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We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval [ 0 , b ]. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on (0 , b ] , with possible singularities of the derivatives of the solution at the left endpoint of the interval [ 0 , b ]. [ABSTRACT FROM AUTHOR]

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]

Introduction/purpose: This paper introduces the concept of β - F-weak contraction by using the concepts of F-weak contraction and a - pscontraction. Methods: The use of the β - F-weak contraction proves some fixed points theorems in the framework of S-metric spaces. Results: The obtained results on fixed points in S-metric spaces generalize some known results in the literature. Conclusions: The β - F-weak contraction generalizes some important contraction types and examines the existence of a fixed point in S-metric spaces. The results are used to solve a non-linear Fredholm integral equation.

Introduction/purpose: This paper introduces the concept of β − F-weak contraction by using the concepts of F−weak contraction and α − ψ− contraction. Methods: The use of the β − F-weak contraction proves some fixed points theorems in the framework of S−metric spaces. Results: The obtained results on fixed points in S−metric spaces generalize some known results in the literature. Conclusions: The β − F−weak contraction generalizes some important contraction types and examines the existence of a fixed point in S−metric spaces. The results are used to solve a non-linear Fredholm integral equation. [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]

Bu çalışmada Urysohn tür integral denklem ile verilen control systemin yörüngeler kümesinin özellikleri incelenmektedir. Kontrol fonksiyonların integral kısıtı sağladığı varsayılmaktadır. Yörüngeler kümesinin control kaynağın sınırını belirleyen parametreye göre Lipschitz sürekli olduğu kanıtlanmıştır. Yörüngeler kümesinin çapı için bir üst değerlendirme verilmiştir. Sistemin yörüngesinin, kalan control kaynağına göre robastlığı tartışılmaktadır.

In this manuscript, the concept of $ \digamma $-contraction is applied to extend the notion of Jaggi-Suzuki-type hybrid contraction in the framework of $ G $-metric space, which is termed Jaggi-Suzuki-type hybrid $ \digamma $-($ G $-$ \alpha $-$ \phi $)-contraction, and invariant point results which cannot be inferred from their cognate ones in metric space are established. The results obtained herein provide a new direction and are generalizations of several well-known results in fixed point theory. An illustrative, comparative example is constructed to give credence to the results obtained. Furthermore, sufficient conditions for the existence and uniqueness of solutions of certain nonlinear polynomial and integral equations are established. For the purpose of future research, an open problem is highlighted regarding discretized population balance model whose solution may be investigated from the techniques proposed herein.

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]

In this manuscript, the concept of ϝ-contraction is applied to extend the notion of Jaggi-Suzuki-type hybrid contraction in the framework of G-metric space, which is termed Jaggi-Suzukitype hybrid ϝ-(G-α-φ)-contraction, and invariant point results which cannot be inferred from their cognate ones in metric space are established. The results obtained herein provide a new direction and are generalizations of several well-known results in fixed point theory. An illustrative, comparative example is constructed to give credence to the results obtained. Furthermore, sufficient conditions for the existence and uniqueness of solutions of certain nonlinear polynomial and integral equations are established. For the purpose of future research, an open problem is highlighted regarding discretized population balance model whose solution may be investigated from the techniques proposed herein. [ABSTRACT FROM AUTHOR]

The objective of the present work is to develop a numerical method for solving a nonlinear integral equation given by y (t) = f (t) + ∫ 0 1 k (t , s) 1 [ y (s) ] α d s , where y(t)∈C2[0,1] with y(t)>0 , f(t)∈C[0,1] with f(t)≥0 , the kernel function k(t,s) is non-negative and continuous on [0,1]×[0,1] and α>0. The existence of a continuous positive solution of this equation is well-established in the literature. However, there is no reported method to solve it numerically for any α>0. To attain the desired objective, the renowned Chebyshev collocation method is used. Having unknown Chebyshev coefficients, this method converts the integral equation into a matrix equation that produces a set of nonlinear algebraic equations. To solve these equations, computationally, the well-established Newton's method is employed. To validate the effectiveness and precision of the method, various numerical examples with well-defined exact solutions are examined. Obtained numerical solutions confirm the accuracy and validity of the numerical method. [ABSTRACT FROM AUTHOR]

