Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications

1.   Introduction

The classical Hermite-Hadamard (H-H) inequality, serving as a litmus test for convexity, formally establishes that if the function $\mathscr{F}:[\Im_1, \Im_2]\to\mathbb{R}$ is a convex function satisfying the essential containment relationship between its midpoint value and integral mean. Specifically, the following inequalities are satisfied:

Convexity inequalities have numerous applications in the study of different models from real-world applications ^[1,2]. Because the H-H inequality is of great significance in convex analysis, it is widely applied in various fields such as integral inequalities, information theory, and optimization theory. In recent years, it has been extended and generalized through various forms of convexity, such as log-convexity ^[3,4], harmonic convexity ^[5,6], h-convexity ^[7,8,9], convexity in $q$-calculus ^[10,11,12] and especially s-convexity ^[13]. Since 1994, s-convexity has been a significant development and widespread application and various generalizations and results regarding H-H inequalities related to s-convex mappings have been established in ^[14,15,16].

From another perspective, interval analysis provides an effective numerical tool for solving uncertain and nonlinear problems. Since the publication of the first monograph in 1966 ^[17], it has evolved into a distinct branch of mathematics, with applications spanning data mining, machine learning, and various other fields. A key focus has been interval-valued (IV) function inequalities. Recently, based on interval calculus and generalized convexity, some authors like Ali et al. ^[18,19,20], Budak et al. ^[21,22], Costa et al. ^[23,24,25], Du et al. ^[26,27], Khan et al.^[28,29], Sarikaya et al. ^[30,31,32], and Zhao et al. ^[33,34,35] established the interval versions of the Chebyshev's inequality, Jensen's inequality, and H-H inequality. Furthermore, with the successive introduction of the bilateral Riemann-Liouville (R-L) fractional integral operators (left-hand and right-hand variants) for IV functions, the results related to inequalities for IV functions are more extensive and profound ^[36,37]. Especially, in 2023, Budak et al. ^[38] introduced a novel generalized integral to demonstrate the generalized H-H-type inclusion of IV convex functions.

Motivated by the above-mentioned literature, we have established some novel interval forms of H-H inequalities by using generalized fractional integral operators and combining them with IV s-convex functions. The classical convexity theory is extended to the generalized s-convex framework, some interesting theorems are proved, and the exact inequality representation of the product of two s-convex functions is given. Our findings not only extend the main conclusions in ^[36,38,39], but also provide new insights for the study of IV inequalities.

The organization of this work is outlined below: Section 2 introduces essential background concepts, while Section 3 establishes a series of H-H-type inequalities for IV s-convex mappings using generalized fractional integral operators. Section 4 gives the conclusion.

2.   Preliminaries

Let $\mathcal{A}$ denote a compact real interval, mathematically expressed as

where $\underline{\mathcal{A}}, \overline{\mathcal{A}}\in\mathbb{R}$ satisfy $\underline{\mathcal{A}}\le\overline{\mathcal{A}}$. If $\underline{\mathcal{A}} = \overline{\mathcal{A}}$, the interval becomes degenerate. The intervals discussed in this paper are all non-degenerate intervals. An interval $[\mathcal{A}]$ is termed positive when $\underline{\mathcal{A}} > 0$, and negative if $\overline{\mathcal{A}} < 0$. The sets $\mathbb{R}_{I}^{-}$, $\mathbb{R}_{I}^{+}$, and $\mathbb{R}_{I}$, respectively, denote all negative, positive, and arbitrary real-valued intervals. Additionally, we adopt the partial ordering "$\supseteq$" defined by

For $\eta\in\mathbb{R}$ and $\mathcal{A}\in\mathbb{R}_I$, then

For arbitrary $\mathcal{A}, \mathcal{B}\in\mathbb{R}_I$, the following four arithmetic operations are given:

For additional information on interval arithmetic, refer to ^[40].

Definition 2.1. ^[13] A function $\mathscr{F}:[\Im_1, \Im_2]\to \mathbb{R}$ is classified as s-convex if it satisfies the convexity condition:

for all $\hbar_1, \hbar_2\in [\Im_1, \Im_2]$ and $\iota\in[0, 1]$ with $s\in(0, 1]$.

