On Ekeland Variational Principle in Asymmetric b-Metric Spaces

Parana Journal of Science and Education, v.11, n.5, (21-27), October 12, 2025 PJSE, ISSN 2447-6153, © 2015-2025 https://sites.google.com/site/pjsciencea/ Received: September 24, 2025; Accepted: October 11, 2025; Published: October 12, 2025. On Ekeland Variational Principle in Asymmetric b-Metric Spaces Anas Yusuf1* and Junaidu Nafiu Mallam2 Abstract This paper extends the Ekeland Variational Principle (EVP) to the setting of asymmetric b-metric spaces, which generalize both asymmetric metric and b-metric spaces. By distinguishing forward, backward and bicompleteness, we establish corresponding forward, backward and bi-complete version of EVP. Examples are provided to demonstrate that forward and backward principles may yield different minimizers while the bi-complete EVP reduces to the classical EVP when symmetry holds. As an application, we obtain a forward and backward Cartisi-type fixed point theorem. These results broaden the scope of EVP and offer a new tools for optimization and fixed-point theory in asymmetric frameworks. Keywords fixed point, b-metric space, quasi-metric space, Ekeland variational principle, Cartisi fixed point. 1,2Department of Mathematics, Federal University, Birnin Kebbi, Nigeria. *Corresponding author: [email protected] Contents 1Inroduction 21 2Preliminaries 22 3Results 23 4Conclusion 27 References 27 1. Inroduction Since its introduction in the 1970s by Ivar Ekeland [ 1 ], the Ekeland Variational Principle (EVP) has become a cornerstone of modern optimization theory and nonlinear analysis. The principle not only provides a variational characterization of approximate minimizers but also serves as a unifying tool for deriving fixed point theorems, Caristi’s theorem, and critical point results in Banach space settings. Its applications span diverse fields including optimal control, equilibrium problems, convex analysis, and partial differential equations. Following Ekeland’s original work, many generalizations of the principle have been established. Kirk [ 2 ] and Takahashi [ 3 ] extended EVP to complete metric spaces and normed linear spaces, while Borwein and Preiss [ 4 ] developed smooth variational principles applicable in infinite-dimensional settings. Wilson [ 11 ] introduced asymmetric metric spaces as quasi-metric spaces and Cobza s¸ [ 5 ] studied quasi-metric versions of the Ekeland variational principle and its connections with completeness properties of the underlying quasi-metric space. Farkas et al. [ 13 ] established a generalized variational On Ekeland Variational Principle in Asymmetric b-Metric Spaces — 22/27 principle for b-metric spaces. On another front, Bakhtin [ 6 ] introduced the concept of a b-metric space, where the triangle inequality is relaxed by a constant s≥1 . This generalization has attracted considerable interest in fixed point theory, as shown in the works of Czerwik [ 7 ] and subsequent authors, leading to numerous results on contractive mappings, Caristi-type theorems, and variational inequalities in b-metric spaces. Bota, Moln ´ ar and Varga [ 8 ] investigated EVP in b-metric spaces and showed that