System Of Nonlinear Integral Equations Research Articles

On infinite systems of nonlinear integral equations in two variables in Banach Space $BC(\mathbb{R_+}\times \mathbb{R_+},c_0$)

In this paper, the solvability of an infinite system of integral equations of the Volterra-Hammerstein type in Banach space $BC(\mathbb{R_+}\times \mathbb{R_+},c_0$) is examined. Technique associated with the measure of noncompactness plays the most important role in adopted analysis and authors present an example to validate the applicability of the result.

On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations

On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations

On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations

Исследуется одна краевая задача для системы нелинейных интегральных уравнений на полуоси, матричное ядро которой имеет единичный спектральный радиус. эта краевая задача имеет приложения в различных областях физики и биологии. в частности, такие задачи возникают в динамической теории $p$-адических струн для скалярного поля тахионов, в математической теории распространения эпидемий, в кинетической теории газов и в теории переноса излучения. обсуждаются вопросы существования, отсутствия и единственности нетривиального решения этой краевой задачи. доказано, что краевая задача с нулевыми краевыми условиями на бесконечности имеет только тривиальное решение в классе неотрицательных и ограниченных функций. также доказано, что если хотя бы одно из краевых значений на бесконечности положительно, то эта задача имеет нетривиальное решение, которое является выпуклым, неотрицательным, ограниченным и непрерывным. приведены примеры матричных ядер и нелинейностей, удовлетворяющих всем условиям доказанных теорем.

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

We consider a special system of integral equations of convolution type with a monotone convex nonlinearity naturally arising when searching for stationary or limit states in various dynamic models of applied nature, for example, in models of the spread of epidemics, and prove theorems stating the existence or absence of a nontrivial bounded solution with limits at +@ depending on the values of these limits and on the structure of the matrix kernel of the system. We also study the uniqueness of such a solution assuming that it exists. Specific examples of systems whose parameters satisfy the conditions stated in our theorems are given.

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

Coupled fixed point results on orthogonal metric spaces with application to nonlinear integral equations

In this article, we prove some well-known coupled fixed point theorems in 0-complete metric spaces. Also, we present some corollaries related to our study. In addition to this, we give an example showing that our results successfully obtain the existence and uniqueness of the coupled fixed point for 0-complete metric spaces, but the results are not valid for complete metric spaces. Finally, we apply our results to examine the existence and uniqueness of a solution of the system of nonlinear integral equations.

On the solvability of a nonlinear optimization problem with boundary vector control of oscillatory processes

In the paper, the solvability of the nonlinear boundary optimization problem has been investigated for the oscillation processes, described by the integro-differential equation in partial derivatives with Fredholm integral operator. It has been established that the components of the boundary vector control are defined as a solution to a system of nonlinear integral equations of a specific form, and the equations of this system have the property of equal relations. An algorithm for constructing a solution to the problem of nonlinear optimization has been developed.

Fixed Point Theorems in Generalized Banach Algebras and an Application to Infinite Systems in 𝒞([0, 1], c 0) × 𝒞([0, 1], c 0)

The purpose of this paper is to extend Boyd and Wong’s fixed point theorem in complete generalized metric spaces and to apply this result under the so-called G-weak topology in generalized Banach algebras. Some tools will be introduced and involved in our work as the generalized measures of weak non-compactness and the sequential condition Our results will extend many known theorems in the literature. Also, we will give an application for an infinite system of nonlinear integral equations defined on the generalized Banach algebra where c 0 is the space of all real sequences converging to zero.

Predicting the last zero before an exponential time of a spectrally negative Lévy process

AbstractGiven a spectrally negative Lévy process, we predict, in an $L_1$ sense, the last passage time of the process below zero before an independent exponential time. This optimal prediction problem generalises [2], where the infinite-horizon problem is solved. Using a similar argument as that in [24], we show that this optimal prediction problem is equivalent to solving an optimal prediction problem in a finite-horizon setting. Surprisingly (unlike the infinite-horizon problem), an optimal stopping time is based on a curve that is killed at the moment the mean of the exponential time is reached. That is, an optimal stopping time is the first time the process crosses above a non-negative, continuous, and non-increasing curve depending on time. This curve and the value function are characterised as a solution of a system of nonlinear integral equations which can be understood as a generalisation of the free boundary equations (see e.g. [21, Chapter IV.14.1]) in the presence of jumps. As an example, we numerically calculate this curve in the Brownian motion case and for a compound Poisson process with exponential-sized jumps perturbed by a Brownian motion.

On non-trivial solvability of one system of non-linear integral equations on the real axis

Рассматривается система сингулярных интегральных уравнений с монотонной и выпуклой нелинейностью на всей числовой прямой. Данная система имеет приложения во многих направлениях естествознания. В частности, такие системы встречаются в теории $p$-адических открыто-замкнутых струн, в математической теории пространственно-временного распространения эпидемии в рамках известной модели Дикмана-Капера, в кинетической теории газов, в теории переноса излучения. Доказывается теорема существования нетривиального и ограниченного решения. Исследуется также асимптотическое поведение построенного решения на $\pm\infty$. Приводятся конкретные примеры нелинейностей и ядерных функций, имеющих прикладной характер. Библиография: 18 наименований.

