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Original Research ARTICLE

Front. Appl. Math. Stat., 31 August 2015 | https://doi.org/10.3389/fams.2015.00009

Wardowski conditions to the coincidence problem

  • 1Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain
  • 2Departament d'Anàlisi Matemàtica, Universitat de València, València, Spain
  • 3Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

In this article we first discuss the existence and uniqueness of a solution for the coincidence problem: Find pX such that Tp = Sp, where X is a nonempty set, Y is a complete metric space, and T, S:XY are two mappings satisfying a Wardowski type condition of contractivity. Later on, we will state the convergence of the Picard-Juncgk iteration process to the above coincidence problem as well as a rate of convergence for this iteration scheme. Finally, we shall apply our results to study the existence and uniqueness of a solution as well as the convergence of the Picard-Juncgk iteration process toward the solution of a second order differential equation.

1. Introduction

Let X, Y be two nonempty sets and let T, S : XY be two arbitrary mappings. The coincidence problem determined by the mappings T and S consists in

Find pX such that Tp=Sp.    (1)

Quite often to solve problem (1), we have to assume that Y is a complete metric space, and T, S : XY are two mappings satisfying some type of contractivity, for instance see [15]. Some nonlinear problems arising from many areas of applied sciences can be formulated, from a mathematical point of view, as a coincidence problem (see, [13, 6, 7] and references within).

Once the existence of a solution to problem (1) is known, a central question consists to study if there exists an approximating sequence (xn) ⊆ X generated by an iterative procedure f(T, S, xn) such that the sequence (xn) converges to the coincidence point of T and S. Jungck [8] introduced the following iterative scheme: given x1X, there exists a sequence (xn) in X such that Txn+1 = Sxn. This procedure becomes the Picard iteration when X = Y and T = Id, where Id is the identity map on X. In Jungck [8], the author proved that if (X, d) and (Y, ρ) are two complete metric spaces and T and S satisfy both that S(X) ⊆ T(X) and that for every x, yX the inequality d(Sx, Sy) ≤ κ d(Tx, Ty), with 0 ≤ κ < 1 holds, then (xn) converges to the unique coincidence point of T and S. Later, this type of convergence results were generalized for more general classes of contractive type mappings, see [6, 7, 9, 10] (to see another type of iterative schemes we can quote [10, 11]).

Since it is well known that the existence of a solution to problem (1) is, under appropriate conditions, equivalent to the existence of a fixed point for a certain mapping. In this article, we will use the Wardowski fixed point theorem [12] in order to show that problem (1) has a unique solution and that the Picard-Jungck iterative scheme converges to the unique coincidence point, moreover a rate of convergence for this scheme will also be given. Finally, we will apply these results to a general second order differential equation.

2. Notations and Preliminaries

Throughout this article ℝ+ and ℕ will denote the set of all non-negative real numbers and the set of all positive integers respectively.

Definition 2.1. Let X and Y be two nonempty sets and T, S:XY two mappings. If there exists xX such that Sx = Tx then x is said to be a coincidence point of S and T.

Definition 2.2. Let S and T be two self-mappings of a nonempty set X. The pair of mappings S and T is said to be weakly compatible if they commute at their coincidence points, that is, TSx = STx whenever Tx = Sx.

The following straightforward result states a relationship between coincidence points and common fixed points of two weakly compatible mappings, see Proposition 1.4 in Abbas and Jungck [9].

Lemma 2.1. Let S and T be weakly compatible self-mappings of a nonempty set X. If S and T have a unique coincidence point x, then x is the unique common fixed point of S and T.

Given k ∈ (0, 1), by Fk denote the set of all strictly increasing real functions f:(0, ∞) → ℝ satisfying the following conditions:

(F1) For each sequence {αn}n∈ℕ of positive numbers, limnαn=0limnf(αn)=-.

(F2) limα0+αkf(α)=0.

Definition 2.3. Let (X, d) be a complete metric space. A mapping T : XX is said to be an F-contraction if there exist τ > 0 and fFk such that, for all x, yX,

d(x,y)>0τ+f(d(x,y))f(d(Tx,Ty)).    (2)

The following result will be the key in the proof of our results. This result was proved by Wardowski [12].

