Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo
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We present sufficient conditions for the existence of solutions of Fredholm integral inclusion equations using new sort of contractions, named as multivalued almost F -contractions and multivalued almost F -contraction pairs under ı-distance, defined in b-metric spaces. We give its relevance to fixed point results in orbitally complete b-metric spaces. To rationalize the notions and outcome, we illustrate the appropriate examples.
Keywords: Almost contraction; Multivalued mapping; b-metric space; F -contraction; Integral inclusion equation
Integral equations appear in numerous scientific and engineering problems. A large class of initial and boundary value problems can be transformed to Volterra or Fredholm integral equations. Mathematical physics models, such as diffraction problems, scattering in quantum mechanics, conformal mapping, and water waves contributed to the creation of integral equations, as well. These equations represent a significant part of mathematical analysis and have various applications in real-world problems. Numerous studies have considered the integral inclusions that arise in the study of problems in applied mathematics, engineering and economics, since some mathematical models utilize multivalued maps instead of single-valued maps, see, e.g., [1-4] and references cited therein.
The advancement of geometric fixed point theory for multivalued mappings was initiated in the work of Nadler, Jr. in 1969 [5]. He used the concept of Hausdorff-Pompeiu metric to establish the multivalued contraction principle containing the Banach contraction principle as a special case. Since then, this discipline has been more developed, and many profound concepts and results have been set up in more generalized spaces.
We construct in this paper a new notion—almost F -contraction for multivalued mappings, by considering the ı-distance in the frame-work of b-metric spaces [6-8] and a concept of F -contractions which was introduced by Wardowski [9]. The paper is organized as follows. In Section 3, we introduce the notion of almost F-contraction for a multivalued mapping under δ-distance in a b-metric space and originate fixed point results in orbitally complete b-metric spaces. In Section 4 we introduce the concept of almost F -contraction pair of multivalued mappings and under δ-distance. The existence and uniqueness of their common fixed point is obtained under additional assumptions on the mappings. We also furnish suitable examples to demonstrate the validity of our results and
to distinguish them from some known ones. In Section 5 we deal with solutions of a Fredholm integral inclusion equation, based on the results of Section 3.
Our work improves and extends the works done in the papers [10-13] with the consideration of orbitally complete b-metric space.
The notion of b-metric space as an extension of metric space was introduced by Bakhtin in [6] and then extensively used by Czerwik in [7, 8, 14]. Since then, a lot of papers have been published on the fixed point theory of various classes of single-valued and multi-valued operators in this type of spaces. We recall here just some basic definitions and notation that we are going to use. ℝ+ and will denote the set of all positive, resp. nonnegative real numbers and ℕ will be the set of positive integers.
A b-metric on a nonempty set ε is a function such that for a constant s ≥ 1 and all x, y, z ∈ ε the following three conditions hold true:
(Ml) db(x,y) = 0 ⟺ x = y,
(M2) db(x,y) = db(y,x),
(M3) db(x,y)≤s(db(x,z)+ db(z,y)).
The triple (𝜀,db, s)is called a b-metric space.
Obviously, each metric space is a b-metric space (for s = 1), but the converse need not be true. Standard examples of b-metric spaces that are not metric spaces are the following:
1. 𝜀 = ℝ and db : 𝜀 × 𝜀 → ℝ defined by db(x, y) = |x – y|2for all x, y ∈ 𝜀, with s = 2.
2. given by
for all {xn}, {yn} ∈ 𝓁p(ℝ).Here s = 21/p. 3. Lp([0,1]) ϶ ƒ : [0,1] → ℝ such that given by
forali f,g ∈ Lp([0, l]);here s = 2p-1.
The topology on b-metric spaces and the notions of convergent and Cauchy sequences, as well as the completeness of the space are defined similarly as for standard metric spaces. However, one has to be aware of some differences. For instance, a b-metric need not be a continuous mapping in both variables (see, e.g., [15]).
Now, we give a brief background for multivalued mappings defined in a b-metric space (𝜀,db,s).