the principle continues to hold under suitable completeness assumptions. Fixed point results in asymmetric metric spaces were studied by Khorshidvandpour et al. [ 14 ]. Recent developments in b-metric and asymmetric metric spaces can be found in [ 15 , 16 ]. Studies in asymmetric b-metric (quasib-metric) spaces appear in [17,18,19]. Despite these advances, relatively little attention has been paid to asymmetric b-metric spaces, which combine both asymmetry and the relaxed triangle inequality. These spaces arise naturally in applications where distances are not symmetric and triangular estimates are approximate, such as computer science, decision theory, and models of directed networks. To the best of our knowledge, no systematic treatment of EVP in asymmetric b-metric spaces has yet been given. Our work fills this gap by formulating and proving forward, backward, and bi-complete versions of the Ekeland Variational Principle in asymmetric b-metric spaces. In doing so, we unify and extend the previous lines of research: when the coefficient s=1 , our results reduce to the asymmetric metric setting, and when symmetry is imposed, we recover the classical EVP in b-metric and metric spaces. 2. Preliminaries Definition 2.1 (b-metric [ 6 , 7 ]).Let X be a non-empty set and s≥1 a real number. A function d:X×X→R+ is called a b-metric provided that for all x,y,z∈X: (i) d(x,y) = 0if and only if x =y, (ii) d(x,y) = d(y,x), (iii) d(x,z)≤sd(x,y)+d(y,z). A pair (X,d)is called a b-metric space. It is clear that the definition of b-metric space extends the usual metric space; indeed if s=1 we recover the classical metric space. Example 2.1 ([9]).The space `p(R)with 0<p<1, `p(R) = {(xn)⊂R: ∞ ∑ n=1 |xn|p<∞}, together with d(x,y) = ∞ ∑ n=1 |xn−yn|p1/p, is a b-metric space. By an elementary calculation we obtain that d(x,z)≤21/pd(x,y)+d(y,z), so one may take s =21/p. Definition 2.2 (Asymmetric metric [ 10 ]).A function d:X× X→R+ is called an asymmetric metric and (X,d) is an asymmetric metric space if for all x,y,z∈X: (i) d(x,y) = 0if and only if x =y, (ii) d(x,z)≤d(x,y)+d(y,z). Example 2.2 ([10]).Let α>0. Define d :R×R→R+by d(x,y) = (y−x,y≥x, α(x−y),y<x. Then d is an asymmetric metric on R. Definition 2.3. Let ρ be an asymmetric metric on X . The conjugate of ρis the mapping ¯ ρdefined by ¯ ρ(x,y) = ρ(y,x),x,y∈X. The mapping ds(x,y) = max{ρ(x,y),¯ ρ(x,y)}=max{ρ(x,y),ρ(y,x)} is called the symmetrized metric on X , and ds is a metric on X if and only if ρis an asymmetric metric on X. Definition 2.4 (Forward/backward topology [ 10 ]).The forward topology τ+ induced by d is generated by forward open balls B+(x,ε) = {y∈X:d(x,y)<ε},x∈X,ε>0. Likewise, the backward topology τ− induced by d is generated by backward open balls B−(x,ε) = {y∈X:d(y,x)<ε}. Definition 2.5 (Forward/backward convergence [ 10 ]).A sequence (xn)n∈N is said to be forward convergent to x0∈X if lim n→∞d(x0,xn) = 0, and backward convergent to x0if lim n→∞d(xn,x0) = 0. We denote forward convergence by xn f →x0 and backward convergence by xn b →x0. On Ekeland Variational Principle in Asymmetric b-Metric Spaces — 23/27 Definition 2.6 (Forward/backward completeness [ 10 ]).A space (X,d) is