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Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation κ (u) = v , where κ : D (κ) ⊆ X ⟶ X is a nonlinear monotone operator and X is a Hilbert space. The operator in the example used in Tautenhahn's paper was not a monotone operator. So, the following question arises. Can we use LRM for ill-posed nonlinear equations when the involved operator is not monotone? This paper provides a sufficient condition to employ the Lavrentiev regularization technique to such equations whenever the operator involved is non-monotone. Under certain assumptions, the error analysis and adaptive parameter choice strategy for the method are discussed. Moreover, the developed theory is applied to two well-known ill-posed problems—inverse gravimetry and growth law problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a noncoercive nonlinear elliptic operator with a drift term in an unbounded domain. The singular first-order term grows like | E (x) | | ∇ u | , where E (x) is a vector field belonging to a suitable Morrey-type space. Our operator arises as a stationary equation of diffusion–advection problems. We prove existence, regularity, and uniqueness theorems for a Dirichlet problem. To obtain our main results, we use the weak maximum principle and the same a priori estimates. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a new concept of Z-Z-B spaces, which is a generalization of Z-C-X spaces. Meanwhile, the new concept of the superior cone is introduced. Secondly, we study some new problems for semi-closed 1-set-contractive operators in the Z-Z-B space and obtain some new results. These new theorems are proven by combining partial order theory with fixed point index theory. Regarding these theorems, in the latter part of the paper, the proofs are omitted since the methods of proving these theorems are similar. Moreover, two important inequality lemmas are proven. [ABSTRACT FROM AUTHOR]

In this paper, we study the Liouville-type behaviors of the generalized Greiner operators with nonlinear boundary value conditions. We use the technique based upon the method of moving planes. [ABSTRACT FROM AUTHOR]

The goal of the present study is to characterize solutions under a travelling wave formulation to a degenerate Fisher-KPP problem. With the degenerate problem, we refer to the following: a heterogeneous diffusion that is formulated with a high order operator; a non-linear advection and non-Lipstchitz spatially heterogeneous reaction. The paper examines the existence of solutions, uniqueness and travelling wave oscillatory properties (also called instabilities). Such oscillatory behaviour may lead to negative solutions in the proximity of zero. A numerical exploration is provided with the following main finding to declare: the solutions keeps oscillating in the proximity of the null stationary solution due to the high order operator, except if the reaction term is quasi-Lipschitz, in which it is possible to define a region where solutions are positive locally in time. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example. [ABSTRACT FROM AUTHOR]

We consider a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. We use a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we establish convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. These criteria provide the necessary and sufficient conditions on a sequence { u n } , which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result. [ABSTRACT FROM AUTHOR]

This paper features the search for common fixed points of two operators in the nonlinear metric setting provided by CAT(0) spaces. The analysis is performed for the generalized nonexpansivity condition known as condition (E), Garcia-Falset et al., and relies on the three step iteration procedure S n by Sintunavarat and Pitea. The convergence analysis reveals the approximate solutions as limit points for an iteration sequence, where both the nonexpansive mappings to be analyzed and the specific curved structure of the framework interfere. To point out properly the meaning of this approach, we provide also examples accompanied by numerical simulations. The Poincaré half-plane is one of the non-positively curved setting to be used. [ABSTRACT FROM AUTHOR]

Making use of the Boyd-Wong fixed point theorem, we establish a new existence and uniqueness result and an approximation process of the fixed point for the product of two nonlinear operators in Banach algebras. This provides an adequate tool for deriving the existence and uniqueness of solutions of two interesting type of nonlinear functional equations in Banach algebras, as well as for developing an approximation method of their solutions. In addition, to illustrate the applicability of our results we give some numerical examples. [ABSTRACT FROM AUTHOR]

In the given study, we investigate the three-step NTS's ball convergence for solving nonlinear operator equations with a convergence order of five in a Banach setting. A nonlinear operator's first-order derivative is assumed to meet the generalized Lipschitz condition, also known as the κ-average condition. Furthermore, several theorems on the convergence of the same method in Banach spaces are developed with the conditions that the derivative of the operators must satisfy the radius or center-Lipschitz condition with a weak κ-average and that κ is a positive integrable but not necessarily non-decreasing function. This novel approach allows for a more precise convergence analysis even without the requirement for new circumstances. As a result, we broaden the applicability of iterative approaches. The theoretical results are supported further by illuminating examples. The convergence theorem investigates the location of the solution ϵ* and the existence of it. In the end, we achieve weaker sufficient convergence criteria and more specific knowledge on the position of the ϵ* than previous efforts requiring the same computational effort. We obtain the convergence theorems as well as some novel results by applying the results to some specific functions for κ(u). Numerical tests are carried out to corroborate the hypotheses established in this work. [ABSTRACT FROM AUTHOR]

In this article, we study the JF iterative algorithm to approximate the fixed points of a non-linear operator that satisfies condition (E) in uniformly convex Banach spaces. Further, some weak and strong convergence results are presented for the same operator using the JF iterative algorithm. We also demonstrate that the JF iterative algorithm is weakly w 2 G -stable with respect to almost contractions. In connection with our results, we provide some illustrative numerical examples to show that the JF iterative algorithm converges faster than some well-known iterative algorithms. Finally, we apply the JF iterative algorithm to estimate the solution of a functional non-linear integral equation. The results of the present manuscript generalize and extend the results in existing literature and will draw the attention of researchers. [ABSTRACT FROM AUTHOR]

Morphological operators are nonlinear transformations commonly used in image processing. Their theoretical foundation is based on lattice theory, and it is a well-known result that a large class of image operators can be expressed in terms of two basic ones, the erosions and the dilations. In practice, useful operators can be built by combining these two operators, and the new operators can be further combined to implement more complex transformations. The possibility of implementing a compact combination that performs a complex transformation of images is particularly appealing in resource-constrained hardware scenarios. However, finding a proper combination may require a considerable trial-and-error effort. This difficulty has motivated the development of machine-learning-based approaches for designing morphological image operators. In this work, we present an overview of this topic, divided in three parts. First, we review and discuss the representation structure of morphological image operators. Then we address the problem of learning morphological image operators from data, and how representation manifests in the formulation of this problem as well as in the learned operators. In the last part we focus on recent morphological image operator learning methods that take advantage of deep-learning frameworks. We close with discussions and a list of prospective future research directions. [ABSTRACT FROM AUTHOR]