Summary of the Thesis

Transcription

1 Summary of the Thesis Fixed point theory is an exciting branch of mathematics. It is a mixture of analysis, topology and geometry. Over the last 50 years or so the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. It has numerous applications in almost all areas of mathematical sciences. For example, proving the existence of solutions of ordinary and partial differential equations, integral equations, system of linear equations, closed orbit of dynamical systems and of equilibria in economics. In particular fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, engineering and physics. It has very fruitful applications in control theory, game theory, category theory, functional equations, integral equations, mathematical physics, mathematical chemistry, mathematical biology, mathematical economics, W* algebra, functional analysis and many other areas. The concept of fixed point plays a key role in analysis. Also, fixed point theorems are mainly used in existence theory of random differential equations, numerical methods like Newton-Rapshon method and Picard s Existence Theorem and in other related areas. Fixed point theorems based on the consideration of order have importance in algebra, the theory of automata, mathematical linguistics, linear functional analysis, approximation theory and theory of critical points. Fixed point theorems play a major role in many applications, such as variational and linear inequalities, optimization and applications in the field of approximation theory. Thus the study of the fixed point theory has been researched extensively. The space X is said to be have the fixed point property if for any continuous function T: X X, there exists x X such that T(x) = x, that is, a point which remains invariant under the mapping T. Geometrically representation of fixed point is diagonal line. In example of fixed point, we can say identity mappings have all its points as fixed point as well as translation mapping, i.e., Tx= x-a, where a is a constant has no fixed point. Fixed Point Theory is divided into three major areas: 1. Topological Fixed Point Theory, 2. Metric Fixed Point Theory, 3. Discrete Fixed Point Theory. Historically the boundary lines between the three areas was defined by the discovery of three major theorems: 1. Brouwer's Fixed Point Theorem,

2 2. Banach's Fixed Point Theorem, 3. Tarski's Fixed Point Theorem. In this thesis, we will focus mainly on the second area though from time to time we may pay attention to other areas also. Metric fixed point theory has its roots in methods from the late 19th century, when successive approximations were used to establish the existence and uniqueness of solutions to equations, and especially differential equations. This approach is particularly associated with the work of Picard, although it was Stefan Banach who in 1922 (in [9]) developed the ideas involved in an abstract setting. The fixed point theorem, generally known as the Banach contraction principle or Banach s fixed point theorem, appeared in explicit form in Banach s thesis in 1922, which states that every contraction mapping defined on a complete metric space has a fixed point. Banach's contraction principle ensures, under appropriate conditions, the existence and uniqueness of a fixed point. It is the simplest and one of the most versatile results in fixed point theory. Being based on an iterative scheme, it can be implemented on a computer to find the fixed point of a contractive map: it produces approximations of any required accuracy, and moreover, even the number of iterative schemes needed to get a specified accuracy can be determined. It has become a very popular tool in solving existence problems in many branches of mathematics because of its simplicity and usefulness. Many authors ([40, 53, 80, 158] and references there in) have extended, generalized and improved Banach s fixed point theorem in different ways. One of the most important results on this direction has been obtained by S. B. Presic in [158] by generalizing the Banach contraction mapping principle. In 1976, Jungck [80] generalized Banach s contraction principle using the concept of commuting mappings. The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research activity during the last three decades. It was the turning point in the fixed point area" when the notion of weak commutativity was introduced by Sessa [193] as a sharper tool to obtain common fixed points of mappings. As a result, all the results on fixed point theorems for commuting mappings were easily transformed in the setting of the new notion of weak commutativity of mappings. It gives a new impetus to the study of common fixed points of mappings satisfying some contractive type conditions and a number of interesting results have been found by various authors. A bulk of results were produced and it was the centre of vigorous

