Nonlinear Quadratic Integral Equation Research Articles

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

We discuss the existence of a fixed point for a self mapping and its uniqueness satisfying (ϕ˙,η˙)-generalized contractive condition including altering distance functions of rational terms in an ordered b-metric space. It is also discussed whether the two self-maps under the same contraction condition can be coincident and coupled coincident. The results are backed up by a dearth of numerical examples and application to nonlinear quadratic integral equation.

Fixed point and endpoint theorems of multivalued mappings in convex $ b $-metric spaces with an application

<abstract><p>In this paper, we investigated several new fixed points theorems for multivalued mappings in the framework $ b $-metric spaces. We first generalized $ S $-iterative schemes for multivalued mappings to above spaces by means of a convex structure and then we developed the existence of fixed points and approximate endpoints of the multivalued contraction mappings using iteration techniques. Furthermore, we introduced the modified $ S $-iteration process for approximating a common endpoint of a multivalued $ \alpha_{s} $-nonexpansive mapping and a multivalued mapping satisfying conditon $ (E^{'}) $. We also showed that this new iteration process converges faster than the $ S $-iteration process in the sense of Berinde. Some convergence results for this iterative procedure to a common endpoint under some certain additional hypotheses were proved. As an application, we applied the $ S $-iteration process in finding the solution to a class of nonlinear quadratic integral equations.</p></abstract>

Investigation of the F* Algorithm on Strong Pseudocontractive Mappings and Its Application

In the context of uniformly convex Banach space, this paper focuses on examining the strong convergence of the F* iterative algorithm to the fixed point of a strongly pseudocontractive mapping. Furthermore, we demonstrate through numerical methods that the F* iterative algorithm converges strongly and faster than other current iterative schemes in the literature and extends to the fixed point of a strong pseudocontractive mapping. Finally, under a nonlinear quadratic Volterra integral equation, the application of our findings is shown.

The Study of the Solvability of Infinite Systems of Integral Equations via Measures of Noncompactness

The paper is devoted to investigate the existence of solutions of infinite systems of nonlinear quadratic integral equations of Volterra type. We look for solutions of those infinite systems of integral equations which are defined, continuous and bounded on the real half-axis and take real values. In our considerations the basic role will be played by the technique of measures of noncompactness defined in the space of functions defined, continuous and bounded on the real half-axis with values in the sequence space consisting of real sequences equipped with the standard supremum norm. We are going to prove an existence result which generalizes all recently obtained results for infinite systems of integral equations. An example illustrating the obtained result will be included.

Existence of solutions to non-linear quadratic integral equations via measure of non-compactness

The objective of this manuscript is to enquire for the solvability of a specific type of non-linear quadratic integral equations via the interesting notion of measure of non-compactness. Firstly, we inquire into couple of exciting fixed point theorems involving a measure of non-compactness in the setting of a Banach space. Subsequently, bringing into play a suitable measure of non-compactness and the acquired results, we discuss the existence of solutions to the aforementioned kind of non-linear quadratic integral equations.

ON SOLUTIONS OF INFINITE SYSTEMS OF INTEGRAL EQUATIONS COORDINATEWISE CONVERGING AT INFINITY

<abstract> In the paper we are going to prove the existence of solutions of an infinite system of nonlinear quadratic integral equations of Volterra-Hammerstein type. Those solutions are continuous and bounded functions defined on the real half-axis $ \mathbb{R}_+ $ and created by function sequences which are coordinatewise converging to proper limits at infinity. Considerations of the paper are located in the Banach space consisting of functions defined, continuous and bounded on $ \mathbb{R}_+ $ with values in the space of real bounded sequences. The main tool applied in the paper is the technique of measures of noncompactness. </abstract>

Approximations to the solution of quadratic fractional integral equation (QFIE) with generalized Mittag-Leffler Q function

In this paper, we will find the solution to the quadratic fractional integral equation involving the Q function which is the generalization of Mittag-Leffler function with the help of forming the sequence of solutions converging to the solution of the fractional integral equation involving the Q function. We will study in this paper the existence and convergence of a nonlinear quadratic fractional integral equation with the new Q function which is the generalization of Mittag-Leffler function, on a closed and bounded interval of the real line with the help of some conditions.

Approximations to the solution of quadratic fractional integral equation (QFIE) with generalized Mittag-Leffler Q function

In this paper, we will find the solution to the quadratic fractional integral equation involving the Q function which is the generalization of Mittag-Leffler function with the help of forming the sequence of solutions converging to the solution of the fractional integral equation involving the Q function. We will study in this paper the existence and convergence of a nonlinear quadratic fractional integral equation with the new Q function which is the generalization of Mittag-Leffler function, on a closed and bounded interval of the real line with the help of some conditions.

Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra

Here, we investigate the existence result for a nonlinear quadratic functional integral equation of fractional order using a fixed point theorem of Dhage. The continuous dependence of solution on the delay functions will be studied. As an application, an existence theorem for the fractional hybrid differential equations is proved. Also, we study a general quadratic integral equation of fractional order.

On the Solution of Quadratic Nonlinear Integral Equation with Different Singular Kernels

All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.

Convergence, stability, and data dependence of a new iterative algorithm with an application

The purpose of this article is to introduce a new two-step iterative algorithm, called $$F^{*}$$ algorithm, to approximate the fixed points of weak contractions in Banach spaces. It is also showed that the proposed algorithm converges strongly to the fixed point of weak contractions. Furthermore, it is proved that $$F^{*}$$ iterative algorithm is almost-stable for weak contractions, and converges to a fixed point faster than Picard, Mann, Ishikawa, S, normal-S, and Varat iterative algorithms. Moreover, a data dependence result is obtained via $$F^{*}$$ algorithm. Some numerical examples are presented to support the main results. Finally, the solution of the nonlinear quadratic Volterra integral equation is approximated by utilizing our main result. The results of the paper are new and extend several relevant results in the literature.

On the existence of continuous solutions of a nonlinear quadratic fractional integral equation

We prove an existence theorem for a nonlinear quadratic integral equation of fractional order, in the Banach space of real functions defined and continuous on a closed interval. This equation contains as a special case numerous integral equation studied by other authors. Finally, we give an example for indicating the natural realizations of our abstract result presented in this paper.

Existence of Solutions for an Infinite System of Quadratic Integral Equations in the Banach Algebra $$C(I,c_0)$$

The paper is devoted to prove the existence of solutions for an infinite system of nonlinear quadratic integral equations. This system is investigated in the WC–Banach algebra $$C_0=C(I,c_0)$$, consisting of all continuous functions acting from an interval I into the sequence space $$c_0$$. The assumptions imposed on the operators involving the system are formulated in terms of a generalized Lipschitz continuity.

Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result

The existence of \begin{document}$ L^{2} $\end{document} -nonnegative solutions for nonlinear quadratic integral equations on a bounded closed interval is investigated. Two existence results for different classes of functions are shown. As a consequence an existence theorem for the Chandrasekhar integral quadratic equation, well-known in theory of radiative transfer, is obtained. The aim is achieved by means of a new fixed point theorem for multimaps in locally convex linear topological spaces.

Solving two-dimensional non-linear quadratic integral equations of fractional order via operational matrix method

Purpose The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically. Design/methodology/approach For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time. Findings By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated. Originality/value Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

Solvability of nonlinear quadratic integral equation by using simulation type condensing operator and measure of noncompactness

The aim of this article is to prove the existence of solution for nonlinear quadratic integral equations using the generalized form of Darbo fixed point theorem with the help of measure of noncompactness and simulation type condensing operator in the Banach space L2[0, 1]. To illustrate validity of the analytical results, we present a nonlinear integral equation as an application. Finally, we introduce an iteration algorithm by modified homotopy perturbation and Adomian decomposition method to find solution of the above problem with a high accuracy.

Existence and asymptotically stable solution of a Hammerstein type integral equation in a Hölder space

The following nonlinear quadratic integral equation of Hammerstein type is studied. $$x(t)=p(t)+x(t)\int_0^{q(t)} H(t,\tau,x(\tau)){\rm d}\tau.$$ The methodology relies on the measure of noncompactness in the space of functions with tempered increments, namely the space of $\alpha$-Hölder continuous functions. The results follow from the Darbo fixed point theorem. Some examples are included to show the applicability of the main results.

On generalized coupled fixed points with applications to the solvability of coupled systems of nonlinear quadratic integral equations

On generalized coupled fixed points with applications to the solvability of coupled systems of nonlinear quadratic integral equations

Approximate solution of nonlinear quadratic integral equations of fractional order via piecewise linear functions

In this paper, the main objective is to describe an approximate scheme for solving nonlinear quadratic integral equations (NQIEs) of fractional order. The method is based on piecewise linear functions that are called hat functions (HFs). By applying the HFs approximation, their properties and operational matrices, nonlinear equation becomes to a nonlinear system of equations which can be solved by using simple arithmetic methods. Also, the error analysis of HFs method and convergence analysis of proposed model are given. At the end, two nonlinear examples are presented to illustrate the validity and applicability of the explained approach.

Metric-like spaces to prove existence of solution for nonlinear quadratic integral equation and numerical method to solve it

The purpose of this paper is to obtain some common fixed point results for two mappings satisfying various contractive conditions in metric-like spaces. These results extend some previous results in the literature, since the condition under which the operator admits common fixed points is more general than the others in literature. Therefore, several well known results are generalized. As an application we use these results to existence of solution for nonlinear quadratic integral equation. To credibility, we apply modified homotopy and Adomian decomposition method to find solution of the above problem with high accuracy.