Random Integral Equations Research Articles

A fixed point result on a DCMLS and applications to matrix equations and random integral equations

In this work, we present a fixed point result in the context of a double controlled metric-like space involving its control functions. A matrix equation has been solved via a fixed point technique. We also give some concrete and numerical examples. Finally, we ensure the existence of a solution of a random integral equation of Fredholm type.

A boundary value problem for a random-order fractional differential equation

In this paper, we define a new Riemann–Liouville fractional integral with random order, from this Caputo and Riemann–Liouville fractional derivatives are straightforward obtained, where the fractional order of these operators is a simple random variable. We derive useful properties analogous to those of the fractional operators with constant order, such as the semigroup property. As an application, we study a boundary value problem for the fractional oscillator with random order, using a random integral equation of Fredholm type. Finally, in order to solve this problem, we adapt the Nystrom method to get a numerical solution.

Global stability results for Volterra–Hadamard random partial fractional integral equations

Global stability results for Volterra–Hadamard random partial fractional integral equations

Random fixed point theorems under weak topology features and application to random integral equations with lack of compactness

We present several random fixed point theorems for random operators with deterministic or stochastic domains. The main assumptions of our results are formulated in terms of the weak topology and an axiomatic definition of the measure of weak noncompactness. The results herein extend in a broad sense some new and classical results in the literature. As an application, we discuss the solvability of a random Hammerstein integral equation with lack of compactness.

C^{*}-Algebra valued fuzzy normed spaces with application of Hyers\u2013Ulam stability of a random integral equation

In this paper, we consider C^{*}-algebra valued fuzzy normed spaces. We study the random integral equation (frac{1}{2c})int _{x-cd}^{x+cd}u(gamma ,tau ,d_{0}),dtau =u( gamma ,x,d) which is related to the stochastic wave equation. In addition, using a C^{*}-algebra valued fuzzy controller function, we consider its C^{*}-algebra valued fuzzy Hyers–Ulam stability.

Investigation of the Behaviour of Volterra Integral Equations with Random Effects

In this study, random Volterra integral equations obtained by transforming components of deterministic Volterra integral equations to random variables are analysed. Beta, Normal (Gaussian), Gamma, Geometric and Uniform distributions are used to investigate the random behaviour of the solutions for Volterra integral equations under random effects. The random version of Differential Transformation Method (RDTM) is used to obtain an approximation to the solution of the random Volterra integral equation. Using the approximate solutions, approximate expected values and approximate variances are calculated. Some integro-differential equations, obtained by using random components with the above mentioned distributions, are solved as numerical examples. Results are obtained in MAPLE and shown in graphs. It is seen that random Differential Transformation Method is effective for the examination of random Volterra integral equations. Comparison of the solutions is given to underline the accuracy of the method.

The Existence of Approximate Solutions for Nonlinear Volterra Type Random Integral Equations

The Existence of Approximate Solutions for Nonlinear Volterra Type Random Integral Equations

Random fixed point theorems in Banach spaces applied to a random nonlinear integral equation of the Hammerstein type

The purpose of this paper is to define a new random operator called the generalized ϕ-weakly contraction of the rational type. This new random operator includes those studied by Khan et al. (Filomat 31(12):3611–3626, 2017) and Zhang et al. (Appl. Math. Mech. 32(6):805–810, 2011) as special cases. We prove some convergence, existence, and stability results in separable Banach spaces. Moreover, we produce some numerical examples to demonstrate the applicability of our analytical results. Furthermore, we apply our results in proving the existence of a solution of a nonlinear integral equation of the Hammerstein type.

Coupled Random Fixed Point Theorems for Mixed Monotone Nonlinear Operators

In this paper, we prove the existence of a random coupled coincidence and coupled random fixed point theorems in complete separable metric space without the mixed \(g\)-monotone property. The results are used to prove existence of random solutions for random integral equation.

