Multiplicity results for some fourth-order elliptic equations with combined nonlinearities

1.   Introduction

The convexity of function is a classical concept, since it plays a fundamental role in mathematical programming theory, game theory, mathematical economics, variational science, optimal control theory and other fields, a new branch of mathematics, convex analysis, appeared in the 1960s. However, it has been noticed that the functions encountered in a large number of theoretical and practical problems in economics are not classical convex functions, therefore, in the past decades, the generalization of function convexity has attracted the attention of many scholars and aroused great interest, such as $h$-convex functions ^[1,2,3,4,5], $\log$-convex functions ^[6,7,8,9,10], $\log$-$h$-convex functions ^[11], and especially for coordinated convex ^[12]. Since 2001, various extensions and generalizations of integral inequalities for coordinated convex functions have been established in ^{[12,13,14,15,16,17]}.

On the other hand, calculation error has always been a troublesome problem in numerical analysis. In many problems, it is often to speculate the accuracy of calculation results or use high-precision operation as far as possible to ensure the accuracy of the results, because the accumulation of calculation errors may make the calculation results meaningless, interval analysis as a new important tool to solve uncertainty problems has attracted much attention and also has yielded fruitful results, we refer the reader to the papers ^[18,19]. It is worth notion that in recent decades, many authors have combined integral inequalities with interval-valued functions(IVFs) and obtained many excellent conclusions. In ^[20], Costa gave Opial-type inequalities for IVFs. In ^[21,22], Chalco-Cano investigated Ostrowski type inequalities for IVFs by using generalized Hukuhara derivative. In ^[23], Román-Flores derived the Minkowski type inequalities and Beckenbach's type inequalities for IVFs. Very recently, Zhao ^[5,24] established the Hermite-Hadamard type inequalities for interval-valued coordinated functions.

Motivated by these results, in the present paper, we introduce the concept of coordinated log-$h$-convex for IVFs, and then present some new Jensen type inequalities and Hermite-Hadamard type inequalities for interval-valued coordinated functions. Also, we give some examples to illustrate our main results.

2.   Preliminaries

Let $\mathbb{R}_{\mathcal{I}}$ the collection of all closed and bounded intervals of $\mathbb{R}$. We use$\; \mathbb{R}_{\mathcal{I}}^{+}\;$and$\; \mathbb{R}^{+}$ to represent the set of all positive intervals and the family of all positive real numbers respectively. The collection of all Riemann integrable real-valued functions on $[a, b]$, IVFs on $[a, b]$ and IVFs on $\triangle = [a, b]\times[c, d]$ are denoted by $\mathcal{R}_{([a, b])}$, $\mathcal{IR}_{([a, b])}$ and $\mathcal{ID}_{(\triangle)}$. For more conceptions on IVFs, see ^[4,25]. Moreover, we have

Theorem 1. ^[4] Let $f:[a, b]\rightarrow \mathbb{R}_{\mathcal{I}}$ such that $f = [\underline{f}, \overline{f}]$. Then $f\in\mathcal{IR}_{([a, b])}$ iff $\underline{f}$, $\overline{f}\in\mathcal{R}_{([a, b])}$ and

Theorem 2. ^[25] Let $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}$. If $\mathcal{F}\in\mathcal{ID}_{(\triangle)}$, then

Definition 1. ^[26] Let $h:[0, 1]\rightarrow \mathbb{R}^{+}$. We say that $f:[a, b]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ is interval $\log$-$h$-convex function or that $f\in SX(\log\text{-}h, [a, b], \mathbb{R}_{\mathcal{I}}^{+}),$ if for all $x, y\in[a, b]$ and $\vartheta\in[0, 1],$ we have

$h$ is called supermultiplicative if

for all $\vartheta, \tau\in[0, 1]$. If "$\geq$" in (2.1) is replaced with "$\leq$", then $h$ is called submultiplicative.

Theorem 3. ^[26] Let $\mathcal{F}:[a, b]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}, h\left(\frac{1}{2}\right)\not = 0.$ If $\mathcal{F}\in SX(\log\mathit{\text{-}}h, [a, b], \mathbb{R}_{\mathcal{I}}^{+})$ and $\mathcal{F}\in\mathcal{IR}_{([a, b])}$, then

Theorem 4. ^[27] Let $\mathcal{F}:[a, b]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}, h\left(\frac{1}{2}\right)\not = 0.$ If $\mathcal{F}\in SX(\log\mathit{\text{-}}h, [a, b], \mathbb{R}_{\mathcal{I}}^{+})$ and $\mathcal{F}\in\mathcal{IR}_{([a, b])}$, then

3.   Main results

In this section, we define the coordinated $\log$-$h$-convex for IVFs and prove some new Jensen type inequalities and Hermite-Hadamard type inequalities by using this new definition.

