Nonlinear Integral Operators Research Articles

An eigenvalue problem for self-similar patterns in Hele-Shaw flows

Hele-Shaw problems are prototypes to study the interface dynamics. Linear theory suggests the existence of self-similar patterns in a Hele-Shaw flow. That is, with a specific injection flux the interface shape remains unchanged while its size increases. In this paper, we explore the existence of self-similar patterns in the nonlinear regime and develop a nonlinear theory characterizing their fundamental features. Using a boundary integral formulation, we pose the question of self-similarity as a generalized nonlinear eigenvalue problem, involving two nonlinear integral operators. The nonlinear flux constant Cf is the eigenvalue and the corresponding self-similar pattern x̃ is the eigenvector. We develop a quasi-Newton method to solve the problem and show the existence of nonlinear shapes with k-fold dominated symmetries. Nonlinear results are compared with the established linear theory, demonstrating a divergence between the two due to non-linear effects absent in the linear stability analysis. Further, we investigate how sensitive the shape of the interface is to the viscosity. Additionally, we conduct numerous numerical experiments using a wide range of initial guesses and initial flux constants. Through these experiments, one is able to obtain a diagram of self-similar shapes and the corresponding flux. It could be used to verify possible self-similar shapes with a proper initial guess and initial flux constant. Our results go beyond the predictions of linear theory and establish a bridge between the linear theory and simulations.

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.

Novel Multivariate Integral Operators Incorporating Trigonometric Transformations

The creation of new nonlinear multivariate integral operators is motivated by the need for mathematical tools that can handle the complex interdependencies that naturally arise in contemporary applications. From an abstract scientific point of view, it is also necessary to develop new operator theories beyond existing ones to offer original research perspectives. This article contributes to these complementary aspects. We present two nonlinear multivariate integral operators that have the particularity of incorporating trigonometric transformations of the main function. Thanks to their trigonometric nature, they completely stand out from existing operators, offering a new and complete framework. Therefore, we take advantage of advanced mathematical techniques for trigonometric functions to address the challenges they pose. In particular, we show that they have manageable integrals and series expansions, that they are solutions of specific differential and functional equations, and that they are involved in general inequalities of various types (Hölder-type, convex-type, etc.). In the application part, we use some of these properties to propose a wide collection of trigonometric inequalities that are both original and precise. Figures are produced to illustrate them for a direct visual check.

Convergence in $$\varphi $$-variation for Mellin-type nonlinear integral operators

Convergence in $$\varphi $$-variation for Mellin-type nonlinear integral operators

Learning nonlinear integral operators via recurrent neural networks and its application in solving integro-differential equations

In this paper, we propose using LSTM-RNNs (Long Short-Term Memory-Recurrent Neural Networks) to learn and represent nonlinear integral operators that appear in nonlinear integro-differential equations (IDEs). The LSTM-RNN representation of the nonlinear integral operator allows us to turn a system of nonlinear integro-differential equations into a system of ordinary differential equations for which many efficient solvers are available. Furthermore, because the use of LSTM-RNN representation of the nonlinear integral operator in an IDE eliminates the need to perform a numerical integration in each numerical time evolution step, the overall temporal cost of the LSTM-RNN-based IDE solver can be reduced to O(nT) from O(nT2) if a nT-step trajectory is to be computed. We illustrate the efficiency and robustness of this LSTM-RNN-based numerical IDE solver with a model problem. Additionally, we highlight the generalizability of the learned integral operator by applying it to IDEs driven by different external forces. As a practical application, we show how this methodology can effectively solve the Dyson’s equation for quantum many-body systems.

Subordinations by η-convex functions for a class of nonlinear integral operators

The aim of the present paper is to investigate some subordination implications for a class of nonlinear integral operators associated with multivalent functions in the open unit disk. Some known results associated with a class of nonlinear averaging integral operators are also pointed out as special cases of the main findings which are presented here.

Propagation of minima for nonlocal operators

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$ -th fractional truncated Laplacian or the $k$ -th fractional eigenvalue which are fully nonlinear integral operators whose nonlocality is somehow $k$ -dimensional.

Positivity and discretization of Fredholm integral operators

We provide sufficient conditions for vector-valued Fredholm integral operators and their commonly used spatial discretizations to be positive in terms of an order relation induced by a corresponding order cone. It turns out that reasonable Nyström methods preserve positivity. Among the projection methods, persistence is obtained for the simplest ones based on polynomial, piecewise linear or specific cubic interpolation (collocation), as well as for piecewise constant basis functions in a Bubnov-Galerkin approach. However, for semi-discretizations using quadratic splines or sinc-collocation we demonstrate that positivity is violated. Our results are illustrated in terms of eigenpairs for Krein-Rutman operators and form the basis of corresponding investigations for nonlinear integral operators.

Two-phase Stokes flow by capillarity in the plane: The case of different viscosities

We study the two-phase Stokes flow driven by surface tension for two fluids of different viscosities, separated by an asymptotically flat interface representable as graph of a differentiable function. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single and double layer potential, spectral results on the corresponding integral operators, and abstract results on nonlinear parabolic evolution equations.

