Infinitely many solutions for operator equations involving duality mappings on Orlicz-Sobolev spaces

Let $X$ be a real reflexive, smooth and separable Banach space having the Kadeč-Klee property and compactly imbedded in the real Banach space $Y$ and let $G:Y\rightarrow \mathbb{R} $ be a differentiable functional. By using the "fountain theorem" (Bartsch [3]), we will study the multiplicity of solutions for the operator equation \[ J_{\varphi}u=G^{\prime}(u)\text{,} \] where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Geometric duality theory of cones in dual pairs of vector spaces

The purpose of this paper is to study an inclusion problem which involves the sum of two monotone operators in a real reflexive Banach space. Using the technique of Bregman distance, we study the operator Res f T A f which is the composition of the resolvent of a maximal monotone operator T and the antiresolvent of a Bregman inverse strongly monotone operator A and prove that 0 T x+Ax if and only if x is a fixed point of the composite operator Res f T A f . Consequently, weak and strong convergence results are given for the inclusion problem under study in a real reflexive Banach space. We apply our results to convex optimization and mixed variational inequalities in a real reflexive Banach space. Our results are new, interesting and extend many related results on inclusion problems from both Hilbert spaces and uniformly smooth and uniformly convex Banach spaces to more general reflexive Banach spaces.

A group G G is representable in a Banach space X X if G G is isomorphic to the group of isometries on X X in some equivalent norm. We prove that a countable group G G is representable in a separable real Banach space X X in several general cases, including when G ≃ { − 1 , 1 } × H G \simeq \{-1,1\} \times H , H H finite and dim ⁡ X ≥ | H | \dim X \geq |H| , or when G G contains a normal subgroup with two elements and X X is of the form c 0 ( Y ) c_0(Y) or ℓ p ( Y ) \ell _p(Y) , 1 ≤ p > + ∞ 1 \leq p >+\infty . This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space X X and a countable bounded group G G of isomorphisms on X X containing − I d -Id , there exists an equivalent norm on X X for which G G is equal to the group of isometries on X X . We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least 2 2 may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries. It follows that every real Banach space of dimension at least 4 4 and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.

Selfadjoint operators on real or complex Banach spaces

In this paper, we consider a generalized system in real Banach spaces. Using Brouwer’s fixed-point theorem, we establish some existence theorems for generalized system without monotonicity. Further, we extend the concept of C-strong pseudomonotonicity for a bifunction and extend Minty’s lemma for a generalized system. Furthermore, using the Minty lemma and KKM-Fan lemma, we establish an existence theorem for a generalized system with monotonicity in real reflexive Banach spaces.

In this paper, we introduce a generalized system (for short, GS) in real Banach spaces. Using Brouwer’s fixed point theorem, we establish some existence theorems for the generalized system without monotonicity. Further, we extend the concept of C-strong pseudomonotonicity and extend Minty’s lemma for the generalized system. And using the Minty lemma and KKM-Fan lemma, we establish an existence theorem for the generalized system with monotonicity in real reflexive Banach spaces. As the continuation of existing studies, our paper present a series of extended results based on existing corresponding results.

We prove a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces. Furthermore, we apply our method to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our theorems complement some recent results that have been proved for this important class of nonlinear mappings.

A real Banach space E of dimension _3 is an inner product space iff there exists a bounded smooth convex subset of E which is the range of a nonexpansive retraction. De Figueiredo and Karlovitz [3] have shown that if E is a strictly convex real finite-dimensional Banach space and dim E> 3 then there can exist no bounded smooth nonexpansive retract of E unless E is a Hilbert space. (A subset F of E is a nonexpansive retract of E if it is the range of a nonexpansive retraction r: E-F.) This is a consequence of their more general result that if E is reflexive and a convex nonexpansive retract of E has at a boundary point xo a unique supporting hyperplane xo+H then H is the range of a projection of norm 1. As they have pointed out, the latter theorem fails in nonreflexive spaces (the unit ball of C[O, 1] furnishes a counterexample). Nevertheless, their first result is true in general: THEOREM. Suppose E is a real Banach space with dim E> 3. Then E is an inner product space iff there exists a bounded smooth nonexpansive retract of E with nonempty interior. We separate out of the proof of the theorem a lemma, valid in all real Banach spaces: LEMMA. Suppose F is a bounded smooth closed convex subset of a real Banach space E and F has nonempty interior. Then given disjoint bounded closed convex sets M and K in E with K compact, there exist p E E and 2>0 such that Kcp+)LF and (p+ 2F) rnM= 0. PROOF OF LEMMA. Clearly the hypotheses and conclusions of the lemma are invariant if K and M are translated by the same vector; thus without loss of generality we may assume 0 E K. Similarly, we may also assume 0 E int F. Since K is compact and M is closed, a basic separation theorem for convex sets assures the existence of a closed hyperplane H which strictly separates M and K; that is, there exist we E*, c e R' Received by the editors June 26, 1972 and, in revised form, August 21, 1973. AMS (MOS) subject classifications (1970). Primary 46C05.

