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The primary objective of this article is to devis an effective method for solving a high-degree nonlinear Volterra integro-differential equation. First, we conduct a comprehensive literature review to underscore the significance of such mathematical equations in scientific research. Subsequently, we establish sufficient conditions to demonstrate the existence and uniqueness of the exact solution. Then, we employ two different discretization methods, namely Nyström and collocation. Then, we obtain two approximate systems, each of which yields a solution converging to the exact solution. The convergence of these systems is rigorously demonstrated through two theorems. Finally, we present numerical tests to evaluate the effectiveness and simplicity of the proposed numerical processes. [ABSTRACT FROM AUTHOR]

We use methods of the theory of monotone (in the Browder–Minty sense) operators to prove global theorems on the existence, uniqueness, and solution norm estimates for various classes of nonlinear integral equations with power sum kernel in the real Lebesgue spaces. Corollaries illustrating the obtained results are presented. Under additional assumptions, it is shown that solutions can be found by the method of successive approximations of the Picard type without restrictions on the values of the scalar parameter. These theorems cover the corresponding linear integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear parabolic differential equation with involution. With respect to spatial variable is used Dirichlet boundary value conditions and spectral problem with involution is obtained. Eigenvalues and eigenfunctions of the spectral problems are found. The Fourier series method of separation of variables is applied. The countable system of nonlinear integral equations is obtained. Theorem on a unique solvability of the countable system of nonlinear integral equations is proved. The method of successive approximations is used in combination with the method of contraction mapping. The generalized solution of the nonlinear mixed problem is obtained in the form of Fourier series. Absolutely and uniformly convergence of Fourier series is proved. [ABSTRACT FROM AUTHOR]

The problem of a solitary surface gravity wave in a flow of an inviscid incompressible fluid in a channel of constant depth is considered. The problem is solved in two-dimensional formulation. The wave moves at a constant speed. In a coordinate system moving along with the wave, the flow is stationary. Its mathematical model is reduced to a boundary value problem for a strip in the complex potential plane. This is converted to a boundary value problem for a half-plane by conformal mapping. The solution is obtained using a Cauchy-type integral for the density of which a nonlinear integral equation is derived. Its solution is found with the Galerkin method and the Newton–Raphson technique. The calculated results are compared with the experimental data and the calculations by other researchers. The lower limit of the speed of a solitary wave is found. The advantage of the proposed method is the simplicity of the resulting integral equation, which makes it possible to effectively apply numerical methods of solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the controllability of two problems involving the same Chandrasekhar-type integral equation, but under different kinds of controls. A viability condition is imposed as well. We provide existence results of continuous trajectories coupled to continuous controls. Then, in the non-viable case, we investigate the optimal estimates to be taken in view of the existence of solutions for both problems. The last part of the paper deals with the application of the previous results to the classical Chandrasekhar equation, first showing the existence of a viable continuous solution, then providing also uniqueness and approximability. Two examples of controllability problems governed by this equation are given. [ABSTRACT FROM AUTHOR]

We find conditions under which a generalized potential-type operator acts continuously from a Lebesgue space with a general weight to its dual space and possesses the positivity property. Based on these conditions, we prove the global existence and uniqueness theorems for various classes of nonlinear integral equations of convolution type in real weighted Lebesgue spaces using the method of monotonic (in the Browder–Minty sense) operators. Also we obtain estimates of the norms of solutions, which imply that the corresponding homogeneous equations have only a trivial solution. [ABSTRACT FROM AUTHOR]

In this article, we propose a faster iterative scheme, called the AH iterative scheme, for approximating fixed points of contractive-like mappings and Reich–Suzuki-type nonexpansive mappings. We show that the AH iterative scheme converges faster than a number of existing iterative schemes for contractive-like mappings. The w 2 -stability result of the new iterative scheme is established and a supportive example is provided to illustrate the notion of w 2 -stability. Then, we prove weak and several strong convergence results of our new iterative scheme for fixed points of Reich–Suzuki-type nonexpansive mappings. Further, we carry out a numerical experiment to illustrate the efficiency of our new iterative scheme. As an application, we use our main result to prove the existence of a solution of a mixed-type nonlinear integral equation. An illustrative example to validate the result in our application is also given. Our results extend and generalize several related results in the existing literature. [ABSTRACT FROM AUTHOR]

