Fixed point results for weakly compatible mappings in cone metric spaces and C*-algebra-valued metric spaces: Applications to functional equations

Introduction

The study of common fixed points for pairs of commuting mappings was initiated by Jungck [1] in 1976, who introduced the concept of compatibility to weaken the commutativity condition. This breakthrough led to extensive research in metric fixed point theory. The notion of weak compatibility, introduced by Jungck and Rhoades [2] in 1998, further generalized these ideas and has become a fundamental concept in modern fixed-point theory. The generalization of metric spaces to cone metric spaces by Huang and Zhang [3] in 2007 opened new avenues for research. In these spaces, the distance between two points is measured in an ordered Banach space rather than in the real line, providing a framework for multi-criteria distance measurements. Subsequently, Ma et al. [4] introduced C*-algebra-valued metric spaces in 2014, where distances take values in C*-algebras, offering applications in quantum mechanics and operator theory.

Research Motivation

While numerous fixed point results exist for single mappings in these generalized metric spaces, the theory of common fixed points for pairs of mappings, particularly under weak compatibility conditions, remains less developed. This paper addresses this gap by: 1. Establishing common fixed point theorems for weakly compatible mappings in cone metric spaces 2. Developing analogous results in C*-algebra-valued metric spaces 3. Investigating rational and integral-type contractive conditions 4. Providing applications to functional equations and dynamic programming

Structure of the Paper

Section 2 provides essential preliminaries and definitions. Section 3 presents main results for weakly compatible mappings in cone metric spaces. Section 4 develops theorems for C*-algebra-valued spaces. Section 5 discusses applications to functional equations, and Section 6 concludes with future research directions.

Preliminaries 2.1 Cone Metric Spaces

Definition 2.1 [3]: Let 𝐸be a real Banach space. A subset 𝑃 ⊆ 𝐸is called a cone if:  (i) 𝑃is closed, non-empty, and 𝑃 ≠ {0}  (ii) 𝑎, 𝑏 ∈ ℝ + , 𝑥, 𝑦 ∈ 𝑃implies 𝑎𝑥 + 𝑏𝑦 ∈ 𝑃  (iii) 𝑃 ∩ (-𝑃) = {0}

The partial ordering ⪯with respect to 𝑃is defined by: 𝑥 ⪯ 𝑦 iff 𝑦 -𝑥 ∈ 𝑃. We write 𝑥 ≪ 𝑦if 𝑦 -𝑥 ∈ int(𝑃). Definition 2.2 [3]: A cone 𝑃is normal if there exists 𝐾 ≥ 1such that for all 𝑥, 𝑦 ∈ 𝐸: 0 ⪯ 𝑥 ⪯ 𝑦 ⟹ ∥ 𝑥 ∥≤ 𝐾 ∥ 𝑦 ∥ Definition 2.3 [3]: Let 𝑋be a non-empty set. A mapping 𝑑: 𝑋 × 𝑋 → 𝐸is a cone metric if:  (CM1) 0 ⪯ 𝑑(𝑥, 𝑦)and 𝑑(𝑥, 𝑦) = 0iff 𝑥 = 𝑦  (CM2) 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥)  (CM3) 𝑑(𝑥, 𝑦) ⪯ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦) Definition 2.4: A sequence {𝑥 𝑛 }in cone metric space (𝑋 , 𝑑)converges to 𝑥if for every 𝑐 ∈ 𝐸with 0 ≪ 𝑐, there exists 𝑁such that 𝑑(𝑥 𝑛 , 𝑥) ≪ 𝑐for all 𝑛 ≥ 𝑁. Lemma 2.1 [5]: Let 𝑃be a normal cone with normal constant 𝐾. Then:  If 𝑎 ⪯ 𝜆𝑎where 0 ≤ 𝜆 < 1, then 𝑎 = 0  If 0 ⪯ 𝑢 𝑛 and 𝑢 𝑛 → 0, then for every 𝑐 ∈ int(𝑃), there exists 𝑁such that 𝑢 𝑛 ≪ 𝑐for 𝑛 ≥ 𝑁 2.2 C*-Algebra-Valued Metric Spaces Definition 2.5 [4]: A C*-algebra 𝒜is a complex Banach algebra with involution * satisfying:

Definition 2.6 [4]: An element 𝑎 ∈ 𝒜is positive (written 𝑎 ≥ 0) if 𝑎 = 𝑎 * and 𝜎(𝑎) ⊆ [0, ∞). Definition 2.7 [4]: A C*-algebra-valued metric on 𝑋is a mapping 𝑑: 𝑋 × 𝑋 → 𝒜 + satisfying:  (CA1) 𝑑(𝑥, 𝑦) = 0iff 𝑥 = 𝑦  (CA2) 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥)  (CA3) 𝑑(𝑥, 𝑧) ≤ 𝑑(𝑥, 𝑦) + 𝑑(𝑦, 𝑧)

Weak Compatibility

Definition 2.8 [2]: Mappings 𝑓, 𝑔: 𝑋 → 𝑋are weakly compatible if they commute at coincidence points, i.e., if 𝑓𝑥 = 𝑔𝑥for some 𝑥 ∈ 𝑋, then 𝑓𝑔𝑥 = 𝑔𝑓𝑥. Definition 2.9 [6]: Mappings 𝑓, 𝑔: 𝑋 → 𝑋are occasionally weakly compatible (OWC) if there exists a point 𝑥 ∈ 𝑋such that 𝑓𝑥 = 𝑔𝑥 and 𝑓𝑔𝑥 = 𝑔𝑓𝑥. Remark 2.1: Commutativity ⟹ Compatibility ⟹ Weak compatibility ⟹ Occasional weak compatibility. For 𝑚 > 𝑛:

Main Results in Cone Metric

By normality and Lemma 2.1, {𝑥 𝑛 }is Cauchy, converging to some 𝑧 ∈ 𝑋.