We denote the class of non-empty and bounded subsets of 𝜀 by 𝒫 b(𝜀) and the class of non-empty, closed and bounded subsets of 𝜀 by , we define:
with 𝒟b,(w,𝒲) = 𝒟b({w},𝒲) = inf{db(w,x): x ∈𝒲}.
The following are some easy properties of 𝒟b and δb (see, e.g. [7, 8, 14]):
(i) if 𝒰 = {w}and 𝒱 = {v}then 𝒟b(𝒰,𝒱) = δb(𝒰,𝒱) = db(u,v);
(ii) 𝒟b(𝒰,𝒱)≤δb(𝒰,𝒱);
(iii) 𝒟b(x, 𝒱) ≤ db(x, b) for any b ∈ 𝒱;
(iv) δb(𝒰,𝒱)≤ s[δb(𝒰,𝒱) + δb(𝒲,𝒱)];
(v) δb(𝒰,𝒱) = 0 iff 𝒰 = 𝒱 = {υ}.
Moreover, we will always suppose that
(vi) the function 𝒟b is continuous in its variables.
Recall that z ∈ 𝜀 is called a fixed point of a multivalued mapping if z ∈ 𝒥z.
The concepts of orbit, orbitally complete space and orbitally continuous mapping given in [16-18] for metric spaces can be extended to the case of b-metric spaces, as follows:
Let (𝜀,db, s) be a b-metric space and be three mappings.
1. An orbit 𝒪(x0; 𝒥) of 𝒥 at a point x0 ∈ 𝜀 is any sequence {xn} such that for n = 1,2,....
2. If for a point x0 ∈ 𝜀, there exists a sequence {xn} in 𝜀 such that then the set 𝒪(x0; 𝒥1, 𝒥2) = {xn : n = 1,2,...} is called an orbit of (𝒥1, 𝒥2) at x0.
3. The space (𝜀, db,s) is said to be (𝒥1, 𝒥2) -orbitally complete if any Cauchy subsequence of 𝒪(x0; 𝒥1,𝒥2) (for some x0 in 𝜀) converges in 𝜀. In particular, for 𝒥1 = 𝒥2 = 𝒥, we say that 𝜀 is 𝒥-orbitally complete.
4. The mapping 𝒥 is said to be orbitally continuous at a point x0 ∈ 𝜀 if for any sequence {xn }n≥0 ⊂ 𝒪(x0; 𝒥) and z ∈ 𝜀, d(xn, z) → 0 as n → ∞ implies as n → ∞. 𝒥 is called orbitally continuous in 𝜀 if it is orbitally continuous at every point of 𝜀.
5. The graph G (𝒥) of 𝒥 is defined as G (𝒥) = {(x,y): x ∈ 𝜀, y ∈ 𝒥x}. The graph G (𝒥) of 𝒥 is called 𝒥-orbitally closed if for any sequence {xn}, we have (x, x) ∈ G (𝒥) whenever (xn,xn+1) ∈ G (𝒥) and
limn→∞ xn= x.
In his paper [9], Wardowski introduced a new type of contractions which he called F-contractions. Several authors proved various variants of fixed point results using such contractions. In particular, Acar and Altun proved in [10] a fixed point theorem for multivalued mappings under δ-distance.
Adapting Wardowski’s approach to b-metric space, Cosentino et al. used in [13] the set of functions 𝔉s defined as follows
Let s ≥ 1 be a real number. We denote by 𝔉s the family of all functions F : ℝ+ → ℝ with the
following properties:
(F1) F is strictly increasing;
(F2) for each sequence {αn} of positive numbers, limn→∞ αn = 0 if and only if limn→∞ F(αn) = —∞
(F3) for each sequence {αn} of positive numbers with limn→∞ αn = 0, there exists k ∈ (0, 1) such that
(F4) there exists τ ∈ ℝ+ such that for each sequence {αn} of positive numbers, if τ + F(sαn) ≤ F (αn-1) for all
n ∈ Ν, then for all n ∈ ℕ.
Let F : ℝ+ →ℝ be defined by F (α) = Inα or F (α) = α + In α. It can be easily checked [13, Example 3.2] that F satisfies the properties (F1)-(F4).
They proved the following (note that here denotes the b-Hausdorff-Pompeiu metric).
