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The main objective of this paper is to investigate the existence, uniqueness, and Ulam–Hyers stability of positive solutions for fractional integro-differential boundary values problem. Uniqueness result is obtained by using the Banach principle. For obtaining two positive solutions, we apply another fixed point criterion due to Avery–Anderson–Henderson on cones by establishing some inequalities. An illustrative example is presented to indicate the validity of the obtained results. The results are new and provide a generalization to some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of a solution for a class of ψ-Hilfer implicit fractional integro-differential equations with mixed nonlocal conditions. The arguments are based on Banach's, Schaefer's, and Krasnosellskii's fixed point theorems. Further, applying the techniques of nonlinear functional analysis, we establish various kinds of the Ulam stability results for the analyzed problem, that is, the Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we provide some examples to illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Liouville–Caputo type fractional integro-differential equations with multi-point and sub-strip boundary conditions. The fractional integro-differential equations involve Caputo derivative operators of different orders and finitely many Riemann–Liouville fractional integral and non-integral type nonlinearities. The boundary conditions at the terminal position t = 1 involve sub-strips and multi-point contributions. The Banach fixed point theorem and the Leray–Schauder alternative are used to establish our results. The obtained results are illustrated with the aid of examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dissipativity of Runge-Kutta methods for a class of nonlinear functional-integro-differential equations (FIDEs). The dissipativity results of Runge-Kutta methods for the FIDEs are given. It is shown under a suitable condition that an algebraically stable Runge-Kutta method is dissipative when applied to the FIDEs. Numerical examples are given to illustrate the correctness of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we show a connection between Levin-Nohel integro-differential equations and ordinary functional differential equations. Based on this connection, we obtain several new conditions for the stability of the solution, including the famous 3/2 stability criterion. [ABSTRACT FROM AUTHOR]

In this paper, by means of the Krasnoselskii fixed point theorem, the existence of solutions for a boundary value problem of nonlinear sequential fractional integro-differential equations are investigated. Two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (J. Comput. Appl. Math. 290:633–640, 2015). [ABSTRACT FROM AUTHOR]

New nonlinear integral inequalities (NII) are presented in this paper. Based on mathematical analysis technique, several estimation results are obtained, which not only complement the aforementioned results, but also generalize the inequalities to the more general nonlinearities. As an application, they can be employed to estimate the bound on the solutions of power integro-differential equations (IDE). [ABSTRACT FROM AUTHOR]

In the present paper, by using the variational and topological methods, we obtain a multiplicity and bifurcation result for the following fractional problems involving critical nonlinearities and a lower order perturbation: − L K v = μ v + | v | 2 s ∗ − 2 v + g (x , v) in Ω , v = 0 in R N ∖ Ω , where Ω is an open and bounded domain with Lipschitz boundary, N > 2 s , with s ∈ (0 , 1) , g is a lower order perturbation of the critical power | v | 2 s ∗ − 2 v and μ is a positive real parameter, 2 s ∗ = 2 N N − 2 s is the fractional critical Sobolev exponent, while L K is an integro-differential operator. Precisely, we show that the number of nontrivial solutions for this equation under suitable assumptions is at least twice the multiplicity of the eigenvalue. Our conclusions improve the related results in some respects. [ABSTRACT FROM AUTHOR]

It is important that we increase our ability for studying of complicate fractional integro-differential equation. In this paper, we investigates the existence of solutions for a pointwise defined multi-singular fractional differential equation under some integral boundary conditions. We provide an example to illustrate our main result. [ABSTRACT FROM AUTHOR]

