Class Of Nonlinear Integral Equations Research Articles

Integral solution of a class of nonlinear integral equations

This paper is concerned with the existence of integral solutions to a general nonlinear integral equationx(t)=f1(t,x(ϕ1(t)))+f2t,∫0ϕ2(t)k(t,s)f3(s,x(ϕ3(s)))ds,t∈R+.With the help of Krasnoselskii’s fixed point theorem and the theory of measure of weak noncompactness, we establish a new and general existence theorem for the nonlinear functional integral equation. Moreover, an example, which can not be treated by the related theorems in [5,20,22], is given to illustrate the new existence theorem.

Cyclic generalized contractions and fixed point results with applications to an integral equation

Abstract We set up a new variant of cyclic generalized contractive mappings for a map in a metric space and present existence and uniqueness results of fixed points for such mappings. Our results generalize or improve many existing fixed point theorems in the literature. To illustrate our results, we give some examples. At the same time as applications of the presented theorems, we prove an existence theorem for solutions of a class of nonlinear integral equations. MSC:47H10, 54H25.

Common fixed point theorems under implicit relations on ordered metric spaces and application to integral equations

In this paper, we prove existence results for common fixed points of two or three relatively asymptotically regular mappings satisfying the orbital continuity of one of the involved maps under implicit relation on ordered orbitally complete metric spaces. We furnish suitable examples to demonstrate the validity of the hypotheses of our results. At the end of the results, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented.

Existence, uniqueness and stability of solutions for a class of nonlinear integral equations under generalized Lipschitz condition

In this paper, we prove the existence, uniqueness and the stability of solutions for some nonlinear functional-integral equations by using generalized Lipschitz condition. We prove a fixed point theorem to obtain the mentioned aims in Banach space X:= C([a, b],R). As application we study some Volterra integral equations with linear, nonlinear and singular kernel.

Cyclic generalized [formula omitted]-weakly contractive mappings and fixed point results with applications to integral equations

We introduce a cyclic generalized ψ-weakly contractive mappings for a map in a metric space and originate existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. At the end of the results, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented.

Jacob’s ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations

In this paper we obtain new formulae for short and microscopic parts of the Hardy-Littlewood integral, and the first asymptotic formula for the sixth-order expression \(\left| {\zeta \left( {\tfrac{1} {2} + i\phi _1 \left( t \right)} \right)} \right|^4 \left| {\zeta \left( {\tfrac{1} {2} + it} \right)} \right|^2\). These formulae cannot be obtained in the theories of Balasubramanian, Heath-Brown and Ivić.

Coincidence points of mappings and relations with applications

We obtain coincidence points of mappings and relations under a contractive condition in a metric space. As applications, we achieve an existence and uniqueness theorem of solution for a general class of nonlinear integral equations.2010 Mathematics Subject Classification: 47H10; 54H25; 54C60.

Fixed point theorems on ordered gauge spaces with applications to nonlinear integral equations

We establish coincidence and fixed point theorems for mappings satisfying generalized weakly contractive conditions on the setting of ordered gauge spaces. Presented theorems extend and generalize many existing studies in the literature. We apply our obtained results to the study of existence and uniqueness of solutions to some classes of nonlinear integral equations.

SEMI-ORTHOGONAL SPLINE SCALING FUNCTIONS FOR SOLVING HAMMERSTEIN INTEGRAL EQUATIONS

We developed a new numerical procedure based on the quadratic semi-orthogonal B-spline scaling functions for solving a class of nonlinear integral equations of the Hammerstein-type. Properties of the B-spline wavelet method are utilized to reduce the Hammerstein equations to some algebraic equations. The advantage of our method is that the dimension of the arising algebraic equation is 10 × 10. The operational matrix of semi-orthogonal B-spline scaling functions is sparse which is easily applicable. Error estimation of the presented method is analyzed and proved. To demonstrate the validity and applicability of the technique the method applied to some illustrative examples and the maximum absolute error in the solutions are compared with the results in existing methods.20,25,27,29

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225–234, 2005; Appl. Math. E-Notes 7:214–221, 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian’s accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.

On a class of nonlinear integral equations with a noncompact operator

The paper is devoted to investigation of the solvability of a class of nonlinear integral equations on the semiaxis in a critical case, which possess a noncompact operator of almost Hammerstein type. Under some conditions on the equation kernel, the existence of a bounded, monotone increasing, positive solution is proved. The asymptotic behavior of the solution near infinity is studied.

Existence of Nontrivial Global Solution for a Class of Nonlinear Volterra Integral Equations in Banach Space

We get a global existence result for a class of nonlinear integral equations by applying a new fixed point thereom in weaker condition.Because there is a nonlinear integral operator in nonlinear term,compared with linear integral operator,some results in knowed papers are expanded and enriched.

Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs

We study the blowing-up behavior of solutions of a class of nonlinear integral equations of Volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities. We present some analytic results and, in the case of the kernel of Abel-kind with power nonlinearity and fixed initial data, we give a numerical approximation by using one-point collocation methods. By means of the numerical simulations, we give the dependence of the blow-up time from the parameters of the equation.

Asymptotic properties of solutions to some nonlinear integral equations of convolution type

For a class of nonlinear integral equations of convolution type we give necessary and sufficient conditions for the boundedness of nonnegative solutions. Moreover, conditions for the solution to converge asymptotically to a determined limit are obtained.

Convergence of the Adomian method applied to a class of nonlinear integral equations

In this work, a reliable approach for convergence of the Adomian method when applied to a class of nonlinear Volterra integral equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.

A Mixed Mann Iteration Process for Fixed Points of Mixed Increasing Operators in Ordered Banach Spaces

In this paper, we develop a new mixed Mann iteration process to approximate the fixed points of mixed increasing operators in ordered Banach spaces and discuss the convergence analysis of the iterative sequences generated by the iteration process. As an application, we develop a new iterative method for solving a class of nonlinear integral equations.

New approach for numerical solution of Hammerstein integral equations

In this paper, we use a numerical procedure for solving a class of nonlinear integral equations of the Hammerstein-type, using the globally defined Sinc basis functions. Properties of the Sinc procedure are utilized to reduce the computation of the Hammerstein equations to some algebraic equations. We used some numerical examples to illustrate the accuracy and the implementation of the method.

On the RF‐pair operations for the exact solution of some classes of nonlinear Volterra integral equations

We study the exact solution of some classes of nonlinear integral equations by series of some invertible transformations and RF‐pair operations. We show that this method applies to several classes of nonlinear Volterra integral equations as well and give some useful invertible transformations for converting these equations into differential equations of Emden‐Fowler type. As a consequence, we analyze the effect of the proposed operations on the exact solution of the transformed equation in order to find the exact solution of the original equation. Some applications of the method are also given. This approach is effective to find a great number of new integrable equations, which thus far, could not be integrated using the classical methods.

A Class of Partial Integro-Differential Equations with Correlation-Convolution Integral I

By means of the iteration method with weighted norms existence and uniqueness theorems are proved for three classes of nonlinear integral equations and first order integro-differential equations in two variables. The quadratic non-linearity is given by the correlation-convolution integral. Existence and uniqueness of the solutions are shown in Lebesgue spaces with mixed norms in rectangle and strip.

On the problem of dissipative perturbations of nonexpansive mappings

Some fixed point theorems for mappings of the type — A+T are established, where P is a cone in a Hilbert space,A:P→2P is an accretive mappings andT:P→P is a nonexpansive mappings. In application, the results presented in the paper are used to study the existence problem of solutions for a class of nonlinear integral equations in L2(Ω).