In this article, we study an inverse fractional problem that is ill-posed in the sense of Hadamard. The goal is to recover a source term in a time-fractional parabolic equation with a time-dependent coefficient, using Tikhonov regularization. A no-regret control approach is employed to address the regularization problem when the initial condition is unknown. The source term is characterized through the corresponding optimality system.
There are various known refinements to Becker-Stark inequality. In this paper, we present a more rigorous refinement of Becker-Stark inequality which is based on the result of Zhu [15]. Zhu gave a double inequality with quadratic estimations appear on both the left and right sides. In this paper, based on the inequality given by Zhu, we establish a new double inequality with quadratic estimations. The double inequality can be said to be an improvement of Zhu's inequality, as the left side is a larger function and the right side is a smaller function than Zhu's ones.
The aim of this paper is to apply the Norlund sum to Bessel function and generalized Bernoulli polynomials involving this function, we derive some new relations and integral formulas. By applying the Laplace transform with its inverse to generating function for the generalized Bernoulli polynomials involving the Bessel function, we also derive infinite series representation for these polynomials. Furthermore, some relationship between the Bessel function and the Norlund sum is found. We also give some new relations by comparing specific values to the Bessel function and modified Bessel functions with the aid of their Laplace transforms by its inverse.
The aim of this paper is to establish new Stirling-type approximation formulas in terms of the digamma and trigamma functions. Our new formulas offer approximations of order n(-2), more accurate than the classical Stirling approximation of order n(-1). Finally, some inequalities are given.
In this paper, we present a new Wilker-type inequality using cos x/1-x(2)/3 instead of sin x /x . We prove that (cos x/1-x(2)/3)(2)+ tan x/x in the new inequality is monotonically increasing for 0 < x < pi/2 . Furthermore, similar to the ( cosx)2 result of Sumner et al., we present an inequality in which (cos x/1-x(2)/3)(2)+ tan x /x - 2 is evaluated at a constant multiple of x(3) tan x.
Recently, Adell introduced the probabilistic Stirling numbers of the second kind, which has led to a growing body of research focused on the probabilistic versions of various special functions and their properties. In this study, we introduce the probabilistic degenerate Daehee polynomials, a generalization of the classical Daehee polynomials, and explore their connections with other special functions. In particular, we derive several interesting identities involving these polynomials under gamma random variable with parameter alpha, beta, Poisson random variable with parameter alpha, Bernoulli random variable, binomial distribution with parameter m, p, or uniform distribution.
In 1935, Lehmer defined certain generalization of Euler numbers, as a natural generalization of Bernoulli and Euler numbers. In this paper, we set a new polynomial related to the higher-order generalized Lehmer-Euler numbers and determine its a q-supercongruence.
In this paper, we establish several new lower and upper bounds for the (p) pound norms of some Cauchy-Toeplitz matrices. Moreover, we compare our results with others that appear in the literature.
In this paper, we consider the elastic problem with mixed boundary conditions (Dirichlet-Fourier-maximal monotone graph) in a smooth domain. We prove the existence and the regularity results for the weak solution of this problem. The proof is based on the approach of maximal monotone graph by its Yosida regularization and the contraction method of Brezis.
The main objective of the present paper is to investigate a generalization of (k)-H.M.F. using (k)-M.L.F. with two parameters. Many identities of these extensions are studied, such as derivative formulas, generating relations, symmetric relations, functional relations, summation formulas, integral representations, Mellin transform, and generalizing these identities that satisfy (k)-G.H.M.F. and (k)-C.H.M.F..
In this study, we define the generalized degenerate Fubini-Euler-Genocchi polynomials. Then, we derive some of their properties, including addition formulas, summation formulas, identities, and relations, utilizing some series manipulation methods and analyzing their generating function. Also, we provide several symmetric properties for the generalized degenerate Fubini-Euler-Genocchi polynomials. In addition, we consider the generalized degenerate Fubini-Euler-Genocchi polynomials of the order beta and then we investigate implicit summation formulas, correlations, addition formulas, and their derivative and difference properties.
. The present investigation enhances a basic theme in geometric function theory, namely introducing new operators and establishing geometric characterization properties of starlikeness and convexity for these operators. This paper presents a new integral operator defined by applying the fractional integral of the Bessel function of the first kind and order nu >_ 0. The investigation establishes sufficient and necessary conditions for starlikeness and convexity of the newly introduced operator by applying specific techniques of the differential subordination theory and certain properties previously obtained for the fractional integral of the Bessel function of the first kind and order nu >_ 0. Although a simple application example is provided, the starlikeness and convexity properties given here could motivate additional investigation on this operator for further applications in other lines of research in geometric function theory.
