The fixed point theory is an important tool not only of nonlinear analysis, but also for many other branches of modern mathematics. It has numerous applications within mathematics and has been applied in diverse fields such as medical sciences, chemistry, economics, management, engineering, game theory, and physics.

Historically the beginning of metric fixed point theory goes back two centuries, but its name was coined only in 1922 after the pioneer work of Polish mathematician Stefan Banach in his Ph.D. dissertation. Many remarkable results of fixed point theory have been obtained during nineteen sixtees and nineteen seventies such as Caristi's fixed point theorem, Nadlar's fixed point theorem, etc. A large number of research papers have already appeared in the literature on extensions and generalizations of the Banach contraction principle.

On the other hand, Ivar Ekeland established a result on the existence of an approximate minimizer of a bounded below and lower semicontinuous function in 1972. Such a result is now known as Ekeland's variational principle. It is one of the most elegant and applicable results that appeared in the area of nonlinear analysis with diverse applications in fixed point theory, optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, etc. Later, it was found that several well-known results, namely, the Caristi–Kirk fixed point theorem, Takahashi's minimization theorem, the Petal theorem, and the Daneš drop theorem, from nonlinear analysis are equivalent to Ekeland's variational principle in the sense that one can be derived by using the other results.

The set-valued maps, also called multivalued maps or point-to-set maps or multifunctions, are first considered in the famous book on topology by Kuratowski. Other eminent mathematicians, namely, Painlevé, Hausdorff, and Bouligand, have also visualized the vital role of set-valued maps as one often encounters such objects in concrete and real-life problems.

During the last decade of the last century, the theory on equilibrium problems emerged as one of the popular and hot topics in nonlinear analysis, optimization, optimal control, game theory, mathematical economics, etc. The equilibrium problem is a unified model of several fundamental mathematical problems, namely, optimization problems, saddle point problems, fixed point problems, minimax inequality problems, Nash equilibrium problem, complementarity problems, variational inequality problems, etc.

Equilibrium Problems

The mathematical formulation of the equilibrium problem is to find an element x̄ of a set K such that

where FK × K →is a bifunction such that F(x, x) = 0 for all x ∈ K. It is an unified model of several fundamental mathematical problems, namely, optimization problems, saddle point problems, fixed point problems, minimax inequality problems, Nash equilibrium problem, complementarity problems, variational inequality problems, etc. In 1955, Nikaido and Isoda [134] first considered equilibrium problem (5.1) as an auxiliary problem to establish the existence results for Nash's equilibrium points in noncooperative games. In the theory of equilibrium problems, the key contribution was made by Ky Fan [79], whose new existence results contained the original techniques which became a basis for most further existence theorems in the setting of topological vector spaces. That is why equilibrium problem (5.1) is also known as Ky Fan type inequality. Within the context of calculus of variations, motivated mainly by the works of Stampacchia [160], there arises the work of Brézis, Niremberg, and Stampacchia [45] establishing a more general result than that in [79]. In the last three decades, the theory of equilibrium problems emerges as a new direction of research in nonlinear analysis, optimization, optimal control, game theory, mathematical economics, etc. Most of the results on the existence of solutions for equilibrium problems are studied in the setting of topological vector spaces by using some kind of fixed point (Fan-Browder type fixed point) theorem or KKM type theorem. The term “equilibrium problem” was first used by Muu and Oettli [130] and later adopted by Blum and Oettli [38]. For further details, we refer to [3–5, 7–10, 12, 13, 32– 36, 38, 54–57, 62–64, 83, 84, 96, 97, 106, 107, 125, 130, 135] and the references therein. In most of the existence results for a solution of equilibrium problems, the convexity of the underlying set K and the bifunction F is assumed; see, for example, [10, 13, 35, 36, 45, 56, 57, 83, 84, 107] and the references therein. Inspired by the work of Blum and Oettli [38] and Oettli and Théra [135], the existence theory for solutions of equilibrium problems has been developed by many researchers in the setting of metric spaces and without any convexity assumption on the underlying set K and bifunction F; see, for example, [4, 7–9, 12, 32, 38, 54, 55, 63, 107, 106, 111, 135] and the references therein.

Definition B.1 (Partial Ordering) A binary relation ≼ on a nonempty set X is called partial ordering if the following conditions hold:

(i) For all a ∈ X, a ≼ a; (Reflexivity)

(ii) If a ≼ b and b ≼ a, then a = b; (Antisymmetry)

(iii) If a ≼ b and b ≼ c, then a ≼ c. (Transitivity)

Definition B.2 (Partially Ordered Set) A nonempty set X is called partially ordered if there is a partial ordering on X.

Remark B.1 The word ‘partially’ emphasizes that X may contain elements a and b for which neither a ≼ b nor b ≼ a holds. In this case, a and b are called incomparable elements. If a ≼ b or b ≼ a (or both), then a and b are called comparable elements.