3 research activity in Fixed point theory and its application in various other Branches of Mathematical Sciences" in last two decades. A major breakthrough was done by Jungck [77] when he proclaimed the new notion what he called compatibility" of mapping and its usefulness for obtaining common fixed points of mappings was shown by him. Thereafter a flood of common fixed point theorems were produced by various researchers by using the improved notion of compatibility of mappings. In fact, every weak commutative pair of mappings is compatible but the converse is not true ([79]). The study of noncompatible mappings is also very interesting. The study of common fixed points of noncompatible maps was initiated by Pant [151] by using the notion of pointwise R-weak commutativity. In 1998, Jungck and Rhoades [79] introduced the notion of weakly compatible and showed that compatible maps are weakly compatible but not conversely. Brian Fisher [58] proved an important common fixed point theorem in this regard. During the sixties, the notion of 2-metric spaces was introduced by Gahler [59, 60, 61], as a generalization of usual notion of metric space (X, d). But other authors proved that there is no relation between these two functions. Many applications of fixed point theory in 2-metric spaces can be found, as well as applications in Medicine and Economics [19,119]. A new structure of a generalized metric space called D-metric space was introduced by Dhage [48], on the lines of ordinary metric spaces. Generally the usual ordinary metric is called the distance function. D-metric is called the diameter function of the points of spaces. Unfortunately, it was shown that certain theorems involving Dhage s D-metric spaces are flawed, and most of the results claimed by Dhage and others are invalid. In 2005, Mustafa and Sims [115] introduced a new structure of generalized metric spaces, which are called G-metric spaces as generalization of metric space (X, d), to develop and introduce a new fixed point theory for various mappings in this new structure. Papers with more details on G -metric spaces are [114]-[117]. Fixed point theory was extended to multivalued mappings in 1941 with the fixed point theorems of Nadler[118] and Markin [110]. If f is a multivalued map from X to a collection of non empty subsets of X, then a point x X is a fixed point of f if x f(x). Fixed point theory for multivalued maps find special significance in the theory of inclusions, differential equations for multivalued operators, game theory, fractal theory, discrete dynamics and other areas of mathematical sciences (see for instance[86,110,185]).

4 Some mathematical problems, such as the problems of metrization of convergence with respect to measure, leads to a generalization of metric. S. Czerwik [42] introduced the idea of b- metric space. The concept of b-metric space appeared in some works, such as I. A. Bakhtin [8], S. Czerwik [42, 43]. Motivated by G-metric spaces as well as b-metric spaces, we define the concept of generalized b-metric spaces and then prove the existence of fixed points for multivalued contraction mappings in generalized b-metric spaces using Picard iterative scheme and also Jungck iterative scheme. Our results extend, improve and unify a multitude of classical results in fixed point theory of single and multivalued contraction mappings. B. E. Rhoades and S. M. Solutz showed that the convergence of Mann iterative scheme is equivalent to the convergence of Ishikawa iterative scheme for various classes of functions at [179], [180] and [181]. A reasonable conjecture is that, whenever T is a function for which Mann iterative scheme converges, so does the Ishikawa and Noor iterative schemes. Given the large variety of functions and spaces, such a global statement is, of course, not provable. In this regard, Ozdemir et.al. [147] showed that Mann, Ishikawa and Noor iterative schemes are equivalent for a special class of Lipschitzian operators defined in a closed, convex subset of an arbitrary Banach space. S. M. Solutz [201, 202] proved that Picard, Mann, Ishikawa and Noor iterative schemes are equlivalent for quasi-contractive operators. In 2009, Isa Yildirim et. al. [215] proved that Picard, new two step, Mann and Ishikawa iterative schemes are equivalent for certain quasi- contractive operators. Motivated by the above facts, we study the equivalence between the convergence of already existing and newly introduced iterative schemes for quasi-contractive operators. A fixed point theorem is valuable from a numerical point of view if it satisfies several requirements, among which we mention: (a) it helps to provide an error estimate for the iterative scheme used to approximate the fixed point and (b) it can give concrete information on the stability of this iterative scheme or alternatively on the data dependence of the fixed point.

