Nonlinear integral inequalities of Gronwall-Bihari type for piecewise continuous functions are solved. Inequalities for functions with delay as well as functions without delays are considered. Some of the obtained results are applied in the deriving of estimates for the solutions of impulsive integral, impulsive integro-differential and impulsive differential-difference equations.

By using the concept of integrable dichotomy, the fixed point theory, functional analysis methods and some new technique of analysis, we obtain new criteria for the existence and uniqueness of bounded and periodic solutions of general and periodic systems of nonlinear integro-differential equations with infinite delay.

In the present work we discuss the existence of solutions for a system of nonlinear fractional integro-differential equations with initial conditions. This system involving the Caputo fractional derivative and Riemann−Liouville fractional integral. Our results are based on a fixed point theorem of Schauder combined with the diagonalization method.

In this paper, the method of upper and lower solutions and monotone iterative technique are employed to the study of nonlinear three-point boundary value problems for a class of first order impulsive integro-differential equations of mixed type. Sufficient conditions for the existence of extreme solutions are obtained. Mathematics Subject Classification: 34B37

Abstract: In this paper, Laplace decomposition method is developed to solve linear and nonlinear fractional integrodifferential equations. The proposed method is based on the application of Laplace transform to nonlinear fractional integro-differential equation. The nonlinear term can easily be handled with the help of Adomian polynomials. The fractional derivative is described in the Caputo se...

In the present article, we implement the new iterative method with a reliable algorithm to solve the nth-order linear/ nonlinear integro-differential equations. The obtained results confirm the power of the method in reducing the size of calculations compared with the other methods.

This paper is concerned with the nonlinear boundary value problems for first order integro-differential equations with impulsive integral conditions. By using of the method of lower and upper solutions coupled with the monotone iterative technique, we give conditions for the existence of extremal solutions.

Ito’s construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy-Khintchin...

We develop a general construction for nonlinear Lévy processes with given characteristics. More precisely, given a set Θ of Lévy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and a nonlinear generator corresponding to the supremum of all generators of classical Lévy processes with triplets in Θ. The nonli...

We predict theoretically and numerically the existence of incoherent dispersive shock waves. They manifest themselves as an unstable singular behavior of the spectrum of incoherent waves that evolve in a noninstantaneous nonlinear environment. This phenomenon of "spectral wave breaking" develops in the weakly nonlinear regime of the random wave. We elaborate a general theoretical formulation of...