Theorem 2.1. [[12]] Let (X, d) be a complete metric space and let T : XX be an F-contraction. Then T has a unique fixed point x* ∈ X and for every x0X the sequence {Tn(x0)}n is convergent to x*.

3. Main results

3.1. Existence and Uniqueness

In this subsection we present a result which guarantees the existence and uniqueness of a solution to problem (1) when the mappings T and S satisfy a Wardowski's contractivity type condition.

Theorem 3.1. Let X be a nonempty set and let (Y, ρ) be a complete metric space. Assume that T, S:XY are two mappings satisfying the following conditions:

(i) T(X) is closed;

(ii) S(X) ⊆ T(X);

(iii) There exist τ > 0 and fFk such that, for all x, yX,

ρ(Sx,Sy)>0τ+f(ρ(Sx,Sy))f(ρ(Tx,Ty)).    (3)

Then, T and S have at least one coincidence point in X. If, moreover, T is one-to-one, then this coincidence point is unique.

Proof. Consider h : T(X) → 2T(X) given by h(x): = S(T−1x), where T−1x: = {ξ ∈ X:T(ξ) = x}. Notice that h is single-valued. Indeed, if u, vh(x) with uv, then by definition we know that there exists ξu,ξvT-1x such that u = u and v = v. Since ρ(u, v) = ρ(u, v) > 0, from (3), we have that

τ+f(ρ(Sξu,Sξv))f(ρ(Tξu,Tξv))=f(ρ(x,x))=f(0),

which is a contradiction, because f is not defined at 0.

Therefore, h : T(X) → T(X) is a single valued map from T(X) into itself. Furthermore, h verifies Wardowski's contractive condition [12], since if 0 < ρ(h(x), h(y)) = ρ(S(T−1x), S(T−1y)), then by (3) we have that

τ+f(ρ(S(T1x),S(T1y)))f(ρ(T(T1x),T(T1y))),

that is, τ + f(ρ(h(x), h(y))) ≤ f(ρ(x, y)).

Bearing in mind that (Y, ρ) is complete and T(X) is closed, Wardowski's Theorem states that h has a unique fixed point y* ∈ T(X). Consider x* ∈ T−1y*. Then, by definition, we have that Sx* = S(T−1y*) = h(y*) = y* = Tx*, that is, x* is a coincidence point of T and S.

Now suppose that T is injective. If there exist x*, x′ ∈ X such that Sx* = Tx*, Tx′ = Sx′ and x* ≠ x′, then Sx* = Tx* ≠ Tx′ = Sx′ because T is injective. From (3), we obtain

τ+f(ρ(Sx*, Sx))f(ρ(Tx*, Tx))=f(ρ(Sx*, Sx)),

i.e., τ ≤ 0 which is a contradiction.            □

Corollary 3.1. Let X be a nonempty set and (Y, ρ) be a complete metric space. Assume that T, S : XY are two mappings such that:

(a) T(X) is closed;

(b) S(X) ⊆ T(X);

(c) There exist τ > 0 such that, for all x, yX,

ρ(Sx,Sy)>0ρ(Sx,Sy)ρ(Tx,Ty)(1+τρ(Tx,Ty))2.    (4)

Then, T and S have at least one coincidence point in X. If, moreover, T is one to one, then this coincidence point is unique.

Proof. From (c) it follows that

ρ(Sx,Sy)ρ(Tx,Ty)1+τρ(Tx,Ty),

that is,

1+τρ(Tx,Ty)ρ(Tx,Ty)1ρ(Sx,Sy),

therefore

τ+1ρ(Tx,Ty)1ρ(Sx,Sy).

The above inequality can be written as

τ1ρ(Sx,Sy)1ρ(Tx,Ty).

The last inequality means that T and S satisfy the conditions of Theorem 3.1 with respect to the function f(t)=-1t, which belongs to Fk for some k(12,1).            □

3.2. Picard-Juncgk's Iteration Process

In this subsection we present the results on the convergence for the Picard-Jungck scheme when the conditions of Theorem 3.1 are satisfied. Before giving our convergence result, we state the following lemma proved implicitly in the proof of Wardowski's Theorem [12].

