fractional sum boundary value conditions, by using the Banach contraction principle and Schaefer’s fixed point theorem. Our problem contains different numbers of order in fractional difference and fractional sums. Finally, we present some examples to show the importance of these results.

Nonlinear discrete fractional mixed type sum-difference equation boundary value problems in Banach spaces

This paper is concerned with the existence of a unique solution to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Under certain suitable nonlinear growth conditions imposed on the nonlinear term, the existence and uniqueness result is established by using the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main result. MSC:26A33, 39A05, 39A10, 39A12.

Some new generalized Volterra–Fredholm type finite difference inequalities involving four iterated sums

In this paper, based on some new generalized Volterra–Fredholm type finite difference inequalities involving four iterated sums, we derive some new explicit bounds for the unknown functions concerned. These inequalities are of more general forms than some existing results in the literature, and are generalization of the main results of Ma [J. Comput. Appl. Math. 233 (2010) 2170–2180]. As application of the established results, we present a Volterra–Fredholm sum-difference equation, and research the qualitative as well as quantitative properties for it.