Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations

Abstract

In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Fréchet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Hölder condition; however, these operators verify the omega condition established. Then, the semilocal convergence balls are obtained and the R-order of convergence and error bounds can be obtained by following thee main theorem. Finally, we perform a numerical experience by solving a nonlinear Hammerstein integral equations in order to show the applicability of the theoretical results by obtaining the existence and uniqueness balls.

Extended domain for fifth convergence order schemes

“…Then, as in [13,14], let r 0 = M β 0 η 0 , s 0 = β 0 η 0 ω(η 0 ) and define sequences {r k }, {s k } and {η k } for k = 0, 1, 2, . .…”

“…Obviously G (x) is not bounded on Ω. So, the convergence of scheme (1.2) is not guaranteed by the analysis in [13,14]. In this study we use only assumptions on the first derivative to prove our results.…”

“…The explosion of technology requires the development of higher convergence schemes. Starting from the quadratically convergent Newton's method higher order schemes develop all the time [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19].…”

“…Later in [14] the applicability of scheme (1.2) was extended using w-continuity conditions. In general, the convergence domain is small.…”

“…4. It is worth noticing that method (1.2) is not changing when we use the conditions of Theorem 2.1 instead of the stronger conditions used in [13,14]. Moreover, we can compute the computational order of convergence (COC) defined by…”