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On a discrete version of Fejér inequality for α-convex sequences without symmetry condition

Jleli Mohamed, Samet Bessem

Demonstratio Mathematica

2024

Vol. 57, nr 1

art. no. 20240055

In this study, we introduce the notion of α-convex sequences which is a generalization of the convexity concept. For this class of sequences, we establish a discrete version of Fejér inequality without imposing any symmetry condition. In our proof, we use a new approach based on the choice of an appropriate sequence, which is the unique solution to a certain second-order difference equation. Moreover, we obtain a refinement of the standard (right) Fejér inequality for convex sequences.

On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group

Jleli Mohamed

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220193

We investigate the existence and nonexistence of nonnegative radial solutions to exterior problems of the form ΔHmu(q)+λψ(q)K(r(q))f(r2−Q(q),u(q))=0 in Bc1 , under the Dirichlet boundary conditions u=0 on ∂B1 and limr(q)→∞u(q)=0 . Here, λ≥0 is a parameter, ΔHm is the Kohn Laplacian on the Heisenberg group Hm=R2m+1 , m>1 , Q=2m+2 , B1 is the unit ball in Hm, Bc1 is the complement of B1 , and ψ(q)=∣∣z∣∣2r2(q) . Namely, under certain conditions on K and f , we show that there exists a critical parameter λ∗∈(0,∞] in the following sense. If 0≤λ<λ∗ , the above problem admits a unique nonnegative radial solution uλ ; if λ∗<∞ and λ≥λ∗ , the problem admits no nonnegative radial solution. When 0≤λ<λ∗ , a numerical algorithm that converges to uλ is provided and the continuity of uλ with respect to λ , as well as the behavior of uλ as λ→λ∗− , are studied. Moreover, sufficient conditions on the the behavior of f(t,s) as s→∞ are obtained, for which λ∗=∞ or λ∗<∞ . Our approach is based on partial ordering methods and fixed point theory in cones.