Common fixed point theorems for generalized $\mathcal{J}\mathcal{H}$-operator classes and invariant approximations

Abstract

In this article, we introduce two new different classes of noncommuting selfmaps. The first class is more general than $\mathcal{J}\mathcal{H}$-operator class of Hussain et al. (Common fixed points for $\mathcal{J}\mathcal{H}$-operators and occasionally weakly biased pairs under relaxed conditions. Nonlinear Anal. 74(6), 2133-2140, 2011) and occasionally weakly compatible class. We establish the existence of common fixed point theorems for these classes. Several invariant approximation results are obtained as applications. Our results unify, extend, and complement several well-known results.

2000 Mathematical Subject Classification: 47H09; 47H10.

1. Introduction

The fixed point theorem, generally known as the Banach contraction principle, appeared in explicit form in Banach's thesis in 1922 [1], where it was used to establish the existence of a solution for an integral equation. Since its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions. Many authors established fixed point theorems involving more general contractive conditions.

In 1976, Jungck [2] extend the Banach contraction principle to a common fixed point theorem for commuting maps. Sessa [3] defined the notion of weakly commuting maps and established a common fixed point for this maps. Jungck [4] coined the term compatible mappings to generalize the concept of weak commutativity and showed that weakly commuting maps are compatible but the converse is not true. Afterward, many authors studied about common fixed point theorems for noncommuting maps (see [5–14]).

In 1996, Al-Thagafi [15] established some theorems on invariant approximations for commuting maps. Shahzad [16], Al-Thagafi and Shahzad [17, 18], Hussain and Jungck [19], Hussain [20], Hussain and Rhoades [21], Jungck and Hussain [22], O'Regan and Hussain [23], and Pathak and Hussain [24] extended the result of Al-Thagafi [15] and Ciric [25] for pointwise R-subweakly commuting maps, compatible maps, C _q -commuting maps, and Banach operator pairs. Pathak and Hussain [26] introduced two new classes of noncommuting selfmaps, so-called $\mathcal{P}$-operator and $\mathcal{P}$-suboperator pair class. Recently, Hussain et al. [27] introduced $\mathcal{J}\mathcal{H}$-operator and occasionally weakly g-biased class which are more general than above classes and established common fixed point theorems for these class.

In this article shall introduce two new classes of noncommuting selfmaps. First class, generalized $\mathcal{J}\mathcal{H}$-operator class, contains $\mathcal{J}\mathcal{H}$-operator classes of Hussain et al. [27] and occasionally weakly compatible classes. Second class is the so-called generalized $\mathcal{J}\mathcal{H}$-suboperator class. We will be present some common fixed point theorems for these classes and the existence of the common fixed points for best approximation. Our results improve, extend, and complement all the results in literature.

2. Preliminaries

Let M be a subset of a norm space X. We shall use cl(A) and wcl(A) to denote the closure and the weak closure of a set A, respectively, and d(x, A) to denote inf{||x-y|| : y ∈ A} where x ∈ X and A ⊆ X. Let f and T be selfmaps of M. A point x ∈ M is called a fixed point of f if fx = x. The set of all fixed points of f is denoted by F(f). A point x ∈ M is called a coincidence point of f and T if fx = Tx. We shall call w = fx = Tx a point of coincidence of f and T. A point x ∈ M is called a common fixed point of f and T if x = fx = Tx. Let C(f, T), PC(f, T), and F(f, T) denote the sets of all coincidence points, points of coincidence, and common fixed points, respectively, of the pair (f, T).

The map T is called contraction [resp. f-contraction] on M if ||Tx-Ty|| ≤ k||x-y|| [resp. ||Tx - Ty|| ≤ k||fx - fy||] for all x, y ∈ M and for some k ∈ [0, 1). The map T is called nonexpansive [resp. f-nonexpansive] on M if ||Tx - Ty|| ≤ ||x - y|| [resp. ||Tx - Ty|| ≤ ||fx - fy||] for all x, y ∈ M. The pair (f, T) is called:

(i): commuting if Tfx = fTx for all x ∈ M;

(ii): R-weakly commuting[8] if for all x ∈ M, there exists R > 0 such that

$$\parallel fTx-Tfx\parallel \le R\parallel fx-Tx\parallel .$$

If R = 1, then the maps are called weakly commuting;

(iii): compatible[28] if $\underset{n\to \infty }{lim}\parallel Tf{x}_n-fT{x}_n\parallel =0$ when {x _n } is a sequence such that

$$\underset{n\to \infty }{lim}T{x}_n=\underset{n\to \infty }{lim}f{x}_n=t$$

for some t ∈ M;