In ^[41], Breckner introduced the IV s-convex functions.

Definition 2.2. ^[41] $\mathfrak{F}:[\Im_1, \Im_2]\to\mathbb{R}_{I}^{+}$ is defined as an IV s-convex function if

for $\hbar_1, \hbar_2\in[\Im_1, \Im_2]$ and $\iota\in[0, 1]$.

Consider $\mathfrak{F}:[\Im_1, \Im_2]\to\mathbb{R}_{I}^{+}$, we define $\int_{\Im_1}^{\Im_2}\mathfrak{F}(\iota)d\iota$ by

and say that the function $\mathfrak{F}$ is interval Lebesgue integrable on $[\Im_1, \Im_2]$ (or that $\mathfrak{F}\in IL_{[\Im_1, \Im_2]}$).

Definition 2.3. ^[42] Let $\mathscr{F}\in L[\Im_1, \Im_2]$. The bilateral R-L fractional integral operators, namely the left-sided $\mathcal{J}_{\Im_1+}^{\alpha}\mathscr{F}$ and the right-sided $\mathcal{J}_{\Im_2-}^{\alpha}\mathscr{F}$, are defined as

and

respectively, where $\alpha > 0$, $\Im_1\ge0$, $\mathcal{J}_{\Im_1+}^{0}\mathscr{F}(\sigma) = \mathcal{J}_{\Im_2-}^{0}\mathscr{F}(\sigma) = \mathscr{F}(\sigma)$ and $\Gamma(\alpha)$ is the Gamma function.

The H-H inequality in the form of R-L fractional integrals was proved by Sarikaya et al. in ^[32] as follows:

Theorem 2.4. Let $\mathscr{F}\in L[\Im_1, \Im_2]$ be a mapping from $[\Im_1, \Im_2]$ to $\mathbb{R}^{+}$. If $\mathscr{F}$ satisfies the convexity condition, then

with $\alpha > 0$.

Definition 2.5. ^[36,37] Let $\mathfrak{F}:[\Im_1, \Im_2]\to\mathbb{R}_{I}^{+}$. The IV R-L fractional integral operators, including both left-sided and right-sided variants, are defined as follows:

and

respectively. Here, $\mathcal{I}_{\Im_1+}^{\alpha}\mathfrak{F}(\sigma) = \bigl [\mathcal{J}_{\Im_1+}^{\alpha}\underline{\mathfrak{F}}(\sigma), \mathcal{J}_{\Im_1+}^{\alpha}\overline{\mathfrak{F}}(\sigma)\bigr]$, and $\alpha > 0$.

In ^[36], Budak et al. gave the fractional H-H inequality for IV convex functions as follows:

Theorem 2.6. Let $\mathfrak{F}:[\Im_1, \Im_2]\to \mathbb{R}_{I}^{+}$, and $\alpha > 0$. If $\mathfrak{F}$ is an IV convex function, then

Definition 2.7. ^[31] Let $\varphi:[\Im_1, \Im_2]\to\mathbb{R}^{+}$ be a monotonically increasing mapping, and suppose $\mathscr{F}, \varphi\in L[\Im_1, \Im_2]$. The generalized R-L fractional integral operators $\mathcal{J}_{\Im_1+, \varphi}^{\alpha, k}\mathscr{F}$ and $\mathcal{J}_{\Im_2-, \varphi}^{\alpha, k}\mathscr{F}$ are defined by

and

respectively, where $k\in\mathbb{N}\cup\{0\}$, $\alpha > 0$, and $\Im_1\ge 0$.

In 2023, Budak et al. introduced the generalized fractional integral for IV function as follows:

Definition 2.8. ^[38] Let $\varphi:[\Im_1, \Im_2]\to\mathbb{R}^{+}$ be a monotonically increasing function, and $\mathfrak{F}:[\Im_1, \Im_2]\to\mathbb{R}_{I}^{+}$. The generalized R-L fractional integrals $\mathcal{I}_{\Im_1+, \varphi}^{\alpha, k}\mathfrak{F}$ and $\mathcal{I}_{\Im_2-, \varphi}^{\alpha, k}\mathfrak{F}$ of interval-valued functions are defined by

and

respectively, where $\alpha > 0$, $\Im_1\ge0$ and $k\in\mathbb{N}\cup\{0\}$.

