Search Results (26)

Let $\mathcal{E}$ and $\mathcal{F}$ be nonempty disjoint subsets of a metric space $(M,d)$. For a non-self-mapping $\phi :\mathcal{E}\to \mathcal{F}$, which is fixed-point free, a point $\varkappa \in \mathcal{E}$ is said to be a best proximity point for the mapping $\phi$ whenever the distance of the point $\varkappa$ to its image under $\phi$ is equal to the distance between the sets, $\mathcal{E}$ and $\mathcal{F}$. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations. Full article

We give a variational characterization of the Born rule. For each measurement context, we project a quantum state ρ onto the corresponding abelian algebra by minimizing Umegaki relative entropy; Petz’s Pythagorean identity makes the dephased state the unique local minimizer, so the Born weights $p_C\left(i\right)=\mathrm{Tr}\left(\rho P_i\right)$ arise as a consequence, not an assumption. Globally, we measure contextuality by the minimum classical Kullback–Leibler distance from the bundle $\left\{p_C\left(\rho \right)\right\}$ to the noncontextual polytope, yielding a convex objective $\Phi \left(\rho \right)$. Thus, $\Phi \left(\rho \right)=0$ exactly when a sheaf-theoretic global section exists (noncontextuality), and $\Phi \left(\rho \right)>0$ otherwise; the closest noncontextual model is the classical I-projection of the Born bundle. Assuming finite dimension, full-rank states, and rank-1 projective contexts, the construction is unique and non-circular; it extends to degenerate PVMs and POVMs (via Naimark dilation) without change to the statements. Conceptually, the work unifies information-geometric projection, the presheaf view of contextuality, and categorical classical structure into a single optimization principle. Compared with Gleason-type, decision-theoretic, or envariance approaches, our scope is narrower but more explicit about contextuality and the relational, context-dependent status of quantum probabilities. Full article

We introduce a quantum integrated-information measure $\Phi$ for multipartite states within the Relational Quantum Dynamics (RQD) framework. $\Phi \left(\rho \right)$ is defined as the minimum quantum Jensen–Shannon distance between an n-partite density operator $\rho$ and any product state over a bipartition of its subsystems. We prove that its square root induces a genuine metric on state space and that $\Phi$ is monotonic under all completely positive trace-preserving maps. Restricting the search to bipartitions yields a unique optimal split and a unique closest product state. From this geometric picture, we derive a canonical entanglement witness directly tied to $\Phi$ and construct an integration dendrogram that reveals the full hierarchical correlation structure of $\rho$. We further show that there always exists an “optimal observer”—a channel or basis—that preserves $\Phi$ better than any alternative. Finally, we propose a quantum Markov blanket theorem: the boundary of the optimal bipartition isolates subsystems most effectively. Our framework unites categorical enrichment, convex-geometric methods, and operational tools, forging a concrete bridge between integrated information theory and quantum information science. Full article

The focus of this paper is on three types of convexity: generalized uniform convexity, $\Phi$-convexity and superquadracity. The similar structures of these types of convexity are such that the same processes can be applied to each one of them to obtain further refinements of known inequalities. Full article

In this work, we describe the $\mathfrak{q}$-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\tau )$, and we obtain findings related to geometric function theory (GFT) by utilizing approaches established through subordination and knowledge of $\mathfrak{q}$-calculus operators. By using this operator, we develop generalized classes of quasi-convex and close-to-convex functions in this paper. Additionally, the classes ${\mathfrak{K}}_{\mathfrak{q},\rho }^s(\nu ,\tau )\left(\phi \right)$, ${\mathfrak{Q}}_{\mathfrak{q},\rho }^s(\nu ,\tau )\left(\phi \right)$ are introduced. The invariance of these recently formed classes under the $\mathfrak{q}$-Bernardi integral operator is investigated, along with a number of intriguing inclusion relationships between them. Additionally, several unique situations and the beneficial outcomes of these studies are taken into account. Full article

