1. Introduction

The study of differential equations is a wide field in pure and applied mathematics, chemistry, physics, engineering and biological science. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics investigated the existence and uniqueness of solutions, but applied mathematics focuses on the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 4 Differential Equations - Theory and Current Research

Following the ordinary differential equations with boundary value condition

$$\frac{d^n x}{dt^n} = f\left(t, x, \frac{d x}{dt}, \dots, \frac{d^{n-1} x}{dt^{n-1}}\right).$$

where <sup>f</sup> : ½ �� <sup>0</sup>; <sup>1</sup> ½ Þ! <sup>0</sup>; <sup>∞</sup> ½ Þ <sup>0</sup>; <sup>∞</sup> is a continuous function and <sup>D</sup><sup>α</sup>

One method for existence and uniqueness of solution of difference equation due to fixed point theory. The primary result in fixed point theory which is known as Banach's contraction principle

Fixed Point Theory Approach to Existence of Solutions with Differential Equations

Theorem 1.1. Let Xð Þ ; d be a complete metric spaces and T : X ! X be a contraction mapping (that is,

d Tx ð Þ ; Ty ≤ αd xð Þ ; y

Since Banach contraction is a very popular and important tool for solving many kinds of mathematics problems, many authors have improved, extended and generalized it (see in [2–4])

In this chapter, we discuss on the existence and uniqueness of the differential equations by

Throughout the rest of the chapter unless otherwise stated ð Þ X; d stands for a complete metric

Definition 2.1. Let <sup>X</sup> be a nonempty set and <sup>T</sup> : <sup>X</sup> ! <sup>X</sup> be a mapping. A point <sup>x</sup><sup>∗</sup> <sup>∈</sup> <sup>X</sup> is said to

Definition 2.2. Let ð Þ X; d be a metric space. The mapping T : X ! X is said to be Lipschitzian if

d Tx ð Þ ; Ty ≤ αd xð Þ ; y for all x, y∈ X:

Definition 2.3. Let F and X be normed spaces over the field K, T : F ! X an operator and c∈F. We say that T is continuous at c if for every ε > 0 there exists δ > 0 such that ∥T xð Þ� T cð Þ∥ < e whenever ∥x � c∥ < δ and x∈ F. If T is continuous at each x ∈F, then T is said to be continuous

Definition 2.4. Let X and Y be normed spaces. The mapping T : X ! Y is said to be completely

continuous if T Cð Þ is a compact subset of Y for every bounded subset C of X.

there exists a constant α > 0 (called Lipschitz constant) such that

A mapping T with a Lipschitz constant α < 1 is called contraction.

Liouville fractional derivative.

there exists 0 ≤ α < 1) such that

and references therein.

2. Basic results

2.1. Fixed point

be a fixed point of <sup>T</sup> if T x<sup>∗</sup> ð Þ¼ <sup>x</sup><sup>∗</sup>:

space.

on T.

was introduced by Banach [1] in 1922.

for all x, y∈ X, then T has a unique fixed point.

using fixed point theory to approach the solution.

<sup>0</sup><sup>þ</sup> is the standard Riemann-

5

http://dx.doi.org/10.5772/intechopen.74560

where y xð Þ¼ <sup>0</sup> 0, y<sup>0</sup> ð Þ¼ <sup>x</sup><sup>1</sup> <sup>c</sup>1, …, yð Þ <sup>n</sup>�<sup>1</sup> ð Þ¼ xn�<sup>1</sup> cn�<sup>1</sup> the positive integer <sup>n</sup> (the order of the highest derivative). This will be discussed. Existence and uniqueness of solution for initial value problem (IVP).

$$\begin{aligned} \mu'(t) &= f(t, \mu(t)), \\ \mu(t\_0) &= \mu\_0. \end{aligned}$$

Differential equations contains derivatives with respect to two or more variables is called a partial differential equation (PDEs). For example,

$$A\frac{\partial^2 u}{\partial x^2} + B\frac{\partial^2 u}{\partial x \partial y} + C\frac{\partial^2 u}{\partial y^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial y} + Fu = G$$

where u is dependent variable and A, B, C, D, E, F and G are function of x, y above equation is classified according to discriminant <sup>B</sup><sup>2</sup> � <sup>4</sup>AC � � as follows,


This will be discussed. Existence of solution for semilinear elliptic equation. Consider a function <sup>u</sup> : <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> that solves,

$$- \Delta u = f(u) \quad \text{in} \quad \Omega$$

$$u = u\_0 \qquad \text{on} \quad \partial \Omega$$

where <sup>f</sup> : <sup>R</sup><sup>m</sup> ! <sup>R</sup><sup>m</sup> is a typically nonlinear function. And fractional differential equations. This will be discussed. Fractional differential equations are of two kinds, they are Riemann-Liouville fractional differential equations and Caputo fractional differential equations with boundary value.

$${}^{c}D\_{t}^{\alpha}u(t) = Bu(t); t > 0$$

$$u(0) = u\_0 \in X$$

where <sup>c</sup> Dα <sup>t</sup> is the Caputo fractional derivative of order α ∈ð Þ 0; 1 , and t ∈½ � 0; τ , for all τ > 0.

The following fractional differential equation will boundary value condition.

$$D\_{0+}^{\\\alpha}u(t) + f(t, u(t)) = 0, \ 0 < t < 1, \ 1 < \alpha \le 2$$

$$u(0) = 0, \ \mu(1) = \int\_0^1 u(s)ds,$$

where <sup>f</sup> : ½ �� <sup>0</sup>; <sup>1</sup> ½ Þ! <sup>0</sup>; <sup>∞</sup> ½ Þ <sup>0</sup>; <sup>∞</sup> is a continuous function and <sup>D</sup><sup>α</sup> <sup>0</sup><sup>þ</sup> is the standard Riemann-Liouville fractional derivative.

One method for existence and uniqueness of solution of difference equation due to fixed point theory. The primary result in fixed point theory which is known as Banach's contraction principle was introduced by Banach [1] in 1922.

Theorem 1.1. Let Xð Þ ; d be a complete metric spaces and T : X ! X be a contraction mapping (that is, there exists 0 ≤ α < 1) such that

$$d(Tx, Ty) \le \alpha d(x, y)$$

for all x, y∈ X, then T has a unique fixed point.

Since Banach contraction is a very popular and important tool for solving many kinds of mathematics problems, many authors have improved, extended and generalized it (see in [2–4]) and references therein.

In this chapter, we discuss on the existence and uniqueness of the differential equations by using fixed point theory to approach the solution.
