Existence theorem for integral inclusions by a fixed point theorem for multivalued implicit-type contractive mappings

The contraction mapping principle proposed by Banach is an important and significant inception in functional analysis due to its applicability in other areas of mathematics and applied sciences. With the passage of time, this result has become a solid base for metric fixed point theory. Thus, several researchers generalized it by stating fixed point, common fixed point, coincidence point, couple fixed point theorems regarding mappings satisfying certain type of contractive conditions on metric spaces or on various abstract spaces. In [31], nonlinear contractions in ordered metric spaces are studied in their cyclic form. In [13], metric spaces endowed with partial order proved to be a suitable setting to develop fixed point theorems for adequate mappings and also an application regarding the existence and uniqueness of a solution to a periodic boundary value problem. Generalized weak Berinde contractions on partial metric spaces are studied in [30]. In [15], a quasicontractivity-type condition, which entails the conclusions from Banach principle, is presented. In [34], a Banach-type condition is studied in connection with the completeness of the underlying metric space. [16] and [32] refer to some Prešic´-type generalization of the Banach principle. Fixed point properties in the context of α ψ-contractions are developed in [25]. In [8], closed multivalued mappings are in view with respect to α φcontractive conditions. In [9], weakly compatible mappings, which satisfy an implicit relation, are studied. In [12] and [14], it is stated a generalization of some weak contraction principle. In [10], b-metric-like spaces feature common fixed point properties. In [29], cone b-metric spaces are an adequate setting to develop generalized Hardy– Rogers-type contractions. [20] focuses on Mizoguchi–Takahashi-type fixed point theorems. In [23], approximate fixed point theorems are stated in the context of α-contractive mappings. The α-admissibility was used to prove interesting fixed point results in [2, 26, 28] or [27]. In [35], implicit contraction mappings are studied. Later on, the combination of metric fixed point theory and optimization theory enabled discussions on best proximity points of nonself mappings satisfying certain types of proximal contractive conditions on metric spaces or on abstract spaces. Browder theorems are extended in [17]. In [1], multivalued operators are studied from the point of view of their best proximity points. [3] has in view Kakutani multimappings, while [4] equilibrium pairs for finite families of multivalued mappings are presented. Work [7] is dedicated to the study of proximal contractions by means of suitable simulation functions. Controlled contractions are used in [5] in order to obtain best proximity properties. [6] is devoted to best proximity results for Prešic´-type operators. Generalized proximal contractive mappings are developed in [11], while [18] refers to global optimal solutions. The existence of best proximity points for generalized classes of contractions is performed in [21]. Applications of best proximity points associated with α ψ-proximal contractions are presented in [19]. Hyperconvex spaces proved to be an adequate framework to develop best proximity theorems for mappings endowed with suitable continuity properties in [22].

In this article, we will use the following type of implicit functions presented in [7]. Here κψ represents the set of functions κ : (R+)4 → R+ = [0,∞ ) endowed with the following properties:

(K1) κ is continuous and nondecreasing in each coordinate; (K2) If l ≥ j and l ≤ κ(l, j, l, l), then l = 0;

(K3) If l < j and l ≤ κ(j, j, l, j), then l ≤ ψ(j), where ψ : : [0, ∞) → [0, ∞) is a nondecreasing mapping with ∑∞n=1ᵠn (t) <∞ for all t ≥ 0.

Definition 1. (See [7].) A mapping χ : R+ x R+ → R is known as an implicit simulation function with respect to κψ if the following conditions hold:

(C1) χ(al, κ(m, n, o, p)) ≤ κ(m, n, o, p) − al for any a, m, n, o, p ≥ 0;

(C2) χ(j, κ(0, 0, j, j/2)) ≥ a if and only if κ(0, 0, j, j/2) - j ≥ a for any real number a;

(C3) χ(j, κ(0, 0, j, j/2)) ≥ 0 implies j = 0.

The next example illustrates such a kind of mappings.