Lemma 3.1. Let τ > 0 and fFk with k ∈ (0, 1). Ifn}n∈ℕ is a sequence of real non-negative numbers satisfying τ + fn+1) ≤ fn) for all n ∈ ℕ, then the series i=0γi is convergent.

Theorem 3.2. Let X be a nonempty set and (Y, ρ) be a complete metric space. If T, S : XY satisfy the three conditions of Theorem 3.1 and T is one-to-one, then given x1X the iterative scheme Txn+1 = Sxn satisfies that the sequences {Txn}n∈ℕ and {Sxn}n∈ℕ converge to Tp = Sp, where pX is the unique coincidence point of T and S.

Proof. Notice that under these assumptions, Theorem 3.1 guarantees the existence and uniqueness of a coincidence point of T and S. Let x1X. It is worth pointing out that the sequence {xn}n∈ℕ, implicitly defined as

Txn+1=Sxnfor all n,    (5)

is well-defined, since S(X) ⊆ T(X). Furthermore, from the injectiveness of T, there exists T−1:T(X) → X and, therefore, the sequence {xn}n∈ℕ can be explicitly defined by xn+1=T-1(Sxn) for all n ∈ ℕ.

If there exists n0 ∈ ℕ such that Sxn0 = Sxn0+1, then by (5) xn0+1 is a coincidence point of T and S. But, in this case, we have that Txn0+2 = Sxn0+1 = Txn0+1, which implies xn0+2 = xn0+1 because T is injective. Again applying (5), we deduce that Txn0+3 = Sxn0+2 = Txn0+1. Bearing in mind the injectiveness of T, we get xn0+3 = xn0+1. Hence, {xn}n>n0 is a constant sequence.

Thus, we can assume that SxnSxn+1 for all n ∈ ℕ. For each n ∈ ℕ, we define γn: = ρ(Sxn, Sxn+1). Thus, γn>0 for all n ∈ ℕ. Moreover, from (3) and (5), τ+fn+1) ≤ fn) for all n ∈ ℕ. By Lemma 3.1, the series i=0γi is convergent. Then, {Sxn}n∈ℕ is a Cauchy sequence, since for mn,

ρ(Sxm,Sxn)γm1+γm2++γn<i=nγi.

Since T(X) is complete, there exists qT(X) such that Sxnq as n → ∞. By (5), we also deduce that Txnq as n → ∞. Since qT(X), there exists pX such that q = Tp. Let us see that Tp = Sp.

Notice that there exists n1 ∈ ℕ such that SxnSp for all nn1. Otherwise there exists a subsequence {Sxnk}nk∈ℕ such that Sxnk = Sp for all nk ∈ ℕ. In this case, Sp = Tp since SxnTp as n → ∞.

Therefore, we can assume that SxnSp for all nn1. By the contractive condition (3), for each nn1,

τ+f(ρ(Sxn,Sp))f(ρ(Txn,Tp)).

Since τ > 0 and f is strictly increasing, we have that ρ(Sxn, Sp) < ρ(Txn, Tp) for all nn1. Taking limits and bearing in mind that TxnTp as n → ∞, we infer that SxnSp as n → ∞. Then, Tp = Sp.            □

We now state the convergence of the sequence {xn}n∈ℕ to the unique coincidence point of T and S.

Theorem 3.3. Let (X, d) and (Y, ρ) be two metric spaces, with Y being complete. Suppose that T, S : XY satisfy the three conditions of Theorem 3.1. If T is injective and T−1 is continuous, then the sequence {xn}n∈ℕ, defined by xn+1=T-1Sxn for each n ∈ ℕ, converges to the unique coincidence point of T and S.

Proof. Let p be the unique coincidence point of T and S, whose existence and uniqueness is guaranteed by Theorem 3.1. Fix x1X. By Theorem 3.2, we know that {Txn}n∈ℕ and {Sxn}n∈ℕ converge to Tp = Sp. From the continuity of T−1 we conclude that

limnxn+1=limnT1Sxn=T1(Tp)=p.

