Existence of unique common solution to the system of non-linear integral equations via fixed point results in incomplete metric spaces

Journal of Inequalities and Applications, Jan 2017

In this article, we apply common fixed point results in incomplete metric spaces to examine the existence of a unique common solution for the following systems of Urysohn integral equations and Volterra-Hammerstein integral equations, respectively: u ( s ) = ϕ i ( s ) + ∫ a b K i ( s , r , u ( r ) ) d r , where s ∈ ( a , b ) ⊆ R ; u , ϕ i ∈ C ( ( a , b ) , R n ) and K i : ( a , b ) × ( a , b ) × R n → R n , i = 1 , 2 , … , 6 and u ( s ) = p i ( s ) + λ ∫ 0 t m ( s , r ) g i ( r , u ( r ) ) d r + μ ∫ 0 ∞ n ( s , r ) h i ( r , u ( r ) ) d r , where s ∈ ( 0 , ∞ ) , λ , μ ∈ R , u, p i , m ( s , r ) , n ( s , r ) , g i ( r , u ( r ) ) and h i ( r , u ( r ) ) , i = 1 , 2 , … , 6 , are real-valued measurable functions both in s and r on ( 0 , ∞ ) . MSC: 47H10, 54H25.

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Existence of unique common solution to the system of non-linear integral equations via fixed point results in incomplete metric spaces

Bahadur Zada et al. Journal of Inequalities and Applications Existence of unique common solution to the system of non-linear integral equations via fixed point results in incomplete metric Dir 0 Pakistan 0 Mian Bahadur Zada 0 1 3 Muhammad Sarwar 0 1 3 Stojan Radenovi c´ 0 2 0 University of Malakand , Chakdara 1 Department of Mathematics 2 Faculty of Mechanical Engineering, Universitry of Belgrade , Kraljice Marije 16, Beograd, 11 120 , Serbia 3 Department of Mathematics, University of Malakand , Chakdara, Dir(L) , Pakistan In this article, we apply common fixed point results in incomplete metric spaces to examine the existence of a unique common solution for the following systems of Urysohn integral equations and Volterra-Hammerstein integral equations, respectively: and; common fixed point; weakly compatible maps; common (CLR)-property - → Rn, i = 1, 2, . . . , 6 Ki(s, r, u(r)) dr, n(s, r)hi(r, u(r)) dr, MSC: 47H10; 54H25 1 Introduction and preliminaries © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. In this setting, Azam et al. [] generalized the Banach contraction principle for two selfmaps under rational type contraction. Inspired by the impact of a complex-valued metric space, several authors [, –] proceeded with the investigation of common fixed point results. Many mathematicians applied fixed point methods to the existence of unique solutions to non-linear integral equations, for example, refer to [, , , –]. Particularly, Sintunavarat et al. [] and Rashwan and Saleh [] established fixed point results to find the existence of a unique common solution to a system of Urysohn integral equations. On the other hand, Pathak et al. [] and Rashwan and Saleh [] studied the existence of unique common solution to the system of Volterra-Hammerstein non-linear integral equations. Throughout this manuscript Y represents a complex-valued metric space, unless otherwise specified. For two self-maps f and f defined on a non-empty set Y , w ∈ Y is a common fixed point of f and f if fw = fw = w. To study common fixed points, Jungck [] initiated the concept of weak compatibility of maps thus: f and f on Y are weakly compatible maps if ffw = ffw whenever fw = fw, for some w ∈ Y . In the study of common fixed point results of weakly compatible mappings we often require the assumption of the continuity of mappings or the completeness of the underlying space. Regarding this Aamri and Moutawakil [] relaxed