The families of all functions that are Riemann integrable and interval Riemann integrable over the closed interval $[\Im_1, \Im_2]$ are respectively represented by the notations $\mathcal{R}_{[\Im_1, \Im_2]}$ and $\mathcal{IR}_{[\Im_1, \Im_2]}$.

Theorem 2.9. ^[38] Suppose $\mathfrak{F}\in\mathcal{IR}_{[\Im_1, \Im_2]}$, $\varphi:[\Im_1, \Im_2]\to\mathbb{R}$ is a monotonically increasing function on $(\Im_1, \Im_2)$ with $\varphi\in L[\Im_1, \Im_2]$. Letting $\Phi(\iota) = \mathfrak{F}(\iota)+\mathfrak{F}(\Im_1+\Im_2-\iota)$, then $\Phi\in \mathcal{IR}_{[\Im_1, \Im_2]}$, and we have

where $\alpha > 0$ and $k\in\mathbb{N}\cup\{0\}$.

3.   Main results

Let $\Phi(\varpi) = \mathfrak{F}(\varpi)+\mathfrak{F}(\Im_1+\Im_2-\varpi)$ and $\Lambda(\varpi) = \mathcal{G}(\varpi)+\mathcal{G}(\Im_1+\Im_2-\varpi)$ for $\varpi\in [\Im_1, \Im_2]$. It is straightforward to demonstrate that if $\mathfrak{F}, \mathcal{G}\in\mathcal{IR}_{[\Im_1, \Im_2]}$, then $\Phi(\varpi), \Lambda(\varpi)\in\mathcal{IR}_{[\Im_1, \Im_2]}$.

Theorem 3.1. Let $\mathfrak{F}:[{\Im_1}, {\Im_2}]\to\mathbb{R}_{I}^{+}$, $\mathfrak{F} \in\mathcal{IR}_{[\Im_1, \Im_2]}$, and $\varphi:[\Im_1, \Im_2]\to\mathbb{R}^{+}$ be a monotonically increasing function. If $\mathfrak{F}$ is an IV s-convex function, then

where $\alpha > 0$ and $k\in \mathbb{N}\cup\{0\}$.

Proof. By the assumption, we have

Letting $\hbar_1 = \iota\Im_1+(1-\iota)\Im_2$, $\hbar_2 = (1-\iota)\Im_1+\iota\Im_2$ for $\iota \; \in [0, 1]$, we obtain

Then, multiplying the two sides of (3.3) by $\iota^{\alpha-1}\bigl(\varphi(\Im_2)-\varphi(\iota\Im_1 +(1-\iota)\Im_2) \bigr)^{k}$ and integrating on [0, 1], we obtain

Letting $y = \iota\Im_1+(1-\iota\Im_2)$, we have

That is,

By multiplying the two sides of (3.3) by $\iota^{\alpha-1}\Bigl(\varphi\bigl((1-\iota)\Im_1+\iota\Im_2\bigr) -\varphi(\Im_1)\Bigr)^{k}$ and integrating on [0, 1], we obtain

Letting $y = (1-\iota)\Im_1+\iota\Im_2$, we have

Combining with conclusions (3.4) and (3.5), we obtain

According to (2.2), we have

That is,

Multiplying the two sides of (3.6) by $\iota^{\alpha-1}\Bigl(\varphi(\Im_2)-\varphi\bigl(\iota\Im_1 +(1-\iota)\Im_2\bigr)\Bigr)^{k}$ and integrating on [0, 1], we obtain

Letting $y = \iota\Im_1+(1-\iota)\Im_2$, we have

That is,

Similarly, multiplying the two sides of $(3.6)$ by $\iota^{\alpha-1}\Bigl(\varphi\bigl((1-\iota)\Im_1+\iota\Im_2\bigr) -\varphi(\Im_1)\Bigr)^{k}$ and integrating on [0, 1], let $y = \iota\Im_2+(1-\iota)\Im_1$, we have

Then, we obtain

By (3.7) and (3.8), we have

Hence, Theorem 3.1 is verified. □

Corollary 3.2. If $\varphi(\iota) = \iota-\omega$ for $\omega\in\mathbb{R}$ and $s = 1$, then

Remark 3.3. If $s = 1$, Theorem 3.1 is reduced to Theorem 4.1 obtained by Budak et al. in^[38].

