On T-Stability of the Picard Iteration for Generalized φ-Contraction Mappings

Abstract

We introduce some results on T-stability of the Picard iteration for φ-contraction and generalized φ-contraction mappings on metric spaces.

1. Introduction

It is known that iteration methods are numerical procedures which compute a sequence of gradually accurate iterates to approximate the solution of a class of problems. Such methods are useful tools of applied mathematics for solving real life problems ranging from economics and finance or biology to transportation, network analysis, or optimization. An iteration method is considered to be sound if possesses some qualitative properties such as convergence and stability. That is why several scientists paid and still pay attention to the qualitative study of iteration methods; please, see [1–7].

There are some papers about the stability or different iteration methods. In [3], Harder and Hicks studied the stability of Picard iteration for several contractivity conditions [7], while in [6] Rhoades introduced a contractivity condition independent of that in [7] to obtain stability results for Mann, Kirk, or Massa iteration processes. Meantime, Bosede and Rhoades [2] introduced stability results of Picard and Mann iteration for a general class of functions; also, see [4], while Rezapour et al. [5] studied the almost stability of Mann iteration for 𝜑-contraction mappings and the stability of Picard iteration for mappings satisfying a contractive condition of integral type. In the present paper, we introduce our new results on stability of Picard iteration for 𝜑-contraction and generalized 𝜑-contraction mappings on metric spaces.

2. Previous Notation and Definitions

Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋 a map and 𝑥_𝑛+1=𝑓(𝑇,𝑥_𝑛) an iteration procedure. Suppose that 𝑇 has at least one fixed point and that sequence {𝑥_𝑛} converges to a fixed point 𝑥^∗∈𝑋. We denote the set of fixed points of mapping 𝑇 by 𝐹_𝑇. Let {𝑦_𝑛} be an arbitrary sequence in 𝑋 and 𝜖_𝑛=𝑑(𝑦_𝑛+1,𝑓(𝑇,𝑦_𝑛)).

If lim_𝑛→∞𝜖_𝑛=0 implies that lim_𝑛→∞𝑦_𝑛=𝑥^∗, then the iteration procedure 𝑥_𝑛+1=𝑓(𝑇,𝑥_𝑛) is said to be T-stable (e.g., [1, 6]).

If {𝑦_𝑛} is a bounded sequence and lim_𝑛→∞𝜖_𝑛=0 implies that lim_𝑛→∞𝑦_𝑛=𝑥^∗, then the iteration procedure 𝑥_𝑛+1=𝑓(𝑇,𝑥_𝑛) is said to be boundedly T-stable.

In most papers on T-stability, some authors consider the notion of boundedly T-stability instead of T-stability. Here, we mention the Picard iteration methods. Let 𝑥₀∈𝑋. The Picard iteration is given by 𝑥_𝑛+1=𝑇𝑥_𝑛.

The following example illustrates that the notion of T-Stability is different from the notion of boundedly T-stability.

Example 2.1. Consider mapping 𝑇∶[0,∞)→[0,∞) given by 𝑇𝑥=(1/2)(𝑥+1) whenever 𝑥∈[0,1] and 𝑇𝑥=𝑥+1 whenever 𝑥>1. Put 𝑦_𝑛=𝑛+(1/𝑛) for all 𝑛≥1. Note that {𝑦_𝑛} is unbounded, while lim_𝑛→∞|𝑦_𝑛+1−𝑇𝑦_𝑛|=0.

3. Main Results

Now, we are ready to state and prove our main results.

Definition 3.1 (see [1]). A function 𝜑∶[0,∞)→[0,∞) is said to be comparison if 𝜑 is increasing and 𝜑^𝑛(𝑡) converges to 0 for all 𝑡≥0.

Note that if 𝜑 is comparison, then 𝜑(𝑡)<𝑡 for all 𝑡>0 and 𝜑(0)=0.

Definition 3.2 (see [1]). Let (𝑋,𝑑) be a metric space, and let 𝜑∶[0,∞)→[0,∞) be a comparison function. A mapping 𝑇∶𝑋→𝑋 is called 𝜑-contraction whenever 𝑑(𝑇𝑥,𝑇𝑦)≤𝜑(𝑑(𝑥,𝑦)),(3.1) for all 𝑥,𝑦∈𝑋.

We say that 𝜑∶[0,∞)→[0,∞) is a subadditive comparison function whenever 𝜑 is comparison and 𝜑(𝑡+𝑠)≤𝜑(𝑡)+𝜑(𝑠) for all 𝑡,𝑠∈[0,∞).

There are many subadditive comparison mappings.

