# Fixed-Point Theorems in Complete Gauge Spaces and Applications to Second-Order Nonlinear Initial-Value Problems

Authors: Meryam Cherichi; Bessem Samet; Calogero Vetro
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We establish fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Our theorems generalize and extend some fixed-point results in the literature. We apply our obtained results to the study of existence and uniqueness of solution to a second-order nonlinear initial-value problem.


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## Body

## 1. Introduction

In the context of fixed-point theory, the metric fixed-point theory is the branch where metric conditions on the involved spaces and mappings play a crucial role in establishing theoretical results. In a certain sense, this theory is a far-reaching outgrowth of a well-known theorem of Banach [1] which states that if (X,d) is a complete metric space and if T:X→X is a mapping which satisfies the condition
(1)d(Tx,Ty)≤kd(x,y),∀x,y∈X,
where 0≤k<1 is a constant, then T has a unique fixed point. Since most of the spaces studied in mathematical analysis share many algebraic and topological properties as well as metric properties, there is no clear line separating the metric fixed-point theory from the topological or the set-theoretic branch of the theory. Consequently, several authors considered the problem of existence (and uniqueness) of a fixed point for generalized contractions in a metric space (see, e.g., [2–9]). On the other hand, many definitions and theorems in the literature do not require that all of the properties of a metric hold true. For this reason, in the last decades, various concepts of generalized metrics were introduced (see, e.g., [10, 11]). Here, we are interested in the so-called gauge spaces that are characterized by the fact that the distance between two points of the space may be zero even if the two points are distinct. For instance, Frigon [12] and Chiş and Precup [13] gave generalizations of the Banach contraction principle on gauge spaces (see also [14–16]). Consistent with this line of research, the aim of this paper is to present some fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Then, to illustrate the usefulness of our theorems, we apply our results to the study of existence and uniqueness of solutions to a second-order nonlinear initial-value problem.


## 2. Preliminaries

In this section, we recall some preliminaries on gauge spaces and introduce some basic definitions.Definition 1.
LetX be a nonempty set. A function d:X×X→[0,+∞) is called a pseudometric in X whenever the following hold:(i)
d(x,x)=0 for all x∈X,(ii)
d(x,y)=d(y,x) for all x,y∈X,(iii)
d(x,y)≤d(x,z)+d(z,y) for all x,y,z∈X.Definition 2.
LetX be a nonempty set endowed with a pseudometric d. The d-ball of radius ε>0 centered at x∈X is the set
(2)B(x;d,ε)={y∈X∣d(x,y)<ε}.Definition 3.
A familyℱ={dλ∣λ∈𝒜} of pseudometrics is called separating if, for each pair (x,y) with x≠y, there is a dλ∈ℱ such that dλ(x,y)≠0.Definition 4.
LetX be a nonempty set, and let ℱ={dλ∣λ∈𝒜} be a separating family of pseudometrics on X. The topology 𝒯(ℱ) having for a subbasis the family
(3)ℬ(ℱ)={B(x;dλ,ε)∣x∈X,dλ∈ℱ,ε>0}
of balls is called the topology in X induced by the family ℱ. The pair (X,𝒯(ℱ)) is called a gauge space. Note that (X,𝒯(ℱ)) is Hausdorff because we require ℱ to be separating.Definition 5.
Let(X,𝒯(ℱ)) be a gauge space with respect to the family ℱ={dλ∣λ∈𝒜} of pseudometrics on X. Let {xn} be a sequence in X, and x∈X. Then, the following are considered.(a)
The sequence{xn} converges to x if and only if
(4)∀λ∈𝒜,∀ε>0,∃N∈ℕsuch  thatdλ(xn,x)<ε,∀n≥N.
In this case, we denote that xn→ℱx.(b)
The sequence{xn} is Cauchy if and only if
(5)∀λ∈𝒜,∀ε>0,∃N∈ℕsuch  thatdλ(xn+p,xn)<ε,∀n≥N,p∈ℕ.(c)
(X,𝒯(ℱ)) is complete if and only if any Cauchy sequence in (X,𝒯(ℱ)) is convergent to an element of X.(d)
A subset ofX is said to be closed if it contains the limit of any convergent sequence of its elements.For more details on gauge spaces, we refer the reader to [17]. Obviously, every metric space is automatically a pseudometric space. On the contrary, if a pseudometric d is not a metric, it is because there are at least two points x≠y for which d(x,y)=0. In most situations, this does not happen; indeed, metrics come up in mathematics more often than pseudometrics. However, pseudometrics arise in a natural way in functional analysis and in the theory of hyperbolic complex manifolds [18].


