Solution to Fredholm integral inclusions via (F, δ_b)-contractions

Let (𝜀,d_b, s) be a b-metric space and J,J1,J2:E→Pb(E) be three mappings.

1. An orbit 𝒪(x₀; 𝒥) of 𝒥 at a point x₀ ∈𝜀 is any sequence {x_n} such that xn∈Jxn−1 for n = 1,2,....

2. If for a point x₀ ∈𝜀, there exists a sequence {x_n} in 𝜀 such thatx2n+1∈J2x2n,x2n+2∈J1x2n+1,n=0,1,2,…,then the set 𝒪(x₀; 𝒥₁, 𝒥₂) = {x_n : n = 1,2,...} is called an orbit of (𝒥₁, 𝒥₂) at x₀.

3. The space (𝜀, d_b,s) is said to be (𝒥₁, 𝒥₂) -orbitally complete if any Cauchy subsequence {xni}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 {x_n }_n≥0 ⊂𝒪(x₀; 𝒥) and z ∈𝜀, d(x_n, z) → 0 as n → ∞ implies δb(Jxn,Jz)→0as 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 {x_n}, we have (x, x) ∈G (𝒥) whenever (x_n,x_n+1) ∈G (𝒥) and

lim_n→∞x_n= 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 𝔉_s defined as follows

Let s ≥ 1 be a real number. We denote by 𝔉_s the family of all functions F : ℝ⁺ → ℝ with the

following properties:

(F1) F is strictly increasing;

(F2) for each sequence {α_n} of positive numbers, lim_n→∞ α_n = 0 if and only if lim_n→∞F(α_n) = —∞

(F3) for each sequence {α_n} of positive numbers with lim_n→∞ α_n = 0, there exists k ∈ (0, 1) such thatlimn→∞αnkF(αn)=0;

(F4) there exists τ ∈ ℝ+ such that for each sequence {α_n} of positive numbers, if τ + F(sα_n) ≤F (α_n-1) for all

n ∈ Ν, then τ+F(snαn)≤F(sn−1αn−1) 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 Hb here denotes the b-Hausdorff-Pompeiu metric).

[13, Theorem 3.4] Let (𝜀,d_b, s) be a complete b-metric space and let J:E→Pcb(E). 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.

Let (𝜀, d_b, s) be a b-metric space with s > 1. We say that a multivalued mapping J:E→Pb(E) is a multivalued almost (F, δ_b)-contraction if F ∈ 𝔉_s (with parameter τ) and there exists λ ≥ 0 such that

for all x, y ∈𝜀 with min{δ_b,(𝒥x, 𝒥y), d_b (x, y)} > 0, where

and

If (2) is satisfied just for x,y∈O(x0;J)¯ (for some x₀ ∈ε), we say that 𝒥 is a multivalued almost orbitally (F, δ_b)-contraction.

We are equipped now to state our first main result.

Let (ε, d_b, s) be a b-metric space with s > 1 and let J:E→Pb(E) be a multivalued almost orbitally (F, δ_b)-contraction. Suppose that (ε, d_b, s) is 𝒥-orbitally complete (for the same x₀ ∈ε). If F is continuous and 𝒥x is closed for all x,y∈O(x0;J)¯, or 𝒥 has 𝒥-orbitally closed graph, then 𝒥 has a fixed point in ε.

Starting from the given point x₀, choose a sequence {x_n}in ε such that x_n+1 ∈ 𝒥x_n, for all n ≥ 0. Now, if xn0∈Jxn0for some n₀, then the proof is finished. Therefore, we assume x_n ≠ x_n+1for all n ≥ 0. So

d_b(x_n+1, x_n+2) > 0 andδb(𝒥x_n, 𝒥x_n+1) > 0 for all n ≥ 0.

Using the condition (2) for elements x = x_n, y = x_n+1, for arbitrary n ≥ 0 we have

where

and

As 12sdb(xn,xn+2)≤max{db(xn,xn+1),db(xn+1,xn+2)},it follows that

Suppose that d_b(x_n, x_n+1)≤ d_b(x_n+1, x_n+2),for some positive integer n.Then from (4), we have

a contradiction with (Fl). Hence,

and consequently

It follows by (5) and the property (F4) that

Denote ϱ_n = d(x_n, x_n+1) for n = 0,1,2,.... Then, ϱ_n > 0 for all n and, using (6), the following holds:

for all n ∈ℕ. From (7), we get F(sⁿ ϱ_n)→ —∞as n →∞. Thus, from (F2), we have

Now, by the property (F3) there exists k ∈(0, 1) such that

By (7), the following holds for all n ∈ℕ:

Passing to the limit as n →∞in (10) and using (8) and (9), we obtain

and hence lim_n→∞n^1/k sⁿ ϱ_n = 0. Now, the last limit implies that the series ∑n=1∞snϱn is convergent and hence {x_n}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 {n_k}⊂ ℕ such that {nk}⊂Nfor all k ∈ℕ,since 𝒥 z is closed and limk→∞xnk=z, we deduce that z ∈ 𝒥 z and hence the proof is completed. Then we assume that there exists n₀∈ℕ such that x_n ∉𝒥 z for all n ∈ℕ with n ≥ n₀. It follows that δb(𝒥x_n,𝒥z) > 0 for all n ≥ n₀. Using the condition (2) for x = x_n, y = z, we have

where

and

Since F and D_b are continuous, if D_b(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, D_b(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 (x_n,x_n+1) ∈ G(𝒥) for all n ∈ ℕ ∪ {0} and lim_{n → ∞}x_n = 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).

