Functional Integral Equations Research Articles

Numerical Analysis of the Chebyshev Collocation Method for Functional Volterra Integral Equations

The collocation method based on Chebyshev basis functions, coupled Picard iterative process, is proposed to solve a functional Volterra integral equation of the second kind. Using the Banach Fixed Point Theorem, we prove theorems on the existence and uniqueness solutions in the L2-norm. We also provide the convergence and stability analysis of the proposed method, which indicates that the numerical errors in the L2-norm decay exponentially, provided that the kernel function is sufficiently smooth. Numerical results are presented and they confirm the theoretical prediction of the exponential rate of convergence.

Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra

Here, we investigate the existence result for a nonlinear quadratic functional integral equation of fractional order using a fixed point theorem of Dhage. The continuous dependence of solution on the delay functions will be studied. As an application, an existence theorem for the fractional hybrid differential equations is proved. Also, we study a general quadratic integral equation of fractional order.

Existence of solutions of infinite systems of nonlinear functional integral equations of N-variables in C(I × I ×⋯ × I,m(ϕ))

The aim of this paper is to investigate the solvability of infinite systems of nonlinear functional integral equations of [Formula: see text]-variables in [Formula: see text] by using the Hausdorff measure of noncompactness with the help of Meir–Keeler condensing operators. We also provide an illustrative example in support of our existence theorems.

On the solvability of a nonlinear functional integral equations via measure of noncompactness in

Using the technique of a suitable measure of non-compactness and the Darbo fixed point theorem, we investigate the existence of a nonlinear functional integral equation of Urysohn type in the space of Lebesgue integrable functions Lp(RN). In this space, we show that our functional-integral equation has at least one solution. Finally, an example is also discussed to indicate the natural realizations of our abstract result.

Multivariate generalized Meir–Keeler condensing operators and their applications to systems of integral equations

In this work, we introduce the concepts of multivariate generalized Meir–Keeler condensing operator, and multivariate L-function and prove some new fixed point theorems with the aid of the measure of non-compactness. Our results generalize and extend a lot of comparable results in the literature. Also, we use these results to discuss the solvability for a system of functional integral equations of Volterra type in three variables. Finally, we present an example to support our results.

On the existence of solutions of some functional integral equations in L^p(R_+)

In this paper, we study the existence of solutions for a functional integral equations by using Darbo's fixed point theorem and a measure of noncompactness on the spaces Lp(R+), 1 <p<+∞. The obtained result generalizes and extends several one obtained earlier in many papers and monographs. An example which shows the applicability of our result is also included.

Influence of the illumination direction on the optical extinction by a scatterer close to a dielectric interface

The intrinsic optical response of a scattering object is known to be modified when it is placed on a substrate. Moreover, the total extinction through ohmic losses and Rayleigh scattering is basically dependent on the reversal of the direction of the excitation wave. Considering, as stated by the generalized optical theorem, that the total extinction of the incident wave is shared among the extinction of the waves transmitted and reflected by the non-absorbing substrate, we show that only the contribution of the transmitted wave is insensitive to the direction of illumination, by analogy with the textbook case of transmission through a planar stratified medium. This property was recently confirmed experimentally [J. Phys. Chem. C 123, 15217 (2019)] and is established here on theoretical grounds in the frameworks of the Green's function integral equation method and the Lorentz reciprocity theorem. Some of its limitations with regard to the incidence and the polarization of the excitation wave are also discussed in detail.

On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces

AbstractThe existence of a.e. monotonic solutions for functional quadratic Hammerstein integral equations with the perturbation term is discussed in Orlicz spaces. We utilize the strategy of measure of noncompactness related to the Darbo fixed point principle. As an application, we discuss the presence of solution of the initial value problem with nonlocal conditions.

New Generalization of Darbo's Fixed Point Theorem via $alpha$-admissible Simulation Functions with Application

In this paper, at first, we introduce $alpha_{mu}$-admissible, $Z_mu$-contraction and $N_{mu}$-contraction via simulation functions. We prove some new fixed point theorems for defined class of contractions via $alpha$-admissible simulation mappings, as well. Our results can be viewed as extension of the corresponding results in this area. Moreover, some examples and an application to functional integral equations are given to support the obtained results.

Measure of noncompactness and a generalized Darbo\u2019s fixed point theorem and its applications to a system of integral equations

Işık et al. (Mathematics 7:862, 2019) presented an interesting generalization of the Banach contraction principle. In this paper, motivated by Işık et al., we give a new extension of the well-known Darbo inequality in a Banach space. Our results provide several generalizations of the Darbo inequality. As an application, we study the existence of solutions for a system of functional integral equations in C[0,T]. Finally, we expose a genuine example to support the effectiveness of our results.