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The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmetrical kernel is solved in the Banach space L 2 [ 0 , 1 ] × L 2 [ 0 , 1 ] . Here, the NIE's existence and singular solution are described in this passage. Additionally, we use a numerical strategy that uses hybrid and block-pulse functions to obtain the approximate solution of the NIE in a two-dimensional problem. For this aim, the two-dimensional NIE will be reduced to a system of nonlinear algebraic equations (SNAEs). Then, the SNAEs can be solved numerically. This study focuses on showing the convergence analysis for the numerical approach and generating an estimate of the error. Examples are presented to prove the efficiency of the approach. [ABSTRACT FROM AUTHOR]

The motive of this article is to study a modified iteration scheme for monotone nonexpansive mappings in the class of uniformly convex Banach space and establish some convergence results. We obtain weak and strong convergence results. In addition, we present a nontrivial numerical example to show the convergence of our iteration scheme. To demonstrate the utility of our scheme, we discuss the solution of nonlinear integral equations as an application, which is again supported by a nontrivial example.

The aim of this manuscript is to introduce a novel concept called Jaggi-type hybrid (ϕ - F) -contraction and establish some fixed point results for this class of contractions in the framework of G-metric space. The validity of the main result is shown by a suitable example and the realized improvements with respect to the corresponding literature are highlighted. By using the constructed example, it is observed that the results established herein cannot be deduced from their analogs in previously announced results in the literature. As an application, the existence and uniqueness of solutions to certain nonlinear Volterra integral equations are investigated to illustrate the utility of our obtained results. [ABSTRACT FROM AUTHOR]

This article implements the idea of introducing the concept of C-weak-fuzzy contraction mapping in the framework of fuzzy cone metric spaces, and deriving some unique common fixed point results. A nontrivial example is enunciated to uphold our results. Our results unify, enrich and extend some pioneer results in the existing literature in the fuzzy setting. In the end, an application to Fredholm nonlinear integral equation is offered showing the superiority of our results, which is in turn supported by an example. [ABSTRACT FROM AUTHOR]

In this paper, our purpose is to study the following weighted integral equation with negative power$ \begin{equation} f^{q-1}(x) = \int_{\Omega}\dfrac {|x|^{\alpha}|y|^{\alpha}f(y)}{|x-y|^{N-\mu}} \ dy \ +\lambda\int_{\Omega}\dfrac {f(y)}{|x-y|^{N-\mu-1}} \ dy \ , f\geq0, x\in\overline{\Omega}\;(0.1) \end{equation} $where $ \Omega $ is a bounded domain with smooth boundary in $ \mathbb{R}^N $, $ 0\in \Omega $, $ \mu>N $, $ \alpha<0 $ and $ \lambda $ is a real parameter. We prove that there exists a positive solution of problem (0.1) if $ 0

The purpose of this paper is to present some new contraction mappings via control functions. In addition, some fixed point results for Θ , α , θ , Ψ contraction, rational Θ , α , θ , Ψ contraction and almost Θ , α , θ , Ψ contraction mappings are obtained. Moreover, under contraction mappings of types (I), (II), and (III) of Θ , θ , Ψ υ 0 , several fixed circle solutions are provided in the setting of a G-Metric space. Our results extend, unify, and generalize many previously published papers in this direction. In addition, some examples to show the reliability of our results are presented. Finally, a supporting application that discusses the possibility of a solution to a nonlinear integral equation is incorporated. [ABSTRACT FROM AUTHOR]

We study an optimal consumption, investment, life insurance, and retirement decision of an economic agent who has an option to retire early any time before the mandatory retirement date. We conduct a thorough theoretical analysis for the optimal retirement problem with general utility function in the presence of a mandatory retirement date, which leads to the optimal stopping problem in finite horizon. Furthermore, the different marginal utility of consumption before and after retirement is considered, which can provide an explanation for the retirement-consumption puzzle, while it makes the problem technically more challenging. Based on the theory of partial differential equation, we analyze the variational inequality arising from the dual problem and establish the duality theorem. We show that the optimal retirement decision is determined by the time-varying optimal retirement wealth boundary, and we provide an integral equation representation for the optimal retirement wealth boundary, which can be solved accurately and efficiently by using recursive integration method. As an extension, the case with stochastic labor income is also considered. The properties of the optimal strategies are provided with emphasis on the role of the mandatory retirement date and the impact of having a retirement option on the optimal financial decisions. [ABSTRACT FROM AUTHOR]

Abstract 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 $w^{2}$ -stability result of the new iterative scheme is established and a supportive example is provided to illustrate the notion of w 2 $w^{2}$ -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.