5 The data dependence abounds in literature of fixed point theory when dealing with Picard iterative scheme [12, 188], but is quasi-inexistent when dealing with Mann-Ishikawa iterative scheme. As far as we know, the only data-dependence result concerning Mann-Ishikawa iterative scheme is in [203, 204, 205]. There, Solutz proved the data dependence of Ishikawa iterative scheme when applied to contractions and contractive-like operators. Inspired by the work of Solutz, we prove the data dependence results for Noor and SP iterative schemes using certain quasi-contractive operators. Although fixed point iterative schemes are used in solving problems of industrial and applied mathematics still there is no systematic study of numerical aspects of these iterative schemes. In computational mathematics, it is important to compare the iterative schemes with regard to their rate of convergence. By using computer programs, perhaps for the first time, B. E. Rhoades [175] illustrated the difference in convergence rate of Mann and Ishikawa iterative schemes for increasing and deceasing functions through examples. S. L. Singh [195] extended the work of Rhoades. In [13], Berinde showed that Picard iterative scheme converges faster than Mann iterative scheme for quasi-contractive operators. Recently, Nawab Hussian et al. [68] provide an example of a quasi-contractive operator for which the iterative scheme due to Agarwal et al. is faster than Mann and Ishikawa iterative schemes. By providing examples, Phuengrattana and Suantai [155] showed that SP iterative scheme converges faster than Mann, Ishikawa and Noor iterative scheme for nondecreasing and continuous functions on real line intervals. Moreover, Rana, Dimri and Tomar [166] showed that Ishikawa iterative scheme converges faster than Mann iterative scheme while Picard iterative scheme converges faster than both in complex space. Motivated by the above facts on rate of convergence, we introduce some new iterative schemes namely CR iterative scheme, SP iterative scheme mixed errors, Kirk-Noor iterative scheme, Kirk-SP iterative scheme, Kirk-CR iterative scheme, Jungck-SP iterative scheme and Jungck- Kirk-Noor iterative scheme. We prove that newly introduced iterative schemes are equivalent to and converges faster as compared to the already existing iterative schemes in the literature using certain quasi-contractive operators. Moreover with the help of C++/Matlab programs, we compare rate of convergence of newly introduce iterative schemes with already existing iterative schemes through examples.

6 Most of the modeled formulations can easily be expressed as fixed point equations and the solution of this equation is approximated by a sequence using an iterative scheme that converges to a fixed point of the function. But due to rounding off or discretization in the function, we obtain a approximate sequence instead of actual sequence. We say that an iterative scheme is stable iff the actual sequence converges to a fixed point of the function, and then the approximate sequence also converges to the same fixed point. Intuitively, a fixed point iterative scheme is numerically stable if, small modification in the initial data or in the data that are involved in the computation process, will produce a small influence on computed value of the fixed point. The study of stability of iterative schemes enjoy a celebrated place in applied sciences and engineering due to chaotic behavior of functions in discrete dynamics and other numerical computations. The revolution in computational mathematics through computer programming has accelerated the increasing interest in the stability theory of iterative schemes. In [67], Harder and Hicks gave the concept of T-stability as follows: Let X be a Banach space, T a self map of X and assume that x n+1= f(t, x n ) define some iterative scheme involving T. Suppose that { x n } converges to a fixed point p of T. Let { y } be an arbitrary sequence in X and define n yn+1 f(t, y n) for n = 0, 1, 2.If lim ε n = p implies that lim y n = p, then iterative scheme x n+1= f(t, x n ) is said to be T-stable. The iterative scheme defined by x n+1 = f(t, x n ) is said to be almost T-stable if n n n 0 ε n n n implies that lim y = p. Clearly, any T-stable iterative scheme is almost T-stable. But converse is not true. The stability of several iterative schemes in metric spaces and normed linear spaces for certain contractive operators has been studied by several authors. Some of the various contributors in the study of stability of the fixed point iterative schemes are Ostrowski [145], Harder and Hicks [67], Rhoades [170], Osilike [144], Berinde [12] and Singh et al. [198]. The first stability result on T-stable mappings was due to Ostrowski [145] where he established the stability of the Picard iterative scheme by using Banach contraction condition. Harder and Hicks [67], Rhoades [170], Osilike [144] and Singh et al. [198] used the method of the summability theory of infinite matrices to prove various stability results for certain contractive definitions. Osilike and

7 Udomene [140] introduced a shorter method of proof of stability results and this has also been employed by Berinde [12], Imoru and Olatinwo [69], Olatinwo et al. [135] and some others. Osilike [139] proved that Ishikawa iterative scheme is almost T- stable when the operator is a Lipschitz strongly pseudocontractive operator. Olaleru and Mogbademu [124] considered the stability of Mann, Ishikawa and Kirk iterative schemes with errors and showed that they are almost stable with respect to some classes of quasi-contractive maps. Z. Liu and J.S. Ume[103] proved that Noor iterative scheme with errors both converges strongly to a unique fixed point and is almost T-stable for local pseudo-contractions and local strongly accretive operators in an arbitrary Banach space. Imoru and Olatinwo [69] proved the stability of the Picard and the Mann iterative scheme for the following operator which is more general than the one introduced by Osilike [138]. The operator satisfies the following contractive definition: there exists a [0, 1) and a monotone increasing function : R+ R+ with (0) = 0, such that d(tx, Ty) (d(x, Tx)) + ad(x, y) x, y X. Recently, Olatinwo [130] also considered the stability of the Kirk-Mann and Kirk-Ishikawa iterative schemes when the operator satisfies above inequality. We extend the above results of Olatinwo by proving the stability of the Kirk-Noor iterative scheme when the operator satisfies the same inequality. Also, we show that SP iterative scheme with mixed errors is almost stable for the accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces. Also, we prove the (S,T)-stability of Jungck-Kirk- Noor and Jungck-SP iterative schemes using certain quasi-contractive nonself operators. Objectives of Research The specific objectives of this research work are to introduce some new iterative schemes and establish some convergence and stability results for these new iterative schemes using various operators. study the equivalence between the convergence of already existing and newly introduced iterative schemes for quasi-contractive operators.