This paper is concerned with the semilinear fractional integrodifferential system with Riemann–Liouville fractional derivative. Firstly, we introduce the suitable C 1 − α -solution to Riemann–Liouville fractional integrodifferential equations in the new frame of fractional resolvents. Some properties of fractional resolvents are given. Then we discuss the sufficient conditions for the existence of solutions without the Lipschitz assumptions to nonlinear item. Finally, an example on fractional partial differential equations is presented to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence of solutions for a class of functional integro-differential equations with nonlinear boundary conditions. New comparison principles are obtained. By using the comparison principles and quasilinearization method, we present two monotone iterative sequences uniformly and monotonically converging to the unique solution with rate of order 2. Meanwhile, an example is given to demonstrate applications of the result reported. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of mild solutions for nonlocal stochastic integro-differential equations driven by fractional Brownian motions with Hurst parameter $H>\frac{1}{2}$ in a Hibert space. Sufficient conditions for the existence of mild solutions are derived by means of the Leray-Schauder nonlinear alternative. A special case of this result is given and an example is provided to illustrate the effectiveness of the proposed result. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of weighted pseudo almost periodic mild solutions for the class of integro-differential equations in the form $u'(t)=Au(t)+\alpha\int_{-\infty}^{t}e^{-\beta(t-s)}Au(s) \,ds+f(t,u(t)) $ where $f(\cdot,u(\cdot))$ is a Stepanov-like weighted pseudo almost periodic function and A generates an immediately norm continuous $C_{0}$-semigroup on a Banach space X. Also, we give a short proof to show that the vector-valued space of Stepanov-like weighted pseudo almost periodic functions is a Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of mild solutions to impulsive integrodifferential evolution equations in Banach spaces. Based on a measure of noncompactness and important properties of semicompact sets, new existence results are obtained. Here the evolution system is only supposed to be strongly continuous, without any compact or equicontinuous assumptions. Some applications are given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

This paper concerns the dissipativity of nonlinear Volterra functional differential equations (VFDEs) in Banach space and their numerical discretization. We derive sufficient conditions for the dissipativity of nonlinear VFDEs. The general results provide a unified theoretical treatment for dissipativity analysis to ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type appearing in practice. Then the dissipativity property of the backward Euler method for VFDEs is investigated. It is shown that the method can inherit the dissipativity of the underlying system. The close relationship between the absorbing set of the numerically discrete system generated by the backward Euler method and that of the underlying system is revealed. [ABSTRACT FROM AUTHOR]

This paper discusses different types of Ulam stability of first-order nonlinear Volterra delay integro-differential equations with impulses. Such types of equations allow the presence of two kinds of memory effects represented by the delay and the kernel of the used fractional integral operator. Our analysis is based on Pachpatte's inequality and the fixed point approach represented by the Picard operators. Applications are provided to illustrate the stability results obtained in the case of a finite interval. [ABSTRACT FROM AUTHOR]

In this paper, by using some fixed point theorems and the measure of noncompactness, we discuss the existence of solutions for a boundary value problem of impulsive integrodifferential equations of fractional order $\alpha\in(1,2]$ . Our results improve and generalize some known results in (Zhou and Chu in Commun. Nonlinear Sci. Numer. Simul. 17:1142-1148, 2012; Bai et al. in Bound. Value Probl. 2016:63, 2016). Finally, an example is given to illustrate that our result is valuable. [ABSTRACT FROM AUTHOR]

An integro-differential model of competition is studied in this paper. We show that the system always admits a unique globally attractive positive equilibrium. Our result complements and supplements the main result in the literature. [ABSTRACT FROM AUTHOR]

By formulating a new contraction mapping on a product space, the authors originally employed Banach fixed point theorem to derive the LMI-based robust exponential stability criterion for impulsive BAM neural networks with distributed delays and uncertain parameters. Numerical example illuminates that the new criterion is not worse than the existing results derived by Lyapunov functional method. Hence both the methods and the results of this paper are really novel to a certain extent. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of solutions for nonlinear fractional differential equations of Volterra type with nonlocal fractional integro-differential boundary conditions on an infinite interval. The results are obtained by using the Altman fixed point theorem. An example is presented in order to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of the unique solution for a Hadamard fractional integral boundary value problem of a Hadamard fractional integro-differential equation with the monotone iterative technique. Two examples to illustrate our result are given. [ABSTRACT FROM AUTHOR]

