Search Results (120)

This research explores two novel geometric concepts—nearly convex points and locally nearly uniformly convex points within the frameworks of Banach spaces and Orlicz spaces equipped with the Luxemburg norm. First, we establish the general characterization criteria for nearly convex points in Banach spaces. Then, we analyze the intrinsic connection between locally nearly uniformly convex points and nearly extreme points in Banach spaces. Additionally, we provide comprehensive characterizations of locally nearly uniformly convex points in both Orlicz function spaces and Orlicz sequence spaces under the Luxemburg norm. These findings enrich the geometric theory system of Banach and Orlicz spaces, offering new theoretical support for related research directions. Full article

We study the Sneddon $\mathcal{R}$-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel $xcos\left(xt\right)+hsin\left(xt\right)$, the $\mathcal{R}$-transform acts as a unifying operator for sine- and cosine-type integral transforms. Boundedness, continuity, and weighted $L^p$-estimates are established in an appropriate Banach space framework, together with Parseval–Goldstein type identities. Initial and final value theorems are derived for generalized functions in Zemanian-type spaces, yielding precise asymptotic behaviour at the origin and at infinity. A finite-interval theory is also developed, leading to polynomial growth estimates and final value theorems for the finite $\mathcal{R}$-transform. Full article

This paper introduces new generalized forms of contractive mappings in the framework of complete metric spaces. By extending the classical Reich and Sehgal contractions to their iterated counterparts in Singh’s sense, we establish unified fixed-point theorems that ensure both existence and uniqueness under constant and variable contractive parameters. The proposed p-Reich and p-Sehgal contractions encompass several well-known results, including those of Banach, Kannan, Chatterjea, Reich, and Sehgal, as special cases. Convergence of the associated Picard iterative process is rigorously analyzed, revealing deeper insights into the iterative stability and asymptotic behavior of nonlinear mappings in metric spaces. The practical utility of our unified fixed-point theorems is illustrated through concrete applications in fractal and fractional calculus. Full article

This paper introduces a novel analytical and numerical framework for studying nonlinear fractional Volterra integral equations by employing an intuitionistic fuzzy metric structure over a Banach algebra. The principal contribution of this work is the development of fixed-point theory for a new class of intuitionistic fuzzy Z-contractions in $\mathcal{IFM}$-spaces over $\mathcal{BA}$, which extends existing fuzzy and algebra-valued metric frameworks. Within this setting, we established existence, uniqueness, and convergence results for solutions of fractional integral equations of the Caputo type by proving that the associated fractional integral operator satisfies the proposed contractive conditions. Furthermore, we demonstrated how the algebra-valued intuitionistic fuzzy structure enhances the analytical flexibility and robustness of the model. To support the theoretical findings, a numerical simulation based on a discretized iterative scheme is presented, illustrating the rapid convergence of the approximating sequence together with the monotone behavior of intuitionistic fuzzy nearness and non-nearness measures. The numerical results are consistent with the analytical theory and confirm the effectiveness of the proposed $\mathcal{IFM}$-spaces over the $\mathcal{BA}$ approach for fractional dynamical systems. Full article

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter $p\ge 1$, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated mappings ${\mathit{ħ}}^p$. We establish fixed-point theorems, ensuring both existence and uniqueness of fixed points for continuous self-maps on complete metric spaces that satisfy these p-contractive conditions. The proofs are constructed via geometric estimates on the iterates and by transferring the fixed point from the p-th iterate ${\mathit{ħ}}^p$ to the original mapping ħ. Our results unify and broaden several well-known fixed-point theorems reported in previous studies, including those of Banach, Hardy–Rogers, and Zamfirescu as special cases. Full article