called forward complete (resp. backward complete) if every forward (resp. backward) Cauchy sequence in X forward( resp.backward) converges to a point in X. Definition 2.7 (Forward/backward Cauchy [ 10 ]).A sequence (xk)k∈N⊂X is forward Cauchy if for every ε>0 there exists N such that for all m≥n≥N , d(xn,xm)<ε . The sequence is backward Cauchy if for all m≥n≥N we have d(xm,xn)<ε . Definition 2.8 (Asymmetric b-metric / quasi-b-metric [ 17 ]). Let X be a non-empty set. A function ρ:X×X→[0,∞) is called an asymmetric b-metric (also called quasi-b-metric) on X if there exists a constant s ≥1such that for all x,y,z∈X: (i) ρ(x,y) = 0iff x =y, (ii) ρ(x,z)≤sρ(x,y)+ρ(y,z). Here s is called the coefficient of the asymmetric b-metric. Remark 2.1. • If symmetry is added, ρ(x,y) = ρ(y,x) , then ρ reduces to a standard b-metric. •If s =1, we get an asymmetric metric space. So the hierarchy is metric ⊂b-metric ⊂asymmetric b-metric. Example 2.3. Define ρ:R×R→[0,∞)by ρ(x,y) = max{0,y−x}2= (y−x)2 +. Then ρ(x,x) = 0 , and in general ρ(x,y)6=ρ(y,x) . One can check the b-triangle inequality with s =2: ρ(x,z)≤2ρ(x,y)+ρ(y,z), by taking ρ(x,y)=(max{0,y−x})2= (y−x)2 + . This can easily be seen as ((z−x)+)2≤2((z−y)2 ++(y−x)2 +) , hence (R,ρ) is an asymmetric b-metric space with coefficient s=2 . Theorem 2.1 (Classical EVP [ 1 ]).Let (X,d) be a complete metric space. Let f:X→R be proper, lower semicontinuous and bounded below. Then for every ε>0 and every x∈X with f(x)≤inf Xf+ε and every λ>0there exists y ∈X such that f(y)≤f(x),d(x,y)≤λ, and for all z 6=y, f(z)>f(y)−ε λd(y,z). Theorem 2.2 (Caristi-type fixed point theorem [ 12 ]).Let (X,d) be a complete metric space and let T:X→X be a mapping such that d(x,Tx)≤f(x)−f(Tx)for all x ∈X, where f:X→[0,+∞) is lower semicontinuous. Then T has at least one fixed point. 3. Results Definition 3.1 (Forward/backward lower semicontinuity).A function f :X→Ris forward-lsc if xn f →x implies f(x)≤liminf n→∞f(xn), and backward-lsc if xn b →x implies f(x)≤liminf n→∞f(xn), . Theorem 3.1 (Forward Ekeland Variational Principle).Let (X,ρ) be a forward complete asymmetric b-metric space with coefficient s≥1 . Let f:X→R be proper, bounded below and forward-lsc. Fix ε>0. Suppose x0∈X satisfies f(x0)≤inf Xf+ε. For every λ>0 set α:=ε/λ>0 . Then there exists x∗∈X such that: (1) f (x∗)≤f(x0), (2) ρ(x0,x∗)≤sλ, (3) For every y ∈X with y 6=x∗, f(y)>f(x∗)−α ρ(x∗,y). Equivalently, f(y)+α ρ(x∗,y)≥f(x∗)for all y ∈X. Proof. Fix λ>0 and set α=ε/λ . We construct inductively a sequence (xn) as follows. Choose a summable sequence (εn) with εn↓0 and ∑∞ n=0εn<∞ . Define for each n≥0 the perturbed functional ϕn(y):=f(y)+α ρ(xn,y). Choose xn+1∈Xsuch that f(xn+1)+α ρ(xn,xn+1)≤inf y∈Xϕn(y)+εn.(1) Putting y=xnin (1) gives f(xn+1)+α ρ(xn,xn+1)≤f(xn)+ εn, hence α ρ(xn,xn+1)≤f(xn)−f(xn+1)+ εn. Therefore (f(xn)) is nonincreasing and bounded below, so it converges to some L≥infXf . Summing the inequality from n=0 to N−1 yields α N−1 ∑ n=0 ρ(xn,xn+1)≤f(x0)−f(xN)+ N−1 ∑ n=0 εn. On Ekeland Variational Principle in Asymmetric b-Metric Spaces — 24/27 Let N→∞. Since f(xN)→Land ∑εn<∞we obtain α ∞ ∑ n=0 ρ(xn,xn+1)≤f(x0)−L+ ∞ ∑ n=0 εn≤ε+ ∞ ∑ n=0 εn<∞. Thus ∑∞ n=0ρ(xn,xn+1)<∞and ρ(xn,xn+1)→0. Forward Cauchy property. For m>n , repeated use of the b-triangle inequality gives ρ(xn,xm)≤s(ρ(xn,xm−1)+ρ(xm−1,xm)) ≤s(ρ(xn,xn+1)+ρ(xn+1,xn+2)+...