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Abstract: Let be a reflexive Banach space with the dual space and be a nonempty closed convex subset of . Let us have and . We introduce the class of generalized -monotone multifunctions with respect to and where . By using the KKM technique and the concept of the Hausdorff metric, we establish some existence results for generalized variational-like inequalities with generalized monotone multivalued mappings in . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the smoothness of generalized solutions of the Cauchy–Dirichlet problem for the second-order hyperbolic equation in domains with a conical point. [Copyright &y& Elsevier]

Abstract: We introduced and studied the concept of well-posedness to a generalized mixed variational inequality. Some characterizations are given. Under suitable conditions, we prove that the well-posedness of the generalized mixed variational inequality is equivalent to the well-posedness of the corresponding inclusion problem. We also discuss the relations between the well-posedness of the generalized mixed variational inequality and the well-posedness of the corresponding fixed-point problem. Finally, we derive some conditions under which the generalized mixed variational inequality is well-posed. [Copyright &y& Elsevier]

Based on an approximation to Cayley transform, we propose an approximate Cayley transform method and its inexact version for solving inverse eigenvalue problems, which has the advantage over other known methods in the sense that it avoids solving systems in obtaining the approximate eigenvectors. Under the nonsingular condition used in D. Sun and J. Sun [29] , we show that the proposed methods converge at least superlinearly. Moreover, numerical experiments are given which illustrate that, comparing with the Cayley transform methods, our methods need much less inner iterations and CPU time. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to study the solvability for vector mixed variational inequalities (for short, VMVI) in Banach spaces. Utilizing Ky Fan’s Lemma and Nadler’s theorem, we derive the solvability for VMVIs with compositely monotone vector multifunctions. On the other hand, we first introduce the concepts of compositely complete semicontinuity and compositely strong semicontinuity for vector multifunctions. Then we prove the solvability for VMVIs without monotonicity assumption by using these concepts and by applying Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to investigate the asymptotic behavior of algorithms for finding solutions for a certain class of variational inequalities involving nonexpansive type mappings in smooth Banach spaces. We study the existence of solutions of variational inequalities when is the set of solutions of zeros of accretive operators or the set of fixed points of nonexpansive mappings or the set of fixed points of pseudocontractive mappings. Our convergence analysis covers proximal point algorithm for finding zeros of accretive operators as well as functional Halpern algorithm for finding fixed points of nonexpansive mappings in Banach spaces. Our results improve a number of results concerned with viscosity approximation methods in the context of weakly contraction mappings. [Copyright &y& Elsevier]

Abstract: We suggest the oriented coincidence index theory for pairs consisting of nonlinear zero-index Fredholm operators and multivalued maps which may be represented as compositions of multimaps with aspheric values. We consider, sequentially, finite dimensional, compact, and condensing cases. The theory developed is applied to the study of a feedback impulsive control system. [Copyright &y& Elsevier]

Abstract: In this paper we introduce an implicit iterative scheme for finding a common element of the set of common fixed points of nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The implicit iterative scheme is based on two well-known methods: extragradient and approximate proximal. We obtain a weak convergence theorem for three sequences generated by this implicit iterative scheme. On the basis of this theorem, we also construct an implicit iterative process for finding a common fixed point of mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mappings are nonexpansive. [Copyright &y& Elsevier]

Abstract: We propose an approximate proximal algorithm for solving generalized variational inequalities in Hilbert space. Extension to Bregman-function-based approximate proximal algorithm is also discussed. Weak convergence of these two algorithms are established under the paramonotonicity and pseudomonotonicity assumptions of the operators. [Copyright &y& Elsevier]

Abstract: Let be a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. Recently, Takahashi and Shimoji [W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling 32 (2000) 1463–1471] introduced an iterative scheme given by finitely many nonexpansive mappings in and proved weak convergence theorems which are connected with the problem of image recovery. In this paper we introduce a new iterative scheme which includes their iterative scheme as a special case. Under the assumption that is a reflexive Banach space whose norm is uniformly Gâteaux differentiable and which has a weakly continuous duality mapping, we prove strong convergence theorems which are connected with the problem of image recovery. Using the established results, we consider the problem of finding a common fixed point of finitely many nonexpansive mappings. [Copyright &y& Elsevier]

Abstract: In this paper, we consider equilibrium problems with differentiable bifunctions in aBanach space setting and investigate properties of gap functions for such problems. We suggest a derivative-free descent method and give conditions which provide strong convergence of the method. [Copyright &y& Elsevier]

In this paper, we first derive the intersection, theorems on the pseudo H-spaces, and then discuss the fixed-point results by using the intersection theorems to establish some existence results of the equilibrium point in the abstract economics. [Copyright &y& Elsevier]

In this paper, we obtain some existence results of equilibrium problems with lower and upper bounds by employing a fixed-point theorem due to Ansari and Yao [1] and Ky Fan Lemma [2], respectively. Our results give answers to the open problem raised by Isac, Sehgal and Singh [3]. [Copyright &y& Elsevier]

• In this phase II trial of atezolizumab and bevacizumab in advanced MCC patients, the ORR was 64 %, with 27 % complete responses. • Atezolizumab and bevacizumab was well tolerated, grade 3 TRAEs were hypertension (18 %), proteinuria (9 %), and hepatitis (9 %). • Patients with CR had a predicted inflamed tumor microenvironment based on the RNA signature. • Atezolizumab plus bevacizumab demonstrated encouraging activity in patients with advanced MCC. [ABSTRACT FROM AUTHOR]