## 3. Main Results

LetΨ denote the set of all functions ψ:[0,+∞)→[0,+∞) which satisfy the following conditions. (i)
ψ is continuous and nondecreasing.(ii)
ψ(t)=0 if and only if t=0.Similarly, let Φ denote the set of all functions φ:[0,+∞)→[0,+∞) which satisfy the following conditions.  (i)
ϕ is lower semicontinuous.(ii)
ϕ(t)=0 if and only if t=0.Definition 6.
Let(X,𝒯(ℱ)) be a gauge space, and let T:X→X be a mapping. For x∈X, let 𝒪(x) denote the orbit of x; that is, 𝒪(x)={Tnx,n=0,1,…}.Our first result is the following theorem.Theorem 7.
Let(X,𝒯(ℱ)) be a complete gauge space, and let T:X→X be a mapping satisfying the following conditions.(i)
For allλ∈𝒜, for all u∈X, for all x∈𝒪(u), and for all y∈𝒪(u)¯,dλ(x,y)=0⇒dλ(Tx,Ty)=0.(ii)
For allλ∈𝒜 and for all 0<a<b<∞, there exist ψλ,a,b∈Ψ and φλ,a,b∈Φ such that
(6)∀x,y∈X,a≤dλ(x,y)≤b⟹ψλ,a,b(dλ(Tx,Ty))≤ψλ,a,b(dλ(x,y))-φλ,a,b(dλ(x,y)).Then, T has a unique fixed point.Proof.
First, we will remark that(7)∀λ∈𝒜,∀x,y∈X,dλ(x,y)≠0⟹dλ(Tx,Ty)<dλ(x,y).
In fact, let λ∈𝒜, and let x,y∈X such that dλ(x,y)≠0; by taking a=dλ(x,y) and b=2dλ(x,y), we get, by applying assumption (ii) of the theorem, the existence of ψλ,a,b∈Ψ and φλ,a,b∈Φ such that
(8)ψλ,a,b(dλ(Tx,Ty))≤ψλ,a,b(dλ(x,y))-φλ,a,b(dλ(x,y)).
Since dλ(x,y)≠0, we have φλ,a,b(dλ(x,y))>0, and so we get
(9)ψλ,a,b(dλ(Tx,Ty))<ψλ,a,b(dλ(x,y)).
Using the monotony of ψλ,a,b, we obtain the desired result (7). Moreover, assumption (i) implies that
(10)∀λ∈𝒜,∀u∈X,∀x∈𝒪(u),∀y∈𝒪(u)¯,dλ(Tx,Ty)≤dλ(x,y).
Now, let us takex0∈X, and let {xn} be the sequence in X given by
(11)xn+1=Txn,∀n≥0.
Thus, we complete the proof in the following four steps.
Step
1. We will prove that
(12)dλ(xn+1,xn)⟶0asn⟶+∞,∀λ∈𝒜.
Let λ∈𝒜. Since xn,xn+1∈𝒪(x0), by applying (10), we obtain
(13)dλ(xn+1,xn)=dλ(Txn,Txn-1)≤dλ(xn,xn-1),∀n≥1.
It follows that {dλ(xn+1,xn)} is a decreasing sequence of nonnegative real numbers, and, hence, there is r=r(λ)≥0 such that
(14)dλ(xn+1,xn)⟶rasn⟶+∞.
Suppose that r>0; this implies that r≤dλ(xn,xn-1)≤dλ(x1,x0),for alln≥1. Now, taking a=r/2 and b=dλ(x1,x0), by applying (ii), we get the existence of ψλ,a,b∈Ψ and φλ,a,b∈Φ such that
(15)ψλ,a,b(d(xn+1,xn))≤ψλ,a,b(d(xn,xn-1))-φλ,a,b(d(xn,xn-1)),∀n≥1.
Letting n→+∞ in the above inequality, using the continuity of ψλ,a,b and the lower semicontinuity of φλ,a,b, we get
(16)ψλ,a,b(r)≤ψλ,a,b(r)-φλ,a,b(r),
which implies that r=0. This is a contradiction, and, therefore, our claim (12) holds.
Step