+ ρ(xm−1,xm)) ≤s m−1 ∑ k=n ρ(xk,xk+1) Thus, for every η>0 there exists N such that for all m>n≥ N we have ρ(xn,xm)<η . Hence (xn) is forward Cauchy. By forward completeness there exists x∗∈Xwith ρ(x∗,xn)→0 (i.e. xn f →x∗). Distance bound. For every m, ρ(x0,xm)≤s(ρ(x0,x1)+ρ(x1,x2)+...+ρ(xm−1,xm)) =s m−1 ∑ k=0 ρ(xk,xk+1) Letting m→∞and using the summability bound gives ρ(x0,x∗)≤s ∞ ∑ k=0 ρ(xk,xk+1)≤sε+∑εn α. Choosing ∑εn arbitrarily small and recalling α=ε/λ yields ρ(x0,x∗)≤sλ. Thus (2) holds. Showing f(x∗)≤f(x0) . Since f is forward-lsc and xn f →x∗ we get f(x∗)≤liminf n→∞f(xn) = L≤f(x0), so (1) holds. Variational inequality. From (1) with an arbitrary y∈X we have f(xn+1)+α ρ(xn,xn+1)≤f(y)+ α ρ(xn,y)+εn.(2) Rearrange: f(y)≥f(xn+1)+αρ(xn,xn+1)−ρ(xn,y)−εn. Letting n→∞ , using ρ(xn,xn+1)→0 , ρ(xn,y)→ρ(x∗,y) (by forward convergence of (xn) to x∗ ), and f(xn+1)→f(x∗) , we obtain f(y)≥f(x∗)−α ρ(x∗,y). Hence f(y)+ α ρ(x∗,y)≥f(x∗) for all y . If equality occurs for some y6=x∗ then by using inequality 3and letting n→∞ gives equality in the limit. That is lim n→∞((f(y)+α ρ(xn,y)+εn)−(f(xn+1)+α ρ(xn,xn+1))) = 0. (3) Because each term xn+1 is an approximate minimizer of ϕn(y) , the only way the values at xn+1 and at y can be arbirarily close is if the points themselves become arbitrarily close in the forward sense: ρ(xn+1,y)→0 , but we already have xn+1→x∗ . So xn+1 would be converging to both x∗ and y , forcing y=x∗ , a contradiction. Therefore strict inequality holds for y6=x∗ , proving (3). Corollary 3.1 (Backward EVP).Let (X,ρ) be backward complete asymmetric b-metric space with coefficient s≥1 . Let f:X→R be proper, bounded below and backward-lsc. Fix ε>0and suppose x0∈X satisfies f(x0)≤inf Xf+ε. For every λ>0 , set α:=ε/λ>0 . Then there exists x∗∈X such that: (1) f (x∗)≤f(x0), (2) ρ(x∗,x0)≤sλ, (3) For every y ∈X with y 6=x∗, f(y)>f(x∗)−α ρ(y,x∗). Equivalently, f (y)+α ρ(y,x∗)≥f(x∗)for all y. Proof. The proof parallels that of Theorem 3.1 with the roles of arguments in ρ reversed: at each step one minimizes the backward-perturbed functionals and uses backward-Cauchy and backward completeness. The order of limits uses backwardlsc of f. Theorem 3.2 (Bi-complete (symmetrized) EVP).Let (X,ρ) be an asymmetric b-metric space with coefficient s≥1 . Define the symmetrized metric ds(x,y):=max{ρ(x,y),ρ(y,x)},x,y∈X. Assume (X,ρ) is bi-complete (i.e. (X,ds) is complete). Let f: X→R be proper, bounded below and lower semicontinuous with respect to ds. Fix ε>0and suppose x0∈X satisfies f(x0)≤inf Xf+ε. Then for every λ>0there exists x∗∈X such that: (1) f (x∗)≤f(x0), (2) ds(x0,x∗)≤sλ, On Ekeland Variational Principle in Asymmetric b-Metric Spaces — 25/27 (3) For every y ∈X with y 6=x∗, f(y)>f(x∗)−ε λds(x∗,y). Proof. Put α:=ε/λ>0. For each z∈Xdefine Gz(w):=f(w)+αds(w,z),w∈X. Each Gz is ds -lsc because f is ds -lsc and w7→ ds(w,z) is continuous. By completeness of (X,ds) any minimizing sequence for Gz is ds -Cauchy and has a ds -limit where the infimum is attained. Thus we may