We consider the solution of a class of nonlinear Volterra integral equations, $x(t)=g(t)+{\int }_{t_0}^{t}k(t,s;x(s))ds$, where $t\ge t_0$ and the kernel function $k$ is finitely decomposable, and derive variation of parameters formulae that provide the solution of corresponding perturbed nonlinear equations. We attain this by relating the integral equations to a certain class of initial value problems for ODEs, for which variation of parameters are also formulated.

A new measure of noncompactness (Mnc) and Darbo fixed point theorem are utilized on the space $L^{\infty }({ℝ}_{+})$ to prove the existence of solution for functional Volterra–Fredholm integral equations. An example is given to confirm the validity of results. Furthermore, we propound an iterative algorithm by sinc interpolation to find the solution with an acceptable accuracy. In this algorithm, it is not necessary for the problem to be discretized to an algebraic system with unknown coefficients. We also use an iterative process to approximate a solution with exponential convergence.

Using Petryshyn’s fixed point theorem associated with the measure of noncompactness, we discuss the existence result for functional integral equations in Banach algebra, which covers many existence results for functional integral equations as a particular case under some weaker conditions. Further, we provide some examples of functional integral equations to illustrate our analytical findings.

We investigate semilinear evolution equations of the form $u(t)=U(t,s)u(s)+{\int }_s^{t}U(t,\xi )f(\xi ,u(\xi ))d\xi$ for $t\ge s$ and $s\in ℝ$ in Banach space $X$. Under the assumptions that the evolution family ${(U(t,s))}_{t\ge s}$ has the exponential dichotomy and the function $f:ℝ\times X\to X$ has the Carathéodory property, we show that the semilinear evolution equations on the line has a unique admissible solution, bounded solution, periodic solution when the function $f$ satisfies the condition $\varphi$-Lipschitz and there exists a periodic solution when the function $f$ satisfies the condition $\parallel f(t,x)\parallel \le \phi (t)(1+\parallel x\parallel )$ for all $x\in X$ and almost everywhere $t\in ℝ$.

We study the solution of a backward stochastic differential equation driven by an inhomogeneous simple Lévy process with a rcll reflecting barrier. We show the existence and uniqueness of solution by means of the Snell envelope and the fixed point theorem when the coefficient is stochastic Lipschitz. In terms of application, we provide the fair price of the American option in a Lévy market.

A probabilistic interpretation of the exact solution of a certain inhomogeneous Fredholm integral equation of the second kind is given. Then, this interpretation is used to obtain an approximate solution of a generalization of the integral equation. The approximate solutions are compared with the corresponding Neumann series solutions in various particular cases.

In this paper we are concerned with the robust stability of Volterra equations. We consider the conditions preserving the stability of these systems under perturbations. Also, we study the so-called Bohl–Perron type stability theorems.

We establish that Nyström discretizations of linear Fredholm integral operators on Hölder spaces converge in the operator norm while preserving the consistency order of the quadrature or cubature rule. This allows to employ tools from classical perturbation theory, rather than collective compactness, when studying numerical approximations of integral operators, as well as applications in for instance the field of nonautonomous dynamical systems.

We prove some new existence and uniqueness theorems of fixed points for a contractive type self-mapping in generalized convex metric spaces. Moreover, an application to the solution of a Volterra integral equation is given to illustrate the usability of the results. Our results improve and extend the corresponding results of M. Moosaei ( Fixed Point Theory Appl. (2012), art. id. 164) and several known results.