Ahmed Zeghal

In this work, we derive sufficient conditions on a linear operator L and a nonlinear mapping K that ensure the existence of a unique fixed point of the sum L +K within the framework of Krasnosel’skii’s fixed point theorem. As a special case, when L is the zero operator, our result reduces to the well-knownclassical Kellogg uniqueness theorem. Moreover, we extend Talman’s uniqueness theorem by employing the concept of the measure of noncompactness. In addition, we investigate the asymptotic behavior of the unique fixed point in connection with the Belitski ̆ı–Lyubich conjecture. Finally, we present an illustrative application that demonstrates the applicability and effectiveness of the main theoretical result.

In this work we establish some fixed point theorems for a class of mappings on Banach spaces with the Dunford-Pettis property. Since L 1 -spaces possess the Dunford-Pettis property, we apply our result to discuss existence of solutions to a nonlinear stationary transport equation with delayed neutrons and abstract boundary conditions in slab geometry the space on the space L 1 (Ω; dxdξ) (cf. Section 4.1) which is the convenient setting of the problem.

In this paper, we prove the existence of solutions to a nonlinear singular transport equation (ie, transport equation with unbounded collision frequency and unbounded collision operator) with vacuum boundary conditions in bounded spatial domain on L^p‐spaces with 1 ≤ p<+∞. This problem was already considered in^6,8,9 under the hypothesis that the collision frequency σ(·) and the collision operator are bounded. In this work, we show that these hypotheses are not necessary; it suffices to assume that σ(·) is locally bounded, and the collision operator is bounded between (a weighted space) and (cf Section 2). Although the analysis for p∈(1,+∞) is standard in the sense that it uses the Schauder fixed point theorem, the compactness of the involved operator is not easy to derive. However, the analysis in the case p=1 uses the concept of Dunford‐Pettis operators and a new version of the Darbo fixed point theorem for a measure of weak noncompactness introduced in the paper.