1 Introduction

In a fast developing area of nonlinear theory of differential equation, control theory, image recovery, game theory, etc,. many indispensable results have been established by the use of nonlinear functional analysis based on fixed point theory. In the recent past, fixed point theory has grown into a full fledged research area. Several notions associated with fixed point theory, which can be used to generate a particular elegant approach for the solution of nonlinear problems arising in mathematics, statistics, engineering, economics, approximation theory, theory of differential equations, theory of integral equations, etc., has been established in the contemporary literature (see, for example, [1] and the references therein).

After the initial impetus provided by Nadler [2] in 1969, unwavering attention has been given to the study of fixed point theorems for multivalued mappings. This stem from the significance of fixed point theory for this class of nonlinear mappings in different fields. Other interesting results that followed the remarkable conclusion obtained in [2] with respect to multivalued contraction mappings include, but not limited to the following: Markin [3] initiated the concept of employing Hausdorff metric to examine the fixed points of certain multivalued contraction and nonexpansive mappings, Hu et al [4] proved theorems concerning common fixed points of two multivalued nonexpansive mappings satisfying appropriate contractive inequalities, Bunyawat and Suntain [5] originated a method of establishing common element of solution for a countable family of multivalued quasi-nonexpansive mappings in a uniformly convex Banach space, Isogugu [6] initiated the concept of type-one ϑ-SPM which assures strong convergence without imposing any condition on the fixed point set, Agwu and Igbokwe [7] introduced a technique for obtaining common element of solution for minimization problems with fixed point constraint, etc.

However, we were being captivated by the following techniques studied in [9]: Let H be a real Hilbert space and ∅ ≠ Q ⊂ H be convex and closed. Given the point ℘₀ ∈ Q and for each , let ( is a countable family of type-one -SPM (defined below). The Mann-type technique developed by {℘_q} is (1) where for each ξ^o. If is a bifunction fulfilling (B1) − (B4) and be as described above for each , then the modified lshikawa technique developed from {℘_q} is given by (2) where for some a > 0, {r_q} ⊂ [a, ∞).

Our captivation is basically because of the introduction of the new schemes ((1) and (2)) that address the setbacks (sum conditions) which restricted the application of many results published in this direction.

Subsequently, an unwavering attention has be drawn to methods incorporating several auxiliary maps (see, for example, [8] for details) which is known to be more robust against certain numerical errors as compared to those that involve only one auxiliary mapping. In view of this, the following question becomes necessary:

Question 1.1 Can we obtain a method involving several auxiliary mapping which guarantees strong convergence for certain class of multivalued mappings?

Moltivated and inspired by several works studied, and in particular the remarkable conclusions in [9], our focus in this paper are the following:

1. (a) To intiate the notion of ϑ-SAPM in a real Hilbert space domain;
2. (b) To address the request of Question 1.1 above.
3. (c) To establish strong convergence theorem involving equilibrium problems and mixed-type fixed point problems.

2 Relevant preliminaries

In what follows, the following concepts and known results will be required in order to prove our main results: Let H be a real Hilbert space H with the inner product 〈, ., 〉 and the norm ‖.‖ and ∅ ≠ Q ⊂ H be a convex and closed. Throughout the remaining sections in this paper, the following symbols shall be used: will represent the set of natural numbers, will represent the set of real numbers and ⇀ and → will represent weak and strong convergence of any sequence in H, respectively.

Let ℑ, ð: Q → Q be two nonlinear mappings. We shall use F(ℑ), F(ð) and to denote the set of fixed points of ℑ and ð and the set common fixed point of ℑ and ð, respectively.

