\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 160, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/160\hfil Ume's u-distance] {Ume's u-distance and its relation with both (PS)-condition and coercivity} \author[G. Goga\hfil EJDE-2011/160\hfilneg] {Georgiana Goga} \address{Georgiana Goga \newline Nicolae Rotaru College, 2, Ion Corvin St., Constanta, Romania} \email{georgia\_goga@yahoo.com} \thanks{Submitted August 15, 2011. Published November 30, 2011.} \subjclass[2000]{49K27, 54E35, 54E50, 58E30} \keywords{$u$-distance; Palais-Smale condition; coercivity; \hfill\break\indent Zhong's variational principle} \begin{abstract} In this article, we study the connection between the $u$-distance and a new Palais-Smale condition of compactness. We compare this Palais-Smale condition with the coercivity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction and preliminaries} In 1997, Zhong \cite{z1,z2} generalized the Ekeland variational principle and proved the existence of minimal points for G\^ateaux-differentiable functions under weak (PS) conditions. The following theorem is well-known and we name it Zhong's variational principle (ZVP). \begin{theorem}[\cite{z1,z2}] \label{thm1.1} Let $(X,d)$ be a complete metric space, $x_0\in X$ fixed and $f:X\to (-\infty ,\infty ]$ a proper lower semicontinuous function which is bounded from below. Let $h:[0,\infty )\to [0,\infty )$ be a nondecreasing continuous function such that \begin{equation*} \int_0^{\infty }\frac{1}{1+h(r)}\,dr=+\infty . \end{equation*} Then, for every $\varepsilon >0$, every $y\in X$ such that \begin{equation*} f(y)<\int_{x\in X} f(x) +\varepsilon , \end{equation*} and $\lambda >0$, there exists some point $z\in X$ such that \begin{itemize} \item[(i)] $f(z)\leq f(y)$, \item[(ii)] $d(x_0,z)\leq r_0+r^{\ast}$, \item[(iii)] $f(x)\geq f(z)-\frac{\varepsilon }{\lambda (1+h(d(x_0,z)) )}\cdot d(z,x)$, for all $x\in X$, where $r_0=d(x_0,y)$, and $r^{\ast }$ is such that \begin{equation*} \int_{r_0}^{r_0 +r^{\ast }}\frac{1}{1+h(t)}\,dt\geq \lambda . \end{equation*} \end{itemize} \end{theorem} In 2010, Ume \cite{u1} introduced a new concept of distance called $u$-distance, which generalizes some distances anterior studied (see e.g., $\omega$-distance \cite{k1,w1}, Tataru's distance \cite{t1}, $\tau$-distance \cite{s2}) and expanded the celebrated Ekeland's variational principle. In Section 2, we present a generalization of Zhong's variational principle using Ume's $u$-distance. In Section 3, we define a new Palais-Smale condition related to above variational principle and we study the existence of the minimal point for G\^{a}teaux-differentiable functions. In the last section, we deal with the relation between new Palais-Smale condition and the coercivity, following a techniques which is based on $u$-distance. Our results extend and improve other known results due to Zhong \cite{z1,z2}, Ekeland \cite{e1,e2} and Costa \& Silva \cite{c3}. For the beginning, we present some results needed in our approach. First, we recall Ume's \cite{u1} concept of generalized distance in metric spaces. \begin{definition} \label{def1.2} \rm Let $(X,d)$ be a metric space. A function $p:X\times X\to\mathbb{R}_{+}$ is called $u$-distance on $X$ if there exists a map $\Theta :X\times X\times\mathbb{R}_{+}\times\mathbb{R}_{+} \to\mathbb{R}_{+}$ such that the following conditions hold: \begin{itemize} \item[(U1)] $p(x,z)\leq p(x,y) +p(y,z)$, for all $x,y,z\in X$; \item[(U2)] $\Theta (x,y,0,0)=0$ and $\Theta (x,y,s,t)\geq \min \{ s,t\}$ for all $x,y\in X$, $s,t\in\mathbb{R}_{+}$, and for every $x\in X$ and $\varepsilon >0$, there is $\delta >0$ such that \begin{equation*} | \Theta (x,y,s,t)-\Theta ( x,y,s_0,t_0)| <\varepsilon \end{equation*} if $| s-s_0| <\delta$, $|t-t_0| <\delta$, $s,s_0,t,t_0\in\mathbb{R}_{+}$ whenever $y\in X$; \item[(U3)] $\lim_nx_n=x$ and $\limsup_n \{ \Theta (w_n,z_n,p(w_n,x_{m}) ,p(z_n,x_{m})):m\geq n\} =0$ imply \begin{equation*} p(y,x)\leq \liminf_{n\to \infty }p(y,x_n)\quad\text{ for } y\in X; \end{equation*} \item[(U4)] The four equalities \begin{gather*} \limsup_n \{ p(x_n,w_{m}):m\geq n\} =0, \quad \limsup_n \{ p(y_n,z_{m}):m\geq n\} =0, \\ \lim_n\Theta (x_n,w_n,s_n,t_n)=0, \quad \lim_n\Theta (y_n,z_n,s_n,t_n)=0 \end{gather*} imply $\lim_n\Theta (w_n,z_n,s_n,t_n)=0$; or the four equalitires \begin{gather*} \limsup_n \{ p(w_{m},x_n):m\geq n\} =0, \quad \limsup_n \{ p(z_{m},y_n):m\geq n\} =0, \\ \lim_n\Theta (x_n,w_n,s_n,t_n)=0,\quad \lim_n\Theta (y_n,z_n,s_n,t_n)=0 \end{gather*} imply $\lim_n\Theta (w_n,z_n,s_n,t_n)=0$; \item[(U5)] The two equalities \begin{gather*} \lim_n\Theta (w_n,z_n,p(w_n,x_n),p( z_n,x_n))=0, \\ \lim_n\Theta (w_n,z_n,p(w_n,y_n),p( z_n,y_n))=0 \end{gather*} imply $\lim_nd(x_n,y_n)=0$; or the two equalities \begin{gather*} \lim_n\Theta (a_n,b_n,p(x_n,a_n),p( x_n,b_n))=0, \\ \lim_n\Theta (a_n,b_n,p(z_n,a_n),p( y_n,b_n))=0 \end{gather*} imply $\lim_nd(x_n,y_n)=0$. \end{itemize} \end{definition} \begin{example}[\cite{u1}] \label{examp1.3} \rm Let $X$ be a space with norm $\|\cdot\|$. Then the function $p:X\times X\to\mathbb{R}_{+}$ defined by $p(x,y)=\|x\|$ is a $u$-distance on $X$, but it is not a $\tau$-distance on $X$, in Suzuki's sense \cite{s2}. \end{example} \begin{example}[\cite{u1}] \label{examp1.4}\rm Let $p$ be a $u$-distance on a metric space $(X,d)$ and let $c$ be a real positive number. Then a function $q:X\times X\to \mathbb{R}_{+}$ defined by $q(x,y)=cp(x,y)$ for every $x,y\in X$ is also a $u$-distance on $X$. \end{example} By means of the generalized $u$-distance, Ume obtained in \cite{u1} the following version of Ekeland's variational principle. This result will play a crucial role in the proof of our variational principle. \begin{theorem}[\cite{u1}] \label{thm1.5} Let $(X,d)$ be a complete metric space, let $f:X\to (-\infty ,\infty ]$ be a proper lower semicontinuous function which is bounded from below, and let $p:X\times X\to\mathbb{R}_{+}$ be a $u$-distance on $X$. Then the following two statements hold: \begin{itemize} \item[(1)] For each $x\in X$ with $f( x)<\infty$, there exists $v\in X$ such that $f(v)\leq f(x)$ and $f(w) >f(v)-p(v,w)$, for all $w\in X\backslash \ v\}$. \item[(2)] For each $\varepsilon >0$, $\lambda>0$ and $x\in X$ with $p(x,x)=0$ and $f(x)<\inf_{a\in X} f(a)+\varepsilon$, there exists $v\in X$ such that \begin{gather*} f(v)\leq f(x), \quad p(x,v)\leq \lambda, \\ f(w)>f(v)-\frac{\varepsilon }{\lambda } \cdot p(v,w),\quad \text{for all } w\in X\backslash \{v\} . \end{gather*} \end{itemize} \end{theorem} \section{A generalization of Zhong's variational principle} We start this section by extending a result by Suzuki \cite{s3}, using the $u$-distance. \begin{proposition} \label{prop2.1} Let $(X,d)$ be a complete metric space and let $p:X\times X\to\mathbb{R}_{+}$ be a $u$-distance on $X$. Let $q:X\times X\to\mathbb{R}_{+}$ be a function such that \begin{itemize} \item[(a)] $q(x,z)\leq q(x,y) +q(y,z)$ for all $x,y,z\in X$; \item[(b)] $q$ is lower