
In recent years, numerous scholars have investigated the relationship between symmetry and generalized convexity. Due to this close relationship, generalized convexity and symmetry have become new areas of study in the field of inequalities. With the help of fuzzy up and down relation, the class of up and down λ-convex fuzzy-number valued mappings is introduced in this study; and weighted Hermite-Hadamard type fuzzy inclusions are demonstrated for these functions. The product of two up and down λ-convex fuzzy-number valued mappings also has Hermite-Hadamard type fuzzy inclusions, which is another development. Additionally, by imposing some mild restrictions on up and down λ-convex (λ-concave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down λ-convexity (λ-concavity), referred to as lower up and down λ-convex (lower up and down λ-concave) and upper up and down λ-convex (λ-concave) fuzzy number valued mappings. Using these definitions, we have amassed many classical and novel exceptional cases that implement the key findings. Our proven results expand and generalize several previous findings in the literature body. Additionally, we offer appropriate examples to corroborate our theoretical findings.
Citation: Muhammad Bilal Khan, Hakeem A. Othman, Gustavo Santos-García, Muhammad Aslam Noor, Mohamed S. Soliman. Some new concepts in fuzzy calculus for up and down λ-convex fuzzy-number valued mappings and related inequalities[J]. AIMS Mathematics, 2023, 8(3): 6777-6803. doi: 10.3934/math.2023345
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In recent years, numerous scholars have investigated the relationship between symmetry and generalized convexity. Due to this close relationship, generalized convexity and symmetry have become new areas of study in the field of inequalities. With the help of fuzzy up and down relation, the class of up and down λ-convex fuzzy-number valued mappings is introduced in this study; and weighted Hermite-Hadamard type fuzzy inclusions are demonstrated for these functions. The product of two up and down λ-convex fuzzy-number valued mappings also has Hermite-Hadamard type fuzzy inclusions, which is another development. Additionally, by imposing some mild restrictions on up and down λ-convex (λ-concave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down λ-convexity (λ-concavity), referred to as lower up and down λ-convex (lower up and down λ-concave) and upper up and down λ-convex (λ-concave) fuzzy number valued mappings. Using these definitions, we have amassed many classical and novel exceptional cases that implement the key findings. Our proven results expand and generalize several previous findings in the literature body. Additionally, we offer appropriate examples to corroborate our theoretical findings.
In this paper, we consider the following initial-boundary value problem
{ut−Δut−Δu=|x|σ|u|p−1u,x∈Ω,t>0,u(x,t)=0,x∈∂Ω,t>0,u(x,0)=u0(x),x∈Ω | (1) |
and its corresponding steady-state problem
{−Δu=|x|σ|u|p−1u,x∈Ω,u=0,x∈∂Ω, | (2) |
where
1<p<{∞,n=1,2;n+2n−2,n≥3,σ>{−n,n=1,2;(p+1)(n−2)2−n,n≥3. | (3) |
(1) was called homogeneous (inhomogeneous) pseudo-parabolic equation when
The homogeneous problem, i.e.
Li and Du [12] studied the Cauchy problem of equation in (1) with
(1) If
(2) If
Φα:={ξ(x)∈BC(Rn):ξ(x)≥0,lim inf|x|↑∞|x|αξ(x)>0}, |
and
Φα:={ξ(x)∈BC(Rn):ξ(x)≥0,lim sup|x|↑∞|x|αξ(x)<∞}. |
Here
In view of the above introductions, we find that
(1) for Cauchy problem in
(2) for zero Dirichlet problem in a bounded domain
The difficulty of allowing
σ>(p+1)(n−2)2−n⏟<0 if n≥3 |
for
The main results of this paper can be summarized as follows: Let
(1) (the case
(2) (the case
(3) (arbitrary initial energy level) For any
(4) Moreover, under suitable assumptions, we show the exponential decay of global solutions and lifespan (i.e. the upper bound of blow-up time) of the blowing-up solutions.
The organizations of the remain part of this paper are as follows. In Section 2, we introduce the notations used in this paper and the main results of this paper; in Section 3, we give some preliminaries which will be used in the proofs; in Section 4, we give the proofs of the main results.
Throughout this paper we denote the norm of
‖ϕ‖Lγ={(∫Ω|ϕ(x)|γdx)1γ, if 1≤γ<∞;esssupx∈Ω|ϕ(x)|, if γ=∞. |
We denote the
Lp+1σ(Ω):={ϕ:ϕ is measurable on Ω and ‖u‖Lp+1σ<∞}, | (4) |
where
‖ϕ‖Lp+1σ:=(∫Ω|x|σ|ϕ(x)|p+1dx)1p+1,ϕ∈Lp+1σ(Ω). | (5) |
By standard arguments as the space
We denote the inner product of
(ϕ,φ)H10:=∫Ω(∇ϕ(x)⋅∇φ(x)+ϕ(x)φ(x))dx,ϕ,φ∈H10(Ω). | (6) |
The norm of
‖ϕ‖H10:=√(ϕ,ϕ)H10=√‖∇ϕ‖2L2+‖ϕ‖2L2,ϕ∈H10(Ω). | (7) |
An equivalent norm of
‖∇ϕ‖L2≤‖ϕ‖H10≤√λ1+1λ1‖∇ϕ‖L2,ϕ∈H10(Ω), | (8) |
where
λ1=infϕ∈H10(Ω)‖∇ϕ‖2L2‖ϕ‖2L2. | (9) |
Moreover, by Theorem 3.2, we have
for p and σ satisfying (4), H10(Ω)↪Lp+1σ(Ω) continuously and compactly. | (10) |
Then we let
Cpσ=supu∈H10(Ω)∖{0}‖ϕ‖Lp+1σ‖∇ϕ‖L2. | (11) |
We define two functionals
J(ϕ):=12‖∇ϕ‖2L2−1p+1‖ϕ‖p+1Lp+1σ | (12) |
and
I(ϕ):=‖∇ϕ‖2L2−‖ϕ‖p+1Lp+1σ. | (13) |
By (3) and (10), we know that
We denote the mountain-pass level
d:=infϕ∈NJ(ϕ), | (14) |
where
N:={ϕ∈H10(Ω)∖{0}:I(ϕ)=0}. | (15) |
By Theorem 3.3, we have
d=p−12(p+1)C−2(p+1)p−1pσ, | (16) |
where
For
Jρ={ϕ∈H10(Ω):J(ϕ)<ρ}. | (17) |
Then, we define the set
Nρ={ϕ∈N:‖∇ϕ‖2L2<2(p+1)ρp−1},ρ>d. | (18) |
For
λρ:=infϕ∈Nρ‖ϕ‖H10,Λρ:=supϕ∈Nρ‖ϕ‖H10 | (19) |
and two sets
Sρ:={ϕ∈H10(Ω):‖ϕ‖H10≤λρ,I(ϕ)>0},Sρ:={ϕ∈H10(Ω):‖ϕ‖H10≥Λρ,I(ϕ)<0}. | (20) |
Remark 1. There are two remarks on the above definitions.
(1) By the definitions of
(2) By Theorem 3.4, we have
√2(p+1)dp−1≤λρ≤Λρ≤√2(p+1)(λ1+1)ρλ1(p−1). | (21) |
Then the sets
‖sϕ‖H10≤√2(p+1)dp−1⇔s≤δ1:=√2(p+1)dp−1‖ϕ‖−1H10,I(sϕ)=s2‖∇ϕ‖2L2−sp+1‖ϕ‖p−1Lp+1σ>0⇔s<δ2:=(‖∇ϕ‖2L2‖ϕ‖p+1Lp+1σ)1p−1,‖sϕ‖H10≥√2(p+1)(λ1+1)ρλ1(p−1)⇔s≥δ3:=√2(p+1)(λ1+1)ρλ1(p−1)‖ϕ‖−1H10,I(sϕ)=s2‖∇ϕ‖2L2−sp+1‖ϕ‖p−1Lp+1σ<0⇔s>δ2. |
So,
{sϕ:0<s<min{δ1,δ2}}⊂Sρ,{sϕ:s>max{δ2,δ3}}⊂Sρ. |
In this paper we consider weak solutions to problem (1), local existence of which can be obtained by Galerkin's method (see for example [22,Chapter II,Sections 3 and 4]) and a standard limit process and the details are omitted.