Definition 2.1 Recall that

1. (a) ℑ is known as an asymptotically strict pseudocontraction (ASPM, for short) if and a ϑ ∈ [0, 1) that guarantees (3) The class of mappings represented by (3) is a superclass of the class of asymptotically nonexpansive mappings (ANM, for short) (where ℑ is known as ANM if for all ℘, ℏ ∈ Q, which assures the inequality ‖ℑ^q℘ − ℑ^qℏ‖ ≤ ν_q‖℘ − ℏ‖, ∀q ≥ 1) studied in [10].
   Remark 2.1 It is worthy to mention that if F(ℑ) ≠ ∅, then (3) becomes an asymptotically demicontractive mapping (ADM, for short).
2. (b) ℑ is known as k-strictly pseudocontractive if there exists a constant ϑ ∈ [0, 1) such that for all ℘, ℏ ∈ Q, we have (4) This class of k-strictly pseudocontractive has been extensively studied by several authors (see, for example, [7, 8, 11, 12] and the reference therein). It is shown in [13] that a strictly pseudocontractive map is L Lipschitzian (i.e., ‖ℑ℘ − ℑℏ‖ ≤ L‖℘ − ℏ‖ for all ℘, ℏ ∈ D(ℑ)) in [14] that the class of k-strictly asymptotically pseudocontractive maps and the class of strictly pseudocontractive maps are independent.
3. (c) ℑ is called uniformly L-Lipschitzian if there exists a constant L > 0 such that and is said to be demiclosed at a point ν if whenever is a sequence in D(ℑ) such that converges weakly to ℘^⋆ ∈ D(ℑ) and converges strongly to ν, then ℑ℘^⋆ = ν.

Let be a bifunction. An EP for ℧ is to search for an ω ∈ Q that assures the inequality (5) A point z ∈ Q is referred to as an equilibrium point if it solves problem (5).

We shall use EP(℧) to indicate the solution set of problem (5); that is, (6) Considering the invaluable position of equilibrium problems in real life applications, several methods have been deployed to approximate the solution of problem (5); see [15] for more detail. In recent past, different authors have investigated joint problems involving equilibrium and fixed point problem of one mapping in the Hilbert space domain; see, for instance, [5, 9, 15–19] and the references contained in them.

Let B denote a strong positive bounded linear operator on a real Hilbert space domain H; that is, it is possible to get a constant which assures that inequality The problem here is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping τ in a real Hilbert space domain: given that b ∈ H.

In view of the above, and motivated the results in [20], Marino and Xu [21] initiated the following method for approximating the fixed point of nonexpansive mapping via viscosity technique initiated by Moudafi [22]: (7) where ℑ and f represent nonexpansive and contraction mappings, respectively. Using (7), they obtained a strong convergence to the unique solution of the variational inequality problem (8) which represents the optimality condition for the minimization problem with denoting the potential function of γf (i.e., .

Generally, approximating fixed points of single-valued mappings is simpler compared to its multivalued counterpart. However, several researchers have continued to investigate different methods of obtaining invariant point of multivalued mappings, reasons basically contained in their involvement in several real world applications including optimisation and variational inequalities problems (see [2, 3, 23–28]).

A subset Q of a normed space Δ is considered as being proximinal if it is possible to find a point ϕ ∈ Q which assures (9) for each ℘ ∈ Δ. It has been established that a subset of a real uniformly convex Banach space admitting closedness and convexity properties and a subset of a real Banach space guaranteeing convexity and weakly compactness properties are both proximinal.

In what follows, CB(Δ), C(Q) and shall represent the family of nonempty bounded closed subsets of Δ, the family of nonempty compact subsets of Q and the family of nonempty bounded proximinal subsets of Q, respectively. The Hausdorff metric induced by the metric ρ of Δ for all A, B ∈ CB(Δ) is given as (10) where ρ(℘, B) = inf{‖℘ − ℏ‖ : ℏ ∈ B} denotes the distance from the point ℘ to the subset B. A point ℘^⋆ ∈ Q is said to be a fixed point of the multivalued mapping ℑ if ℘^⋆ ∈ ℑ℘^⋆. T Denote by F(ℑ) = {℘ ∈ Q : ℘ ∈ ℑ℘} the set of fixed points of ℑ

Definition 2.2 The ℑ : D(ℑ) ⊆ Δ → 2^Δ is known as:

1. uniformly L-Lipschitzian it it is possible to get an L ≥ 0 which assures (11) If L = ν_q in (11), where , then ℑ becomes ANM.
2. type-one [6] if ∀℘, ℏ ∈ D(ℑ), we get (12) where .
3. ϑ-strictly asymptotically pseudocontraction (ϑ-SAPM) if it is possible to find a sequence and a k ∈ [0, 1) in which, for any pair ℘, ℏ ∈ D(ℑ) and an a ∈ ℑ^q℘, ∃b ∈ ℑ^qℏ assuring ‖a − b‖ ≤ Θ(ℑⁿ℘, ℑ^qℏ) and (13) If ϑ = 1 in (13) then ℑ becomes asymptotically pseudocontractive; whereas ℑ reduces to ANM if ϑ = 0 in (13).

