Band 43, Heft 2

In the recent years, various authors introduced different generalizations of the Hurwitz–Lerch zeta function and discussed its various properties. The main aim of our study is to introduce a new bicomplex generalization of the Hurwitz–Lerch zeta function using the new generalized form of the beta function that involves the Appell series and Lauricella functions. The new bicomplex generalization of the Hurwitz–Lerch zeta function reduces to some already known functions like the Hurwitz–Lerch zeta function, Hurwitz zeta function, Riemann zeta function and polylogarithmic function. Its different properties such as recurrence relation, summation formula, differentiation formula, generating relations and integral representations are investigated. All results induced are general in nature and reducible to already known results. As an application of the new bicomplex generalization of the Hurwitz–Lerch zeta function, we have developed a new generalized form of fractional kinetic equation and obtained its solution using the natural transform.

In this study, we propose an inertial extragradient algorithm for solving a split generalized equilibrium problem as well as a split feasibility and common fixed point problem. We demonstrate that, under certain reasonable assumptions, the sequences induced by the proposed algorithm converge strongly to a solution of the corresponding problem. In addition, with the help of a numerical example, we demonstrate the efficiency of proposed algorithm. As a result of this paper, some recent well-known results in this area have been improved, generalized, and extended.

This paper deals with the evaluation of certain integral transforms involving the product of certain Appell and Bessel functions with a weight e - γ ⁢ t 2 {e^{-\gamma t^{2}}} . The transformations of these integrals are evaluated in terms of the Appell, Kampé de Fériet and the triple hypergeometric functions. As an application, we studied propagation of generalized Humbert–Gaussian beams (GHGBs) and hypergeometric-Gaussian beams (HyGGBs) in turbulent atmosphere and through an ABCD paraxial optical system. The evaluation of these integral transforms has initiated a great interest in mathematical physics and its applications to laser physics and linear or non-linear optics.

Natural phenomena as well as problems encountered in pure and applied sciences are modeled by ordinary, partial or integral differential equations. Most of these problems have a nonlinear aspect which makes their studies difficult, or even impossible. For this, they must resort to other alternatives; among the methods used is the integral inequalities approach, which allows the study of quantitative and qualitative properties of solutions such as existence, uniqueness, delimitation, oscillation, and stability. In this study, we present some new integral inequalities of the Gronwall–Bellman–Bihari type associated with the fractional derivative of ψ-Hilfer, which represents a strong tool and is applicable in the study of certain differential equations. Several known results are derived and some applications to ordinary differential equations are provided to demonstrate the effectiveness of our finding.

This paper deals with the existence and uniqueness results for a class of problems for nonlinear Caputo tempered implicit fractional differential equations in b -metric spaces with initial and nonlocal conditions. The arguments are based on some fixed point theorems. Furthermore, two illustrations are presented to demonstrate the plausibility of our results.