8 study the comparison between rate of convergence of newly introduced iterative schemes and already existing iterative schemes. study the data dependence of iterative schemes. endow the space in which the problems are being considered with some rich structure. Thesis at a Glance The research work reported in this is covered into five chapters and each chapter is subdivided into the sections depending on the diversity of the subject matter. Chapter 1 is introductory in nature which include origin, development of the subject and some basic concepts so as to make it convenient for understanding the rest of the research work. Chapter 2 is composed of two sections. In the first section, we define the concept of generalized b-metric space and give an example of this newly defined space. In the second section, we prove the existence of fixed points for multivalued contraction mappings in generalized b-metric spaces using Picard and also Jungck iterative schemes. Our results extend, improve and unify a multitude of classical results in fixed point theory of single and multivalued contraction mappings. We obtain more general results than those of Nadler [118], Berinde and Berinde [10], M.O. Olatinwo [125, 126] and Daffer and Kaneko [45]. The findings of this chapter have been published in International Journal of Mathematical Archive-3(3) (2012), Chapter 3 deals with the study the strong convergence and stability of SP type iterative schemes for quasi-contractive and accertive Lipschitzian type operators in Banach spaces. Equivalence and rate of convergence of SP type iterative schemes are also discussed as compared to other iterative schemes. Chapter 3 comprises six sections. In section 3.1 introduction is given. In section 3.2, we show the strong convergence of SP iterative scheme using quasi-contractive self operators. Also in this section, we show that the SP iterative scheme is equivalent to and faster than the other existing iterative schemes. The results included in this section are published in International Journal of Computer Applications, volume 31, no.5, October In section 3.3, we introduce Jungck-SP iterative scheme and study the stability of this iterative scheme using nonself quasi-contractive operators. Also, we show that Jungck- SP iterative scheme have better convergence rate as compared to the other Jungck type iterative

9 schemes. The results of this section have been published in International Journal of Computer Applications, volume 36, no.12, December In section 3.4, we introduce a modified iterative scheme called the SP iterative scheme with mixed errors and study the strong convergence of this iterative scheme for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces. We show that SP iterative scheme is almost stable for both types of operators. Moreover, with the help of computer programs in C++, comparison between SP, Ishikawa and Noor iterative schemes is also shown for both types of operators through examples. The findings of this section have been submitted to International Journal of Computer Mathematics. In chapter 4, we introduce a new iterative scheme called the CR iterative and compare the rate of convergence of already existing iterative schemes with newly introduced iterative scheme with the help of examples by using C++ / Matlab programs. Data dependence of some iterative schemes is also discussed by providing examples. Chapter 4 comprises six sections. In section 4.1 introduction is given. In the section 4.2, we analyze the rate of convergence of three iterative schemes namely-agarwal et al., Noor and SP iterative schemes for complex space by using Matlab programs. The results obtained are extensions of some recent results of Rana, Dimri and Tomar [166]. The results of this section have been published in International Journal of Computer Applications, volume 41, no.11, March In the section 4.3, we suggest a new type of three step iterative scheme called the CR iterative scheme and study the strong convergence of this iterative scheme for a certain class of quasi-contractive operators in Banach spaces. We show that for the aforementioned class of operators, the CR iterative scheme is equivalent to Picard, Mann, Ishikawa, Agarwal et. al, Noor and SP iterative schemes. In section 4.4, we show that for quasi-contractive operators, the CR iterative converges faster than Picard, Mann, Ishikawa, Agarwal et. al, Noor and SP iterative schemes. Moreover, in section 4.5 we also present various numerical examples using computer programming in C++ for the CR iterative scheme to compare it with the other above mentioned iterative schemes. Results of section 4.4 and 4.5 show that as far as the rate of convergence is concerned (i) for increasing functions the CR iterative scheme is best, while for decreasing functions the SP iterative scheme is best (ii) CR iterative scheme is best for a certain class of quasi-contractive operators. The results of sections 4.3, 4.4 and 4.5 have been submitted to the journal Mathematical Reports. In the

