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A common fixed point theorem for expansive mappings under strict implicit conditions on b-metric spaces. (English) Zbl 1246.54035
Let $(X,d)$ be a $b$-metric space with parameter $s$, in the sense of {\it S. Czerwik} [Acta Math. Inform. Univ. Ostrav. 1, 5--11 (1993; Zbl 0849.54036)]. Let $S$ and $T$ be two weakly compatible self-mappings of $X$ such that: (1) $S$ and $T$ satisfy property (E.A) of {\it M. Aamri} and {\it D. El Moutawakil} [J. Math. Anal. Appl. 270, No. 1, 181--188 (2002; Zbl 1008.54030)]; (2) $T(X)\subset S(X)$; and (3) $$G(d(Tx,Ty),d(Sx,Sy),d(Sx,Tx),\break d(Sy,Ty),d(Sx,Ty),d(Sy,Tx))>0$$ for all $x,y\in X$ such that $x\ne y$, where $G:\Bbb{R}_+^6\to\Bbb{R}$ is continuous and satisfies: (a) $G$ is nondecreasing in variable $t_1$ and nonincreasing in variable $t_2$; (b) $G(st,0,0,t,\frac1st,0)<0$ for all $t>0$; and (c) $G(t,t,0,0,t,t)\leq0$ for all $t>0$. The author proves that if $S(X)$ or $T(X)$ is a closed subspace of $X$, then $T$ and $S$ have a unique common fixed point. If the $b$-metric $d$ is weakly continuous (i.e., if $\lim_{n\to\infty}d(x_n,x)=0$ implies $\lim_{n\to\infty}d(x_n,y)=d(x,y)$ for every sequence $\{x_n\}$ in $X$ and all $x,y\in X$), then the same conclusion holds with weaker assumptions for the function $G$.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: EuDML
[1] Aamri, M., El Moutawakil, D.: Some new common fixed point theorems under strict contractive conditions. Math. Anal. Appl. 270 (2002), 181-188. · Zbl 1008.54030 · doi:10.1016/S0022-247X(02)00059-8
[2] Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostraviensis 1 (1993), 5-11. · Zbl 0849.54036 · eudml:23748
[3] Czerwik, S.: Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Fis. Univ. Modena 46, 2 (1998), 263-276. · Zbl 0920.47050
[4] Djoudi, A.: General fixed point theorems for weakly compatible maps. Demonstratio Math. 38 (2005), 195-205. · Zbl 1065.54020
[5] Imdad, M., Khan, T. I.: Fixed point theorems for some expansive mapping via implicit relations. Nonlinear Analysis Forum 9, 2 (2004), 209-218. · Zbl 1066.54038
[6] Imdad, M., Ali, J.: Jungck’s Common Fixed Point Theorem and E.A Property. Acta Mathematica Sinica, English Series 24, 1 (2008), 87-94. · Zbl 1158.54021 · doi:10.1007/s10114-007-0990-0
[7] Jungck, G.: Compatible mappings and common fixed points. Internat. J. Math. Math. Sci. 9 (1986), 771-779. · Zbl 0613.54029 · doi:10.1155/S0161171286000935 · eudml:46151
[8] Jungck, G.: Common fixed points for noncontinuous nonself mappings on nonnumeric spaces. Far. East, J. Math.Sci. 4 (1996), 199-215. · Zbl 0928.54043
[9] Pant, R. P.: Common fixed points of contractive maps. J. Math. Anal. Appl. 226 (1998), 251-258. · Zbl 0916.54027 · doi:10.1006/jmaa.1998.6029
[10] Pant, R. P.: R-weak commutativity and common fixed points of noncompatible maps. Ganita 49, 1 (1998), 19-27. · Zbl 0977.54039
[11] Pant, R. P.: R-weak commutativity and common fixed points. Soochow J. Math. 25 (1999), 37-42. · Zbl 0918.54038
[12] Popa, V.: Fixed point theorems for implicit contractive mappings. Stud. Cerc. St. Ser. Mat. Univ. Bacău 7 (1997), 127-133. · Zbl 0967.54041
[13] Popa, V.: Some fixed point theorems for compatible mappings satisfying an implicit relation. Demonstratio Math. 32 (1999), 157-163. · Zbl 0926.54030
[14] Popa, V.: A General fixed point theorem for expansive mappings under strict implicit conditions. Stud. Cerc. St. Ser. Mat. Univ. Bacău 17 (2007), 197-200. · Zbl 1199.54242