(iv): weakly compatible[29] if Tfx = fTx for all x ∈ C(f, T);

(v): occasionally weakly compatible[18, 30] if fTx = Tfx for some x ∈ C(f, T);

(vi): Banach operator pair[31] if f(F(T)) ⊆ F(T);

(vii):$\mathcal{P}$-operator[26] if ||u - Tu|| ≤ diam (C(f, T)) for some u ∈ C(f, T);

(viii):$\mathcal{J}\mathcal{H}$-operator[27] if there exist a point w = fx = Tx in PC(f, T) such that

$$\parallel w-x\parallel \le \text{diam}\left(PC\left(f,T\right)\right).$$

The set M is called convex if kx + (1 - k)y ∈ M for all x, y ∈ M and all k ∈ [0, 1]; and q-starshaped with q ∈ M if the segment [q, x] = {kx + (1 - k)q : k ∈ [0, 1]} joining q to x is contained to M. The map f : M → M is called affine if M is convex and f(kx + (1 - k)y) = kfx + (1 - k)fy for all x, y ∈ M and all k ∈ [0, 1]; and q-affine if M is q-starshaped and f(kx + (1 - k)q) = kfx + (1 - k)fq for all x, y ∈ M and all k ∈ [0, 1].

A map T : M → X is said to be semicompact if a sequence {x _n } in M such that (x _n - Tx _n ) → 0 has a subsequence {x _j } in M such that x _j → z for some z ∈ M. Clearly if cl(T(M)) is compact, then T(M) is complete, T(M) is bounded, and T is semicompact. The map T : M → X is said to be weakly semicompact if a sequence {x _n } in M such that (x _n - Tx _n ) → 0 has a subsequence {x _j } in M such that x _j → z weakly for some z ∈ M. The map T : M → X is said to be demiclosed at 0 if, for every sequence {x _n } in M converging weakly to x and {Tx _n } converges to 0 ∈ X, then Tx = 0.

3. Generalized $\mathcal{J}\mathcal{H}$-operator classes

We begin this section by introduce a new noncommuting class.

Definition 3.1. Let f and T be selfmaps of a normed space X. The order pair (f, T) is called a generalized$\mathcal{J}\mathcal{H}$-operator with order n if there exists a point w = fx = Tx in PC(f, T) such that

$$\parallel w-x\parallel \le {\left(\text{diam}\left(PC\left(f,T\right)\right)\right)}^n$$

(3.1)

for some n ∈ ℕ.

It is obvious that a $\mathcal{J}\mathcal{H}$-operator pair (f, T) is generalized $\mathcal{J}\mathcal{H}$-operator with order n. But the converse is not true in general, see Example 3.2.

Example 3.2. Let X = ℝ with usual norm and M = [0, ∞). Define f, T : M → M by

$$fx=\left\{\begin{array}{cc}\hfill 3,\hfill & \hfill x=0;\hfill \\ \hfill 5,\hfill & \hfill x=2;\hfill \\ \hfill 2x,\hfill & \hfill anotherpoint,\hfill \end{array}\right.Tx=\left\{\begin{array}{cc}\hfill 3,\hfill & \hfill x=0;\hfill \\ \hfill 5,\hfill & \hfill x=2;\hfill \\ \hfill x^2,\hfill & \hfill anotherpoint.\hfill \end{array}\right.$$

Then C(f, T) = {0, 2} and PC(f, T) = {3, 5}. Obvious (f, T) is a generalized $\mathcal{J}\mathcal{H}$-operator with order n ≥ 2 but not a $\mathcal{J}\mathcal{H}$-operator and so not a occasionally weakly compatible and not weakly compatible. Moreover, note that F(T) = {1} and f 1 = 2 ∉ F(T) which implies that (f, T) is not a Banach operator pair.

Theorem 3.3. Let f and T be selfmaps of a nonempty subset M of a normed space X and (f, T) be a generalized$\mathcal{J}\mathcal{H}$-operator with order n on M. If f and T satisfying the following condition:

$$\parallel Tx-Ty\parallel \le kmax\left\{\parallel fx-fy\parallel ,\parallel fx-Tx\parallel ,\parallel fy-Ty\parallel ,\parallel fx-Ty\parallel ,\parallel fy-Tx\parallel \right\},$$

(3.2)

for all x, y ∈ M and 0 ≤ k < 1, then f and T have a unique common fixed point.