[13, Theorem 3.4] Let (𝜀,db, s) be a complete b-metric space and let Assume that there exists a continuous from the right function F ∈ 𝔉 and τ ∈ ℝ+ such that
for all x, y ∈ 𝜀, 𝒥x ≠ 𝒥y. Then 𝒥 has a fixed point.
We first introduce the notion of multivalued almost (F, δb)-contraction in a b-metric space and give relevance to fixed point results.
Let (𝜀, db, s) be a b-metric space with s > 1. We say that a multivalued mapping is a multivalued almost (F, δb)-contraction if F ∈ 𝔉s (with parameter τ) and there exists λ ≥ 0 such that (2)
for all x, y ∈ 𝜀 with min{δb,(𝒥x, 𝒥y), db (x, y)} > 0, where (3)
and
If (2) is satisfied just for (for some x0 ∈ ε), we say that 𝒥 is a multivalued almost orbitally (F, δb)-contraction.
We are equipped now to state our first main result.
Let (ε, db, s) be a b-metric space with s > 1 and let be a multivalued almost orbitally (F, δb)-contraction. Suppose that (ε, db, s) is 𝒥-orbitally complete (for the same x0 ∈ ε). If F is continuous and 𝒥x is closed for all or 𝒥 has 𝒥-orbitally closed graph, then 𝒥 has a fixed point in ε.
Starting from the given point x0, choose a sequence {xn}in ε such that xn+1 ∈ 𝒥xn, for all n ≥ 0. Now, if for some n0, then the proof is finished. Therefore, we assume xn ≠ xn+1for all n ≥ 0. So
db(xn+1, xn+2) > 0 andδb(𝒥xn, 𝒥xn+1) > 0 for all n ≥ 0.
Using the condition (2) for elements x = xn, y = xn+1, for arbitrary n ≥ 0 we have
where
and
As it follows that
Suppose that db(xn, xn+1)≤ db(xn+1, xn+2),for some positive integer n.Then from (4), we have
a contradiction with (Fl). Hence,
and consequently (5)
It follows by (5) and the property (F4) that (6)
Denote ϱn = d(xn, xn+1) for n = 0,1,2,.... Then, ϱn > 0 for all n and, using (6), the following holds: (7)
for all n ∈ℕ. From (7), we get F(sn ϱn)→ —∞as n →∞. Thus, from (F2), we have (8)
Now, by the property (F3) there exists k ∈(0, 1) such that (9)
By (7), the following holds for all n ∈ℕ: (10)
Passing to the limit as n →∞in (10) and using (8) and (9), we obtain
and hence limn →∞ n1/k sn ϱn = 0. Now, the last limit implies that the series is convergent and hence {xn}is a Cauchy sequence in 𝒪(x0;𝒥). Since 𝜀is 𝒥-orbitally complete, there exists a z ∈ 𝜀 such that
Suppose that 𝒥 z is closed.
We observe that if there exists an increasing sequence {nk}⊂ ℕ such that for all k ∈ℕ,since 𝒥 z is closed and , we deduce that z ∈ 𝒥 z and hence the proof is completed. Then we assume that there exists n0∈ℕ such that xn ∉𝒥 z for all n ∈ℕ with n ≥ n0. It follows that δb(𝒥xn,𝒥z) > 0 for all n ≥ n0. Using the condition (2) for x = xn, y = z, we have
where
and
Since F and Db are continuous, if Db(z, 𝒥 z) > 0, passing to the limit as n → ∞ in (11), we obtain
which is impossible since τ > 0, s ≥ 1 and F is strictly increasing. Hence, Db(z, 𝒥 z) = 0 and, since 𝒥 z is closed, we have z ∈ 𝒥 z. Thus, z is a fixed point of 𝒥.
Suppose that G(𝒥) is 𝒥-orbitally closed.
Since (xn,xn+1) ∈ G(𝒥) for all n ∈ ℕ ∪ {0} and limn → ∞ xn = z, we have (z, z) ∈ G(𝒥) by the 𝒥-orbitally closedness. Hence, z ∈ 𝒥 z.