Definition 2. Let $h:[0, 1]\rightarrow \mathbb{R}^{+}$. Then $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ is called a coordinated $\log$-$h$-convex IVFs on $\triangle$ if the partial mappings

are $\log$-$h$-convex for all $y\in[c, d]$ and $x\in[a, b]$. Then the set of all coordinated $\log$-$h$-convex IVFs on $\triangle$ is denoted by $SX(\log\text{-}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$.

Definition 3. Let $h:[0, 1]\rightarrow \mathbb{R}^{+}$. Then $\mathcal{F}:\triangle\rightarrow \mathbb{R}^{+}$ is called a coordinated $\log$-$h$-convex function in $\triangle$ if for any $(x_{1}, y_{1}), (x_{2}, y_{2})\in\triangle$ and $\vartheta\in[0, 1]$ we have

The set of all $\log$-$h$-convex functions in $\triangle$ is denoted by $SX(\log\text{-}h, \triangle, \mathbb{R}^{+}).$ If inequality (3.1) is reversed, then $\mathcal{F}$ is said to be a coordinated $\log$-$h$-concave function, the set of all $\log$-$h$-concave functions in $\triangle$ is denoted by $SV(\log\text{-}h, \triangle, \mathbb{R}^{+}).$

Definition 4. Let $h:[0, 1]\rightarrow \mathbb{R}^{+}$. Then $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ is called a coordinated $\log$-$h$-convex IVF in $\triangle$ if for any $(x_{1}, y_{1}), (x_{2}, y_{2})\in\triangle$ and $\vartheta\in[0, 1]$ we have

The set of all $\log$-$h$-convex IVFs in $\triangle$ is denoted by $SX(\log\text{-}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+}).$

Theorem 5. Let $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ such that $\mathcal{F} = [\mathcal{\underline{F}}, \mathcal{\overline{F}}]$. If $\mathcal{F}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ iff $\mathcal{\underline{F}}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}^{+})$ and $\mathcal{\overline{F}}\in SV(\log\mathit{\text{-}}h, \triangle, \mathbb{R}^{+})$.

Proof. The proof is completed by combining the Definitions 3 and 4 above and the Theorem 3.7 of ^[4].

Theorem 6. If $\mathcal{F}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$, then $\mathcal{F}\in SX(\log\mathit{\text{-}}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$.

Proof. Assume that $\mathcal{F}\in SX(\log\text{-}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$. Let $\mathcal{F}_{x}:[c, d]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}, \mathcal{F}_{x}(y) = \mathcal{F}(x, y).$ Then for all $\vartheta\in[0, 1]$ and $y_{1}, y_{2}\in[c, d]$, we have

Hence $\mathcal{F}_{x}(y) = \mathcal{F}(x, y)$ is $\log$-$h$-convex on $[c, d]$. The fact that $\mathcal{F}_{y}(x) = \mathcal{F}(x, y)$ is $\log$-$h$-convex on $[a, b]$ goes likewise.

Remark 1. The converse of Theorem 6 is not generally true. Let $h(\vartheta) = \vartheta$ and $\vartheta\in[0, 1]$, $\triangle_{1} = \left[\frac{\pi}{4}, \frac{\pi}{2}\right]\times\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, and $\mathcal{F}:\triangle_{1}\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ be defined:

Obviously, we have that $\mathcal{F}\in SX(\log\text{-}ch, \triangle_{1}, \mathbb{R}_{\mathcal{I}}^{+})$ and $\mathcal{F}\not\in SX(\log\text{-}h, \triangle_{1}, \mathbb{R}_{\mathcal{I}}^{+}).$ Indeed, if $\left(\frac{\pi}{4}, \frac{\pi}{2}\right), \left(\frac{\pi}{2}, \frac{\pi}{4}\right)\in\triangle_{1}$, we have

If $\vartheta = 0,$ then

Thus, $\mathcal{F}\not\in SX(\log\text{-}h, \triangle_{1}, \mathbb{R}_{\mathcal{I}}^{+}).$

In the following, Jensen type inequalities for coordinated $\log$-$h$-convex functions in $\triangle$ is considered.