inductively choose a sequence (xn) with x0given and, for each n≥0, Gxn(xn+1) = inf w∈XGxn(w). Putting w=xnin the minimization yields f(xn+1)+αds(xn+1,xn)≤f(xn). Hence αds(xn+1,xn)≤f(xn)−f(xn+1), so (f(xn)) is nonincreasing and bounded below and therefore convergent. Summing the inequality gives α ∞ ∑ n=0 ds(xn+1,xn)≤f(x0)−lim n→∞f(xn)≤ε, so ∑ds(xn+1,xn)≤ε/α=λ(4) . ds -Cauchy and limit. For m>n , the the b-triangle inequality gives ds(xn,xm)≤s m−1 ∑ k=n ds(xk+1,xk) Since the series of increments converges, the right handside tends to 0 as n→∞ . Hence (xn) is ds -Cauchy. By completeness there exists x∗∈Xwith ds(xn,x∗)→0 as n→∞. From lower semi-continuity of f and monotonicity of f(xn),we get f(x∗)≤liminf n→∞f(xn) = L≤f(x0), so property (1) holds. To show ds(x0,x∗)≤sλ . Use the b-triangle inequality repeatedly: for evey m, ds(x0,xm)≤s(ds(x0,x1)+ds(x1,x2)+...+ds(xm−1,xm)) =s m−1 ∑ k=0 ds(xk,xk+1) Letting m→∞and equation (4) we obtain ds(x0,x∗)≤s ∞ ∑ k=0 ds(xk,xk+1)s·ε α=sλ. Thus (2) holds. Variational inequality. For arbitrary x∈X , the minimizer property yields for each n f(xn+1)+αds(xn+1,xn)≤f(x)+αds(x,xn).(5) Rearrange: f(x)≥f(xn+1)+αds(xn+1,xn)−ds(x,xn). Letting n→∞ . Since xn→x∗ in ds we have ds(xn+1,xn)→0 , ds(x,xn)→ds(x,x∗) , while f(xn+1)→f(x∗) . Passing to the limit in (5) we obtain f(x)≥f(x∗)−αds(x,x∗). Hence f(x)+αds(x,x∗)≥f(x∗). Strict inequality. To see strict inequality when y6=x∗ , suppose f(y)+αds(y,x∗)≥ f(x∗) for some y6=x∗ . Then from (5) with x=y equality must hold in the limit, forcing ds(xn+1,y)→0 , but we already have xn+1→x∗ in ds . So ds(y,x∗) = 0 , hence y=x∗ which is a contradiction. Therefore strict inequality holds for y6=x∗ , proving (3). Corollary 3.2 (Reduction to classical EVP).Let (X,ρ) be an asymmetric b-metric space with coefficient s=1 . Suppose ρ is symmetric (so ρ(x,y) = ρ(y,x) for all x,y ). Then ds=ρ is a complete metric on X . Let f:X→R be proper, bounded below and lower semicontinuous with respect to ρ . Fix ε>0 and suppose x0∈X satisfies f(x0)≤inf Xf+ε. Then for every λ>0there exists x∗∈X such that: f(x∗)≤f(x0),ρ(x0,x∗)≤λ, and for every y 6=x∗, f(y)>f(x∗)−ε λρ(x∗,y). Proof. If ρ is symmetric and s=1 then ds(x,y) = ρ(x,y) . Applying Theorem 3.2 with ds=ρ yields the classical Ekeland variational principle; the factor s disappears from the distance bound. Example 3.1 (Forward vs Backward EVP selecting different minimizers).Take X = [0,3]and define ρ(x,y) = (y−x,y≥x, 2(x−y),y<x. On Ekeland Variational Principle in Asymmetric b-Metric Spaces — 26/27 Then ρ is asymmetric and satisfies the b-triangle inequality with s =2: ρ(x,z)≤2ρ(x,y)+ρ(y,z). Define f(x)=(x−0.5)2 for x∈[0,3] and choose EVP parameters x0=2,ε=2.25,λ=2.25 so that ε/λ=1. Forward penalized function. The forward penalized function is gf(y) = f(y)+ ε λρ(x0,y) = (y−0.5)2+ρ(2,y). Compute ρ(2,y): ρ(2,y) = (4−2y,y≤2, y−2,y≥2. On [0,2], gf(y)=(y−0.5)2+4−2y,g0f(y) = 2(y−0.5)−2=2y−3, so the critical point is y=1.5 and g00 f(y) = 2>0 , hence y= 1.5 is the minimizer on [0,2] . On [2,3] the function increases. Thus the global minimizer of gfon [0,3]is x∗=1.5. Check the distance bound: ρ(x0,x∗) = ρ(2,1.5) = 2(2− 1.5) = 1≤sλ=2(2.25) = 4.5 . Also f(1.5) = 1≤f(2) = 2.25 so f (x∗)≤f(x0). The forward variational inequality requires for x∗=1.5 and any y 6=1.5, f(y)>f(1.5)−ρ(1.5,y). or equivalently ∆(y):=f(y)−(f(1.5)−ρ(1.5,y)). Computing ∆(y)seperately on the two regions. 