It is proved that z is a fixed point of 𝒥. D
The following corollaries follow from Theorem 3.2 by taking F(α) = In α, resp. F(α) = α + In α in (2).
In this section, we prove a common fixed point theorem for a pair of multivalued mappings satisfying certain conditions.
First we introduce the notion of multivalued almost (F, δb)-contraction pair in b-metric spaces.
Let (𝜀, db, s) be a b-metric space with s > 1. Two multivalued mappings are said to form a multivalued almost (F, δb)-contraction pair, if F ∈ 𝔉s and there exist τ > 0, λ ≥ 0 such that (13)
for all x, y ∈ 𝜀 with min{δb (𝒥1 x, 𝒥2 y), dp(x, y)} > 0 where
and
If (13) is satisfied just for (for some x0 ∈ 𝜀), we say that (𝒥1, 𝒥2) is a multivalued almost orbitally (F, δb)-contraction pair.
The main result of this section is the following theorem.
Let (𝜀, db, s) be a b-metric space with s > 1 and let form a multivalued almost orbitally (F, δb)-contraction pair (for some x0). Assume that 𝜀 is (𝒥1, 𝒥2)-orbitally complete at x0. If F is continuous and 𝒥1 and 𝒥2 are orbitally continuous at x0, then 𝒥1 and 𝒥2 have a common fixed point in 𝜀.
Starting with the given point x0, choose a sequence {xn} ⊂ 𝜀 satisfying
and let an = db (xn, xn + 1)· If for some n0, then the proof is finished. So assume xn ≠ xn + 1 for all n ≥ 0. We claim that
Suppose that n is an odd number. Substituting x = xn and y = xn+ 1 in (13), we obtain (14)
where
as and
Therefore it follows from (14) that (15)
Suppose that db(xn-1, xn) ≤ db(xn, xn+1). Then from (15), we have
a contradiction, which means that
Consequently, τ + F(sdb(xn, xn+1)) ≤ F(db(xn-1, xn)),that is (16)
In a similar way, we can establish inequality (16) when n is an even number.
It follows by (16) and property (F4) that
Similarly as in Theorem 3.2, we can prove that the sequence {xn} is a b-Cauchy sequence in Since 𝜀 is (𝒥1, 𝒥2)-multivalued orbitally complete at x0, there exists a z ∈ 𝜀 such that
If 𝒥1 and 𝒥2 are orbitally continuous, then clearly 𝒥2z = 𝒥1z = z.
Consequences similar to Corollaries 3.3 and 3.4 can be formulated in an obvious way.
If in Theorem 4.2, 𝒥1 and 𝒥2 are single-valued mappings, we deduce the following result.
Let (𝜀, db, s) be α b-metric space with s > 1 and let 𝒥1, 𝒥2 :𝜀 → 𝜀 be self-mappings such that 𝜀 is (𝒥1, 𝒥2)-orbitally complete (at some x0). Suppose that F ∈ 𝔉s and there exist τ > 0, λ ≥ 0 such that
for all (for the same x0) with min{db (𝒥x, 𝒥y), db(x, y)} > 0, where
and
If F is continuous and 𝒥1 and 𝒥2 are (𝒥1, 𝒥2)-orbitally continuous at x0, then 𝒥1 and 𝒥2 have a common fixed point.
We illustrate the preceding result with the following example (inspired by [20, Example 2.10]).
Let the set 𝜀 = [0, +∞) be equipped with b-metric db(x, y) = (x — y)2 (s = 2) and define 𝒥1, 𝒥2 : 𝜀 → 𝜀 by
Take Then it is easy to show that
We will check that the contractive condition of Corollary 4.3 is ful filled for x, y ∈ 𝓞(x0 ; 𝒥1, 𝒥2) with and F(α) = In α. Indeed, it takes the form
which, after the substitution y = tx, t ≥ 0 reduces to
The last inequality can be easily checked by considering possible values of the parameter t > 0.
All other conditions are also fulfilled, and hence, by Corollary 4.3, we conclude that 𝒥1 and 𝒥2 have a common fixed point (which is z = 0).
