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The application of Caputo-Katugampola gH−differentiability to solve systems of fractional partial differential equations is investigated in this work. The existence and uniqueness of two types of gH−weak solutions for fuzzy fractional coupled partial differential equations are established. Lipschitz conditions are employed, and the Banach fixed-point theorem and mathematical induction are used in our approach. To address the challenges of the initial value problems, a matrix-form Cornwall’s inequality is developed. Additionally, a novel analysis of the continuous dependence of the coupled system’s solutions on the given conditions and approximate solutions is provided. An illustrative example is presented to validate the findings.

Abstract This paper presents a systematic approach to investigating the existence of solutions for fractional integro-differential equation systems incorporating delay and relaxation impulsive terms. By employing suitable definitions of fractional derivatives, we establish physically interpretable boundary conditions. To account for abrupt state changes, impulsive conditions are integrated into the model. The system is transformed into an equivalent integral equation, facilitating the application of Banach and Schaefer fixed-point theorems to prove the existence and uniqueness of solutions. The practical applicability of our findings is demonstrated through an illustrative example.

Abstract This manuscript is concerned with obtaining results for fixed points that arise from new contractive mappings on controlled metric spaces. These mappings are a mixture of Wardowski’s contractions with both multivalued, nonlinear mappings and auxiliary functions. It is also proved that the obtained fixed-point outcomes are well-posed. Additionally, a data-dependence result for fixed points is given. To aid with understanding, several illustrative examples are also provided. Numerous findings that are currently in the literature are specific instances of the findings that were made.

The concept of contraction mappings plays a significant role in mathematics, particularly in the study of fixed points and the existence of solutions for various equations. In this study, we described two types of enriched contractions: enriched $ F $-contraction and enriched $ F^{\prime } $-contraction associated with $ u $-fold averaged mapping, which are involved with Kirk's iterative technique with order $ u $. The contractions extracted from this paper generalized and unified many previously common super contractions. Furthermore, $ u $-fold averaged mappings can be seen as a more general form of both averaged mappings and double averaged mappings. Moreover, we demonstrated that the $ u $-fold averaged mapping with enriched contractions has a unique fixed point. Our work examined the necessary conditions for the $ u $-fold averaged mapping and weak enriched contractions to have equal sets of fixed points. Additionally, we illustrated that an appropriate Kirk's iterative algorithm can effectively approximate a fixed point of a $ u $-fold averaged mapping as well as the two enriched contractions. Also, we delved into the well-posedness, limit shadowing property, and Ulam-Hyers stability of the $ u $-fold averaged mapping. Furthermore, we established necessary conditions that guaranteed the periodic point property for each of the illustrated strengthened contractions. To underscore the generality of our findings, we presented several examples that aligned with comparable results found in the existing literature.

This work primarily delves into three key areas: the presence of mild solutions, exploration of the topological and geometrical makeup of solution sets, and the continuous dependency of solutions on a second-order semilinear integro-differential inclusion. The Bohnenblust-Karlin fixed-point method has been integrated with Grimmer's theory of resolvent operators. Ultimately, the study delves into a mild solution for a partial integro-differential inclusion to showcase the achieved outcomes.

This study focused on introducing a fresh model of fractional operators incorporating multiple delays, termed fractional integro-differential Langevin equations with multiple delays. Additionally, the research evaluated the relative controllability of this model within finite-dimensional spaces. Employing fixed-point theory yields the desired outcomes, with the controllability assessment facilitated by Schauder's fixed point and the Grammian matrix defined through the Mittag-Leffler matrix function. Validation of the results was conducted through an application.

Many scholars have lately explored fractional-order boundary value issues with a variety of conditions, including classical, nonlocal, multipoint, periodic/anti-periodic, fractional-order, and integral boundary conditions. In this manuscript, the existence and uniqueness of solutions to sequential fractional differential inclusions via a novel set of nonlocal boundary conditions were investigated. The existence results were presented under a new class of nonlocal boundary conditions, Carathéodory functions, and Lipschitz mappings. Further, fixed-point techniques have been applied to study the existence of results under convex and non-convex multi-valued mappings. Ultimately, to support our findings, we analyzed an illustrative example.

This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erdélyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying periodic contraction rate has been explored by using the Banach contraction principle. Finally, two numerical examples demonstrate the practical application of these findings in the realm of fractional integral equation theory.

Developing a model of fractional differential systems and studying the existence and stability of a solution is considered one of the most important topics in the field of analysis. Therefore, this manuscript was dedicated to presenting a hybrid system of delay fractional differential equations with boundary integral conditions, namely Atangana-Baleanu-Caputo (ABC) differential equations. Also, the fixed point (FP) technique has been applied to investigate the existence of solutions for the proposed hybrid system. Moreover, stability results have been studied for the solution of the desired system in the sense of Hyers-Ulam (HU). Finally, to support our findings, we provide two examples with different parameter values.

This article delves deeply into some mathematical basic theorems and their diverse applications in a variety of domains. The major issue of interest is the Banach Fixed Point Theorem (BFPT), which states the existence of a unique fixed point in fractional metric spaces. The significance of this theorem stems from its utility in a variety of mathematical situations for approximating solutions and resolving iterative problems. On this foundational basis, the study expands by introducing the concept of fractional geometric contraction mappings, which provide a new perspective on how convergence develops in fractional metric spaces.

In this article, we present two inertial modifications of regularized algorithms for the split variational inclusion problem (SVIP, for short) in real Hilbert spaces (RHSs). When the circumstances are right, strong convergence theorems are demonstrated. The major findings are used to solve the split minimization, split common fixed point problem (SCFPP), split minimization problem (SMP), and split feasibility problems (SFP) in applications. The proposed algorithms are contrasted with a number of other existing algorithms in the literature in order to test their numerical performances. Finally, the computer tests demonstrate that the suggested algorithms outperform alternative strategies in terms of speed and efficiency.

In this corrigendum, we would like to emphasize that the findings in paper [1] are a generalization of the results presented by Zhou et al. in [2]. This remark highlights key elements of prior research that are relevant to our work [1]. This correction does not alter any results or the conclusion of the article.

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we used the fixed-point method to examine the existence of a solution to the proposed system, mainly focusing on the theorems of Leray, Schauder, and Perov in generalized metric spaces. Finally, an example has been provided to support and underscore our results.