            □

Notice that it is not easy to check the continuity of T−1. However, one can give some metric type condition for T which implies the continuity of T−1. In order to do this, we denote by G the set of functions g:ℝ+ → ℝ+ such that, for any sequence {tn}n∈ℕ, limng(tn)=0 implies limntn=0. On one hand, it is easily seen that if gG then g(t) > 0 for all t > 0. On the other hand, G contains a large number of functions, because G contains the set of all monotone nondecreasing real functions g:ℝ+ → ℝ+ such that g(t) = 0 if and only if t = 0, see [10, Lemma 2.2].

Corollary 3.2. Let (X, d) and (Y, ρ) be two metric spaces, with Y being complete. Suppose that T, S : XY satisfy the three conditions of Theorem 3.1. If there exists gG such that

g(d(x,y))ρ(Tx,Ty), for all x,yX,    (6)

then the sequence {xn}n∈ℕ, defined by xn+1=T-1Sxn for each n ∈ ℕ, converges to the unique coincidence point of T and S.

Proof. It is sufficient to prove that T is one to one and T−1 is continuous. Notice first that (6) implies that T is one to one. Indeed, if Tx = Ty then g(d(x, y)) = 0 which implies that d(x, y) = 0, since gG. Then, T−1 : T(X) → X is well-defined. We now see that T−1 is continuous. Let {un}n∈ℕT(X) be a sequence converging to uT(X). From (6), we have g(d(T−1v, T−1w)) ≤ ρ(v, w) for all v, wT(X). Then,

0limng(d(T1un,T1u))limnρ(un,u)=0.

Since gG, we deduce that T-1unT-1u as n → ∞.            □

Remark 3.1. It is worth pointing out that the continuity of T−1 does not imply that (6) holds: Just take T:ℝ+ → ℝ+ defined by Tx=x.

Remark 3.2. Since the identity mapping is weakly compatible with respect to any mapping, from Corollary 3.2, we recapture Theorem 2.1.

Corollary 3.3. Let (X.d) and (Y, ρ) be two metric spaces, with Y being complete. Assume that T, S : XY are two mappings satisfying the conditions of Corollary 3.1 and, in addition, that there exists gG such that T : XY satisfies inequality (Equation 6). Then the sequence {xn}, defined by xn+1=T-1(Sxn), converges to the unique coincidence point of T and S.

Proof. The proof of Corollary 3.1 shows that T and S satisfy the hypotheses of Corollary 3.2 with f(t)=-1t and therefore we obtain the result.            □

3.3. Rate of Convergence

The idea given in Kohlenbach [13] allows us to introduce the concept of modulus of uniqueness for the coincidence problem as follows.

Definition 3.1. Let (X, d) and (Y, ρ) be two metric spaces and let T, S : XY be two mappings. A function ψ : (0, ∞) → (0, ∞) is said to be a modulus of uniqueness for the coincidence problem defined by T and S if, for any ε > 0, max{ρ(Tx, Sx), ρ(Ty, Sy)} < ψ(ε) implies that d(x, y) < ε.

Theorem 3.4. Let (X, d) and (Y, ρ) be two metric spaces. Suppose that T, S : XY satisfy the three conditions of Theorem 3.1 and also that there exists an increasing function g : (0, +∞) → (0, +∞) such that

g(d(x,y))ρ(Tx,Ty), for all x,yX.    (7)

If the function β : (0, +∞) → (0, +∞) defined by β(t): = tf−1(f(t)−τ) is increasing then ψ:=12βg is a modulus of uniqueness for the coincidence problem defined by T and S.

Proof. Let ε > 0 and x, yX such that max {ρ(Tx, Sx), ρ(Ty, Sy)} < ψ(ε). Notice that

ρ(Tx,Ty)ρ(Tx,Sx)+ρ(Sx,Sy)+ρ(Sy,Ty)<2ψ(ε)                                                 +f1(f(ρ(Tx,Ty))τ).

Then, β(ρ(Tx, Ty)) < 2ψ(ε) = β(g(ε)). Since β is increasing, we get ρ(Tx, Ty) < g(ε). From (7), we deduce that d(x, y) < ε because g is increasing.            □

Remark 3.3. As a direct consequence of the above theorem, we can get a new result on generalized Ulam-Hyers stability of the coincidence problem (1).

