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 paper presents some results on the existence of solutions vanishing at infinity of a quadratic Hammerstein integral equation of Stieltjes type. We investigate solutions of the mentioned integral equation which are defined, continuous and bounded on the real half-axis ℝ + . The main tools used in our investigations are the technique of measures of noncompactness and a fixed point theorem of Darbo type. Moreover, we introduce the concept of the variation of a function on an unbounded interval and the integral of Stieltjes type defined on such an interval. Our result will be illustrated by an appropriate example.

This study establishes rigorous existence criteria for positive continuous solutions to a significant class of nonlinear quadratic integral equations of Hammerstein type. The investigation employs Schauder's fixed-point theorem as the principal analytical tool to derive the main existence result. Furthermore, under appropriately formulated monotonicity constraints, the existence of both maximal and minimal solutions is demonstrated. These theoretical contributions extend the existing mathematical literature on quadratic integral equations by introducing novel methodological frameworks and broadening the domain of applicability for this important category of nonlinear functional equations.

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

Chelyshkov matrix-collocation method for solving nonlinear quadratic integral equations

In this manuscript, the measure of noncompactness ($\mathbf{\mathcal{MNC}}$), Darbo and Banach contraction fixed point theorems ($\mathbf{\mathcal{FPT}}$), as well as fractional calculus, are used to carry out the analysis of the solvability of a general but abstract coupled system of quadratic Hadamard-fractional integral equations in Orlicz spaces $L_\Theta$. Several qualitative properties of the solution to the studied coupled system are established, such as the existence, monotonicity, and uniqueness, in addition to continuous dependence on the data. We conclude with some examples that illustrate our hypothesis.

Existence of the solution for the nonlinear quadratic integral equation of the Hammerstein type in the Banach space BC(R+) has been proved by using the technique of measure of noncompactness and fixed-point theorem. In this article, we obtain an approximate solution for the quadratic integral equation by using the Sinc method and the fixed-point technique. Moreover, the convergence of the numerical scheme for the solution of the integral equation is demonstrated by a theorem, and numerical experiments are presented to show the accuracy of the numerical scheme and guarantee the analytical results.

This paper investigates the well-posedness of analytical solutions to fractional quadratic differential equations, which involve generalized fractional integrals with respect to other functions. The nonlinear components f and h depend on spatial variables and the general fractional integral, respectively. We use the operator equation T1ωT2ω+T3ω=ω to investigate the existence of solutions, marking the first study of its kind. Using an auxiliary function and Boyd and Wang’s fixed-point theorem, the uniqueness and continuous dependence of the solution are obtained. In particular, we applied nonlinear functional analysis to investigate Hyers-Ulam and Hyers-Ulam-Rassias stabilities for fractional quadratic integral equations. New results are provided for specific values of the parameter z, and a fundamental inequality is formulated to ensure the existence of maximal and minimal solutions. Some examples are given to illustrate our results.

Numerical solution and dynamical studies for solving system of Quadratic integral equations

This work includes the study of a quadratic functional integral equation by utilizing the generalized Riemann–Liouville fractional integral of order $$\lambda > 0$$ with respect to an increasing and positive function $$\psi $$ . The first aim of this study is to obtain the asymptotic stability of solutions for this generalized framework. The concepts of measure of noncompactness and fixed point theorem are used to prove this result. Moreover, the second aim of this study is to introduce a novel polynomial-based computational method to obtain approximate solutions for the considered problem. Besides, the results of the error analysis with discussions of the accuracy of solutions are presented in this article. To the best knowledge of the authors, this paper presents the first reference regarding the numerical methods for this generalized problem to obtain approximate solutions. Finally, two examples are discussed with the computational tables and convergence graphs to interpret the efficiency and applicability of the presented method.

Background. The paper is devoted to the numerical study of integral equations of the first kind with quadratic nonlinearity, which are part of the generalized Volterra integropower series and describe dynamical systems with one input and one output. Such equations are widely used in the modeling of stationary systems with constant dynamic characteristics during the transfer process. Materials and methods. The proposed iterative numerical methods are based on the preliminary linearization of the integral operator according to the modified Newton-Kantorovich scheme and the use of the regularization parameter to ensure stability to fluctuations in the input data. To solve linear equations ateach iteration, the method of successive approximations is used in combination with the approximation of the exact solution by a polynomial spline constructed for each segment on the zeros of Legendre polynomials. The Gauss compound quadrature formula is used to calculate integrals. Results and conclusions. A number of iterative numerical schemes for solving quadratic Volterra integral equations are proposed. The convergence theorems of the modified Newton-Kantorovich method are formulated. Numerical results confirming the convergence of the methods are presented.