A class of nonlinear integral equations on the whole axis with a noncompact integral operator of Hammerstein type is investigated. This class of equations has applications in various fields of natural science. In particular, such equations are found in mathematical biology, in the kinetic theory of gases, in the theory of radiation transfer, etc. The existence of a nonnegative nontrivial and bounded solution is proved. The asymptotic behavior of the constructed solution on ±∞ is studied. In one important special case, the uniqueness of the constructed solution in a certain weighted space is established. At the end of the work, specific applied examples of the equations under study are given. [ABSTRACT FROM AUTHOR]

We consider the corrosion detection problem in terms of the Laplace equation and study a simply connected bounded domain with Wentzell-type GIBC boundary condition. We derive the systems of integral equations and establish the equivalence to the inverse shape problem in a Sobolev space setting. For the direct problem, we use potential theory to simulate the Neumann data from Dirichlet data on Dirichlet boundary. Then we propose a Newton iterative approach based on the boundary integral equations derived from Green's representation theorem. After describing the linearization and the iteration scheme for the inverse shape, we compute the Fréchet derivatives with respect to the unknowns. We conclude by presenting several numerical examples for shape reconstructions to show the validity of the proposed method. [ABSTRACT FROM AUTHOR]

We prove local solvability and stability for the inverse problem of recovering a complex-valued square integrable potential in the Sturm–Liouville equation on a finite interval from spectra of two boundary value problems with one common boundary condition. For this purpose we generalize classical Borg's method to the case of multiple spectra. [ABSTRACT FROM AUTHOR]

Very Recently, Das et al. (J Math Chem 54:527-551, 2016) proposed a method based on variational iteration method for solving Bratu-type problems. In this work, we design a fast iterative scheme based on the optimal homotopy analysis method to obtain series solution for such problem. The proposed method contains a convergence control parameter which adjusts the interval of convergence of the series solution. This parameter is computed by minimizing the sum of the squared residual of the governing equation and with the help of which we obtain an optimal series solution to the problem. Further, our method avoids the usual time-consuming procedure to find the root of a nonlinear algebraic or transcendental equation for the undetermined coefficients. Besides the design of the method, convergence analysis and error estimate of the proposed method are supplemented. Two Bratu type problems are solved to demonstrate the efficiency and accuracy of the proposed method. Numerical results reveal the superiority of the proposed iterative scheme over a newly developed iterative method based on the variational iteration method (Das et al. 2016). [ABSTRACT FROM AUTHOR]

In this paper we provide for a first time, to our knowledge, a mathematical model for imaging an anisotropic, orthotropic medium with polarization-sensitive optical coherence tomography. The imaging problem is formulated as an inverse scattering problem in three dimensions for reconstructing the electrical susceptibility of the medium using Maxwell’s equations. Our reconstruction method is based on the second-order Born-approximation of the electric field. [ABSTRACT FROM AUTHOR]

In this article, the thin plate splines are given to obtain the numerical solution of nonlinear Fredholm integral equations of the second kind. The scheme approximates the solution using the discrete collocation method based on the shape functions of thin plate splines constructed on a set of scattered points. The thin plate splines can be seen as a type of the free shape parameter radial basis functions which are used to establish an effective technique for the interpolation of an unknown function. The numerical method developed in the current paper utilizes the non-uniform Gauss-Legendre quadrature rule to compute its integrals. Since the proposed scheme does not require any background mesh for approximations and numerical integrations, it is meshless. This approach can be easily implemented and its algorithm is simple and effective to solve nonlinear integral equations. Moreover the error estimate and the convergence rate of the method are presented. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

