Search results for "Monotone operator"
showing 10 items of 12 documents
Solution of an initial-value problem for parabolic equations via monotone operator methods
2014
We study a general initial-value problem for parabolic equations in Banach spaces, by using a monotone operator method. We provide sufficient conditions for the existence of solution to such problem.
Variable exponent p(x)-Kirchhoff type problem with convection
2022
Abstract 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.
An abstract doubly nonlinear equation with a measure as initial value
2007
Abstract The solvability of the abstract implicit nonlinear nonautonomous differential equation ( A ( t ) u ( t ) ) ′ + B ( t ) u ( t ) + C ( t ) u ( t ) ∋ f ( t ) will be investigated in the case of a measure as an initial value. It will be shown that this problem has a solution if the inner product of A ( t ) x and B ( t ) x + C ( t ) x is bounded below.
Anisotropic Navier Kirchhoff problems with convection and Laplacian dependence
2022
We consider the Navier problem-Delta(2)(k,p)u(x)=f(x,u(x), del u(x), Delta u(x)) in Omega, u vertical bar(partial derivative Omega) =Delta u vertical bar(partial derivative Omega) = 0,driven by the sign-changing (degenerate) Kirchhoff type p(x)-biharmonic operator, and involving a (del u, Delta u)-dependent nonlinearity f. We prove the existence of solutions, in weak sense, defining an appropriate Nemitsky map for the nonlinearity. Then, the Brouwer fixed point theorem assessed for a Galerkin basis of the Banach space W-2,W-p(x)(Omega)boolean AND W-0(1,p(x))(Omega) leads to the existence result. The case of nondegenerate Kirchhoff type p(x)-biharmonic operator is also considered with respec…
The Existence of Solutions for Local Dirichlet (r(u),s(u))-Problems
2022
In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the spac…
Anisotropic elliptic equations with gradient-dependent lower order terms and L^1 data
2023
<abstract><p>We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $ \mathcal Au+\Phi(x, u, \nabla u) = \mathfrak{B}u+f $ in $ \Omega $, where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ and $ f\in L^1(\Omega) $ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $ \mathcal A $, the prototype of which is $ \mathcal A u = -\sum_{j = 1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u) $ with $ p_j &gt; 1 $ for all $ 1\leq j\leq N $ and $ \sum_{j = 1}^N (1/p_j) &gt; 1 $. As a novelty in this paper, our lower order terms involve a new class of operators $ \mathfrak B $ such…
A singular (p,q)-equation with convection and a locally defined perturbation
2021
We consider a parametric Dirichlet problem driven by the (p,q)-Laplacian and a reaction which is gradient dependent (convection) and the competing effects of two more terms, one a parametric singular term and a locally defined perturbation. We show that for all small values of the parameter the problem has a positive smooth solution.
Doubly nonlinear periodic problems with unbounded operators
2004
Abstract The solvability of the evolution system v ′( t )+ B ( t ) u ( t )∋ f ( t ), v ( t )∈ A ( t ) u ( t ), 0 t T , with the periodic condition v (0)= v ( T ) is investigated in the case where A (t) are bounded, possibly degenerate, subdifferentials and B (t) are unbounded subdifferentials.
Strengthened splitting methods for computing resolvents
2021
In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the “strengthening” of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs. FJAA and RC were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund …
Convergence rate of a relaxed inertial proximal algorithm for convex minimization
2018
International audience; In a Hilbert space setting, the authors recently introduced a general class of relaxed inertial proximal algorithms that aim to solve monotone inclusions. In this paper, we specialize this study in the case of non-smooth convex minimization problems. We obtain convergence rates for values which have similarities with the results based on the Nesterov accelerated gradient method. The joint adjustment of inertia, relaxation and proximal terms plays a central role. In doing so, we highlight inertial proximal algorithms that converge for general monotone inclusions, and which, in the case of convex minimization, give fast convergence rates of values in the worst case.