1. If y>1.5; then, ρ(1.5,y) = y−1.5 . So ∆(y):= (y− 0.5)2−1+ (y−1.5) = y2−y+0.25 −1+y−1.5= y2−2.25, so from y >1.5, y2>2.25,so ∆(y)>0 2. For y<1.5; then, ρ(1.5,y) = 2(1.5−y) = 3−2y . So ∆(y):= (y−0.5)2−1+(3−2y) = y2−y+0.25−1+ 3−2y=y2−3y+2.25 = (y−0.5)2>0 Thus, the variational inequality holds strictly for every y6=x∗ . So the forward EVP conclusion holds and x∗=1.5 is the forward EVP point. Backward penalized function. The backward penalized function is gb(y) = f(y)+ρ(y,2). Compute ρ(y,2): ρ(y,2) = (2−y,y≤2, 2(y−2),y≥2. On [0,2], gb(y)=(y−0.5)2+2−y,g0 b(y) = 2(y−0.5)−1=2y−2, so the critical point is y=1 (with g00 b=2>0 ). On [2,3] the function increases. Hence the global minimizer is xb=1 . Checking the backward variational inequality shows xb=1 satisfies it strictly. Therefore the forward EVP and backward EVP select different minimizers ( 1.5 vs 1 ) for the same starting data. Theorem 3.3 (Forward Caristi’s theorem).Let (X,ρ) be an asymmetric b-metric space with coefficient s≥1 which is forward complete. Let f:X→R be proper, bounded below and forward-lsc. Let T:X→X be a map satisfying the Caristi-type condition ρ(x,Tx)≤f(x)−f(Tx)for every x ∈X.(6) Then T has a fixed point. Proof. Fix arbitrary x0∈X . Since f is bounded below, infXf is finite; choose ε>0 and λ>0 such that f(x0)≤infXf+ε . Set α=ε/λ and apply the forward EVP (Theorem 3.1) to obtain x∗∈Xsatisfying f(x∗)≤f(x0),ρ(x0,x∗)≤sλ, and for every y6=x∗, f(y)>f(x∗)−α ρ(x∗,y). If Tx∗=x∗ we are done. Suppose Tx∗6=x∗ . Put y=Tx∗ in the variational inequality to get f(Tx∗)>f(x∗)−α ρ(x∗,Tx∗). Rearrange: f(Tx∗)+ α ρ(x∗,T x∗)>f(x∗). On the other hand, the Caristi condition (6) at x∗implies ρ(x∗,Tx∗)≤f(x∗)−f(Tx∗), i.e. f(Tx∗)+ ρ(x∗,Tx∗)≤f(x∗). Because α>0 and ρ(x∗,Tx∗)≥0 , the two inequalities contradict each other, hence Tx∗=x∗ . Therefore T has a fixed point. On Ekeland Variational Principle in Asymmetric b-Metric Spaces — 27/27 Corollary 3.3 (Backward Caristi’s theorem).Let (X,ρ) be an asymmetric b-metric space with coefficient s ≥1which is forward complete. Let f:X→R be proper, bounded below and forward-lsc. Let T :X→X be a map satisfying ρ(Tx,x)≤f(x)−f(Tx)for every x ∈X. Then T has a fixed point. Proof. The proof follows by applying the backward EVP to the backward-perturbed functionals and duplicating the contradiction argument used in Theorem 3.3. 4. Conclusion In this paper we established new extensions of the Ekeland Variational Principle within the framework of asymmetric b-metric spaces. By introducing forward, backward, and bicomplete versions of EVP, we showed that the asymmetry of the distance leads to distinct variational inequalities, which in turn may yield different minimizers. An illustrative example was provided to highlight the differences between forward and backward EVP. As an application, we derived a forward and backward Caristitype fixed point theorem in asymmetric b-metric spaces. 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