10 section 4.6, we prove results concerning data dependence of Noor and SP iterative schemes using certain quasi-contractive operators in real Banach spaces. Our results reveal that by choosing an approximate quasi-contractive operator (for which it is possible to compute the fixed point), we can approximate the fixed point of the given operator. An example is also provided to explain the validity of our results. The findings of this section have been published in International Journal of Computer Applications, volume 40, no.15, February The purpose of chapter 5 is to introduce some new hybrid fixed point iterative schemes of Kirk-Noor type, to prove convergence as well as stability results for these iterative schemes and to compare these iterative schemes with other iterative schemes. Chapter 5 covers six sections. In section 5.1 introduction is given. In section 5.2, we introduce new iterative schemes namely Kirk-Noor, Kirk-SP and Kirk-CR iterative scheme and prove strong convergence of these iterative schemes for self operators by employing certain quasicontractive operators while in section 5.3, we prove stability of Kirk-Noor iterative scheme for self and nonself operators by using aforementioned class of operators. The results obtained are improvements, generalization, and extensions of the works of M.O. Olatinwo [130] and many others in the literature. Some results of section 5.2 have been published in Journal of Applied Mathematics, volume 2012 (2012) and results of section 5.3 have been published in International Journal of Contemporary Mathematical Sciences, vol 7, no. 24 (2012), In section 5.4, we introduce an another fixed point iterative scheme namely Kirk-Noor iterative scheme with errors and establish a general theorem to approximate the unique common fixed point of three certain quasi-contractive operators in an arbitrary Banach space using this iterative scheme. We use a more general contractive condition and prove a convergence theorem under weaker conditions on parameters than those of Rashwan et al. [167]. Our results, therefore, are improvements, generalization and extensions of the works of Rhoades [175], Berinde [11], Rashwan et al. [167] and many others in the literature. An example to illustrate the validity of our results is also provided. The findings of this section have been accepted for publication in International Journal of Pure and Applied Mathematics. In section 5.5, by taking an example of certain quasi-contractive operator, we compare Kirk-SP, Kirk-CR, Kirk-Mann, Kirk-Ishikawa and Kirk-Noor iterative schemes. Moreover, in section 5.6, using computer programs in C++, we compare above mentioned iterative schemes through examples of

11 increasing, decreasing, sublinear, superlinear and oscillatory functions. Results of section 5.5 and 5.6 have been published in Journal of Applied Mathematics, volume 2012 (2012). At the end of the thesis, observations and conclusions of the thesis as well as bibliography of references are given. References 1. Bakhtin, I. A., The contraction mapping principle in almost metric spaces, Funct. Anal., Gos. Ped. Inst. Unianowsk, 30(1989), Banach, S., Surles operations dans les ensembles abstraites et leurs applications, Fund. Math., 3(1922), Berinde, M. and Berinde V., On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl., 326(2007), Berinde, V., On the convergence of the Ishikawa iteration in the class of quasicontractive operators, Acta Math. Univ. Comenianae LXXIII(2004), Berinde, V., Iterative Approximation of Fixed Points, Springer-Verlag, Berlin Heidelberg, Berinde, V., Picard iteration converges faster than Mann iteration for a class of quasicontractive operators, Fixed Point Theory and Applications, volume 2(2004), Berinde, V., On the stability of some fixed point procedures, Bul. S tiint. Univ. Baia Mare, Ser. B, Matematica-Informatica, 18 (2002), Bosede, A. O., Strong convergence results for the Jungck-Ishikawa and Jungck-Mann iteration processes, Bulletin of Mathematical Analysis and Applications, 2(3)(2010), Bosede, A. O. and Rhoades, B. E., Stability of Picard and Mann iterations for a general class of functions. Journal of Advanced Mathematical Studies, 3, 2 (2010), Browder, F., Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54(1965), Browder, F., Nonlinear mapping of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc., 73 (1967), Chang, S.S., Cho, Y.J., Lee, B. S. and Kang, S.M., Iterative approximations of fixed

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