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces $L^{\Phi }$ generated by the b-Orlicz function $\Phi$ equipped with a Luxemburg quasi-norm contain both classical spaces $L^p(p\ge 1)$ and $L^p(0<p<1)$. The Orlicz spaces $L^{\Phi }$ are quasi-Banach spaces. Some basic properties in quasi-normed Orlicz spaces are discussed, and the criteria that a quasi-normed Orlicz space is strictly monotonic and lower (upper) locally uniformly monotonic are given. Full article

Let $M_g^{+}f$ be the one-sided Hardy–Littlewood maximal function, ${\phi }_1$ be a nonnegative and nondecreasing function on $[0,\infty )$, $\gamma$ be a positive and nondecreasing function defined on $[0,\infty )$; let ${\phi }_2$ be a quasi-convex function and $u,v,w$ be three weight functions. In this paper, we present necessary and sufficient conditions on weight functions $(u,v,w)$ such that the inequality ${\phi }_1\left(\lambda \right){\int }_{\{M_g^{+}f>\lambda \}}u\left(x\right)g\left(x\right)dx\le C{\int }_{-\infty }^{+\infty }{\phi }_2\left(C{\displaystyle \frac{\left|f\right(x\left)\right|v\left(x\right)}{\gamma \left(\lambda \right)}}\right)w\left(x\right)g\left(x\right)dx$ holds. Then, we unify the weak and extra-weak-type one-sided Hardy–Littlewood maximal inequalities in the above inequality. Full article

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In addition to a detailed derivation of a few special situations, certain known findings are summarized. We also point out that some results in this study, in some special cases, such as setting $\alpha =0=\phi ,\gamma =1,$ and $w=0,\sigma \left(0\right)=1,\lambda =1$, are more reasonable than those obtained. Finally, it is believed that the technique presented in this paper will encourage additional study in this field. Full article

In this article, we examine variational inequalities of the form $\left\{\begin{array}{c}⟨\mathcal{A}\left(u\right),v-u⟩+⟨\mathcal{F}\left(u\right),v-u⟩\ge 0,\forall v\in K\hfill \\ u\in K,\hfill \end{array}\right.$, where $\mathcal{A}$ is a generalized fractional $\mathsf{\Phi }$-Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and $\mathcal{F}$ is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term $\mathcal{F}$ such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions. Full article

Let ${\displaystyle E}$ be a Hausdorff locally convex space. We investigate the space ${\displaystyle {\mathsf{\Lambda }}_{\phi }\left[E\right]}$ of weakly Köthe–Orlicz summable sequences in ${\displaystyle E}$ with respect to an Orlicz function ${\displaystyle \phi }$ and a perfect sequence space ${\displaystyle \mathsf{\Lambda }}$. We endow ${\displaystyle {\mathsf{\Lambda }}_{\phi }\left[E\right]}$ with a Hausdorff locally convex topology and determine the continuous dual of the so-obtained space in terms of strongly Köthe–Orlicz summable sequences from the dual space ${\displaystyle E^{\prime }}$ of ${\displaystyle E}$. Next, we give necessary and sufficient conditions for ${\displaystyle {\mathsf{\Lambda }}_{\phi }\left[E\right]}$ to be barrelled or quasi-barrelled. This contributes to the understanding of different spaces of vector-valued sequences and their topological properties. Full article