In this correspondence, complete convergence theorems are established for weighted row sums from arrays of random elements taking values in real separable Rademacher type Banach spaces as well as real separable martingale type Banach spaces. It is assumed that A version of the Rademacher type p complete convergence theorem is also established with random variable weights. Illustrative examples are included.

Introduction.Let I bea reflexive real Banach space, X* its conjugate space, (w, u) the pairing between w in X* and u in X.We consider multi-valued mappings T of X into X* (i.e, mappings in the ordinary sense of X into 2**) which are monotone, i.e., if veT(u), Vy eT(ux) for u and ux in X, then (V -Vy, U -Uy) ^ 0.It is our object in the present paper to generalize to the multi-valued case the results obtained in a number of recent papers by the author and G. J. Minty for single-valued mappings T (cf.[2]-[14]).The first results for multi-valued mappings for X a Hubert space have been obtained in an unpublished paper of Minty [15].The methods of [15] are not directly extendable to more general spaces, but our discussion of the finite-dimensional case (Lemma 2.1) has been very much influenced by the manuscript of [15] which Minty has recently transmitted to the author.(The basic result of [15] is stated at the end of 2 below.)Our results for general multi-valued monotone mappings have an interesting specific application given in 3 below to the generalization of a theorem of Beurling and Livingston [1] on duality mappings in Banach spaces.In a previous paper [12], we showed that for strictly convex reflexive spaces, this theorem could be obtained from results on single-valued monotone mappings.In 3 below we give a generalization of this theorem to general reflexive Banach spaces which runs as follows: Let X be a reflexive Banach space, ej)(r) a non-negative nondecreasing function on P1 with ej)(0) =0.The duality map Tof X with respect to c> is defined by T{u) m \v\veX*> M-*M>.> \(V,U) = \\v\\ || M I .

Introduction.Let I bea reflexive real Banach space, X* its conjugate space, (w, u) the pairing between w in X* and u in X.We consider multi-valued mappings T of X into X* (i.e, mappings in the ordinary sense of X into 2**) which are monotone, i.e., if veT(u), Vy eT(ux) for u and ux in X, then (V -Vy, U -Uy) ^ 0.It is our object in the present paper to generalize to the multi-valued case the results obtained in a number of recent papers by the author and G. J. Minty for single-valued mappings T (cf.[2]-[14]).The first results for multi-valued mappings for X a Hubert space have been obtained in an unpublished paper of Minty [15].The methods of [15] are not directly extendable to more general spaces, but our discussion of the finite-dimensional case (Lemma 2.1) has been very much influenced by the manuscript of [15] which Minty has recently transmitted to the author.(The basic result of [15] is stated at the end of §2 below.)Our results for general multi-valued monotone mappings have an interesting specific application given in §3 below to the generalization of a theorem of Beurling and Livingston [1] on duality mappings in Banach spaces.In a previous paper [12], we showed that for strictly convex reflexive spaces, this theorem could be obtained from results on single-valued monotone mappings.In §3 below we give a generalization of this theorem to general reflexive Banach spaces which runs as follows: Let X be a reflexive Banach space, ej)(r) a non-negative nondecreasing function on P1 with ej)(0) =0.The duality map Tof X with respect to c¡> is defined by T{u) m \v\veX*> M-*M>.> \(V,U) = \\v\\ • || M I .

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

In this paper, we propose an inertial algorithm for solving split equality of monotone inclusion and ‐fixed point of Bregman relatively ‐nonexpansive mapping problems in reflexive real Banach spaces. Using the Bregman distance function, we prove a strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. In addition, we provide some applications of our method and give numerical results to demonstrate the applicability and efficiency of the proposed method.

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).