Nonlinear Inclusion Problems Research Articles

Existence of solution for nonlinear elliptic inclusion problems with degenerate coercivity and $$L^{1}$$-data

Existence of solution for nonlinear elliptic inclusion problems with degenerate coercivity and $$L^{1}$$-data

System of generalized nonlinear variational-like inclusion problems in 2-uniformly smooth Banach spaces

In this manuscript, we introduce and study the existence of a solution of a system of generalized nonlinear variational-like inclusion problems in 2-uniformly smooth Banach spaces by using $H(.,.)$-$eta$-proximal mapping. The method used in this paper can be considered as an extension of methods for studying the existence of solutions of various classes of variational inclusions considered and studied by many authors in 2-uniformly smooth Banach spaces. Some important results, theorems and the existence of solution of the proposed system of generalized nonlinear variational-like inclusion problems have been derived.

On a class of differential inclusions in the frame of generalized Hilfer fractional derivative

<abstract><p>In the present paper, we extend and develop a qualitative analysis for a class of nonlinear fractional inclusion problems subjected to nonlocal integral boundary conditions (nonlocal IBC) under the $ \varphi $-Hilfer operator. Both claims of convex valued and nonconvex valued right-hand sides are investigated. The obtained existence results of the proposed problem are new in the frame of a $ \varphi $-Hilfer fractional derivative with nonlocal IBC, which are derived via the fixed point theorems (FPT's) for set-valued analysis. Eventually, we give some illustrative examples for the acquired results.</p></abstract>

Eigenvalue Problems for Fredholm Operators with Set-Valued Perturbations

Abstract By means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.

Evolutionary variational\u2013hemivariational inequalities with applications to dynamic viscoelastic contact mechanics

The purpose of this work is to introduce and investigate a complicated variational–hemivariational inequality of parabolic type with history-dependent operators. First, we establish an existence and uniqueness theorem for a first-order nonlinear evolution inclusion problem, which is driven by a convex subdifferential operator for a proper convex function and a generalized Clarke subdifferential operator for a locally Lipschitz superpotential. Then, we employ the fixed point principle for history-dependent operators to deliver the unique solvability of the parabolic variational–hemivariational inequality. Finally, a dynamic viscoelastic contact problem with the nonlinear constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, where normal contact and friction are described, respectively, by two nonconvex and nonsmooth multi-valued terms.

Positive solutions to nonlinear inclusion problems in Orlicz–Sobolev spaces

In this paper, we study the existence of positive solutions to a nonlinear elliptic inclusion problem driven by an elliptic differential operator with Dirichlet boundary and a multivalued term depending on the solution and its gradient in reflexive Orlicz–Sobolev spaces by using a linear functional analysis method and a subsolution–supersolution method. We provide some sufficient conditions to ensure that a subsolution and a supersolution exist. An existence theorem for positive solutions of the problem is established.

Extended Newton-type method for nonlinear functions with values in a cone

In this paper, we consider the problem of finding solutions of nonlinear inclusion problems in Banach space. Using convex optimization techniques introduced by Robinson (Numer Math 19:341–347, 1972), a convergence theorem for Kantorovich-like methods is given, which improves the results of Yamamoto (Jpn J Appl Math 3(2):295–313, 1986; Numer Math 51(5):545–557, 1987) and Robinson (Numer Math 19:341–347, 1972). The result is compared with previously known results. Numerical examples further justify the theoretical results.

Positive solutions to nonlinear nonhomogeneous inclusion problems with dependence on the gradient

The goal of the paper is to study a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with the Dirichlet boundary condition and a convection multivalued term. An existence theorem for positive solutions of the problem is established by applying the method of subsolution–supersolution, together with truncation and comparison techniques.

Inexact Newton method for non-linear functions with values in a cone

ABSTRACTThe problem of finding a solution of non-linear inclusion problems in Banach space is considered in this paper. Using convex optimization techniques introduced by Robinson (Numer Math. 1972;19:341–347), a robust convergence theorem for the inexact Newton method is proved. As an application, an affine invariant version of Kantorovich’s theorem and Smale’s -theorem for the inexact Newton method is obtained.

Existence of solutions to bilinear problems with a closed-loop control

Here we prove the existence of solutions to nonlinear differential inclusion problems with closed-loop control z + A (z ) = B (u,z ) + f, u ∈ U (t,z ),z (0) = z 0 where the operator B is bilinear with respect to the control u and the state z in reflexive, separable Banach spaces denoted Y and V , respectively. The operator A is nonlinear in V , and given a positive real number T , the set-valued map U is defined in [ 0,T ] × V . Without making any assumptions about the convexity of U , its values are taken to be non-empty closed, decomposable subsets of Y .