On non-trivial solvability of one system of non-linear integral equations on the real axis

A system of singular integral equations with monotonic and convex non-linearity on the entire real line is considered. System of this form have applications in many areas of natural science. In particular, such systems arise in the theory of $p$-adic open-closed strings, in the mathematical theory of spatial-temporal epidemic spread within the framework of the well known Diekmann-Kaper model, in the kinetic theory of gases, in the radiative transfer theory. An existence theorem for a non-trivial and bounded solution is proved. The asymptotic behaviour of the constructed solution at $\pm\infty$ is also studied. Specific examples of non-linearities and kernel functions having an applied character are given.

On Solution of Nonlinear Integral and Fractional Differential Equations via Discontinuous Nonlinear Contractions

The aim of this research work is to demonstrate the survival of exactly one common fixed point for nonlinear contractions without assuming containment of range spaces of an involved pair of mappings, commutativity, and continuity (or any of their weaker forms). It is interesting to note that these findings involve different techniques of proof and comparatively fewerassumptions related to underlying mappings to find the existence of common fixed points of certain mappings. In this way, such investigations are generalizations and improvements over the several prominent recent results of the existing literature. For the application part, we solve the system of nonlinear integral equations of Fredholm and Volterra type and fractional differential equation of Caputo type to validate our results. Moreover, we validate the main conclusion with the help of an example.

Stability Analysis of the Solution to a System of Nonlinear Integral Equations Arising in a Logistic Dynamics Model

Stability Analysis of the Solution to a System of Nonlinear Integral Equations Arising in a Logistic Dynamics Model

On Solvability for Some Classes of System of Non-Linear Integral Equations in Two Dimensions via Measure of Non-Compactness

In this paper, we present some results of coupled fixed points for the system of non-linear integral equations in Banach space. Our results enlarge the results of newer papers. Additionally, we prove the applicability of those results to the solvability of the system of non-linear integral equations. Finally, we give an example to validate the applicability of our results.

Application of a Generalized Fixed Point Principle to the Study of a System of Nonlinear Integral Equations Arising in the Population Dynamics Model

Application of a Generalized Fixed Point Principle to the Study of a System of Nonlinear Integral Equations Arising in the Population Dynamics Model

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel\Abstracteng{In this article the problems of the unique classical solvability and theconstruction of the solution of a nonlinear boundary value problem for a fifth orderpartial integro-differential equations with degenerate kernel are studied. Dirichletboundary conditions are specified with respect to the spatial variable. So, the Fourierseries method, based on the separation of variables is used. A countable system of~thesecond order ordinary integro-differential equations with degenerate kernel is obtained.The method of degenerate kernel is applied to this countable system of ordinaryintegro-differential equations. A system of~countable systems of algebraic equations isderived. Then the countable system of nonlinear Fredholm integral equations is obtained.Iteration process of solving this integral equation is constructed. Sufficient coefficientconditions of the unique solvability of the countable system of nonlinear integralequations are established for the regular values of parameter. In proof of uniquesolvability of the obtained countable system of nonlinear integral equations the method ofsuccessive approximations in combination with the contraction mapping method is used.In the proof of the convergence of Fourier series the Cauchy--Schwarz and Besselinequalities are applied. The smoothness of solution of the boundary value problemis also proved.

Existence of Solutions for Nonlinear Integral Equations in Tempered Sequence Spaces via Generalized Darbo-Type Theorem

Two concepts—one of Darbo-type theorem and the other of Banach sequence spaces—play a very important and active role in ongoing research on existence problems. We first demonstrate the generalized Darbo-type fixed point theorems involving the concept of continuous functions. Keeping one of these theorems into our account, we study the existence of solutions of system of nonlinear integral equations in the setting of tempered sequence space. Moreover, a very interesting and illustrative example is designed to visualize our findings.

Positive Solutions for a System of Fractional Boundary Value Problems with r-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters

In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator A, the solutions of the integral system are the fixed points of A. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator A has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution.

Existence results for a system of nonlinear operator equations and block operator matrices in locally convex spaces

Abstract The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space ( X , { | ⋅ | p } p ∈ Λ ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 𝜏-Krein–Šmulian property, where 𝜏 is a weaker Hausdorff locally convex topology of 𝑋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 𝜏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ ( R + ) × C ⁢ ( R + ) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result.

New iterative scheme for fixed point results of weakly compatible maps in multiplicative $ {G_{\boldsymbol{M}}}- $metric space via various contractions with application

<abstract><p>In this manuscript, we induced several vivid common fixed point theorems for four maps in the setting of complete multiplicative $ {G_{\boldsymbol{M}}}- $metric space via various contractive conditions such as $ \Delta $-implicit contractions. Some new definitions and results are introduced in multiplicative $ {G_{\boldsymbol{M}}}- $ metric space.Moreover, illustrative examples are given to validate our obtained results and an application to a system of nonlinear integral equations are provided to show the novelty of our new results.</p></abstract>