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• W2227779008 abstract "We discuss derivatives of the solution of the second order parameter dependent boundary value problem with an integral boundary condition ( ) ′′ ′ = ,      1 1 ( ) =    , 2 2 ( ) ( )            and its relationship to a second order nonhomogeneous differential equation which corresponds to the traditional variational equation. Specifically, we show that given a solution ( )   of the boundary value problem, the derivative of the solution with respect to the parameter  is itself a solution to the aforementioned nonhomogeneous equation with interesting boundary conditions. Introduction In this paper, we differentiate, with respect to  , the solution of the nonlocal second order differential equation ( ) in ′′ ′ = ,       (1) with parameter  and boundary conditions 1 1 2 2 ( ) ( ) ( )                  (2) where       in  ; note the integral boundary condition. In particular, we show the Corresponding author: Jeffrey W. Lyons, Department of Mathematics, Halmos College of Natural Sciences and Oceanography, Nova Southeastern University, USA. E-mail: jlyons@nova.edu. The Derivative of a Solution to a Second Order Parameter Dependent Boundary Value Problem with a Nonlocal Integral Boundary Condition 44 relationship between the derivative of the solution ( )   to (1), (2) and the nonhomogeneous equation along ( )   ′′ ′ = + + .            (3) The nonhomogeneous equation (3) is a corresponding equation of the traditional variational equation along ( )   ′′ ′ = + .          (4) The study of the relationship between the derivative of a solution to a differential equation and its associated variational equation is first attributed to Peano by Hartman in [5]. The theory presented in Hartman’s book was developed for initial value problems. Since then, much research has been done to find analogues of Peano’s work to various types of boundary value problems. One can find examples with a wide variety of domains and boundary conditions as can be seen in [3, 4, 11, 14] for differential equations, [1, 2, 7, 8, 12] for difference equations, and [13] for dynamic equations on time scales. The preceding papers were written with a parameter independent differential equation. The primary motivation for the research presented here is from two articles [10] and [6]. In the former, the authors studied differentiation of solutions of a differential equation with an integral boundary condition and parameter independence. The second article covers a discrete version of the topic but introduces the parameter  . The work presented here expands directly upon that found in [10] by adding parameter dependence and studying the derivative of the solution with respect to theparameter. One interesting theme in each of the preceding works is the method of writing the boundary value problem in terms of an initial value problem which in turn allows the author to apply a version of Peano’s theorem for initial value problems which leads to the desired result. The same technique is employed to achieve our results here. The remainder of the paper is arranged as follows. Section 2 develops the conditions necessary for our theorems and introduces the theorems employed to obtain our results. To conclude, in Section 3 the main result of the paper is presented and followed with its proof. Assumptions and Theorems Before presenting the results of the paper, we will impose the following hypotheses. First, we require The Derivative of a Solution to a Second Order Parameter Dependent Boundary Value Problem with a Nonlocal Integral Boundary Condition 45 that the differential equation (1) and its derivatives are continuous: (i) 3 ( ) ( ) : × →          is continuous, (ii) 3 ( ) ( ) ∂ / ∂ : × →            is continuous ( 1 2) = ,  , and (iii) 3 ( ) ( ) ∂ / ∂ : × →           is continuous. Second, we require that solutions of (1) be unique implying the condition: (iv) Given , ∈   and       in  , if ( )   and ( )   are solutions of (1), (2), then on ( ) " @default.
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• W2227779008 title "The Derivative of a Solution to a Second Order Parameter Dependent Boundary Value Problem with a Nonlocal Integral Boundary Condition" @default.
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