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Abstract: In this paper, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study the strong and weak convergence of a finite step iteration process with errors for two finite families of asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in the paper improve and extend some results in Sitthikul and Saejung (2009) . [Copyright &y& Elsevier]

Abstract: In this paper, some existence theorems of nonzero solutions for a system of bilinear variational inequalities are proved by using the coincidence degree theory in reflexive Banach spaces. The results presented in this paper improve and extend some known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of solutions of the operator equations in the Banach space . It is assumed the vector-valued function is nonlinear Pettis-integrable. Some additional assumptions imposed on are expressed in terms of a weak measure of noncompactness. To encompass the full scope of the paper, we investigate the existence of pseudo-solutions for the nonlinear boundary value problem of fractional type under the Pettis integrability assumption imposed on . [Copyright &y& Elsevier]

Abstract: Our purpose in this paper is to construct a new iterative scheme by the hybrid method and prove the strong convergence theorem using our new iterative scheme for approximation of a common fixed point of a countable family of uniformly quasi--asymptotically nonexpansive mappings which is also a common solution to infinite systems of equilibrium and variational inequality problems in a uniformly smooth and strictly convex real Banach space with the Kadec–Klee property using the properties of the generalized -projection operator. Our results extend many known recent results in the literature. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence, uniqueness and continuous dependence of mild solution for an impulsive neutral fractional order differential equation with infinite delay. The results are obtained by using the fixed point technique and solution operator on a complex Banach space. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new class of set-valued vector implicit quasi complementarity problems with corresponding set-valued implicit quasi variational inequality problems. By means of the Fan–KKM theorem, we investigate the nonemptiness and compactness of solution sets of these problems. Our work generalizes and improves some results appeared recently in the literature. [Copyright &y& Elsevier]

Abstract: Zamfirescu [T. ZamfirescuFix point theorems in metric spaces Arch. Math. (Basel) 23 (1972) 292-298], obtained a very interesting fixed point theorem on complete metric spaces by combining the results of S. Banach, R. Kannan and S.K. Chatterjea. In [S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, in: Ann. New York Acad. Sci., vol. 728, 1994, pp. 183-197], the author introduced and studied the concept of partial metric spaces, and obtained a Banach type fixed point theorem on complete partial metric spaces. In this paper, we study new extensions of the Zamfirescu theorem to the context of partial metric spaces, and among other things, we give some generalized versions of the fixed point theorem of Matthews. The theory is illustrated by some examples. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to introduce a hybrid algorithm for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings and the set of solutions of a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space which also has Kadec–Klee property using the properties of generalized -projection operator. Our results extend important recent results. [Copyright &y& Elsevier]

Abstract: In this paper, we construct a new iterative scheme by hybrid method and prove strong convergence theorems for approximation of a common element of set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings, set of solutions to a variational inequality problem and set of common solutions to a system of generalized mixed equilibrium problems in a 2-uniformly convex real Banach space which is also uniformly smooth. Furthermore, using our iterative scheme, we prove strong convergence theorem to a common element of the set of fixed point of a weak relatively nonexpansive mapping, set of solutions to a variational inequality problem and the set of common solutions to a system of generalized mixed equilibrium problems. Finally, we give applications of our results in a Banach space. Our results extend many important recent results in the literature. [Copyright &y& Elsevier]

Abstract: Let be a nonempty closed convex subset of a real reflexive Banach space that has weakly sequentially continuous duality mapping for some gauge . Let be a family of multivalued nonexpansive mappings with which is a sunny nonexpansive retract of with a nonexpansive retraction. It is our purpose in this paper to prove the convergence of two viscosity approximation schemes to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, is the unique solution in to a certain variational inequality. [Copyright &y& Elsevier]

Abstract: In this paper, we study the fractional functional differential equations with causal operators in an arbitrary separable Banach space. By means of the techniques of the measure of noncompactness, the existence and continuation of solutions are given. Further, some topological properties of solution sets are discussed and the existence of optimal solutions of the associated control problem are presented. An example is given to illustrate the result. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to construct a new iterative scheme by hybrid methods to approximate a common element of the common fixed points set of a finite family of -asymptotically nonexpansive mappings, the solutions set of a variational inequality problem and the solutions set of a system of generalized mixed equilibrium problems in a 2-uniformly convex real Banach space which is also uniformly smooth. Then, we prove strong convergence of the scheme to a common element of the three sets. Our results extend many known recent results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a one-step iterative process to approximate common fixed points of a finite family of generalized nonexpansive multivalued mappings and prove some weak and strong convergence theorems for such mappings in uniformly convex Banach spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we construct a new simple algorithm for solving some variational inequality in Banach spaces. Furthermore, we prove that the proposed algorithm has strong convergence. [Copyright &y& Elsevier]

Abstract: In this paper, the system of mixed variational inequalities is introduced and considered in Banach spaces, which includes some known systems of variational inequalities and the classical variational inequalities as special cases. Using the projection operator technique, we suggest some iterative algorithms for solving the system of mixed variational inequalities and prove the convergence of the proposed iterative methods under suitable conditions. Our theorems generalize some known results shown recently. [Copyright &y& Elsevier]

