APPLICATIONS OF FIXED POINT THEOREMS TO SO-LUTIONS OF OPERATOR EQUATIONS IN BANACH SPACES

In this paper we use Browder’s and Gohde’s fixed point theorem, Kirk’s fixed point theorem and the Sadovskii fixed point theorem to obtain solutions of operator equations in Banach spaces.


INTRODUCTION
Perhaps the most famous fixed-point theorem is the Banach's contraction principle which has several applications.Motivated by this we have considered in this review article, applications of other wellknown fixed-point theorems in various kinds of Banach spaces.This article should be of interest to mathematicians working in the fields of fixed-point theory and functional analysis.In Section 1 we apply the Browder's and Göhde's fixed point theorem for the existence of solutions of operator equations involving asymptotically nonexpansive mappings in uniformly convex Banach spaces.In Section 2 we apply Kirk's fixed point theorem for the existence of solutions of the operator equation x − T x = f in reflexive Banach spaces and in Section 3 we apply the Sadovskii fixed point theorem for existence of solutions of the operator equation x − T x = f in arbitrary Banach spaces.
2 Application of Browder's and Göhde's fixed point theorem Definition 1. [1] A mapping T from a metric space (X, d) into another metric space (Y, ρ) is said to satisfy Lipschitz condition on X if there exists a constant L > 0 such that ρ(T x, T y) ≤ Ld(x, y) for all x, y ∈ X.If L is the least number for which Lipschitz condition holds, then L is called Lipschitz constant.If L = 1, the mapping is said to be nonexpansive.Definition 2. [2] Let K be a nonempty subset of a Banach space X.A mapping T : K → K is said to be asymptotically nonexpansive if for each n ∈ N there exists a positive constant k n ≥ 1 with The Browder's and Göhde's fixed point theorem is as follows: Theorem 1. [3] Let X be a uniformly convex Banach space and C a nonempty, closed, convex and bounded subset of X.Then every nonexpansive mapping T : C → C has a fixed point in C.
We now state the main theorem of Section 1.
Theorem 2. Let X be a uniformly convex Banach space and K a nonempty subset of X.Let T : K → K be an asymptotically nonexpansive mapping and f n ∈ K, then the operator equation where n ∈ N and k n is the Lipschitz constant of the iterates T n , has a solution if and only if, for any x 1 ∈ K, the sequence of iterates {x n } in K defined by Proof.For every n ∈ N, let T fn be defined to be a mapping from K into K by Suppose T fn has a fixed point u n ∈ K. Then Let C = n∈N C n .Since C n increases with n, C is a closed, convex and bounded subset of K. We can easily see that T fn maps C into C.
Applying the Browder's and Göhde's theorem to T fn and C we get a fixed point of T fn in C. Since C ⊂ K, we obtain a fixed point of T fn in K.

Application of Kirk's Fixed Point Theorem
Let us recall some definitions and results that we shall require for the proof of the Main Theorem of Section 2.
Definition 4. [1] A convex subset K of a Banach space X is said to have normal structure if each bounded, convex subset S of K with diam S > 0 contains a nondiametral point.
The following theorem gives application of the Browder-Göhde-Kirk's theorem for the existence of solutions of the operator equation It is known that every uniformly convex Banach space is reflexive.We generalize the theorem below to reflexive Banach spaces using Kirk's fixed point theorem Theorem 3. [3] Let X be a uniformly convex Banach space, f an element in X and T : X → X a nonexpansive mapping, then the operator equation has a solution x if and only if for any x 0 ∈ X, the sequence of Picard iterates {x n } in X defined by A Banach space X is said to satisfy the Opial condition if whenever a sequence {x n } in X converges weakly to x 0 ∈ X, then Let X be a reflexive Banach space with the Opial condition.Then X has normal structure.