Another consequence of Theorem 3.4 is the following result that states a rate of convergence for Picard-Juncgk's iteration process.

Theorem 3.5. Under the hypotheses of Theorem 3.4. Let {xn}n∈ℕ be the sequence defined by xn+1=T-1Sxn for each n ∈ ℕ. Let pX be some coincidence point of T and S. Then, for all n ≥ Φ(ε), we have that d(xn, p) < ε, where Φ : (0, +∞) → ℕ is given as

Φ(ε):={f(ρ(Sx1,Tx1))f(ψ(ε))τ+2if ψ(ε)ρ(Sx1,Tx1),1if ρ(Sx1,Tx1)<ψ(ε).

Proof. Fix ε > 0. By Theorem 3.4, if we prove that ρ(Txn, Sxn) < ψ(ε) for all n ≥ Φ(ε), then we are done, since in this case it is enough to take x = xn and y = p.

Let us prove that ρ(Txn, Sxn) < ψ(ε) for all n ≥ Φ(ε), i.e., ρ(Sxn−1, Sxn) < ψ(ε) for all n ≥ Φ(ε). From the proof of Theorem 3.2 we know that the sequence {γn}n∈ℕ, defined by γn: = ρ(Sxn, Sxn+1), satisfies

τ+f(γn+1)f(γn)  for all n.    (8)

Since f is increasing and τ > 0, we have that {γn}n∈ℕ is strictly decreasing.

If γ1: = ρ(Sx1, Tx1) < ψ(ε), then γn < ψ(ε) for all n ≥ 1 = Φ(ε). Thus, we can assume that ψ(ε) ≤ γ1.

We claim that γΦ(ε) < ψ(ε). By contradiction, suppose that ψ(ε) ≤ γΦ(ε). Using (8), we obtain that (Φ(ε)−1)τ + fΦ(ε)) ≤ f1). Bearing in mind that f is increasing, we deduce that (Φ(ε)−1)τ+f(ψ(ε)) ≤ f1), which contradicts the definition of Φ(ε). Therefore, γΦ(ε) < ψ(ε). Since {γn}n∈ℕ is decreasing, we conclude that γn < ψ(ε) for all n ≥ Φ(ε).            □

Corollary 3.4. Let (X, d) and (Y, ρ) be two metric spaces. If T, S:XY satisfy the condition of Corollary 3.3, then the function ψ(ε)=12(βg)(ε), where β(t)=t-1(τ+1t)2, is a modulus of uniqueness for the coincidence problem defined by T and S.

Proof. In this case, the proof of Corollary 3.1 shows that f(t) = -1t, and then it is clear that f-1(t)=1t2. The above facts imply that β(t)=t-1(τ+1t)2 and then its derivative is β(t)=1-1(1+τt)30, which says that β is an increasing function. Finally, by Theorem 3.4, we infer that ψ(ϵ)=12(βg)(ϵ) is a modulus of uniqueness.            □

4. An Application to Differential Equations

We consider the following problem associated to a general differential equation of second order with homogeneous Dirichlet condition:

(P) {u(t)=G(t,u(t),u(t)),  for t[a,b]u(a)=0,u(b)=0,

where G:[a, b] × ℝ × ℝ → ℝ is certain known function satisfying the following two general conditions:

(H1) G is continuous in [a, b] × ℝ × ℝ;

(H2) there exist τ, μ > 0 and fFk, for some k ∈ (0, 1), such that

|G(t,x1,x2)G(t,y1,y2)|f1(f(μmax1=1,2αi|xiyi|)τ)

for all t ∈ [0, 1], xi, yi ∈ ℝ, with i = 1, 2; where,

0α18μ(ba)2  and    0α22μ(ba).

Let Y = (C[a, b], ||·||) be the Banach space of the continuous functions u : [a, b] → ℝ, with its norm ||u||: = max{|u(t) : atb}. In the linear space C2[a, b]: = {u : [a, b] → ℝ : u″ ∈ C[a, b]} we consider the linear subspace X: = {uC2[a, b] : u(a) = u(b) = 0}. Notice that X endowed with the norm ||u||*:=max{||u||,||u||,||u||} is a Banach space.

