A general variational principle and some of its applications

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.

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).

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

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

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.

Existence of infinitely many weak solutions for a Neumann problem

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

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

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.

In a real reflexive Banach space we consider a problem on existence of solutions to a problem with a spectral parameter for equations with discontinuous operators. By the variational approach we obtain theorems on the number of the solutions to the considered problems. As an application, we consider main boundary value problems for elliptic equations with a spectral parameter and discontinuous nonlinearities.

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.

This paper discovers that current variational principle and Noether theorem for different physics systems with (in) finite freedom systems have missed the double extremum processes of the general extremum functional that both is deduced by variational principle and is necessarily taken in deducing all the physics laws, but these have not been corrected for over a century since Noether's proposing her famous theorem, which result in the crisis deducing relevant mathematical laws and all physics laws. This paper discovers there is the hidden logic cycle that one assumes Euler-Lagrange equations, and then he finally deduces Euler-Lagrange equations via the equivalent relation in the whole processes in all relevant current references. This paper corrects the current key mistakes that when physics systems choose the variational extreme values, the appearing processes of the physics systems are real physics processes, otherwise, are virtual processes in all current articles, reviews and (text) books. The real physics should be after choosing the variational extreme values of physics systems, the general extremum functional of the physics systems needs to further choose the minimum absolute extremum zero of the general extremum functional, otherwise, the appearing processes of physics systems are still virtual processes. Using the double extremum processes of the general extremum functionals, the crisis and the hidden logic cycle in current variational principle and current Noether theorem are solved. Furthermore, the new mathematical and physical double extremum processes and their new mathematical pictures and physics for (in) finite freedom systems are discovered. This paper gives both general variational principle and general Noether theorem as well as their classical and quantum new physics, which would rewrite all relevant current different branches of science, as key tools of studying and processing them.