Step 2: The subsequences {𝑓𝑥 2𝑛 }and {𝑔𝑥 2𝑛+1 }both converge to 𝑧.

Step 3: We prove 𝑓𝑧 = 𝑧 and 𝑔𝑧 = 𝑧. From the contraction:

Since 𝛾 < 1, by Lemma 2.1(i), 𝑑(𝑧, 𝑔𝑧) = 0, thus 𝑔𝑧 = 𝑧.

Step 4: Since 𝑓𝑧 = 𝑔𝑧 = 𝑧, by weak compatibility, 𝑓𝑔𝑧 = 𝑔𝑓𝑧. But 𝑓𝑧 = 𝑧implies 𝑓𝑔𝑧 = 𝑓𝑧 = 𝑧 and 𝑔𝑓𝑧 = 𝑔𝑧 = 𝑧. Thus 𝑧is a common fixed point.

Step 5: Uniqueness follows by assuming two common fixed points and applying the contraction condition.

Corollary 3.1:

If 𝑓 = 𝑔in Theorem 3.1, we obtain a fixed point theorem for a single mapping with 𝛼 + 𝛽 + 2𝛾 < 1. . The condition becomes Φ(∥ 𝑑(𝑓𝑥, 𝑔𝑦) ∥) ≤ 𝛼Φ(∥ 𝑑(𝑥, 𝑦) ∥).

Rational Expression

Since Φis strictly increasing, Φ -1 exists. Thus:

Where the last inequality uses monotonicity of Φ. The result follows from Theorem 3. then 𝑓and 𝑔have a unique common fixed point.

Proof: For sequences defined as before:

∥ 𝑑(𝑥 𝑛+1 , 𝑥 𝑛 ) ∥≤∥ 𝜓(𝑑(𝑥 𝑛 , 𝑥 𝑛-1 )) ∥<∥ 𝑑(𝑥 𝑛 , 𝑥 𝑛-1 ) ∥

Thus {∥ 𝑑(𝑥 𝑛 , 𝑥 𝑛+1 ) ∥}is strictly decreasing and bounded below, converging to some 𝑟 ≥ 0.

𝑟 ≤∥ 𝜓(𝑑(𝑥 𝑛 , 𝑥 𝑛-1 )) ∥<∥ 𝑑(𝑥 𝑛 , 𝑥 𝑛-1 ) ∥→ 𝑟 yielding a contradiction. Thus 𝑟 = 0and {𝑥 𝑛 }is Cauchy. The limit is the common fixed point. □

Applications to Functional Equations 5.1 Dynamic Programming

Consider the functional equation arising in dynamic programming [7] :

References

1. Jungck G. Commuting mappings and fixed points. Am Math Mon. 1976;83(4):261-263.
2. Jungck G, Rhoades BE. Fixed points for set valued functions without continuity. Indian J Pure Appl Math. 1998;29(3):227238.
3. Huang LG, Zhang X. Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl. 2007;332(2):1468-1476.
4. Ma Z, Jiang L, Sun H. C*-algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory Appl. 2014;2014:206.
5. Rezapour S, Hamlbarani R. Some notes on the paper Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl. 2008;345(2):719-724.
6. Al-Thagafi MA, Shahzad N. Generalized I-nonexpansive selfmaps and invariant approximations. Acta Math Sin. 2008;24(5):867-876.
7. Bellman R, Lee ES. Functional equations in dynamic programming. Aequationes Math. 1978;17(1):1-18.
8. Abbas M, Jungck G. Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J Math Anal Appl. 2008;341(1):416-420.
9. Kadelburg Z, Radenović S, Rakočević V. Remarks on Quasi-contraction on a cone metric space. Appl Math Lett. 2011;22(11):1674-1679.
10. Branciari A. A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J Math Sci. 2002;29(9):531-536.
11. Ma Z, Jiang L. C*-algebra-valued b-metric spaces and related fixed point theorems. Fixed Point Theory Appl. 2015;2015:222.
12. Sintunavarat W, Kumam P. Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. J Appl Math. 2011;2011:637958.
13. Boyd DW, Wong JSW. On nonlinear contractions. Proc Am Math Soc. 1969;20(2):458-464.
14. Ćirić LB. A generalization of Banach's contraction principle. Proc Am Math Soc. 1974;45(2):267-273.
15. Dhage BC. Generalized metric spaces and mappings with fixed point. Bull Calcutta Math Soc. 1992;84:329-336.