2. We will prove that {xn} is a Cauchy sequence in (X,𝒯(ℱ)). Suppose that {xn} is not a Cauchy sequence. Then, there exists (λ,ε)∈𝒜×(0,+∞) for which we can find two sequences of positive integers {m(k)} and {n(k)} such that, for all positive integers k,
(17)n(k)>m(k)>k,dλ(xn(k),xm(k))≥ε,dλ(xn(k)-1,xm(k))<ε.
Using (17) and the triangular inequality, we get
(18)ε≤dλ(xn(k),xm(k))≤dλ(xn(k),xn(k)-1)+dλ(xn(k)-1,xm(k))<dλ(xn(k),xn(k)-1)+ε.
Thus, we have
(19)ε≤dλ(xn(k),xm(k))<d(xn(k),xn(k)-1)+ε.
Letting k→+∞ in the above inequality and using (12), we obtain
(20)dλ(xn(k),xm(k))⟶εask⟶+∞.
By the triangular inequality, we have
(21)|dλ(xn(k)+1,xm(k)+1)-dλ(xn(k),xm(k))|≤dλ(xn(k)+1,xn(k))+dλ(xm(k),xm(k)+1).
Letting k→+∞ in the above inequality, using (12) and (20), we get
(22)dλ(xn(k)+1,xm(k)+1)⟶εask⟶+∞.
On the other hand, from (20), there exists a positive integer k0 such that
(23)ε2≤dλ(xn(k),xm(k))≤2ε,∀k≥k0.
Applying the hypothesis of the theorem by taking a=ε/2 and b=2ε, we get that there exist ψλ,a,b∈Ψ and φλ,a,b∈Φ such that
(24)ψλ,a,b(dλ(Txn(k),Txm(k)))≤ψλ,a,b(dλ(xn(k),xm(k)))-φλ,a,b(dλ(xn(k),xm(k))),∀k≥k0;
that is,
(25)ψλ,a,b(dλ(xn(k)+1,xm(k)+1))≤ψλ,a,b(dλ(xn(k),xm(k)))-φλ,a,b(dλ(xn(k),xm(k))),∀k≥k0.
Letting k→+∞ in the above inequality, from (20), (22), the continuity of ψλ,a,b, and the lower semicontinuity of φλ,a,b, we obtain
(26)ψλ,a,b(ε)≤ψλ,a,b(ε)-φλ,a,b(ε),
which implies that φλ,a,b(ε)=0, which leads to the contradiction ε=0. Finally, we deduce that {xn} is a Cauchy sequence.
Step
3 (existence of fixed point). As {xn} is a Cauchy sequence in the complete gauge space (X,𝒯(ℱ)), then there exists x*∈X such that
(27)xn⟶ℱx*asn⟶+∞.
This implies that x*∈𝒪(x0)¯. On the other hand, by (10), for all xn∈𝒪(x0), we get
(28)dλ(Txn,Tx*)≤dλ(xn,x*),∀λ∈𝒜,∀n≥0.
Now, from
(29)dλ(x*,Tx*)≤dλ(x*,xn+1)+dλ(Txn,Tx*)≤dλ(x*,xn+1)+dλ(xn,x*)
for all λ∈𝒜, letting n→+∞, we obtain dλ(x*,Tx*)≤0, for all λ∈𝒜.
In the virtue of the separating structure ofℱ, we deduce that Tx*=x*, and, hence, the existence of the fixed point is proved.
Step
4 (uniqueness). Suppose that there exist x*,y* such that x*≠y*, Tx*=x*, and Ty*=y*. Then, there exists λ∈𝒜 such that dλ(x*,y*)≠0, and so, by using (7), we get a contradiction.Remark 8.
Theorem7 is a generalization of the main result of [6].
By takingψλ,a,b(t)=t in Theorem 7, we get the following corollary.Corollary 9.
Let(X,𝒯(ℱ)) be a complete gauge space, and let T:X→X be a mapping satisfying the following conditions.(i)