Solvability and approximation of a nonlinear Hammerstein type Volterra quadratic integral equation

The primary objective of this paper is to comprehensively establish the Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam-Rassias, and generalized Hyers-Ulam stability properties for (k, ?)-fractional order quadratic integral equations. These stability concepts play a crucial role in understanding the persistence, resilience, and response of solutions to small perturbations, providing insight into the behavior and reliability of solutions within complex systems. Our analysis is grounded in the application of Gronwall?s lemma, an essential tool that we adapt specifically for the unique structure of (k, ?)-fractional order systems. This approach not only enriches the theoretical understanding of stability within these fractional order integral equations but also broadens the applicability of Gronwall?s lemma to new contexts. To substantiate our findings, we provide two illustrative examples, carefully chosen to demonstrate the stability characteristics across a range of conditions and parameter settings. These examples are further supplemented by detailed 2D and 3D graphical representations generated in MATLAB, allowing for a visual examination of stability and solution dynamics. These visualizations not only complement the analytical proofs but also offer an intuitive validation of the stability results. Through this integrated approach the paper aims to present a well-rounded and thorough assessment of stability in (k, ?)-fractional order quadratic integral equations.

We prove a hybrid fixed point theorem for the product of two operators in a latticeordered Banach algebra and apply to nonlinear hybrid quadratic integral equations of mixed type for proving the existence of maximal and minimal positive integrable solutions under certain mixed conditions of Lipschitzicity and monotonicity of the nonlinear functions. Our main existence result is illustrated with a numerical example as well as with an application to IVPs of nonlinear first-order discontinuous quadratically perturbed ordinary differential equations.

In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions C[a,b] equipped with norms stronger than the supremum norm. We are particularly interested in Hölder spaces since they are natural ranges of integral operators of fractional order. Our goal is to preserve the expected regularity of the fixed points (solutions of the equations) when investigating their uniqueness, without assuming a contraction condition on the space under study. We claim some symmetry between the case of the obtained results and the case of the classical Banach fixed-point theorem in such spaces, even for operators which are not necessarily contractions in the sense of the norm of these subspaces. This result is of particular interest for the study of quadratic integral equations, and as an application example we prove the uniqueness theorem for such a kind equations with general fractional order integral operators, which are not necessarily contractions, in a suitably constructed generalized Hölder space.

We derive an analog of Mirzakhani’s recursion relation for hyperbolic string vertices and investigate its implications for closed string field theory. Central to our construction are systolic volumes: the Weil-Petersson volumes of regions in moduli spaces of Riemann surfaces whose elements have systoles L ≥ 0. These volumes can be shown to satisfy a recursion relation through a modification of Mirzakhani’s recursion as long as L ≤ 2 sinh−1 1. Applying the pants decomposition of Riemann surfaces to off-shell string amplitudes, we promote this recursion to hyperbolic string field theory and demonstrate the higher order vertices are determined by the cubic vertex iteratively for any background. Such structure implies the solutions of closed string field theory obey a quadratic integral equation. We illustrate the utility of our approach in an example of a stubbed scalar theory.

A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space C([0,T]×[0,T]). The existence of at least one solution to the QIE is discussed in this article. Our evidence depends on a suitable combination of the measures of the noncompactness approach and the fixed-point principle of Darbo. The quadratic integral equation can be used to derive a system of integral equations of the second kind using the quadrature method. With the aid of two different polynomials, Laguerre and Hermite, the system of integral equations is solved using the collocation method. In each numerical approach, the estimation of the error is discussed. Finally, using some examples, the accuracy and scalability of the proposed method are demonstrated along with comparisons. Mathematica 11 was used to obtain all of the results from the techniques that were shown.

Numerical solution, convergence and stability of error to solve quadratic mixed integral equation

Results on fixed points in b-metric space by altering distance functions