Example 1. (See [7].) Define the mappings χ : R+ × R+ → R and κ : (R+)4 → R+ = [0, ∞),

where ζ ϵ [0, 1). One can check that the above defined χ is an implicit simulation function with respect to the above defined function κ.

The following result was presented in [7] (along the paper, N = {1, 2, . . . }).

Theorem 1. (See [7].) Let T be a mapping from a complete metric space (X, dm) into itself, which satisfies

for all j, l ϵ X, where χ is an implicit simulation function with respect to κψ. Further, assume that the following conditions hold:

1. (i) T is α-admissible, that is, for j, l ∈ X, α(j, l) ≥ 1 implies α(Tj, Tl) ≥ 1;

2. (ii) There exists j1 ∈ X satisfying α(j1, Tj1) ≥ 1;

3. (iii) For all sequences {jn} in X with α(jn, jn+1) ≥ 1, n ∈ N and jn → j, we have

   α(jn, j) ≥ 1 for each n ∈ N.

Then T has a fixed point.

Consider (X, dm) a metric space and CL(X) the collection of all nonempty closed subsets of X. For J ∈ CL(X) and l ∈ X, dm(l, J) = inf{dm(l, j), j ∈ J}. For each J, L ∈ CL(X), let

The function Hm is known as the generalized Hausdorff metric induced by dm.

Here Γ represents the set of all functions η : (R+)5 → R+ = [0, ∞) having the following properties:

(E1) η(m, n, o, p, q) = 0 if and only if at least one of m, n, o, p, q is zero; (E2) η is continuous.

We now present the first result of this section.

Theorem 2. Let X be a nonempty set and dm a distance on it. Assume that T is a mapping from X into CL(X) that satisfies

for all j, l ∈ X with j ≠ l, where χ is an implicit simulation function with respect to κψ, η ∈ Γ and L ≥ 0. Further, assume that the following conditions hold:

1. (i) T is α-admissible, that is, for j, l ∈ X, α(j, l) ≥ 1 implies α(a, b) ≥ 1 for all a ∈ Tj, b ∈ Tl;

2. (ii) There exist j1 ∈ X and j2 ∈ Tj1 satisfying α(j1, j2) ≥ 1;

3. (iii) For all sequences {jn} in X with α(jn, jn+1) ≥ 1, n ∈ N and jn → j, we have α(jn, j) ≥ 1 for each n ∈ N.

Then T has a fixed point.

Proof. Hypothesis (ii) ensures the existence of two elements j1, j2 ∈ X with α(j1, j2) ≥ 1 and j2 ϵ Tj1. Without loss of generality, we may presume that j1 j2. Inequality (1) and condition (C1) imply that

As dm(j2, Tj1) = 0, we get by the use of (E1) that

Thus, inequality (2) implies

There is ε1 > 0 such that

Since ε1 > 0, then we have an element j3 ∈ Tj2 satisfying

If j2 = j3, there is nothing left to be proved. Therefore, we may consider j2 = j3. By using relations (3), (4) and the nondecreasing behavior of κ we get

We now claim that dm(j2, j3) < dm(j1, j2). Suppose this inequality does not hold; then we have dm(j2, j3) ≥ dm(j1, j2). By using this in inequality (5) it follows

From this inequality and condition (K2) we get dm(j2, j3) = 0, which is not possible due to our assumption. Thus, dm(j2, j3) < dm(j1, j2). Hence, inequality (5) yields

Note that hypothesis (i) compels that α(j2, j3) ≥ 1.

Continuing the proof after the above pattern, we get a sequence {jn} satisfying jn+1 ∈ Tjn, α(jn, jn+1) ≥ 1 and dm(jn+1, jn+2) < ψ(dm(jn, jn+1)) < dm(jn, jn+1) for n ∈ N. Having in mind the monotone behavior of ψ and inequality (6), we are led to

Next, we will prove that {jn} is a Cauchy sequence. Consider the natural numbers q, p, q > p. By using the triangular inequality and relation (7) we obtain

This shows that {jn} is a Cauchy sequence in X. By the completeness of X there is an element j∗ ∈ X so that jn → j∗. Hypothesis (iii) implies α(jn, j∗) ≥ 1 for each n ∈ N.