Definition B.3 (Totally Ordered Set) A partially ordered set X is said to be totally ordered if every two elements of X are comparable.

In other word, partially ordered set X is totally ordered if it has no incomparable elements.

Definition B.4 Let M be a nonempty subset of a partially ordered set X.

(a) The element x ∈ X is called an upper bound of M if a ≼ x for all a ∈ M.

(b) The element x ∈ X is called a lower bound of M if x ≼ a for all a ∈ M.

(c) The element x ∈ M is called a maximal element of M if x ≼ a implies x = a.

Remark B.2 A subset of a partially ordered set M may or may not have an upper or lower bound. Also, M may or may not have maximal elements. Note that a maximal element need not be an upper bound.

Example B.1 (a) Let X be the set of all real numbers and let a ≤ b have its usual meaning. Then, X is totally ordered and it has no maximal element.

Some Basic Concepts and Inequalities

The first natural instance when set-valued maps occur is the inverse f −1 of a single-valued map f from a set X to another set Y. We can always define f −1 as a set-valued map which associates with any data y the (possibly empty) set of solutions

f −1(y) = ﹛x ∈ Xf (x) = y﹜

to the equation f (x) = y. Kuratowski realized the importance of set-valued maps, also called multivalued maps or point-to-set maps or multifunctions, and devoted considerable space in his famous book on topology. Other eminent mathematicians, namely, Painlevé, Hausdorff, and Bouligand, have also visualized the vital role of set-valued maps as one often encounters such objects in concrete and real-life problems. However, the authors of Bourbaki's volume Topologie Génerale emphasized the study of properties of set-valued maps as single-valued maps from a set to the power set of another set, or factorizing single-valued maps to make them bijective. This came as a stumbling block in the progress of this field. The set-valued maps started becoming popular among mathematicians working in the areas of game theory and economics when Kakutani's fixed point theorem [109], an extension of Brouwer's fixed point theorem to set-valued maps, came in existence, which was motivated by the work of Von Neumann. In 1969, Nadler [131] extended the Banach contraction principle to set-valued maps which plays an important and vital role in the theory of variational inequalities, differential inclusions, control theory, fractal geometry, etc. Since then, the theory of set-valued maps became an important subject for mathematicians working in nonlinear analysis, optimization theory, game theory, control theory, economics, fractal geometry, etc.

In view of such a wide variety of applications, most of the basic notions and results of singlevalued maps have been extended to set-valued maps. These include:

• • Limits and continuity

• • Nonlinear functional analysis (existence and approximation of solutions to equations and inclusions; fixed point theory; etc.)

• • Tangents and normals

• • Differentiation of set-valued maps

• • Convergence of a sequence of set-valued maps

• • Measures and integration

• • Differential inclusions

In 1972, Ekeland [74] (see also [75, 76]) established a theorem on the existence of an approximate minimizer of a bounded below and lower semicontinuous function. This theorem is now known as Ekeland's variational principle (in short, EVP). It is one of the most applicable results from nonlinear analysis and used as a tool to study the problems from fixed point theory, optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, etc.; see, for example, [15, 17, 18, 40, 74–77, 70, 135, 141] and the references therein. Several well-known fixed point results, namely, Banach contraction principle, Clarke's fixed point theorem, Caristi's fixed point theorem, Nadler's fixed point theorem, etc., can be easily derived by using EVP. Later, it was found that several well-known results, namely, the Caristi-Kirk fixed point theorem [52, 53], Takahashi's minimization theorem [168], the Petal theorem [141], and the Daneš drop theorem [68], from nonlinear analysis are equivalent to Ekeland's variational principle. Since the discovery of EVP, there have also appeared many of its extensions or equivalent formulations; see, for example, [4, 9, 12, 18, 23, 41, 66, 80, 85, 90–93, 105, 116–119, 127, 128, 135, 137–139, 141, 159, 161, 162, 165, 168, 169, 178, 181, 182] and the references therein. Sullivan [161] established that the conditions of EVP on a metric space (X, d) imply the completeness of the metric space (X, d). In 1982, McLinden [127] showed how EVP, or more precisely the augmented form of it provided by Rockafellor [149], can be adapted to extremum problems of minimax type. For further detail on the equivalence of these results, we refer to [4, 69, 85, 91, 138] and the references therein.

In this chapter, we present several forms of Ekeland's variational principle, its equivalence with Takahashi's minimization theorem and the Caristi-Kirk fixed point theorem, and some applications to fixed point theory and weak sharp minima. A variational principle given by Borwein and Preiss [39] and further revised by Li and Shi [116] is discussed. By using EVP, a fixed point theorem due to Uderzo [171] for ℋ-continuous and directional Ψ-contraction set-valued mappings is also presented.

The fixed point theory has numerous applications within mathematics and also in the diverse fields as biology, chemistry, economics, engineering, game theory, management, social sciences, etc. The Banach contraction principle is one of the most widely applicable fixed point theorems in all of analysis.

