Common fixed point and invariant approximation results

Abstract

Some common fixed point results for Banach operator pairs in strongly M-starshaped metric spaces are obtained. As application, invariant approximation theorems are derived.

MSC:47H10, 54H25.

1 Introduction and preliminaries

We first review needed definitions. Let X be a metric space with metric d, $M\subset X$ and $J=[0,1]$. The space X is called

• (1) M-starshaped [1] if there exists a continuous mapping $W:X\times M\times J\to X$ satisfying

  $$d(x,W(y,q,\lambda ))\le \lambda d(x,y)+(1-\lambda )d(x,q)$$

  for all $x,y\in X$, $q\in M$ and all $\lambda \in J$;

• (2) strongly M-starshaped [2, 3] if it is M-starshaped and satisfies the property $(I)$, that is,

  $$d(W(x,q,\lambda ),W(y,q,\lambda ))\le \lambda d(x,y)$$

  for all $x,y\in X$, $q\in M$ and all $\lambda \in J$;

• (3) (strongly) convex if it is (strongly) X-starshaped;

• (4) starshaped if it is $\{q\}$-starshaped for some $q\in X$.

A strongly convex metric space is also said to be a metric space of hyperbolic type (see Ciric [4]). Obviously, every normed space X is a strongly convex metric space with W defined by $W(x,q,\lambda )=\lambda x+(1-\lambda )q$ for all $x,q\in X$ and all $\lambda \in J$. More generally, if X is a linear space with a translation invariant metric satisfying $d(\lambda x+(1-\lambda )y,0)\le \lambda d(x,0)+(1-\lambda )d(y,0)$, then X is a strongly convex metric space. A subset D of an M-starshaped metric space X is called q-starshaped if there exists $q\in D\cap M$ such that $W(x,q,\lambda )\in D$ for all $x\in D$ and all $\lambda \in J$. For details, we refer the reader to Al-Thagafi [2], Guay et al. [5] and Takahashi [1].

Let $I,T:X\to X$ be two mappings and $D\subset X$. Then T is called

• (5) I-nonexpansive on D if $d(Tx,Ty)\le d(Ix,Iy)$ for all $x,y\in D$;

• (6) I-contraction on D if there exists $k\in [0,1)$ such that $d(Tx,Ty)\le kd(Ix,Iy)$ for all $x,y\in D$.

A point $x\in D$ is a coincidence point (common fixed point) of I and T if $Ix=Tx$ ($x=Ix=Tx$). The set of coincidence points of I and T is denoted by $C(I,T)$. The mappings I and T are called

• (7) commuting on D if $ITx=TIx$ for all $x\in D$;

• (8) weakly compatible if they commute at their coincidence points, i.e., if $ITx=TIx$ whenever $Ix=Tx$.

The ordered pair $(I,T)$ of two self-maps of a metric space X is called a Banach operator pair if the set $Fix(T)$ is I-invariant, namely $I(Fix(T))\subseteq Fix(T)$. Obviously, a commuting pair $(I,T)$ is a Banach operator pair but not conversely in general, see [6–8].

Let $S\subset X$ and $\hat{x}\in X$. Then $P_S(\hat{x})=\{x\in S:d(x,\hat{x})=d(\hat{x},S)\}$ is called the set of best S-approximants to $\hat{x}$, where $d(\hat{x},S)=inf\{d(\hat{x},y):y\in S\}$ and $C_S^I(\hat{x})=\{x\in S:Ix\in P_S(\hat{x})\}$.

In 1963, Meinardus [9] employed the Schauder fixed point theorem to prove a result regarding invariant approximation. In 1979, Singh [10] proved the following extension of the result of Meinardus.

Theorem 1.1 Let T be a nonexpansive operator on a normed space X, let M be a nonempty subset of X, $T(M)\subset M$ and $u\in F(T)$. If $P_M(u)$ is nonempty compact and starshaped, then $P_M(u)\cap F(T)\ne \mathrm{\varnothing }$.

Hicks and Humphries [11] found that Singh’s results remain true if $T(M)\subset M$ is replaced by $T(\partial M)\subset M$. In 1988, Sahab et al. [12] established the following result which contains the result of Hicks and Humphries and Theorem 1.1.

Theorem 1.2 Let I and T be self-maps of a normed space X with $u\in F(I)\cap F(T)$, $M\subset X$ with $T(\partial M)\subset M$, and $q\in F(I)$. If $D=P_M(u)$ is compact and q-starshaped, $I(D)=D$, I is continuous and linear on D, I and T are commuting on D and T is I-nonexpansive on $D\cup \{u\}$, then $P_M(u)\cap F(T)\cap F(I)\ne \mathrm{\varnothing }$.

Invariant approximation results for commuting maps due to Al-Thagafi [13] extended and generalized Theorems 1.1-1.2 and the works of [11, 14, 15]. Al-Thagafi results were further extended by [7, 8, 16–26] to R-subweakly commuting, pointwise R-subweakly commuting and a Banach operator pair.