For allλ∈𝒜, for all u∈X, for all x∈𝒪(u), and for all y∈𝒪(u)¯,dλ(x,y)=0⇒dλ(Tx,Ty)=0.(ii)
For allλ∈𝒜 and for all 0<a<b<+∞, there exists φλ,a,b∈Φ such that
(30)∀x,y∈X,a≤dλ(x,y)≤b⟹dλ(Tx,Ty)≤dλ(x,y)-φλ,a,b(dλ(x,y)).Then, T has a unique fixed point.By takingφλ,a,b(t)=(1-k)t in Corollary 9, we get the following result.Corollary 10.
Let(X,𝒯(ℱ)) be a complete gauge space, and let T:X→X be a mapping satisfying the following conditions.(i)
For allλ∈𝒜, for all u∈X, for all x∈𝒪(u), and for all y∈𝒪(u)¯,dλ(x,y)=0⇒dλ(Tx,Ty)=0.(ii)
For allλ∈𝒜 and for all 0<a<b<∞, there exists k=k(λ,a,b)∈(0,1) such that
(31)∀x,y∈X,a≤dλ(x,y)≤b⟹dλ(Tx,Ty)≤kdλ(x,y).Then, T has a unique fixed point.Next, we give other consequences of Theorem7.Corollary 11.
Let(X,𝒯(ℱ)) be a complete gauge space, and let T:X→X be a mapping satisfying the following condition.
For allλ∈𝒜, there exist ψλ∈Ψ and φλ∈Φ such that
(32)ψλ(dλ(Tx,Ty))≤ψλ(dλ(x,y))-φλ(dλ(x,y))∀x,y∈X.
Then, T has a unique fixed point.Proof.
Condition (i) of Theorem7 is satisfied from inequality (32), and taking ψλ,a,b(t)=ψλ(t) and φλ,a,b(t)=φλ(t), we get Corollary 11.By takingψλ(t)=t in Corollary 11, we get the following result.Corollary 12.
Let(X,𝒯(ℱ)) be a complete gauge space, and let T:X→X be a mapping satisfying the following condition.
For allλ∈𝒜, there exists φλ∈Φ such that
(33)∀x,y∈X,dλ(Tx,Ty)≤dλ(x,y)-φλ(dλ(x,y)).
Then, T has a unique fixed point.Also, by takingφλ(t)=(1-k)t in Corollary 12, we get the following result.Corollary 13.
Let(X,𝒯(ℱ)) be a complete gauge space, and let T:X→X be a mapping satisfying the following condition.
For allλ∈𝒜, there exists k=k(λ)∈(0,1) such that
(34)∀x,y∈X,dλ(Tx,Ty)≤kdλ(x,y).
Then, T has a unique fixed point.Finally, we give a result involving a condition of integral type. LetΓ denote the set of all functions α:[0,+∞)→[0,+∞) which satisfy the following conditions.(c1)
α is Lebesgue integrable on [0,+∞).(c2)
For allε>0, we have ∫0εα(t)dt>0.Corollary 14.
Let(X,𝒯(ℱ)) be a complete gauge space, and let T:X→X be a mapping satisfying the following conditions.(i)
For allλ∈𝒜, for all u∈X, for all x∈𝒪(u), and for all y∈𝒪(u)¯,dλ(x,y)=0⇒dλ(Tx,Ty)=0.(ii)
For all0<a<b<∞ and for all λ∈𝒜, there exist αλ,a,b,βλ,a,b∈Γ such that
(35)∀x,y∈X,a≤dλ(x,y)≤b⟹∫0dλ(Tx,Ty)αλ,a,b(t)dt≤∫0dλ(x,y)αλ,a,b(t)dt-∫0dλ(x,y)βλ,a,b(t)dt.Then, T has a unique fixed point.Proof.
It follows from Theorem7 that, by taking
(36)ψλ,a,b(x)=∫0xαλ,a,b(t)dt,φλ,a,b(x)=∫0xβλ,a,b(t)dt,x≥0.Remark 15.
Corollary14 is a generalization of the main result of [19].