A nonlinear inclusion problem involving (α,λ)-NODM set-valued mappings in ordered Hilbert space

The purpose of this work is as follows: First, a class of nonlinear inclusion problems involving (α,λ)-NODM set-valued mappings in an ordered Hilbert space is introduced, and then, applying the resolvent operator associated with (α,λ)-NODM set-valued mappings, an existence theorem on the solvability of this class of nonlinear inclusion problems is established. Furthermore, on the basis of the existence theorem and the new (α,λ)-NODM model, a general algorithm is introduced and applied to the approximation solvability of the nonlinear inclusion problem at hand. The results obtained seem to be general in nature.

On general over-relaxed proximal point algorithm and applications

A general framework to the over-relaxed proximal point algorithm based on H-maximal monotonicity is introduced, and then it is applied to the approximation solvability of a general class of nonlinear inclusion problems based on the generalized resolvent operator technique. This form of the proximal point algorithm seems to be more application-oriented to inclusion problems.

GENERALIZED RELAXED PROXIMAL POINT ALGORITHMS INVOLVING RELATIVE MAXIMAL ACCRETIVE MODELS WITH APPLICATIONS IN BANACH SPACES

General models for the relaxed proximal point algorithm using the notion of relative maximal accretiveness (RMA) are developed, and then the convergence analysis for these models in the context of solving a general class of nonlinear inclusion problems differs significantly than that of Rockafellar (1976), where the local Lipschitz continuity at zero is adopted instead. Moreover, our approach not only generalizes convergence results to real Banach space settings, but also provides a suitable alternative to other problems arising from other related fields.

A new relaxed proximal point procedure and applications to nonlinear variational inclusions

A new relaxed algorithmic procedure based on the notion of A -maximal relaxed monotonicity is developed, and then the convergence analysis in the context of solving a general class of nonlinear inclusion problems is examined. Furthermore, some results involving A -maximal relaxed monotone mappings in a Hilbert space setting are included.

General [formula omitted]-monotone operators and perturbed iterations for nonlinear set-valued relaxed cocoercive operator inclusion problems

In this paper, we introduce the notion of general A -monotone operators in Banach spaces, study some properties of general A -monotone operator and the new proximal mapping associated with the general A -monotone operators. By using Alber’s inequalities, Nalder’s results and the new proximal mapping technique, we also construct two new class of perturbed iterative algorithms with mixed errors for solving a new class of nonlinear set-valued relaxed cocoercive operator inclusions and study applications of general A -monotone operators to the approximation-solvability of the nonlinear set-valued relaxed cocoercive operator inclusion problems in Banach spaces. Furthermore, the general A -monotone operators are illustrated by some examples. The results presented in this paper improve and generalize the corresponding results on strongly monotone quasi-variational inclusions and nonlinear implicit quasi-variational inclusions.

General implicit variational inclusion problems based on A-maximal ( m)-relaxed monotonicity (AMRM) frameworks

A general framework for an algorithmic procedure based on the variational convergence of operator sequences involving A-maximal ( m)-relaxed monotone (AMRM) mappings in a Hilbert space setting is developed, and then it is applied to approximating the solution of a general class of nonlinear implicit inclusion problems involving A-maximal ( m)-relaxed monotone mappings. Furthermore, some specializations of interest on existence theorems and corresponding approximation solvability theorems on H-maximal monotone mappings are included that may include several other results for general variational inclusion problems on general maximal monotonicity in the literature.

The generalized relaxed proximal point algorithm involving [formula omitted]-maximal-relaxed accretive mappings with applications to Banach spaces

A general design for the relaxed proximal point algorithm using the notion of A -maximal-relaxed accretiveness is developed, and then the convergence analysis for this algorithm in the context of solving a general class of nonlinear inclusion problems is examined along with some other results of interest involving A -maximal-relaxed accretive mappings in a real Banach space setting.

General over-relaxed proximal point algorithm involving A-maximal relaxed monotone mappings with applications

A general framework for the over-relaxed proximal point algorithm based on the notion of A-maximal relaxed monotonicity is developed, and then the convergence analysis in the context of solving a general class of nonlinear implicit inclusion problems is explored. Furthermore some auxiliary results of interest involving A-maximal relaxed monotone mappings in a Hilbert space setting are included.

General Proximal Point Algorithmic Models and Nonlinear Variational Inclusions Involving RMM Mappings

The proximal point algorithms based on relative $A$-maximal monotonicity (RMM) is introduced, and then it is applied to the approximation solvability of a general class of nonlinear inclusion problems using the generalized resolvent operator technique. This algorithm seems to be more application-oriented to solving nonlinear inclusion problems. Furthermore, the obtained result could be applied to generalize the Douglas-Rachford splitting method to the case of RMM mapping based on the generalized proximal point algorithm.

On the generalized proximal point algorithm with applications to inclusion problems

A class of generalizedproximal point algorithms based on the $A-$ maximal monotonicity isintroduced, and then it is applied to the approximation solvabilityof a general class of nonlinear inclusion problems using thegeneralized resolvent operator technique. This seems to be ofinterest in the sense that it is application-oriented.