semicontinuous in its second argument; \item[(c)] $q(x,y)\geq p(x,y)$ for all $x,y\in X$. \end{itemize} Then $q$ is also a $u$-distance. \end{proposition} \begin{proof} Assumption (a) is equivalently with (U1)$_q$. Let $\Theta :X\times X\times\mathbb{R}_{+}\times\mathbb{R} _{+}\to\mathbb{R}_{+}$ be a function satisfying (U2)--(U5). Clearly, (U3)$_q$ follows from (b). Now, we assume that $$\begin{gathered} \limsup_n \{ q(x_n,w_{m}):m\geq n\} =0, \\ \limsup_n \{ q(y_n,z_{m}):m\geq n\} =0, \\ \lim_n\Theta (x_n,w_n,s_n,t_n)=0, \\ \lim_n\Theta (y_n,z_n,s_n,t_n)=0. \end{gathered} \label{e2.1}$$ By \eqref{e2.1} and (c), we have \begin{gather*} \limsup_n\{ p(x_n,w_{m}):m\geq n\}=0,\\ \limsup_n\{ p(y_n,z_{m}):m\geq n\}=0. \end{gather*} Therefore, by (U4), we find $\lim_n\Theta(w_n,z_n,s_n,t_n)=0$, and derive (U4)$_q$. Next, we assume that \begin{gather} \lim_n\Theta (w_n,z_n,q(w_n,x_n),q( z_n,x_n))=0, \label{e2.2} \\ \lim_n\Theta (w_n,z_n,q(w_n,y_n),q( z_n,y_n))=0. \label{e2.3} \end{gather} Applying again (c) in \eqref{e2.2} and \eqref{e2.3}, we obtain \begin{gather*} \lim_n\Theta (w_n,z_n,p(w_n,x_n),p(z_n,x_n))=0,\\ \lim_n\Theta (w_n,z_n,p(w_n,y_n),p(z_n,y_n))=0. \end{gather*} By (U5), we have $\lim_n d(x_n,y_n)=0$, and (U5)$_q$, is also verified. \end{proof} Next, we establish a more general variational principle \cite{b1,t2}, which is an extension of both Ekeland's and Zhong's variational principles. \begin{theorem} \label{thm2.2} Let $(X,d)$ be a complete metric space, $a\in X$ be a fixed point and let $p:X\times X\to\mathbb{R}_{+}$ be a $u$-distance on $X$ lower semicontinuous in its second argument. Let $f:X\to (-\infty ,\infty ]$ be a proper lower semicontinuous function which is bounded from below and let $b:[0,\infty )\to (0,\infty )$ be a non-increasing continuous function such that \begin{equation*} B(t)=\int_0^{t}b(r)dr, \end{equation*} where $B$ is a $C^{1}$ function from $\mathbb{R}_{+}$ to itself and $B(\infty )=+\infty$. Let $y\in X$ be such that $p(y,y)=0$ and $$f(y)>\inf_{x\in X} f(x). \label{e2.4}$$ Then, for $\epsilon _0>0$, there exists $z\in X$ such that \begin{itemize} \item[(i)] $f(z)\leq f(y)$, \item[(ii)] $p(a,z)\leq \beta (y)+\beta ^{\ast }$, \item[(iii)] $f(x)>f(z)-\frac{\epsilon _0}{\lambda }b(\beta (z))p( z,x)$, for all $x\in X$ where $\beta (.)=p(a,.)$, and $\beta ^{\ast }$ is such that $$\int_{\beta (y)}^{\beta (y) + \beta^{\ast }}b(t)\,dt \geq \alpha (y), \label{e2.5}$$ with $\alpha (y)=f(y)-\inf{x\in X} f(x)\geq \lambda >0$. \end{itemize} \end{theorem} \begin{proof} First, we define the function $q:X\times X\to \mathbb{R}_{+}$ by \begin{equation*} q(x,y):=\int_{p(a,x)}^{p(a,x) + p(x,y)}b(t)\,dt. \end{equation*} Since $b$ is non-increasing, for $(x,z)\in X\times X$, we deduce \begin{align*} q(x,z)&= \int_{p(a,x)}^{p(a,x) + p(x,z)}b(t)\,dt\\ &\leq \int_{p(a,x)}^{p(a,x) + p(x,y) + p(y,z)}b(t)\,dt \\ &= \int_{p(a,x)}^{p(a,x) + p(x,y)}b(t)\,dt+\int_{p(a,x) + p(x,y)}^{p(a,x) + p(x,y) + p(y,z)}b(t)\,dt \\ &\leq \int_{p(a,x)}^{p(a,x) + p(x,y)}b(t)\,dt+\int_{p(a,y)}^{p(a,y) + p(y,z)}b(t)\,dt \\ &= q(x,y)+q(y,z). \end{align*} In addition, $q$ is obviously lower semicontinuous in its second variable. On the other hand, we have \label{e2.6} \begin{aligned} q(x,y)&= \int_{p(a,x)}^{p(a,x) + p(x,y)}b(t)\,dt \\ &= B(p(a,x) + p(x,y)) -B(p(a,x)) \\ &\geq b(p(a,x) + p(x,y))p(x,y). \end{aligned} Taking into account the definition of function $b$, we obtain boundedness from below, $$b(p(a,x) + p(x,y)) >b(\infty )\geq M\geq 0 \label{e2.7}$$ Combining \eqref{e2.6} and \eqref{e2.7}, we deduce \begin{equation*} q(x,y)\geq Mp(x,y). \end{equation*} Since $Mp(x,y)$ is a $u$-distance and the assumptions of Proposition \ref{prop2.1} are verified, $q(x,y)$ is also $u$-distance. Now, from \eqref{e2.4} and \eqref{e2.5}, we obtain $$\label{e2.8} \begin{split} 0 &< \lambda \leq f(y)-\underset{x\in X}{\inf }f(x) =\alpha (y)\\ &\leq \int_{\beta (y)}^{\beta (y) + \beta ^{\ast }}b(t)\,dt = \int_0^{ \beta ^{\ast }}b(u+\beta (y)) \,du\\ &\leq \int_0^{ \beta ^{\ast }}b(u)du=B(\beta^{\ast }). \end{split}$$ So by the above inequality, \begin{equation*} f(y)\leq \inf_{x\in X} f(x) +B(\beta ^{\ast }), \end{equation*} and the Theorem \ref{thm1.5} is applicable to $q(x,y)$ for $\varepsilon =B(\beta ^{\ast })>0$ and $\lambda =\alpha ( y)>0$. Therefore, there exists $z\in X$ such that \begin{gather} f(z)\leq f(y), \label{e2.9}\\ q(y,z)\leq \alpha (y)\label{e2.10} \\ f(x)>f(z)-\frac{B(\beta ^{\ast })}{ \alpha (y)}\cdot q(z,x),\quad \forall x\neq z,\; x\in X. \label{e2.11} \end{gather} By (U1), we know that $$\ p(a,z)\leq p(a,y)+p(y,z) = \beta (y)+p(y,z). \label{e2.12}$$ On the other hand, from \eqref{e2.5} and \eqref{e2.10} it follows that $B(\beta (y) +p(y,z))-B(\beta (y)) \leq \alpha (y) \leq B(\beta (y) +\beta ^{\ast } )-B(\beta (y)).$ Thereby, we find that $$p(y,z)\leq \beta ^{\ast }, \label{e2.13}$$ because $B$ is a nondecreasing function. Thus, (ii) follows from \eqref{e2.12} and \eqref{e2.13}. Moreover, since $$q(z,x)= \int_{p(a,z)}^{p(a,z)+ p(z,x)}b(t)\,dt \leq b(p(a,z))p(z,x) = b((\beta (z)))p(z,x); \label{e2.14}$$ multiplying by $(-1)$ and, using \eqref{e2.8} and \eqref{e2.11}, for 0&f(z)-\frac{B(\beta ^{\ast })}{\alpha (y)}\cdot q(z,x) &\geq f(z)- \frac{\epsilon _0}{\lambda }q(z,x) &\geq f(z)-\frac{\epsilon }{\lambda }b((\beta(z)))p(z,x), \end{align*} for allx\in X$, and (iii) is verified. This completes the proof. \end{proof} \begin{remark} \label{rmk3.3} \rm Let$a$,$f$,$b$,$p$,$\alpha (y)$,$\beta (y)$,$\beta ^{\ast }$, and$X$be as in Theorem \ref{thm2.2}. \begin{itemize} \item[(i)] When$a=y$,$b(t)\equiv 1$,$\beta^{\ast }=\lambda $,$\epsilon _0>\alpha (y)\geq \lambda >0$, and$p(x,y)=d(x,y)$, Theorem \ref{thm2.2} reduces to Ekeland's variational principle (EVP) \cite{e1,e2}. \item[(ii)] Take$a=x_0$, \begin{equation*} b(t)=\frac{1}{1+h(t)}, \end{equation*} where$h:[0,\infty )\to [0,\infty )$is a continuous nondecreasing function such that \begin{equation*} \int_0^{\infty }\frac{1}{1+h(r)}dr=+\infty , \end{equation*}$\epsilon _0>\alpha (y)\geq \lambda >0$,$\beta (y)=d(x_0,y)=r_0$,$\beta ^{\ast }=r^{\ast }$and$p(x,y)=d(x,y)$. Therefore, Theorem \ref{thm2.2} implies Theorem \ref{thm1.1}. \end{itemize} \end{remark} \section{The b-(PS) condition and the existence of a minimal point} Throughout this section$X$denotes a Banach space. We recall that a function$f:X\to (-\infty ,\infty ]$is called G\^{a}teaux differentiable at$x\in X$with$f(x)<\infty $if there exists a continuous linear functional$f'(x)$such that \begin{equation*} \lim_{t\to 0} \frac{f(x+ty)-f(x) }{t}=\langle f'(x),y\rangle \end{equation*} holds for every$y\in X$. In the following, we assume that$f:X\to (-\infty ,\infty ]$is G\^{a}teaux differentiable. \begin{theorem} \label{thm3.1} Let$a\in X$be fixed and$p:X\times X\to\mathbb{R}_{+}$a$u$-distance on$X$lower semicontinuous in its second argument. Let$f:X\to (-\infty ,\infty ]$be a proper lower semicontinuous function which is bounded from below and let$b:[0,\infty )\to(0,\infty)$be