Fixed Point Theory Approach to Existence of Solutions with Differential Equations Fixed Point Theory Approach to Existence of Solutions with Differential Equations

In this chapter, we introduce a generalized contractions and prove some fixed point theorems in generalized metric spaces by using the generalized contractions. Moreover, we will apply the fixed point theorems to show the existence and uniqueness of solution to the ordinary difference equation (ODE), Partial difference equation (PDEs) and fractional boundary value problem.


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.
Following the ordinary differential equations with boundary value condition d n x dt n ¼ f t; x; where y x 0 ð Þ ¼ 0, y 0 x 1 ð Þ ¼ c 1 , …, y nÀ1 ð Þ x nÀ1 ð Þ¼c nÀ1 the positive integer n (the order of the highest derivative). This will be discussed. Existence and uniqueness of solution for initial value problem (IVP).
Differential equations contains derivatives with respect to two or more variables is called a partial differential equation (PDEs). For example, 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 B 2 À 4AC À Á as follows, This will be discussed. Existence of solution for semilinear elliptic equation. Consider a function u : Ω ⊂ R n ! R n that solves, where f : R m ! R m 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.
The following fractional differential equation will boundary value condition. 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 [ 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][3][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.

Basic results
Throughout the rest of the chapter unless otherwise stated X; d ð Þstands for a complete metric space.

Fixed point
Definition 2.1. Let X be a nonempty set and T : X ! X be a mapping. A point x * ∈ X is said to be a fixed point of T if T x * ð Þ ¼ A mapping T with a Lipschitz constant α < 1 is called contraction. 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 x ð Þ À T c ð Þ∥ < e whenever ∥x À c∥ < δ and x ∈ F. If T is continuous at each x ∈ F, then T is said to be continuous on T.
Definition 2.4. Let X and Y be normed spaces. The mapping T : X ! Y is said to be completely continuous if T C ð Þ is a compact subset of Y for every bounded subset C of X.
Definition 2.5. Compact operator is a linear operator L form a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y such an operator is necessarily a bounded operator, and so continuous.
Definition 2.6. A subset C of a normed linear space X is said to be convex subset in X if λx þ 1 À λ ð Þy ∈ C for each x, y ∈ C and for each scalar λ ∈ 0; 1 ½ .
Theorem 2.8. (Schauder's Fixed Point Theorem) Let X be a Banach space, M ⊂ X be nonempty, convex, bounded, closed and T : M ⊂ X ! M be a compact operator. Then T has a fixed point.
Proof. Form to Jensen's inequality where C is a positive constant depending on a, b, p and r only, since u ∈ L p Ω ð Þ, we have ð Let u n be a sequence converging to u in L p Ω ð Þ. There exists a subsequence u n , and a function g ∈ L p Ω ð Þ such that set, u n 0 ! u x ð Þ, and |u n 0 x ð Þ| ≤ g x ð Þ, almost everywhere. This is sometimes called the generalized DCT, or the partial converse of the DCT, or the Riesz-Fisher Theorem. From the continuity of f , |f u x ð Þ ð ÞÀf u n 0 ð Þ| ! 0 on Ω\ℕ, and where C is another positive constant depending on a, b, p and r only, the left-hand-side is independent of n 0 and is in L 1 Ω ð Þ. We can apply the Dominated Convergence Theorem to conclude the ð ð Þ ! 0: Since the limit does not depend on the subsequence this convergence u holds for u n . □ Corollary 2.10. ref. [5] Let μ ≥ 0. Then the map g ↦ ÀΔ þ μI d À Á À1 g is ii. compact as map form Proof. The first part is due to the fact that A key tool to obtain the compactness of the fixed point maps.