these conditions by introducing the notion of the (E.A)-property. In , the new notion of Common Limit in the Range property (for short (CLR)-property) was given by Sintunavarat and Kumam [], which does not enforce the above mention conditions. Liu et al. [] extended the (E.A)-property [] to the common (E.A)-property and Imdad et al. [] extended the (CLR)-property [] to common (CLR)-property. Sarwar and Bahadur Zada [] defined these views in the complex-valued metric space as follows. Definition . Let f, f, f, f : Y → Y be four maps. If there are two sequences {zn} and {wn} in Y . Then the pairs (f, f) and (f, f) satisfy () the common (E.A)-property if () the common (CLRff )-property if lim fzn = lim fzn = lim fwn = lim fwn = t ∈ Y ; n→∞ n→∞ n→∞ n→∞ lim fzn = lim fzn = lim fwn = lim fwn = t ∈ f(Y ) ∩ f(Y ). n→∞ n→∞ n→∞ n→∞ Note that the (E.A)-property tolerates the condition of closeness of the range of subspaces of the involved mappings. However, the significance of the (CLR)-property reveals that closeness of the range of subspaces is not essential. Sarwar and Bahadur Zada [] established the following common fixed point results. Theorem . Let f, f, f, f, f, and f be six maps on Y such that () f(Y ) ⊆ f(Y ), f(Y ) ⊆ f(Y ), f(Y ) ⊆ f(Y ) and f(Y ) ⊆ f(Y ); () for all u, v ∈ Y and  < k < , d(fu, fv) d(fu, fu)d(fu, fu)d(fu, fv)d(fu, fv)  + d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu) d(fv, fv)d(fv, fv)d(fv, fu)d(fv, fu)  + d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu) () the pairs (f, f), (f, f), (f, f), and (f, f) are weakly compatible; () either both the pairs (f, f) and (f, f) satisfies common (CLRf )-property or both the pairs (f, f) and (f, f) satisfies common (CLRf )-property. Then f, f, f, f, f, and f have a unique common fixed point in Y . Theorem . Let f, f, f, f, f, and f be six maps on Y such that all the conditions of Theorem . except condition () holds. In addition if either the pairs (f, f) and (f, f) or the pairs (f, f) and (f, f) satisfy the common (E.A)-property such that either f(Y ) and f(Y ) or f(Y ) and f(Y ) are closed subspaces of Y , then f, f, f, f, f, and f have a unique common fixed point in Y . 2 Existence of unique common solution to the systems of Urysohn integral equations Ki s, r, u(r) dr, Let us denote u(s) = Assume that the following conditions hold: where i = , , . . . , . u(s) + φ (s) + φ (s) –  i u(s) + φ (s) + φ (s) = ,  i u(s) + φ (s) + φ (s) –  j u(s) + φ (s) + φ (s) = ,  j φ (s) + φ (s) +   j u(s) – φ (s) = u(s), j φ (s) + φ (s) +   i u(s) – φ (s) = u(s). i Let Y = C((a, b), ), a >  be an incomplete complex-valued metric space with metric d(u, v) = s∈(a,b) u(s) – v(s) for all u, v ∈ Y . Define six operators f, f, f, f, f, f : Y → Now, we are in a position to formulate the existence results. () there exist two sequences {zn} and {wn} in Y such that n→∞ n→∞ n→∞ n→∞ f w = z ∈ f(Y );  n s∈(a,b) ϒ) · (max s∈(a,b) ϒ) · (max u(s) – v(s) + φ (s) – φ (s)     ϒ = u(s) – u(s) – u(s) – φ (s) – φ (s)      ϒ = u(s) – u(s) – u(s) – φ (s) – φ (s)      ϒ = u(s) – u(s) – v(s) – φ (s) – φ (s)      ϒ = u(s) – u(s) – v(s) – φ (s) – φ (s)      ϒ = v(s) – v(s) – v(s) – φ (s) – φ (s)      ϒ = v(s) – v(s) – v(s) – φ (s) – φ (s)      ϒ = v(s) – v(s) – u(s) – φ (s) – φ (s)      ϒ = v(s) – v(s) – u(s) – φ (s) – φ (s)      () f (Y) ⊆ f(Y), f (Y) ⊆ f(Y), f (Y) ⊆ f(Y), and f (Y) ⊆ f(Y) such that (f ,f ), (f ,f ),         (f ,f ), and (f ,f ) are weakly compatible.     