In the present paper, we first investigate the construction of compact sets of $\mathit{DC}^{n}[J,E]$ and $C^{n}_{0}[J,E]$, and then we introduce new measures of noncompactness on these spaces. In addition, as an application, we discuss the existence of solutions of initial value problems for nth-order nonlinear integro-differential equations of mixed type on an infinite interval in Banach spaces. We will also state an interesting example which shows that our results can apply for solving infinite systems of integro-differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability of partial neutral stochastic functional integro-differential inclusions with infinite delay and impulsive effects in Hilbert spaces is considered. By using Hölder's inequality, stochastic analysis and fixed point strategy with the properties of analytic resolvent operator, a new set of sufficient conditions is formulated, which guarantees the approximate controllability of the nonlinear impulsive stochastic system. The results are obtained under the assumption that the associated linear system is approximately controllable. An example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability of fractional impulsive partial neutral integrodifferential inclusions with infinite delay in a Hilbert space is studied. By using the nonlinear alternative of Leray-Schauder type for multivalued maps due to O'Regan and properties of the α-resolvent operator combined with approximation techniques, we derive a new set of sufficient conditions for the approximate controllability of fractional impulsive evolution system under the assumption that the corresponding linear system is approximately controllable. An example is provided to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

This paper investigates the approximate controllability and optimal controls of fractional dynamical systems of order $1< q<2$ in Banach spaces. We research a class of fractional dynamical systems governed by fractional integrodifferential equations with nonlocal initial conditions. Using the Krasnosel'skii fixed point theorem and the Schauder fixed point theorem, the approximate controllability results are obtained under two cases of the nonlinear term. We also present the existence results of optimal pairs of the corresponding fractional control systems with a Bolza cost function. Finally, an application is given to illustrate the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and the uniqueness of solutions for coupled and uncoupled systems of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions. The existence and the uniqueness of the solutions are established by using the Banach contraction principle, while the existence of solutions is derived by applying Leray-Schauder's alternative. Examples illustrating our results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential inclusions with infinite delays in Hilbert spaces. The results are obtained by using the fixed-point theorem for multi-valued operators due to Dhage. An example is provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

In this research article, we turn to studying the existence and different types of stability such as generalized Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability of solutions for a new modeling of a boundary value problem equipped with the fractional differential equation which contains the multi-order generalized Caputo type derivatives furnished with four-point mixed generalized Riemann–Liouville type integro-derivative conditions. At the end of the current paper, we formulate two illustrative examples to confirm the correctness of theoretical findings from computational aspects. [ABSTRACT FROM AUTHOR]

In the present paper, starting with the Black–Scholes equations, whose solutions are the values of European options, we describe the exponential jump-diffusion model of Levy process type. Here, a jump-diffusion model for a single-asset market is considered. Under this assumption the value of a European contingency claim satisfies a general "partial integro-differential equation" (PIDE). With a combined compact difference (CCD) scheme for the spatial discretization, a high-order method is proposed for solving exponential jump-diffusion models. The method is sixth-order accurate in space and second-order accurate in time. A known analytical solution to the model is used to evaluate the performance of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we prove existence and uniqueness results for a mixed initial value problem which contains Hadamard derivative and Riemann-Liouville fractional integrals by using standard fixed point theorems. Examples illustrating the results are also presented. [ABSTRACT FROM AUTHOR]

We study the extremal solutions of a class of fractional integro-differential equation with integral conditions on infinite intervals involving the p-Laplacian operator. By means of the monotone iterative technique and combining with suitable conditions, the existence of the maximal and minimal solutions to the fractional differential equation is obtained. In addition, we establish iterative schemes for approximating the solutions, which start from the known simple linear functions. Finally, an example is given to confirm our main results. [ABSTRACT FROM AUTHOR]