a nonincreasing continuous function such that \begin{equation*} B(t)=\int_0^{t}b(r)dr, \end{equation*} where$B$is a$C^{1}$function from$\mathbb{R}_{+}$to itself such that$B(\infty )=+\infty$. Let$y\in X$be such that$p(y,y)=0$and \begin{equation*} f(y)>\inf_{x\in X} f(x). \end{equation*} Then, for every$\epsilon >0$, there exists$z\in X$such that \begin{itemize} \item[(i')]$f(z)\leq f(y)$, \item[(ii')]$\beta (z)\leq \beta (y)+\beta ^{\ast }$, \item[(iii')]$\|f'(z)\| /b(\beta (z))\leq \epsilon $for all$x\in X$, where$\beta (.)=p(a,.)$, and$\beta ^{\ast }$is a real number such that \begin{equation*} \int_{\beta (y)}^{\beta (y) + \beta ^{\ast }}b(t)\,dt\geq \alpha (y), \end{equation*} with$\alpha (y)=f(y)-\inf_{x\in X} f(x)>0$. \end{itemize} \end{theorem} \begin{proof} We have the hypotheses of Theorem \ref{thm2.2}. So, applying this theorem, we obtain (i') and (ii') from (i) and (ii). Moreover, (iii) guaranties that there exists$z\in X$such that $$f(x)\geq f(z)-\frac{\epsilon }{\lambda }b( \beta (z))p(z,x),\quad \text{for all } x\in X, \label{e3.1}$$ where$0<\lambda \leq \alpha (y)$. Choose$x=z+ty$with$ \| y\| =1$in \eqref{e3.1} and obtain $$\frac{f(z+ty)-f(z)}{t}\geq - \frac{\epsilon }{\lambda }\frac{b(\beta (z))p(z,z+ty)}{t}, \label{e3.2}$$ for every$t>0$. Let$\lambda $be such that $$\lim_{t\to 0} \frac{p(z,z+ty)}{t}\leq \lambda. \label{e3.3}$$ Then, letting$t\to 0$in \eqref{e3.2} and using \eqref{e3.3}, we conclude that $$\langle f'(z),y\rangle \geq -\epsilon \cdot b(\beta (z)), \label{e3.4}$$ for all$y\in X$with$\| y\| =1$. Since \eqref{e3.4} is true for$\pm y$, we deduce that $$| \langle f'(z),y\rangle| \leq \epsilon \cdot b(\beta (z)). \label{e3.5}$$ Now, from \eqref{e3.5}, we obtain \begin{equation*} \| f'(z)\| =\sup_{y\in X,\| y\| =1} \frac{| \langle f'(z),y\rangle | }{\| y\| }\leq \epsilon \cdot b(\beta (z)), \end{equation*} and the claim (iii') holds. \end{proof} \begin{corollary} \label{coro3.2} Suppose that the hypotheses of Theorem \ref{thm3.1} are verified. Then there exists a minimizing sequence$\{z_n\} _n$of$f$such that \begin{gather*} f(z_n)<\inf_{x\in X} f(x)+\epsilon,\\ \| f'(z_n)\| /b(\beta (z_n))\to 0. \end{gather*} \end{corollary} The proof follows form taking$\epsilon =\frac{1}{n}$,$n=1,2,\dots$in Theorem \ref{thm3.1}. Let$\mathcal{B}$be the set of all non-increasing and strictly positive continuous functions$b:[0,\infty )\to(0,\infty )$such that \begin{equation*} \int_0^{\infty }b(t)\,dt=\infty . \end{equation*} Let$p:X\times X\to\mathbb{R}_{+}$be a$u$-distance on$X$lower semicontinuous in its second variable with$p(x,x)=0\forall x\in X$,$a\in X$a fixed point and$\beta :X\to \mathbb{R}_{+}$defined by$\beta (x)=p(a,x)$. \begin{definition} \label{def3.3} \rm Let$f:X\to (-\infty ,+\infty ]$be a$C^{1}$function,$c\in \mathbb{R}$and$b\in\mathcal{B}$. \begin{itemize} \item$f$is said to satisfy the b-(PS) condition if any sequence$\{ x_n\} _n$in$X$such that$\{ f(x_n)\} $is bounded and$\| f'(x_n)\| /b( \beta (x_n))\to 0$has a convergent subsequence. \item$f$is said to satisfy the b-(PS)$_c$condition if any sequence$\{ x_n\} _n$in$X$such that$f(x_n)\to c$and$\| f'(x_n)\| /b(\beta (x_n))\to 0$has a convergent subsequence. \end{itemize} \end{definition} \begin{remark} \label{rmk3.4} \rm Suppose that$\beta (x)=d(a,x)$. \begin{itemize} \item Then the b-(PS) condition is the Schechter-(PS) condition \cite{s1}. \item If$b$is constant, then the b-(PS) condition is the usual$(PS)$condition. \item If$b(t)=1/(1+t)$, then the