Fuzzy
A fuzzy set in X is a function with domain X and values in 0; 1 ½ . If A is a fuzzy set on X and x ∈ X, then the functional value Ax is called the grade of membership of x in A. The αÀ level set of A, denoted by A α is defined by where denotes by A the closure of the set A. For any A and B are subset of X we denote by H A; B ð Þthe Huasdorff distance.
Definition 2.12. A fuzzy set A in a metric linear space is called an approximate quantity if and only if A α is convex and compact in X for each α ∈ 0; 1 ½ and sup x ∈ X Ax ¼ 1: Let I ¼ 0; 1 ½ and W X ð Þ⊂ I X be the collection of all approximate in X. For α ∈ 0; 1 ½ , the family W α X ð Þ is given by A ∈ I X : A α È is nonempty and compact}.
For a metric space X; d ð Þwe denoted by V X ð Þ the collection of fuzzy sets A in X for which A α is compact and supAx ¼ 1 for all α ∈ 0; 1 ½ . Clearly, when X is a metric linear space W X ð Þ⊂ V X ð Þ: where H is the Hausdorff distance. Denote with Φ, the family of nondecreasing function ϕ : 0; þ∞ ½ Þ! 0; þ∞ ½ Þ such that P ∞ n¼1 ϕ n t ð Þ < ∞ for all t > 0: Þbe a complete ordered metric space and T 1 , T 2 : X ! W α X ð Þ be two fuzzy mapping satisfying for all comparable element x, y ∈ X, where L ≥ 0 and Also suppose that ii. if x, y ∈ X are comparable, then every u ∈ T 1 x ð Þ α and every v ∈ T 2 y ð Þ α are comparable, iii. if a sequence x n f g in X converges to x ∈ X and its consecutive terms are comparable, then x n and x are comparable for all n.
Then there exists a point x ∈ X such that x α ⊂ T 1 x and x α ⊂ T 2 x: Proof. See in [6]. ð Þbe a complete ordered metric space and T 1 , T 2 : X ! W α X ð Þ be two fuzzy mappings satisfying for all comparable elements x, y ∈ X. Also suppose that ii. if x, y ∈ X are comparable, then every u ∈ T 1 x ð Þ α and every v ∈ T 2 y ð Þ α are comparable, iii. if a sequence x n f g in X converges to x ∈ X and its consecutive terms are comparable, then x n and x are comparable for all n.
Then there exists a point x ∈ X such that x α ⊂ T 1 x and x α ⊂ T 2 x:

Metric-like space
Definition 2.17. [7] Let X be nonempty set and function p : X Â X ! R þ be a function satisfying the following condition: for all x, y, z ∈ X, Then p is called a partial metric on X, so a pair X; p ð Þis said to be a partial metric space.
If for all x, y, z ∈ X, the following conditions hold: Then a pair X; σ ð Þis called a metric-like space.
It is easy to see that a metric space is a partial metric space and each partial metric space is a metric-like space, but the converse is not true but the converse is not true as in the following examples: Þis a metric-like space, but it is not a partial metric space, cause σ 0; 0 ð Þ≰σ 0; 1 ð Þ: Lemma 2.20. ref. [9] Let X; p ð Þbe a partial metric space. Then i.
x n f g is a Cauchy sequence in X; p ð Þ if and only if it is a Cauchy sequence in the metric ii. X is complete if and only if the metric space X; d p À Á is complete. ii.
A sequence x n f g in X; σ ð Þis said to be a 0 À σÀ Cauchy sequence if lim n, m!∞ σ x n ; x m ð Þ¼0: The space X; σ ð Þ is said to be 0 À σÀ complete if every 0 À σÀ Cauchy sequence in X converges (in τ σ ) to a point x ∈ X such that σ x; x ð Þ ¼ 0: iii. A mapping T : X ! X is continuous, if the following limits exist (finite) and lim n!∞ σ x n ; x ð Þ¼σ Tx; x ð Þ: Following Wardowski [11], we denote by F the family of all function, F : R þ ! R satisfying the following conditions: for all x, y ∈ X and α, β, γ, η, δ ≥ 0 with α þ β þ γ þ 2η þ 2δ < 1.
Theorem 2.25. ref. [12] Let X; σ ð Þ be 0 À σÀ complete metric-like spaces and T : X ! X be a generalized Roger Hardy type FÀ contraction. Then T has a unique fixed point in X, either T or F is continuous.