Proof. By the notation of generalized $\mathcal{J}\mathcal{H}$-operator, we get that there exists a point w ∈ M such that w = fx = Tx and

$$\parallel w-x\parallel \le {\left(\text{diam}\left(PC\left(f,T\right)\right)\right)}^n$$

(3.3)

for some n ∈ ℕ. Suppose there exists another point y ∈ M for which z = fy = Ty. Then from (3.2), we get

$$\begin{array}{ll}\hfill \parallel Tx-Ty\parallel & \le kmax\left\{\parallel fx-fy\parallel ,\parallel fx-Tx\parallel ,\parallel fy-Ty\parallel ,\parallel fx-Ty\parallel ,\parallel fy-Tx\parallel \right\}\\ =kmax\left\{\parallel Tx-Ty\parallel ,0,0,\parallel Tx-Ty\parallel ,\parallel Ty-Tx\parallel \right\}\\ \le k\parallel Tx-Ty\parallel .\end{array}$$

(3.4)

Since 0 ≤ k < 1, the inequality (3.4) implies that ||Tx - Ty|| = 0, which, in turn implies that w = fx = Tx = z. Therefore, there exists a unique element w in M such that w = fx = Tx. So diam(PC(f, T)) = 0. Using (3.3), we have

$$d\left(w,x\right)\le {\left(\text{diam}\left(PC\left(f,T\right)\right)\right)}^n=0.$$

Thus w = x, that is x is a unique common fixed point of f and T. □

Definition 3.4. Let M be a q-starshaped subset of a normed space X and f, T selfmaps of a normed space M. The order pair (f, T) is called a generalized$\mathcal{J}\mathcal{H}$-suboperator with order n if for each k ∈ [0, 1], (f, T _k ) is a generalized $\mathcal{J}\mathcal{H}$-operator with order n that is, for k ∈ [0, 1] there exists a point w = fx = T _k x in PC(f, T _k ) such that

$$d\left(w,x\right)\le {\left(\text{diam}\left(PC\left(f,T_k\right)\right)\right)}^n$$

(3.5)

for some n ∈ ℕ, where T _k is selfmap of M such that T _k x = kTx + (1 - k)q for all x ∈ M.

Clearly, a generalized $\mathcal{J}\mathcal{H}$-suboperator with order n is generalized $\mathcal{J}\mathcal{H}$-operator with order n but the converse is not true in general, see Example 3.5.

Example 3.5. Let X = ℝ with usual norm and M = [0, ∞). Define f, T : M → M (see Example 3.2). Then M is q-starshaped for q = 0 and C(f, T) = {0, 2}, $C\left(f,T_k\right)=\left\{\frac{2}{k}\right\}$, and $PC\left(f,T_k\right)=\left\{\frac{4}{k}\right\}$ for k ∈ (0, 1). Obvious (f, T) is a generalized $\mathcal{J}\mathcal{H}$-operator with n = 2 but not a generalized $\mathcal{J}\mathcal{H}$-suboperator for every n ∈ ℕ as

$$∥\frac{2}{k}-T_k\left(\frac{2}{k}\right)∥=∥\frac{2}{k}-\frac{4}{k}∥=\frac{2}{k}>0={\left(\text{diam}\left(PC\left(f,T_k\right)\right)\right)}^n$$

(3.6)

for each k ∈ (0, 1).

Theorem 3.6. Let f and T be selfmaps on a q-starshaped subset M of a normed space X. Assume that f is q-affine, (f, T) is a generalized$\mathcal{J}\mathcal{H}$-suboperator with order n₀, and for all x, y ∈ M,

$$\parallel Tx-Ty\parallel \le max\left\{\parallel fx-fy\parallel ,d\left(fx,\left[q,Tx\right]\right),d\left(fy,\left[q,Ty\right]\right),d\left(fx,\left[q,Ty\right]\right),d\left(fy,\left[q,Tx\right]\right)\right\}.$$

(3.7)

Then F(f, T) ≠ ∅ if one of the following conditions holds:

(a): cl(T(M)) is compact and f and T are continuous;

(b): wcl(T(M)) is weakly compact, f is weakly continuous and (f - T) is demiclosed at 0;

(c): T(M) is bounded, T is semicompact and f and T are continuous;

(d): T(M) is bounded, T is weakly semicompact, f is weakly continuous and (f - T) is demiclosed at 0.

