In this article, we have established some new bounds of Fejér-type Hermite-Hadamard inequality for k -fractional integrals involving r -times differentiable preinvex functions. It is noteworthy that in the past, there was no weighted version of the left and right sides of the Hermite-Hadamard inequality for k -fractional integrals for generalized convex functions available in the literature.

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an $\alpha -\theta -$Geraghty–Pata–Suzuki-type proximal contraction; an $\alpha -\theta -$generalized Geraghty–Pata–Suzuki-type proximal contraction; and an $\alpha -\theta -$modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches.

This paper retrieves bright 1-soliton solution to the perturbed concatenation model by the application of the semi-inverse variational principle. The perturbation terms are from inter-modal dispersion and self-steepening effect that is considered with maximum allowable intensity. The parameter constraints that naturally emerged from the integration scheme are enumerated.