On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain

We study the wave inequality with a Hardy potential (Eqation Presented), where Ω is the exterior of the unit ball in ℝN, N ≥ 2, p > 1, and λ ≥ - (N-2/2)2, under the inhomogeneous boundary condition 'Equation Presented', where α, β ≥ 0 and (α, β) ≠ (0, 0). Namely, we show that there exists a critical exponent pc(N, λ) ∈ (1, ∞] for which, if 1 < p < pc(N, λ), the above problem admits no global weak solution for any w ∈ L1 (∂Ω) with ∫∂Ω w(x) dσ > 0, while if p > pc(N, λ), the problem admits global solutions for some w > 0. To the best of our knowledge, the study of the critical behavior for wave inequalities with a Hardy potential in an exterior domain was not considered in previous works. Some open questions are also mentioned in this paper

Nonexistence results for higher order fractional differential inequalities with nonlinearities involving Caputo fractional derivative

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function

Novel results on Hermite-Hadamard kind inequalities for η\eta-convex functions by means of (k,r)(k,r)-fractional integral operators

We establish new integral inequalities of Hermite-Hadamard type for the recent class of $\eta$-convex functions. This is done via generalized $(k,r)$-Riemann-Liouville fractional integral operators. Our results generalize some known theorems in the literature. By choosing different values for the parameters $k$ and $r$, one obtains interesting new results.Comment: This is a preprint of a paper whose final and definite form is a Springer chapter in the Book 'Advances in Mathematical Inequalities and Applications', published under the Birkhauser series 'Trends in Mathematics', ISSN: 2297-0215 [see http://www.springer.com/series/4961]. Submitted 02-Jan-2018; Revised 10-Jan-2018; Accepted 13-Feb-201