Volterra-type Integral Equations Research Articles

On existence of extremal integrable solutions and integral inequalities for nonlinear Volterra type integral equations

We prove the existence of maximal and minimal integrable solutions of nonlinear Volterra type integral equations. Two basic integral inequalities are obtained in the form of extremal integrable solutions which are further exploited for proving the boundedness and uniqueness of the integrable solutions of the considered integral equation.

Multivalued nonlinear dominated mappings on a closed ball and associated numerical illustrations with applications to nonlinear integral and fractional operators

In the context of complete strong b-metric-like spaces, we prove new multi-fixed point solutions for the pair of multivalued, dominated operators that fulfill the generalized nonlinear contractions on a closed ball. We employ a mix of two different types of mappings in our approach: one is a class of multi-dominated mappings, while the other is a weaker class of strictly increasing mappings. Additionally, some new fixed-point results concerning the multi-graph-dominated structure in graph contraction are presented. To validate the hypothesis of acquired results, a few sample cases are presented. A numerical experiment has been carried out to approximate the fixed point. In addition, to demonstrate the originality of our findings, we proposed simple and efficient solutions to the system of fractional differential equations and nonlinear Volterra-type integral equations.

Hyers–Ulam–Rassias Stability of Nonlinear Implicit Higher-Order Volterra Integrodifferential Equations from above on Unbounded Time Scales

In this paper, we present sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function.

THE DEVELOPMENT OF A PIEZOELECTRIC DEFECT DETECTION DEVICE THROUGH MATHEMATICAL MODELING APPLIED TO POLYMER COMPOSITE MATERIALS

In this research article, the focus is on examining the strength characteristics of polymer composite materials under various stress states. The investigation into the material dissolution process is conducted in two distinct stages. Initially, the article employs a criterion approach to derive explicit formulas for the latent decay time (incubation period) of materials, considering singular, exponential and Abelian kernels of the damage operator. In the subsequent stage, the classical strength condition is applied to establish a Volterra type integral equation, which characterizes the damage process for hereditary materials. These resulting differential equations are then solved with appropriate initial conditions. The study utilizes the obtained results to create time-dependent graphs of the damage parameter and performs comparisons. A piezoelectric defect detection device for polymer composite materials has been proposed through the application of mathematical modeling.

Optimal exercise of American options under time-dependent Ornstein–Uhlenbeck processes

We study the barrier that gives the optimal time to exercise an American option written on a time-dependent Ornstein–Uhlenbeck process, a diffusion often adopted by practitioners to model commodity prices and interest rates. By framing the optimal exercise of the American option as a problem of optimal stopping and relying on probabilistic arguments, we provide a non-linear Volterra-type integral equation characterizing the exercise boundary, develop a novel comparison argument to derive upper and lower bounds for such a boundary, and prove its Lipschitz continuity in any closed interval that excludes the expiration date and, thus, its differentiability almost everywhere. We implement a Picard iteration algorithm to solve the Volterra integral equation and show illustrative examples that shed light on the boundary's dependence on the process's drift and volatility.

A Novel Approach with Comparative Computational Simulation of Analytical Solutions of Fractional Order Volterra Type Integral Equations

The main purpose of this article is to reconsider the theory of variational iteration method and embed it with Laplace transform to construct the conjoined technique variational iteration Laplace transform method. And then, by using the proposed method we solve a class of integral equations, especially fractional order Volterra type integro‐partial‐differential equations occurring in different fields of natural and social sciences. For this, we first turn to some prelude concepts and different types of equations especially integral equations of different kinds from the literature. Then, we will present step by step the embedded technique. Next, some fractional order model integral equations will be tackled using the proposed method along with some existing techniques, like the Adomian decomposition method and successive approximations method. The efficacy and authentication of the proposed technique will be analysed with the help of the obtained results and graphical exhibitions of the particular solved examples. Finally, use of the initiated method in the future for different types of equations occurring in different fields of science will be presented.

