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In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 풟 ⁢ ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 풟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and interior W 2 , p -regularity of the solution, and the regularity of the free boundary ∂ ⁡ { u > ϕ } to the obstacle problem of the porous medium equation, u t = Δ ⁢ u m ( m > 1 ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ ⁡ { u > ϕ } and ∂ ⁡ { u > 0 } , we consider two cases on the initial data which make the free boundary ∂ ⁡ { u > ϕ } separate from the free boundary ∂ ⁡ { u > 0 } . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 -regularity of the free boundary ∂ ⁡ { u > ϕ } is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator. [ABSTRACT FROM AUTHOR]

In this paper, we consider the iteratively regularized Gauss–Newton method with frozen derivative and formulate its convergence rates in the settings of Banach spaces. The convergence rates of iteratively regularized Gauss–Newton method with frozen derivative are well studied via generalized source conditions. We utilize the recently developed concept of conditional stability of the inverse mapping to derive the convergence rates. Also, in order to show the practicality of this paper, we show that our results are applicable on an ill-posed inverse problem. Finally, we compare the convergence rates derived in this paper with the existing convergence rates in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse coefficient problem of identifying both the mass density ρ ⁢ (x) > 0 and flexural rigidity r ⁢ (x) > 0 of a damped Euler–Bernoulli (cantilever) beam governed by the equation ρ ⁢ (x) ⁢ u t ⁢ t + μ ⁢ (x) ⁢ u t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x = 0 , (x , t) ∈ (0 , ℓ) × (0 , T) , subject to boundary conditions u ⁢ (0 , t) = u x ⁢ (0 , t) = 0 , u x ⁢ x ⁢ (ℓ , t) = 0 , - (r ⁢ (x) ⁢ u x ⁢ x ⁢ (x , t)) x | x = ℓ = g ⁢ (t) , from the available measured boundary deflection ν ⁢ (t) := u ⁢ (ℓ , t) and rotation θ ⁢ (t) := u x ⁢ (ℓ , t) at the free end of the beam. The distinctive feature of the considered inverse coefficient problem is that not one, but two Neumann-to-Dirichlet operators have to be formally defined. The inverse problem is hence formulated as a system of nonlinear Neumann-to-Dirichlet operator equations with the right-hand sides consisting of the measured outputs. As a natural consequence of this approach, a vector-form Tikhonov functional is introduced whose components are squares of the L 2 -norm differences between predicted and measured outputs. We then prove existence of a quasi-solution of the inverse problem and derive explicit gradient formulae for the Fréchet derivatives of both components of the Tikhonov functional. These results are instrumental to any gradient based algorithms for reconstructing the two unknown coefficients of the considered damped Euler–Bernoulli beam. [ABSTRACT FROM AUTHOR]

In this paper, we studied the separability of the non-linear Schrodinger operator of the form S u x = − Δ u x + V x , u u x , where Δ u x = ∑ i = 1 n ∂ 2 u x ∂ x i 2 , with the non-linear matrix potential V x , u . We obtained the sufficient conditions for separability of this operator in the space L 2 R n ℓ and we established the suitable coercive inequalities. [ABSTRACT FROM AUTHOR]

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method. [ABSTRACT FROM AUTHOR]

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 휏-Krein–Šmulian property, where 휏 is a weaker Hausdorff locally convex topology of 푋. 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 휏-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. [ABSTRACT FROM AUTHOR]

This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors' recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search. [ABSTRACT FROM AUTHOR]

We consider an elliptic variational-hemivariational inequality 𝓟 in a reflexive Banach space, governed by a set of constraints K, a nonlinear operator A, and an element f. We associate to this inequality a sequence {𝓟n} of variational-hemivariational inequalities such that, for each n ∈ ℕ, inequality 𝓟n is obtained by perturbing the data K and A and, moreover, it contains an additional term governed by a small parameter εn. The unique solvability of 𝓟 and, for each n ∈ ℕ, the solvability of its perturbed version 𝓟n, are guaranteed by an existence and uniqueness result obtained in literature. Denote by u the solution of Problem 𝓟 and, for each n ∈ ℕ, let un be a solution of Problem 𝓟n. The main result of this paper states the strong convergence of un → u in X, as n → ∞. We show that the main result extends a number of results previously obtained in the study of Problem 𝓟. Finally, we illustrate the use of our abstract results in the study of a mathematical model which describes the contact of an elastic body with a rigid-deformable foundation and provide the corresponding mechanical interpretations. [ABSTRACT FROM AUTHOR]

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ (ℝ n) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1

In this paper, we study the relationship between the class of weakly demicompact linear operators, introduced in [KRICHEN, B.—O'REGAN, D.: On the class of relatively weakly demicompact nonlinear operators, Fixed Point Theory 19 (2018), 625–630], and measures of weak noncompactness of linear operators with respect to an axiomatic one. Moreover, some Fredholm and perturbation results involving the class of weakly demicompact linear operators are investigated. Our results are then used to investigate the relationship between the relative essential spectrum of the sum of two linear operators and the relative essential spectrum of each of these operators. [ABSTRACT FROM AUTHOR]

In this paper, we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue λ F ⁢ (p , Ω) {\lambda_{F}(p,\Omega)} of the anisotropic p-Laplacian, 1 < p < + ∞ {1

In this paper by using of Suzuki-type approach we introduce the new contractive condition in the framework of non-Archimedean fuzzy metric spaces. We prove also the corresponding coincidence fixed point theorem for two mappings in this framework. Finally, two examples are presented to verify the effectiveness and applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we prove some fixed point theorems of Ćirić type for mappings having non-unique fixed points by employing rational-type contractive conditions and the notion of comparison functions. Our results are generalizations, extensions and improvements of some previous and corresponding results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider the concept of the so-called generalized metric space and give some results on the existence, uniqueness and estimation of the solutions of Fredholm type integro- differential equations in two variables using Perov's fixed point theorem. Furthermore, we give some illustrative examples to verify the effectiveness and applicability of our main result. [ABSTRACT FROM AUTHOR]

We discuss the existence of solutions for a differential equation of fractional order with multi-point nonlocal boundary conditions via standard tools of fixed point theory. The paper concludes with some illustrative examples. [ABSTRACT FROM AUTHOR]