Modular metric space
Let X be a nonempty set. Throughout this paper, for a function ω : for all λ > 0 and x, y ∈ X: Definition 2.26 [13,14] Let X be a nonempty set. A function ω : 0; ∞ ð ÞÂX Â X ! 0; ∞ ½ is called a metric modular on X if satisfying, for all x, y, z ∈ X the following conditions hold: i.
If instead of (i) we have only the condition (i') then ω is said to be a pseudomodular (metric) on X: A modular metric ω on X is said to be regular if the following weaker version of (i) is satisfied: x ¼ y if and only if ω λ x; y ð Þ ¼ 0 for some λ > 0: Note that for a metric (pseudo)modular ω on a set X, and any x, y ∈ X, the function λ ↦ ω λ x; y ð Þis nonincreasing on 0; ∞ ð Þ: Indeed, if 0 < μ < λ, then Note that every modular metric is regular but converse may not necessarily be true.
it is easy to verify that ω is regular modular metric but not modular metric.
Definition 2.28. [13,14] Let X ω be a (pseudo)modular on X: Fix x 0 ∈ X: The two sets are said to be modular spaces (around x 0 ).
Throughout this section we assume that X; ω ð Þ is a modular metric space, D be a nonempty subset of X ω and G≔ G ω f is a directed graph with V G ω ð Þ ¼ D and Δ⊆E G ω ð Þg: Definition 2.29. [15,16] The pair D; G ω ð Þhas Property (A) if for any sequence x n f g n ∈ ℕ in D, with x n ! x as n ! ∞ and x n ; ii. there exists a number τ > 0 such that for all x, y ∈ D with Rx; Ry ð Þ∈ E G ω ð Þ: Example 2.31. ref. [17] Let F ∈ F be arbitrary. Then every F-contractive mapping w.r.t. R is an We denote C T; R ð Þ≔ x ∈ D : Tx ¼ Rx f g the set of all coincidence points of two self-mappings T and R, defined on D.
Theorem 2.32. ref. [17] Let X; ω ð Þbe a regular modular metric space with a graph G ω : Assume that D ¼ V G ω ð Þ is a nonempty ω-bounded subset of X ω and the pair D; G ω ð Þhas property (A) and also satisfy Δ M -condition. Let R, T : D ! D be two self mappings satisfying the following conditions: Proof. See in [17]. □

Fixed point approach to the solution of differential equations
Next, we will show a differential equation which solving by fixed point theorem in suitable spaces.
Proof. Suppose that u is a solution of u 0 t ð Þ ¼ f t; u t ð Þ ð Þdefined on an interval I and satisfying u t 0 ð Þ ¼ u 0 . We integrate both sides of the equation Þds: Þds, t ∈ I: (3.1) We will show that, conversely, any function which satisfies this integral equation satisfies both the differential equation and the initial condition. Suppose that u is a function defined on an interval I and satisfies (3.1). Setting t ¼ t 0 yields u t 0 ð Þ ¼ u 0 , so that u satisfies the initial condition. Next, we note that an integral is always a continuous function, so that a solution of (3.1) is automatically continuous. Since both u and f are continuous, it follows that the integrand f s; u s ð Þ ð Þis continuous. We may therefore apply the fundamental theorem of calculus to (3.1) and conclude that u is differentiable, and that is The contraction mapping theorem may by used to prove the existence and uniqueness of the initial problem for ordinary differential equations. We consider a first-order of ODEs for a function u t ð Þ that take value in R n 3) The function f t; u t ð Þ ð Þalso take value in R n and is assumed to be a continuous function of t and a Lipschitz continuous function of u on suitable domain.
We say that f t; u t ð Þ ð Þis a globally Lipschitz continuous function of u uniformly in t if there is a constant C > 0 such that for all x, y ∈ R n and all t ∈ I.
The initial value problem can be reformulated as an integral equation. Proof. We will show that T is a contraction on the space of continuous function defined on a time interval t 0 ⩽ t ⩽ t 0 þ δ, for sufficiently small δ.
Suppose that u, v : t 0 ; t 0 þ δ ½ !R n are two continuous function. Then, form (3.4), (3.5) we estimate, Let f x; y ð Þ be a continuous real-valued function on a; b ½ Â c; d ½ . The Cauchy initial value problem is to find a continuous differentiable function y on a; b ½ satisfying the differential equation Consider the Banach space C a; b ½ of continuous real-valued functions with supremum norm defined by ∥y∥ ¼ sup y x ð Þj : x ∈ a; b ½ f g : Integrating (3.6), that yield an integral equation The problem (3.6) is equivalent the problem solving the integral Eq. (3.7).
We define an integral operator T : C a; b ½ ! C a; b ½ by Therefore, a solution of Cauchy initial value problem (3.6) corresponds with a fixed point of T.
One may easily check that if T is contraction, then the problem (3.6) has a unique solution.
Then C is a closed subset of the complete metric space C x 0 À h; x 0 þ h ½ and hence C is complete. Note T : C ! C is a contraction mapping. Indeed, for x ∈ x 0 À h; x 0 þ h ½ and two continuous functions y 1 , y 2 ∈ C, we have Therefore, T has a unique fixed point implying that the problem (3.6) has a unique fixed point.