Remark 3.4. If $k = 0$, then

Remark 3.5. If $s = 1$ and $k = 0$, then Theorem 3.1 simplifies to Theorem 3.2 as presented by Budak et al. in ^[36].

Example 3.6. Consider $\mathfrak{F}:[0, 2]\to\mathbb{R}_I^{+}$, where $\mathfrak{F}(x) = \bigl[2x^2-4, -8x^2 + 20x - 3\bigr]$, $\varphi(\iota) = \iota-\omega$ for $\omega\in\mathbb{R}$, $\alpha = 2, s = 1$, and $k = 1$. Then we have

and

Then, we obtain

The graphical representation (Figure 1) confirms the results. $

Theorem 3.7. Assuming that the conditions of Theorem 3.1 are met, then

where $\alpha > 0$ and $k\in \mathbb{N}\cup\{0\}.$

Proof. First, consider $\hbar_1 = \frac{1}{2}\iota\Im_1 +\frac{1}{2}(2-\iota)\Im_2$ and $\hbar_2 = \frac{1}{2}\iota\Im_2+\frac{1}{2}(2-\iota)\Im_1$ for $\iota\in[0, 1]$ in the inclusion (3.2). Then,

Then, multiplying the two sides of (3.10) by $\iota^{\alpha-1} \Bigl(\varphi(\Im_2)- \varphi\bigl(\frac{1}{2}\iota\Im_1+\frac{1}{2}(2-\iota)\Im_2\bigr) \Bigr)^{k}$ and integrating on [0, 1], we obtain

Letting $y = \frac{1}{2}\iota\Im_1 +\frac{1}{2}(2-\iota)\Im_2$, we have

that is,

Similarly, multiplying the two sides of (3.10) by $\iota^{\alpha-1}\biggl(\varphi\Bigl(\frac{1}{2}(2-\iota)\Im_1 +\frac{1}{2}\iota\Im_2\Bigr)-\varphi(\Im_1)\biggr)^{k}$ and integrating on [0, 1], we have

Letting $y = \frac{1}{2}(2-\iota)\Im_1+\frac{1}{2}\iota\Im_2$, we have

that is,

By (3.11) and (3.12), we obtain the result. □

Corollary 3.8. If $\varphi(\iota) = \iota-\omega$ for $\omega\in\mathbb{R}$ and $s = 1$, then

Remark 3.9. If $s = 1$, then Theorem 3.7 simplifies to Theorem 4.4 given by Budak et al. in^[38].

Remark 3.10. If $k = 0$, then

Remark 3.11. If $s = 1$ and $k = 0$, then Theorem 3.7 which has been obtained by Zhao et al. in ^[39].

Example 3.12. Consider $\mathfrak{F}:[0, 1]\to\mathbb{R}_I^{+}$, where $\mathfrak{F}(x) = \bigl[x^2, 5-(\frac{1}{2})^x\bigr]$, $\varphi(\iota) = \iota-\omega$ for $\omega\in\mathbb{R}$, $\alpha = 2, s = 1$, and $k = 1$. Then we have

and

Then, we obtain

Theorem 3.13. Given that the assumptions of Theorem 3.1 hold, we can obtain

for $\alpha > 0$ and $k\in\mathbb{N}\cup\{0\}.$

Proof. Considering $\hbar_1 = \frac{1}{2}(1+\iota)\Im_1+\frac{1}{2}(1-\iota)\Im_2$ and $\hbar_1 = \frac{1}{2}(1-\iota)\Im_1+\frac{1}{2}(1+\iota)\Im_2$ for $\iota\in[0, 1]$ in the inclusion (3.2). Then we obtain

Then, multiplying two sides of (3.14) by

and integrating on [0, 1], we obtain

Letting $z = \frac{1}{2}(1+\iota)\Im_1+\frac{1}{2}(1-\iota)\Im_2$, we have

that is,

Multiplying two sides of (3.14) by

and integrating on [0, 1], we obtain

Letting $z = \frac{1}{2}(1-\iota)\Im_1+\frac{1}{2}(1+\iota)\Im_2$, we obtain

Adding the inclusion (3.15) to (3.16), we complete the proof. □

Corollary 3.14. If $\varphi(\iota) = \iota-\omega$ for $\omega\in\mathbb{R}$ and $s = 1$, then

Remark 3.15. If $s = 1$, Theorem 3.13 simplifies to Theorem 4.7 presented by Budak et al. in^[38].

