On the Existence of Positive Solution for nth Order Differential Equation for Boundary Value Problems

: We considering the problem of solving a nonlinear differential equation in the Banach space of real functions and continuous on a bounded and closed interval. By means of the fixed point theory for a strict set contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a nonlinear higher order boundary value problem.


Introduction
In the current paper, we considering the problem of solving a nonlinear differential equation of nth order. We will try to find the solutions of this equation in the Banach space. The main tool used in our investigations for existence of positive solutions for the nonlinear nth order boundary value problem. Let us mention that the theory of nonlinear differential equations has many useful applications in describing numerous events and problems of the real world. On the other hand, the existence results of positive solutions for integer order differential equations have been studied by several researchers (see [6][7][8][9] and the references therein), but, as far as we know, only a few papers consider the BVP for higher order nonlinear differential equations in Banach space of real functions and continuous on a bounded and closed interval, (see [1,3,5], and the references therein), So, the aim of this paper is to fill this gap. In this paper, we will obtain the existence and nonexistence of positive solution for the BVP (1), (2) and (3) in Banach space. The results presented in this paper seems to be new and original. The generalize equations are often applicable in engineering, mathematical physics, economies, and biology.

Main Result
In this section, we will study the existence and nonexistence of positive solutions for the nonlinear boundary value problem: This is equivelent for an integral equation: (2) and (3), equation (1) has a unique solution.
Proof. Applying the Laplace transform to equation (1) we get Where ( ) and y(s) u s is the Laplace transform of ( ) and y(t) u t respectively. The laplace inversion of Eq.(3.4) gives the final solution as: The proof is complete. Defining : T X X → as:   ( 1)! ! , , , t -t , t t Then the boundary value problem (1), (2) and (3) has a solution.
Proof: Following [2, 4 and 10], we will apply the nonlinear alternative theorem to prove that T has one fixed point. Let , be open subset of X , where We suppose that there is a point u ∈ ∂Ω and 1 (0,1) c ∈ such that u Tu = . So, for u ∈ ∂Ω , we have:  R R R R R R < + + + + + = which implies that T R u ≠ = , that is a contraction. Then the nonlinear alternative theorem implies that T has a fixed point u ∈ Ω , that is, problem(1), (2) and (3) has a solution u ∈ Ω . Finally, we give an example to illustrate the results obtained in this paper.

Conclusion
In This Paper we investigated the existence, nonexistence, and multiplicity of positive solutions for a nonlinear higher order boundary value problem on a bounded and closed interval by means of the fixed point theory for a strict set contraction operator. Let us mention that the theory of nonlinear differential equations has many useful applications in describing numerous events and problems of the real world. The results presented in this paper seem to be new and original. They generalize equations are often applicable in engineering, mathematical physics, economies, and biology.