Let (𝜀, d_b, s) be a b-metric space with s > 1. Two multivalued mappings J1,J2:E→Pb(E) are said to form a multivalued almost (F, δ_b)-contraction pair, if F ∈ 𝔉_sand there exist τ > 0, λ ≥ 0 such that

for all x, y ∈ 𝜀 with min{δ_b (𝒥₁x, 𝒥₂y), d_p(x, y)} > 0 where

and

If (13) is satisfied just for x,y∈O(x0;J1,J2)¯ (for some x₀ ∈ 𝜀), we say that (𝒥₁, 𝒥₂) is a multivalued almost orbitally (F, δ_b)-contraction pair.

The main result of this section is the following theorem.

Let (𝜀, d_b, s) be a b-metric space with s > 1 and let J1,J2:E→Pb(E) form a multivalued almost orbitally (F, δ_b)-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 {x_n} ⊂ 𝜀 satisfying

and let a_n = d_b (x_n, x_{n + 1})· If xn0∈J2xn0orxn0∈J1xn0 for some n₀, then the proof is finished. So assume x_n ≠ x_{n + 1} for all n ≥ 0. We claim that

Suppose that n is an odd number. Substituting x = x_n and y = x_{n+ 1} in (13), we obtain

where

as 12sδb(xn,xn+2)≤max{δb(xn,xn+1),db(xn+1,xn+2)} and

Therefore it follows from (14) that

Suppose that d_b(x_n-1, x_n) ≤ d_b(x_n, x_n+1). Then from (15), we have

a contradiction, which means that

Consequently, τ + F(sd_b(x_n, x_n+1)) ≤ F(d_b(x_n-1, x_n)),that is

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 {x_n} is a b-Cauchy sequence in O(x0;J1,J2) 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, d_b, 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 ∈ 𝔉_s and there exist τ > 0, λ ≥ 0 such that

for all x,y∈O(x0;J1,J2)¯ (for the same x₀) with min{d_b (𝒥x, 𝒥y), d_b(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 d_b(x, y) = (x — y)² (s = 2) and define 𝒥₁, 𝒥₂ : 𝜀 → 𝜀 by

Take x0=12.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 τ=ln⁡98and 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).

Under the conditions (I)-(III), the integral inclusion (17) has a solution in C [a, b].

Let 𝜀 = C [a, b] (with b-metric d_b as defined in (18)) and consider the set-valued operator 𝒥 : 𝜀 → 𝒫_cb (𝜀) defined by

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 𝒦_x(t, s) : [a, b] × [a, b] → 𝒫_cb (ℝ), it follows from the Michael's selection theorem that there exists a continuous operator k_x : [a, b] x [a, b] → ℝ such that k_x(t, s) ∈ 𝒦_x(t, s) for each (t, s) ∈ [a, b] × [a,b]. It follows that f(t)+∫abkx(t,s)ds∈Jx. Hence, 𝒥 x ≠ 0. Since ƒ and 𝒦_x are continuous on [a, b], resp. [a, b]², their ranges are bounded and hence 𝒥 x is bounded, i.e., 𝒦 : 𝜀 → 𝒫_Cb (𝜀).

We will check that the contractive condition (2) holds for 𝒥 in 𝜀 with some τ > 0, λ ≥ 0 and F ∈ 𝔉_s, 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 kx1(t,s)∈Kx1(t,s)=K(t,s,x1(s)) such that y1(t)=f(t)+∫abkx1(t,s)ds for t ∈ [a,b].

For all x₁, x₂ ∈ 𝜀, it follows from (II) that

It means that there exists z(t,s)∈Kx2(t,s) such that

for all t, s ∈ [a,b].

Denote by 𝒰(t, s) : [a, b] × [a, b] → 𝒫_cb(∞) the operator defined by

Since, by (I), 𝒰 is lower semicontinuous, it follows that there exists a continuous operator kx2:[a,b]×[a,b]→R such that kx2(t,s)∈U(t,s) for t ,s ∈ [a, b]. Theny2(t):=f(t)+∫abkx2(t,s)dsds satisfies that

i.e., y₂ ∈ 𝒥 x₂ and

for all t, s ∈ [a,b].

Thus, we obtain that