Very recently, Isogugu [29] introduced the following nonlinear map in the Hilbert space domain:

Definition 2.3 Let X be a normed space and ℑ : D(ℑ) ⊆ X → 2^X be a given map. Then ℑ is known as ϑ-strictly pseudocontractive-type in the sense of Browder and Petryshyn [30] if there exists ϑ ∈ [0, 1) such that given any ℘, ℏ ∈ D(ℑ), and a ∈ ℑ℘, we can find b ∈ ℑℏ satisfying ‖a − b‖ ≤ Θ(ℑ℘, ℑℏ) and (14) Note that ℑ in (14) becomes pseudocontractive-type if ϑ = 1 and nonexpansive-type if ϑ = 0. It is not hard to see from (14) that every nonexpansive-type multivalued mapping is ϑ-strictly pseudocontractive-type and every ϑ-strictly pseudocontractive type multivalued mapping is pseudocontractive-type. It is shown in [29] that the class of nonexpansive-type and ϑ-strictly pseudocontractive-type multivalued mappings are properly contained in the class of ϑ-strictly pseudocontractive-type and pseudocontractive-type multivalued mappings, respectively.

Definition 2.4 [6] Let E be a Banach space and ℑ : D(ℑ) ⊆ E → 2^E be a multivalued mapping. I − ℑ is said to be weakly demiclosed at zero if for any sequence such that {℘_n} converges weakly to ν and a sequence with ℏ_n ∈ ℑ℘_n for all such that {℘_n − ℏ_n} strongly converges to zero. Then, ν ∈ ℑν(i.e., 0 ∈ (I − ℑ)ν).

Lemma 2.1 1001 [21] Consider a bounded linear mapping A on H which assures strongly positive self adjoint (with the coefficient ϰ > 0 and 0 < ϱ ≤ ‖A‖⁻¹), then ‖1 − ϱA‖ ≤ 1 − ϱϰ.

Lemma 2.2 1001 [12] Let H be as described above. Then

Lemma 2.3 (see 1001 [20]) Let {φ_n} ⊂ [0, ∞) with φ_n+1 = (1 − α_n)φ_n + σ_n, n ≥ 0, where {α_n} ⊂ (0, 1) and {σ_n} is a sequence in R such that and . Then, lim_n→∞ φ_n = 0.

Lemma 2.4 1001 [31] For each ℘₁, ℘₂, ⋯, ℘_m and α₁, α₂, ⋯, α_m ∈ [0, 1] with , we have (15)

Lemma 2.5 1001 [32] Let {ℏ_r}_r≥1 be a sequence of real numbers that does not decrease at infinity. In addition, consider the sequence of integers defined by Then, is a nondecreasing sequence verifying and for all r ≥ r₀, the following two inequalities hold: For solving the equilibrium problem, we take the following assumptions into consideration: the function ℧ : Q × Q → R satisfies the following conditions:

1. (M1)
2. (M2) ℧ is monotone, i.e,
3. (M3) ℧ is upper hemicontinuous, i.e., for each ℘, ℏ, z ∈ Q,
4. (B4) is convex and lower semicontinuous for each ℘ ∈ Q.