In this paper,a new type of invariant mean "mu -limit" is introduced and a new set of some specific bounded sequences is determined by using a bijective shift operator mu where mu (-1) of each term of a sequence of this set is present in the same sequence so that mu -limits of these sequences can be evaluated. Using this mu , some new matrix summability methods are developed that are applied here as matrix transformations on the aforesaid sequences to generate new sublinear functionals which can form new I mu -core of sequences. Lastly some important I-mu-core theorems are established in our main results by showing inequalities between sublinear functionals used in Knopps core and I mu -core itself .
Our study reveals intriguing convergence patterns within the Fibonacci double sequences in picture fuzzy normed spaces. We demonstrate that while each component sequence exhibits unique convergence characteristics, collectively they converge towards a distinctive ideal represented by Fibonacci numbers. This ideal convergence aligns with the inherent properties of Fibonacci sequences, offering insights into the interplay between fuzzy normed spaces and the Fibonacci sequence.Also, the idea of Fibonacci Z(2)-convergence on picture fuzzy normed spaces is explained in this article. With regard to picture fuzzy normed spaces, we define the Fibonacci Z(2)-Cauchy sequences and Z(2)-completeness.
In this work, we establish new criteria for the oscillation of solutions for second-order Emden-Fowler differential equations. New prerequisites are presented in order to analyze the oscillatory features of the analyzed equation. To support these findings, we employed a range of analysis tools, creating new linkages to address some of the problems that have hindered previous research. We were therefore able to obtain results that both build upon and improve those discovered in earlier studies by applying the Riccati transformation and the principles of comparison. Several examples are given to illustrate the significance of our results.
The results obtained in this work contribute to one of the important lines of study in the geometric functions theory, which is the study of certain geometric properties of different types of operators. In the present study, sufficient univalence conditions for the confluent hypergeometric function's fractional integral are highlighted using methods that are specific to the differential sub ordinations theory, the method of the subordination chains, as well as the properties of the class of functions with positive real part of the first derivative. The subordination results are contained in the theorem demonstrated in this study which is further used in the statement of two univalence criteria for the confluent hypergeometric function's fractional integral established in the two corollaries associated to this theorem. An example is given as basic application but the criteria are expected to generate interest for further investigations on this significant operator.
We introduce a new sequence of unsigned degenerate Stirling numbers of the first kind. Following the work of Adell-Lekuona, who represented unsigned Stirling numbers of the first kind as multiples of the expectations of specific random variables, we express our new numbers as finite sums of multiples of the expectations of certain random variables. We also provide a representation of these new numbers as finite sums involving the classical unsigned Stirling numbers of the first kind. As an inversion formula, we define a corresponding sequence of new type degenerate Stirling numbers of the second kind. We derive expressions for these numbers as finite sums that involve the Stirling numbers of the second kind.
This paper presents the mixed additive-Jensen equations and shows that a solution of it is a complex linear mapping. We also define the Lie bracket of triple Jordan derivations related to this equation in triple Banach algebra and investigate their properties. Lastly, we show the Hyers-Ulam (HU) stability of the mixed additive-Jensen equations and the Lie bracket of triple Jordan derivations on triple Banach algebra through the fixed point method, with control functions introduced by Gavruta and Rassias.
In this paper, we introduce a class of polynomials called type 2 poly-Frobenius-Euler polynomials, defined using the polyexponential function. We derive explicit expressions and identities for these polynomials and explore their connections with Stirling numbers of the first and second kinds. Additionally, we introduce type 2 unipoly-poly-Frobenius-Euler polynomials using the unipoly function and examine various properties, including derivatives and integrals. Furthermore, we establish a link between the unipoly-Frobenius-Euler polynomials and the classical Frobenius-Euler polynomials.
In this paper, we present a new hybrid steepest decent algorithm to solve the bilevel variational inequality problems, where the feasible set is the intersection of a variational inequality problem and the fixed point set of a nonexpansive mapping in a real Hilbert space. An iterative algorithm is introduced by using the hybrid steepest descent method and fixed-point formulation techniques. The strong convergence of the iterative sequences generated by the algorithm is shown under some suitable assumptions. In addition, the efficiency of our suggested algorithm is established by a numerical example.