We deal with a class of coupled systems of neutral fractional integro-differential equations with infinite delay in a Banach space in this paper. Based on the Banach contraction principle and Krasnoselskii's fixed point theorem, some new sufficient criteria are established to guarantee the existence of mild solutions. As an application, an example is given to illustrate the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we present sufficient criteria ensuring the existence and uniqueness of solutions for nonlinear impulsive multi-order Caputo-type generalized fractional differential equations supplemented with nonlocal integro-initial value conditions involving generalized fractional integrals. Extremal solutions for the given problem are also discussed. The main tools of our study include Krasnoselskii's fixed point theorem, Banach contraction mapping principle and monotone iterative technique. Examples are constructed for illustrating the obtained results. [ABSTRACT FROM AUTHOR]

Volterra integro-differential equations arise in the modeling of natural systems where the past influence the present and future, for example pollution, population growth, mechanical systems and financial market. Furthermore, as many real-world phenomena are subject to perturbations or random noise, it is natural to move from deterministic models to stochastic models. Generally exact solutions of such models are not available and numerical methods are used to obtain the approximate solutions. Therefore the efficiency and long-term behavior of approximate solutions for these systems is an important area of investigation. This paper presents a new numerical approach for the approximate solution of stochastic Volterra integro-differential (SVID) equations based on the Legendre-spectral collocation method. In order to fully use the properties of orthogonal polynomials, we use some function and a variable transformation to change the given SVID equation into a new equation, which is defined on the standard interval [ − 1 , 1 ] . For the evaluation of the integral term efficiently a Legendre–Gauss quadrature formula will be used. A rigorous error analysis of the proposed scheme will be provided under the assumption that the solution of the given SVID is sufficiently smooth. For the illustration of our theoretical results a number of numerical experiments will be performed. [ABSTRACT FROM AUTHOR]

One of the interesting fractional integro-differential equations is the three step crisis equation which has been reviewed recently. In this paper, we investigate the existence of solutions for a three step crisis fractional integro-differential equation under some boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, by employing fixed point theory, we investigate the existence and uniqueness of solutions for a class of nonlinear fractional integro-differential equations on semi-infinite domains in a Banach space. [ABSTRACT FROM AUTHOR]

A new approximate technique is introduced to find a solution of FVFIDE with mixed boundary conditions. This paper started from the meaning of Caputo fractional differential operator. The fractional derivatives are replaced by the Caputo operator, and the solution is demonstrated by the hybrid orthonormal Bernstein and block-pulse functions wavelet method (HOBW). We demonstrate the convergence analysis for this technique to emphasize its reliability. The applicability of the HOBW is demonstrated using three examples. The approximate results of this technique are compared with the correct solutions, which shows that this technique has approval with the correct solutions to the problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a new boundary value problem of arbitrary order fractional differential equations equipped with new integro-multipoint boundary conditions. Existence and uniqueness results for the given problem are obtained by applying the standard tools of fixed point theory. We also extend the problem at hand to its inclusions case and prove an existence result for it by applying a fixed point theorem due to Bohnenblust and Karlin. [ABSTRACT FROM AUTHOR]

In this article, we first create a new comparison principle for a nonlinear impulsive boundary problem involving different deviating arguments. Then we employ the new result and iterative method to study the existence of the max-minimal solution of a second-order impulsive functional integro-differential equation. The results achieved in this paper are more general and complement many previously known results. [ABSTRACT FROM AUTHOR]

In this paper, a new set of functions called fractional-order Euler functions (FEFs) is constructed to obtain the solution of fractional integro-differential equations. The properties of the fractional-order Euler functions are utilized to construct the operational matrix of fractional integration. By using the matrix and the functions approximation, the fractional integro-differential equations are reduced to systems of algebraic equations. The convergence analysis of fractional-order Euler functions approximation is given. Illustrative examples are included to demonstrate the high precision and good performance of the new scheme. [ABSTRACT FROM AUTHOR]