For example, if we consider 𝜆<1 and 𝑔∶[0,∞)→[0,𝜆) is a decreasing function, then 𝜑(𝑡)=∫^𝑡0𝑔(𝑥)𝑑𝑥 is a comparison function. In fact, 𝜑 is increasing because 𝑔>0. Also, 𝜑(𝑡)<min{𝑡,𝜆}. Hence, 𝜑^𝑛(𝑡) converges to 0 for all 𝑡≥0. Since 𝑔 is decreasing, we have 𝜑(𝑢+𝑣)≤=^0𝑢+𝑣𝑔(𝑥)𝑑𝑥=^𝑢0𝑔(𝑥)𝑑𝑥+^{𝑢𝑢+𝑣}𝑔(𝑥)𝑑𝑥_𝑢0𝑔(𝑥)𝑑𝑥+^𝑣0𝑔(𝑥)𝑑𝑥=𝜑(𝑢)+𝜑(𝑣),(3.2) for all 𝑢,𝑣≥0.

In particular, if we consider 𝑔(𝑡)=𝜆𝑒^−𝑡, it follows that 𝜑(𝑡)=∫^𝑡0𝑔(𝑥)𝑑𝑥 is a subadditive comparison function.

Theorem 3.3. Let (𝑋,𝑑) be a complete metric space, and, 𝜑∶[0,∞)→[0,∞) a subadditive comparison function. If 𝑇∶𝑋→𝑋 is a 𝜑-contraction, then the Picard iteration is T-stable.

Proof. By using Theorem 2.7 in [1], we conclude that 𝑇 has a unique fixed point 𝑞.
Let {𝑦_𝑛} be a sequence in 𝑋 with lim_𝑛→∞𝑑(𝑦_𝑛+1,𝑇𝑦_𝑛)=0.
First, we show that {𝑦_𝑛} is bounded. If {𝑦_𝑛} is not bounded, then there exist subsequence {𝑧_𝑛} of {𝑦_𝑛} for which 𝑑(𝑧_𝑛,𝑞)≥𝑛. Since lim_𝑛→∞𝑑(𝑦_𝑛+1,𝑇𝑦_𝑛)=0, we can take a subsequence {𝑥_𝑛} of {𝑧_𝑛} such that 𝑑(𝑥_𝑛+1,𝑇𝑥_𝑛)≤1/𝑛². Now, we have 𝑑𝑇𝑥_𝑛,𝑞≤≤𝜑1+𝑑1𝑥_𝑛,𝑞≤𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛−1+𝜑²𝑑𝑥_𝑛−1,𝑞2²+⋯+1𝑛²+𝜑^𝑛𝑑𝑥₁,𝑞≤^∞_𝑖=11𝑖²+𝑑𝑥₁,𝑞.(3.3) Thus, {𝑇𝑥_𝑛} is bounded and so is {𝑥_𝑛}. This is a contradiction. Therefore {𝑦_𝑛} is bounded.
Now, choose 𝑀>0 such that 𝑑(𝑦_𝑛,𝑞)<𝑀 for all 𝑛≥1. For each 𝜀>0 there exist natural numbers 𝑝₀ and 𝑁 such that 𝜑^𝑝0(𝑀)<𝜀,𝑑𝑦_𝑛+1,𝑇𝑦_𝑛<𝜀,(3.4) for all 𝑛≥𝑁. But we have 𝑑𝑑𝑦𝑦_𝑛+2,𝑞≤𝑑𝑦_𝑛+2,𝑇𝑦_𝑛+1+𝑑𝑇𝑦_𝑛+1,𝑞≤𝑑𝑦_𝑛+2,𝑇𝑦_𝑛+1+𝜑𝑑𝑦_𝑛+1,𝑞_𝑛+3,,𝑞≤≤𝑑𝑑𝑦𝑦_𝑛+3𝑛+3,𝑇𝑦_𝑛+2+𝑑𝑇𝑦_𝑛+2,𝑞≤𝑑𝑦_𝑛+3,𝑇𝑦_𝑛+2+𝜑𝑑𝑦_𝑛+2,𝑞,𝑇𝑦_𝑛+2+𝜑𝑑𝑦_𝑛+2,𝑇𝑦_𝑛+1+𝜑²𝑑𝑦_𝑛+1,𝑞.(3.5) By continuing this process, we obtain 𝑑𝑦_𝑛+𝑝0+1,𝑞≤𝑑+⋯𝑦+𝜑_𝑛+𝑝0+1,𝑇𝑦_{𝑛+𝑝0𝑝0−1}+𝜑𝑑𝑦_𝑛+𝑝0,𝑇𝑦_{𝑛+𝑝0−1}<𝑑𝑦_𝑛+𝑝0+1𝑑𝑦_𝑛+1,𝑇𝑦_𝑛+𝜑^𝑝0𝑑𝑦_𝑛,𝑞+⋯+𝜑,𝑇𝑦_{𝑛+𝑝0𝑝0−1}+𝜑𝑑𝑦_𝑛+𝑝0,𝑇𝑦_{𝑛+𝑝0−1}𝑑𝑦_𝑛+1,𝑇𝑦_𝑛+𝜀.(3.6) Hence, lim_𝑛→∞sup𝑑(𝑦_𝑛,𝑞)≤𝜀. Since 𝜀>0 was arbitrary, lim_𝑛→∞𝑑(𝑦_𝑛,𝑞)=0.