Lemma 2.6 1001 [33] Let H be a real Hilbert space H, ∅ ≠ Q ⊂ H be closed and convex and let ℧ be a bifunction of Q × Q assuring (M1) − (M4). For r > 0, and given , we can find that guarantees the inequality (16)

Lemma 2.7 1001 [15] Assume that assures (B1) − (B4). Define an a operator ℑ_r : H → Q as where r > 0. Subsequently,

1. (i) ℑ_r is single-valued;
2. (ii) for any ℘, ℏ ∈ H,
3. (iii)

Proposition 2.1 1001 [9] Let be a countable subset of , where s is a fixed nonnegative integer and υ is any integer with s + 1 ≤ υ. Then, the following identity holds: (17)

Proposition 2.2 1001 [9] Let t, u, v ∈ H be arbitrary. Let s be any fixed nonnegetive integer and be such that s + 1 ≤ υ. Let and . Define Then, where and w_q = (1 − c_q)v.

Recently, Rizwan et al. [34–38] worked on several types of fixed point algorithms, HR-Ciric-Reich-Rus contractions, generalized enriched contractions, and MR-Kannan-type interpolative contractions. They provide very important applications of fixed point theory including activation functions through fixed-circle problems.

3 Main results

Definition 3.1 Let X be a normed space and ℑ : D(ℑ) ⊆ X → 2^X be a given map. Then, ℑ is k-ASPM in the thought of Isogugu et al. [9] if there exists μ ∈ [0, 1) such that given any ℘, ℏ ∈ D(ℑ) and u_q ∈ ℑ^q℘, we can find with and v_q ∈ ℑ^qℏ satisfying for which (18)

Remark 3.1 From Definition 3.1, it is not difficult to see that every multivalued nonexpansive-type mapping is strictly asymptotically pseudocontrctive-type mapping. The examples below show that the class multivalued nonexpansive-type mapping is properly included into the class of multivalued strictly asymptotically pseudocontrctive-type mapping and the class of multivalued strictly asymptotically pseudocontractive-type mapping is properly included into the class of asymptotically pseudocontrctive-type mapping.

Example 3.1 (see [39]) Give the usual metric and let the map be given as Then, for n odd (q ≥ 2), we obtain Now, (19) Also, for each . Choose v_q = −δ^qℏ. Then and (20) From (19) and (20), we obtain

The following example shows that the class of θ-strictly asymptotically pseudocontractivetype multivalued mapping is more general than the class of asymptotically nonexpansive-type mappings.

Example 3.2 Let be endowed with the usual metric and define the mapping by Then, for n odd q ≥ 2, we get Then, for all ℘ ∈ [−1.5, 1] and hence it is not ANM. Indeed,

Observe that for each . Choose v_q = −δ^qℏ so that and (21) Now,

Therefore, ℑ is k-SAPM with k_n = 1 and . Note that ℑ, not being ANM, demonstrates the conclusion that the class of ANM mappings is properly included into the class of k-SAPM.

Now, we show with the following examples that the class of multivalued asymptotically strictly pseudocontractive-type mappings and the class of multivalued strictly pseudocontractive-type mappings are independent.

Example 3.3 Let be endowed with the usual metric and define by It is shown in [29] that ℑ is a strictly pseudocontractive-type mapping.

For q even (q > 1), we have Observe that for each . Choose v_q = δ^qℏ so that and (22) Now, where k = 0 and ν_q = 1. Hence, ℑ is not asymptotically k-strictly pseudocontractive-type.mapping.

Example 3.4 Let and let . Define by where is a real sequence satisfying a₂, a₃ > 0, 0 < a_t < 1, t ≠ 2, 3 and . Then, for all k ∈ (0, 1), n ≥ 1 and , where . Since , it follows that ℑ is asymptotically pseudocontractive-type.

Now, choose and a₃ = 4, then we get where and . Hence, ℑ is not strictly pseudocontractive-type.

Now, we shall prove the strong convergence of the new method to the solution set of an equilibrium problem (EP) and the set of common fixed points of two finite families of type-one (θ-SAPM) and θ-strictly pseudocontractive-type multivalued mapping (θSPM).