Proof. Let {k _n } ⊆ (0, 1) such that k _n → 1 as n → ∞. For n ∈ ℕ, we define T _n : M → M by T _n x = k _n Tx + (1 - k _n )q for all x ∈ M. Since (f, T ) is a generalized $\mathcal{J}\mathcal{H}$-suboperator with order n₀, (f, T _n ) is a generalized $\mathcal{J}\mathcal{H}$-operator order n₀ for all n ∈ ℕ. Using inequality (3.7) it follows that

$$\begin{array}{ll}\hfill \parallel T_{n}x-T_{n}y\parallel & =k_n\parallel Tx-Ty\parallel \\ \le k_{n}max\left\{\parallel fx-fy\parallel ,d\left(fx,\left[q,Tx\right]\right),d\left(fy,\left[q,Ty\right]\right),d\left(fx,\left[q,Ty\right]\right),d\left(fy,\left[q,Tx\right]\right)\right\}\\ \le k_{n}max\left\{\parallel fx-fy\parallel ,\parallel fx-T_{n}x\parallel ,\parallel fy-T_{n}y\parallel ,\parallel fx-T_{n}y\parallel ,\parallel fy-T_{n}x\parallel \right\},\end{array}$$

for all x, y ∈ M. By Theorem 3.3, there exists x _n ∈ M such that x _n = fx _n = T _n x _n for every n ∈ ℕ.

(a): As cl(T(M)) is compact, there exists a subsequence {Tx _m } of {Tx _n } such that $\underset{m\to \infty }{lim}T{x}_m=y$ for some y ∈ M. By the definition of T _m , we get

$$\underset{m\to \infty }{lim}{x}_m=\underset{m\to \infty }{lim}{T}_m{x}_m=\underset{m\to \infty }{lim}\left(k_{m}T{x}_m+\left(1-k_m\right)q\right)=\underset{m\to \infty }{lim}T{x}_m=y.$$

Since f and T are continuous, y = fy = Ty that is y ∈ F(f, T) and then F(f, T) ≠ ∅.

(b): From weakly compact of wcl(T(M)) there exist a subsequence {x _m } of {x _n } in M converging weakly to y ∈ M as m → ∞. Since f is weakly continuous, fy = y that is $\underset{m\to \infty }{lim}\left(f{x}_m-T{x}_m\right)=0$. It follows from (f - T) is demiclosed at 0 and $\underset{m\to \infty }{lim}\left(f{x}_m-T{x}_m\right)=0$ that fy - Ty = 0. Therefore, y = fy = Ty that is F(f, T) ≠ ∅.

(c): Since T(M) is bounded, k _n → 1, and

$$\begin{array}{ll}\hfill \parallel x_n-T{x}_n\parallel & =\parallel T_n{x}_n-T{x}_n\parallel \\ =\parallel k_{n}T{x}_n+\left(1-k_n\right)q-T{x}_n\parallel \\ =\parallel \left(1-k_n\right)\left(q-T{x}_n\right)\parallel \\ \le \left(1-k_n\right)\left(\parallel q\parallel +\parallel T{x}_n\parallel \right)\end{array}$$

for all n ∈ ℕ, we get $\underset{m\to \infty }{lim}\left(x_n-T{x}_n\right)=0$. As T is semicompact, there exist a subsequence {x _m } of {x _n } in M such that $\underset{m\to \infty }{lim}{x}_m=y$ for some y ∈ M. By definition of T _m , we get

$$y=\underset{m\to \infty }{lim}{x}_m=\underset{m\to \infty }{lim}{T}_m{x}_m=\underset{m\to \infty }{lim}\left(k_{m}T{x}_m+\left(1-k_m\right)q\right)=\underset{m\to \infty }{lim}T{x}_m.$$

By the continuous of both f and T, we have y = fy = Ty. Therefore F(f, T) ≠ ∅.

(d): Similarly case (c), we have $\underset{m\to \infty }{lim}\left(x_n-T{x}_n\right)=0$. Since T is weakly semicompact, there exist a subsequence {x _m } of {x _n } in M such that converging weakly to y ∈ M as m → ∞. By weak continuity of f, we get fy = y. It follows from $\underset{m\to \infty }{lim}\left(f{x}_m-T{x}_m\right)=\underset{m\to \infty }{lim}\left(x_m-T{x}_m\right)=0$, x _m converging weakly to y, and f - T is demiclosed at 0 that (f - T)(y) = 0 which implies that fy = Ty. Therefore y = fy = Ty and hence y ∈ F(f, T).

□

Remark 3.7. We can replace assumption of f being q-affine by q ∈ F(f) and f(M) = M in Theorem 3.6.

If f is identity mapping in Theorem 3.6, then we get the following corollary.