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]

The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmetrical kernel is solved in the Banach space L2[0,1]×L2[0,1]. Here, the NIE’s existence and singular solution are described in this passage. Additionally, we use a numerical strategy that uses hybrid and block-pulse functions to obtain the approximate solution of the NIE in a two-dimensional problem. For this aim, the two-dimensional NIE will be reduced to a system of nonlinear algebraic equations (SNAEs). Then, the SNAEs can be solved numerically. This study focuses on showing the convergence analysis for the numerical approach and generating an estimate of the error. Examples are presented to prove the efficiency of the approach.

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]

The aim of this manuscript is to introduce a novel concept called Jaggi-type hybrid (ϕ -F)-contraction and establish some fixed point results for this class of contractions in the framework of G-metric space. The validity of the main result is shown by a suitable example and the realized improvements with respect to the corresponding literature are highlighted. By using the constructed example, it is observed that the results established herein cannot be deduced from their analogs in previously announced results in the literature. As an application, the existence and uniqueness of solutions to certain nonlinear Volterra integral equations are investigated to illustrate the utility of our obtained results.

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]

We consider a real options model for the optimal irreversible investment problem of a profit-maximizing company. The company has the opportunity to invest in a production plant capable of producing two products, of which the prices follow two independent geometric Brownian motions. After paying a constant sunk investment cost, the company sells the products on the market and thus receives a continuous stochastic revenue flow. This investment problem is set as a two-dimensional optimal stopping problem. We find that the optimal investment decision is triggered by a convex curve, which we characterize as the unique continuous solution to a nonlinear integral equation. Furthermore, we provide analytical and numerical comparative statics results of the dependency of the project's value and investment decision with respect to the model's parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider Galerkin and multi-Galerkin methods and their iterated versions for solving the nonlinear Hammerstein-type integral equation on the half-line with sufficiently smooth kernels, using Laguerre polynomials as basis functions. We obtain optimal convergence results in iterated-Galerkin method in both infinity and weighted L 2 -norms. We also obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods, respectively, in weighted L 2 -norm. Numerical results are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of solution of the non-local problem for the parabolic-hyperbolic type equation with non linear loaded term involving Caputo derivative \[ f(x) = \begin{cases} {u_{xx}}-_CD_{0y}^\alpha u+a_1(x)u^{p_1}(x,0), y 0, \\ {u_{xx}}-{u_{yy}}+a_2(x)u^{p_2}(x,0), y0, \\ \end{cases} \] where \[ {}_CD_{0y}^{\alpha }f(y) = \frac{1}{{\Gamma (1-\alpha )}}\int_0^y {(y - t)}^{-\alpha}f'(t)\,dt, \quad 0 \alpha 1, \] \(a_i(x)\) are given functions, \(p_i\), \(\alpha=\mathrm{const}\), besides \(p_i0\) \((i=1,2)\), \(0 \alpha 1\) in the domain \(\Omega\) bounded with segments: \[ A_1 A_2 = \{ (x,y): x = 1, 0 y h\},\quad B_1 B_2 = \{ (x,y): x = 0, 0 y h\}, \] \[ B_2 A_2 = \{ (x,y): y = h, 0 x 1\} \] at the \(y 0\), and characteristics: \[ A_1C: x - y = 1,\quad B_1C: x + y = 0 \] of the considered equation at \(y 0\), where \(A_1 (1, 0)\), \(A_2 (1, h)\), \(B_1( 0, 0)\), \(B_2( 0, h)\), and \(C(1/2, -1/2)\). Uniqueness of solution of the investigated problem was proved by an integral of energy. The existence of solution of the problem was proved by the method of integral equations. The theory of the second kind Fredholm type integral equations and the successive approximations method were widely used. We notice, that boundary value problems for the mixed type equations of fractional order with non linear loaded term have not been investigated.

