Bu çalışmada, zenginleştirilmiş (Cγ) şartını sağlayan dönüşümlerin bir sınıfı tanıtılmaktadır. Zenginleştirilmiş (Cγ) şartını sağlayan dönüşümlerin sabit noktaları, Banach uzayında modifiye edilmiş üç adımlı F-iterasyon yöntemi kullanılarak incelenmiştir. Giriş bölümünde, ana sonucumuzu elde etmemize yardımcı olan önceden tanıtılmış ve ispatlanmış bilgiler temel kavramlar olarak sunuldu, sonra sabit noktanın ne olduğu hakkında bilgi verildi ve genişlemeyen dönüşüm, Suzuki genişlemeyen dönüşüm, (Cγ) şartı, (C) şartı, (E) şartı ve (Eu) şartı tanımları verildi. Daha sonra zenginleştirilmiş genişlemeyen dönüşüm ve zenginleştirilmiş Suzuki genişlemeyen dönüşümler verildi. Ayrıca, zenginleştirilmiş (Cγ) şartını sağlayan dönüşümler tanıtıldı. Son olarak, sayısal örneklerde yakınsama hızını karşılaştırdığımız dört iterasyon yöntemi (Agarwal iterasyonu, Thakur iterasyonu, M-iterasyonu ve F-iterasyonu) verildi. İkinci bölümde, zenginleştirilmiş Suzuki genişlemeyen dönüşümler ve bu dönüşümlerle ilgili önemli teoremler incelenmiştir. Hilbert uzaylarında tanımlanan bu dönüşümlerin asimptotik düzenlilik ve yakınsaklık özellikleri hakkında teorik sonuçlar sunulmuştur. Krasnoselskii iterasyon dizileri kullanılarak, dönüşümlerin sabit noktalarına zayıf ve kuvvetli yakınsaklıkları ile bu dönüşümlerin konveks, kapalı ve sınırlı alt kümeler üzerindeki sonuçları incelendi. Üçüncü bölümde, ilk olarak zenginleştirilmiş (Cγ) şartını sağlayan dönüşüm tanımı verilerek, bir dönüşümün zenginleştirilmiş (Cγ) şartını sağlaması durumunda, onun ortalama dönüşümünün de (Cγ) şartını sağladığını ispat eden bir lemma sunulmuştur. Daha sonra, F-iterasyonu kullanılarak zayıf ve kuvvetli yakınsama teoremleri ispatlanmıştır. Ayrıca, (I) şartı tanımı kullanılarak kuvvetli yakınsaklık teoremi verilmiştir. Dördüncü bölümde, iki sayısal örnek verilerek üçüncü bölümde elde edilen sonuçlar ile tanımlanan F- iterasyon yönteminin diğer yöntemlere göre sabit noktaya daha hızlı yakınsadığı gösterilmiştir. Sayısal örnekler ve hesaplamalar ile çeşitli iterasyon yöntemlerinden bulunan veriler kullanılarak tablolar ve grafikler yapılmıştır. Örnek 4.1'de hem (Cγ) şartını hem de zenginleştirilmiş (Cγ) şartını sağlayan bir dönüşüm örneği verilmiş, örnek 4.2'de zenginleştirilmiş (Cγ) şartını sağlayan ancak (Cγ) şartını sağlamayan başka bir dönüşüm verilmiştir. Bu iki örnek yardımıyla dört çeşit iterasyonun sabit noktaya nasıl yakınsadığını gösteren tablolar ve grafikler, Python dili kullanılarak kodlar yazılıp hesaplanmış ve grafikleri çizilmiştir. Son bölümde ise tüm tezde elde edilen sonuçlar kısaca özetlenmiş ve sonrasında bu konu üzerinde nasıl çalışmalar geliştirilebileceği ile ilgili bir açık problem ortaya konulmuştur.