In this section we apply the obtained results to achieve the existence of solutions for a certain Fredholm-type integral inclusion. The application is inspired by [3, 21].
Consider the following integral inclusion of Fredholm type. (17)
Here, ƒ ∈ C[a, b] is a given real function and a given set-valued operator; x ∈ C [a, b] is the unknown function.
Now, for ρ ≥ 1, consider the b-metric db, on C [a, b] defined by (18)
for all x, y ∈ C[a,b]. Then (C[a, b], db, 2ρ–1) is a complete b-metric space. Let 𝓓b and δb, have the respective meanings.
We will assume the following:
(I) For each x ∈ C[a, b], the operator 𝒦x(t, s) := 𝒦,(t, s, x(s)), (t,s) ∈ [a,b] × [a, b] is continuous.
(II) there exists a continuous function Υ : [a, b]2 ∈ [0, +∞) such that
for all t, s ∈ [a, b], all u, ύ ∈ C[a, b] and all ku(t, s) ∈ 𝒦u(t, s), 𝒦v(t, s) ∈ 𝒦v(t, s), where λ > 0, p > 1; (III) there exists τ ∈ [1, +∞) such that
Under the conditions (I)-(III), the integral inclusion (17) has a solution in C [a, b].
Let 𝜀 = C [a, b] (with b-metric db as defined in (18)) and consider the set-valued operator 𝒥 : 𝜀 → 𝒫cb (𝜀) defined by (19)
It is clear that the set of solutions of the integral inclusion (17) coincides with the set of fixed points of the operator 𝒥. Hence, we have to prove that under the given conditions, 𝒥 has at least one fixed point in 𝜀. For this, we shall check that the conditions of Theorem 3.2 hold true.
Let x ∈ 𝜀 be arbitrary. For the set-valued operator 𝒦x(t, s) : [a, b] × [a, b] → 𝒫cb (ℝ), it follows from the Michael's selection theorem that there exists a continuous operator kx : [a, b] x [a, b] → ℝ such that kx(t, s) ∈ 𝒦x(t, s) for each (t, s) ∈ [a, b] × [a,b]. It follows that Hence, 𝒥 x ≠ 0. Since ƒ and 𝒦x are continuous on [a, b], resp. [a, b]2, their ranges are bounded and hence 𝒥 x is bounded, i.e., 𝒦 : 𝜀 → 𝒫Cb (𝜀).
We will check that the contractive condition (2) holds for 𝒥 in 𝜀 with some τ > 0, λ ≥ 0 and F ∈ 𝔉s, i.e.,
for elements x1, x2 ∈ 𝜀 Let y1 ∈ 𝒥x1 be arbitrary, i.e.,
holds true. This means that for all t,s ∈ [a,b] there exists such that for t ∈ [a,b].
For all x1, x2 ∈ 𝜀, it follows from (II) that
It means that there exists such that
for all t, s ∈ [a,b].
Denote by 𝒰(t, s) : [a, b] × [a, b] → 𝒫cb(∞) the operator defined by
Since, by (I), 𝒰 is lower semicontinuous, it follows that there exists a continuous operator such that for t ,s ∈ [a, b]. Then ds satisfies that
i.e., y2 ∈ 𝒥 x2 and
for all t, s ∈ [a,b].
Thus, we obtain that
(This shows again that the sets 𝒥 x1 and 𝒥 x2 are bounded.) By passing to logarithms, we write
Taking the function F ∈ 𝔉s defined by F (α) = In α, we obtain that the condition (19) is fulfilled. Using Theorem 3.2, we conclude that the given integral inclusion has a solution.
We are indebted to the learned referee for his/her valuable comments that helped us to improve the text in several places.
The first author is thankful to the United State-India Education Foundation, New Delhi, India and IIE/CIES, Washington, DC, USA for Fullbright-Nehru PDF Award (No. 2052/FNPDR/2015).
The third author is thankful to Ministry of Education, Science and Technological Development of Serbia, Grant No. 174002.
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Received: 2016-03-22
Accepted: 2016-11-15
Published Online: 2016-12-31
Published in Print: 2016-01-01
Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 1053–1064, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0092.
© 2016 Nashine et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0
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