Ordinary fuzzy differential equation
Now, we consider the existence of solution for the second order nonlinear boundary value problem: where the Green's function G is given by If necessary, for a more detailed explanation of the background of the problem, the reader can refer to the reference [21,22]. Here, we will prove our results, by establishing the existence of a common fixed point for pair of integral operators defined as Theorem 3.5 ref. [6] Assume that the following conditions are satisfied:
Then the pair of nonlinear integral equations has a common solution in The C; D ð Þ is a complete metric space, which can also be equipped with the partial ordering given by x, y ∈ C, ⇔ x t ð Þ ≤ y t ð Þ for all t ∈ 0; Λ ½ : In [23], it is proved that C; ≼ ð Þsatisfies the following condition: (r) for every nondecreasing sequence x n f g in C convergent to some x ∈ C, we have x n ≼x for all n ∈ ℕ ∪ 0 f g.
Let T 1 , T 2 : C ! C be two integral operators defined by (3.10); clearly, T 1 , T À 2 are well defined since k 1 , k 2 , and β are continuous functions. Now, x * is a solution of (3.9) if and only if x * is a common fixed point of T 1 and T 2 .
Next, for all comparable x, y ∈ C, From hypothesis (c) we obtain successively and (3.13) From (3.12) and (3.13), we obtain easily Consequently, in view of hypothesis (d), the contractive condition (5) is satisfied with Therefore, Corollary 2.16 applied to T 1 and T 2 , which have common fixed point x * ∈ C, that is, x * is a common solution of (3.9). □

Second-order differential equation
Now, we consider the boundary value problem for second order differential equation where I ¼ 0; 1 ½ and f : I Â R ! R: is a continuous function.
It is known, and easy to check, that the problem (3.14) is equivalent to the integral equation Þds, for t ∈ I, (3.15) where G is the Green's function define by x is a solution of problem (3.14) iff x is a solution of the integral Eq. (3.15).
Let X ¼ C I ð Þ be the space of all continuous functions defined on I. Consider the metric-like σ on X define by Note that σ is also a partial metric on X and since d σ x; y ð Þ≔2σ x; y ð ÞÀσ x; x ð ÞÀσ y; y ð Þ ¼ 2∥x À y∥ ∞ : By Lemma 2.20, hence X; σ ð Þis complete since the metric space X; ∥ Á ∥ ð Þis complete. ii. there exist continuous functions q : for all s ∈ I and a ∈ R; iii. max s ∈ I p s ð Þ ¼ αλ 1 < 1 49 , which is 0 ≤ α < 1 7 ; iv. max s ∈ I q s ð Þ ¼ αλ 2 < 1 49 which is 0 ≤ α < 1 7 : Then problem (3.14) has a unique solution u ∈ X ¼ C I; R ð Þ.
Proof. Define the mapping T : X ! X by for all x ∈ X and t ∈ T: Then the problem (3.14) is equivalent to finding a fixed point u of T in X. Let x, y ∈ X, we obtain |Tx t ð Þ À Ty t ð Þ| ¼ | ÞÀf ðs, y s ð Þ|ds In the above equality, we used that for each t ∈ I, we have  where α þ β þ γ þ 2η þ 2δ < 1. Taking the function F : Therefore all hypothesis of Theorem (2.25) are satisfied, and so T has a unique fixed point u ∈ X, that is, the problem (3.14) has a unique solution u ∈ C 2 I ð Þ: □

Partial differential equation
Consider the Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence ∇Á ð Þ of the gradient ∇f ð Þ. Thus if f is a twicedifferentiable real-valued function, then the Laplacian of f is defined by where the latter notations derive from formally writing ∇ ¼ ∂ ∂x1 ; ∂ ∂x2 ; ⋯; ∂ ∂xn . Equivalently, the Laplacian of f the sum of all the unmixed As a second-order differential operator, the Laplace operator maps C k functions to C kÀ2 functions for k ≥ 2. the expression (3.21)(or equivalently(3.22)) defines an operator Δ : for any open set Ω Consider semilinear elliptic equation. Look for a function u : Ω ⊂ R n ! R m that solves where f : R n ! R m is a typically nonlinear function. Equivalently look for a fixed point of Theorem 3.7. ref. [5] Let f ∈ C R ð Þ and sup x ∈ R |f x ð Þ| ¼ a < ∞. then (3.23) has a weak solution Proof. Our strategy is to apply Schauder's Fixed Point Theorem to the map where T is continuous. Lemma (2.9) show that u ! f u ð Þ is continuous form L 2 Ω ð Þ into itself.
Corollary (2.10) shows that ÀΔ ð Þ À1 is continuous form Find a closed, non-empty bounded convex set such that T : M ! M.
Cauchy-Schwarz. T here fore, using Ponincare's inequality We have established that T : M ! M, T is compact. Using Poincare's inequality on the right-hand-side in (3.25), we obtain. ∥∇Tu∥ 2 , and since the embedding