This study delves into the fixed point theory of various classes of nonlinear maps in normed and Banach spaces. Let W be a nonempty subset of a normed space \mathcal{Z}, and D:W\rightarrow W be a map. A point u\in W is a fixed point of D if Du=u. Fixed point theory is a crucial mathematical concept with applications in various fields such as economics, engineering, and the sciences. Nonexpansive maps, characterized by the property \parallel Du-Dv\parallel\le\parallel u-v\parallel for all u,v\in W, have been extensively studied. This condition ensures that the distance between mapped points does not exceed the distance between the original points, which is essential for convergence analysis. Suzuki (2008) introduced generalized nonexpansive maps, satisfying condition (C), which later inspired further generalizations, such as condition (E), (E_\mu), and (C_\gamma). A map D:W\rightarrow W satisfies condition (C) if \frac{1}{2}\parallel u-Du\parallel\le\parallel u-v\parallel implies \parallel Du-Dv\parallel\le\parallel u-v\parallel for all u,v\in W. This condition ensures that the map D does not increase distances significantly under certain constraints, allowing for more general mappings while maintaining some control over their behavior. Another generalization is condition (E_\mu), where \parallel u-Dv\parallel\le\mu\parallel u-Du\parallel+\parallel u-v\parallel for \mu\geq1. This condition allows for a bounded increase in distance based on a multiplicative factor \mu, providing a framework for analyzing maps that are not strictly nonexpansive but still exhibit controlled behavior. Additionally, condition (C_\gamma) for \gamma\in(0,1) is defined as \gamma\parallel u-Du\parallel\le\parallel u-v\parallel implying \parallel Du-Dv\parallel\le\parallel u-v\parallel. This condition further generalizes nonexpansiveness by introducing a parameter \gamma that scales the distance condition, allowing for finer control over the map's behavior. In 2019, Berinde introduced enriched nonexpansive maps, where D is enriched nonexpansive if there exists b\in[0,\infty) such that \parallel b(u-v)+Du-Dv\parallel\le(b+1)\parallel u-v\parallel. This class of maps incorporates an additional parameter b that enriches the nonexpansive condition, providing a broader framework for analyzing nonlinear maps. Later, in 2021, Ullah et al. introduced enriched Suzuki nonexpansive maps, further studied in Banach spaces by Abdeljawad et al. in 2022. These maps combine the ideas of enrichment and Suzuki's condition, offering a versatile tool for fixed point analysis. Iterative methods play a crucial role in approximating fixed points. These methods generate sequences that converge to fixed points, which are essential for practical applications. Notable methods include Picard, Mann, Ishikawa, Noor, Agarwal, Thakur, M-iterative, and F-iterative methods. These methods are essential for maps ensuring fixed points under certain conditions. The iterative methods vary in their approach and convergence properties, making them suitable for different types of maps and applications. This paper focuses on the convergence of the F-iterative method for maps satisfying an enriched version of condition (C_\gamma). We establish both weak and strong convergence results in Banach spaces using the F-iterative method. Moreover, we present numerical examples demonstrating the efficiency of our method compared to others. The F-iterative method is particularly effective in handling enriched conditions, providing faster convergence and greater accuracy in approximating fixed points. We introduce the class of maps satisfying the enriched condition (C_\gamma) and prove convergence results for the F-iterative method. Theorem 3.3 demonstrates that {u_l} is bounded in W and \lim_{l\rightarrow\infty}ul-Drul=0 if and only if F_D\neq\emptyset. Theorem 3.4 establishes weak convergence under Opial's condition. Strong convergence results are proved under conditions such as compactness of W (Theorem 3.5), {lim\,\inf}_{l\rightarrow\infty}\hairspdist\funcapply(u_l,F_D)=0 (Theorem 3.6), and condition (I) (Theorem 3.8). These theorems provide a comprehensive framework for analyzing the convergence behavior of the F-iterative method under various conditions, ensuring robust and reliable approximations of fixed points. Numerical examples illustrate the superior convergence rate of the F-iterative method compared to other iterative methods. For instance, Example 4.1 presents a map satisfying both the enriched condition (C_\gamma) and the ordinary condition (C_\gamma). The map D on W=[0,1] defined by Du=\frac{u}{2} for u\neq1 and D(1)=\frac{11}{19} satisfies both conditions. The F-iterative method again demonstrates faster convergence compared to other iterative processes, validating its effectiveness. Example 4.2 demonstrates a map satisfying the enriched condition (C_\gamma) but not the