In this chapter, we focus on applications of the Banach contraction principle and its variants to the following problems:

• • System of linear equations

• • Differential equations and second-order two-point boundary-value problems

• • Linear and nonlinear Volterra integral equations, Fredholm integral equations, and mixed Volterra–Fredholm integral equations

Application to System of Linear Equations

In this section, we present an application of the Banach contraction principle to the system of linear equations (2.1).

As we have seen in Example 2.3, the system of linear equations (2.1) can be reformulated as follows:

and I is the identity matrix. If Tn → n is a matrix transformation defined by

then finding a solution of the system (6.1) is equivalent to find a fixed point of T. So, if T, defined by (6.2), can be proved to be a contraction mapping, then one can use the Banach contraction principle to obtain a unique fixed point of T, and hence, a unique solution of the system (6.1).

The conditions under which T is a contraction mapping depend on the choice of the metric on X = n. Here we discuss only one case and have left two others for exercise.

then the linear system (6.1) of n linear equations in n unknowns has a unique solution.

Proof Since X = n with respect to the metric d∞ is complete, it is sufficient to prove that the mapping T defined by (6.2) is a contraction. Indeed,

Thus, T is a contraction mapping. By Banach contraction principle (Theorem 2.1), the linear systems (6.1) has a unique solution.

Example 6.1 Consider the following system of linear equations:

the system of linear equations (6.4) can be written as Note that Here, Thus, the matrix transformation Tdefined by

is a contraction under the metric d∞. Therefore, the system of linear equations given by (6.4) has a unique solution.

The origin of metric fixed point theory goes back to the remarkable work of Polish mathematician Stefan Banach in his Ph.D. dissertation in 1920. The scholarly outcome of the dissertation is now known as the Banach contraction principle. The beauty of the Banach contraction principle is that it requires only completeness and contraction condition on the underlying metric space and mapping, respectively. With these conditions it provides the following assertions:

• • The existence and uniqueness of a fixed point.

• • The method to compute the approximate fixed points.

• • The error estimates for approximate fixed points.

A large number of research papers have already appeared in the literature on extensions and generalizations of the Banach contraction principle. One of the most important generalizations is due to Boyd and Wong [43] for 𝜓-contraction mappings. Another important generalization of the Banach contraction principle is given by Rhoades [148] for weakly contraction mappings introduced by Alber and Guerre-Delabriere [2]. However, a remarkable generalization of the Banach contraction principle is the Caristi’s fixed point theorem [52].

In this chapter, the Banach contraction principle and some of its extensions, namely, Boyd–Wong fixed point theorem for 𝜓-contraction mappings, a fixed point theorem for weakly contraction mappings, and Caristi’s fixed point theorem, are presented. The converse of the Banach contraction principle that provides the characterization of completeness of the metric space is also discussed.

Fixed Points

Definition 2.1 Let X be a nonempty set and TX → X be a mapping.

• A point x ∈ X is said be a fixed point of T if T(x) = x.

• The problem of finding a point x ∈ X such that T(x) = x is called a fixed point problem.

• We denote by Fix(T) = {x ∈ XT(x) = x} the set of all fixed points of T.

Example 2.1 (a) Let T →be a mapping defined by T(x) = x + a for some fixed number a ≠ 0. Then, T has no fixed point.

In this chapter, we gather some basic definitions, concepts, and results from metric spaces which

are required throughout the book. For detail study of metric spaces, we refer to [8, 46, 61, 95, 110,

150, 154].

Definitions and Examples

Definition 1.1 Let X be a nonempty set. A real-valued function dX × X →is said to be a

metric on X if it satisfies the following conditions:

The set X together with a metric d on X is called a metric space and it is denoted by (X, d). If there

is no confusion likely to occur we, sometime, denote the metric space (X, d) by X.

Example 1.1 Let X be a nonempty set. For any x, y ∈ X, define

Then d is a metric, and it is called a discrete metric. The space (X, d) is called a discrete metric space.

The above example shows that on each nonempty set, at least one metric that is a discrete metric

can be defined.

Example 1.2 Let X = n, the set of ordered n-tuples of real numbers. For any x = (x1, x2, … , xn) ∈

X and y = (y1, y2, … , yn) ∈ X, we define

Then, d1, d2, dp (p ≥ 1), d∞ are metrics on n.

Example 1.3 Let ℓ∞ be the space of all bounded sequences of real or complex numbers, that is,

is a metric on ℓ∞ and (ℓ∞, d∞) is a metric space.

Example 1.4 Let s be the space of all sequences of real or complex numbers, that is,

is a metric on s.

Example 1.5 Let ℓp, 1 ≤ p < ∞, denote the space of all sequences ﹛xn﹜ of real or complex numbers such that that is,

is a metric on ℓp and (ℓp, d) is a metric space.

Example 1.6 Let B[a, b] be the space of all bounded real-valued functions defined on [a, b], that is,