Operators Of Monotone Type Research Articles

Singular double phase problems with convection

We consider a Dirichlet problem driven by the double phase differential operator and a parametric reaction which has the combined effects of a singular term and of a convective perturbation. Using nonlinear operators of monotone type, truncation and comparison techniques, and fixed point theory, we show that for all small values of the parameter, the problem has a bounded positive solution.

Positive solutions for singular problems with multivalued convection

We consider an elliptic Dirichlet problem, driven by a general nonlinear, nonhomogeneous differential operator and a reaction that has the competing effects of a parametric singular term and of a multivalued perturbation which is gradient dependent (convection). Using truncations, comparisons and the theory of nonlinear operators of monotone type, we show that for all small values of the parameter λ>0, the problem has a positive and smooth solution.

Dirichlet Problems with Unbalanced Growth and Convection

We consider a double phase Dirichlet problem with a gradient dependent reaction term (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a bounded strictly positive solution. Moreover, we show that the set of these solutions is compact in the corresponding generalized Sobolev–Orlicz space.

Variable exponent p(x)-Kirchhoff type problem with convection

We study a nonlinear p(x)-Kirchhoff type problem with Dirichlet boundary condition, in the case of a reaction term depending also on the gradient (convection). Using a topological approach based on the Galerkin method, we discuss the existence of two notions of solutions: strong generalized solution and weak solution. Strengthening the bound on the Kirchhoff type term (positivity condition), we establish existence of weak solution, this time using the theory of operators of monotone type.

Dirichlet (p,q)-equations with gradient dependent and locally defined reaction

We consider a Dirichlet (p,q)-equation, with a gradient dependent reaction which is only locally defined. Using truncations, theory of nonlinear operators of monotone type, and fixed point theory (the Leray-Schauder Alternative Theorem), we show the existence of a positive smooth solution.
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The Obstacle Problem at Zero for the Fractional p-Laplacian

In this paper we establish a multiplicity result for a class of unilateral, nonlinear, nonlocal problems with nonsmooth potential (variational-hemivariational inequalities), using the degree map of multivalued perturbations of fractional nonlinear operators of monotone type, the fact that the degree at a local minimizer of the corresponding Euler functional is equal one, and controlling the degree at small balls and at big balls.

Nonlinear vector Duffing inclusions with no growth restriction on the orientor field

We consider nonlinear multivalued Dirichlet Duffing systems. We do not impose any growth condition on the multivalued perturbation. Using tools from the theory of nonlinear operators of monotone type, we prove existence theorems for the convex and the nonconvex problems. Also we show the existence of extremal trajectories and show that such solutions are $C_0^1(T,\mathbb{R}^N)$-dense in the solution set of the convex problem (strong relaxation theorem).

Multiple solutions with sign information for semilinear Neumann problems with convection

We consider a semilinear Neumann problem with convection. We assume that the drift coefficient is indefinite. Using the theory of nonlinear operators of monotone type, together with truncation and comparison techniques and flow invariance arguments, we prove a multiplicity theorem producing three nontrivial smooth solutions (positive, negative and nodal).

On the convergence of sequence of maximal monotone operators of type (D) in Banach spaces

Within the setting of general real Banach spaces, we prove that the sequence of maximal monotone operators of type (D) graphically converges provided, their corresponding class of representative functions converge epigraphically. Moreover, we provide a condition to guarantee that the lower limit of a sequence of maximal monotone operators of type (D) is a maximal monotone operator of type (D) in real Banach spaces.

Periodic solutions for implicit evolution inclusions

We consider a nonlinear implicit evolution inclusion driven by a nonlinear, nonmonotone, time-varying set-valued map and defined in the framework of an evolution triple of Hilbert spaces. Using an approximation technique and a surjectivity result for parabolic operators of monotone type, we show the existence of a periodic solution.

Nonlinear systems with Hartman-type perturbations

We consider a nonlinear Lienard-type system driven by a nonlinear, nonhomogeneous differential operator and a maximal monotone map. On the Caratheodory perturbation we do not impose any global growth condition. Instead we employ a Hartman-type hypotheses. Using tools from fixed point theory and the theory of operators of monotone type, we prove two existence theorems.