Definition 2.1. Assume
∫Ω(utv+∇ut⋅∇v+∇u⋅∇v−|x|σ|u|p−1uv)dx=0 | (22) |
holds for any
u(⋅,0)=u0(⋅) in H10(Ω). | (23) |
Remark 2. There are some remarks on the above definition.
(1) Since
(2) Denote by
(3) Taking
‖u(⋅,t)‖2H10=‖u0‖2H10−2∫t0I(u(⋅,s))ds,0≤t≤T, | (24) |
where
(4) Taking
J(u(⋅,t))=J(u0)−∫t0‖us(⋅,s)‖2H10ds,0≤t≤T, | (25) |
where
Definition 2.2. Assume (3) holds. A function
∫Ω(∇u⋅∇v−|x|σ|u|p−1uv)dx=0 | (26) |
holds for any
Remark 3. There are some remarks to the above definition.
(1) By (10), we know all the terms in (26) are well-defined.
(2) If we denote by
Φ={ϕ∈H10(Ω):J′(ϕ)=0 in H−1(Ω)}⊂(N∪{0}), | (27) |
where
With the set
Definition 2.3. Assume (3) holds. A function
J(u)=infϕ∈Φ∖{0}J(ϕ). |
With the above preparations, now we can state the main results of this paper. Firstly, we consider the case
(1)
(2)
(3)
Theorem 2.4. Assume (3) holds and
‖∇u(⋅,t)‖L2≤√2(p+1)J(u0)p−1,0≤t<∞, | (28) |
where
V:={ϕ∈H10(Ω):J(ϕ)≤d,I(ϕ)>0}. | (29) |
In, in addition,
‖u(⋅,t)‖H10≤‖u0‖H10exp[−λ1λ1+1(1−(J(u0)d)p−12)t]. | (30) |
Remark 4. Since
J(u0)>p−12(p+1)‖∇u0‖2L2>0. |
So the equality (28) makes sense.
Theorem 2.5. Assume (3) holds and
limt↑Tmax∫t0‖u(⋅,s)‖2H10ds=∞, |
where
W:={ϕ∈H10(Ω):J(ϕ)≤d,I(ϕ)<0} | (31) |
and
Tmax≤4p‖u0‖2H10(p−1)2(p+1)(d−J(u0)). | (32) |
Remark 5. There are two remarks.
(1) If
(2) The sets
f(s)=J(sϕ)=s22‖∇ϕ‖2L2−sp+1p+1‖ϕ‖p+1Lp+1σ,g(s)=I(sϕ)=s2‖∇ϕ‖2L2−sp+1‖ϕ‖p+1Lp+1σ. |
Then (see Fig. 2)
(a)
maxs∈[0,∞)f(s)=f(s∗3)=p−12(p+1)(‖∇ϕ‖L2‖ϕ‖Lp+1σ)2(p+1)p−1≤d⏟By (14) since s∗3ϕ∈N, | (33) |
(b)
maxs∈[0,∞)g(s)=g(s∗1)=p−1p+1(2p+1)2p−1(‖∇ϕ‖L2‖ϕ‖Lp+1σ)2(p+1)p−1, |
(c)
f(s∗2)=g(s∗2)=p−12p(p+12p)2p−1(‖∇ϕ‖L2‖ϕ‖Lp+1σ)2(p+1)p−1, |
where
s∗1:=(2‖∇ϕ‖2L2(p+1)‖ϕ‖p+1Lp+1σ)1p−1<s∗2:=((p+1)‖∇ϕ‖2L22p‖ϕ‖p+1Lp+1σ)1p−1<s∗3:=(‖∇ϕ‖2L2‖ϕ‖p+1Lp+1σ)1p−1<s∗4:=((p+1)‖∇ϕ‖2L22‖ϕ‖p+1Lp+1σ)1p−1. |
So,
Theorem 2.6. Assume (3) holds and
G:={ϕ∈H10(Ω):J(ϕ)=d,I(ϕ)=0}. | (34) |
Remark 6. There are two remarks on the above theorem.
(1) Unlike Remark 5, it is not easy to show
(2) To prove the above Theorem, we only need to show
Theorem 2.7. Assume (3) holds and let
Secondly, we consider the case
Theorem 2.8. Assume (3) holds and the initial value
(i): If
(ii): If
Here
Next, we show the solution of the problem (1) can blow up at arbitrary initial energy level (Theorem 2.10). To this end, we firstly introduce the following theorem.
Theorem 2.9. Assume (3) holds and
Tmax≤8p‖u0‖2H10(p−1)2(λ1(p−1)λ1+1‖u0‖2H10−2(p+1)J(u0)) | (35) |
and
limt↑Tmax∫t0‖u(⋅,s)‖2H10ds=∞, |
where
ˆW:={ϕ∈H10(Ω):J(ϕ)<λ1(p−1)2(λ1+1)(p+1)‖ϕ‖2H10}. | (36) |
and
By using the above theorem, we get the following theorem.
Theorem 2.10. For any
The following lemma can be found in [11].
Lemma 3.1. Suppose that
F″(t)F(t)−(1+γ)(F′(t))2≥0 |
for some constant
T≤F(0)γF′(0)<∞ |
and
Theorem 3.2. Assume
Proof. Since
We divide the proof into three cases. We will use the notation
Case 1.
\begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{p+1}( \Omega)$ continuously and compactly.} \end{equation} | (37) |
Then we have, for any
\begin{equation*} \|u\|_{L^{p+1}_{\sigma}}^{p+1} = \int_ \Omega|x|^\sigma|u|^{p+1}dx\leq R^\sigma\|u\|_{L^{p+1}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, \end{equation*} |
which, together with (37), implies
Case 2.
\begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{\frac{(p+1)r}{r-1}}( \Omega)$ continuously and compactly,} \end{equation} | (38) |
for any
\begin{align*} \|u\|_{L^{p+1}_{\sigma}}^{p+1}& = \int_ \Omega|x|^\sigma|u|^{p+1}dx\\&\leq \left(\int_{B(0,R)}|x|^{\sigma r}dx\right)^{\frac1r}\left(\int_ \Omega|u|^{\frac{(p+1)r}{r-1}}dx\right)^{\frac{r-1}{r}}\\ &\leq\left\{ \begin{array}{ll} \left(\frac{2}{\sigma r+1}R^{\sigma r+1}\right)^{\frac 1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, & n = 1;\\ \left(\frac{2\pi}{\sigma r+2}R^{\sigma r+2}\right)^{\frac 1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, & n = 2, \end{array} \right. \end{align*} |
which, together with (38), implies
Case 3.