A non-homogeneous linear parabolic partial differential equation
We consider the following initial value problem where H is continuous and φ assume to be continuously differentiable such that φ and φ 0 are bounded.
By a solution of the problem (3.26), we mean a function u u x; t ð Þ defined on R Â I, where I≔ 0; T ½ , satisfying the following conditions: ii. u and u x are bounded in R Â I, It is important to note that the differential equation problem (3.26) is equivalent to the following integral equation problem for all x ∈ R and 0 < t ≤ T, where The problem (3.26) admits a solution if and only if the corresponding problem (3.27) has a solution. Let Obviously, the function ω : R þ Â Ω Â Ω ! R þ given by is a metric modular on Ω. Clearly, the set Ω ω is a complete modular metric space independent of generators.
Theorem 3.8. ref. [17] Consider the problem (3.26) and assume the following: i. for c > 0 with |s| < c and |p| < c, the function F x; t; s; p ð Þis uniformly Hölder continuous in x and t for each compact subset of R Â I, iii. H is bounded for bounded s and p: Then the problem (3.26) admits a solution.
Proof. It is well known that u ∈ Ω ω is a solution (3.26) iff u ∈ Ω ω is a solution integral Eq. (3.27). Now, from example 2.22 (i) and taking T ¼ Λ and R ¼ I (Identity map), we deduce that the operator T satisfies all the hypothesis of theorem 2.32.
Therefore, as an application of theorem 2.32 we conclude the existence of u * ∈ Ω ω such that u * ¼ Λu * and so u * is a solution of the problem 3.26.

Fractional differential equation
Before we will discuss the source of fractional differential equation.
Cauchy's formula for repeated integration. Let f be a continuous function on the real line. Then the n th repeated integral of f based at a, ð σnÀ1 a f σ n ð Þdσ n …dσ 3 dσ 2 dσ 1 is given by single integration A proof is given by mathematical induction. Since f is continuous, the base case follows from the fundamental theorem of calculus.
Now, suppose this is true for n, and let us prove it for n þ 1.
Firstly, using the Leibniz integral rule. Then applying the induction hypothesis This completes the proof. In fractional calculus, this formula can be used to construct a notion of differintegral, allowing one to differentiate or integrate a fractional number of time.
Integrating a fractional number of time with this formula is straightforward, one can use fractional n by interpreting n À 1 ð Þ! as Γ n ð Þ, that is the Riemann-Liouville integral which is defined by This also makes sense if a ¼ À∞, with suitable restriction on f . The fundamental relation hold the latter of which is semigroup properties. These properties make possible not only the definition of fractional differentiation by taking enough derivative of I α f . One can define fractional-order derivative of as well by where Á ½ denote the ceilling function. One also obtains a differintegral interpolation between differential and integration by defining An alternative fractional derivative was introduced by Caputo in 1967, and produce a derivative that has different properties it produces zero from constant function and more importantly the initial value terms of the Laplace Transform are expressed by means of the value of that function and of its derivative of integer order rather than the derivative of fractional order as in the Riemann-Liouville derivative. The Caputo fractional derivative with base point x is then Lemma 3.9. ref. [24] Let u : 0; ∞ ½ !X be continuous function such that u ∈ C 0; τ ½ ; X ð Þfor all τ > 0. Then u is a global solution of if and only if u the integral equation (ii) It is trivial that U 0 t ð Þ ¼ u 0 . So we compute, using the gamma function properties, that By a simple induction process, we conclude that □ From the above works, we can see a fact, although the fractional boundary value problems have been studied, to the best of our knowledge, there have been a few works using the lower and upper solution method. However, only positive solution are useful for many application, motivated by the above works, we study the existence and uniqueness of positive solution of the following integral boundary value problem. where f : 0; 1 ½ Â 0; ∞ ½ Þ! 0; ∞ ½ Þ is a continuous function and D α 0þ is the standard Riemann-Liouville fractional derivative.
In the following, we present the Green function of fractional differential equation with integral boundary value condition.