## 4. Results for Cyclic Mappings

In [20], Kirk et al. obtained extensions of the Banach contraction principle for cyclic mappings, by considering a cyclical contractive condition as given by the following theorem.Definition 16.
Let{Ai}i=1p be nonempty subsets of a nonempty set X. Then, T:X→X is called a cyclic mapping associated with (A1,…,Ap) if the following conditions hold:(i)
X=⋃i=1pAi,(ii)
T(Ai)⊆Ai+1for1≤i≤p, where Ap+1=A1.Theorem 17.
Let{Ai}i=1p be nonempty closed subsets of a metric space (X,d), and suppose that T:X→X is a cyclic mapping associated with (A1,…,Ap) satisfying the following:
(37)∃k∈(0,1)suchthatd(Tx,Ty)≤kd(x,y),∀x∈Ai,y∈Ai+1,for1≤i≤p.
Then, T has a unique fixed point.Inspired by this result, other fixed-point theorems with cyclical contractive conditions were obtained (see [21–30]). Our aim in this section is to extend Theorem 7 for cyclic mappings.Theorem 18.
Let{Ai}i=1p be nonempty closed subsets of a complete gauge space (X,𝒯(ℱ)), and suppose that T:X→X is a cyclic mapping associated with (A1,…,Ap) satisfying the following conditions.(i)
For allλ∈𝒜, for all u∈X, for 1≤i≤p, and for all x∈𝒪(u)⋂Ai, y∈𝒪(u)¯⋂‍Ai+1,
(38)dλ(x,y)=0⟹dλ(Tx,Ty)=0.(ii)
For allλ∈𝒜 and for all 0<a<b<∞, there exist ψλ,a,b∈Ψ and φλ,a,b∈Φ such that, for 1≤i≤p,
(39)∀x∈Ai,y∈Ai+1,a≤dλ(x,y)≤b⟹ψλ,a,b(dλ(Tx,Ty))≤ψλ,a,b(dλ(x,y))-φλ,a,b(dλ(x,y)).Then, T has a unique fixed point.Proof.
As in the proof of Theorem7, one can prove that
(40)∀λ∈𝒜,∀x∈Ai,y∈Ai+1,dλ(x,y)≠0⟹dλ(Tx,Ty)<dλ(x,y),(41)∀λ∈𝒜,∀u∈X,∀x∈𝒪(u)∩Ai,y∈𝒪(u)¯∩Ai+1,dλ(Tx,Ty)≤dλ(x,y).
Next, in establishing the existence of a fixed point, since T(⋂i=1p‍Ai)⊂⋂i=1p‍Ai, one only needs to prove that ⋂i=1p‍Ai≠∅. In fact, applying Theorem 7 to the restriction of T on the complete gauge space (⋂i=1p‍Ai,𝒯(ℱ)), we obtain that T has a fixed point in ⋂i=1p‍Ai.
Now, letx0∈A1, and we construct a sequence {xn} in X such that
(42)xn+1=Txn,∀n≥0.
Remark that, for all n≥0, there exists i=i(n)∈{1,…,p} such that (xn,xn+1)∈Ai×Ai+1. Then, as in the proof of Theorem 7, we obtain that
(43)dλ(xn-1,xn)⟶0asn⟶+∞.
We will prove that {xn} is a Cauchy sequence in the complete gauge space (X,𝒯(ℱ)). Suppose that {xn} is not a Cauchy sequence. Then, there exists (λ,ε)∈𝒜×(0,+∞) for which we can find two sequences of positive integers {m(k)} and {n(k)} such that, for all positive integers k,
(44)n(k)>m(k)>k,dλ(xm(k),xn(k))≥ε,dλ(xm(k),xn(k)-1)<ε.
Using (44) and the triangular inequality, we get
(45)ε≤dλ(xn(k),xm(k))≤dλ(xm(k),xn(k)-1)+dλ(xn(k)-1,xn(k))<ε+dλ(xn(k),xn(k)-1).
Thus, we have
(46)ε≤dλ(xn(k),xm(k))<ε+dλ(xn(k),xn(k)-1).
Letting k→+∞ in the above inequality and using (43), we obtain
(47)dλ(xn(k),xm(k))⟶εask⟶+∞.
Let k be a positive integer, and let j(k)∈{1,…,p} be the integer such that