In order to prove the existence and uniqueness of a solution of (P) in C2[a, b], we need the following result attributed to Tumura [14], see [15, p. 80].

Lemma 4.1. For any uX we have that ||u||(ba)28||u|| and ||u||(ba)2||u||. Moreover, the above inequalities are sharp, since they become equalities for the function u(t) = (ta)(bt).

Now we are able to state the main result of this section on the existence and uniqueness of a solution of (P).

Theorem 4.1. With the previous notation, suppose that: G : [a, b] × ℝ × ℝ → ℝ satisfies conditions (H1) and (H2). Then, problem (P) has a unique solution usC2[a,b].

Proof. We define T, S : XY as Tu(t) = u″(t) and Su = G(t, u(t), u′(t)). In order to obtain the existence and uniqueness of the solution to the problem (P), we will see that T and S satisfy the conditions of Theorem 3.1. Notice that T is onto. Indeed, given wY it is enough to consider

u(t):=atv(s)dstabaabv(s)ds,   where v(s):=asw(r)dr,

since in this case uX and Tu = w. Thus, assumptions (i) and (ii) in Theorem 3.1 hold. Let us prove that T and S satisfy (iii). Assume that u, vX with ||uv|| ≠ 0. Then, there exists at least one t ∈ [a, b] such that u(t) ≠ v(t). Hence, by (H2),

|Su(t)Sv(t)|=|G(t,u(t),u(t))G(t,v(t),v(t))|                         f1(f(μmax{α1|u(t)v(t)|,                          α2|u(t)v(t)|})τ)                         f1(f(μmax{α1u(t)v(t),                          α2u(t)v(t)})τ)                         f1(f(uv)τ),

the last inequality is obtained from Lemma 4.1 and because f is increasing. Thus, ||SuSv||f1(f(||TuTv||)τ), that is, T and S satisfy (iii). From Theorem 3.1, T and S have a unique coincidence point in X, i.e., problem (P) has a unique solution usC2[a,b].

Remark 4.1. Under the conditions of Theorem 4.1, applying Lemma 4.1 we obtain that ||u||*M ||u||, where

M:=max{(ba)28,ba2,1}.

Then

(a) If we define g(t) = t/M, it is clear that T satisfies inequalities (Equations 6, 7). Therefore, by Corollary 3.2, we infer that for each u1X, the sequence {un}n∈ℕ defined by

un+1(t):=at(asGn(r)dr)dstabaab(asGn(r)dr)ds,    (9)

where Gn(r):=Gr,un(r),un(r), converges to us,

(b) If the function β : (0, ∞) → ℝ, defined by β(t): = tf−1 (f(t) − τ), is increasing, Theorem 3.5 yields that for any ε > 0, ||unus||* < ε for all n ≥ Φ(ε), where

Φ(ε):= {f(u2u1)f(β(g(ε))/2)τ+2, if β(g(ε))2u2u1,                           1,                               if 2u2u1<β(g(ε)),    (10)

which means that Φ given by (10) is a rate of convergence for {un} to us.

4.1. A Particular Case

Let G : [0, 1] × ℝ × ℝ → ℝ be a continuous function such that for every t ∈ [0, 1], and for all x, y ∈ ℝ the following inequality holds for some τ > 0,

|G(t,x,y)G(t,u,v)| max{|xu|,|yv|}(1+τmax{|xu|,|yv|})2.    (11)

Let us check that G satisfies condition (H2). Indeed, consider the function f:(0, ∞) → (−∞, 0) defined by f(t)=-1t. It is clear that f−1 : (−∞, 0) → (0, ∞) is given by f-1(s)=1s2. Therefore, taking μ = α1 = α2 = 1 we have:

f1(f(max{|xu|,|yv|})τ))         =1(f(max{|xu|,|yv|})τ)2         =max{|xu|,|yv|}(1+τmax{|xu|,|yv|})2,

which means that G satisfies condition (H2).