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]

In this paper, using the classical Schauder fixed point theorem, we prove that a nonlinear class of integral equations has at least one solution in the space of functions satisfying the Hölder condition. Also, some important examples showing the applicability of the existence theorem presented in this study are given. [ABSTRACT FROM AUTHOR]

Abstract 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.

The purpose of this paper is to present some new contraction mappings via control functions. In addition, some fixed point results for Θ,α,θ,Ψ contraction, rational Θ,α,θ,Ψ contraction and almost Θ,α,θ,Ψ contraction mappings are obtained. Moreover, under contraction mappings of types (I), (II), and (III) of Θ,θ,Ψυ0, several fixed circle solutions are provided in the setting of a G-Metric space. Our results extend, unify, and generalize many previously published papers in this direction. In addition, some examples to show the reliability of our results are presented. Finally, a supporting application that discusses the possibility of a solution to a nonlinear integral equation is incorporated.

The purpose of this paper is to introduce the notion of a Suzuki-Gerghaty type contractive mapping via the simulation functions along with the C -class functions and prove the existence of the fixed point result. An example is given to show the validity of our results. Moreover, we prove the existence of solution of a nonlinear Hammerstein integral equation.

The aim of the present paper is to study the existence and asymptotic behavior of the solutions of some functional integral equations which contain a number of classical nonlinear integral equations as special cases. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. The applicability of the results is illustrated by examples. [ABSTRACT FROM AUTHOR]

Recently, there appeared new stability results in the inverse Sturm–Liouville theory that, unlike the usual local stability, involve uniform estimates. Besides more advanced justification of numerical simulations, the uniform stability can reveal the nature of an inverse problem to a significantly greater extent. For example, it allows obtaining global solvability of an inverse problem as a corollary of its solvability for any dense subclass of input data. In this paper, we obtain uniform stability of an inverse problem for one important class of integro-differential operators, which is a natural generalization of the classical Sturm–Liouville operator. For this purpose, we develop a new method, which is different from that used for the classical problem. Moreover, the considered situation arose relevance of the so-called full stability, i.e. the stability with respect to the full set of the input data including all a priori known components of the operator. The obtained results illustrate some essential difference from the inverse Sturm–Liouville problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of nonlinear integral operators of mixed type of integral equations with kernels depending on two variables. The results concerning the continuity and compactness of these operators are obtained.

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite‐dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators. [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]

An integro-differential Dirac system with a convolution kernel consisting of four independent functions is considered. We prove that the kernel is uniquely determined by specifying the spectra of two boundary value problems with one common boundary condition. The proof is based on the reduction of this nonlinear inverse problem to solving some nonlinear integral equation, which we solve globally. On this basis we also obtain a constructive procedure for solving the inverse problem along with necessary and sufficient conditions for its solvability in an appropriate class of kernels. [ABSTRACT FROM AUTHOR]

In the present work a generalized epidemic model containing a system of integral-differential equations is described. Using different transformations the system is reduced to a single nonlinear multidimensional integral equation. For the obtained equation the existence and uniqueness results are proved. Based on theoretical convergence results several application examples are presented with corresponding numerical results. [ABSTRACT FROM AUTHOR]

A tripled fixed point theorems in ordered metric spaces is used in order to prove the existence and uniqueness of a solution for a class of integral equations. The conditions of the theorem are much weaker than those existing in literature and the theorem is useful for solving some general problems. An example to illustrate our theoretical results is also given. [ABSTRACT FROM AUTHOR]

In this work, the main concept of the homotopy perturbation method (HPM) was outlined and convergence theorems of the HPM for solving some classes of nonlinear integral, integro-differential and differential equations were proved. A theorem for estimating the error in the approximate solution was proved as well. The proposed HPM convergence theorems were confirmed and the efficiency of the technique was explored by applying the HPM for solving several classes of nonlinear integral/integro-differential equations.