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We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…

An explicit perturbative solution to all orders is given for a general class of nonlinear differential equations. This solution is written as a sum indexed by rooted trees and uses the Green function of a linearization of the equations. The…

We study the spectra for a class of differential operators with asymptotically constant coefficients.These operators widely arise as the linearizations of nonlinear partial differential equations about patterns or nonlinear waves. We…

We consider a terminal control problem for processes governed by a nonlinear system of fractional ODEs. In order to show existence of the control, we first consider the linear counterpart of the system and reprove a number of classical…

We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The…

We prove the existence of non-trivial solutions for a fractional Schr$\ddot{o}$dinger-Poisson equation in $\mathbb{R}^{3}$. The proof is based on the perturbation method and the mountain pass theorem.

We study a nonlinear, nonlocal Dirichlet problem driven by the degenerate fractional p-Laplacian via a combination of topological methods (degree theory for operators of monotone type) and variational methods (critical point theory). We…

In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and…

We prove some existence results for a class of nonlinear fractional equations driven by a nonlocal operator.

Nowadays, fractional differential equations are a well established tool to model phenomena from the real world. Since the analytical solution is rarely available, there is a great effort in constructing efficient numerical methods for their…

In this paper we present a theory for the existence of multiple nontrivial solutions for a class of perturbed Hammerstein integral equations. Our methodology, rather than to work directly in cones, is to utilize the theory of fixed point…

By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical…

The main objective of this paper is to study the existence of solutions to some basic fractional difference equations. The tools employed are Krasnosel'skii fixed point theorem which guarantee at least two positive solutions.

We present a new view onto the successive approximations' approach in study of the two-point nonlinear fractional boundary value problems. In order to reduce the original problem and further construct its approximate solution we use the…

In this work, we consider a boundary value problem for nonlinear triharmonic equation. Due to the reduction of nonlinear boundary value problems to operator equation for nonlinear terms we establish the existence, uniqueness and positivity…

We prove the existence of solution for a class of $p(x)$-Laplacian equations where the nonlinearity has a critical growth. Here, we consider two cases: the first case involves the situation where the variable exponents are periodic…

This paper systematically treats the asymptotic behavior of many (linear/nonlinear) classes of higher-order fractional differential equations with multiple terms. To do this, we utilize the characteristics of Caputo fractional…

In the present study, a numerical method, perturbation-iteration algorithm (shortly PIA), have been employed to give approximate solutions of nonlinear fractional-integro differential equations (FIDEs). Comparing with the exact solution,…

We prove existence of solutions for a nonlinear fractional oscillator equation with both left Riemann-Liouville and right Caputo fractional derivatives subject to natural boundary conditions. The proof is based on a transformation of the…

In this paper, we consider a class of nonlinear fractional differential equations involving Hilfer derivative with boundary conditions. First, we obtain an equivalent integral for the given boundary value problem in weighted space of…