Definition 3.4 (see [1]). A function 𝜑∶ℝ⁵⁺→ℝ₊ is called (5-dimensional) comparison function whenever 𝜑(𝑢)≤𝜑(𝑣), for each 𝑢,𝑣∈ℝ⁵⁺ with 𝑢≤𝑣, and the function 𝜓∶ℝ₊⟶ℝ₊,𝜓(𝑡)=𝜑(𝑡,𝑡,𝑡,𝑡,𝑡)(3.7) satisfies lim_𝑛→∞𝜓^𝑛(𝑡)=0, for all 𝑡>0.
Note that 𝜓 is a comparison function, while the following are 5-dimensional comparison functions:(i)𝜑(𝑡)=𝑎max{𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅} for each 𝑡=(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅), where 𝑎∈[0,1),(ii)𝜑(𝑡)=𝑎max{𝑡₁,𝑡₂,𝑡₃,𝑡₄,(1/2)(𝑡₄+𝑡₅)}, 𝑎∈[0,1),(iii)𝜑(𝑡)=𝑎𝑡₁+𝑏(𝑡₂+𝑡₃), 𝑎,𝑏∈ℝ₊ with 𝑎+2𝑏<1,(iv)𝜑(𝑡)=𝑎max{𝑡₂,𝑡₃}, 𝑎∈(0,1).
In the previous four examples, function 𝜓 given by (3.7) is a subadditive comparison function.

Definition 3.5. Let (𝑋,𝑑) be a metric space, and, 𝜑∶ℝ⁵⁺→ℝ₊, a 5-dimensional comparison function. A mapping 𝑇∶𝑋→𝑋 is called generalized 𝜑-contraction whenever 𝑑(𝑇𝑥,𝑇𝑦)≤𝜑(𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦),𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥)),(3.8) for all 𝑥,𝑦 in 𝑋.

In the sequel, we will use functions 𝜑 such that 𝜓 is subadditive.

Lemma 3.6. Let (𝑋,𝑑) be a metric space, and let 𝑇∶𝑋→𝑋 be a generalized 𝜑-contraction map. Suppose {𝑦_𝑛} is a bounded sequence in 𝑋 such that lim_𝑛→∞𝑑(𝑦_𝑛+1,𝑇𝑦_𝑛)=0. Let 𝑝_𝑛 be the diameter of the set 𝐴_𝑛={𝑦_𝑖}_𝑖≥𝑛⋃{𝑇𝑦_𝑖}_𝑖≥𝑛. Then, lim_𝑛→∞𝑝_𝑛=0. In particular, lim_𝑛→∞𝑑(𝑦_𝑛,𝑇𝑦_𝑛)=0.