\begin{equation*} -\frac{\sigma}{n} < \frac1r < 1-\frac{(p+1)(n-2)}{2n}. \end{equation*} |
By the second inequality of the above inequalities, we have
\begin{equation*} \frac{(p+1)r}{r-1} = \frac{p+1}{1-\frac1r} < \frac{p+1}{\frac{(p+1)(n-2)}{2n}} = \frac{2n}{n-2}. \end{equation*} |
So,
\begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{\frac{(p+1)r}{r-1}}( \Omega)$ continuously and compactly.} \end{equation} | (39) |
Then by Hölder's inequality, for any
\begin{align*} \|u\|_{L^{p+1}_{\sigma}}^{p+1}& = \int_ \Omega|x|^\sigma|u|^{p+1}dx\\&\leq \left(\int_{B(0,R)}|x|^{\sigma r}dx\right)^{\frac1r}\left(\int_ \Omega|u|^{\frac{(p+1)r}{r-1}}dx\right)^{\frac{r-1}{r}}\\ &\leq\left(\frac{\omega_{n-1}}{\sigma r+n}R^{\sigma r+n}\right)^{\frac1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, \end{align*} |
which, together with (39), implies
Theorem 3.3. Assume
\begin{equation*} d = \frac{p-1}{2(p+1)}C_{p\sigma}^{\frac{2(p+1)}{p-1}}, \end{equation*} |
where
Proof. Firstly, we show
\begin{equation} \inf\limits_{\phi\in N}J(\phi) = \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi), \end{equation} | (40) |
where
\begin{equation} s_\phi^*: = \left(\frac{\|\nabla \phi\|_{L^2}^2}{\|\phi\|_{L_\sigma^{p+1}}^{p+1}}\right)^{\frac{1}{p-1}}. \end{equation} | (41) |
By the definition of
On one hand, since
\begin{equation*} \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi)\leq\min\limits_{\phi\in N}J(s_\phi^*\phi) = \min\limits_{\phi\in N}J(\phi). \end{equation*} |
On the other hand, since
\begin{equation*} \inf\limits_{\phi\in N}J(\phi)\leq \inf\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi). \end{equation*} |
Then (40) follows from the above two inequalities.
By (40), the definition of
\begin{align*} d& = \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi)\\ = &\frac{p-1}{2(p+1)}\min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}\left(\frac{\|\nabla \phi\|_{L^2}}{\|\phi\|_{L_\sigma^{p+1}}}\right)^{\frac{2(p+1)}{p-1}}\\ = &\frac{p-1}{2(p+1)}C_{p\sigma}^{-\frac{2(p+1)}{p-1}}. \end{align*} |
Theorem 3.4. Assume (3) holds. Let
\begin{equation} \sqrt{\frac{2(p+1)d}{p-1}}\leq \lambda_\rho\leq \Lambda_\rho\leq \sqrt{\frac{2(p+1)( \lambda_1+1)\rho}{ \lambda_1(p-1)}}. \end{equation} | (42) |
Proof. Let
\begin{equation} \lambda_\rho\leq\Lambda_\rho. \end{equation} | (43) |
Since
\begin{align*} d& = \inf\limits_{\phi\in N}J(\phi)\\ & = \frac{p-1}{2(p+1)}\inf\limits_{\phi\in N}\|\nabla\phi\|_{L^2}^2\\ &\leq\frac{p-1}{2(p+1)}\inf\limits_{\phi\in N^\rho}\|\phi\|_{H_0^1}^2\\ & = \frac{p-1}{2(p+1)} \lambda_\rho^2, \end{align*} |
which implies
\begin{equation*} \lambda_\rho\geq\sqrt{\frac{2(p+1)d}{p-1}} \end{equation*} |
On the other hand, by (8) and (18), we have
\begin{equation*} \begin{split} \Lambda_\rho& = \sup\limits_{\phi\in N^\rho}\|\phi\|_{H_0^1}\\&\leq\sqrt{\frac{ \lambda_1+1}{ \lambda_1}}\sup\limits_{\phi\in N^\rho}\|\nabla\phi\|_{L^2}\\&\leq\sqrt{\frac{ \lambda_1+1}{ \lambda_1}}\sqrt{\frac{2(p+1)\rho}{p-1}}. \end{split} \end{equation*} |
Combining the above two inequalities with (43), we get (42), the proof is complete.
Theorem 3.5. Assume (3) holds and
Proof. We only prove the invariance of
For any
\begin{equation*} \|\nabla\phi\|_{L^2}^2 < \|\phi\|_{L_\sigma^{p+1}}^{p+1}\leq C_{p\sigma}^{p+1}\|\nabla \phi\|_{L^2}^{p+1}, \end{equation*} |
which implies
\begin{equation} \|\nabla\phi\|_{L^2} > C_{p\sigma}^{-\frac{p+1}{p-1}}. \end{equation} | (44) |
Let
\begin{equation} I(u(\cdot,t)) < 0,\; \; \; t\in[0, \varepsilon]. \end{equation} | (45) |
Then by (24),
\begin{equation} J(u(\cdot,t)) < d\hbox{ for }t\in(0, \varepsilon]. \end{equation} | (46) |
We argument by contradiction. Since
\begin{equation} J(u(\cdot,t_0)) < d \end{equation} | (47) |
(note (25) and (46),
\begin{equation*} \|\nabla u(\cdot,t_0)\|_{L^2}\geq C_{p\sigma}^{-\frac{p+1}{p-1}} > 0, \end{equation*} |
which, together with
\begin{equation*} J(u(\cdot,t_0))\geq d, \end{equation*} |
which contradicts (47). So the conclusion holds.
Theorem 3.6. Assume (3) holds and
\begin{equation} \|\nabla u(\cdot,t)\|_{L^2}^2\geq\frac{2(p+1)}{p-1}d,\; \; \; 0\leq t < {T_{\max}}, \end{equation} | (48) |
where
Proof. Let
By the proof in Theorem 3.3,
\begin{align*} d& = \min\limits_{\phi \in H_0^1( \Omega)\setminus\{0\}}J(s^*_\phi\phi)\\ &\leq\min\limits_{\phi\in N^-}J(s_\phi^*\phi)\\ &\leq J(s_u^*u(\cdot,t))\\ & = \frac{(s_u^*)^2}{2}\|\nabla u(\cdot,t)\|_{L^2}^2-\frac{(s_u^*)^{p+1}}{p+1}\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\\ &\leq\left(\frac{(s_u^*)^2}{2}-\frac{(s_u^*)^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2, \end{align*} |
where we have used
\begin{align*} d& = \min\limits_{\phi \in H_0^1( \Omega)\setminus\{0\}}J(s^*_\phi\phi)\\ &\leq\min\limits_{\phi\in N^-}J(s_\phi^*\phi)\\ &\leq J(s_u^*u(\cdot,t))\\ & = \frac{(s_u^*)^2}{2}\|\nabla u(\cdot,t)\|_{L^2}^2-\frac{(s_u^*)^{p+1}}{p+1}\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\\ &\leq\left(\frac{(s_u^*)^2}{2}-\frac{(s_u^*)^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2, \end{align*} |
Then
\begin{equation*} \begin{split} d&\leq\max\limits_{0\leq s\leq 1}\left(\frac{s^2}{2}-\frac{s^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ & = \left(\frac{s^2}{2}-\frac{s^{p+1}}{p+1}\right)_{s = 1}\|\nabla u(\cdot,t)\|_{L^2}^2\\ & = \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t)\|_{L^2}^2, \end{split} \end{equation*} |
and (48) follows from the above inequality.
Theorem 3.7. Assume (3) holds and
Proof. Firstly, we show
\begin{align*} \frac12\|\nabla u_0\|_{L^2}^2-\frac{1}{p+1}\|u_0\|_{L_\sigma^{p+1}}^{p+1}& = J(u_0)\\ & < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2\\ &\leq\frac{p-1}{2(p+1)}\|\nabla u_0\|_{L^2}^2, \end{align*} |
which implies
\begin{equation*} I(u_0) = \|\nabla u_0\|_{L^2}^2-\|u_0\|_{L_\sigma^{p+1}}^{p+1} < 0. \end{equation*} |
Secondly, we prove
\begin{equation} \begin{split} J(u_0)& < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2\\ & < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u(\cdot,t_0)\|_{H_0^1}^2\\ &\leq \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t_0)\|_{L^2}^2. \end{split} \end{equation} | (49) |
On the other hand, by (24), (12), (13) and
\begin{equation*} J(u_0)\geq J(u(\cdot,t_0)) = \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t_0)\|_{L^2}^2, \end{equation*} |
which contradicts (49). The proof is complete.
