Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities.

In this work, we introduced a matrix-induced framework for quantum calculus on discrete forward orbits generated by affine contractions of the form $T(x)=a+Q(x-a)$ in ${\mathbb{R}}^N$, where the matrix $Q$ has a spectral radius less than one. The proposed approach extends the classical one-dimensional $q$-calculus to higher dimensions by replacing scalar dilations with matrix-driven dynamics. We defined matrix-induced directional derivatives and integrals on the associated forward orbit and established their fundamental properties, including integration by parts, and first and second fundamental theorems of calculus. As applications, we proved a Hermite–Hadamard-type inequality for convex functions under a segment-preserving condition on $Q$, derived a Poincaré-type inequality on the orbit, and obtained an existence–uniqueness result for a nonlinear first-order problem driven by the new derivative. In dimension one, the framework reduces to the standard $q$-calculus.

We establish Lyapunov-type inequalities for Dirichlet boundary value problems driven by the modified discrete Helmholtz operator on finite balls of the integer lattice $\mathbb{Z}$. This yields weighted inequalities for scalar problems and a spectral-radius criterion for coupled systems, as well as sharpness results. As an application, we study a weighted eigenvalue problem and obtain explicit two-sided bounds for the first eigenvalue by combining a Lyapunov-type lower estimate with a variational upper bound. Numerical illustrations are provided for localized weights.

A class of nonlinear integral equations with delay, related to infectious diseases, is studied. Making use of some tools from operators theory, we deal with the well-posedness in an adequate functional space, approximation of solution, estimates of lower/upper solutions and the data dependence of solutions.

We consider a fractional differential inequality involving $\psi$-Caputo fractional derivatives of different orders, with a polynomial nonlinearity and a singular potential term. Using the test function method and some integral inequalities, we establish nonexistence criteria of global solutions in both cases: $\underset{t\to \mathrm{\infty }}{lim}\psi (t)=\mathrm{\infty }$ and $\underset{t\to \mathrm{\infty }}{lim}\psi (t)<\mathrm{\infty }$.

We study the existence and nonexistence of weak solutions to a semilinear higher order (in time) evolution inequality involving a convection term in the exterior of the half-ball, under Dirichlet-type boundary conditions. A weight function of the form $|x{|}^a$ is allowed in front of the power nonlinearity. When $a>-2$, we show that the dividing line with respect to existence or nonexistence is given by a critical exponent (Fujita critical exponent), which depends on the parameters of the problem, but independent of the order of the time-derivative. Our study yields naturally optimal nonexistence results for the corresponding stationary problem.

We are concerned with the study of existence and nonexistence of weak solutions for a class of hyperbolic inequalities with a Hardy potential singular on the boundary $\mathrm{\partial }{B}_1$ of the annulus $A=\left\{x\in {\mathbb{R}}^3:1<|x|\le 2\right\}$, where $\mathrm{\partial }{B}_1=\left\{x\in {\mathbb{R}}^3:|x|=1\right\}$. A singular potential function of the form $(|x|-1{)}^{-\rho }$, $\rho \ge 0$, is considered in front of the power nonlinearity. Two types of inhomogeneous boundary conditions on $(0,\mathrm{\infty })\times \mathrm{\partial }{B}_2$, $\mathrm{\partial }{B}_2=\left\{x\in {\mathbb{R}}^3:|x|=2\right\}$, are studied: Dirichlet and Neumann. We use a unified approach to show the optimal criteria of Fujita-type for each case.

A nonlinear hyperbolic polyharmonic system in an exterior domain of ${\mathbb{R}}^N$ is considered under inhomogeneous Navier-type boundary conditions. Using nonlinear capacity estimates specifically adapted to the polyharmonic operator $(-\mathrm{\Delta }{)}^m$, the geometry of the domain, and the boundary conditions, a sharp criterium for the nonexistence of weak solutions is obtained. Next, an optimal nonexistence result for the corresponding stationary problem is deduced.

In this paper, we first introduced the notion of $\theta$-hyperbolic sine distance functions on a metric space and studied their properties. We investigated the existence and uniqueness of fixed points for some classes of single-valued mappings defined on a complete metric space and satisfying contractions involving the $\theta$-hyperbolic sine distance function.

A nonlinear time-fractional cable equation posed on the interval $(0,1)$ under a homogeneous Dirichlet boundary condition is investigated in this work. The considered equation reflects the anomalous electro-diffusion in nerve cells. Using nonlinear capacity estimates specifically adapted to the considered problem, we establish sufficient conditions for the nonexistence of weak solutions.