b-(PS) condition is the Cerami-(PS) condition \cite{c2}. \item If$b(t)=1/(1+h(t))$, where$h:[0,\infty )\to [0,\infty )$is a non-decreasing function, then the b-(PS) condition is the Zhong-(WPS) condition \cite{z1,z2}. \end{itemize} \end{remark} \begin{theorem} \label{thm3.5} If$f$is bounded below and satisfying the {\rm b-(PS)} condition, then$f$has a minimal point. \end{theorem} \begin{proof} By Corollary \ref{coro3.2}, there is a minimizing sequence$\{ z_n\} _n$in$X$such that$f(z_n)<\inf_{x\in X} f(x)+\epsilon $and$\| f'(z_n) \| /b(\beta (z_n))\to 0$. The b-(PS) condition implies that$\{z_n\} _n$has a subsequence$\{ z_{n_{k}}\} _{k}$convergent to some point$z^{\ast }$. Since$f$is lower semicontinuous, we obtain \begin{equation*} \inf_{X} f\leq f(z^{\ast })\leq \liminf_{k\to \infty } f(z_{n_{k}})\leq \inf_{X} f. \end{equation*} Therefore,$f(z^{\ast })=\inf_{X}f$. \end{proof} \section{The b-(PS) condition versus coercivity} Using the method of gradient flows, Li \cite{l1} first observed that the (PS) condition implies the coercivity for$C^{1}$functionals bounded from below. Using Ekeland's variational principle, Caklovic, Li and Willem \cite{c1} proved the same result for a G\^ateaux differentiable functional which is lower semicontinuous. The same conclusion was also proved by Costa and Silva \cite{c3} and Brezis and Nirenberg \cite{b2} for$C^{1}$functionals by also employing Ekeland's principle. Using ZVP, Zhong \cite{z1} studied the connection between (WPS) and coercivity. A similar result was established by Suzuki \cite{s2}, using$\tau$-distance. In this section, we discuss the relation between the b-(PS) condition and coercivity. We recall that a function$f:X\to (-\infty ,\infty ]$is said to be coercive if \begin{equation*} \underset{r\to \infty }{\lim }\underset{\| x\| \geq r}{\inf }f(x)=\infty. \end{equation*} For our aim, we first prove the following lemma. \begin{lemma} \label{lem4.1} Let$p:X\times X\to \mathbb{R}_{+}$be a$u$-distance on$X$and$f:X\to\mathbb{R}$is a G\^{a}teaux differentiable function. Suppose that there are$\xi \geq 0$,$\delta >0$and either of the following conditions is satisifed: \begin{itemize} \item$f(y)\geq f(x)-\xi p(x,y)$for all$y\in X$with$0-\xi \delta \label{e4.2} Then, $$\frac{f(x+\delta z)-f(x)}{\delta }>-\xi . \label{e4.3}$$ Taking the limit as $\delta \to 0$, we obtain $$\langle f'(x),y\rangle \geq -\xi . \label{e4.4}$$ As \eqref{e4.4} holds for both of $\pm y$, we derive $$| \langle f'(x),y\rangle | \leq \xi . \label{e4.5}$$ Then, for all $y\in X$ with $\| y\| =1$, the inequality \eqref{e4.5} implies that $\| f'(x)\| = \sup_{y\in X,\| y\| =1} \frac{| \langle f'(x),y\rangle | } {\| y\| } = \sup_{y\in X,\| y\| =1} |\langle f'(x),y\rangle | \leq \xi,$ and the desired claim holds. \end{proof} Next, we consider a more suitable version of Theorem \ref{thm1.5}, for our purpose. \begin{theorem} \label{thm4.2} Let $(X,d)$ be a complete metric spaces, let $f:X\to (-\infty ,\infty ]$ be a proper lower semicontinuous function which is bounded from below, and let $p:X\times X\to\mathbb{R}_{+}$ be a $u$-distance on $X$ lower semicontinuous in its second argument. Then for $\varepsilon >0$ and $x\in X$ with $f(x)<\infty$ and $p(x,x)=0$, there exists $v\in X$ such that \begin{itemize} \item[(i)] $f(v)\leq f(x)-\varepsilon p(x,v)$; \item[(ii)] $f(w)>f(v)-\varepsilon p(v,w)$, for all $w\in X\backslash \{ v\}$. \end{itemize} \end{theorem} For the sake of completeness, we supply a proof of the equivalence between