ordinary condition (C_\gamma). The map D on W=[-0.5,-2]∪[0.5,2] defined by Du=u^{-1} satisfies the enriched condition with b=1.5 but not the ordinary condition. The numerical results show that the F-iterative method converges faster than other methods, highlighting its efficiency. In conclusion, this study contributes to the understanding of fixed point theory for various classes of nonlinear maps in Banach spaces, presenting significant theoretical results and practical applications through iterative methods. The introduction of the enriched condition (C_\gamma) and the development of the F-iterative method provide powerful tools for analyzing and approximating fixed points. The convergence theorems and numerical examples illustrate the robustness and efficiency of our approach, making it a valuable addition to the field of fixed point theory. This work lays the foundation for further research and applications in diverse areas, emphasizing the importance of iterative methods in solving complex nonlinear problems. The fixed point theory has profound implications in various scientific disciplines. In economics, fixed point theorems are used in game theory and equilibrium analysis. For instance, Nash equilibrium in game theory is a solution concept where no player can benefit by unilaterally changing their strategy, and it can be proven using fixed point theorems. In engineering, fixed point methods are employed in signal processing, control theory, and the design of algorithms for solving differential equations. The ability to ensure the existence and uniqueness of solutions is fundamental in these applications. The generalizations introduced in this paper, particularly the enriched condition (C_\gamma), extend the applicability of fixed point theory to a broader class of problems. By incorporating parameters such as \mu and \gamma, we can handle maps that exhibit more complex behaviors, making the theory more versatile. This is particularly important in real-world applications where ideal conditions (such as strict nonexpansiveness) are rarely met. The enriched condition (C_\gamma), as introduced in this study, offers a nuanced approach to handling nonlinear maps. The condition \gamma\parallel u-Du\parallel\le\parallel u-v\parallel implies \parallel Du-Dv\parallel\le\parallel u-v\parallel allows for a controlled relaxation of the nonexpansive condition. This is crucial for dealing with maps that are not strictly nonexpansive but still exhibit convergence properties under certain conditions. The F-iterative method leverages this enriched condition to provide robust convergence results. By iteratively applying the map and adjusting parameters, the method ensures that the sequence generated converges to a fixed point. The convergence theorems established in this study provide a rigorous mathematical foundation for the method, ensuring its reliability and effectiveness. The numerical examples provided in this study highlight the practical implications of the theoretical results. By demonstrating the convergence behavior of the F-iterative method on specific maps, we provide concrete evidence of its efficiency. The comparative analysis with other iterative methods, such as the Agarwal and Thakur methods, further underscores the advantages of the F-iterative method. The results of this study open up several avenues for future research. One potential direction is the extension of the enriched condition (C_\gamma) to other types of spaces, such as CAT(0) spaces. These spaces generalize the concept of non-positive curvature and have applications in various areas of mathematics and computer science. Exploring the applicability of the F-iterative method in these spaces could lead to new insights and broader applications. Another area for future research is the development of new iterative methods based on the enriched condition (C_\gamma). By further refining the parameters and conditions, it may be possible to design methods that offer even faster convergence and greater accuracy. Additionally, investigating the stability and robustness of these methods in the presence of perturbations or noise would be valuable for practical applications. In conclusion, this study presents significant advancements in the fixed point theory for nonlinear maps in Banach spaces. The introduction of the enriched condition (C_\gamma) and the development of the F-iterative method provide powerful tools for analyzing and approximating fixed points. The comprehensive theoretical framework and numerical examples illustrate the robustness and efficiency of our approach, making it a valuable addition to the field of fixed point theory. This work lays the foundation for further research and applications, emphasizing the importance of iterative methods in solving complex nonlinear problems.
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