In this paper, we develop the existence theory for nonlinear second-order ordinary differential equations equipped with new kinds of nonlocal non-separated type integral multi-point boundary conditions on an arbitrary domain. Existence results are proved with the aid of fixed point theorems due to Schaefer, Krasnoselskii, and Leray-Schauder, while the uniqueness of solutions for the given problem is established by means of contraction mapping principle. Examples are constructed for the illustration of the obtained results. Ulam-stability is also discussed for the given problem. A variant of the problem involving different boundary data is also discussed. Finally, we introduce an associated boundary value problem involving integro-differential equations and discuss the uniqueness of its solutions. [ABSTRACT FROM AUTHOR]

In this work, we solve the system of integro-differential equations (in terms of Caputo–Fabrizio calculus) using the concepts of the best proximity pair (point) and measure of noncompactness. We first introduce the concept of cyclic (noncyclic) Θ-condensing operator for a pair of sets using the measure of noncompactness and then establish results on the best proximity pair (point) on Banach spaces and strictly Banach spaces. In addition, we have illustrated the considered system of integro-differential equations by three examples and discussed the stability, efficiency, and accuracy of solutions. [ABSTRACT FROM AUTHOR]

The aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used fractional operator is generated by iterating a local integral of the form (I a ρ f) (t) = ∫ a t f (s) s ρ − 1 d s . Our reported results are obtained in the Banach space of absolutely continuous functions that rely on Babenko's technique and Banach's fixed point theorem. Besides, our main findings are illustrated by some examples. [ABSTRACT FROM AUTHOR]

There are different approaches that indicate the dynamic of the growth of microbe. In this research, we simulate the growth by utilizing the concept of fractional calculus. We investigate a fractional system of integro-differential equations, which covers the subtleties of the diffusion between infected and asymptomatic cases. The suggested system is applicable to distinguish the presentation of growth level of the infection and to approve if its mechanism is positively active. An optimal solution under simulation mapping assets is considered. The estimated numerical solution is indicated by employing the fractional Tutte polynomials. Our methodology is based on the Atangana–Baleanu calculus (ABC). We assess the recommended system by utilizing real data. [ABSTRACT FROM AUTHOR]

The shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations. [ABSTRACT FROM AUTHOR]

In this article, first, we present an example of fuzzy normed space by means of the Mittag-Leffler function. Next, we extend the concept of fuzzy normed space to matrix valued fuzzy normed space and also we introduce a class of matrix valued fuzzy control functions to stabilize a nonlinear ϕ-Hadamard fractional Volterra integro-differential equation. In this sense, we investigate the Ulam–Hyers–Rassias stability for this kind of fractional equations in matrix valued fuzzy Banach space. Finally, as an application, we investigate the Ulam–Hyers–Rassias stability using matrix valued fuzzy control function obtained through the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

In this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integro-differential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method. [ABSTRACT FROM AUTHOR]

This work is mainly concentrated on finite-time stability of multiterm fractional system for 0 < α 2 ≤ 1 < α 1 ≤ 2 with multistate time delay. Considering the Caputo derivative and generalized Gronwall inequality, we formulate the novel sufficient conditions such that the multiterm nonlinear fractional system is finite time stable. Further, we extend the result of stability in the finite range of time to the multiterm fractional integro-differential system with multistate time delay for the same order by obtaining some inequality using the Gronwall approach. Finally, from the examples, the advantage of presented scheme can guarantee the stability in the finite range of time of considered systems. [ABSTRACT FROM AUTHOR]

To stabilize a nonlinear system d x (t) = f (t , x (t)) d t , we stochastically perturb the deterministic model by using two types of aperiodic intermittent stochastic noise driven by G-Brownian motion. We demonstrate quasi-sure exponential stability for the perturbed system and give the convergence rate, which is related to the control intensity. An application to SIS epidemic model is presented to confirm the theoretical results. [ABSTRACT FROM AUTHOR]