Some new weakly singular versions of discrete nonlinear inequalities are established, which generalize some existing weakly singular inequalities and can be used in the analysis of nonlinear Volterra type difference equations with weakly singular kernels. A few applications to the upper bound and the uniqueness of solutions of nonlinear difference equations are also involved.

By using a generalized Riccati transformation technique and an inequality, we establish some oscillation theorems for the fractional differential equation $[a\left(t\right)$ ${\left(p\left(t\right)+q\left(t\right)\left(D_{-}^{\alpha }x\right)\left(t\right){)}^{\gamma }\right]}^{\mathrm{\prime }}$ − $b(t)f\left({\int }_t^{\mathrm{\infty }}\mathrm{‍}(s-t{)}^{-\alpha }x(s)ds\right)$ = $0$, for $t⩾t_0>0$, where $D_{-}^{\alpha }x$ is the Liouville right-sided fractional derivative of order $\alpha \in (\mathrm{0,1})$ of $x$ and $\gamma$ is a quotient of odd positive integers. The results in this paper extend and improve the results given in the literatures (Chen, 2012).

By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for nonlinear fractional differential systems with coupled four-point boundary value problems.

This paper is concerned with the existence of solutions to a discrete three-point boundary value problem at resonance involving the Riemann-Liouville fractional difference of order $\alpha \in (\mathrm{0,1}]$. Under certain suitable nonlinear growth conditions imposed on the nonlinear term, the existence result is established by using the coincidence degree continuation theorem. Additionally, a representative example is presented to illustrate the effectiveness of the main result.

This paper investigates the robust delay-dependent stability problem for neutral system with mixed delays and nonlinear perturbations. A delay decomposition approach is used in this paper in which the information of the delayed plant states can be taken into full consideration. Then, based on a special Lyapunov functional approach, the novel delay-dependent stability criteria are obtained in terms of linear matrix inequalities (LMIs). A numerical example illustrates the effectiveness of the derived method and the improvement over some existing methods.

In this paper, we are concerned with the multiplicity of nontrivial radial solutions for the following elliptic equations $(\mathrm{P}{)}_{\lambda }$: $-\mathrm{\Delta }u+V\left(x\right)u=\lambda Q\left(x\right){\left|u\right|}^{q-\mathrm{2}}u+Q\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N$; $u(x)\to \mathrm{0},$ as $\left|x\right|\to +\mathrm{\infty }$, where , $V$, and $Q$ are radial positive functions, which can be vanishing or coercive at infinity, and $f$ is asymptotically linear at infinity.

We introduce a new Dadras system with complex variables which can exhibit both four-wing hyperchaotic and chaotic attractors. Some dynamic properties of the system have been described including Lyapunov exponents, fractal dimensions, and Poincaré maps. More importantly, we focus on a new type of synchronization method of modified hybrid project synchronization with complex transformation matrix (CMHPS) for different dimensional hyperchaotic and chaotic complex systems with complex parameters, where the drive and response systems can be asymptotically synchronized up to a desired complex transformation matrix, not a diagonal matrix. Furthermore, CMHPS between the novel hyperchaotic Dadras complex system and other two different dimensional complex chaotic systems is provided as an example to discuss increased order synchronization and reduced order synchronization, respectively. Numerical results verify the feasibility and effectiveness of the presented schemes.

We firstly develop the high-order numerical algorithms for the left and right Riemann-Liouville derivatives. Using these derived schemes, we can get high-order algorithms for the Riesz fractional derivative. Based on the approximate algorithm, we construct the numerical scheme for the space Riesz fractional diffusion equation, where a fourth-order scheme is proposed for the spacial Riesz derivative, and where a compact difference scheme is applied to approximating the first-order time derivative. It is shown that the difference scheme is unconditionally stable and convergent. Finally, numerical examples are provided which are in line with the theoretical analysis.

The paper is concerned with eigenvalues of complex Sturm-Liouville boundary value problems. Lower bounds on the real parts of all eigenvalues are given in terms of the coefficients of the corresponding equation and the bound on the imaginary part of each eigenvalue is obtained in terms of the coefficients of this equation and the real part of the eigenvalue.

We discuss oscillation criteria for second-order half-linear neutral delay dynamic equations on time scales by using the generalized Riccati transformation and the inequality technique. Under certain conditions, we establish four new oscillation criteria. Our results in this paper are new even for the cases of $𝕋=ℝ$ and $𝕋=\mathbb{Z}$.

This paper investigates the existence of positive solutions for a class of singular $p$-Laplacian fractional differential equations with integral boundary conditions. By using the Leggett-Williams fixed point theorem, the existence of at least three positive solutions to the boundary value system is guaranteed.

A remarkably large number of fractional integral formulas involving the number of special functions, have been investigated by many authors. Very recently, Agarwal (National Academy Science Letters) gave some integral transform and fractional integral formulas involving the $F_p^{\left(\alpha ,\beta \right)}\left(·\right)$. In this sequel, here, we aim to establish some image formulas by applying generalized operators of the fractional integration involving Appell’s function $F_3(·)$ due to Marichev-Saigo-Maeda. Some interesting special cases of our main results are also considered.

We construct a variational functional of a class of three-point boundary value problems with impulse. Using the critical points theory, we study the existence of solutions to second-order three-point boundary value problems with impulse.