s∈(a,b) ⎧ d(f u,f v) = max u(s) – v(s) + φ (s) – φ (s)       s∈(a,b) ⎪⎪ d(f v,f u) = max v(s) – v(s) – u(s) – φ (s) – φ (s)       ⎩ s∈(a,b) ⎪ ⎪ ⎪ ⎪ ⎪ ιarctana ⎪  ˙  + a e ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ιarctana ⎪  ˙  + a e ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ιarctana ⎪  ˙  + a e From condition () of Theorem ., we have which implies that sm∈(aa,xb)ϒ√ + aeι˙arctana using (.), we obtain λ (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ)  + (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ) + (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ) d(fu,fu)d(fu,fu)d(fu,fv)d(fu,fv)  + d(fu,fv)d(fu,fv) + d(fv,fu)d(fv,fu) d(fv,fv)d(fv,fv)d(fv,fu)d(fv,fu) +  + d(fu,fv)d(fu,fv) + d(fv,fu)d(fv,fu) . Using (C), we get f(fu(s) + φ(s)) = fu(s), which implies that f(Y) ⊆ f(Y). Similarly, one can prove that f(Y) ⊆ f(Y), f(Y) ⊆ f(Y) and f(Y) ⊆ f(Y). Next, we need to show the weak compatibility of the pair (f,f). For this, we have with the help of (C), we get ffu(s) – ffu(s) = , which implies that ffu(s) = ffu(s), whenever fu(s) = fu(s). Thus (f, f) is weakly compatible. In a similarly way one can easily show the weakly compatibility of the pairs (f, f), (f, f) and (f, f). Also, from condition () of Theorem ., the pairs (f, f) and (f, f) satisfy the common (CLRf )-property. Thus by Theorem . we can find a unique common fixed point of f, f, f, f, f, and f in Y , that is, the system (.) of Urysohn integral equations has a unique common solution in Y . Theorem . Under the assumptions (C)-(C) and the conditions (), () of Theorem ., if there exist two sequences {zn} and {wn} in Y such that lim fzn = lim fzn = lim fwn = lim fwn = z, for some z ∈ Y , n→∞ n→∞ n→∞ n→∞ and both f(Y ) and f(Y ) are closed subspaces of Y , then the system (.) of Urysohn integral equations has a unique common solution. 3 Existence of unique common solution to the systems of Volterra-Hammerstein integral equations In this section, we present the real-valued metric version of Theorem . and Theorem . and the proof can easily be obtained, so we omit its proof here. d(fv, fv)d(fv, fv)d(fv, fu)d(fv, fu)  + d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu) () the pairs (f, f), (f, f), (f, f) and (f, f) are weakly compatible; () either both the pairs (f, f) and (f, f) satisfies common (CLRf )-property or both the pairs (f, f) and (f, f) satisfies common (CLRf )-property. Then f, f, f, f, f, and f have a unique common fixed point in Z. Corollary . Let f, f, f, f, f, f be six maps on a metric space (Z, d) such that all the conditions of corollary . except condition () holds. In addition if either the pairs (f, f) and (f, f) or (f, f) and (f, f) satisfy the common (E.A)-property such that either f(Z) and f(Z) or f(Z) and f(Z) are closed subspaces of Z, then f, f, f, f, f, and f have a unique common fixed point in Z. We apply the above results to study the existence of unique common solution to the following system (.) of non-linear Volterra-Hammerstein integral equations. n(s, r)hi r, u(r) dr, where i = , , . . . , . Assume that (C∗) for i = , , iu(s) = m(s, r)gi r, u(r) dr ∇iu(s) = n(s, r)hi r, u(r) dr, (C∗) for j = , , (C∗) for j = , , (C∗) for i = , , u(s) + ∇u(s) + p(s) + pi(s) – u(s) + ∇u(s) + p(s) + pi(s) – ∇i u(s) + ∇u(s) + p(s) + pi(s) = , u(s) + ∇u(s) + p(s) + pj(s) – u(s) + ∇u(s) + p(s) + pj(s) – ∇j u(s) + ∇u(s) + p(s) + pj(s) = , p(s) + pj(s) +  ju(s) + ∇ju(s) + u(s) + ∇u(s) + p(s) j u(s) – ju(s) – ∇ju(s) – pj(s) + ∇ u(s) + ∇u(s) + p(s) + ∇j u(s) – ju(s) – ∇ju(s) – pj(s) = u(s), p(s) + pi(s) +  iu(s) + ∇iu(s) + u(s) + ∇u(s) + p(s) i u(s) – iu(s) – ∇iu(s) – pi(s) + ∇ u(s) + ∇u(s) + p(s) + ∇i u(s) – iu(s) – ∇iu(s) – pi(s) = u(s). u(s) – v(s) , for