О числе решений в задачах со спектральным параметром для уравнений с разрывными операторами

Consider the equation Au = λTu (1) with parameter λ > 0, where A is a linear self-adjoint operator from E into E∗ (E is a real reflexive Banach space) and T : E → E∗ is an antimonotone (possibly discontinuous) mapping bounded on E, T (0) = 0. Obviously, there exists a zero solution of Eq. (1) for any λ. We seek those λ > 0 for which Eq. (1) has nonzero solutions (such λ are called the eigenvalues of Eq. (1)). In this case, the kernel of the operator A can be zero. Using a variational method, one proves a theorem on the existence of a semiaxis of eigenvalues and the existence for them of an eigenvector such that it is a radial continuity point of the operator T . Let E be a real reflexive Banach space, and let E∗ be the space dual to E. By (z, x) we denote the value of a functional z ∈ E∗ on an element x ∈ E. Definition 1. A mapping T : E → E∗ is said to be quasi-potential if there exists a functional f : E → R such that

We prove theorems on the existence of fixed points and the structure of fixed point sets for asymptotic 1-set contraction mappings T on certain subsets of Banach spaces by assuming some condition on T. We also prove some fixed point theorems for a sum of asymptotic 1-set contraction and compact (strongly continuous) mappings in real Banach spaces (reflexive real Banach spaces).

In this paper, we propose an iterative algorithm for approximating a common fixed point of an infinite family of quasi-Bregman nonexpansive mappings which is also a solution to finite systems of convex minimization problems and variational inequality problems in real reflexive Banach spaces. We obtain a strong convergence result and give applications of our result to finding zeroes of an infinite family of Bregman inverse strongly monotone operators and a finite system of equilibrium problems in real reflexive Banach spaces. Our result extends many recent corresponding results in literature.

An existence result for a mixed variational problem arising from Contact Mechanics

Semiregular solutions of integral equations with discontinuous nonlinearities were introduced byKrasnosel'skii and Pokrovskii in [i], where they also obtained sufficient conditions for the existence of such solutions. In the present paper we obtain a theorem on the existence of semiregular solutions of the Dirichlet problem for quasilinear equations of elliptic type with discontinuous weak nonlinearities. The proof is based on a general result of the author in [2] concerning equations with discontinuous operators in Banach spaces. In contrast to [I], it is not assumed that a nonlinearity of f(x, y) appearing in the equation is monotone in u; in addition, the restrictions on the growth of f(x, u) are weakened. i. Formulation of the Principal Result. We consider the Dirichlet problem

This paper considers an iterative approximation of a common solution of a finite family of variational inequailties in a real reflexive Banach space. By employing the Bregman distance and projection methods, we propose an iterative algorithm which uses a newly constructed adaptive step size to avoid a dependence on the Lipschitz constants of the families of the cost operators. The algorithm is carefully constructed so that the need to find a farthest element in any of its iterate is avoided. A strong convergence theorem was proved using the proposed method. We report some numerical experiments to illustrate the performance of the algorithm and also compare with existing methods in the literature.

Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space

Let f be a real-valued differentiable function defined on the real reflexive Banach space X. The problem of minimizing f over a subset of X is investigated under the following mild monotonicity assumption on the derivative f ′ f’ of f: if { u n } \{ {u_n}\} is a sequence in X converging weakly to some u ∈ X u \in X , then lim sup ( f ′ u n , u n − u ) ≧ 0 \lim \sup (f’{u_n},{u_n} - u) \geqq 0 . The eigenvalue problem f ′ u = λ g ′ u f’u = \lambda g’u for some λ ∈ R 1 \lambda \in {R^1} , with g ′ g’ being the derivative of a further function g, is then reduced to that first question.

We introduce a new parallel hybrid subgradient extragradient method for solving the system of the pseudomonotone equilibrium problem and common fixed point problem in real reflexive Banach spaces. The algorithm is designed such that its convergence does not require prior estimation of the Lipschitz-like constants of the finite bifunctions underlying the equilibrium problems. Moreover, a strong convergence result is proven without imposing strong conditions on the control sequences. We further provide some numerical experiments to illustrate the performance of the proposed algorithm and compare with some existing methods.

In this paper, using Bregman distance technique, we introduce aN inertial type algorithm with self - adaptive step size for approximating a common element of the set of solutions of pseudomonotone variational inequality problem and the set of common fixed point of a finite family of generic generalized Bregman nonspreading mapping in a real reflexive Banach space. Furthermore, we prove a strong convergence theorem of our algorithm without prior knowledge of the Lipschitz constant of the operator under some mild assumptions. We also give a numerical example to illustrate the performance of our algorithm. Our result generalize and improve many existing results in the literature.

Let X be a real reflexive Banach space with dual X* and G C X open and bounded and such that 0 ∈ G. Let T: X ⊃ D(T) → 2 X* be maximal monotone with 0 ∈ D(T) and 0 ∈ T(0), and C: X ⊃ D(C) → X* with 0 E D(C) and C(0) ≠ 0. A general and more unified eigenvalue theory is developed for the pair of operators (T, C). Further conditions are given for the existence of a pair (A, x) ∈ (0, ∞ ) x (D(T + C) n ∂G) such that (**) Tx+λCx? 0. The implicit eigenvalue problem, with C(λ, x) in place of λCx, is also considered. The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators T, C. No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.

A multiplicity result for p-Lapacian boundary value problems via critical points theorem

Existence of infinitely many weak solutions for a Neumann problem

A general variational principle and some of its applications

LetV be a reflexive Real Banach space and letA be an operator (not necessarily linear) fromV into its dualV′. We shall suppose thatX is a closed convex subset ofV. We shall denote by (,) the duality relation betweenV andV′. For a fixed elementf∈V′ we shall try to determineu∈X such that $$(Au,v - u) \geqslant (f,v - u)\rlap{--} \vee v\varepsilon X.$$ Problems of type (1) occur in optimal control theory and contain a fairly large class of type of linear and non linear partial differential equations (and inequations). The main method is based on the calculation of the operatorPX (the projector on the closed convex setX) i) the operatorPX is accessible, in which case we indicate how to obtain a numerical solution. ii) the operatorPX is not accessible, in which case we indicate a variaty of methods to approximate the solution