On random fixed point theorems with applications to integral equations

This particular research establishes some random fixed point theorems for general nonlinear random contractive operators in the context of partially ordered separable metric spaces. The existence and uniqueness of the random solution for the nonlinear integral equation is obtained by applying the result of the random fixed point. The results generalize and improve on some related works in the literature.•Our theorems are proved in the context of metric space while Saluja and Tripathi [8] proved in the context of partial metric spaces.•Nieto, Ouahab and. Rodriguez-Lopez [9] proved their theorem using Banach contraction mappings while we proved our theorems using Hardy and Rogers contraction mappings.•Rashwan and Albaqeri [3] proved the solution of the random integral equation using the Banach contraction operator while we proved our solution to the random integral equations employing more general contractive operator.

On stochastic solutions of nonlocal random functional integral equations

In this paper, we use Schauder’s fixed point to establish the existence of at least one solution for a functional nonlocal stochastic differential equation under sufficient conditions in the space of all square integrable stochastic processes with a finite second moment. We state and prove the conditions which guarantee the uniqueness of the solution. We solve a nonlinear example analytically and obtain the initial condition which makes the solution passes through a random position with a given normal distribution at a specified time. Also, the Milstein scheme to this example is studied.

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of ϑ -methods for Volterra integral equations.

A Coupled Random Fixed Point Result With Application in Polish Spaces

In this paper, we present a new concept of random contraction and prove a coupled random fixed point theorem under this condition which generalizes stochastic Banach contraction principle. Finally, we apply our contraction to obtain a solution of random nonlinear integral equations and we present a numerical example.

Some existence results and stability concepts for partial fractional random integral equations with multiple delay

Abstract In this paper, we present some results concerning the existence and the stability of solutions for some functional integral equations of Riemann–Liouville fractional order with random effects and multiple delay, by applying a random fixed point theorem with stochastic domain and the measure of noncompactness.

Weak solutions for partial Pettis Hadamard fractional integral equations with random effects

In this article, we apply M\onch and Engl's fixed point theorems associated with the technique of measure of weak noncompactness to investigate the existence of random solutions for a class of partial random integral equations via Hadamard's fractional integral, under the Pettis integrability assumption.

Random fixed point theorems for Hardy-Rogers self-random operators with applications to random integral equations

In this paper, we prove some random fixed point theorems for Hardy–Rogers self-random operators in separable Banach spaces and, as some applications, we show the existence of a solution for random nonlinear integral equations in Banach spaces. Some stochastic versions of deterministic fixed point theorems for Hardy–Rogers self mappings and stochastic integral equations are obtained.

Convergence and (S, T)-stability almost surely for random Jungck-type iteration processes with applications

The purpose of this paper is to introduce the random Jungck–Mann-type and the random Jungck–Ishikawa-type iterative processes. We prove some convergence and stability results for these random iterative processes for certain random operators. Furthermore, we apply our results to study random non-linear integral equation of the Hammerstein type. Our results generalize, extend and unify several well-known deterministic results in the literature. Moreover, our results generalize recent results of Okeke and Abbas and Okeke and Kim .

Anticipating random periodic solutions—I. SDEs with multiplicative linear noise

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify them as solutions of coupled forward–backward infinite horizon stochastic integral equations (IHSIEs), using the “substitution theorem” of stochastic differential equations with anticipating initial conditions. In general, random periodic solutions and the solutions of IHSIEs, are anticipating. For the linear noise case, with the help of the exponential dichotomy given in the multiplicative ergodic theorem, we can identify them as the solutions of infinite horizon random integral equations (IHSIEs). We then solve a localised forward–backward IHRIE in C(R,Lloc2(Ω)) using an argument of truncations, the Malliavin calculus, the relative compactness of Wiener–Sobolev spaces in C([0,T],L2(Ω)) and Schauder's fixed point theorem. We finally measurably glue the local solutions together to obtain a global solution in C(R,L2(Ω)). Thus we obtain the existence of a random periodic solution and a periodic measure.

Random compact operators

Random compact operators are useful to study random differentiation and random integral equations. In this paper, we define the random norm of R-bounded operators and study random norms of differentiation operators and integral operators. The definition of random norm of R-bounded operators led us to study the random operator theory.

Existence and Attractivity Results for Volterra Type Nonlinear Perturbed Random Integral Equations

Existence and Attractivity Results for Volterra Type Nonlinear Perturbed Random Integral Equations