Corollary 3.8. Let T be selfmaps on a q-starshaped subset M of a normed space X. Assume that for all x, y ∈ M,

$$\parallel Tx-Ty\parallel \le max\left\{\parallel x-y\parallel ,d\left(x,\left[q,Tx\right]\right),d\left(y,\left[q,Ty\right]\right),d\left(x,\left[q,Ty\right]\right),d\left(y,\left[q,Tx\right]\right)\right\}.$$

(3.8)

Then F(T) ≠ ∅ if one of the following conditions holds:

(a): cl(T(M)) is compact and T is continuous;

(b): wcl(T(M)) is weakly compact and (I - T) is demiclosed at 0, where I is identity on M;

(c): T(M) is bounded, T is semicompact and T is continuous;

(d): T(M) is bounded, T is weakly semicompact and (I - T) is demiclosed at 0, where I is identity on M.

4. Invariant approximations

In 1999, invariant approximations for noncommuting maps were considered by Shahzad [32]. As M is a subset of a normed space X and p ∈ X, let

$$\begin{array}{c}{B}_M\left(p\right):=\left\{x\in M:\parallel x-p\parallel =d\left(p,M\right)\right\},\\ C_M^f\left(p\right):=\left\{x\in M:fx\in B_M\left(p\right)\right\},\\ D_M^f\left(p\right):=B_M\left(p\right)\cap C_M^f\left(p\right),\end{array}$$

and

$$M_p:=\left\{x\in M:\parallel x\parallel \le 2\parallel p\parallel \right\}.$$

The set B _M (p) is called the set of best approximants to p ∈ X out of M. Let ${\mathcal{C}}_0$ denote the class of closed convex subsets M of X containing 0. It is known that B _M (p) is closed, convex, and contained in $M_p\in {\mathcal{C}}_0$.

Theorem 4.1. Let M be a subset of a normed space X, f and T be selfmaps of X with T(∂M ∩ M) ⊆ M, p ∈ F(f, T), B _M (p) be a closed q-starshaped. Assume that f(B _M (p)) = B _M (p), q ∈ F (f ), (f, T ) is a generalized$\mathcal{J}\mathcal{H}$-suboperator with order n₀on B _M (p), and for all x, y ∈ B _M (p) ∪ {p},

$$\parallel Tx-Ty\parallel \le \{\begin{array}{ll}\parallel fx-fp\parallel \hfill & ify=p;\hfill \\ \max \{\parallel fx-fy\parallel ,d(fx,[q,Tx]),d(fy,[q,Ty]),\hfill \\ d(fx,[q,Ty]),d(fy,[q,Tx])\}\hfill & ify\in B_M(p).\hfill \end{array}$$

(4.1)

If cl(T(B _M (p))) is compact, f and T are continuous on B _M (p), then F (f, T )∩B _M (p) ≠ ∅.

Proof. Let x ∈ B _M (p). It follows from ||kx + (1 - k)p - p)|| = k||x - p|| < d(p, M) for all k ∈ (0, 1) that {kx+(1 - k)p : k ∈ (0, 1)}∩M ≠ ∅ which implies that x ∈ ∂M ∩ M. So B _M (p) ⊆ ∂M ∩ M and hence T(B _M (p)) ⊆ T (∂M ∩ M ). As T (∂M ∩ M ) ⊆ M that T(B _M (p)) ⊆ M. Now the result follows from Theorem 3.6 (a) with M = B _M (p). Therefore, F(f, T) ∩ B _M (p) ≠ ∅. □

Theorem 4.2. Let M be a subset of a normed space X, f and T be selfmaps of X with T(∂M ∩ M) ⊆ M, p ∈ F(f, T), $C_M^f\left(p\right)$be a closed q-starshaped. Assume that$f\left(C_M^f\left(p\right)\right)=C_M^f\left(p\right)$, q ∈ F (f ), (f, T ) is a generalized$\mathcal{J}\mathcal{H}$ -suboperator with order n₀on$C_M^f\left(p\right)$, and for all$x,y\in C_M^f\left(p\right)\cup \left\{p\right\}$,

$$\parallel Tx-Ty\parallel \le \{\begin{array}{ll}\parallel fx-fp\parallel \hfill & ify=p;\hfill \\ \max \{\parallel fx-fy\parallel ,d(fx,[q,Tx]),d(fy,[q,Ty]),\hfill \\ d(fx,[q,Ty]),d(fy,[q,Tx])\}\hfill & ify\in C_M^f(p).\hfill \end{array}$$

(4.2)

If$cl\left(T\left(C_M^f\left(p\right)\right)\right)$is compact, f and T are continuous on$C_M^f\left(p\right)$, then F (f, T)∩B _M (p) ≠ ∅.