Example 4.1. The second order differential equation with homogeneous Dirichlet condition

{u(t)=et+|u(t)|(2+2|u(t)|)2+|u(t)|(2+2|u(t)|)2,   for t[0,1],u(0)=0,u(1)=0,    (12)

has a unique classical solution.

To see that Equation (12) has a unique classical solution it is enough to show that the conditions of Theorem 4.1 are satisfied. Since Equation (12) can be rewritten as

{u(t)=G(t,u(t),u(t)),  for t[0,1],u(0)=0,u(1)=0,    (13)

where G : [0, 1] × ℝ × ℝ → ℝ is defined by

G(t,x,y)=et+12[|x|(1+|x|)2+|y|(1+|y|)2],

we are going to prove that G satisfies inequality (Equation 11). To do this, we notice first that the following elementary properties hold:

(1) the function φ : [0, ∞) → [0, ∞), φ(t)=t(1+t)2, is increasing since φ(t)=1(1+t)3>0,

(2) φ is concave since φ(t)=-32t(1+t)4<0,

(3) since φ(0) = 0 and φ is concave, then it is sub-additive, that is φ(t + s) ≤ φ(t) + φ(s).

Since

|G(t,x,y)G(t,u,v)|12||x|(1+|x|)2|u|(1+|u|)2|                                       +12||y|(1+|y|)2|v|(1+|v|)2|

With the above three properties, the above inequality can be written as follows

|G(t,x,y)G(t,u,v)| 12|φ(|x|)φ(|u|)|+12|φ(|y|)φ(|u|)|                                    12|φ(|x||u|)|+12|φ(|y||v|)|                                    12φ(|xu|)+12φ(|yv|)                                    φ(max{|xu|,|yv|})                                    =max{|xu|,|yv|}(1+max{|xu|,|yv|})2,

which means that the conditions of Theorem 4.1 are satisfied and therefore Equation (12) admits a unique classical solution.

Finally, let us give, by using expression (10), a rate of convergence for the iterative scheme given in Equation (9) concerning Equation (12). To apply Theorem 3.5, first we have to notice that the following facts hold:

1. f : (0, +∞) → (−∞, 0) is given by f(t)=-1t,

2. f−1 : (−∞, 0) → (0, +∞) is f-1(s)=1s2.

3. g:[0, ∞) → [0, ∞) is given by g(t) = t,

4. τ = 1,

5. β(t)=t-t(1+t)2,

6. ψ(ϵ)=12β(ϵ).

In Figure 1, we use the above facts and expression (Equation 10) to compute the number of iterations that we have to do to obtain an error less than ϵ = 10k for k = 1, ⋯ , 6 and taking as starting points u1 = 0 and u1=t2-t respectively.

FIGURE 1
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Figure 1. (A) Red line is the ratio taking as a starting point u1 = 0 and blue line is the ratio for the starting point u1=t2-t. (B) (The ratio of convergence Φ0 corresponds to u1 = 0, while Φ2 corresponds to u1=t2-t).

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

Part of this work was carried out while the first author was visiting the Department of Mathematical Analysis of the University of Valencia.

The research of the first author has been partially supported by MTM 2012-34847-C02-01 and P08-FQM-03453. The second author has been partially supported by MTM 2012-34847-C02-02.

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Keywords: coincidence points, iterative methods, rate of convergence, common fixed points

MSC: 54H25, 47J25

Citation: Ariza-Ruiz D, Garcia-Falset J and Sadarangani K (2015) Wardowski conditions to the coincidence problem. Front. Appl. Math. Stat. 1:9. doi: 10.3389/fams.2015.00009

Received: 14 May 2015; Accepted: 31 July 2015;
Published: 31 August 2015.

Edited by:

Alexander Zaslavski, The Technion - Israel Institute of Technology, Israel

Reviewed by:

Yekini Shehu, University of Nigeria, Nigeria
Suthep Suantai, Chiang Mai University, Thailand
Prasit Cholamjiak, University of Phayao, Thailand

Copyright © 2015 Ariza-Ruiz, Garcia-Falset and Sadarangani. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Jesus Garcia-Falset, Departament d'Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, València, Spain, garciaf@uv.es