The aim of this paper is to establish certain common fixed point theorem for a Banach operator pair in the setup of strongly M-starshaped metric spaces. As application, invariant approximation results for this class of maps are derived. Our results extend and unify the work of Al-Thagafi [2, 13], Dotson [27], Habiniak [14], Hicks and Humphries [11], Hussain and Berinde [28], Hussain et al. [22], Naz [3], Latif [29], Sahab et al. [12] and Singh [10, 15].

The following result will be needed.

Lemma 1.3 [2]

Let D be a subset of an M-starshaped metric space $(X,d)$ and $\hat{x}\in X$. Then $P_D(\hat{x})\subset \partial D\cap D$.

2 Main results

The following result will be needed (see Lemma 2.10 [7] and Lemma 2.2 [8]).

Lemma 2.1 Let S be a nonempty subset of a metric space $(X,d)$, and let T, f be self-maps of S. If $F(f)$ is nonempty, $\mathit{clT}(F(f))\subseteq F(f)$, $\mathit{cl}(T(M))$ is complete, and T and f satisfy for all $x,y\in S$ and $0\le h<1$,

$$d(Tx,Ty)\le hmax\{d(fx,fy),d(Tx,fx),d(Ty,fy),d(Tx,fy),d(Ty,fx)\},$$

(2.1)

then $S\cap F(T)\cap F(f)$ is a singleton.

Theorem 2.2 Let S be a nonempty subset of a strongly M-starshaped metric space X and let T, f be self-maps of S. Suppose that $F(f)$ is q-starshaped, $\mathit{clT}(F(f))\subseteq F(f)$, $\mathit{cl}(T(S))$ is compact, T is continuous on S and

$$\begin{array}{rl}\parallel Tx-Ty\parallel \le & max\{\parallel fx-fy\parallel ,dist(fx,[q,Tx]),dist(fy,[q,Ty]),\\ dist(fy,[q,Tx]),dist(fx,[q,Ty])\},\end{array}$$

(2.2)

for all $x,y\in S$, then $S\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$.

Proof Define $T_n:F(f)\to F(f)$ by $T_{n}x=W(Tx,q,k_n)$ for all $x\in F(f)$ and a fixed sequence of real numbers $k_n$ ($0<k_n<1$) converging to 1. Since $F(f)$ is q-starshaped and $\mathit{clT}(F(f))\subseteq F(f)$, therefore ${\mathit{clT}}_n(F(f))\subseteq F(f)$ for each $n\ge 1$. Also, by (2.2),

$$\begin{array}{rcl}d(T_{n}x,T_{n}y)& =& d(W(Tx,q,k_n),W(Ty,q,k_n))\\ =& k_{n}d(Tx,Ty)\\ \le & k_{n}max\{d(fx,fy),dist(fx,[q,Tx]),dist(fy,[q,Ty]),\\ dist(fx,[q,Ty]),dist(fy,[q,Tx])\}\\ \le & k_{n}max\{d(fx,fy),d(fx,T_{n}x),d(fy,T_{n}y),d(fy,T_{n}x),d(fx,T_{n}y)\}\end{array}$$

for each $x,y\in F(f)$ and $0<k_n<1$. If $\mathit{cl}(T(S))$ is compact for each $n\ge 1$, then $\mathit{cl}(T_n(S))$ is compact and hence complete. By Lemma 2.1, for each $n\ge 1$, there exists $x_n\in F(f)$ such that $x_n=f{x}_n=T_n{x}_n$. The compactness of $\mathit{cl}(T(M))$ implies that there exists a subsequence $\{T{x}_m\}$ of $\{T{x}_n\}$ such that $T{x}_m\to z\in \mathit{cl}(T(M))$ as $m\to \mathrm{\infty }$. Since $\{T{x}_m\}$ is a sequence in $T(F(f))$ and $\mathit{clT}(F(f))\subseteq F(f)$, therefore $z\in F(f)$. Further, $x_m=T_m{x}_m=W(T{x}_m,q,k_m)\to z$. By the continuity of T, we obtain $Tz=z=fz$. Thus, $S\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$. □

Corollary 2.3 Let S be a nonempty subset of a strongly M-starshaped metric space X and let T, f be self-maps of S. Suppose that $F(f)$ is q-starshaped, $\mathit{clT}(F(f))\subseteq F(f)$, $\mathit{cl}(T(S))$ is compact, T is continuous on S and T is f-nonexpansive on S, then $S\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$.

Corollary 2.4 Let S be a nonempty subset of a strongly M-starshaped metric space X and let T, f be self-maps of S. Suppose that $F(f)$ is closed and q-starshaped, $(T,f)$ is a Banach operator pair, $\mathit{cl}(T(S))$ is compact, T is continuous on S and T satisfies (2.2) or T is f-nonexpansive on S, then $S\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$.