Proof. Let $x\in C_M^f\left(p\right)$. By definition of $C_M^f\left(p\right)$ and $f\left(C_M^f\left(p\right)\right)=C_M^f\left(p\right)$, we have $C_M^f\left(p\right)\subseteq B_M\left(p\right)$. Using the same argument in the proof of Theorem 4.1 shows that there exists x ∈ ∂M ∩ M. It follows from T(∂M ∩ M) ⊆ f(M) ∩ M that Tx ∈ f(M). Therefore, we can find a point z ∈ M such that Tx = fz. Thus $z\in C_M^f\left(p\right)$ which implies that $T\left(C_M^f\left(p\right)\right)\subseteq f\left(C_M^f\left(p\right)\right)=C_M^f\left(p\right)$. Now the result follows from Theorem 3.6 (a) with $M=B_M^f\left(p\right)$. Therefore, we have F (f, T) ∩ B _M (p) ≠ ∅. □

Theorem 4.3. Let M be a subset of a normed space X, f and T be selfmaps of X with T(∂M ∩ M) ⊆ M, p ∈ F(f, T), B _M (p) be a weakly closed and q-starshaped. Assume that f(B _M (p)) = B _M (p), q ∈ F (f), (f, T) is a generalized$\mathcal{J}\mathcal{H}$-suboperator with order n₀on B _M (p), and for all x, y ∈ B _M (p) ∪ {p},

$$\parallel Tx-Ty\parallel \le \{\begin{array}{ll}\parallel fx-fp\parallel \hfill & ify=p;\hfill \\ \max \{\parallel fx-fy\parallel ,d(fx,[q,Tx]),d(fy,[q,Ty]),\hfill \\ d(fx,[q,Ty]),d(fy,[q,Tx])\}\hfill & ify\in B_M(p).\hfill \end{array}$$

(4.3)

If wcl(T(B _M (p))) is weakly compact, f is weakly continuous on B _M (p) and (f - T) is demiclosed at 0, then F(f, T) ∩ B _M (p) ≠ ∅.

Proof. We use an argument similar to that in Theorem 4.1 and apply Theorem 3.6 (b) instead of Theorem 3.6 (a). □

Theorem 4.4. Let M be a subset of a normed space X, f and T be selfmaps of X with T(∂M ∩ M) ⊆ M, p ∈ F(f, T), $C_M^f\left(p\right)$be a weakly closed and q-starshaped. Assume that$f\left(C_M^f\left(p\right)\right)=C_M^f\left(p\right)$, q ∈ F (f), (f, T) is a generalized$\mathcal{J}\mathcal{H}$-suboperator with order n₀on$C_M^f\left(p\right)$, and for all$x,y\in C_M^f\left(p\right)\cup \left\{p\right\}$,

$$\parallel Tx-Ty\parallel \le \{\begin{array}{ll}\parallel fx-fp\parallel \hfill & ify=p;\hfill \\ \max \{\parallel fx-fy\parallel ,d(fx,[q,Tx]),d(fy,[q,Ty]),\hfill \\ d(fx,[q,Ty]),d(fy,[q,Tx])\}\hfill & ify\in C_M^f(p).\hfill \end{array}$$

(4.4)

If$wcl\left(T\left(C_M^f\left(p\right)\right)\right)$is weakly compact, f is weakly continuous on$C_M^f\left(p\right)$and (f - T) is demiclosed at 0, then F(f, T) ∩ B _M (p) ≠ ∅.

Proof. We use an argument similar to that in Theorem 4.2 and apply Theorem 3.6 (b) instead of Theorem 3.6 (a). □

Theorem 4.5. Let M be a subset of a normed space X, f and T be selfmaps of X, p ∈ F(f, T), $M\in {\mathcal{C}}_0$with T (M _p ) ⊆ f(M ) ⊆ M. Assume that ||fx - p|| = ||x - p|| for all x ∈ M and for all x, y ∈ M _p ∪ {p},

$$\parallel Tx-Ty\parallel \le \{\begin{array}{ll}\parallel fx-fp\parallel \hfill & ify=p;\hfill \\ \max \{\parallel fx-fy\parallel ,d(fx,[q,Tx]),d(fy,[q,Ty]),\hfill \\ d(fx,[q,Ty]),d(fy,[q,Tx])\}\hfill & ify\in M_p.\hfill \end{array}$$

(4.5)

