Fixed point theory for 1-set contractive and pseudocontractive mappings

The purpose of this paper is to study the existence and uniqueness of fixed point for a class of nonlinear mappings defined on a real Banach space, which, among others, contains the class of separate contractive mappings, as well as to see that an important class of 1-set contractions and of pseudocontractions falls into this type of nonlinear mappings. As a particular case, we give an iterative method to approach the fixed point of a nonexpansive mapping. Later on, we establish some fixed point results of Krasnoselskii type for the sum of two nonlinear mappings where one of them is either a 1-set contraction or a pseudocontraction and the another one is completely continuous, which extend or complete previous results. In the last section, we apply such results to study the existence of solutions to a nonlinear integral equation.

[1]  S. Sadiq Basha,et al.  Best proximity point theorems: An exploration of a common solution to approximation and optimization problems , 2012, Appl. Math. Comput..

[2]  The fixed-point index and the fixed-point theorems of 1-set-contraction mappings , 1988 .

[3]  V. Lakshmikantham,et al.  Differential Equations on Closed Subsets of a Banach Space , 1976 .

[4]  R. Martin,et al.  Differential equations on closed subsets of a Banach space , 1973 .

[5]  S. Djebali,et al.  Nonlinear alternatives of Schauder and Krasnosel'skij types with applications to Hammerstein integral equations in L1 spaces , 2010 .

[6]  M. A. Taoudi,et al.  Integrable solutions of a mixed type operator equation , 2010, Appl. Math. Comput..

[7]  T. Burton,et al.  A fixed-point theorem of Krasnoselskii , 1998 .

[8]  L. Kantorovich,et al.  THE FIXED-POINT PRINCIPLE , 1982 .

[9]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .

[10]  Enrique Llorens Fuster,et al.  The Fixed Point Theory for Some Generalized Nonexpansive Mappings , 2011 .

[11]  K. Latrach,et al.  Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations , 2006 .

[12]  S. Ishikawa.  Fixed points and iteration of a nonexpansive mapping in a Banach space , 1976 .

[13]  K. Latrach,et al.  Existence results for a generalized nonlinear Hammerstein equation on L1 spaces , 2007 .

[14]  A topological and geometric approach to fixed points results for sum of operators and applications , 2004, math/0403474.

[15]  W. A. Kirk,et al.  Handbook of metric fixed point theory , 2001 .

[16]  RobertF Brown,et al.  A topological introduction to nonlinear analysis , 1993 .

[17]  J. Garcia-Falset,et al.  Schaefer–Krasnoselskii fixed point theorems using a usual measure of weak noncompactness , 2012 .

[18]  Shaoyuan Xu,et al.  FIXED POINTS OF WEAKLY INWARD 1-SET-CONTRACTION MAPPINGS , 2008 .

[19]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[20]  J. Garcia-Falset,et al.  Existence of fixed points for the sum of two operators , 2010 .

[21]  S. Prus,et al.  Fixed point free mappings which satisfy a Darbo type condition , 2006 .

[22]  A. Rodkina,et al.  Measures of Noncompactness , 1992 .

[23]  J. Garcia-Falset.  Existence of fixed points and measures of weak noncompactness , 2009 .

[24]  Felix E. Browder,et al.  Nonlinear mappings of nonexpansive and accretive type in Banach spaces , 1967 .

[25]  W. Petryshyn.  Structure of the fixed points sets of k-set-contractions , 1971 .

[26]  L. Karlovitz,et al.  On the radial projection in normed spaces , 1967 .

[27]  Yicheng Liu,et al.  Schaefer type theorem and periodic solutions of evolution equations , 2006 .

[28]  J. Appell,et al.  On some Banach space constants arising in nonlinear fixed point and eigenvalue theory , 2004 .

[29]  R. Yuan,et al.  A class of expansive-type Krasnosel’skii fixed point theorems , 2009 .

[30]  M. Crandall,et al.  Some Problems in Nonlinear Analysis , 1990 .

[31]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[32]  G. L. Acedo,et al.  Measures of Noncompactness in Metric Fixed Point Theory , 1997 .

[33]  G. L. Acedo,et al.  Measures of Noncompactness , 1997 .

[34]  A. Rodkina,et al.  Measures of noncompactness and condensing operators , 1992 .

[35]  Tosio Kato,et al.  Nonlinear semigroups and evolution equations , 1967 .

[36]  T. Burton,et al.  A Fixed Point Theorem of Krasnoselskii—Schaefer Type , 1998 .

[37]  D. O’Regan,et al.  Browder-Krasnoselskii-Type Fixed Point Theorems in Banach Spaces , 2010 .