Existence and uniqueness results for non-linear fractional integral equation with reciprocal singularities

In this paper, we investigate a (k,Υ) fractional quadratic integral equation in the Banach space of real-valued continuous functions on [0,1]. By using a measure of noncompactness associated with monotonicity and Darbo’s fixed point theorem, we provide sufficient conditions for the existence of at least one monotonic solution and analyze its stability. Finally, an illustrative example is presented to demonstrate the theoretical results, including several particular cases.

Solvability of a quadratic nonlinear fractional integral equation

The primary objective of this research article is to investigate the concept of extended F-metric spaces and to establish a series of fixed point theorems for generalized contractions within this framework. We further introduce and analyze the notion of interpolative Kannan-type cyclic contractions in extended F-metric spaces, deriving several novel fixed point results associated with these mappings. In addition, we obtain common fixed point theorems for rational contractions, thereby extending and unifying a variety of existing results available in the literature. To highlight the novelty and effectiveness of the proposed results, several illustrative examples are provided. Moreover, the theoretical findings are successfully applied to the solution of Atangana–Baleanu fractional integral equations as well as Volterra integral equation of Hammerstein type, demonstrating their practical significance and wide-ranging applicability.

This paper introduces the Caputo fractional differential equation for interval valued functions. A fractional differential equation incorporates memory sense through an iterated kernel for describing dynamical systems. Impreciseness exists in the process of quantification and analysis of the involved variables influencing such physical processes. In other words, memory and imprecision may coexist, necessitating the study of an imprecise fractional differential equation. Interval numbers and interval valued functions are mathematical tools to manifest uncertainty due to the variance of decision parameters between ranges. In this paper, an imprecise fractional differential equation is studied under Type 2 interval uncertainty, a generalization of interval uncertainty. This paper analyzes conditions for the existence of a unique solution of the Type 2 interval valued Caputo fractional differential equations. Riemann–Liouville fractional integral equations and metric spaces for Type 2 interval numbers and interval valued functions are employed for establishing results. Examples of linear and nonlinear Type 2 interval valued Caputo fractional differential equations are discussed, ensuring a smooth extension of the interval fractional differential equation to a wider domain. Economic and biological models are hinted at as possible applications of this proposed theory.

In this article, we investigate a nonlinear ψ-Hilfer fractional order Volterra integro-delay differential equation (ψ-Hilfer FRVIDDE) and a nonlinear ψ-Hilfer fractional Volterra delay integral equation (ψ-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the ψ-Hilfer FRVIDDE and ψ-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered ψ-Hilfer FRVIDDE and ψ-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature.

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 IFM-spaces over 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 IFM-spaces over the BA approach for fractional dynamical systems.

In this study, we aim to investigate the existence results for a general class of fractional nonlinear integral equations with order α ∈ ( 0 , 1 ) in a continuous function space ( C [ a , b ] , ∥ ⋅ ∥ ) . We use the Schauder fixed point theorem as a tool with providing a compact integral operator from a subset of the Banach space ( C [ a , b ] , ∥ ⋅ ∥ ) into itself. Furthermore, we present an example to illustrate and support the work.

The product Simpson method: excellent estimating solution of fractional integral equations via uniformly modulus of continuity

This paper is dedicated to the investigation of new generalizations of the classical Gronwall–Bellman–Bihari integral inequalities, which are fundamental tools in the qualitative and quantitative analysis of differential, integral, and integro-differential equations. We establish two primary, novel theorems. The first theorem presents a significant generalization for inequalities involving composite nonlinear functions and iterated integrals. This result provides an explicit bound for an unknown function u(t) satisfying an inequality of the form Φ(u(t))≤a(t)+∫t0t f(s)Ψ(u(s))ds+∫t0t g(s)Ω(∫t0s h(τ)K(u(τ))dτ)ds. The proof is achieved by defining a novel auxiliary function and applying a rigorous comparison principle. The second main theorem establishes a new bound for a class of fractional integral inequalities involving the Riemann–Liouville fractional integral operator Iα and a non-constant coefficient function b(t) in the form u(t)≤a(t)+b(t)Iα[ω(u(s))]. This result extends several recent findings in the field of fractional calculus. The mathematical derivations are detailed, and the assumptions on the involved functions are made explicit. To illustrate the utility and potency of our main results, we present two applications. The first application demonstrates how our first theorem can be used to establish uniqueness and boundedness for solutions to a complex class of nonlinear integro-differential equations. The second application utilizes our fractional inequality theorem to analyze the qualitative behavior (specifically, the boundedness of solutions) for a generalized class of fractional integral equations. These new inequalities provide a powerful analytical framework for studying complex dynamical systems that were not adequately covered by existing results.

Theaim of this research is to establish existence and uniqueness results for the Riemann–Liouville fractional integral equation of order αϰ(t)=f(t)+λΓα∫0tt−sα−1gs,ϰ(s)ds,t∈[0,1], by developing common fixed point theorems for generalized contractions involving control functions of two variables in the framework of complex valued suprametric spaces. The proposed results extend and generalize several existing findings in the literature, and some illustrative examples are provided to demonstrate the novelty and applicability of the main theorem.