If cl(f(M _p )) is compact, then B _M (p) is nonempty, closed, and convex and T (B _M (p)) ⊆ f(B _M (p)) ⊆ B _M (p). If in addition, for all x, y ∈ BM (p),

$$\parallel fx-fy\parallel \le max\left\{\parallel x-y\parallel ,d\left(x,\left[q,fx\right]\right),d\left(y,\left[q,fy\right]\right),d\left(x,\left[q,fy\right]\right),d\left(y,\left[q,fx\right]\right)\right\},$$

(4.6)

then F(f) ∩ B _M (p) ≠ ∅ and F(T) ∩ B _M (p) ≠ ∅. Moreover, F(f, T) ∩ B _M (p) ≠ ∅ if for some q ∈ B _M (p), f is q-affine and (f, T) is a generalized$\mathcal{J}\mathcal{H}$suboperator with order n on B _M (p).

Proof. Assume that p ∉ M. If u ∈ M\M _p , then ||u|| > 2||p||. Since 0 ∈ M, we get

$$\parallel x-p\parallel \ge \parallel x\parallel -\parallel p\parallel >\parallel p\parallel \ge d\left(p,M\right).$$

Thus α := d(p, M _p ) = d(p, M). As cl(f (M _p )) is compact and the norm is continuous that there exists z ∈ cl(f(M _p )) such that β := d(p, cl(f (M _p ))) = ||z - p||. So we have

$$d\left(p,cl\left(f\left(M_p\right)\right)\right)\le \parallel fy-p\parallel =\parallel y-p\parallel .$$

for all y ∈ M _p . Therefore, α = β and B _M (p) is nonempty closed and convex such that f(B _M (p)) ⊆ B _M (p). Next step, we show that T (B _M (p)) ⊆ f (B _M (p)). Suppose that w ∈ T(B _M (p)). It follows from T (B _M (p)) ⊆ T (M _p ) ⊆ f (M) that there exists w₁ ∈ M _p and w₂ ∈ M such that w = Tw₁ = fw₂. Using the condition (4.5), we have

$$\parallel w_2-p\parallel =\parallel f{w}_2-Tp\parallel =\parallel T{w}_1-Tp\parallel \le \parallel f{w}_1-fp\parallel =\parallel f{w}_1-p\parallel =\parallel w_1-p\parallel =d\left(p,M\right).$$

Thus, w₂ ∈ B _M (p) and w₁ ∈ f (B _M (p)) which implies that T (B _M (p)) ⊆ f (B _M (p)) ⊆ B _M (p). Now, suppose that f satisfies inequality (4.6) on B _M (p). Therefore, the condition (4.5) on M _p ∪ {p} implies that

$$\parallel Tx-Ty\parallel \le max\left\{\parallel x-y\parallel ,d\left(x,\left[q,Tx\right]\right),d\left(y,\left[q,Ty\right]\right),d\left(x,\left[q,Ty\right]\right),d\left(y,\left[q,Tx\right]\right)\right\},$$

(4.7)

for all x, y ∈ B _M (p). Since f (M _p ) is compact, f (B _M (p)) and T (B _M (p)) are compact. Moreover, f(B _M (p)) ⊆ B _M (p) and T (B _M (p)) ⊆ B _M (p). It follows from Corollary 3.8 that F(f) ∩ B _M (p) ≠ ∅ and F(T) ∩ BM (p) ≠ ∅. Finally, we follow from Theorem 3.6 by replacing M with B _M (p). □

Theorem 4.6. Let M be a subset of a normed space X, f and T be selfmaps of X, p ∈ F(f, T), $M\in {\mathcal{C}}_0$with T (M _p ) ⊆ f (M ) ⊆ M. Assume that ||fx - p|| = ||x - p|| for all x ∈ M and for all x, y ∈ M _p ∪ {p},

$$\parallel Tx-Ty\parallel \le \{\begin{array}{ll}\parallel fx-fp\parallel \hfill & ify=p;\hfill \\ \max \{\parallel fx-fy\parallel ,d(fx,[q,Tx]),d(fy,[q,Ty]),\hfill \\ d(fx,[q,Ty]),d(fy,[q,Tx])\}\hfill & ify\in M_p.\hfill \end{array}$$

(4.8)

If cl(T(M _p )) is compact, then B _M (p) is nonempty, closed, convex, and T (B _M (p)) ⊆ f(B _M (p)) ⊆ B _M (p). If in addition, for all x, y ∈ B _M (p),

$$\parallel fx-fy\parallel \le max\left\{\parallel x-y\parallel ,d\left(x,\left[q,fx\right]\right),d\left(y,\left[q,fy\right]\right),d\left(x,\left[q,fy\right]\right),d\left(y,\left[q,fx\right]\right)\right\},$$

(4.9)

then F(T) ∩ B _M (p) ≠ ∅. Moreover, F(f, T) ∩ B _M (p) ≠ ∅ if for some q ∈ B _M (p), f is q-affine and (f, T) is a generalized$\mathcal{J}\mathcal{H}$suboperator with order n on B _M (p).