Abstract: In this paper we extend and generalize common fixed point theorems for six self-maps of Singh and Jain [B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439–448] from Menger and metric spaces to cone metric spaces. We also extend the notions of compatible and weakly compatible mappings from the setting of Menger and metric spaces to the setting of cone metric spaces. We do not impose the normality property on the cone, but suppose only that the cone , in the ordered Banach space , has a nonempty interior. Examples are given to illustrate the results. [Copyright &y& Elsevier]

Abstract: Let be a real uniformly convex and smooth Banach space with as a sunny nonexpansive retraction, be a nonempty closed convex subset of . Let be three of weakly inward and asymptotically nonexpansive mappings with respect to . It is proved that three step iteration process with errors converges weakly and strongly to a common fixed point of under certain conditions. It presents some new results in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive semilinear differential inclusions with periodic boundary conditions: where and is the infinitesimal generator of a -semigroup on a separable Banach space and is a multi-valued map. The functions characterize the jump of the solutions at impulse points (). We will have to distinguish between the cases when either or neither 1 lies in the resolvent of . Accordingly, the problem is either formulated as a fixed point problem for an integral operator or for a Poincaré translation operator. Our existence results rely on fixed point theory and on a new nonlinear alternative for compact u.s.c. maps respectively. Then, we present some existence results and investigate the topological structure of the solution set. A continuous version of Filippov’s theorem is provided and the continuous dependence of solutions on parameters in both convex and nonconvex cases are examined. [Copyright &y& Elsevier]

Abstract: In this paper we study the global solvability and existence of self-similar solutions to a generalized Davey–Stewartson system. We first use the Banach fixed point theorem to establish a general global existence theorem. By using this general result to a class of special initial data we obtain the existence of global self-similar solutions. [Copyright &y& Elsevier]

Abstract: In this paper, some iterative schemes are given to approximate a fixed point of the nonexpansive non-self-mapping and nonexpansive self-mapping. Furthermore, the strong convergence of the scheme to a fixed point is shown in a Banach space with uniformly Gâteaux differentiable norm. The theorems extend and improve some corresponding results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions, Nonlinear Anal. 68 (2008) 412–419], Chang et al. [S.S. Chang, H.W. Joseph Lee, C.K. Chan, On Reich’s strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. 66 (2007) 2364–2374], Chidume and Chidume [C.E. Chidume, C.O. Chidume, Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal. Appl. 318 (2006) 288–295] and Suzuki [T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed point of nonexpansive mappings, Proc. Amer. Math. Society 135 (1) (2007) 99–106]. [Copyright &y& Elsevier]

Abstract: In this paper we shall study a fractional order semilinear differential equation in an arbitrary Banach space . We used the analytic semigroup theory of linear operators and fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of a global solution. Existence and convergence of an approximate solution to the given problem is also proved in a separable Hilbert space. Finally, we give an example to illustrate the applications of the abstract results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the solvability of a multi-term fractional differential equation in a reflexive Banach space relative to weak topology. A particular case of our main result is treated. [Copyright &y& Elsevier]

Abstract: This paper investigates a class of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces. The existence of mild solutions of these inclusions is determined by using the theory of cosine families of bounded linear operators and a recent fixed point theorem for multivalued maps due to Dhage. [Copyright &y& Elsevier]

Abstract: In this paper, we study the second-order three-point boundary value problem in a Banach space , where is the zero element of . By using the Sadovskii fixed point theorem, we get the existence of at least one (positive) solution. As an application, we give an example to demonstrate our results. [Copyright &y& Elsevier]

Abstract: Let be a real uniformly convex Banach space and be a nonempty closed convex subset of which is also a nonexpansive retract of . Let be nonself asymptotically nonexpansive mappings. It is proved that the modified multistep iterative sequence converges weakly and strongly to some common fixed point of under suitable conditions. The results of this paper improve and extend the corresponding results of [J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991) 407–413; M.O. Osilike, S.C. Aniagbosor, Weak and strong convergence theormes for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32 (2000) 1181–1191; W. Takahashi, G.E. Kim, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces, Nonlinear Anal. 3 (32) (1998) 447–454; C.E. Chidume, E.U. Ofoedu, H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 280 (2003) 364–374; L. Wang, Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl. 323 (2006) 550–557; B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444–453]. On the other hand, it is shown that the necessary and sufficient conditions for the strong convergence of the modified multistep iterative sequence to some common fixed points of . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the solvability of an n-point nonhomogeneous boundary value problem. By using the Krasnoselskii''s fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution are established for the n-point nonhomogeneous boundary value problem. [Copyright &y& Elsevier]

Abstract: This paper is to illustrate that the conditions and proofs of three theorems of S.S. Chang, Y.J. Cho and J.K. Kim [The equivalence between the convergence of modified Picard, modified Mann and modified Ishikawa iterations, Math. Comput. Modelling 37 (2003) 985–991] concerning the equivalence between the convergence of modified Picard, modified Mann and modified Ishikawa iterations for uniformly Lipschitzian operators in arbitrary Banach spaces are incorrect. [Copyright &y& Elsevier]