(48)n(k)-m(k)=j(k)modp.
Using the triangular inequality, we have
(49)dλ(xn(k),xm(k))≤dλ(xn(k),xn(k)+1)+dλ(xn(k)+1,xm(k)+j(k))+∑i=0j(k)-1‍dλ(xm(k)+1+i,xm(k)+i),dλ(xn(k)+1,xm(k)+j(k))≤dλ(xn(k)+1,xn(k))+dλ(xn(k),xm(k))+∑i=0j(k)-1‍dλ(xm(k)+1+i,xm(k)+i).
Thus, we have
(50)|dλ(xn(k)+1,xm(k)+j(k))-dλ(xn(k),xm(k))|≤dλ(xn(k)+1,xn(k))+∑i=0j(k)-1‍dλ(xm(k)+1+i,xm(k)+i)≤dλ(xn(k)+1,xn(k))+∑i=0p-1‍dλ(xm(k)+1+i,xm(k)+i).
Letting k→+∞ in the above inequality, using (43) and (47), we get that
(51)dλ(xn(k)+1,xm(k)+j(k))⟶εask⟶+∞.
Using the same technique, we obtain that
(52)dλ(xn(k),xm(k)+j(k)-1)⟶εask⟶+∞.
On the other hand, from (52), there exists a positive integer k0 such that
(53)ε2≤dλ(xn(k),xm(k)+j(k)-1)≤2ε,∀k≥k0.
Applying the hypothesis of the theorem by taking a=ε/2 and b=2ε, we get the existence of ψλ,a,b∈Ψ and φλ,a,b∈Φ such that
(54)ψλ,a,b(dλ(Txn(k),Txm(k)+j(k)-1))≤ψλ,a,b(dλ(xn(k),xm(k)+j(k)-1))-φλ,a,b(dλ(xn(k),xm(k)+j(k)-1));
that is,
(55)ψλ,a,b(dλ(xn(k)+1,xm(k)+j(k)))≤ψλ,a,b(dλ(xn(k),xm(k)+j(k)-1))-φλ,a,b(dλ(xn(k),xm(k)+j(k)-1)).
Letting k→+∞ in the above inequality, using (51), (52), the continuity of ψλ,a,b, and the lower semicontinuity of φλ,a,b, we obtain
(56)ψλ,a,b(ε)≤ψλ,a,b(ε)-φλ,a,b(ε),
which implies that φλ,a,b(ε)=0, which leads to the contradiction ε=0. Finally, we deduce that {xn} is a Cauchy sequence in the complete gauge space (X,𝒯(ℱ)); therefore, {xn} converges to some x*∈X. In view of condition (ii) of Definition 16, an infinite number of elements of the sequence {xn} lie in each Ai, i∈{1,…,p}. As Ai is a closed subset, we deduce that x*∈Ai, for i∈{1,…,p}. Thus, x*∈⋂i=1p‍Ai, and so ⋂i=1p‍Ai≠∅.
In order to complete the proof, we have to show that uniqueness of the fixed point.
From condition (ii) of Definition16, any fixed point of T lies necessarily in ⋂i=1p‍Ai. Suppose that there exist x*,y*∈X such that x*≠y*, Tx*=x*, and Ty*=y*. Then, there exists λ∈𝒜 such that dλ(x*,y*)≠0, and so, by using (40), we get a contradiction.Remark 19.
Theorem18 is a generalization of the main result of [24].By takingψλ,a,b(t)=t in Theorem 18, we get the following corollary.Corollary 20.
Let{Ai}i=1p be nonempty closed subsets of a complete gauge space (X,𝒯(ℱ)), and suppose that T:X→X is a cyclic mapping associated with (A1,…,Ap) satisfying the following conditions.(i)
For allλ∈𝒜, for all u∈X, for 1≤i≤p, and for all x∈𝒪(u)⋂‍Ai,y∈𝒪(u)¯⋂‍Ai+1,
(57)dλ(x,y)=0⟹dλ(Tx,Ty)=0.(ii)
For allλ∈𝒜 and for all 0<a<b<∞, there exists φλ,a,b∈Φ such that, for 1≤i≤p,
(58)∀x∈Ai,y∈Ai+1,a≤dλ(x,y)≤b⟹dλ(Tx,Ty)≤dλ(x,y)-φλ,a,b(dλ(x,y)).Then, T has a unique fixed point.The analogous of Corollaries10–14 for cyclic mappings can be obtained easily, and so, to avoid repetitions, we omit the details.