This paper develops the theory of strongly continuous semigroups and abstract evolution equations in modular function spaces. We study the autonomous problem $\dot{u}\left(t\right)=Bu\left(t\right)$ with initial condition $u\left(0\right)=u_0\in L_{\rho }$, where B is the infinitesimal generator of a strongly continuous semigroup ${\left(S\left(t\right)\right)}_{t\ge 0}$ on $L_{\rho }$. Within this framework, we establish modular analogues of classical results from Banach-space semigroup theory, including criteria for $\rho$-boundedness and $\rho$-continuity, a Laplace resolvent representation of the generator, and explicit resolvent bounds in terms of the modular growth function ${\omega }_{\rho }$. Under a ${\Delta }_2$-type condition on the modular, we justify Steklov regularization of semigroup orbits, obtain domain inclusion and the resolvent identity, and derive spectral consequences for classes of operators naturally acting on $L_{\rho }$. The results show that the structural features of the classical semigroup framework persist in the modular topology, providing a unified approach to linear evolution in modular function spaces. Full article

This paper is devoted to the study of existence results for a nonlinear Langevin-type fractional $(p,q)$-difference equation in Banach space. The considered model extends the fractional q-difference Langevin equation by introducing two parameters p and q, which provide additional flexibility in describing discrete fractional processes. By using the Kuratowski measure of noncompactness together with Mönch’s fixed-point theorem, we derive sufficient conditions that guarantee the existence of at least one solution. The main idea consists in converting the boundary value problem into an equivalent fractional $(p,q)$-integral equation and verifying that the corresponding operator is continuous, bounded, and condensing. An illustrative example is presented to demonstrate the applicability of the obtained results. Full article

We establish a modular-space framework for the study of tripled fixed points and tripled best proximity points. Under suitable assumptions on the underlying modular (convexity, the ${\Delta }_2$ property, uniform continuity, and uniform convexity-type properties), we prove that Banach theorems guarantee the existence, uniqueness, and convergence of modular iterative schemes. In particular, we develop results for cyclic $\rho$–Kannan contraction maps and pairs, showing that both tripled fixed points and tripled best proximity points arise uniquely and attract all iterative trajectories. An illustrative example in the space $L^2[0,1]$ with integral operators demonstrates the applicability of the theory and the predicted rate of convergence. These results extend classical fixed point methods to a broader modular setting and open the way for applications in nonlinear functional equations. Full article

In this article, we deduce the existence of a solution to the weighted fractional differential equation with functional boundary data involving an $\omega$-weighted fractional derivative with Riemann–Liouville settings, ${\mathfrak{D}}_{0^{+}}^{\alpha ,\psi ,\omega }$ of order $\alpha \in ]n-1,n[$, on certain weighted Banach spaces when the nonlinear term contains the proportional delay term and fractional derivatives of order $(0,1)$. After carefully defining a few weighted spaces and building a few weighted projection operators, we use Mawhin’s coincidence theory to derive a number of existence results at resonance. Furthermore, our method generalizes some prior results because numerous fractional differential operators are specific instances of the operator ${\mathfrak{D}}_{0^{+}}^{\alpha ,\psi ,\omega }$ and represent functional boundary conditions in a highly generic way. Lastly, we illustrate and support our theoretical results with an example. Full article

In this paper, a $k/n\left(G\right)$ retrial system with multiple working vacations is considered, and a mathematical model of the system is established by supplementary variable method, and a dynamic analysis of the system is carried out. Firstly, the model is transformed into an abstract Cauchy problem in Banach space by introducing the state space, main operator and its definition domain. Secondly, the $C_0$-semigroup theory in functional analysis and the spectral theory of linear operators are used to prove that the main operator of the model generates a positive contraction $C_0$-semigroup, which leads to the existence of a unique, non-negative time-dependent solution of the system that satisfies the probabilistic properties. Finally, Greiner’s boundary perturbation idea and the spectral properties of the corresponding operators are used to show that the time-dependent solution strongly converges to its steady-state solution. Full article

This paper analyzes the Bagley–Torvik fractional-order equation with generalized fractional Hilfer derivatives of two orders for functions in Banach spaces under conditions expressed in the language of weak topology. We develop a comprehensive theory of fractional-order differential equations of various orders. Our focus is on the equivalence results (or the lack thereof) of this new class of fractional-order Hilfer operators and on maximizing the regularity of the solution. To this end, we examine the equivalence of differential problems involving pseudo-derivatives and integral problems involving Pettis integrals. Our results are novel, even within the context of integer-order differential equations. Another objective is to incorporate fractional-order problems into the growing research field that uses weak topology and function spaces to study vector-valued functions. The auxiliary results obtained in this article are general and applicable beyond its scope. Full article