Existence of solutions for a class of multivalued functional integral equations of Volterra type via the measure of nonequicontinuity on the Fréchet space [formula omitted

The existence of continuous not necessarily bounded solutions of nonlinear functional Volterra integral inclusions in infinite dimensional setting is shown with the aid of the measure of nonequicontinuity. New abstract topological fixed point results for admissible condensing operators are introduced. Weak compactness criterion in the space of locally integrable functions in the sense of Bochner is set forth. Some examples illustrating the usefulness of the presented approach are also included.

Exact Analytic Representations for the Integral Characteristics of a Four-Point Coherence Function for Laser Beams in Turbulent Media

A new integral-functional equation is derived for the four-dimensional Fourier transform of the four-point coherence function of laser beams in turbulent media and two families of exact analytical solutions of this equation are found. These solutions hold for any level of fluctuations of the refractive index of air. They are used to obtain exact analytic representations of the integral characteristics of the four-point coherence function. In particular, the truncated spectral characteristics of the spatial correlation function of the intensities are found. These representations can be used to test asymptotic, numerical, and other methods for finding this function and to describe its integral properties.

On Certain 3-Dimensional Limit Boundary Value Problems

The paper is devoted to studying limit behavior for a solution of model elliptic pseudo-differential equation with some integral boundary condition in 4-wedge conical canonical 3D singular domain with two parameters. It is shown that the solution of such boundary value problem can have a limit with respect to endpoint values of the parameters in appropriate Sobolev–Slobodetskii space if the boundary function is a solution of a special functional singular integral equation.

Fejér monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued Banach spaces

It is our purpose in this paper to prove some fixed point results and Fej´er monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued Banach spaces. We prove that results in complex valued Banach spaces are valid in cone metric spaces with Banach algebras. Furthermore, we apply our results in solving certain mixed type VolterraFredholm functional nonlinear integral equation in complex valued Banach spaces.

Nonlinear functional integral equation: Existence, global attractivity and positivity of solutions

Nonlinear functional integral equation: Existence, global attractivity and positivity of solutions

On totally global solvability of controlled second kind operator equation

We consider the nonlinear evolutionary operator equation of the second kind as follows $\varphi=\mathcal{F}\bigl[f[u]\varphi\bigr]$, $\varphi\in W[0;T]\subset L_q\bigl([0;T];X\bigr)$, with Volterra type operators $\mathcal{F}\colon L_p\bigl([0;\tau];Y\bigr)\to W[0;T]$, $f[u]$: $W[0;T]\to L_p\bigl([0;T];Y\bigr)$ of the general form, a control $u\in\mathcal{D}$ and arbitrary Banach spaces $X$, $Y$. For this equation we prove theorems on solution uniqueness and sufficient conditions for totally (with respect to set $\mathcal{D}$) global solvability. Under natural hypotheses associated with pointwise in $t\in[0;T]$ estimates the conclusion on univalent totally global solvability is made provided global solvability for a comparison system which is some system of functional integral equations (it could be replaced by a system of equations of analogous type, and in some cases, of ordinary differential equations) with respect to unknown functions $[0;T]\to\mathbb{R}$. As an example we establish sufficient conditions of univalent totally global solvability for a controlled nonlinear nonstationary Navier-Stokes system.

Fuzzy Bezier splines with application to fuzzy functional integral equations

In this paper, we present the description of fuzzy Bezier splines and as an application we propose an iterative numerical method for approximating the solution of fuzzy functional integral equations of Fredholm type. The convergence of the method is proved by providing an error estimate and it is tested on some numerical examples. The numerical stability regarding the choice of the first iteration is investigated.

On the existence of solutions of a set-valued functional integral equation of Volterra\u2013Stieltjes type and some applications

This paper is concerned with the existence of continuous solutions of a set-valued functional integral equation of Volterra–Stieltjes type. The continuous dependence of the solution on the set of selections of the set-valued function will be proven. As an application, we study the existence of solutions to an initial-value problem of arbitrary fractional-order differential inclusion.

Numerical analysis of a high order method for nonlinear delay integral equations

In this paper, we introduce a new class of Multistep Collocation Method (MCM) for solving the non-linear Volterra Functional Integral Equations (VFIEs) including two types of linear and non-linear lag functions. We divide the definition domain into several subintervals according to the primary discontinuous points associated with the delay function. In each subinterval, where the solution is smooth enough, we can apply MCM to approximate the solution. Convergence analysis of the method for both types is also analyzed and illustrative five examples are provided to confirm the applicability of this method.

Solvability of functional-integral equations (fractional order) using measure of noncompactness

We investigate the solutions of functional-integral equation of fractional order in the setting of a measure of noncompactness on real-valued bounded and continuous Banach space. We introduce a new μ-set contraction operator and derive generalized Darbo fixed point results using an arbitrary measure of noncompactness in Banach spaces. An illustration is given in support of the solution of a functional-integral equation of fractional order.