Sahand Communications in Mathematical Analysis
Volume:20 Issue: 4, Autumn 2023

In this paper, we are introducing for the first time a generalized class named the class of $(\alpha,\beta,\gamma,\delta)-$convex functions of mixed kind. This generalized class contains many subclasses including the class of $(\alpha,\beta)-$convex functions of the first and second kind, $(s,r)-$convex functions of mixed kind, $s-$convex functions of the first and second kind, $P-$convex functions, quasi-convex functions and the class of ordinary convex functions. In addition, we would like to state the generalization of the classical Ostrowski inequality via fractional integrals, which is obtained for functions whose first derivative in absolute values is $(\alpha,\beta,\gamma,\delta)-$ convex function of mixed kind. Moreover, we establish some Ostrowski-type inequalities via fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are $(\alpha,\beta,\gamma,\delta)-$ convex functions of mixed kind using different techniques including H\"older's inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, the applications of special means will also be discussed.

The search for contractive definitions which do not compel the mapping to be continuous at fixed points remained an open problem for a long time. Several solutions to this open problem have been obtained in last two decades. The current paper,  we aim to provide another new solution direction for the discontinuity study  at fixed points using $F$-contractive mappings in a complete metric space. Several consequences of those new results are also provided. This manuscript consists of three main parts. In the first part, the notion of $F$-contractive mappings has been described. In the second part, discontinuity at the fixed point assuming continuity of the composition has been investigated, whereas in the third part, discontinuity at a fixed point without assuming continuity of the composition has been illustrated.

In this note, first the better refinements of Young and its reverse inequalities for scalars are given. Then, several operator and norm versions according to these inequalities are established.

In this paper, a special class of relative reproducing kernel Banach spaces a semi-inner product is studied. We extend the concept of relative reproducing kernel Hilbert spaces to Banach spaces. We present these relative reproducing kernel Banach spaces  in terms of the feature maps and establish the separability of the domains when they are  separable. In addition, we prove some theorems concerning feature maps and reproducing kernel Banach spaces. And finally, the relative kernels are compared with the  semi-inner ones.

We study the Schr\"odinger equation   $\left(\mathrm{Q}_{\varepsilon}\right)$: $- \varepsilon^{2(p-1)} \Delta_p v + V(x)\, |v|^{p-2} v - |v|^{q-1}v = 0$, $x \in \mathbb{R}^N$, with $v(x) \rightarrow 0$ as $|x| \rightarrow+\infty$, for the infinite case, as given by Byeon and Wang for a situation of critical frequency,  $\displaystyle \{x\in \mathbb{R}^N \, / \: V(x) = \inf V = 0\} \neq \emptyset$. In the semiclassical limit, $\varepsilon \rightarrow 0$, the corresponding limit problem is $\left(\mathrm{P}\right)$: $\Delta_p w+|w|^{q-1} w=0$, $x \in \Omega$, with $w(x)=0, x \in \partial \Omega$, where $\Omega \subseteq \mathbb{R}^N$ is a smooth bounded strictly star-shaped region related to the potential $V$. We prove  that for $\left(\mathrm{Q}_{\varepsilon}\right)$ there exists a non-trivial solution with any prescribed $\mathrm{L}^{q+1}$-mass.Applying a Ljusternik-Schnirelman scheme, shows  that  $\left(\mathrm{Q}_{\varepsilon}\right)$ and $\left(\mathrm{P}\right)$ have infinitely many pairs of solutions. Fixed a topological level $k \in \mathbb{N}$, we show that a solution of $\left(\mathrm{Q}_{\varepsilon}\right)$, $v_{k, \varepsilon}$, sub converges, in $\mathrm{W}^{1,p}(\mathbb{R}^N)$ and up to scaling, to a corresponding solution of $\left(\mathrm{P}\right)$. We also prove that the energy of each solution, $v_{k,\eps}$ converges to the corresponding energy of the limit problem  $\left(\mathrm{P}\right)$ so that the critical values of the functionals associated, respectively, to  $\left(\mathrm{Q}_{\varepsilon}\right)$ and $\left(\mathrm{P}\right)$ are topologically equivalent.

In this paper, we provide a different  uniqueness results for inverse spectral problems of conformable fractional Sturm-Liouville operators of order $\alpha$ ($0 < \alpha\leq  1$), with  a  jump and eigen-parameter dependent boundary conditions. Further, we study the asymptotic form of solutions, eigenvalues and the corresponding eigenfunctions of the problem. Also, we consider three terms of the inverse problem,  from the Weyl function,  the spectral data and  two spectra. Moreover, we can also extend Hald's theorem to the problem.