all u, v ∈ Z. Define the six operators f, f, f, f, f, and f on Z by fu(s) = u(s) + ∇u(s) + p(s), fu(s) = u(s) + ∇u(s) + p(s), Now, we are in a position to formulate the existence results. Theorem . Under the assumptions (C∗)-(C∗), if () there exist two sequences {zn} and {wn} in Z such that lim fzn = lim fzn = lim fwn = lim fwn = z ∈ f(Z); n→∞ n→∞ n→∞ n→∞ () for each u, v ∈ Z and  < λ < , () f(Z) ⊆ f(Z), f(Z) ⊆ f(Z), f(Z) ⊆ f(Z) and f(Z) ⊆ f(Z) such that the pairs (f, f), (f, f), (f, f) and (f, f) are weakly compatible, then the system (.) of Volterra-Hammerstein equations has a unique common solution. Proof Notice that the system of Volterra-Hammerstein non-linear integral equations (.) has a unique common solution if and only if the system of operators (.) has a unique common fixed point. From condition () of Theorem ., we have ϒ × ϒ × ϒ × ϒ + ϒ × ϒ × ϒ × ϒ  + (max s∈(a,b) ϒ) · (max s∈(a,b) ϒ) · (max  + (max s∈(,∞) ϒ) · (max s∈(,∞) ϒ) · (max  + (max s∈(,∞) ϒ) · (max s∈(,∞) ϒ) · (max which implies that s∈(,∞) using (.), we get d(fu, fu)d(fu, fu)d(fu, fv)d(fu, fv)  + d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu) d(fv, fv)d(fv, fv)d(fv, fu)d(fv, fu)  + d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu) Now, to show that f (Z) ⊆ f(Z), we have  f f u(s) + p (s)    =  f u(s) + p (s) –   = f u(s) + f u(s) + p (s) –    f u(s) + p (s) – ∇ fu(s) + p (s)     – ∇ u(s) + ∇u(s) + p(s) + p(s) u(s) + ∇u(s) + p(s) + p(s) . Using (C∗), we get f(fu(s) + p(s)) = fu(s), which implies that f(Z) ⊆ f(Z). Similarly, one can prove that f(Z) ⊆ f(Z), f(Z) ⊆ f(Z) and f(Z) ⊆ f(Z). Next, we need to show the weak compatibility of the pair (f, f). For this purpose, ffu(s) – ffu(s) = f =  ffu(s) – ffu(s) u(s) + ∇u(s) + p(s) –  u(s) + ∇u(s) + p(s) – ∇ If fu(s) = fu(s), for u(s) ∈ Z. Then u(s) + ∇u(s) + p(s) = u(s) – u(s) – ∇u(s) – p(s), thus the above equation becomes u(s) + ∇u(s) + p(s) – ∇ u(s) + ∇u(s) + p(s) – p(s) u(s) + ∇u(s) + p(s) – ∇ u(s) + ∇u(s) + p(s) –  u(s) – u(s) – ∇u(s) – p(s) – ∇ u(s) – u(s) – ∇u(s) – p(s) , with the help of (C∗), we get ffz(s) – ffz(s) = , which implies that ffz(s) = ffz(s), whenever fz(s) = fz(s). Thus the pair (f, f) is weakly compatible. In a similar way one can easily show the weakly compatibility of the pairs (f, f), (f, f), and (f, f). Also, from condition () of Theorem . the pairs (f, f) and (f, f) satisfy the common (CLRf )-property. Thus by Corollary ., we can find a unique common fixed point of f, f, f, f, f, and f in Z, that is, the system (.) of Volterra-Hammerstein non-linear integral equations has a unique common solution in Z. In the next theorem we use the common (E.A)-property. Theorem . Under the assumptions (C∗)-(C∗) and the conditions (), () of Theorem ., if there exist two sequences {zn} and {wn} in Z such that lim fzn = lim fzn = lim fwn = lim fwn = z, for some z ∈ Z, n→∞ n→∞ n→∞ n→∞ and both f(Z) and f(Z) are closed subspaces of Z, then the system (.) of VolterraHammerstein equations has a unique common solution. Competing interests The authors declare that they have no competing interests regarding this manuscript. Authors’ contributions All authors contributed equally to the writing of this manuscript. All authors read and approved the final version. 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Mian Bahadur Zada, Muhammad Sarwar, Stojan Radenović. Existence of unique common solution to the system of non-linear integral equations via fixed point results in incomplete metric spaces, Journal of Inequalities and Applications, 2017, 22, DOI: 10.1186/s13660-016-1286-7