CONVERGENCE OF URYSOHN-TYPE NONLINEAR INTEGRAL OPERATORS AT p–LEBESGUE POINT

In this paper, a general form of the family of Urysohn-type nonlinear integral operators with kernel $K_{\lambda }\left( x,t,g\right) $ is discussed and theorems about the point convergence of this family at ${p}$--Lebesgue points of a function in $L_{p}$ are given. Here, $\lambda$ is the accumulation point and is a positive parameter that changes in the real numbers. Kernel function $K_{\lambda }\left( x,t,g\left( t\right) \right)$ is an entire analytic function with respect to its third variable.

Interpolation of nonlinear integral Urysohn operators in net spaces

In this paper, we study the interpolation properties of the net spaces N_p,q(M), in the case when M is a sufficiently general arbitrary system of measurable subsets from R^n. The integral Urysohn operator is considered. This operator generalizes all linear, integral operators, and non-linear integral operators. The Urysohn operator is not a quasilinear or subadditive operator. Therefore, the classical interpolation theorems for these operators do not hold. A certain analogue of the Marcinkiewicz-type interpolation theorem for this class of operators is obtained. This theorem allows to obtain, in a sense, a strong estimate for Urysohn operators in net spaces from weak estimates for these operators in net spaces with local nets. For example, in order for the Urysohn integral operator in a net space, where the net is the set of all balls in R^n, it is sufficient for it to be of weak type for net spaces, where the net is concentric balls.

Compactness of nonlinear integral operators with discontinuous and with singular kernels

This paper establishes compactness of nonlinear integral operators in the space of continuous functions. One result deals with operators whose kernel can have jumps across a finite number of curves, which typically arise from the study of ordinary differential equations with boundary conditions of local or nonlocal type. Several other results deal with operators whose kernels have a singularity, which arise from the study of fractional differential equations. We motivate the study of these integral equations by discussing some initial value problems for fractional differential equations of Caputo and Riemann-Liouville type. We prove a compact embedding theorem for fractional integrals in order to give a new treatment for the singular kernel case.

On differential properties of one class of nonlinear singular integral operators

On differential properties of one class of nonlinear singular integral operators.

Multivariate approximation in φ\varphi-variation for nonlinear integral operators via summability methods

We consider convolution-type nonlinear integral operators endowed with Musielak-Orlicz $\varphi$-variation. Our aim is to get more powerful approximation results with the help of summability methods. In this study, we use $\varphi$-absolutely continuous functions for our convergence results. Moreover, we study the order of approximation using suitable Lipschitz class of continuous functions. A general characterization theorem for $\varphi $-absolutely continuous functions is also obtained. We also give some examples of kernels in order to verify our approximations. At the end, we indicate our approximations in figures together with some numerical computations.

Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentials

We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.

Convergence in Phi-Variation and Rate of Approximation for Nonlinear Integral Operators Using Summability Process

We investigate the approximation properties of nonlinear integral operators of the convolution type. In this approximation, we use functions of bounded variation based on the appropriate functionals. To get more general results, we consider Bell-type summability methods in the approximation. Moreover, we examine the rate of approximation. Then, using summability methods, we obtain a characterization for absolute continuity. Our examples at the end of the paper clearly demonstrate why we used summability methods rather than convergence in the conventional sense.

Characterization of Absolute and Uniform Continuity

In the present paper, by considering nonlinear integral operators and using their approximations via regular summability methods, we obtain characterizations for some function spaces including the space of absolutely continuous functions, the space of uniformly continuous functions, and their other variants. We observe that Bell-type summability methods are quite effective to generalize and improve some related results in the literature. At the end of the paper, we discuss some special cases and applications.

Degeneracy results for fully nonlinear integral operators

It is shown that integral operators of the fully nonlinear type K(x)(t)=∫Ωk(t,s,x(t),x(s))ds exhibit similar degeneracy phenomena in a large class of spaces as superposition operators F(x)(t)=f(t,x(t)). In particular, K is Fréchet differentiable in Lp only if it is affine with respect to the “x(t)” argument. Similar degeneracy results hold if K satisfies a local Lipschitz or compactness condition. Also vector functions, infinite measure spaces, and a much richer class of function spaces than only Lp are considered. As a side result, degeneracy assertions for superposition operators are obtained in this more general setting, complementing the known results for scalar functions. As a particular example, it is shown that the operators arising in continuous limits of coupled Kuramoto oscillators fail everywhere to be Fréchet differentiable or locally compact.

Approximation by nonlinear integral operators via summability process

AbstractIn this paper, we study the approximation properties of nonlinear integral operators of convolution‐type by using summability process. In the approximation, we investigate the convergence with respect to both the variation semi‐norm and the classical supremum norm. We also compute the rate of approximation on some appropriate function classes. At the end of the paper, we construct a specific sequence of nonlinear operators, which verifies the summability process. Some graphical illustrations and numerical computations are also provided.

The contractivity of cone-preserving multilinear mappings

With the notion of the mode-j Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff–Hopf and Perron–Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type , with xi being an element of a cone Ci in a Banach space . We then consider a family of nonlinear integral operators f i with positive kernel, acting on the product of spaces of continuous real-valued functions. In this setting we provide an explicit formula for the mode-j contraction ratio, which is particularly relevant in practice, as this type of operator plays a central role in numerous models and applications.