This paper is devoted to prove the existence of continuous solutions for a nonlinear integral equation. Our existence theorem extends in a broad sense the analogous one obtained by L.T.P. Ngoc and N.T. Long. To this aim we first prove new variants of the Krasnosel'skii-Sadowskii and of the Krasnosel'skii-Däher fixed point theorems in Hausdorff locally convex topological vector spaces. These hybrid theorems improve some recently obtained results. Moreover a fixed point theorem for a multimap defined on a cartesian product subset of a Banach space is stated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mixed monotone operators of a new type in a Cartesian product of a Banach space with an order determined by a normal cone. We establish sufficient conditions for such operators to have a unique fixed point and provide monotone iterative techniques which give sequences convergent to the fixed point. We also apply the obtained results to prove the existence and uniqueness of a solution of a nonlinear integral equation. [ABSTRACT FROM AUTHOR]

We consider the well-posedness problem of nonlinear integral and differential equations with delay which arises in the elaboration of mathematical models of living systems. The questions of existence, uniqueness, and nonnegativity of solutions to these systems on an infinite semiaxis are studied as well as continuous dependence of solutions on the initial data on finite time segments. [ABSTRACT FROM AUTHOR]

In this paper, we use the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals to discuss the existence theorem of weak solutions for a class of the nonlinear integral equations and obtain a new result, which improves and extends some relevant results for differential and integral equations in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new approach for discussing the solvability of a class of operator equations by establishing fixed point theorems in locally convex spaces. Our results are obtained extend some Krasnosel'skii type fixed point theorems. As an application, we investigate the existence and global attractivity of solutions for a general nonlinear integral equation of mixed type of Urysohn and Volterra. [ABSTRACT FROM AUTHOR]

In this paper, we give a fixed point theorem in locally convex algebras and present applications to the existence problem for a class of nonlinear integral equations with unbounded deviations. [ABSTRACT FROM AUTHOR]

This paper investigates a numerical method for solving two-dimensional nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The scheme utilizes the shape functions of the moving least squares (MLS) approximation constructed on scattered points as a basis in the discrete collocation method. The MLS methodology is an effective technique for approximating unknown functions which involves a locally weighted least square polynomial fitting. The proposed method is meshless, since it does not need any background mesh or cell structures and so it is independent of the geometry of the domain. The scheme reduces the solution of two-dimensional nonlinear integral equations to the solution of nonlinear systems of algebraic equations. The error analysis of the proposed method is provided. The efficiency and accuracy of the new technique are illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of integrable solution to a nonlinear integral equation, which includes many important integral and functional equations that arise in nonlinear analysis and its applications. Our results are obtained under rather general assumptions. In particular, the solvability of the well known Chandrasekhar equation is discussed under appropriate assumptions. Our analysis uses the technique of measures of weak noncompactness and rely on a variant of Schauder's fixed point theorem. [ABSTRACT FROM AUTHOR]

The paper presents an application of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind. In this method a series is created, sum of which (if the series is convergent) gives the solution of discussed equation. Conditions ensuring convergence of this series are presented in the paper. Error of approximate solution, obtained by considering only partial sum of the series, is also estimated. Examples illustrating usage of the investigated method are presented as well, including the example having practical application for calculating the charge in supply circuit of flash lamps used in cameras. [ABSTRACT FROM AUTHOR]

We prove an inequality for the p-variation, $$ 1\leqslant p<1 $$ , of Picard iterates associated to a nonlinear refinement Riemann–Stieltjes integral equation with respect to a possibly discontinuous function of bounded p-variation. [ABSTRACT FROM AUTHOR]

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with a Kerr nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of a Green's function. The existence of propagating TE waves is proved using the contraction mapping method. For the numerical solution of the problem, two methods are proposed: an iterative algorithm (whose convergence is proved) and a method based on solving an auxiliary Cauchy problem (the shooting method). The existence of roots of the dispersion equation (propagation constants of the waveguide) is proved. Conditions under which k waves can propagate are obtained, and regions of localization of the corresponding propagation constants are found. [ABSTRACT FROM AUTHOR]

We prove an inequality for the p-variation, $$ 1\leqslant p<1 $$ , of Picard iterates associated to a nonlinear refinement Riemann–Stieltjes integral equation with respect to a possibly discontinuous function of bounded p-variation. [ABSTRACT FROM AUTHOR]

