By Monther Alfuraidan, Qamrul Ansari

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*Fixed element idea and Graph Theory* offers an intersection among the theories of fastened element theorems that provide the stipulations lower than which maps (single or multivalued) have suggestions and graph idea which makes use of mathematical buildings to demonstrate the connection among ordered pairs of gadgets when it comes to their vertices and directed edges.

This edited reference paintings could be the 1st to supply a hyperlink among the 2 theories, describing not just their foundational facets, but in addition the newest advances and the interesting intersection of the domain names.

The authors supply resolution tools for mounted issues in numerous settings, with chapters dedicated to the ideas approach for significantly very important non-linear difficulties in engineering, particularly, variational inequalities, fastened aspect, cut up feasibility, and hierarchical variational inequality difficulties. The final chapters are dedicated to integrating fastened element conception in areas with the graph and using retractions within the mounted element concept for ordered sets.

- Introduces either metric mounted aspect and graph concept by way of their disparate foundations and customary software environments
- Provides a distinct integration of in a different way disparate domain names that aids either scholars trying to comprehend both region and researchers attracted to setting up an built-in learn approach
- Emphasizes answer tools for mounted issues in non-linear difficulties corresponding to variational inequalities, cut up feasibility, and hierarchical variational inequality difficulties that's fairly applicable for engineering and center technology applications

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**Additional info for Fixed Point Theory and Graph Theory. Foundations and Integrative Approaches**

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1974;47:324–353. 64. Eshghinezhad S, Fakhar M, Some generalizations of Ekeland’s variational principle with applications to fixed point theory. Math. Comput. Modelling 2013;57:1250–1258. 65. Fang JX, The variational principle and fixed point theorems in certain topological spaces. J. Math. Anal. Appl. 1996;202:398–412. 66. Feng YJ, Liu SY, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 2006;317:103–112. 67. Filip AD, Fixed point theorems in Kasahara spaces with respect to an operator and applications.

Thesis. “Babes¸-Bolyai” University of ClujNapoca, 2011, Cluj-Napoca, Romania. 69. Frigon M, On some generalizations of Ekeland’s principle and inward contractions in gauge spaces. J. Fixed Point Theory Appl. 2011;10:279–298. 70. Glab S, On the converse of Caristi’s fixed point theorem. Bull. Pol. Acad. Sci. Math. 2004;52:411– 416. 71. Goebel K, Kirk WA, Topics in Metric Fixed Point Theory. Cambridge:Cambridge Univ. Press; 1990. 72. Guo TX, Yang YJ, Ekeland’s variational principle for an L0 -valued function on a complete random metric space.

I = 1, 2, . . 14) 35 36 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches Proof. 8) has at least a fixed point in X. 2). 8) to obtain d(T xn−1 , T xn ) ≤ δ · d(xn−1, xn ), which shows that d(xn , xn+1 ) ≤ δ · d(xn−1 , xn ). 15), we obtain by induction d(xn , xn+1 ) ≤ δ n d(x0 , x1 ), n = 0, 1, 2, . . , and then d(xn , xn+p ) ≤ δ n 1 + δ + · · · + δ p−1 d(x0 , x1 ) δn (1 − δ p ) · d(x0, x1 ), n, p ∈ N, p = 1−δ 0. 16) shows that {xn }∞ n=0 is a Cauchy sequence and hence it is convergent.