Proof. By using definition of 𝑇, for each 𝑛 and 𝑖,𝑗≥𝑛 we have 𝑑𝑇𝑦_𝑖,𝑇𝑦_𝑗≤𝜑𝑑𝑦_𝑖,𝑦_𝑗,𝑑𝑦_𝑖,𝑇𝑦_𝑖,𝑑𝑦_𝑗,𝑇𝑦_𝑗,𝑑𝑦_𝑖,𝑇𝑦_𝑗,𝑑𝑦_𝑗,𝑇𝑦_𝑖≤𝜓𝑝_𝑛.(3.9) Let 𝜖_𝑖=𝑑(𝑦_𝑖+1,𝑇𝑦_𝑖). Then 𝑑𝑦_𝑖,𝑦_𝑗≤≤𝑑𝜖_𝑖−1𝑦_𝑖,𝑇𝑦_𝑖−1+𝑑𝑇𝑦_𝑖−1,𝑇𝑦_𝑗−1+𝑑𝑇𝑦_𝑗−1,𝑦_𝑗𝑑𝑦_𝑖+𝜓𝑝_𝑛−1+𝜖_𝑗−1,,𝑇𝑦_𝑗≤𝑑𝑦_𝑖,𝑇𝑦_𝑖−1+𝑑𝑇𝑦_𝑖−1,𝑇𝑦_𝑗≤𝜖_𝑖−1+𝜓𝑝_𝑛−1.(3.10)
Let 𝑎_𝑛=sup_𝑖≥𝑛2𝜖_𝑖. It is easy to see that lim_𝑛→∞𝑎_𝑛=0, and we have 𝑝_𝑛≤𝑎_𝑛+𝜓𝑝_𝑛−1.(3.11)
By using (3.11), we observe that 𝜓𝑝_𝑛≤𝜓𝑎_𝑛+𝜓²𝑝_𝑛−1.(3.12) Since {𝑦_𝑛} is bounded, {𝑇𝑦_𝑛} so is. Choose 𝑀>0 such that 𝑝_𝑛≤𝑀 for all 𝑛≥1. Since 𝜓 is comparison, for each 𝜀>0 there exists a natural number 𝑘₀ such that 𝜓^𝑘0(𝑀)<𝜀/2. But, for each 𝑛≥1 we obtain 𝜓𝑝_𝑛+1≤𝜓𝑎_𝑛+1+𝜓²𝑝_𝑛≤𝜓𝑎_𝑛+1+𝜓²𝑎_𝑛+𝜓³𝑝_𝑛−1.(3.13) Hence, 𝜓𝑝_𝑛+2≤𝜓𝑎_𝑛+2+𝜓²𝑎_𝑛+1+𝜓³𝑎_𝑛+𝜓⁴𝑝_𝑛−1.(3.14) Since 𝜓(𝑡)<𝑡, for all 𝑡>0, and 𝜓 is increasing, then ∑^𝑘+1𝑖=1𝜓^𝑖(𝑎_{𝑛−𝑖+3})→0, for all natural numbers 𝑘. Thus by continuing these relations, for each 𝑘≥𝑘₀ we have 𝜓𝑝_𝑛+𝑘≤^𝑘+1_𝑖=1𝜓^𝑖𝑎_{𝑛−𝑖+3}+𝜓^𝑘+2𝑝_𝑛−1≤^𝑘+1_𝑖=1𝜓^𝑖𝑎_{𝑛−𝑖+3}+𝜀.(3.15) It implies that lim_𝑛→∞sup𝜓(𝑝_𝑛)≤𝜀. Since 𝜀>0 was arbitrary, lim_𝑛→∞sup𝜓(𝑝_𝑛)=0. Therefore by using (3.11), lim_𝑛→∞𝑝_𝑛=0.

Theorem 3.7. Let (𝑋,𝑑) be a metric space, let 𝑇∶𝑋→𝑋 be a generalized 𝜑-contraction map, and let 𝐹_𝑇={𝑞}. Then the Picard iteration is boundedly T-stable.