Proof. We can obtain the result by using an argument similar to that in Theorem 4.5.

□

Theorem 4.7. Let M be a subset of a Banach space X, f and T be selfmaps of X, p ∈ F(f, T), $M\in {\mathcal{C}}_0$with T (Mp) ⊆ f (M ) ⊆ M. Assume that ||fx - p|| = ||x - p|| for all x ∈ M and for all x, y ∈ M _p ∪ {p},

$$\parallel Tx-Ty\parallel \le \{\begin{array}{ll}\parallel fx-fp\parallel \hfill & ify=p;\hfill \\ \max \{\parallel fx-fy\parallel ,d(fx,[q,Tx]),d(fy,[q,Ty]),\hfill \\ d(fx,[q,Ty]),d(fy,[q,Tx])\}\hfill & ify\in M_p.\hfill \end{array}$$

(4.10)

If wcl(f(M _p )) is weakly compact and (f - T) is demiclosed at 0, then B _M (p) is nonempty, (weakly) closed, and convex and T(B _M (p)) ⊆ f(B _M (p)) ⊆ B _M (p). If, in addition, for all x, y ∈ B _M (p),

$$\parallel fx-fy\parallel \le max\left\{\parallel x-y\parallel ,d\left(x,\left[q,fx\right]\right),d\left(y,\left[q,fy\right]\right),d\left(x,\left[q,fy\right]\right),d\left(y,\left[q,fx\right]\right)\right\},$$

(4.11)

then F(f) ∩ B _M (p) ≠ ∅ and F(T) ∩ B _M (p) ≠ ∅. Moreover, F(f, T) ∩ B _M (p) ≠ ∅ if for some q ∈ B _M (p), f is q-affine, weakly continuous on B _M (p) and (f, T) is a generalized$\mathcal{J}\mathcal{H}$suboperator with order n on B _M (p).

Proof. To obtain the result, we use an argument similar to that in Theorem 4.5 and apply Theorem 3.6 (b) instead of Theorem 3.6(a), respectively. Finally, we use Lemma 5.5 of Singh et al. [33] with f(x) = ||x - p|| and C = wcl(T(M _p )) to show that there exists z ∈ C such that d(p, C) = ||z - p||. □

Theorem 4.8. Let M be a subset of a Banach space X, f and T be selfmaps of X, p ∈ F(f, T), $M\in {\mathcal{C}}_0$with T (M _p ) ⊆ f (M ) ⊆ M. Assume that ||fx - p|| = ||x - p|| for all x ∈ M and for all x, y ∈ M _p ∪ {p},

$$\parallel Tx-Ty\parallel \le \{\begin{array}{ll}\parallel fx-fp\parallel \hfill & ify=p;\hfill \\ \max \{\parallel fx-fy\parallel ,d(fx,[q,Tx]),d(fy,[q,Ty]),\hfill \\ d(fx,[q,Ty]),d(fy,[q,Tx])\}\hfill & ify\in M_p.\hfill \end{array}$$

(4.12)

If wcl(f(M _p )) is weakly compact and (f - T) is demiclosed at 0, then B _M (p) is nonempty, (weakly) closed, and convex and T(B _M (p)) ⊆ f (BM (p)) ⊆ B _M (p). If in addition, for all x, y ∈ B _M (p),

$$\parallel Tx-Ty\parallel \le max\left\{\parallel x-y\parallel ,d\left(x,\left[q,Tx\right]\right),d\left(y,\left[q,Ty\right]\right),d\left(x,\left[q,Ty\right]\right),d\left(y,\left[q,Tx\right]\right)\right\},$$

(4.13)

then F(T) ∩ B _M (p) ≠ ∅. Moreover, F(f, T) ∩ B _M (p) ≠ ∅ if for some q ∈ B _M (p), f is q-affine, weakly continuous on B _M (p) and (f, T) is a generalized$\mathcal{J}\mathcal{H}$suboperator with order n on B _M (p).

Proof. We can obtain the result using an argument similar to that in Theorem 4.7. □