Theorems \ref{thm1.5} and \ref{thm4.2}. \begin{proof} $\Leftarrow$ Let the assumptions of Theorem \ref{thm1.5} be satisfied. Obviously, the conclusion of (1) follows by Theorem \ref{thm4.2}. For (2), applying again Theorem \ref{thm4.2} with $\varepsilon =\frac{e}{\lambda }$, we deduce that \begin{equation*} p(x,v)\leq \frac{\lambda }{e}(f(x)-f( v))\leq \frac{\lambda }{e}e\leq \lambda . \end{equation*} Hence the conclusion of Theorem \ref{thm1.5} is valid. $\Rightarrow$ Now, suppose that Theorem \ref{thm1.5} holds. Let $x\in X$ with $f(x)<\infty$ \ and $\varepsilon >0$ be given. Fix any $e>f(x)-\inf_{a\in X} f(a)$ and set $\lambda =\frac{e}{\varepsilon }$. Consider \begin{equation*} M(x)=\{ v\in X\mid f(v)\leq f(x)-\varepsilon p(x,v)\} . \end{equation*} By the lower semicontinuity of $f$ and $p(x,.)$, the set $M(x)$ is closed. Furthermore, $M(x)$ is nonempty as $x\in M(x)$. Applying Theorem \ref{thm1.5} (2) for the chosen $e$, $\lambda$ and for $M(x)$ instead of $X$ one finds $v\in X$ such that \begin{gather*} f(v)\leq f(x), \quad p(x,v)\leq \lambda , \\ f(w)>f(v)-\frac{e}{\lambda }\cdot p(v,w),\quad \text{for all }w\in M(x)\backslash \{v\} . \end{gather*} Since $v\in M(x)$, then (i) holds. To show (ii) it is sufficient to check that \begin{equation*} f(w)>f(v)-\frac{e}{\lambda }\cdot p( v,w),\quad \text{for all }w\notin M(x). \end{equation*} By the definition of $M(x)$, the property $w\notin M(x)$ means that \begin{equation*} f(w)>f(x)-\varepsilon p(x,w). \end{equation*} From this and (i) we easily deduce (ii) and then obtain Theorem \ref{thm4.2}. \end{proof} We are in position to state the main result of this section. The proof follows a technique developed by Suzuki in \cite{s2}. \begin{theorem} \label{thm4.3} Let $X$ be a Banach space, $a\in X$ fixed, and let $p:X\times X\to\mathbb{R}_{+}$ be a symmetric $u$-distance on $X$, lower semicontinuous in its second argument and such that $p(x,x)=0$ for all $x\in X$. Let $f:X\to (-\infty ,\infty ]$ be a proper lower semicontinuous function which is bounded from below and let $b:[0,\infty )\to (0,\infty )$ be a non-increasing continuous function such that \begin{equation*} B(t)=\int_0^{t}b(r)\,dr, \end{equation*} where $B$ is a function from $\mathbb{R}_{+}$ to itself such that $B(\infty )=+\infty$. Let $a\in X$ be fixed and $\beta :X\to\mathbb{R}_{+}$ defined by $\beta (x)=p(a,x)$. Assume that $f$ is G\^{a}teaux differentiable at every point $x\in X$ with $f(x)\in\mathbb{R}$. If \begin{equation*} \alpha =\liminf_{\beta (y)\to \infty } f(y)\in\mathbb{R}, \end{equation*} then there exists a sequence $\{ z_n\} _n$ in $X$ such that \begin{itemize} \item[(a)] $\lim_{n\to \infty } \beta (z_n)=\infty$; \item[(b)] $\lim_{n\to \infty } f(z_n) =\alpha$; \item[(c)] $\lim_{n\to \infty } \| f'(z_n)\| /b(\beta (z_n))=0$. \end{itemize} \end{theorem} \begin{proof} We shall show only the following: for every $\varepsilon >0$, there exists $v\in X$ satisfying $\beta (v)\geq \frac{1}{\varepsilon }$, $| f(v)-\alpha | \leq \varepsilon$ and $\| f'(v)\| /b(\beta(v))\leq \varepsilon$. Fix $\varepsilon >0$ and define a function $\chi :[0,\infty )\to [0,\infty )$ by $$\chi (t)=\frac{1}{2}b(t+1)\label{e4.6}$$ for $t\in [0,\infty )$. Then $\chi$ is non-increasing, and $\int_0^{\infty }\chi (t)\,dt = \frac{1}{2}\int_0^{\infty}b(t+1)\,dt = \frac{1}{2}\int_{1}^{\infty }b(t)\,dt=\infty .