Without loss of generality, we may assume that jn all n ∈ N, it follows that jn ≠

j∗, n ∈ N. From inequality (1), for all n ϵ N, it follows that.

By using hypothesis (C1) and the fact that α(jn, j∗) ≥ 1 in the above inequality we get

Taking the limit n → ∞ leads us to

From the above inequality and (C2) we get

By using condition (C3) it follows dm(j∗, Tj∗) = 0, that is, j∗ ∈ Tj∗.

The next result can be obtained from the above one by considering α(j, l) = 1 for all j, l ∈ X.

Corollary 1. Let T be a mapping from a complete metric space (X, dm) into CL(X), which satisfies

for all j, l ∈ X with j /= l, where χ is an implicit simulation function with respect to κψ, η ∈ Γ and L ≥ 0. Then T has a fixed point.

Example 2. Let X = C[0, 1] be the collection of all continuous functions from [0, 1] into R with the metric d(u, l) = maxp∈[0,1] |u(p) − l(p)|. Define an operator T as

Then we have

Thus, by taking χ(l, j) = j l, κ(m, n, o, p) = (2/3)m and L = 0 in inequality (8) we get the above inequality. Thus, by Theorem 2 and Corollary 1 the mapping T has a fixed point.

In the following theorem, we will use b-metric spaces, where the triangle inequality of metric spaces is replaced by one in which the right-hand side of the classical triangle inequality is multiplied by a constant s, s ≥ 1. Here we denote by κψs the collection of functions κ : (R+)4 → R+ = [0, ∞), which satisfy (K1), (K2) and (K3) with a nondecreasing mapping ψ : [0,∞) → [0,∞) such that ∑∞n=1 sn ψn < ∞ for all t ≥ 0, where s ≥ 1.

Theorem 3. Let X be a nonempty set and bm a continuous b-metric so that (X, bm) is complete. Let T be a mapping from X into CL(X) that satisfies

for all j, l ϵ X, where χ is an implicit simulation function with respect to κψs , η ϵ Γ and L ≥ 0. Further, assume that the following conditions hold:

1. (i) T is αs-admissible, that is, for j, l ∈ X, α(j, l) ≥ s implies α(a, b) ≥ s for each a ∈ Tj, b ∈ Tl;

2. (ii) There exist j1 ∈ X and j2 ∈ Tj1 satisfying α(j1, j2) ≥ s;

3. (iii) For each sequence {jn} in X with α(jn, jn+1) ≥ s, n ∈ N and jn → j, we have α(jn, j) ≥ s for all n ∈ N.

Then T has a fixed point.

Proof. Hypothesis (ii) ensures that there are two elements j1, j2 in X with α(j1, j2) ≥ s and j2 ϵ Tj1. Without loss of generality, we may assume that j1 ≠ j2. Inequality (9) and condition (C1) imply that

Taking account of property (E1) of the functions from Γ , as bm(j2, Tj1) = 0, we get that

Thus, relation (10) compels

As s ≥ 1, there exists an element j3 ∈ Tj2 such that

Again, we may assume j2 = j3. By taking advantage of relations (11), (12) and the nondecreasing behavior of κ it follows

We now claim bm(j2, j3) < bm(j1, j2). By reductio ad absurdum we presume that bm(j2, j3) ≥ bm(j1, j2). Having in mind also inequality (13), we obtain

bm(j2, j3) ≤ κ bm(j2, j3), bm(j1, j2), bm(j2, j3), bm(j2, j3) .

This inequality, jointly with condition (K2), imposes bm(j2, j3) = 0, which is a contradiction to the assumption that j2 ≠ j3. Thus, bm(j2, j3) < bm(j1, j2). Relation (13) yields

bm(j2, j3) ≤ κ bm(j1, j2), bm(j1, j2), bm(j2, j3), bm(j1, j2) .

