Satyendra Narayan

In this article we study semi-Markov decision processes (SMDPs) where the pay-off criterion is limiting ratio average, generally known as undiscounted pay-off. Here we consider the action space of the decision maker to be possibly countably infinite. However, we do not put any restriction on the reward function. We prove the existence of a near-optimal or ϵ-optimal strategy of the decision maker which turns out to be a deterministic semi-stationary. An efficient algorithm is discussed to compute a near-optimal pure semi-stationary strategy for such SMDP model. Also under some standard ergodicity conditions, we propose an optimality equation of these SMDP models.

In this paper, we introduce and systematically study the class of interpolative Geraghty-type contractive mappings within the framework of complete bicomplex-valued metric spaces (bi-CVMS). We prove seven new results: (i) a fixed point theorem for a single interpolative Geraghty contraction; (ii) a common fixed point theorem for a pair of such mappings; (iii) a fixed point theorem for interpolative Reich–Rus–Ćirić type contractions in bi-CVMS; (iv) a coincidence point and common fixed point theorem for weakly compatible maps; (v) a fixed point theorem for Jaggi-type hybrid contractions in bi-CVMS; (vi) a stability result for the Picard iteration associated with the main contraction; and (vii) an application theorem establishing the existence and uniqueness of solutions to a boundary value problem governed by a Caputo fractional differential equation. All results are furnished with complete proofs and non-trivial illustrative examples. Several well-known theorems—including those of Banach, Kannan, Reich, Geraghty, and their complex-valued analogues—follow as special cases. The paper significantly advances the fixed point theory in bicomplex-valued metric spaces.

Introduction:: This paper introduces the Modified Soft Neutrosophic Metric Space (MSNMS), a framework that generalizes classical metric spaces by integrating soft set and neutrosophic set theory. The study aims to establish fundamental properties and investigate the topological structure of MSNMS. Methods:: Core operations and key properties of MSNMS are defined and analyzed using mathematical derivations. Illustrative examples demonstrate the structure, while practical applications, particularly in solving integral equations, validate the theoretical results. objective: see manuscript Results:: The MSNMS framework successfully establishes essential topological properties and verifies critical theorems. Its application to integral equations demonstrates improved accuracy and robustness compared to existing methods. Discussion:: MSNMS effectively handles uncertainty and imprecision, extending traditional metric space concepts. The framework’s practical applicability highlights its potential in mathematical modeling and engineering problems. Conclusion:: MSNMS provides a versatile and powerful tool, combining rigorous theory with practical effectiveness. It generalizes existing metric spaces and offers enhanced methods for solving complex problems. AMS Mathematics Subject Classification: 47H10; 54H25.

In this paper, we introduce and systematically study the class of interpolative Geraghty-type contractive mappings within the framework of complete bicomplex valued metric spaces (bi-CVMS). We prove seven new results: (i) a fixed point theorem for a single interpolative Geraghty contraction; (ii) a common fixed point theorem for a pair of such mappings; (iii) a fixed point theorem for interpolative Reich–Rus–Ćirić type contractions in bi-CVMS; (iv) a coincidence point and common fixed point theorem for weakly compatible maps; (v) a fixed point theorem for Jaggi-type hybrid contractions in bi-CVMS; (vi) a stability result for the Picard iteration associated with the main contraction; and (vii) an application theorem establishing the existence and uniqueness of solutions to a boundary value problem governed by a Caputo fractional differential equation. All results are furnished with complete proofs and non-trivial illustrative examples. Several well-known theorems — including those of Banach, Kannan, Reich, Geraghty, and their complex-valued analogues — follow as special cases. The paper significantly advances the fixed point theory in bicomplex valued metric spaces.

This study presents a comprehensive approach to solving fuzzy multi-objective linear programming problems (FMOLPP) under uncertainty using trapezoidal fuzzy numbers. The authors propose a novel integration of Yager’s ranking method, the Big-M optimization technique, and Chandra Sen’s statistical mean methods to effectively convert fuzzy objectives into crisp values and optimize them. The methodology allows for managing multiple fuzzy objectives by ranking and aggregating them using various statistical means such as arithmetic, geometric, quadratic, harmonic, and Heronian averages. The model is implemented using TORA software and demonstrated through a detailed numerical example. The results validate the robustness and practicality of the proposed approach, showcasing consistent optimal solutions across all statistical methods. This research significantly enhances decision-making processes in uncertain environments by offering a structured, computationally efficient solution strategy for complex real-world optimization problems.

This research paper delves into the application of optimality results in orthogonal fuzzy metric spaces to demonstrate the existence and uniqueness of solutions of nonlinear differential equations with boundary conditions and nonlinear integral equations, emphasizing the importance of orthogonal fuzzy metric spaces in extending fixed-point theory. Through introducing this innovative concept, the study provides a theoretical framework for analyzing mappings in diverse scenarios. In this study, we introduce the concept of best proximity point (BPP) within the framework of orthogonal fuzzy metric spaces by employing orthogonal fuzzy proximal contractive mappings. Moreover, this research explores the implications of the established results, considering both self-mappings and non-self mappings that share the same parameter set. Additionally, some examples are provided to illustrate the practical relevance of the proven results and consequences in various mathematical contexts. The findings of this study can open up avenues for further exploration and application in solving real-world problems.

The present research introduces a novel concept termed “neutrosophic fuzzy metric space”, which extends the traditional metric space framework by incorporating the notion of neutrosophic fuzzy sets. A thorough investigation of various structural and topological properties within this newly proposed generalization of metric space has been conducted. Additionally, counterparts of well-known theorems such as the Uniform Convergence Theorem and the Baire Category Theorem have been established for this generalized metric space. Through rigorous analysis, a detailed understanding of its fundamental characteristics has been attained, illuminating its potential applications and theoretical significance.