The Existence and Uniqueness of Positive Solutions of an Ordinary Differential Equation with a Nonlocal Conditions

Many researchers have studied problems with non-local conditions of the second-order differential equations. In this work we study the ordinary differential equation v″(t) + g(t, v(t)) = 0, t ∈ (0,1), with the nonlocal conditions v’(1) = 0, v(0) = Dαv(t)|t=1,α ∈ (0,1). First, we study the existence of at least one positive continuous solution under some assumptions on the function g. Then we discuss the uniqueness of solution by assume that there exist a constant k > 0 such that |g(t,v)-g(t,ῡ)| ≤ |v-ῡ|, ∀ t ∈ [0,1], ∀v, ῡ C[0,1] for this ordinary differential equation, a clarifying example was given as an application. The main idea in this paper is to study ordinary differential equations with a fractional order condition.


Introduction
The differential equations is an important field in mathematical it is used in the modeling of many different phenomena in sciences. The nonlocal boundary value problems for differential equations or difference equations arise in a variety of different areas of applied mathematics, physics, chemistry, control of dynamical systems etc. Recently, many researchers paid attention to existence result of solution of the nonlocal value problem for ordinary differential equations and they gave a lot of time and effort to discuss it, such as [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. On the other hand, the study of boundary value problems of integral and fractional conditions is also important part of nonlocal boundary value problems. In [8], the authors considered that a second-order differential equation had many positive solutions ′′ ( ) + ( ) ( ( )) = 0, ∈ [0, 1] with the nonlocal conditions (0) = 0, (1) = ∫ ( ) ( ) Where 0 < < < 1 and : [ , ] → is an increasing function. Besides, the existence of non-negative solutions of a second-order ordinary differential equation has also been investigated by the authors [9]. . Where : ⨯ 2 → and , : → [0, ∞) are given functions. In this paper, the existence and uniqueness of a positive solution to the non-local boundary value problem of the ordinary differential equation are studied. Where is the Riemann-Liouville fractional-order derivative of order ∈ (0,1).

Preliminaries
In this part, we bring back some fundamental notes and meanings that will be included in this article. Let

The existence of solution
This section discusses the integral representation of the solution to problems (1) and (2). The presence of a positive solution is investigated. Consider the following assumptions for problems (1) and (2): We now give the integral representation of the problem's solution.
Consider the problem (1) -(2) with boundary value. if we Integrate both sides of equation (1) twice, we obtain And from the relation ′ (1) = 0 , we have Now, if we operate the above equation by 1− on both sides, we could get differentiating the last relation, we have And from the relation (0) = ( )| =1 , we have We illustrate in what follows that T is a completely continuous operator.
second, we going to prove that T is a compact operator. For 1 , 2 ∈ (0,1), 1 < 2 such that | 2 − 1 | < we have That implies that at least one positive continuous solution is found in the integral equation (3). ∈ [0,1]. To finish the proof differentiating equation (3) twice we obtain the differential equation (1). Operating on both sides of equation (3)  The unique value problem (8) has, therefore, a special solution on [0, ∞) by Theorem 4.1.

Conclusion
For an ordinary differential equation with fractional-order derivative condition, we proved the existence and uniqueness of continuous positive solutions. Using fixed point theorem methods such as the Banach contraction principle and the Schauder fixed point theorem. Finally, we give an example to make our results clear.