Proof of Theorem 2.4. Let
\begin{align*} J(u_0)&\geq J(u(\cdot,t))\geq\frac{p-1}{2(p+1)}\|\nabla u(\cdot,t)\|_{L^2}^2,\; \; \; 0\leq t < {T_{\max}}, \end{align*} |
which implies
\begin{equation} \|\nabla u(\cdot,t)\|_{L^2}\leq \sqrt{\frac{2(p+1)J(u_0)}{p-1}},\; \; \; 0\leq t < \infty. \end{equation} | (50) |
Next, we prove
\begin{align*} \frac{d}{dt}\left(\|u(\cdot,t)\|_{H_0^1}^2\right) = &-2I(u(\cdot,t)) = -2\left(\|\nabla u(\cdot,t)\|_{L^2}^2-\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\right)\\ \leq &-2\left(1-C_{p\sigma}^{p+1}\|\nabla u(\cdot,t)\|_{L^2}^{p-1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ \leq &-2\left(1-C_{p\sigma}^{p+1}\left(\sqrt{\frac{2(p+1)J(u_0)}{p-1}}\right)^{p-1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ = &-2\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ \leq&-\frac{2 \lambda_1}{ \lambda_1+1}\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)\|u(\cdot,t)\|_{H_0^1}^2, \end{align*} |
which leads to
\begin{equation*} \|u(\cdot,t)\|_{H_0^1}^2\leq\|u_0\|_{H_0^1}^2\exp\left[-\frac{2 \lambda_1}{ \lambda_1+1}\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)t\right]. \end{equation*} |
The proof is complete.
Proof of Theorem 2.5. Let
Firstly, we consider the case
\begin{equation} \xi(t): = \left(\int_0^t\|u(\cdot,s)\|_{H_0^1}^2ds\right)^{\frac 12},\; \; \; \eta(t): = \left(\int_0^t\|u_s(\cdot,s)\|_{H_0^1}^2ds\right)^{\frac 12},\; \; \; \; 0\leq t < {T_{\max}}. \end{equation} | (51) |
For any
\begin{equation} F(t): = \xi^2(t)+(T^*-t)\|u_0\|_{H_0^1}^2+ \beta(t+ \alpha)^2,\; \; \; 0\leq t\leq T^*. \end{equation} | (52) |
Then
\begin{equation} F(0) = T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2 > 0, \end{equation} | (53) |
\begin{equation} \begin{split} F'(t)& = \|u(\cdot,t)\|_{H_0^1}^2-\|u_0\|_{H_0^1}^2+2 \beta(t+ \alpha)\\ & = 2\left(\frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right),\; \; \; 0\leq t\leq T^*, \end{split} \end{equation} | (54) |
and (by (24), (12), (13), (48), (25))
\begin{equation} \begin{split} F''(t) = &-2I(u(\cdot,t))+2 \beta\\ = &(p-1)\|\nabla u(\cdot,t)\|_{L^2}^2-2(p+1)J(u(\cdot,t))+2 \beta\\ \geq&2(p+1)(d-J(u_0))+2(p+1)\eta^2(t)+2 \beta,\; \; \; 0\leq t\leq T^*. \end{split} \end{equation} | (55) |
Since
\begin{equation*} F'(t)\geq2 \beta(t+ \alpha). \end{equation*} |
Then
\begin{equation} F(t) = F(0)+\int_0^tF'(s)ds\geq T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2+2 \alpha \beta t+ \beta t^2,\; \; \; 0\leq t\leq T^*. \end{equation} | (56) |
By (6), Schwartz's inequality and Hölder's inequality, we have
\begin{align*} \frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds& = \int_0^t(u(\cdot,s),u_s(\cdot,s))_{H_0^1}ds\\ &\leq\int_0^t\|u(\cdot,s)\|_{H_0^1}\|u_s(\cdot,s)\|_{H_0^1}ds\leq \xi(t)\eta(t),\; \; \; 0\leq t\leq T^*, \end{align*} |
which, together with the definition of
\begin{align*} &\ \ \ \left(F(t)-(T^*-t)\|u_0\|_{H_0^1}^2\right)\left(\eta^2(t)+ \beta\right)\\ & = \left(\xi^2(t)+ \beta(t+ \alpha)^2\right)\left(\eta^2(t)+ \beta\right)\\ & = \xi^2(t)\eta^2(t)+ \beta\xi^2(t)+ \beta(t+ \alpha)^2\eta^2(t)+ \beta^2(t+ \alpha)^2\\ &\geq\xi^2(t)\eta^2(t)+2\xi(t)\eta(t) \beta(t+ \alpha)+ \beta^2(t+ \alpha)^2\\ &\geq\left(\xi(t)\eta(t)+ \beta(t+ \alpha)\right)^2\\ &\geq\left(\frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right)^2,\; \; \; 0\leq t\leq T^*. \end{align*} |
Then it follows from (54) and the above inequality that
\begin{equation} \begin{split} \left(F'(t)\right)^2& = 4\left(\frac12 \int\limits_0^t\frac{d}{ds}\|u(s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right)^2\\ &\leq 4F(t)\left(\eta^2(t)+ \beta\right),\; \; \; \; \; \; 0\leq t\leq T^*. \end{split} \end{equation} | (57) |
In view of (55), (56), and (57), we have
\begin{align*} F(t)F''(t)-\frac{p+1}{2}(F'(t))^2\geq F(t)\left(2(p+1)(d-J(u_0))-2p \beta\right),\; \; \; 0\leq t\leq T^*. \end{align*} |
If we take
\begin{equation} 0 < \beta\leq\frac{p+1}{p}(d-J(u_0)), \end{equation} | (58) |
then
\begin{equation*} T^*\leq\frac{F(0)}{\left(\frac{p+1}{2}-1\right) F'(0)} = \frac{T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2}{(p-1) \alpha \beta}. \end{equation*} |
Then for
\begin{equation} \alpha\in\left(\frac{\|u_0\|_{H_0^1}^2}{(p-1) \beta},\infty\right), \end{equation} | (59) |
we get
\begin{equation*} T^*\leq \frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}. \end{equation*} |
Minimizing the above inequality for
\begin{equation*} T^*\leq\left.\frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}\right|_{ \alpha = \frac{2\|u_0\|_{H_0^1}^2}{(p-1) \beta}} = \frac{4\|u_0\|_{H_0^1}^2}{(p-1)^2 \beta}. \end{equation*} |
Minimizing the above inequality for
\begin{equation*} T^*\leq\frac{4p\|u_0\|_{H_0^1}^2}{(p-1)^2(p+1)(d-J(u_0))}. \end{equation*} |
By the arbitrariness of
\begin{equation*} {T_{\max}}\leq\frac{4p\|u_0\|_{H_0^1}^2}{(p-1)^2(p+1)(d-J(u_0))}. \end{equation*} |
Secondly, we consider the case
Proof of Theorems 2.6 and 2.7. Since Theorem 2.6 follows from Theorem 2.7 directly, we only need to prove Theorem 2.7.