The prime goal of this article is to initiate the notion of fuzzy    $\mu^*$-open(closed) sets and fuzzy $\mu^*$-continuous functions and characterize  them. These concepts are defined in a fuzzy topological space in presence of a  generalized fuzzy topology, which becomes a new tool to study fuzzy topological spaces. It is observed that this class of fuzzy sets fail to form a fuzzy topology but it form a generalized fuzzy topology. Furthermore, the relationship of these fuzzy sets and fuzzy continuity with some existing fuzzy notions are established. Also the notion of fuzzy $(\tau, \mu^*)$-open(closed) functions is introduced and their equivalent conditions with  fuzzy $\mu^*$-continuous functions are established.

In this paper, for generalised preinvex functions, new estimates of the Fej\'{e}r-Hermite-Hadamard inequality on fractional sets $\mathbb{R}^{\rho }$ are given in this study. We demonstrated a fractional  integral inequalities based on Fej\'{e}r-Hermite-Hadamard theory. We establish two new local fractional integral identities for differentiable functions. We construct several novel Fej\'{e}r-Hermite-Hadamard-type inequalities for generalized convex function in local fractional calculuscontexts using these integral identities. We provide a few illustrations to highlight the uses of the obtained findings. Furthermore, we have also given a few examples of new inequalities in use.

In this paper, we define the notions of semi-regular operator, analytical core, surjectivity modulus and the injectivity modulus of bounded linear operators on non-Archimedean Banach spaces over $\mathbb{K}.$ We give a necessary and sufficient condition on the range of bounded linear operators to be closed. Moreover, many results are proved.

The fuzzy conformable Laplace transforms proposed in \cite{lp} are used to solve only fuzzy fractional differential equations of order $ 0 < \iota \leq 1$. In this article, under the generalized conformable fractional derivatives notion, we extend and use this method to solve fuzzy fractional differential equations of order $ 0 < \iota \leq 2$.

In this paper, we aim to extend the Darboux frame field into 3-dimensional Anti-de Sitter space and obtain two cases for this extension by considering a parameterized curve on a hypersurface; then we carry out the Euler-Lagrange equations and derive differential equations for non-null elastic curves in AdS$_{3}$ (i.e. 3-dimensional Anti-de Sitter space). In this study, we investigate the elastic curves in AdS$_{3}$ and obtain equations through which elastic curves are found out. Therefore, we solve these equations numerically and finally plot and design some elastic curves.

In this paper, we present a generalization of the Montgomery Identity to various time scale versions, including the discrete case, continuous case, and the case of quantum calculus. By obtaining this generalization of Montgomery Identity  we establish results about the generalization of Ostrowski-Gr\"{u}ss like inequality to the several time scales, namely discrete case, continuous case and the case of quantum calculus. Additionally, we recapture several published results from different authors in various papers, thus unifying the corresponding discrete and continuous versions. Furthermore, we demonstrate the applicability of our derived consequence to the case of quantum calculus.

We give a  method to establish a fixed point via partial $b$-metric  for multivalued mappings.  Since the geometry of multivalued fixed points perform a significant role in numerous real-world problems and is fascinating and innovative, we introduce the notions of fixed circle and fixed disc to frame  hypotheses to establish fixed circle/ disc theorems in a  space that permits non-zero self-distance with a coefficient more significant  than one.  Stimulated by the reality that the fixed point theorem is the frequently used technique for solving boundary value problems, we solve a pair of elliptic boundary value problems.  Our developments cannot be concluded from the current outcomes in related metric spaces. Examples are worked out to substantiate the validity of the hypothesis of our results.

Considering slowly varying functions (SVF), %Seneta (2019) Seneta in 2019 conjectured the following implication, for $\alpha\geq1$,$$\int_0^x y^{\alpha-1}(1-F(y))dy\textrm{\ is SVF}\ \Rightarrow\ \int_{0}^x y^{\alpha}dF(y)\textrm{\ is SVF, as $x\to\infty$,}$$where $F(x)$ is a cumulative distribution function on $[0,\infty)$. By applying the Williamson transform, an extension of this conjecture is proved. Complementary results related to this transform and particular cases of this extended conjecture are discussed.

Assuming that $\Lambda$ is a bounded operator on a Hilbert space $H$, this study investigate the structure of the $g$-frames generated by  iterations of $\Lambda$. Specifically, we provide  characterizations of $g$-frames  in the form of $\{\Lambda^n\}_{n=1}^{\infty}$ and describe some conditions under which the sequence $\{\Lambda^n\}_{n=1}^{\infty}$ forms  a $g$-frame for $H$. Additionally, we verify the properties of the operator $\Lambda$ when $\{\Lambda^n\}_{n=1}^{\infty}$ is a $g$-frame for $H$. Moreover, we study the $g$-Riesz bases and dual $g$-frames which are generated by iterations. Finally, we discuss the stability of these types of $g$-frames under some perturbations.