Proof. Let {𝑦_𝑛} be a bounded sequence in 𝑋 such that lim_𝑛→∞𝑑(𝑦_𝑛+1,𝑇𝑦_𝑛)=0. Choose 𝑀>0 such that 𝑑(𝑦_𝑛,𝑞)<𝑀 for all 𝑛≥1. Observe that 𝑑𝑇𝑦_𝑛,𝑞≤≤𝜑𝜓𝑑ma𝑦x_𝑛,𝑞𝑑,𝑦𝑑_𝑛𝑦_𝑛,𝑇𝑦_𝑛,0,𝑑𝑦_𝑛,𝑞,𝑑𝑞,𝑇𝑦_𝑛,𝑞,𝑑𝑦_𝑛,𝑇𝑦_𝑛,𝑑𝑞,𝑇𝑦_𝑛.(3.16) If max{𝑑(𝑦_𝑛,𝑞),𝑑(𝑦_𝑛,𝑇𝑦_𝑛),𝑑(𝑞,𝑇𝑦_𝑛)}=𝑑(𝑇𝑦_𝑛,𝑞), then 𝑑(𝑇𝑦_𝑛,𝑞)=0.
Without loss of generality, suppose that the last equality does not hold. Therefore, we get 𝑑𝑇𝑦_𝑛,𝑞≤≤𝜓𝜓𝑑max𝑦_𝑛𝑑𝑦_𝑛,𝑞,𝑑𝑦_𝑛,𝑇𝑦_𝑛,𝑞+𝑑𝑦_𝑛,𝑇𝑦_𝑛≤𝜓𝑑𝑦_𝑛,𝑞+𝜓𝑑𝑦_𝑛,𝑇𝑦_𝑛.(3.17)
For any given 𝜀>0, choose 𝑝₀∈ℕ such that 𝜓^𝑝0(𝑀)<𝜀.(3.18)
Now, for each 𝑛≥1 we have 𝑑𝑇𝑦_𝑛+1,𝑞≤≤𝜓𝜓𝑑𝑑𝑦𝑦_𝑛+1𝑛+1,𝑇𝑦_𝑛+1+𝜓𝑑𝑦_𝑛+1,𝑞≤𝜓𝑑𝑦_𝑛+1,𝑇𝑦_𝑛+1+𝜓𝑑𝑦_𝑛+1,𝑇𝑦_𝑛+𝑑𝑇𝑦_𝑛,𝑞,𝑇𝑦_𝑛+1+𝜓𝑑𝑦_𝑛+1,𝑇𝑦_𝑛+𝜓²𝑑𝑦_𝑛,𝑇𝑦_𝑛+𝜓²𝑑𝑦_𝑛,𝑞.(3.19)
Similarly 𝑑𝑇𝑦_𝑛+2,𝑞≤≤𝜓𝜓𝑑𝑑𝑦𝑦_𝑛+2𝑛+2,𝑇𝑦_𝑛+2+𝜓𝑑𝑦_𝑛+2,𝑞≤𝜓𝑑𝑦_𝑛+2,𝑇𝑦_𝑛+2+𝜓𝑑𝑦_𝑛+2,𝑇𝑦_𝑛+1+𝑑𝑇𝑦_𝑛+1,𝑞+𝜓₂,𝑇𝑦_𝑛+2+𝜓𝑑𝑦_𝑛+2,𝑇𝑦_𝑛+1+𝜓²𝑑𝑦_𝑛+1,𝑇𝑦_𝑛+1𝑑𝑦_𝑛+1,𝑇𝑦_𝑛+𝜓³𝑑𝑦_𝑛,𝑇𝑦_𝑛+𝜓³𝑑𝑦_𝑛,𝑞.(3.20)
Now for each 𝑝≥𝑝₀ we obtain 𝑑𝑇𝑦_𝑛+𝑝,𝑞≤^𝑝_𝑖=1𝜓^{𝑝−𝑖+1}𝑑𝑦_𝑛+𝑖,𝑇𝑦_𝑛+𝑖+^𝑝_𝑖=1+𝜓_𝑝+1𝜓^{𝑝−𝑖+1}𝑑𝑦_𝑛+𝑖,𝑇𝑦_{𝑛+𝑖−1}≤𝑑𝑦_𝑛,𝑞+𝜓^𝑝+1𝑑𝑦_𝑛,𝑇𝑦_𝑛_𝑝_𝑖=1𝜓^{𝑝−𝑖+1}𝑑𝑦_𝑛+𝑖,𝑇𝑦_𝑛+𝑖+^𝑝_𝑖=1𝜓^{𝑝−𝑖+1}𝑑𝑦_𝑛+𝑖,𝑇𝑦_{𝑛+𝑖−1}+𝜓^𝑝+1𝑑𝑦_𝑛,𝑇𝑦_𝑛+𝜀.(3.21)
If 𝑛→∞, then by a similar method in Lemma 3.6, lim_𝑛→∞sup𝑑(𝑇𝑦_𝑛,𝑞)≤𝜀. Since 𝜀>0 is arbitrary, lim_𝑛→∞𝑑(𝑇𝑦_𝑛,𝑞)=0.
Finally note that the inequality 𝑑𝑦_𝑛,𝑞≤𝑑𝑦_𝑛,𝑇𝑦_𝑛+𝑑𝑇𝑦_𝑛,𝑞(3.22) implies that lim_𝑛→∞𝑑(𝑦_𝑛,𝑞)=0.
The proof is complete.

Remark 3.8. Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be a mapping for which there exists ℎ∈[0,1) satisfying 𝑑(𝑇𝑥,𝑇𝑦)≤ℎmax{𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦),𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥)},(3.23) for all 𝑥,𝑦 in 𝑋. If we define 𝜑(𝑡)=ℎmax{𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅}, then by using Theorem 3.7, the Picard iteration is boundedly T-stable. Consider that some contractive conditions are special cases of (3.8), and, for each of those, the Picard iteration is boundedly T-stable. For example, Theorem 1 in [6] and Theorems 1 and 2 in [3] are special cases of Theorem 3.7.