## 5. Application to Ordinary Differential Equation

In this section, we present an example where our obtained results can be applied. Precisely, we study the existence of solution for the following second-order nonlinear initial-value problem:(59)x′′(t)=k(t,x(t)),t>0,x(0)=α,x′(0)=β,
where k:[0,+∞)×ℝn→ℝn is a continuous function. This problem is equivalent to the integral equation
(60)x(t)=∫0t(t-s)k(s,x(s))ds+βt+α,t≥0.
Denote by X=C([0,+∞),ℝn) the set of continuous functions defined on [0,+∞). For each positive integer p≥1, we define the function ∥·∥p:X→[0,+∞) by
(61)∥x∥p=max0≤t≤p|x(t)|,∀x∈X.
This function is a seminorm on X. Also, define
(62)dp(x,y)=∥x-y∥p,∀p≥1,x,y∈X.
Then, ℱ={dp}p≥1 is a separating family of pseudometrics on X, and (X,𝒯(ℱ)) is a complete gauge space.We will prove the following result.Theorem 21.
Suppose that, for all0<a<b<∞, for each s≥0, and for all x,y∈X, one has
(63)a<dp(x,y)<b⟹|k(s,x(s))-k(s,y(s))|≤γ(s)(b-a)dp(x,y)21+(b-a)dp(x,y),
where γ:[0,+∞)→[0,+∞) is such that the function t↦∫0t(t-s)γ(s)ds is bounded on [0,+∞) and
(64)supt≥0∫0t(t-s)γ(s)ds≤1.
Then, (59) has a unique solution x*∈C([0,+∞),ℝn).Proof.
Consider the operatorT:X→X given by
(65)Tx(t)=∫0t(t-s)k(s,x(s))ds+βt+α,t≥0,x∈X.
Clearly, T is well defined since k is a continuous function.
Now,x* is a solution of (59) if and only if x* is a fixed point of T. Next, we will show that the two conditions of Corollary 9 hold true.
Condition (i). Let p be a positive integer such that p≥1, and let x,y∈X such that dp(x,y)=0. This means that
(66)y(s)=x(s),∀s∈[0,p].
Let us take t∈[0,p]; then, we have
(67)Ty(t)-Tx(t)=∫0t(t-s)(k(s,y(s))-k(s,x(s)))ds=0.
Hence,
(68)dp(Tx,Ty)=max0≤t≤p|Ty(t)-Tx(t)|=0.
Thus, the condition is satisfied.
Condition (ii). Let a,b∈ℝ such that 0<a<b, and let x,y∈X such that a<dp(x,y)<b. For all t∈[0,p], p≥1, we have
(69)|Tx(t)-Ty(t)|≤∫0t(t-s)|k(s,y(s))-k(s,x(s))|ds≤∫0t(t-s)γ(s)(b-a)dp(x,y)21+(b-a)dp(x,y)ds≤(b-a)dp(x,y)21+(b-a)dp(x,y)∫0t(t-s)γ(s)ds≤(b-a)dp(x,y)21+(b-a)dp(x,y).
Then, for all p≥1, we have
(70)dp(Tx,Ty)≤(b-a)dp(x,y)21+(b-a)dp(x,y).
Hence, for all p≥1, for all 0<a<b<∞, and for all x,y∈X such that a<dp(x,y)<b, we have
(71)dp(Tx,Ty)≤dp(x,y)-φp,a,b(dp(x,y)),
where φp,a,b(t)=t-(b-a)t2/(1+(b-a)t), and, obviously, φp,a,b∈Φ. Finally, applying Corollary 9, we obtain that T has a unique fixed point x*∈X; that is, x*∈C([0,+∞),ℝn) is the unique solution of (59).



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*Source: 293101-2013-09-12.xml*