Generalizations of the Jensen functional involving diamond integrals via Abel–Gontscharoff interpolation

In this paper we obtain several refinements of the Jensen inequality on time scales by generalizing Jensen’s functional for n-convex functions. We also investigate the bounds for the identities related to the new improvements obtained.

Two efficient modifications of AZPRP conjugate gradient method with sufficient descent property

The conjugate gradient method can be applied in many fields, such as neural networks, image restoration, machine learning, deep learning, and many others. Polak–Ribiere–Polyak and Hestenses–Stiefel conjugate gradient methods are considered as the most efficient methods to solve nonlinear optimization problems. However, both methods cannot satisfy the descent property or global convergence property for general nonlinear functions. In this paper, we present two new modifications of the PRP method with restart conditions. The proposed conjugate gradient methods satisfy the global convergence property and descent property for general nonlinear functions. The numerical results show that the new modifications are more efficient than recent CG methods in terms of number of iterations, number of function evaluations, number of gradient evaluations, and CPU time.

Some periodic and fixed point theorems on quasi-b-gauge spaces

The notions of a quasi-b-gauge space $(U,\textsl{Q}_{s ; \Omega })$ ( U , Q s ; Ω ) and a left (right) $\mathcal{J}_{s ; \Omega }$ J s ; Ω -family of generalized quasi-pseudo-b-distances generated by $(U,\textsl{Q}_{s ; \Omega })$ ( U , Q s ; Ω ) are introduced. Moreover, by using this left (right) $\mathcal{J}_{s ; \Omega }$ J s ; Ω -family, we define the left (right) $\mathcal{J}_{s ; \Omega }$ J s ; Ω -sequential completeness, and we initiate the Nadler type contractions for set-valued mappings $T:U\rightarrow Cl^{\mathcal{J}_{s ; \Omega }}(U)$ T : U → C l J s ; Ω ( U ) and the Banach type contractions for single-valued mappings $T: U \rightarrow U$ T : U → U , which are not necessarily continuous. Furthermore, we develop novel periodic and fixed point results for these mappings in the new setting, which generalize and improve the existing fixed point results in the literature. Examples validating our obtained results are also given.

A class of completely monotonic functions involving the polygamma functions

Let $\Gamma (x)$ Γ ( x ) denote the classical Euler gamma function. We set $\psi _{n}(x)=(-1)^{n-1}\psi ^{(n)}(x)$ ψ n ( x ) = ( − 1 ) n − 1 ψ ( n ) ( x ) ($n\in \mathbb{N}$ n ∈ N ), where $\psi ^{(n)}(x)$ ψ ( n ) ( x ) denotes the nth derivative of the psi function $\psi (x)=\Gamma '(x)/\Gamma (x)$ ψ ( x ) = Γ ′ ( x ) / Γ ( x ) . For λ, α, $\beta \in \mathbb{R}$ β ∈ R and $m,n\in \mathbb{N}$ m , n ∈ N , we establish necessary and sufficient conditions for the functions $$ L(x;\lambda ,\alpha ,\beta )=\psi _{m+n}(x)-\lambda \psi _{m}(x+ \alpha ) \psi _{n}(x+\beta ) $$ L ( x ; λ , α , β ) = ψ m + n ( x ) − λ ψ m ( x + α ) ψ n ( x + β ) and $-L(x;\lambda ,\alpha ,\beta )$ − L ( x ; λ , α , β ) to be completely monotonic on $(-\min (\alpha ,\beta ,0),\infty )$ ( − min ( α , β , 0 ) , ∞ ) .As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

An inertial iterative algorithm for generalized equilibrium problems and Bregman relatively nonexpansive mappings in Banach spaces

The aim of this paper is to introduce and study an inertial hybrid iterative method for solving generalized equilibrium problems involving Bregman relatively nonexpansive mappings in Banach spaces. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational example to emphasize the efficiency and relevancy of main result.

Best approximation of $(\mathcal{G}_{1},\mathcal{G}_{2})$-random operator inequality in matrix Menger Banach algebras with application of stochastic Mittag-Leffler and $\mathbb{H}$-Fox control functions

We stabilize pseudostochastic $(\mathcal{G}_{1},\mathcal{G}_{2})$ ( G 1 , G 2 ) -random operator inequality using a class of stochastic matrix control functions in matrix Menger Banach algebras. We get an approximation for stochastic $(\mathcal{G}_{1},\mathcal{G}_{2})$ ( G 1 , G 2 ) -random operator inequality by means of both direct and fixed point methods. As an application, we apply both stochastic Mittag-Leffler and $\mathbb{H}$ H -fox control functions to get a better approximation in a random operator inequality.

Weighted Ostrowski type inequalities for co-ordinated convex functions

In this paper, utilizing an identity given by Yıldız and Sarıkaya in (Yildiz and Sarikaya in Int. J. Anal. Appl. 13(1):64–69, 2017), we establish some weighted Ostrowski type inequalities for co-ordinated convex functions in a rectangle from the plane $\mathbb{R} ^{2}$ R 2 . Moreover, as special cases of our main results, we give some weighted Hermite–Hadamard type inequalities. The results given in this paper provide generalizations of some result established in earlier works.

A note on optimal Hermite interpolation in Sobolev spaces

This paper investigates the optimal Hermite interpolation of Sobolev spaces $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] , $n\in \mathbb{N}$ n ∈ N in space $L_{\infty }[a,b]$ L ∞ [ a , b ] and weighted spaces $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p< \infty $ 1 ≤ p < ∞ with ω a continuous-integrable weight function in $(a,b)$ ( a , b ) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in $L_{\infty }$ L ∞ (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ) are optimal for $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] in $L_{\infty }[a,b]$ L ∞ [ a , b ] (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

On the novel existence results of solutions for a class of fractional boundary value problems on the cyclohexane graph

A branch of mathematical science known as chemical graph theory investigates the implications of connectedness in chemical networks. A few researchers have looked at the solutions of fractional differential equations using the concept of star graphs. Their decision to use star graphs was based on the assumption that their method requires a common point linked to other nodes but not to each other. Our goal is to broaden the scope of the method by defining the idea of a cyclohexane graph, which is a cycloalkane with the molecular formula $C_{6}H_{12}$ C 6 H 12 and CAS number 110-82-7. It consists of a ring of six carbon atoms, each bonded with two hydrogen atoms above and below the plane with multiple junction nodes. This article examines the existence of fractional boundary value problem’ solutions on such graphs in the sense of the Caputo fractional derivative by using the well-known fixed point theorems. In addition, an example is given to support our key findings.

Equalities and inequalities for several variables mappings

In this paper, some special mappings of several variables such as the multicubic and the multimixed quadratic–cubic mappings are introduced. Then, the systems of equations defining a multicubic and a multimixed quadratic–cubic mapping are unified to a single equation. Under some mild conditions, it is shown that a multimixed quadratic–cubic mapping can be multiquadratic, multicubic and multiquadratic–cubic. Furthermore, by applying a known fixed-point theorem, the Hyers–Ulam stability of multimixed quadratic–cubic, multiquadratic, multicubic and multiquadratic–cubic are studied in non-Archimedean normed spaces.