Corollary 2.5 ([13], Theorem 2.1)

Let M be a nonempty closed and q-starshaped subset of a normed space X and let T and f be self-maps of M such that $T(M)\subseteq f(M)$. Suppose that T commutes with f and $q\in F(f)$. If $\mathit{cl}(T(M))$ is compact, f is continuous and linear and T is f-nonexpansive on M, then $M\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$.

Corollary 2.6 (([30], Theorem 3.3))

Let M be a nonempty subset of a normed space X and let T and f be self-maps of M. Suppose that $F(f)$ is q-starshaped, $\mathit{clT}(F(f))\subseteq F(f)$, $\mathit{cl}(T(M))$ is compact, T is continuous on M and (2.2) holds for all $x,y\in M$. Then $M\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$.

Corollary 2.7 ([7], Theorem 2.11)

Let M be a nonempty subset of a normed space X and let T, f be self-maps of M. Suppose that $F(f)$ is q-starshaped and closed $\mathit{cl}(T(M))$ is compact, T is continuous on M, $(T,f)$ is a Banach operator pair and satisfies (2.2) for all $x,y\in M$. Then $M\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$.

Corollary 2.8 Let X be a strongly M-starshaped metric space, let $f,T:X\to X$ be two mappings, S be a subset of X such that $T(\partial S\cap S)\subset S$ and $\hat{x}\in F(T)\cap F(f)$. Suppose that $P_S(\hat{x})$ is nonempty closed and q-starshaped with $q\in F(f)\cap M$ and $\mathit{cl}(T(P_S(\hat{x})))$ is compact and $f(P_S(\hat{x}))=P_S(\hat{x})$. If T is continuous, $\mathit{clT}(F(f))\subseteq F(f)$ and satisfies, for all $x\in P_S(\hat{x})\cup \{\hat{x}\}$,

$$d(Tx,Ty)\le \{\begin{array}{cc}d(fx,fu)\hfill & \mathit{\text{if}}y=u,\hfill \\ max\{d(fx,fy),dist(fx,[q,Tx]),dist(fy,[q,Ty]),\hfill \\ dist(fx,[q,Ty]),dist(fy,[q,Tx])\}\hfill & \mathit{\text{if}}y\in P_S(\hat{x}),\hfill \end{array}$$

(2.3)

then $P_S(\hat{x})\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$.

Proof Let $x\in P_S(\hat{x})$. Then by Lemma 1.3, $x\in \partial S\cap S$ and so $Tx\in S$ since $T(\partial S\cap S)\subset S$. As T satisfies (2.3) on $P_S(\hat{x})\cup \{\hat{x}\}$ and $I(P_S(\hat{x}))=P_S(\hat{x})$, we have

$$d(Tx,\hat{x})=d(Tx,T\hat{x})\le d(Ix,I\hat{x})=d(Ix,\hat{x})=d(\hat{x},S).$$

This implies that $Tx\in P_S(\hat{x})$. Thus $T(P_S(\hat{x}))\subset P_S(\hat{x})=f(P_S(\hat{x}))$. Now Theorem 2.2 implies that $P_S(\hat{x})\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$. □

Theorem 2.9 Let X be a strongly M-starshaped metric space, let$f,T:X\to X$ be two mappings, S be a subset of X such that $T(\partial S\cap S)\subset S$ and $\hat{x}\in F(T)\cap F(f)$. Suppose that $P_S(\hat{x})$ is nonempty closed and q-starshaped with $q\in F(f)\cap M$ and $\mathit{cl}(T(P_S(\hat{x})))$ is compact and $f(P_S(\hat{x}))=P_S(\hat{x})$. If T is continuous, $\mathit{clT}(F(f))\subseteq F(f)$ and T is f-nonexpansive on $P_S(\hat{x})\cup \{\hat{x}\}$, then $P_S(\hat{x})\cap F(T)\cap F(f)\ne \mathrm{\varnothing }$.

Remark 2.10 A subset S of a strongly M-starshaped metric space X is said to have the property $(N)$ w.r.t. T [22, 28] if

• (i) $T:S\to S$,

• (ii) $W(Tx,q,k_n)\in S$ for some $q\in S\cap M$ and a fixed sequence of real numbers $k_n$ ($0<k_n<1$) converging to 1 and for each $x\in S$.

All results of the paper (Theorem 2.2-Theorem 2.9) remain valid provided f is assumed to be surjective and q-starshapedness of the set $F(f)$ is replaced by the property $(N)$ respectively. Consequently, recent results due to Hussain and Berinde [28] and Hussain et al. [22] are improved and extended.

Remark 2.11 Recently, in [31], the author obtained certain fixed point theorems in convex metric spaces. Using Theorems 3.2 and 3.4 [31] and the technique in [7], we can prove more common fixed point and approximation results for Banach pairs satisfying generalized nonexpansive conditions in a strongly M-starshaped metric space X.

Remark 2.12 All results of the paper can be proved for multivalued Banach operator pairs defined and studied in [32].