The first boundary value problem of nonlinear theory of steady-state creep with powerlaw relationship between the stresses and the strain rates is considered for a half-space under conditions of antiplane (out-of-plane) deformation when tangential distributed forces are given on the half-space boundary. By using the introduced harmonic pseudostress function, we reduce solving this problem to solving a nonlinear singular integral equation admitting an exact solution. [ABSTRACT FROM AUTHOR]

For a particular nonlinear integral equation, its solution is constructed using sharp two-sided estimates and their convergence to the solution is proved. [ABSTRACT FROM AUTHOR]

The following delay integral equation has been proposed by Cooke and Kaplan to describe the spread of certain infectious diseases with periodic contact rate that varies seasonally. This mathematical model can also be interpreted as an evolution equation of a single species population. The purpose of this paper is to present an approximating algorithm for the continuous positive solution of this integral equation from the theory of epidemics. This algorithm is obtained by applying the successive approximations method and the rectangle formula, used for the calculation of the approximate value of integrals which appear in the right-hand-side of the terms of the sequence of successive approximations. In order to establish this approximating algorithm, we will suppose that this integral equation has a unique solution. The main result contains also the error of approximation of the solution obtained by applying this approximating algorithm. [ABSTRACT FROM AUTHOR]

In this paper we introduce a nonlinear integral equation such that the system of global solutions to this equation represents the class of a very narrow beam as T → ∞ (an analog of the laser beam) and this sheaf of solutions leads to an almost-exact representation of the Hardy-Littlewood integral (1918). The accuracy of our result is essentially better than the accuracy of related results of Balasubramanian, Heath-Brown, and Ivic. [ABSTRACT FROM AUTHOR]

It is shown that for abstract linear Volterra operators the spectral radius is the essential spectral radius. The result is applied to prove the existence of a global solution of a Volterra-Hammerstein functional inclusion when the nonlinearity has at most affine growth. [ABSTRACT FROM AUTHOR]

The operator of double differentiation perturbed by the composition of a Volterra convolution operator and the differentiation one on a finite interval with Dirichlet boundary conditions is considered. It is proved that the standard asymptotics is necessary and sufficient for an arbitrary sequence of complex numbers to be the spectrum of such an operator, which is determined uniquely. A constructive procedure for solving the inverse problem is given. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ℓ which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L 2[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ℓ and by two pairs of boundary conditions coinciding at one of the finite points. [ABSTRACT FROM AUTHOR]

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem. [ABSTRACT FROM AUTHOR]

We consider a nonlinear integral equation with infinitely many derivatives that appears when a system of interacting open and closed strings is investigated if the nonlocality in the closed string sector is neglected. We investigate the properties of this equation, construct an iterative method for solving it, and prove that the method converges. [ABSTRACT FROM AUTHOR]

We show that the optimal stopping boundary for the Russian option with finite horizon can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation (an explicit formula for the arbitrage-free price in terms of the optimal stopping boundary having a clear economic interpretation). The results obtained stand in a complete parallel with the best known results on the American put option with finite horizon. The key argument in the proof relies upon a local time-space formula. [ABSTRACT FROM AUTHOR]

Some fixed point theorems for mappings of the type − A + T are established, where P is a cone in a Hilbert space, A: P → 2 P is an accretive mappings and T: 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). [ABSTRACT FROM AUTHOR]

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 and T: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). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to improve some recent fixed point results from Refs. Parvaneh et al. (Sahand Commun Math Anal 7(1):85-104, 2017), Paunović et al. (J Nonlinear Sci Appl 10(8):4162-4174, 2017) and Faraji and Nourouzi (Sahand Commun Math Anal 7(1):49-62, 2017) in the context of b-metric spaces. The techniques used in our proofs are quite different and most remarkably some of the proofs become simpler and more straightforward. Some examples are used to show that the improvements are indeed genuine. In addition, by using our results, we obtain the existence and uniqueness of a solution to a class of nonlinear integral equations. Furthermore, we provide the precise mathematical expression of the solution. [ABSTRACT FROM AUTHOR]