This inequality combined with property (K3) compel

bm(j2, j3) ≤ ψ(bm(j1, j2)).

Note that hypothesis (i) ensures that α(j2, j3) ≥ s.

Continuing the above pattern, we get a sequence {jn} endowed with the properties jn+1 ∈ Tjn and α(jn, jn+1) ≥ s, bm(jn+1, jn+2) < bm(jn, jn+1) and

Next, we will prove that {jn} is a Cauchy sequence. Consider the natural numbers q, p, q > p. By using the triangular inequality, relation (14), and the fact that s ≥ 1 we get

This shows that {jn} is a Cauchy sequence in X, whose completeness ensures the existence of an element j∗ ∈ X so that jn → j∗. Hypothesis (iii) implies α(jn, j∗) ≥ s for each n ∈ N since α(jn, jn+1) ≥ s for all n ∈ N. Without loss of generality, it can be presumed that jn ≠ j∗ for all n ∈ N. From (9) we get

By the use of condition (C1) and the relation α(jn, j∗) ≥ s in the above inequality, for n ∈ N, we get

By taking the limit n → ∞, using the continuity of bm, κ and η and the fact that sbm(jn+1, Tj∗) ≥ bm(j∗, Tj∗) − sbm(jn+1, j∗), it follows that

From the above inequality and condition (C2) we get

By using (C3) we obtain bm(j∗, Tj∗) = 0. That is, j∗ ∈ Tj∗.

By defining α : X x X [0,∞), α(j, k) = s for each j, k ϵ X, we get the following corollary.

Corollary 2. Consider X a nonempty set endowed with a continuous b-metric bm so that (X, bm) is complete. Let T be a mapping from X into CL(X) that satisfies

for all j, l ϵX, where χ is an implicit simulation function with respect to κψs , η ϵ Γ and L ≥ 0. Then T has a fixed point.

The following result can be obtained directly from Theorem 3 by considering χ(l, j) = j − l for all l, j ∈ R+, κ(m, n, o, p) = q max{m, n, o, p} and η(m, n, o, p, q) = mnopq for all m, n, o, p, q ∈ R+.

Corollary 3. Consider X a nonempty set endowed with a continuous b-metric bm so that (X, bm) is complete. Let T be a mapping from X into CL(X) that satisfies

where q ∈ [0, 1) and L ≥ 0. Further, assume that the following conditions hold:

1. (i) T is αs-admissible, that is, for j, l ∈ X, α(j, l) ≥ s implies α(a, b) ≥ s for all a ∈ Tj, b ∈ Tl;

2. (ii) There exist j1 ∈ X and j2 ∈ Tj1 satisfying α(j1, j2) ≥ s;

3. (iii) For all {jn} in X with α(jn, jn+1) ≥ s, n ∈ N and jn → j, we have α(jn, j) ≥ s for all n ∈ N.

Then T has a fixed point.

Here we apply our result to prove the existence of a solution to the integral inclusion having the following form:

where l, c, v : [a, b] → R are continuous functions, c(u) ≤ v(u) for all u ϵ [a, b], and W : [a, b] x [a, b] x R → Pcv(R), Pcv(R) is a collection of nonempty, convex and compact subsets of R such that W ( ., ., j) is a lower semicontinuous operator for each j ϵ C([a, b], R), where C([a, b], R) represents the space of all continuous functions from [a, b] into R.

Consider X = C([a, b], R); this space is a complete b-metric space with dm(j, l) = supu [a,b] j(u) - l(u) 2 and s = 2.

We now define an operator T : C([a, b], R) → CL(C([a, b], R)) for the integral inclusion (15) as

We denote by Wj(u, p) := W (u, p, j(p)) for all u, p ∈ [a, b], j ∈ C([a, b], R). The Michael’s selection theorem [24] ensures that for Wj : [a, b] × [a, b] → Pcv(R), there exists a continuous operator wj : [a, b] × [a, b] → R with wj(u, p) ∈ Wj(u, p) for each u, p ∈ [a, b]. By this fact we get . Thus, Tj is nonempty.