Firstly, we show
\begin{equation*} d = \inf\limits_{\phi\in N}J(\phi) = \frac{p-1}{2(p+1)}\inf\limits_{\phi\in N}\|\nabla\phi\|_{L^2}^2. \end{equation*} |
Then a minimizing sequence
\begin{equation} \lim\limits_{k\uparrow\infty}J(\phi_k) = \frac{p-1}{2(p+1)}\lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = d, \end{equation} | (60) |
which implies
(1)
(2)
Now, in view of
\begin{equation} \lim\limits_{k\uparrow\infty}J(\phi_k) = \frac{p-1}{2(p+1)}\lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = d, \end{equation} | (60) |
We claim
\begin{equation} \|\nabla\varphi\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}\hbox{ i.e. }I(\varphi) = 0. \end{equation} | (62) |
In fact, if the claim is not true, then by (61),
\begin{equation*} \|\nabla\varphi\|_{L^2}^2 < \|\varphi\|_{L_\sigma^{p+1}}^{p+1}. \end{equation*} |
By the proof of Theorem 3.3, we know that
\begin{equation} J(s^*_\varphi\varphi)\geq d, \end{equation} | (63) |
where
\begin{equation} J(s^*_\varphi\varphi)\geq d, \end{equation} | (63) |
On the other hand, since
\begin{equation*} \begin{split} J(s^*_\varphi\varphi)& = \frac{p-1}{2(p+1)}(s^*_\varphi)^2\|\nabla \varphi\|_{L^2}^2\\ & < \frac{p-1}{2(p+1)}\|\nabla \varphi\|_{L^2}^2\\ &\leq\frac{p-1}{2(p+1)}\liminf\limits_{k\uparrow\infty}\|\nabla \phi_k\|_{L^2}^2\\& = d, \end{split} \end{equation*} |
which contradicts to (63). So the claim is true, i.e.
\begin{equation*} \lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}, \end{equation*} |
which, together with
Second, we prove
\begin{equation*} \lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}, \end{equation*} |
Then
\begin{equation*} A: = \{\tau(s)(\varphi+sv):s\in(- \varepsilon, \varepsilon)\} \end{equation*} |
is a curve on
\begin{equation*} A: = \{\tau(s)(\varphi+sv):s\in(- \varepsilon, \varepsilon)\} \end{equation*} |
where
\begin{align*} \xi&: = 2\int_ \Omega\nabla(\varphi+sv)\cdot\nabla vdx\|\varphi+sv\|_{L_\sigma^{p+1}}^{p+1},\\ \eta&: = (p+1)\int_ \Omega|x|^\sigma|\varphi+sv|^{p-1}(\varphi+sv)vdx\|\nabla(\varphi+sv)\|_{L^2}^2. \end{align*} |
Since (62), we get
\begin{equation} \tau'(0) = \frac{1}{(p-1)\|\varphi\|_{L_\sigma^{p+1}}^{p+1}}\left(2\int_ \Omega\nabla\varphi\nabla vdx-(p+1)\int_ \Omega|x|^\sigma|\varphi|^{p-1}\varphi vdx\right). \end{equation} | (65) |
Let
\begin{equation*} \varrho(s): = J(\tau(s)(\varphi+sv)) = \frac{\tau^2(s)}{2}\|\nabla(\varphi+sv)\|_{L^2}^2-\frac{\tau^{p+1}(s)}{p+1}\|\varphi+sv\|_{L^{p+1}_{\sigma}}^{p+1},\; \; \; s\in(- \varepsilon, \varepsilon). \end{equation*} |
Since
\begin{align*} 0 = &\varrho'(0) = \left.\tau(s)\tau'(s)\|\nabla(\varphi+sv)\|_{L^2}^2+\tau^2(s)\int_ \Omega\nabla(\varphi+sv)\cdot\nabla vdx\right|_{s = 0}\\ &\left.-\tau^p(s)\tau'(s)\|\varphi+sv\|_{L_\sigma^{p+1}}^{p+1}-\tau^{p+1}(s)\int_ \Omega|x|^\sigma|\varphi+sv|^{p-1}(\varphi+sv)vdx\right|_{s = 0}\\ = &\int_ \Omega\nabla\varphi\cdot\nabla vdx-\int_ \Omega|x|^\sigma|\varphi|^{p-1}\varphi vdx. \end{align*} |
So,
Finally, in view of Definition 2.3 and
\begin{equation} d = \inf\limits_{\phi\in\Phi\setminus\{0\}}J(\phi). \end{equation} | (66) |
In fact, by the above proof and (27), we have
\begin{equation*} d = \inf\limits_{\phi\in N}J(\phi) \end{equation*} |
and
Proof of Theorem 2.8. Let
\begin{equation*} \omega(u_0) = \cap_{t\geq0} \overline{\{u(\cdot,s):s\geq t\}}^{H_0^1( \Omega)} \end{equation*} |
the
(i) Assume
\begin{equation*} v(x,t) = \left\{ \begin{array}{ll} u(x,t), & \hbox{ if }0\leq t\leq t_0; \\ 0, & \hbox{ if }t > t_0 \end{array} \right. \end{equation*} |
is a global weak solution of problem (1), and the proof is complete.
We claim that
\begin{equation} I(u(\cdot,t)) > 0,\; \; \; 0\leq t < {T_{\max}}. \end{equation} | (67) |
Since
\begin{equation} I(u(\cdot,t)) > 0,\; \; \; 0\leq t < t_0 \end{equation} | (68) |
and
\begin{equation} I(u(\cdot,t_0)) = 0, \end{equation} | (69) |
which together with the definition of
\begin{equation} \|u(\cdot,t_0)\|_{H_0^1}\geq \lambda_{\rho}. \end{equation} | (70) |
On the other hand, it follows from (24), (68) and
\begin{equation*} \|u(\cdot,t)\|_{H_0^1} < \|u_0\|_{H_0^1}\leq \lambda_\rho, \end{equation*} |
which contradicts (70). So (67) is true. Then by (24) again, we get
\begin{equation*} \|u(\cdot,t)\|_{H_0^1}\leq\|u_0\|_{H_0^1},\; \; \; 0\leq t < {T_{\max}}, \end{equation*} |
which implies
By (24) and (67),
\begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = c. \end{equation*} |
Taking
\begin{equation*} \int_0^\infty I(u(\cdot,s))ds\leq\frac12\left(\|u_0\|_{H_0^1}^2-c\right) < \infty. \end{equation*} |
Note that
\begin{equation} \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} | (71) |
Let
\begin{equation} u(\cdot,t_n)\rightarrow\omega \hbox{ in }H_0^1( \Omega)\hbox{ as }n\uparrow\infty. \end{equation} | (72) |
Then by (71), we get
\begin{equation} I(\omega) = \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} | (73) |
As the above, one can easily see
\begin{equation*} \|\omega\|_{H_0^1} < \lambda_\rho\leq \lambda_{J(u_0)}, \; \; \; \underbrace{J(\omega) < J(u_0)}_{\Rightarrow\omega\in J^{J(u_0)}}, \end{equation*} |
which implies
\begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = \lim\limits_{n\uparrow\infty}\|u(\cdot,t_n)\|_{H_0^1} = \|\omega\|_{H_0^1} = 0. \end{equation*} |
(ⅱ) Assume
\begin{equation} I(u(\cdot,t)) < 0,\; \; \; 0\leq t < {T_{\max}}. \end{equation} | (74) |
Since
\begin{equation} I(u(\cdot,t)) < 0,\; \; \; 0\leq t < t_0 \end{equation} | (75) |
and
\begin{equation} I(u(\cdot,t_0)) = 0. \end{equation} | (76) |
Since (75), by (44) and
\begin{equation*} \|\nabla u(\cdot,t_0)\|_{L^2}\geq C_{p\sigma}^{-\frac{p+1}{p-1}}, \end{equation*} |
which, together with the definition of
\begin{equation} \|u(\cdot,t_0)\|_{H_0^1}\leq \Lambda_{\rho}. \end{equation} | (77) |
On the other hand, it follows from (24), (75) and
\begin{equation*} \|u(\cdot,t)\|_{H_0^1} > \|u_0\|_{H_0^1}\geq\Lambda_\rho, \end{equation*} |
which contradicts (77). So (74) is true.
