SOME FIXED POINT THEOREMS FOR GENERALIZED $\alpha$-GERAGHTY CONTRACTION TYPE MAPPINGS IN $B$-METRIC~SPACES AND SOME APPLICATIONS TO THE NONLINEAR INTEGRAL EQUATION

Nguyen Trung Hieu, Le Thi Chac

Abstract


The purpose of this paper is to introduce the notion of a generalized $\alpha$-Geraghty contraction type mapping in $b$-metric~spaces and state the existence and uniqueness of a fixed point for this mapping. These results are generalizations of certain the main results in [D.~\DJ uki\'{c}, Z.~Kadelburg, and S.~Radenovi\'{c}, \emph{Fixed points of Geraghty-type mappings in
various generalized metric spaces}, Abstr. Appl. Anal. \textbf{2011} (2011), 13 pages] and [O.~Popescu, \emph{ Some new fixed point theorems for $\alpha$-Geraghty contraction type maps in metric spaces}, Fixed Point Theory Appl. \textbf{2014:190} (2014), 1 -- 12]. Some examples are given to illustrate the obtained results and to show that these results are proper extensions of the existing ones. Then we apply the obtained theorem to study the existence of solutions to the nonlinear integral equation.


Keywords


fixed point, generalized $\alpha$-Geraghty contraction type, $b$-metric space

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References


A.~Aghajani, M.~Abbas, and J.~R. Roshan, emph{Common fixed point of generalized weak contractive mappings in partially ordered $b$-metric spaces}, Math. Slovaca textbf{64} (2014),no.~4, 941 -- 960.

A.~Aghajani and R.~Arab, emph{Fixed points of $(psi,phi, theta )$-contractive mappings in partially ordered $b$-metric spaces and application to quadratic integral equations}, Fixed Point Theory

Appl. textbf{2013:245} (2013), 20 pages.

A.~Amini-Harandi and H.~Emami, emph{A fixed point theorem

for contraction type maps in partially ordered metric paces and applications to ordinary differential equations}, Nonlinear Anal. textbf{72} (2010), 2238 -- 2242.

T.~V. An, N.~V. Dung, Z.~Kadelburg, and S.~Radenovi'{c},

emph{Various generalizations of metric spaces and fixed point theorems}, Rev. R. Acad. Cienc. Exactas Fis. Nat.Ser. A Mat. RACSAM textbf{109} (2014), no.~1, 175 -- 198.

T.~V. An, L.~Q. Tuyen, and N.~V. Dung, emph{Stone-type theorem on $b$-metric spaces and applications}, Topology Appl.textbf{185} (2015), 50 -- 64.

I.~A. Bakhtin, emph{The contraction principle inquasimetric spaces}, Func. An., Ulianowsk, Gos. Fed. Ins. textbf{30}

(1989), 26 -- 37.

M.~Boriceanu, M.~Bota, and A.~Petrusel,emph{Multivalued fractals in $b$-metric spaces}, Cent. Eur. J. Math. textbf{8} (2010), no.~2, 367 -- 377.

J.~Caballero, J.~Harjani, and K.~Sadarangani, emph{A best proximity point theorem for Geraghty-contractions}, Fixed

Point Theory Appl. textbf{2012:231} (2012), 1 -- 9.

S.~H. Cho, J.~S. Bae, and E.~Karapinar, emph{Fixed point theorems for $alpha$-Geraghty contraction type maps in metric spaces}, Fixed Point Theory Appl. textbf{2013:329} (2013), 1 -- 11.

L.~B. '{C}iri'{c}, emph{A generalization of Banach's contraction principle}, Proc. Amer. Math. Soc. textbf{45} (1974), 267 --

S.~Czerwik, emph{Contraction mappings in $b$-metric spaces}, Acta Math. Univ. Ostrav. textbf{1} (1993), 5 -- 11.

S.~Czerwik, emph{Nonlinear set-valued contractionmappings in $b$-metric spaces}, Atti Semin. Mat. Fis. Univ. Modena

textbf{46} (1998), no.~2, 263 -- 276.

D.~DJ uki'{c}, Z.~Kadelburg, and S.~Radenovi'{c},emph{Fixed points of Geraghty-type mappings in various generalized metric spaces}, Abstr. Appl. Anal. textbf{2011} (2011),13 pages.

M.~Geraghty, emph{On contractive mappings}, Proc.

Am. Math. Soc. textbf{40} (1973), 604 -- 608.

M.~E. Gordji, M.~Ramezami, Y.~J. Cho, and S.~Pirbavafa,

emph{A generalization of Geraghty's theorem in partially ordered metric spaces and applications to ordinary differential equations}, Fixed Point Theory Appl. textbf{2012:74} (2012), 1 -- 9.

N.~T. Hieu and N.~V. Dung, emph{ Some fixed point results for generalized rational type contraction mappings in partially ordered $b$-metric spaces}, Facta Univ. Ser. Math. Inform. textbf{30}

(2015), no.~1, 49 -- 66.

H.~Huang and S.~Xu, emph{Fixed point theorems of contractive mappings in cone $b$-metric spaces and applications}, Fixed

Point Theory Appl. textbf{2012} (2012), 8 pages.

N.~Hussain, V.~Parvaneh, J.~R. Roshan, and Z.~Kadelburg,

emph{Fixed points of cyclic weakly $(psi,varphi, L,A,B)$-contractive mappings in ordered $b$-metric spaces with applications}, Fixed Point Theory Appl. textbf{2013:256} (2013), 18 pages.

E.~Karapinar, emph{ $alpha $-$psi $-Geraghty contraction type mappings and some related fixed point results}, Filomat textbf{28} (2014), no.~1, 37 -- 48.

P.~Kumam, N.~V. Dung, and V.~T.~L. Hang, emph{Some equivalences between cone $b$-metric spaces and $b$-metric spaces}, Abstr.Appl. Anal. textbf{2013} (2013), 8 pages.

P.~Kumam, N.~V. Dung, and K.~Sitthithakerngkiet, emph{A generalization of '{C}iri'{c} fixed point theorems}, Filomat textbf{29}(2015), no.~7, 1549 -- 1556.

V.~Parvaneh, J.~R. Roshan, and S.~Radenovi'{c},emph{Existence of tripled coincidence points in ordered $b$-metric spaces and an application to a system of integral equations}, Fixed Point

Theory Appl. textbf{2013:130} (2013), 19 pages.

O.~Popescu, emph{ Some new fixed point theorems for $alpha$-Geraghty contraction type maps in metric spaces}, Fixed Point

Theory Appl. textbf{2014:190} (2014), 1 -- 12.

A.~C.~M. Ran and M.~C.~B. Reurings, emph{A fixed point theorem in partially ordered sets and some applications to matrix

equations}, Proc. Amer. Math. Soc. textbf{132} (2003), no.~5, 1435 -- 1443.

J.~R. Roshan, V.~Parvaneh, S.~Sedghi, N.~Shobkolaei, and W.~Shatanawi, emph{Common fixed points of almost

generalized $(psi,varphi)_s$-contractive mappings in ordered $b$-metric spaces}, Fixed Point Theory Appl. textbf{2013:159} (2013), 23 pages.

W.~Shatanawi, A.~Pitea, and R.~Lazovi'{c}, emph{Contraction conditions using comparison functions on $b$-metric spaces}, Fixed Point Theory Appl. textbf{2014:135} (2014), 11 pages.

T.~Suzuki, emph{A generalized Banach contraction principle that characterizes metric completenes}, Proc. Amer. Math. Soc.

textbf{136} (2008), no.~5, 1861 -- 1869.


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