Application of Hat functions to simulate coupled systems of partial two-dimensional nonlinear fractional Volterra integral equations

This paper investigates fixed point theory for mean nonexpansive mappings in $p$-uniformly convex metric spaces. It first establishes the existence of fixed points together with a demiclosedness principle in this setting. Building on these foundations, the two-step Karakaya iteration scheme is introduced, and a detailed convergence analysis is provided. In particular, both a $\Delta$-convergence theorem and a strong convergence theorem for mean nonexpansive mappings are proved. To illustrate the applicability of the results, new examples are constructed that clarify the scope of the assumptions. Furthermore, a numerical application to a nonlinear fractional Volterra integral equation within the framework of a $p$-uniformly convex metric space is presented. The existence of a Bochner solution is demonstrated and approximated using the Karakaya iteration scheme, with its numerical performance compared to that of the S-iteration and Thakur schemes.

The theory of metric spaces forms the basis of metric fixed point theory, which has varied applications in the domain of various areas such as engineering, economics, medicine and even in space science such as launch of satellites etc. Nevertheless, fractal calculus too has varied applications. Metric spaces have been generalized and the fixed point results established under many contractive conditions in those newly defined spaces in the past few decades. In this work, we introduce neutrosophic bipolar metric spaces and establish fixed point theorems in these spaces. Our main results establish and generalize some proven results in the existing literature. The derived results have been strengthen with non trivial illustrations. Three applications are presented to supplement the derived results.

Existence of a Solution to an Infinite System of Weighted Atangana-Baleanu Fractional Integral Equations via Measure of Non-Compactness on a Tempered Sequence Space

This paper introduces novel formulations in the framework of α−E-contractions within the context of admissible mappings. We establish new fixed point theorems for α−E-Suzuki-type contractions, thereby generalizing and extending the foundational work of Hossein Piri and Poom Kumam. The principal objective of this research is to investigate the existence and uniqueness of solutions to a class of integral equations by leveraging fixed point methodologies in complete metric spaces. By developing these advanced α−E-contraction concepts and analyzing their implications for admissible mappings, this work contributes to the theoretical advancement of fixed point theory. To support our new definition, we illustrate examples. The results demonstrate the efficacy of this approach for addressing nonlinear problems in analysis, specifically enriching the methodology for solving integral equations. The overarching aim is to consolidate the theoretical underpinnings and provide a rigorous analytical framework for the application of α−E-contraction mappings, thereby fostering further progress in mathematical analysis.

This study uses a mathematical model that includes Caputo-Fabrizio fractional differential equations (CF-FDEs) to simulate the infection through the fractional COVID-19 model. We approximate the solution of the corresponding system of fractional integral equations (FIEs) using Simpson's 1/3 rule for numerical integration. We are interested on the stability of the presented approach. The aim of this work is to stop the disease from spreading to all parts of the world. After being initially quarantined, susceptible people may be moved immediately to the confined area or transferred to one of the infected courses for the exposed person. This approach was used by the researchers, who considered both symptomatic and asymptomatic infected patients. The outcomes are contrasted with those discovered utilizing the fourth-order Runge-Kutta method (RK4M).

This paper introduces a novel simulation approach that employs Caputo and Caputo-Fabrizio fractional derivative operators to explore the solution behavior of the fractional pollution model for a network of three lakes jointed by canals. Two input models are addressed by leveraging a purportedly innovative approximation techniques based on Gegenbauer wavelet polynomials (GWPs) and fractional Simpson's 1/3 rule (FSR). The spectral collocation method (SCM), leveraging the distinctive properties of GWPs are utilized to convert the model under consideration into a set of algebraic equations. The measurement of the residual error function (REF) confirms the precision and efficacy of the SCM. Additionally, for the second method, a numerical simulation of the resulting system of fractional integral equations (FIEs) is carried out using the FSR. Comparative analysis with the Runge-Kutta fourth order method (RK4M) highlights the efficacy of the techniques developed to simulate the solution behavior of such models, offering simple and efficient simulation tools.

This paper introduces a new class of third-kind nonlinear fractional integro-differential equations, defined using the Caputo derivative. A numerical method based on the piecewise Chebyshev cardinal functions is proposed to solve these equations. In the developed approach, a substitution is applied to initially transform these integro-differential equations into an equivalent class of third-kind nonlinear fractional integral equations. Next, by approximating the solution of the problem as a linear combination of the given basis functions and applying both the ordinary and fractional integral operational matrices of these functions, the problem solving process is transformed into solving an algebraic system of equations to determine the coefficients of the linear combination. The theoretical convergence of the proposed method is thoroughly examined. The validity of the scheme is numerically evaluated through the solution of several numerical examples.

We establish the existence and construction of maximal solutions for a class of quadratic fractional integral equations (QFIE) whose kernel is a generalized MittagLeffler Q function. Theoretical guarantees are provided under relaxed conditions. The roles of upper and lower solutions are explored, with a monotone iterative scheme yielding maximal bounds. Several concrete examples with specific parameter sets are provided, with graphical illustrations that highlight the typical solution structure. The results are positioned in the context of recent advances in the field