Theorem 7. Let $p_{i}\in\mathbb{R}^{+}, x_{i}\in[a, b], y_{i}\in[c, d], (i = 1, 2, ..., n), \mathcal{F}:\triangle\rightarrow \mathbb{R}^{+}.$ If $h$ is a nonnegative supermultiplicative function and $\mathcal{F}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}^{+})$, then

where $\mathcal{P}_{n} = \sum\limits_{i = 1}^{n}p_{i}$. If $h$ is a nonnegative submultiplicative function and $\mathcal{F}\in SV(\log\mathit{\text{-}}h, \triangle, \mathbb{R}^{+})$, then $(3.2)$ is reversed.

Proof. If $n = 2,$ then from Definition 3, we have

Suppose (3.2) holds for $n = k$, then

Now, let us prove that (3.2) is valid when $n = k+1,$

This completes the proof.

Remark 2. If $h(\vartheta) = \vartheta,$ then the inequality (3.2) is the Jensen inequality for $\log$-convex functions.

Now, we prove the Jensen inequality for $\log$-$h$-convex IVFs in $\triangle$.

Theorem 8. Let $p_{i}\in\mathbb{R}^{+}, x_{i}\in[a, b], y_{i}\in[c, d], i = 1, 2, ..., n, \mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ such that $\mathcal{F} = [\underline{F}, \overline{F}]$. If $h$ is a nonnegative supermultiplicative function and $\mathcal{F}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$, then

where $\mathcal{P}_{n} = \sum\limits_{i = 1}^{n}p_{i}$. If $\mathcal{F}\in SV(\log\mathit{\text{-}}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$, then $(3.3)$ is reversed.

Proof. By Theorem 5 and Theorem 7, we have

and

Thus,

This completes the proof.

Next, we prove the Hermite-Hadamard type inequalities for coordinated $\log$-$h$-convex IVFs.

Theorem 9. Let $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ and $h:[0, 1]\rightarrow \mathbb{R}^{+}$ be continuous. If $\mathcal{F}\in SX(\log\mathit{\text{-}}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$, then

Proof. Since $\mathcal{F}\in SX(\log\text{-}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$, we have

That is,

Moreover, we have

Similarly, we get

Then

That is,

Integrating over $[a, b]$, we have

Similarly, we have

Finally, we obtain

This concludes the proof.

Remark 3. If $\mathcal{\underline{F}} = \mathcal{\overline{F}}$ and $h(\vartheta) = \vartheta$, then Theorem 9 reduces to Corollary 3.1 of ^[13].

Example 1. Let $[a, b] = [c, d] = [2, 3], h(\vartheta) = \vartheta$. We define $\mathcal{F}:[2, 3]\times[2, 3]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ by

From Definition 2, $\mathcal{F}(x, y)\in SX(\log\text{-}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$.

Since

and

It follows that

and Theorem 9 is verified.

Theorem 10. Let $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ and $h:[0, 1]\rightarrow \mathbb{R}^{+}$ be continuous. If $\mathcal{F}\in SX(\log\mathit{\text{-}}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$, then

Proof. Since $\mathcal{F}\in SX(\log\text{-}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$, by using Theorem 6 and (2.3), we have

That is,

Moreover, we have

Similarly, we have

We also from (2.2),

Again from (2.3),

and proof is completed.

Example 2. Furthermore, by Example 1, we have

and

It follows that

and Theorem 10 is verified.

4.   Conclusions

We introduced the coordinated log-$h$-convexity for interval-valued functions, some Jensen type inequalities and Hermite-Hadamard type inequalities are proved. Our results generalize some known inequalities and will be useful in developing the theory of interval integral inequalities and interval convex analysis. The next step in the research direction investigated inequalities for fuzzy-interval-valued functions, and some applications in interval nonlinear programming.

Acknowledgments

The first author was supported in part by the Key Projects of Educational Commission of Hubei Province of China (D20192501), the Natural Science Foundation of Jiangsu Province (BK20180500) and the National Key Research and Development Program of China (2018YFC1508100).

Conflict of interest

The authors declare no conflict of interest.