Theorem 3.1 Let H, Q and ℧ be as described above. Suppose and , υ ≥ 2 are finite families of type-one and -uniformly Lipschitizian strictly asymptotically pseudocontractive-type and type-one strictly pseudocontractive-type multivalued mappings, respectively, with contractive coeficient for each ξ^o. Suppose and for each ς, and are weakly demiclosed at zero. let be a ρ-contraction self map of Q with ρ ∈ (0, 1) and A be a strong positive self adjoint bounded linear operator on H with coeficient such that . Let be a sequence developed from an arbitrary ℘₀ ∈ Q by (23) where and for each ξ^o, {α_q}, {δ_q} ∈ [0, 1], . Suppose the requirements below are fulfilled:

1. (i) for each i;
2. (ii) and
3. (iii) and
4. (iv) and
5. (v) {r_q} ⊂ [a, ∞) for some a > 0.

Then, the sequence given by (23) admits strong convergence to , which provides a solution to .

To start with, we establish the fact that the operator is a self contraction map of Q. Given and for all ℘, ℏ ∈ H, it follows from Lemma 2.5 with and that (24) Therefore, we can find a unique point ℘^⋆ ∈ Q for which which we can write as

Since α_q → 0 as q → ∞, we can take ∀q ≥ 0. Using condition (iv), it is possible to get a constant ϵ with 0 < ϵ < 1 − δ and for each ς. Also, by Lemma 2.1, we get .

Let and . Since and we obtain (using Lemma 2.7) that (25)

Further, we prove that is bounded. Since is k-SAPM and k-SPM, F(ℑ) ≠ ∅ and . Consequently, we can find a sequence and real positive constants such that for any we obtain (26) and (27)

By (23) and Proposition 2.1 with s_q = η, ω_q = t, ℘^⋆ = u, s = 1 and we have (28) (29) (30) Since is type-one k-SAPM, we have, using (30), that From Proposition 2.2, it follows, for s = 1, that (31)

Also, from (23) and Proposition 2.2 with ω_q = η, u_q = t, ℘^⋆ = u, s = 1 and , it is not difficult to see (employing the same approach as in above) that (32) From (31) and (32), we obtain (33) (34) Also, using (23), we obtain the following estimates: (35) By applying conditions [(i) and (iv)] in (33), we get (36) From (35) and (36), we obtain Employing mathematical inductional argument, we have The last inequality implies that the sequence is bounded; and as a consequence, the boundedness of the following sequences: and are assured.

Next, for each i, we prove the following conclusion: ‖ω_n − π_n,i‖ → 0 and as q → ∞. Using Lemma 2.1, (34) and (36), we get the following estimates: (37) (38) (39) By setting The inequality above becomes (40) To established that ℘_q → ℘^⋆ as q → ∞, consider the two Cases below:

Case A: Let be monotonically decreasing. Then, is convergent. Therefore, (41) Hence, (40) and (41) in company with (i), (ii), (iv) and the characteristic property of {ν_q} give (42) Since it follows from (42) that (43) Applying the same line of thought as in above (taking into account (40) and (41), (i), (iii), (iv) and the characteristic property of {ν_q}), it will not be difficult to see that (44) Since employing (44) we have (45)

Next, we prove that . For any we get so that (46) From (31), (32) and (36), we have which by (46) gives (47) where , and .

Using (41), condition (iv) and the characterization of {ν_q}, we get from the last inequality that (48) Furthermore, the following estimates are due to the application of (23) and Proposition 2.2: (49) (50) and by using (23), Proposition 2.2 and (42), we have (51)

Now, observe that which by (48), (49), (50) and (51) yields (52)

Also, observe that (53) which, from (45), (48), (49), (50) and (51), we obtain (54)

Next, we prove that (55) where represents a unique solution of the variational inequality (8). To start with, select a subsequence of such that (56) Now, consequent upon the bounded of the sequence (as shown above), we can find a subsequence of such that as k → ∞. Since ‖u_q − ℘_q‖ → 0 as q → ∞, it follows that . We prove that .