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

<abstract><p>The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $ \delta $, $ 0 < \delta < 1 $.</p></abstract>

Coincidence and common fixed points for F-Contractive mappings

The purpose of this article is to establish the existence and uniqueness of coincidence and common fixed point of discontinuous non-compatible faintly compatible pair of self maps in non-complete metric space without using containment requirement of range space of involved maps satisfying Ciric type F-contraction and Hardy-Roger type F-contraction. Some illustrative examples associated with pictographic validations are provided to demonstrate the main results and to show the genuineness of our results. We consider the application of our results to the study of a two-point boundary value problem related to second order differential equation, solve the two-point boundary value problem of the second-order differential equation arising in electric circuit equation, and also apply our results to Volterra type integral equation using Ciric type F-contraction as well as Hardy Roger type F-contraction. Mathematics subject classification: 47H10; 54H25; 54E50

KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY

This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable \(t\) and \((n-1)\)-dimensional space variable \(x'= \left(x_1,\ldots, x_ {n-1}\right)\) in the \(n\)-dimensional diffusion equation with a time-variable coefficient at the Laplacian of a direct problem solution. Given a known kernel function, a Cauchy problem is investigated as a direct problem. The integral term in the equation has convolution form: the kernel function is multiplied by a solution of the direct problem's elliptic operator. As an overdetermination condition, the result of the direct question on the hyperplane \(x_n = 0\) is used. An inverse question is replaced by an auxiliary one, which is more suitable for further investigation. After that, the last problem is reduced to an equivalent system of Volterra-type integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in Hölder spaces, we prove the main result of the paper, which is a local existence and uniqueness theorem.

Bazı İntegral Denklemlerin Nokta Kollokasyon Yöntemiyle Çözümü

In several engineering or physics problems, particularly those involving electromagnetic theory, thermal and radiation effects, acoustics, elasticity, and some fluid mechanics, it is not always easy or possible to find the analytical solution of integral equations that describe them. For this reason, numerical techniques are used. In this study, Point-collocation method was applied to linear and nonlinear, Volterra and Fredholm type integral equations and the performance and accuracy of the method was compared with several other methods that seem to be popular choices. As the base functions, a suitably chosen family of polynomials were employed. The convergence of the method was verified by increasing the number of polynomial base functions. The results demonstrate that the collocation method performs well even with a relatively low number of base functions and is a good candidate for solving a wide variety of integral equations. Nonlinear problems take longer to calculate approximate solution coefficients than linear problems. Furthermore, it is necessary to use the real and smallest coefficients found in order to obtain a suitable approximate solution to these problems.

THREE OVER DETERMINED SYSTEM VOLTERRA TYPE INTEGRAL EQUATION WITH THREE SINGULAR AND SUPER-SINGULAR DOMAINS

In this paper, for the first time, an overridden system of Volterra-type integral equations with singular or super-singular kernels is considered. It is proved that, when certain conditions are met, the problem of finding a solution to an over determined Volterra type system of integral equations reduces to the problem of finding a solution to one-dimensional Volterra-type integral equations with singular or super-singular kernels, the theory of which is developed in the works of the author of this article. Depending on the sign of the values of functions in the kernels of the system of Volterra type integral equations, found manifold solution consideration system integral equation.

Solution of a two-dimensional parabolic model problem in a degenerate angular domain

In this paper, the boundary value problem of heat conduction in a domain was considered, boundary of which changes with time, as well as there is no the problem solution domain at the initial time, that is, it degenerates into a point. To solve the problem, the method of heat potentials was used, which makes it possible to reduce it to a singular Volterra type integral equations of the second kind. The peculiarity of the obtained integral equation is that it fundamentally differs from the classical Volterra integral equations, since the Picard method is not applicable to it and the corresponding homogeneous integral equation has a nonzero solution.

Численное решение двумерной задачи определения скорости распространения сейсмических волн в неоднородных cредах с памятью

The numerical method for two-dimensional inverse dynamic seismic problem for a viscoelastic isotropic medium is presented. The system of differential equations of elasticity for isotropic medium of memory type is considered as a mathematical model. The unknown values are the displacement, the memory function of the medium (the kernel of the integral term) and the propagation velocity of elastic waves in a weakly horizontally inhomogeneous medium. Additional information for the inverse problem is the response displacement measured on the surface. The method is based on reducing the inverse problem to a system of Volterra-type integral equations and their sequential numerical implementation. The results of the study are analyzed and compared with the analytical solution. It is shown that the results are in satisfactory agreement.