Suppose by contradiction that
\begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = \tilde c, \end{equation*} |
Taking
\begin{equation*} -\int_0^\infty I(u(\cdot,s))ds\leq\frac12\left( \tilde c-\|u_0\|_{H_0^1}^2\right) < \infty. \end{equation*} |
Note
\begin{equation} \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} | (78) |
Let
\begin{equation} u(\cdot,t_n)\rightarrow\omega \hbox{ in }H_0^1( \Omega)\hbox{ as }n\uparrow\infty. \end{equation} | (79) |
Since
\begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = \lim\limits_{n\uparrow\infty}\|u(\cdot,t_n)\|_{H_0^1} = \|\omega\|_{H_0^1}. \end{equation*} |
Then by (78), we get
\begin{equation} I(\omega) = \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} | (80) |
By (24), (25) and (74), one can easily see
\begin{equation*} \|\omega\|_{H_0^1} > \|u_0\|_{H_0^1}\geq\Lambda_\rho\geq\Lambda_{J(u_0)},\; \; \; \underbrace{J(\omega) < J(u_0)}_{\Rightarrow\omega\in J^{J(u_0)}}, \end{equation*} |
which implies
Proof of Theorem 2.9. Let
\begin{equation} \|\nabla u(\cdot,t)\|_{L^2}^2\geq \frac{ \lambda_1}{ \lambda_1+1}\|u(\cdot,t)\|_{H_0^1}^2\geq\frac{ \lambda_1}{ \lambda_1+1}\|u_0\|_{H_0^1}^2,\; \; \; 0\leq t < {T_{\max}}. \end{equation} | (81) |
The remain proofs are similar to the proof of Theorem 2.9. For any
\begin{equation} \begin{split} F''(t) = &-2I(u(\cdot,t))+2 \beta\\ = &(p-1)\|\nabla u(\cdot,t)\|_{L^2}^2-2(p+1)J(u(\cdot,t))+2 \beta\\ \geq&\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)+2(p+1)\eta^2(t)+2 \beta,\; \; \; 0\leq t\leq T^*. \end{split} \end{equation} | (82) |
We also have (56) and (57). Then it follows from (56), (57) and (82) that
\begin{align*} &F(t)F''(t)-\frac{p+1}{2}(F'(t))^2\\ \geq& F(t)\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)-2p \beta\right),\; \; \; 0\leq t\leq T^*. \end{align*} |
If we take
\begin{equation} 0 < \beta\leq\frac{1}{2p}\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right), \end{equation} | (83) |
then
\begin{equation*} T^*\leq\frac{F(0)}{\left(\frac{p+1}{2}-1\right) F'(0)} = \frac{T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2}{(p-1) \alpha \beta}. \end{equation*} |
Then for
\begin{equation} \alpha\in\left(\frac{\|u_0\|_{H_0^1}^2}{(p-1) \beta},\infty\right), \end{equation} | (84) |
we get
\begin{equation*} T^*\leq \frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}. \end{equation*} |
Minimizing the above inequality for
\begin{equation*} T^*\leq\left.\frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}\right|_{ \alpha = \frac{2\|u_0\|_{H_0^1}^2}{(p-1) \beta}} = \frac{4\|u_0\|_{H_0^1}^2}{(p-1)^2 \beta}. \end{equation*} |
Minimizing the above inequality for
\begin{equation*} T^*\leq\frac{8p\|u_0\|_{H_0^1}^2}{(p-1)^2\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right)}. \end{equation*} |
By the arbitrariness of
\begin{equation*} {T_{\max}}\leq\frac{8p\|u_0\|_{H_0^1}^2}{(p-1)^2\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right)}. \end{equation*} |
Proof of Theorem 2.10. For any
\begin{equation} \| \alpha\psi\|_{H_0^1}^2 > \frac{2( \lambda_1+1)(p+1)}{ \lambda_1(p-1)}M. \end{equation} | (85) |
For such
\begin{equation} J(s_3^*\phi)\geq M-J( \alpha \psi), \end{equation} | (86) |
where (see Remark 5)
\begin{equation} J(s_3^*\phi)\geq M-J( \alpha \psi), \end{equation} | (86) |
which can be done since
\begin{equation*} J(s_3^*\phi) = \frac{p-1}{2(p+1)}\left(\frac{\|\nabla \phi\|_{L^2}}{\|\phi\|_{L_\sigma^{p+1}}}\right)^{\frac{2(p+1)}{p-1}} \end{equation*} |
and
By Remark 5 again,
\begin{equation} J(\{s\phi:0\leq s < \infty\}) = (-\infty, J(s_3^*\phi)]. \end{equation} | (87) |
By (87) and (86), we can choose
\begin{equation*} J(u_0) = J(v)+J( \alpha\psi) = M \end{equation*} |
and (note (85))
\begin{align*} J(u_0)& = M < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\| \alpha\psi\|_{H_0^1}^2\\ &\leq\frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\left(\| \alpha\psi\|_{H_0^1}^2+\|v\|_{H_0^1}^2\right)\\ & = \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2. \end{align*} |
Let
[1] |
H. Kalsoom, M. A. Latif, Z. A. Khan, M. Vivas-Cortez, Some new Hermite-Hadamard-Feje´r fractional type inequalities for h-convex and harmonically h-convex interval-valued functions, Mathematics, 10 (2021), 74. https://doi.org/10.3390/math10010074 doi: 10.3390/math10010074
![]() |
[2] |
S. H. Wu, M. Adil Khan, A. Basir, R. Saadati, Some majorization integral inequalities for functions defined on rectangles, J. Inequal. Appl., 2018 (2018), 146. https://doi.org/10.1186/s13660-018-1739-2 doi: 10.1186/s13660-018-1739-2
![]() |
[3] |
M. B. Khan, H. G. Zaini, S. Treant˘a, M. S. Soliman, K. Nonlaopon, Riemann–Liouville fractional integral inequalities for generalized preinvex functions of interval-valued settings based upon pseudo order relation, Mathematics, 10 (2022), 204. https://doi.org/10.3390/math10020204 doi: 10.3390/math10020204
![]() |
[4] |
N. Sharma, S. K. Mishra, A. Hamdi, A weighted version of Hermite-Hadamard type inequalities for strongly GA-convex functions, Int. J. Adv. Appl. Sci., 7 (2020), 113–118. https://doi.org/10.21833/ijaas.2020.03.012 doi: 10.21833/ijaas.2020.03.012
![]() |
[5] |
D. F. Zhao, T. Q. An, G. J. Ye, W. Liu, New Jensen and Hermite–Hadamard type inequalities for h-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), 302. https://doi.org/10.1186/s13660-018-1896-3 doi: 10.1186/s13660-018-1896-3
![]() |
[6] |
Y. C. Kwun, M. S. Saleem, M. Ghafoor, N. Waqas, S. M. Kang, Hermite–Hadamard-type inequalities for functions whose derivatives are η-convex via fractional integrals, J. Inequal. Appl., 2019 (2019), 44. https://doi.org/10.1186/s13660-019-1993-y doi: 10.1186/s13660-019-1993-y
![]() |
[7] |
M. A. Hanson, On sufficiency of the kuhn-tucker conditions, J. Math. Anal. Appl., 80 (1981), 545–550. https://doi.org/10.1016/0022-247X(81)90123-2 doi: 10.1016/0022-247X(81)90123-2
![]() |
[8] |
B. D. Craven, B. M. Glover, Invex functions and duality, J. Aust. Math. Soc., 39 (1985), 1–20. https://doi.org/10.1017/S1446788700022126 doi: 10.1017/S1446788700022126
![]() |
[9] |
A. Ben-Israel, B. Mond, What is invexity? J. Aust. Math. Soc., 28 (1986), 1–9. https://doi.org/10.1017/S0334270000005142 doi: 10.1017/S0334270000005142
![]() |
[10] |
S. R.Mohan, S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901–908. https://doi.org/10.1006/jmaa.1995.1057 doi: 10.1006/jmaa.1995.1057
![]() |
[11] |