$ We also determine a function $h:X\to (-\infty,+\infty ]$ by $$h(x)=\max \{ f(x),\alpha -2\varepsilon\} \label{e4.7}$$ for $x\in X$. Then it is obvious that $h$ is proper lower semicontinuous and bounded from below. We choose $r,r'\in\mathbb{R}$ with $\frac{1}{\varepsilon }\alpha -\varepsilon , \label{e4.8}\\ \int_{r}^{r'}\chi (t)\,dt=3. \label{e4.9} \end{gather} We also choose$u\in X$with $$\beta (u)>r',\quad f(u)<\alpha +\varepsilon . \label{e4.10}$$ We note that$h(u)=f(u)$because$\beta (u)>r'>r$. We know from the earlier that the function$q:X\times X\to\mathbb{R}_{+}$, defined by $$q(u,v)=\int_{\beta (u)}^{\beta (u) + p(u,v)}\chi (t)\,dt \label{e4.11}$$ is a$u$-distance. So, by Proposition \ref{prop2.1}, the function$s:X\times X\to\mathbb{R}_{+}$, defined by $$s(u,v)=q(u,v)+q(v,u)\label{e4.12}$$ is also a$u$-distance. Thereby, by Theorem \ref{thm4.2}, there exists$v\in X$such that \begin{gather} h(v)\leq h(u)-\varepsilon s(u,v), \label{e4.13} \\ h(w)>h(v)-\varepsilon s(v,w),\quad \forall w\neq v. \label{e4.14} \end{gather} Arguing by contradiction, we assume that$\beta (v)\frac{1}{\varepsilon }, \end{equation*} and (a) holds. Thus, we have $h(v)=f(v)$ and \begin{equation*} \alpha -\varepsilon <\underset{\beta (y)\geq r}{\inf }f( y)\leq f(v)\leq f(u)<\alpha +\varepsilon . \end{equation*} This implies \begin{equation*} | f(v)-\alpha | \leq \varepsilon , \end{equation*} that is (b). For $(c)$, from \eqref{e4.11} $, \eqref{e4.12}$ and \eqref{e4.14} and the non-increasing property of $\chi$, we infer $$\label{e4.18} \begin{split} h(w)&> h(v)-\varepsilon \int_{\beta ( v)}^{\beta (v) + p(v,w)}\chi (t)\,dt-\varepsilon \int_{\beta (w)}^{\beta ( w) + p(v,w)}\chi (t)\,dt \\ &\geq h(v)-\varepsilon (\chi (\beta ( v))+\chi (\beta (w)))\cdot p(v,w), \end{split}$$ for $w\in X$, $w\neq v$. Since $f$ is lower semicontinuous and $f(v)>\alpha -2\varepsilon$, there exists $\delta \in (0,1)$ such that $f(w)>\alpha -2\varepsilon$ for $w\in X$ with $p(v,w)<\delta$. Hence, for $w\in X$ with $0\beta (v)-\delta >\beta (v)-1>0, \] we derive $$\label{e4.19} \begin{split} f(w) &> f(v)-\varepsilon (\chi (\beta (v))+\chi (\beta (v)-1)) \cdot p(v,w) \\ &\geq f(v)-2\varepsilon \chi (\beta (v)-1)\cdot p(v,w) \\ &= f(v)-\varepsilon b(\beta (v))\cdot p(v,w). \end{split}$$ By means of Lemma \ref{lem4.1}, we reach \begin{equation*} \| f'(v)\| \leq \varepsilon b(\beta (v)), \end{equation*} and (c) is verified too. The proof is complete. \end{proof} \begin{corollary} \label{coro4.4} Let$X$be a Banach space. Let$f:X\to (-\infty ,\infty ]$be a proper lower semicontinuous function which is bounded from below. Assume that$f$is G\^ateaux differentiable at every point$x\in X$with$f(x)\in\mathbb{R}$. If$f$satisfies the b-(PS)$_c$condition for all$c\in\mathbb{R}$, then$f$is coercive; i.e.,$f(x) \to \infty $as$\beta (x)\to \infty$. \end{corollary} \begin{proof} Suppose the contrary; then$\alpha =\lim\inf_{\beta (x)\to \infty } f(x)\in\mathbb{R}$. By Theorem \ref{thm4.3}, there exists a sequence$\{ z_n\} _n$in$X$such that$\beta (z_n)\to \infty $,$f(z_n)\to \alpha $and$\| f'(z_n)\| /b(\beta (z_n))\to 0$. Then, the b-(PS)$_{\alpha }$condition implies that$\{ z_n\} _n\$ has a convergent subsequence, which clearly leads to a contradiction. \end{proof} \begin{remark} \label{rmk4.5} \rm Corollary \ref{coro4.4} generalizes the result proved by \cite{l1} using a gradient flow, by Costa-Silva \cite{c3}, Caklovic-Li-Willem \cite{c1} and Brezis-Nirenberg \cite{b2} using EVP, and by Zhong \cite{z1} using ZVP. \end{remark} \begin{thebibliography}{99} \bibitem{b1} T. Q. Bao, P. Q. 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