To start with, we prove that ξ^o⋆ ∈ EP(Ψ). By u_q = Tr_q℘_n, we get Using (B2), we also obtain which consequently becomes, Since and , from (B4), we obtain Let ℏ_t = tℏ + (1 − t)ξ^o⋆, where ℏ ∈ Q and t ∈ (0, 1]. Since ℏ, ξ^o⋆ ∈ Q and Q is convex, we get ℏ_t ∈ Q and ℧(y_t, ξ^o⋆) ≤ 0. Therefore, from (B1)and (B4), we get which yields ℧(ℏ_t, ℏ) ≥ 0. Using (B3), we get ℧(ξ^o⋆, ℏ) ≥ 0, ∀ℏ ∈ Q. Thus, ξ^o⋆ ∈ EP(℧).

Now, from , and the demiclosedness property of for each ς, and by applying standard argument, we have that . In addition, since and lim_q→∞ ‖u_q − s_q‖ = 0, we immediately obtained from the demiclosedness property that . Hence, . Since and , we get from (55) that (57) as required.

Since from (23) and Lemma 2.2 (58) it follows from (25), (31) and (32) that where ϖ, ϖ^⋆ and ϖ^⋆⋆ are still as described above.

From the last inequality, we obtain that (59) Set and

Then, from (59), we have that (60) where b_q = ‖℘_q − ℘^⋆‖². It is not difficult to see, from (iv) and the fact that , that Thus, from Lemma 2.3 and (60), .

Case B: Suppose {‖℘_q − q‖} is monotonically increasing. Then, the integer sequence (for some q₀ large enough) can be written as (61) It is easily seen that {τ_q} is nondecreaing sequence and for all q ≥ q₀, we have (62)

From (40), (43), (48) and (43) with (q replaced by τ_(q)), we obtain (63) and (64) By using similar argument as in Case A, we have (65) κ_τ(q) → 0 as and . Therefore, from Lemma 2.3, we obtain lim_n→∞‖℘_τ(q) − ℘^⋆‖ = 0 and .

Hence, by Lemma 2.5, we get Hence, converges to and the proof is completed.

Next, using our main result (Theorem 2.1), we prove strong convergence theorem for finding a solution of the variational inequality problems in the setup of real Hilbert spaces.

Theorem 3.2 Let and f_ς be as given in Theorem 3.1. Let be a sequence developed from an arbitrary ℘₀ ∈ Q by (66) where and for each ξ^o, {α_q}, {δ_q} ∈ [0, 1] and . Suppose the requirements below are fulfilled:

1. (i) for each ξ^o;
2. (ii) and
3. (iii) and
4. (iv) and .

Then, as q → ∞, which provides a solution to the variational inequality .

If ℧(℘, ℏ) = 0 ∀℘, ℏ∈Q, r = 1 ∀q ≥ 0, then u_q = ℘_q. Therefore, with f(℘) = v and A = I, the conclusion is a consequence of Theorem 3.1.

4 Numerical example

Now, we present a numerical example to support to demonstrate the efficiency of our suggested method.

Example 4.1 Let Q = [−3, 3] and . For each ξ^o = 1, 2, 3, let and , be given as (67) and (68) It is shown in [11] that ℑ is a k-SPM. Also, it is easy to see from Example 3.1 above that ð is k-SAPM. In addition, for n odd (q ≥ 2), we obtain (69)

On the other hand, let the bifunction ℧ be given as (70) It is easy to see that ℧ fulfills conditions (B1) − (B4). Set r_q = q + 1, then , where q ≥ 1 (see [40] for more information). For N = 3, (23) becomes Put and . Then, for arbitrary ℘₀ ∈ Q, the above iteration scheme yields: (71) where for ℘_q ∈ (−3, 0] whereas if ℘_q ∈ (−3, 0].

Observe that the sequence ℘_q → 0 as q → ∞. To be precise, .

5 Conclusion

In this manuscript, we introduce a new class of mappings (θ-SAPM) and propose a novel method for solving equilibrium problem with mixed fixed point constraints. We establish strong convergence result of the proposed technique without any imposition of sum conditions on the iteration parameters (hence less computational cost). In addition, we showed that the class of θ-SPM and the class of θ-SAPM are independent. Also, we illustrated the convergence of our method through numerical experiment. Our future project will consider some comparison test of our technique with some existing techniques that probably imposes sum conditions on the iteration parameters.