Remark 3.16. If $s = 1$ and $k = 0$, then

which has been obtained by Zhao et al. in ^[39].

Example 3.17. Consider $\mathfrak{F}:[0, 2]\to\mathbb{R}_I^{+}$, where $\mathfrak{F}(x) = \bigl[x^2, -x^2+8\bigr]$, $\varphi(\iota) = \iota-\omega$ for $\omega\in\mathbb{R}$, $\alpha = 2, s = 1$, and $k = 1$. Then we have

and

Then, we obtain

Theorem 3.18. Let $\mathfrak{F}, \mathcal{G}:[\Im_1, \Im_2]\to \mathbb{R}_{I}^{+}$, $s_1, s_2\in (0, 1]$, and $\varphi:[\Im_1, \Im_2]\to\mathbb{R}$ be a monotonically increasing function. If $\mathfrak{F}$ and $\mathcal{G}$ are both IV s-convex functions, then

where

and

Proof. By hypothesis, then

Considering $\hbar_1 = \iota\Im_1+(1-\iota\Im_2)$ and $\hbar_2 = (1-\iota)\Im_1+\iota\Im_2$, we obtain

By multiplying two sides of (3.18) by $\iota^{\alpha-1}\bigl(\varphi(\Im_2) -\varphi(\iota\Im_1+(1-\iota)\Im_2) \bigr)^{k}$ and integrating on [0, 1], we obtain

Letting $y = \iota\Im_1+(1-\iota)\Im_2$, then

that is,

By multiplying two sides of (3.18) by $\iota^{\alpha-1}\bigl(\varphi((1-\iota)\Im_1+\iota\Im_2) -\varphi(\Im_1)\bigr)^{k}$ and integrating on [0, 1], we have

Letting $y = (1-\iota)\Im_1+\iota\Im_2$, that is,

Adding the inclusion (3.19) and (3.20), we obtained the desired result. □

Example 3.19. Assume $\mathfrak{F}, \mathcal{G}:[0, 2]\to\mathbb{R}_I^{+}$, where $\mathfrak{F}(x) = \bigl[x^2, -x^2 + 10\bigr]$ and

respectively. Let $\varphi(\iota) = \iota-\omega$ for $\omega\in\mathbb{R}$, $\alpha = 2, k = 1$, and $s = 1$. Then we have

and

and

Then, we obtain

Analogously, we can derive the subsequent results.

Theorem 3.20. Assuming that the conditions of Theorem 3.18 hold, we consequently obtain

Theorem 3.21. Assuming that the conditions of Theorem 3.18 hold, we obtain

4.   Conclusions

This study derives novel H-H-type inequalities through generalized fractional integrals applied to IV s-convex functions. These not only generalize but also refine the previously proposed inequalities established by Budak et al. and provide a foundation for further exploration of generalized convexity and IV estimation. The developed techniques offer new tools for uncertainty quantification in convex optimization problems with imprecise measurements. Future research directions include:

(1) Develop new interval H-H inequalities based on more general convexity.

(2) Extension to IV quasi-convex functions using variable-order fractional operators.

(3) Applications in robust portfolio optimization with IV risk measures.

Author contributions

G. Deng and D. Zhao: Conceptualization, Methodology, Formal analysis; S. Etemad: Conceptualization, Software, Formal analysis, Investigation, Writing–original draft preparation; G. Deng: Writing–original draft preparation; D. Zhao: Validation, Investigation, Supervision, Project administration, Writing–review and editing; J. Tariboon: Validation, Formal analysis, Project administration, Writing–review and editing. All authors read and approved the final manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded by the National Science, Research and Innovation Fund (NSRF) and King Mongkut's University of Technology North Bangkok, under Contract No. KMUTNB-FF-68-B-04, and the Foundation of Hubei Normal University (2024Z022).

Conflict of interest

The authors declare no conflict of interest.