Ensuring the long-term, efficient, and safe operation of reservoir dams relies on the slope stability of embankment dams. Periodic fluctuations of the reservoir water level due to reservoir scheduling operations make the slope of the reservoir bank vulnerable to instability. To investigate the influence of various factors and their interactions with embankment dam slope stability under changing reservoir water levels, a global sensitivity analysis method is proposed that accounts for seepage–stress coupling. An embankment dam in Shaanxi Province, China, is studied as an example, with COMSOL Multiphysics software simulating the seepage and slope stability of the dam under fluctuating reservoir water level conditions and seepage–stress coupling. The global sensitivity analysis of factors affecting dam slope stability is accomplished by combining Plackett–Burman and Box–Behnken experimental designs, with ANOVA determining the sensitivity of each factor and interaction term. The results demonstrate that during the impoundment period of the reservoir, the saturation line is concave, and the overall stability safety of the dam slope increases first and then tends to be stable, according to the coefficient. The internal friction angle φ, cohesion c, and soil density ρ_s represent the three most sensitive factors affecting the stability and safety of the dam slope, while c × ρ_s is a second-order interaction term with significant sensitivity to the stability and safety coefficient of the dam slope. The reservoir drainage period infiltration line is convex, and dam slope stability first reduced and then increased. The magnitude of water level change H, internal friction angle φ, cohesion c, and soil density ρ_s are the four most sensitive factors for the coefficient of safety of dam slope stability, while c × ρ_s, H × ρ_s, and φ × ρ_s are the second-order interaction terms with significant sensitivity to the coefficient of safety of dam slope stability. These research findings and methods can offer valuable technical support and reference for the investigation and evaluation of the stability of embankment dam slopes. Full article

In this paper, we propose two novel inertial forward–backward splitting methods for solving the constrained convex minimization of the sum of two convex functions, ${\phi }_1+{\phi }_2,$ in Hilbert spaces and analyze their convergence behavior under some conditions. For the first method (iFBS), we use the forward–backward operator. The step size of this method depends on a constant of the Lipschitz continuity of $\nabla {\phi }_1,$ hence a weak convergence result of the proposed method is established under some conditions based on a fixed point method. With the second method (iFBS-L), we modify the step size of the first method, which is independent of the Lipschitz constant of $\nabla {\phi }_1$ by using a line search technique introduced by Cruz and Nghia. As an application of these methods, we compare the efficiency of the proposed methods with the inertial three-operator splitting (iTOS) method by using them to solve the constrained image inpainting problem with nuclear norm regularization. Moreover, we apply our methods to solve image restoration problems by using the least absolute shrinkage and selection operator (LASSO) model, and the results are compared with those of the forward–backward splitting method with line search (FBS-L) and the fast iterative shrinkage-thresholding method (FISTA). Full article

As a generalization of a geodesic function, this paper introduces the notion of the geodesic ${\phi }_E$-convex function. Some properties of the ${\phi }_E$-convex function and geodesic ${\phi }_E$-convex function are established. The concepts of a geodesic ${\phi }_E$-convex set and ${\phi }_E$-epigraph are also given. The characterization of geodesic ${\phi }_E$-convex functions in terms of their ${\phi }_E$-epigraphs, are also obtained. Full article

In this article, we demonstrated various Hermite–Hadamard and Fejér type inequalities for modified h-convex functions. We showed several inequalities for the products of two modified h-convex functions. New identities related to inequalities in various forms are also established for different values of the $h\left({\phi }_t\right)$ function. We believe that the approach presented in this paper will inspire more research in this area. Full article

Using the direct method, we prove the Ulam stability results for the general linear functional equation of the form ${\sum }_{i=1}^m{A}_i(f\left({\phi }_i\left(\overline{x}\right))\right)=D\left(\overline{x}\right)$ for all $\overline{x}\in X^n$, where f is the unknown mapping from a linear space X over a field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ into a linear space Y over field $\mathbb{K}$; n and m are positive integers; ${\phi }_1,\dots ,{\phi }_m$ are linear mappings from $X^n$ to X; $A_1,\dots ,A_m$ are continuous endomorphisms of Y; and $D:X^n\to Y$ is fixed. In this paper, the stability inequality is considered with regard to a convex modular on Y, which is lower semicontinuous and satisfies an additional condition (the ${\Delta }_2$-condition). Our main result generalizes many similar stability outcomes published so far for modular space. It also shows that there is some kind of symmetry between the stability results for equations in modular spaces and those in classical normed spaces. Full article