Common Fixed Point Theorems With Applications to Nonlinear Integral Equations

Abstract

Various definitions of compatible maps are characterized and compared in terms of continuity of maps, fixed points and certain limits. Common fixed point theorems for a new class of compatible maps on complete metric spaces are proved. Subsequently, the main result is applied to prove the existence of solutions of two systems of nonlinear integral equation.

Remarks on Recent Fixed Point Theorems and Applications to Integral Equations

“…Then, as shown in [15], conditions (i)-(iv) imply that Ai and Ti, ¿ = 1,2, are operators from Z into itself, and Proof. For any x G Z, define Aix(t) = gi(t,x(t)) and Tix(t) as in the proof of Theorem 9, i = 1,2.…”

“…For any x G Z, define Aix(t) = gi(t,x(t)) and Tix(t) as in the proof of Theorem 9, i = 1,2. Then, as shown in [15], A{ and Tj, i = 1,2 are self-operators of Z and \\A\x -A2y\\ < 6\\T\x -T2y\\, where 6 = K2/( 1 -\\\M2KI) < 1. The condition (v*) implies AX{Z) C T2{Z) and A2(Z) C Ti(Z), while (vi*) ensures that any one of A^Z), A2(Z), TX(Z) or T2{Z) is a complete subspace of Z.…”

“…Further, assuming the commutativity at a coincidence point only, we show the existence of a unique common solution of these equations. We generally follow the notations and definitions used in [15]. Let L[a, oo) denote the Banach space of real or complex valued functions that are defined and Lebesgue integrable on [o, oo) equipped with its usual norm, where a is any real number.…”

“…Equations (VH) and (H) are of interest to industrial problems, and for relevant comments, we refer to [15] and [19].…”

On fixed soft element theorems in se-uniform spaces

“…Also, this theory have been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches of mathematics [16,30,31]. A basic result in fixed point theory is the Banach contraction principle.…”