On applications of lattice fixed point theorem (Q1078498)

A lattice fixed point theorem involving two operators on a Banach algebra is proved using the lattice fixed point theorem due to Tarski and nonlinear contraction of Boyd and Wong. The principal result of the paper is the following. Theorem: Let A and B be two operators on a closed and bounded subset S of a Banach algebra X with an order \(\leq\) such that (i) (S,\(\leq)\) is a complete lattice, (ii) A is a nonlinear contraction, (iii) B is bounded on S, (iv) Ax By\(\in S\) for every x,y\(\in S.\) Then the equation Ax Bx\(=x\) has a solution in S and an set of all solutions is a complete lattice. As an application of the above theorem a set of sufficient conditions is given for proving the existence of extremal solutions in the space C of continuous real-valued functions on \(I=[0,1]\), of nonlinear integral equation \[ x(t)=h(t)\int^{a}_{0}k(t,r)g(r,x(r))dr+(\int^{t}_{0}v(t,s)f(s,x(s\;quad))ds)(\int^{a}_{0}k(t,r)g(r,x(r))ds) \] for \(0\leq s\leq t\leq a.\) It is remarked that the results of this paper are useful for obtaining the integral inequalities of Grownwall type for related integral equations which may be used for proving the boundedness and uniqueness of the solution of the related integral equations.