Nonlinear Dirichlet problems with sign changing drift coefficient

We consider a nonlinear Dirichlet problem driven by the p-Laplacian with gradient dependence (convection). We assume that the coefficient of the gradient term (drift coefficient) is sign changing. Using the theory of nonlinear operators of monotone type together with suitable truncation and comparison techniques, we show the existence of a positive smooth solution.

Positive Solutions for Nonlinear Dirichlet Problems with Convection

We consider a nonlinear Dirichlet problem driven by the p-Laplacian, a convection term and a $$(p-1)$$ -sublinear perturbation. First we assume that the coefficient in the convection term (drift coefficient) is sign changing. Using the theory of nonlinear operators of monotone type together with suitable truncation and comparison techniques we prove the existence of a positive smooth solution. When the drift coefficient is nonnegative, we are able to relax the conditions on the data of the problem.

Semilinear systems with a multi-valued nonlinear term

Abstract Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = (N1, N2), provided that L = (L1, L2) satisfies dim Ker L1 = ∞ and dim Ker L2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed.

Mathematical simulation of a steady process of anisotropic filtration

This article discusses the methods of approximate solution of mixed variational inequalities with operators of monotone type. The functional, which is included in this variational inequality, is separable, in other words, it is the sum of a number of nondifferentiable functionals. These variational inequalities appear, in particular, in the description of steady incompressible filtration processes of highly viscous fluids in anisotropic medium.

About One Class of Operators Inclusions

The operator inclusion 0 ∈ A(x)+N(x) is studied. The main results refer to the case, when A – a bounded operator of monotone type from a reflexive space into conjugate to it, N – a conevalued operator. No solution criterion of the viewed inclusion is set up. Integer characteristics of multivalued mappings with homotopy invariance and additivity are introduced. Application to the theory of variational inequalities with multivalued operators is identified.

New surjectivity results for perturbed weakly coercive operators of monotone type in reflexive Banach spaces

Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X∗. Let T:X⊇D(T)→2X∗ be maximal monotone and S:X→2X∗ be bounded pseudomonotone. Let p be a nonnegative integer. Assume, further, that there exist nonnegative constants ai(i=1,2,…,p) such that 〈v∗+w∗,x〉≥−∑i=1pai‖x‖i−α(‖x‖)‖x‖p+2 for all x∈D(T) with sufficiently large ‖x‖, v∗∈Tx and w∗∈Sx, where α:[0,∞)→[0,∞) such that α(t)→0 as t→∞. New surjectivity results are given for the operator T+S along with weakly coercive type hypothesis on T+S. The results are new and improve the corresponding theory for coercive operators of monotone type. The theory developed herein can be suitably applied in the study of partial differential equations, variational and hemi-variational inequality problems in appropriate Sobolev spaces. To demonstrate the applicability of the theory, an example of time periodic parabolic partial differential equation, which models nonmonotone semipermeability problem, is provided.

Nonlinear Operators of Monotone Type and Convergence Theorems with Equilibrium Problems in Banach Spaces

Our purpose in this paper is first to discuss nonlinear operators and nonlinear projections in Banach spaces which are related to the resolvents of $m$-accretive operators and maximal monotone operators. Some of these operators in Banach spaces are new. Next, we discuss some properties for such nonlinear operators and nonlinear projections in Banach spaces. Further, using these properties, we prove strong convergence theorems by hybrid methods for nonlinear operators with equilibrium problems in Banach spaces.

Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints

Introduction Mathematical background Degree theoretic results Variational-hemivariational inequalities Hemivariational inequalities with an asymmetric subdifferential Bibliography.

Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian

In this paper we consider a nonlinear eigenvalue problem driven by the p -Laplacian differential operator and with a nonsmooth potential. Using degree theoretic arguments based on the degree map for certain operators of monotone type, we show that the problem has at least two nontrivial positive solutions as the parameter λ > 0 varies in a half-line.