T. Weir, V. Jeyakumar, A class of nonconvex functions and mathematical programming, Bull. Aust. Math. Soc., 38 (1988), 177–189. https://doi.org/10.1017/S0004972700027441 doi: 10.1017/S0004972700027441
![]() |
[12] |
T. Weir, B. Mond, Preinvex functions in multiple objective optimization, J. Math. Anal. Appl., 136 (1988), 29–38. https://doi.org/10.1016/0022-247X(88)90113-8 doi: 10.1016/0022-247X(88)90113-8
![]() |
[13] | M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory, 2 (2007), 126–131. |
[14] |
S. S. Dragomir, On the Hadamard's inequlality for convex functions on the coordinates in a rectangle from the plane, Taiwanese J. Math., 5 (2001), 775–788. https://doi.org/10.11650/twjm/1500574995 doi: 10.11650/twjm/1500574995
![]() |
[15] | M. A. Latif, S. S. Dragomir, Some Hermite-Hadamard type inequalities for functions whose partial derivatives in absloute value are preinvex on the coordinates, Facta Univ. Ser. Math. Inform., 28 (2013), 257–270. |
[16] |
M. Matłoka, On some Hadamard type inequalities for (h1, h2)-preinvex functions on the coordinates, J. Inequal. Appl., 2013 (2013), 227. https://doi.org/10.1186/1029-242X-2013-227 doi: 10.1186/1029-242X-2013-227
![]() |
[17] |
M. Matłoka, On Some new inequalities for differentiable (h1, h2)-preinvex functions on the coordinates, Math. Stat., 2 (2014), 6–14. https://doi.org/10.13189/ms.2014.020102 doi: 10.13189/ms.2014.020102
![]() |
[18] |
S. Mehmood, F. Zafar, N. Yasmin, Hermite-Hadamard-Fej´er type inequalities for preinvex functions using fractional integrals, Mathematics, 7 (2019), 467. https://doi.org/10.3390/math7050467 doi: 10.3390/math7050467
![]() |
[19] |
M. A. Noor, K. I. Noor, S. Rashid, Some new classes of preinvex functions and inequalities, Mathematics, 7 (2019), 29. https://doi.org/10.3390/math7010029 doi: 10.3390/math7010029
![]() |
[20] |
S. Rashid, M. A. Latif, Z. Hammouch, Y. M. Chu, Fractional integral inequalities for strongly h-preinvex functions for a kth order differentiable functions, Symmetry, 11 (2019), 1448. https://doi.org/10.3390/sym11121448 doi: 10.3390/sym11121448
![]() |
[21] |
N. Sharma, S. K. Mishra, A. Hamdi, Hermite-Hadamard type inequality for y-Riemann-Liouville fractional integrals via preinvex functions, Int. J. Nonlinear Anal. Appl., 13 (2022), 3333–3345. https://doi.org/10.22075/IJNAA.2021.21475.2262 doi: 10.22075/IJNAA.2021.21475.2262
![]() |
[22] | L. Jaulin, M. Kieffer, O. Didrit, É. Walter, Interval analysis, In: Applied interval analysis, London: Springer, 2001. https://doi.org/10.1007/978-1-4471-0249-6_2 |
[23] | R. E. Moore, Methods and applications of interval analysis, PA: Philadelphia, 1979. https://doi.org/10.1137/1.9781611970906 |
[24] |
A. K. Bhurjee, G. Panda, Multi-objective interval fractional programming problems: An approach for obtaining efficient solutions, Opsearch, 52 (2015), 156–167. https://doi.org/10.1007/s12597-014-0175-4 doi: 10.1007/s12597-014-0175-4
![]() |
[25] |
J. K. Zhang, S. Y. Liu, L. F. Li, Q. X. Feng, The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function, Optim. Lett., 8 (2014), 607–631. https://doi.org/10.1007/s11590-012-0601-6 doi: 10.1007/s11590-012-0601-6
![]() |
[26] |
D. F. Zhao, T. Q. An, G. J. Ye, W. Liu, Chebyshev type inequalities for interval-valued functions, Fuzzy Sets Syst., 396 (2020), 82–101. https://doi.org/10.1016/j.fss.2019.10.006 doi: 10.1016/j.fss.2019.10.006
![]() |
[27] |
Y. T. Guo, G. J. Ye, D. F. Zhao, W. Liu, gH-symmetrically derivative of interval-valued functions and applications in interval-valued optimization, Symmetry, 11 (2019), 1203. https://doi.org/10.3390/sym11101203 doi: 10.3390/sym11101203
![]() |
[28] | R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to interval analysis, SIAM: Philadelphia, 2009. https://doi.org/10.1137/1.9780898717716 |
[29] |
E. J. Rothwell, M. J. Cloud, Automatic error analysis using intervals, IEEE T. Educ., 55 (2011), 9–15. https://doi.org/10.1109/TE.2011.2109722 doi: 10.1109/TE.2011.2109722
![]() |
[30] |
J. M. Snyder, Interval analysis for computer graphics, Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, 26 (1992), 121–130. https://doi.org/10.1145/142920.134024 doi: 10.1145/142920.134024
![]() |
[31] |
Y. Chalco-Cano, W. A. Lodwick, W. Condori-Equice, Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput., 19 (2015), 3293–3300. https://doi.org/10.1007/s00500-014-1483-6 doi: 10.1007/s00500-014-1483-6
![]() |
[32] |
N. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets Syst., 48 (1992), 129–132. https://doi.org/10.1016/0165-0114(92)90256-4 doi: 10.1016/0165-0114(92)90256-4
![]() |
[33] |
M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, New Hermite–Hadamard–type inequalities for (h1, h2)–convex fuzzy-interval-valued functions, Adv. Differ. Equ., 2021 (2021), 149. https://doi.org/10.1186/s13662-021-03245-8 doi: 10.1186/s13662-021-03245-8
![]() |
[34] |
M. B. Khan, P. O. Mohammed, M. A. Noor, A. M. Alsharif, K. I. Noor, New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation, AIMS Mathematics, 6 (2021), 10964–10988. https://doi.org/10.3934/math.2021637 doi: 10.3934/math.2021637
![]() |
[35] |
M. B. Khan, M. A. Noor, M. M. Al-Shomrani, L. Abdullah, Some novel inequalities for LR-𝒽-convex interval-valued functions by means of pseudo-order relation, Math. Meth. Appl. Sci., 45 (2022), 1310–1340. https://doi.org/10.1002/mma.7855 doi: 10.1002/mma.7855
![]() |
[36] |
N. Talpur, S. J. Abdulkadir, H. Alhussian, M. H. Hasan, N. Aziz, A. Bamhdi, A comprehensive review of deep neuro-fuzzy system architectures and their optimization methods, Neural Comput. Applic., 34 (2022), 1837–1875. https://doi.org/10.1007/s00521-021-06807-9 doi: 10.1007/s00521-021-06807-9
![]() |
[37] | U. Kulish, W. Miranker, Computer arithmetic in theory and practice, New York: Academic Press, 2014. https://doi.org/10.1016/C2013-0-11018-5 |
[38] |
A. Alsaedi, B. Ahmad, A. Assolami, Sotiris K. Ntouyas, On a nonlinear coupled system of differential equations involving Hilfer fractional derivative and Riemann-Liouville mixed operators with nonlocal integro-multi-point boundary conditions, AIMS Mathematics, 7 (2022), 12718–12741. https://doi.org/10.3934/math.2022704 doi: 10.3934/math.2022704
![]() |
[39] | B. Bede, Mathematics of fuzzy sets and fuzzy logic, Berlin, Heidelberg: Springer, 2013. https://doi.org/10.1007/978-3-642-35221-8 |
[40] | P. Diamond, P. E. Kloeden, Metric spaces of fuzzy sets: Theory and applications, Singapore: World Scientific, 1994. https: //doi.org/10.1142/2326 |
[41] |
