Browsing by Subject "Boundary value problems."
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Item Existence and uniqueness of solutions of boundary value problems by matching solutions.(2013-09-24) Liu, Xueyan, 1978.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.In this dissertation, we investigate the existence and uniqueness of boundary value problems for the third and nth order differential equations by matching solutions. Essentially, we consider the interval [a, c] of a BVP as the union of the two intervals [a, b] and [b, c], analyze the solutions of the BVP on each, and then match the proper ones to be the unique solution on the whole domain. In the process of matching solutions, boundary value problems with different boundaries, especially at the matching point b, would be quite different for the requirements of conditions on the nonlinear term. We denote the missing derivatives in the boundary conditions at the matching point b by k₁ and k₂. We show how y(ᵏ²)(b) varies with respect to y(ᵏ¹)(b), where y is a solution of the BVP on [a, b] or [b, c]. Under certain conditions on the nonlinear term, we can get a monotone relation between y(ᵏ²)(b) and y(ᵏ¹)(b), on [a, b] and [b, c], respectively. If the monotone relations are different on [a, b] and [b, c], then we can finally get a unique value for y(ᵏ¹)(b) where the k₂nd derivative of two solutions on [a, b] and [b, c] are equal and we can join the two solutions together to obtain the unique solution of our original BVP. If the relations are the same, then we will arrive at the situation that the k₂nd order derivatives of two solutions at b on [a, b] and [b, c] are decreasing with respect to the k₁st derivatives at b at different rates, and by analyzing the relations more in detail, we can finally get a unique value for the k₁st derivative of solutions of BVP's on [a, b] and [b, c], which are matched to be a unique solution of the BVP on [a, c]. In our arguments, we use the Mean Value Theorem and the Rolle's Theorem many times. As the simplest models, third order BVP's are considered first. Then, in the following chapters, nth order problems are studied. Lastly, we provide an example and some ideas for our future work.Item Existence of positive solutions to singular right focal boundary value problems.(Orlando, FL : International Publications., 2005-05) Maroun, Mariette.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.In this dissetation, we seek positive solutions for the n^th order ordinary differential equation, y^(n)=f(x,y), satisfying the right focal boundary conditions, y^(i)(0)=y^(n-2)(p)=y^(n-1)(1)=0, i=0,...,n-3, where p is a fixed value between 1/2 and 1, and where f(x,y) has certain singulrities at x=0, at y=0, and possibly at y=infinity.Item A functional approach to positive solutions of boundary value problems.(2007-05-23T19:30:41Z) Ehrke, John E.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.We apply a well-known fixed point theorem to guarantee the existence of a positive solution and bounds for solutions for second, third, fourth, and nth order families of boundary value problems. We begin by characterizing second order problems having left and right focal boundary conditions. Via an appropriate substitution, associated third, fourth, and nth order problems are resolved. Our main result centers on the nth order equation y(n) + f(y(t)) = 0, t [is an element of] [0, 1], (1) having boundary conditions, y(ri−1)(0) = 0, 1 < i < k, (2) y(sj−1)(1) = 0, 1 < j < n − k, (3) where {s1, · · · , sn−k} and {r1, · · · , rk} form a partition of {1, · · · , n} such that r1 < · · · < rk, s1 < · · · < sn−k, and {rk−1 · · · rk} [is not equal to] {n − 1, n} and {sn−k−1, sn−k} [is not equal to] {n − 1, n}. Under these assumptions we show that the differential equation (1) with boundary conditions (2) and (3) has a positive solution for all n [is greater than or equal to] 2.Item Multiplicity of positive solutions of even-order nonhomogeneous boundary value problems.(2009-06-02T17:58:13Z) Hopkins, Britney.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.In this work, we discuss multiplicity results for nonhomogeneous even-order boundary value problems on both discrete and continuous domains. We develop a method for establishing existence of positive solutions by transforming even-order problems into a series of second order problems satisfying homogeneous boundary conditions. We then construct a sequence of lemmas which give contraction and expansion relationships within a cone. This allows us to apply the Guo-Krasnosel'skii Fixed Point Theorem which, in turn, guarantees several positive solutions.Item Positive solutions of singular boundary value problems.(2007-05-23T16:28:06Z) Kunkel, Curtis J.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.In this dissertation, we focus on singular boundary value problems with mixed boundary conditions. We study a variety of types, to all of which we seek a positive solution. We begin by considering the discrete (or difference equation) case, from which we proceed to look at the continuous (or ordinary differential equation) case. In all cases, we make use of a lower and upper solutions method and the Brouwer fixed point theorem in conjunction with perturbation methods to approximate regular problems.