Common fixed point theorems in intuitionistic fuzzy metric spaces with an application for Volterra integral equations

This work aims to demonstrate some common fixed point results in light of certain common limit range properties. Such results are associated with certain weakly compatible mappings in intuitionistic fuzzy metric spaces that meet a specific implicit relation. In particular, the accomplished results in this paper generalize several significant theorems reported in the literature. As an application of our work, we utilize a certain novel result to establish the existence and uniqueness of a common solution for a system of Volterra type integral equations.

Levinson theorem for discrete Schrödinger operators on the line with matrix potentials having a first moment

This paper proves new results on spectral and scattering theory for matrix-valued Schrödinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half-bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations.

On Orthogonal Fuzzy Interpolative Contractions with Applications to Volterra Type Integral Equations and Fractional Differential Equations

In this paper, orthogonal fuzzy versions are reported for some celebrated iterative mappings. We provide various concrete conditions on the real valued functions J,S:(0,1]→(−∞,∞) for the existence of fixed-points of (J,S)-fuzzy interpolative contractions. This way, many fixed point theorems are developed in orthogonal fuzzy metric spaces. We apply the (J,S)-fuzzy version of Banach fixed point theorem to demonstrate the existence and uniqueness of the solution. These results are supported with several non-trivial examples and applications to Volterra-type integral equations and fractional differential equations.

Existence results for the families of multi-mappings with applications to integral and functional equations

The aim of this manuscript is to prove some new fixed-point results for the two families of multivalued dominated-contractive maps defined on a closed ball in a complete multiplicative-metric space. The fixed-point theorems for multigraph-dominated maps are established. We also investigate the existence of fixed points of the two families of graphic contractions in a multiplicative-metric space. For illustration, an example is provided to clarify that the hypothesis of the obtained result is valid. Moreover, the applications of the obtained results to show the existence of solutions to the coupled systems of nonlinear Volterra-type integral equations and mappingal equations are presented as a confirmation of the novelty of our results.

Orbital b-metric spaces and related fixed point results on advanced Nashine–Wardowski–Feng–Liu type contractions with applications

In this article, we prove some novel fixed-point results for a pair of multivalued dominated mappings obeying a new generalized Nashine–Wardowski–Feng–Liu-type contraction for orbitally lower semi-continuous functions in a complete orbital b-metric space. Furthermore, some new fixed-point theorems for dominated multivalued mappings are established in the scenario of ordered complete orbital b-metric spaces. Some examples are offered to demonstrate the validity of our new results’ premise. To demonstrate the applicability of our findings, applications for a system of nonlinear Volterra-type integral equations and fractional differential equations are shown. These results extend the theoretical results of Nashine et al. (Nonlinear Anal., Model. Control 26(3):522–533, 2021).

On Relational Weak Fℜm,η-Contractive Mappings and Their Applications

In this article, we introduce the concept of weak Fℜm,η-contractions on relation-theoretic m-metric spaces and establish related fixed point theorems, where η is a control function and ℜ is a relation. Then, we detail some fixed point results for cyclic-type weak Fℜm,η-contraction mappings. Finally, we demonstrate some illustrative examples and discuss upper and lower solutions of Volterra-type integral equations of the form ξα=∫0αAα,σ,ξσmσ+Ψα,α∈0,1.

Results for retarded nonlinear integral inequalities with mixed powers and their applications to delay integro-differential equations

UDC 517.9 We present new retarded nonlinear integral inequalities with mixеd powers. The obtained inequalities can be used to study the boundedness and global existence of the solutions of integro-differential equation with delay and Volterra-type integral equation with delay. These inequalities extend some results available in the literature. Finally, we present two examples to demonstrate the usefulness of our main results.