Finding Solutions Of Picard Boundary Value Problem Via Homotopy Method

In this paper, we are concerned with the problem of finding solutions of Picard boundary problemwhere is a continuous function.Picard boundary value problem is a classical nonlinear second order differential equation, the existence of solutions for which has been widely studied. CH.Fabry and P.Habets has presented a method of comparison function to investigate the boundary value problem vector differential equations [1].The boundary value problem(1) (2)can be transformed into the integral equationwherethe mappingis defined bythere C₁ⁿ[a,b] is the Banach space of C¹ functions x : [a,b]→Rⁿ. Consequently, finding solutions of boundary value problem consists in finding fixed points for the mapping T. The existence of a fixed point will be based on the following theorem, which is a simple and classical application of Leray-Schauder degree theory.Lemma 1[1] Let X be a Banach space, A : X→X be a compact mapping such that I - A is one to one, andΩan open boundary set such that 0∈(I - A)Ω. Then the compact mapping T :Ω→X has a fixed point inΩif for anyλ∈(0,1), the equationx =λTx + (I-λ)Ax (5)has no solution x on the boundary (?)ΩofΩ.In our approach, by means of the homotopy method[2], the equation considered is linked to a simpler one by introducing a parameter. Then we can find solutions of Picard boundary value problem by following the path of solutions of the homotopy equation in the case that the solutions of the simpler equation have been determined. Finally, we present our results with some numerical experiments.Let x(t) be a probable solution of the considered equation. Define the setΩwhereis some strictly positive C`2 function andρ> 0. It is clear that such a setΩis open and bounded in C[a,b].Consider the auxiliary problem of Picard boundary value prob- lemwhere is a continuous function,Lemma 2[11] Let x : [a, b]→Rⁿ be an absolutely continuous function with an absolutely continuous derivative. Assume that for almost every t∈[a, b], we haswhere h : R⁺→R⁺\{0} is continuous and satisfiesthenwhere g is defined byIt is easy to show that g is continuous and increasing on R⁺. Our main results state as follows :Theorem 1 Assume that there exist a twice differentiable functionand a continuous functionsatisfy the following conditions: for any (t, x, y), such that t∈[a, b],|x|= Assume moreover that there exist numbersα∈[0,1),β≥0, such that, for any (t, x,y),whereis increasing, continuous and satisfies (9). Then, the problem(1) (2)has a solution x*, such thatFinally, we give some numerical experiments. By means of homotopy method, following the path of homotopy of numerical value until the hyperplane ofλ= 1. Concretely, we begin at y⁰ = (x⁰,λ₀), compute the string of points y¹,y².. , on the curve, such that, each point yⁱ⁺¹ is obtained throw guessing yⁱ = (xⁱ,λ_i), to get z₀, and correcting it by Newton's method.