O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301–317. https://doi.org/10.1016/0165-0114(87)90029-7 doi: 10.1016/0165-0114(87)90029-7
![]() |
[42] |
T. M. Costa, H. Roman-Flores, Some integral inequalities for fuzzy-interval-valued functions, Inf. Sci., 420 (2017), 110–125. https://doi.org/10.1016/j.ins.2017.08.055 doi: 10.1016/j.ins.2017.08.055
![]() |
[43] | W. W. Breckner, Continuity of generalized convex and generalized concave set–valued functions, Rev. Anal. Numér. Théor. Approx., 22 (1993), 39–51. |
[44] |
E. Sadowska, Hadamard inequality and a refinement of Jensen inequality for set-valued functions, Result Math., 32 (1997), 332–337. https://doi.org/10.1007/BF03322144 doi: 10.1007/BF03322144
![]() |
[45] |
M. B. Khan, S. Treanțǎ, M. S. Soliman, K. Nonlaopon, H. G. Zaini, Some Hadamard–Fejér type inequalities for LR-convex interval-valued functions, Fractal Fract., 6 (2022), 6. https://doi.org/10.3390/fractalfract6010006 doi: 10.3390/fractalfract6010006
![]() |
[46] | J. P. Aubin, A. Cellina, Differential inclusions: Set-valued maps and viability theory, Berlin Heidelberg: Springer-Verlag, 1984. https://doi.org/10.1007/978-3-642-69512-4 |
[47] | J. P. Aubin, H. Frankowska, Set-valued analysis, Boston: Birkhäuser, 1990. https://doi.org/10.1007/978-0-8176-4848-0 |
[48] |
T. M. Costa, Jensen's inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets Syst., 327 (2017), 31–47. https://doi.org/10.1016/j.fss.2017.02.001 doi: 10.1016/j.fss.2017.02.001
![]() |
[49] |
D. L. Zhang, C. M. Guo, D. G. Chen, G. J. Wang, Jensen's inequalities for set-valued and fuzzy set-valued functions, Fuzzy Sets Syst., 404 (2021), 187–204. https://doi.org/10.1016/j.fss.2020.06.003 doi: 10.1016/j.fss.2020.06.003
![]() |
[50] | S. Pal, T. K. L. Wong, Exponentially concave functions and new information geometry, 2016. Available from: https://doi.org/10.48550/arXiv.1605.05819. |
[51] |
G. Santos-García, M. B. Khan, H. Alrweili, A. A. Alahmadi, S. S. Ghoneim, Hermite–Hadamard and Pachpatte type inequalities for coordinated preinvex fuzzy-interval-valued functions pertaining to a fuzzy-interval double integral operator, Mathematics, 10 (2022), 2756. https://doi.org/10.3390/math10152756 doi: 10.3390/math10152756
![]() |
[52] |
J. E. Macías-Díaz, M. B. Khan, H. Alrweili, M. S. Soliman, Some fuzzy inequalities for harmonically s-convex fuzzy number valued functions in the second sense integral, Symmetry, 14 (2022), 1639. https://doi.org/10.3390/sym14081639 doi: 10.3390/sym14081639
![]() |
[53] |
M. B. Khan, M. A. Noor, J. E. Macías-Díaz, M. S. Soliman, H. G. Zaini, Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation, Demonstr. Math., 55 (2022), 387–403. https://doi.org/10.1515/dema-2022-0023 doi: 10.1515/dema-2022-0023
![]() |
[54] |
M. B. Khan, M. A. Noor, H. G. Zaini, G. Santos-García, M. S. Soliman, The new versions of Hermite–Hadamard inequalities for pre-invex fuzzy-interval-valued mappings via fuzzy Riemann integrals, Int. J. Comput. Intell. Syst., 15 (2022), 66. https://doi.org/10.1007/s44196-022-00127-z doi: 10.1007/s44196-022-00127-z
![]() |
[55] |
M. B. Khan, G. Santos-García, M. A. Noor, M. S. Soliman, New Hermite–Hadamard inequalities for convex fuzzy-number-valued mappings via fuzzy Riemann integrals, Mathematics, 10 (2022), 3251. https://doi.org/10.3390/math10183251 doi: 10.3390/math10183251
![]() |
[56] |
M. B. Khan, S. Treanțǎ, M. S. Soliman, Generalized preinvex interval-valued functions and related Hermite–Hadamard type inequalities, Symmetry, 14 (2022), 1901. https://doi.org/10.3390/sym14091901 doi: 10.3390/sym14091901
![]() |
[57] |
M. B. Khan, G. Santos-García, M. A. Noor, M. S. Soliman, Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities, Chaos, Soliton. Fract., 164 (2022), 112692. https://doi.org/10.1016/j.chaos.2022.112692 doi: 10.1016/j.chaos.2022.112692
![]() |
[58] |
M. B. Khan, G. Santos-García, M. A. Noor, M. S. Soliman, New class of preinvex fuzzy mappings and related inequalities, Mathematics, 10 (2022), 3753. https://doi.org/10.3390/math10203753 doi: 10.3390/math10203753
![]() |
[59] |
M. B. Khan, J. E. Macías-Díaz, S. Treanțǎ, M. S. Soliman, Some Fejér-type inequalities for generalized interval-valued convex functions, Mathematics, 10 (2022), 3851. https://doi.org/10.3390/math10203851 doi: 10.3390/math10203851
![]() |
[60] |
M. B. Khan, G. Santos-García, S. Treanțǎ, M. S. Soliman, New class up and down pre-invex fuzzy number valued mappings and related inequalities via fuzzy Riemann integrals, Symmetry, 14 (2022), 2322. https://doi.org/10.3390/sym14112322 doi: 10.3390/sym14112322
![]() |
[61] |
M. B. Khan, J. E. Macías-Díaz, M. S. Soliman, M. A. Noor, Some new integral inequalities for generalized preinvex functions in interval-valued settings, Axioms, 11 (2022), 622. https://doi.org/10.3390/axioms11110622 doi: 10.3390/axioms11110622
![]() |
[62] |
M. B. Khan, H. G. Zaini, G. Santos-García, M. A. Noor, M. S. Soliman, New class up and down λ-convex fuzzy-number valued mappings and related fuzzy fractional inequalities, Fractal Fract., 6 (2022), 679. https://doi.org/10.3390/fractalfract6110679 doi: 10.3390/fractalfract6110679
![]() |
[63] |
S. Varošanec, On h-convexity, J. Math. Anal. Appl., 326 (2007), 303–311. https://doi.org/10.1016/j.jmaa.2006.02.086 doi: 10.1016/j.jmaa.2006.02.086
![]() |
[64] |
M. Z. Sarikaya, A. Saglam, H. Yildirim, On some Hadamard-type inequalities for h-convex functions, J. Math. Inequal., 2 (2008), 335–341. https://doi.org/10.7153/jmi-02-30 doi: 10.7153/jmi-02-30
![]() |
[65] |
R. Kumar, A. Baz, H. Alhakami, W. Alhakami, M. Baz, A. Agrawal, et al., A hybrid model of hesitant fuzzy decision-making analysis for estimating usable-security of software, IEEE Access, 8 (2020), 72694–72712. https://doi.org/10.1109/ACCESS.2020.2987941 doi: 10.1109/ACCESS.2020.2987941
![]() |
[66] |
M. Ibrahim, S. Nabi, A. Baz, H. Alhakami, M. S. Raza, A. Hussain, et al., An in-depth empirical investigation of state-of-the-art scheduling approaches for cloud computing, IEEE Access, 8 (2020), 128282–128294. https://doi.org/10.1109/ACCESS.2020.3007201 doi: 10.1109/ACCESS.2020.3007201
![]() |
[67] |
N. Talpur, S. J. Abdulkadir, H. Alhussian, M. H. Hasan, N. Aziz, A. Bamhdi, Deep Neuro-Fuzzy System application trends, challenges, and future perspectives: A systematic survey, Artif. Intell. Rev., 2022 (2022), 1–49. https://doi.org/10.1007/s10462-022-10188-3 doi: 10.1007/s10462-022-10188-3
![]() |
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