Lemma 2. [3] A closed subspace of a reflexive Banach space is reflexive.
Now we state the Kirk's fixed point theorem.
Theorem 4. [3] Let X be a Banach space and C a nonempty weakly compact, convex subset of X with normal structure, then every nonexpansive mapping T : C → C has a fixed point.
We state the main theorem of Section 2.
Theorem 5. Let X be a reflexive Banach space satisfying Opial condition.Let f ∈ X and T : X → X be a nonexpansive mapping.Then the operator equation has a solution x if and only if for any x 0 ∈ X, the sequence of Picard iterates {x n } in X defined by Proof.Let T f be the mapping from X into X given by Then u is a solution of Then for all n ∈ N, Hence {x n } is bounded.
Conversely, suppose that {x n } is bounded.Let d = diam({x n }) and Then C n is a nonempty convex set for each n, and Let C be the closure of the union of C n for n ∈ N, Since C n increases with n, C is a closed, convex and bounded subset of X.It is known that [1] bounded, closed and convex subsets of reflexive Banach spaces are weakly compact, hence we get that C is weakly compact.

Now since
we get that T f maps C into itself.By Lemma 1.3.6,C is a reflexive Banach space.Now X satisfies Opial condition and C being a closed subset of X, will also satisfy Opial condition.Hence by Lemma 1.3.5,C has normal structure.Finally, applying Kirk's fixed point theorem we get that T f has a fixed point in C which proves the theorem.

Application of Sadovskii Fixed Point Theorem
We recall some definitions Definition 6. [1] Let (M, ρ) denote a complete metric space and let B denote the collection of nonempty and bounded subsets of M .Define the Kuratowski measure of noncompactness α : B → R + by taking for A ∈ B, α(A)=inf{ϵ > 0 A is contained in the union of a finite number of sets in B each having diameter less than ϵ}.
If M is a Banach space the function α has the following properties for Definition 7. [2] Let K be a subset of a metric space M .A mapping T : K → M is said to be condensing if T is bounded and continuous and if for all bounded subsets D of M for which α(D) > 0.
We state the Sadovskii fixed point theorem.
Theorem 6. [2] Let K be a nonempty, bounded closed and convex subset of a Banach space and let T : K → K be a condensing mapping, then T has a fixed point.
The main result of section 3 is the following: Theorem 7. Let X be an arbitrary Banach space, let f ∈ X and T : X → X be a condensing mapping, then the operator equation has a solution if and only if for any x 0 ∈ X, the sequence of Picard iterates {x n } in X, defined by Proof.Let the mapping T f : X → X be defined by Then u is a solution of the operator equation if and only if u is a fixed point of T f .Since T is bounded and continuous, T f is also bounded and continuous.Using the properties of the Kuratowski measure of noncompactness, for all bounded subsets D of X, we have Since T is condensing mapping and it follows that T f is a condensing mapping.
Suppose T f has a fixed point u in X.Then for all n ∈ N, since T f is a continuous mapping being condensing, we get Hence {x n } is bounded.
Conversely, suppose that then C n is a nonempty convex set for each n.Using that T is a continuous mapping and the given Picard iteration, we have Applying this, we get the following ⊆ i≥n T f (B d [x i ]) = i≥n {T f (y) : ∥y − x i ∥ ≤ d} = i≥n {(T y + f ) : ∥y − x i ∥ ≤ d} ⊆ i≥n+1 B d [x i ] = C n+1 .Let us define C = n∈N C n .Since C n increases with n, C n ⊂ C n+1 ⊂ C n+2 ⊂ .......,it follows that C is a closed, convex and bounded subset of X.Now we haveT f (C) = T f n∈N C n ⊆ T f n∈N C n = n∈N T f (C n ) ⊆ n∈N C n+1 = C giving T f : C → C since T f is continuous mapping.Finally, applying the Sadovskii fixed point theorem to T f and C, we obtain that T f has a fixed point in C which proves the theorem.https://doi.org/10.17993/3ctic.2022.112.72-79 org/10.17993/3ctic.2022.112.72-79 if and only if u n is a fixed point of T fn .Since T is asymptotically nonexpansive it follows that T fn is nonexpansive for all n ∈ N.