Moreover, the arguments provided in [33] confirm that Tj is also a closed operator.

We now state and prove an existence theorem for the integral inclusion (15).

Theorem 4. Let X = C([a, b], R), and let the operator T : X → CL(X),

where l, c, v, j : [a, b] → R are continuous functions, c(u) ≤ v(u) for all u ϵ [a, b], and W : [a, b] x [a, b] x R → Pcv(R) is such that W (. , ., j) is a lower semicontinuous operator for all j. Further, assume that the following conditions hold:

(i) There exists a mapping α : X × X → (0, ∞) satisfying the following:

(i-a) There exist j1 ∈ X and j2 ∈ Tj1 with α(j1, j2) ≥ 2;

(i-b) For j, k ∈ X with α(j, k) ≥ 2, we have α(a, b) ≥ 2 for each a ∈ Tj, b ∈ Tk;

(i-c) For all {jn} in X with α(jn, jn+1) ≥ 2, n ∈ N and jn → j, we have α(jn, j) ≥ 2 for all n ∈ N.

(ii) There exists a continuous mapping q : X × X → [0, ∞) such that

for each u,p ϵ and j,k ϵ X, where ψ:[0,∞) →[0,∞) is a nondecreasing continuous mapping such that ∑∞n=1 2nψn(t) < ∞ for all t ≥ 0. Moreover, the mapping q : X × X → [0, ∞) satisfies

Then the integral inclusion (15) has a solution.

Proof. Our aim is to prove the existence of a fixed point for the above defined operator T by using Theorem 3. For this, we focus on relation (9). Let j, k ϵ X and e ϵ Tj. Let

w (u, p) ∈ W (u, p) for u, p ∈ [a, b]

By using hypothesis (ii) we have r(u, p) ∈ Wk(u, p) such that

Now, consider the operator S,

where u, p ∈ [a, b]. The lower semicontinuity of the operator S yields that there exists wk : [a, b] × [a, b] → R with wk(u, p) ∈ S(u, p) for each u, p ∈ [a, b]. Thus, we get

and, for each u ∈ [a, b], we have

Consequently, it follows that

α(j, k)dm(e, h) ≤ ψ dm(j, k) .

By replacing the role of j and k we conclude that

α(j, k)Hbm (Tj, Tk) ≤ ψ dm(j, k) for each j, k ∈ X.

By taking χ(l, j) = j - l for all l, j ϵ R+, κ(m, n, o, p) = ψ(m), η(m, n, o, p, q) = mnopq for all m, n, o, p, q R+ and L = 0, inequality (9) reduces to the above inequality. Moreover, hypotheses (i-a), (i-b), (i-c) of the result imply hypotheses (i), (ii) and (iii) of Theorem 3. Thus, Theorem 3 ensures that a fixed point of the operator T does exist, that is, the integral inclusion (15) has a solution.

The following existence theorem is obtained by defining α : X x X → [0, ∞ ), α(j, k) = 2 for each j, k X in Theorem 4. Also, note that the following result can be proved by using Corollary 2.

Theorem 5. Let X = C([a, b], R), and let the operator T : X → CL(X),

where l, c, v, j : [a, b] → R are continuous functions, c(u) ≤ v(u) for all u ϵ [a, b], and W : [a, b] x [a, b] R → Pcv(R) is such that W ( , , j) is a lower semicontinuous operator for any j. Further, assume that there exists a continuous mapping q : X x X → [0, ∞ ) such that

u; p ϵ [a; b], j; k 2 X, where Ψ : [0;∞) → [0;∞) is a nondecreasing continuous mapping such that ∑∞n=1 2nΨn(t) <∞ for all t ≥ 0. Moreover, the mapping q : X x X → [0;1) satisfies

Then the integral inclusion (15) has a solution.

Fixed point results with regard to multivalued mappings endowed with implicit-type contractive conditions of a special form of simulation functions are stated and proved. An application of our result in integral inclusions is presented. As further development, we intend to design numerical schemes based on our outcomes.