This paper investigates the existence of square-mean S-asymptotically $(\omega ,c)$-periodic solutions for a class of neutral impulsive stochastic differential equations driven by fractional Brownian motion, addressing the challenge of modeling long-range dependencies, delayed feedback, and abrupt changes in systems like biological networks or mechanical oscillators. By employing semigroup theory to derive mild solution representations and the Banach contraction principle, we establish sufficient conditions–such as Lipschitz continuity of nonlinear terms and growth bounds on the resolvent operator—that guarantee the uniqueness and existence of such solutions in the space ${\mathcal{S}AP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. The important results demonstrate that under these assumptions, the mild solution exhibits square-mean S-asymptotic $(\omega ,c)$-periodicity, enabling robust asymptotic analysis beyond classical periodicity. We illustrate these findings with examples, such as a neutral stochastic heat equation with impulses, revealing stability thresholds and decay rates and highlighting the framework’s utility in predicting long-term dynamics. These outcomes advance stochastic analysis by unifying neutral, impulsive, and fractional noise effects, with potential applications in control theory and engineering. Full article

This paper introduces a new class of generalized metric structures, called interpolative b-metric spaces, which unify and extend both b-metric spaces and interpolative metric spaces in a non-trivial way. By incorporating a nonlinear correction term alongside a multiplicative scaling parameter into the triangle inequality, this framework enables broader contractive conditions and refined control of convergence behavior. We develop the foundational theory of interpolative b-metric spaces and establish a generalized Ćirić-type fixed point theorem, along with Banach, Kannan, and Bianchini-type results as corollaries. To highlight the originality and applicability of our approach, we apply the main theorem to a nonlinear Volterra-type integral equation, demonstrating that interpolative b-metrics effectively accommodate nonlinear solution structures beyond the scope of traditional metric models. This work offers a unified platform for fixed point analysis and opens new directions in nonlinear and functional analysis. Full article

This paper investigates the boundedness of multiplication operators $M_{\psi }$ between natural $\mu$-Bloch-type spaces $B_{\mu ,\mathrm{nat}}(B_X)$ (or their little $\mu$-Bloch counterparts) and natural $\omega$-Bloch-type spaces $B_{\omega ,\mathrm{nat}}(B_X)$ on the unit ball $B_X$ of a complex Banach space X. We establish complete characterizations for the boundedness of $M_{\psi }$ under varying conditions on the weight functions $\mu$ and $\omega$, including specific cases such as logarithmic and power-weighted Bloch spaces. The results extend classical operator theory to infinite-dimensional settings, unifying prior work on finite-dimensional domains. Full article

Recent developments in fixed-point theory have focused on iterative techniques for approximating solutions, yet there remain important questions about whether different methods are equivalent and how well they resist perturbations. In this study, two recently proposed algorithms, referred to as the DF and AR iteration methods, are shown to be connected by proving that they converge similarly when applied to contraction mappings in Banach spaces, provided that their control sequences meet specific, explicit conditions. This work extends previous research on data dependence by removing restrictive assumptions related to both the perturbed operator and the algorithmic parameters, thereby increasing the range of situations where the results are applicable. Utilizing a non-asymptotic analysis, the authors derive improved error bounds for fixed-point deviations under operator perturbations, achieving a tightening of these estimates by a factor of 3–15 compared to earlier results. A key contribution of this study is the demonstration that small approximation errors lead only to proportionally small deviations from equilibrium, which is formalized in bounds of the form $\parallel s^{*}-{\tilde{s}}^{*}\parallel \le O(\epsilon /(1-\lambda ))$. These theoretical findings are validated through applications involving integral equations and examples from function spaces. Overall, this work unifies the convergence analysis of different iterative methods, enhances guarantees regarding stability, and provides practical tools for robust computational methods in areas such as optimization, differential equations, and machine learning. By relaxing structural constraints and offering a detailed sensitivity analysis, this study significantly advances the design and understanding of iterative algorithms in applied mathematics. Full article