Solution of fractional differential equations via α ψ-geraghty type mappings

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1 Afshari et al. Advances in Difference Equations (8) 8:347 REVIEW Open Access Solution of fractional differential equations via α ψ-geraghty type mappings Hojjat Afshari*, Sabileh Kalantari and Dumitru Baleanu * Correspondence: hojat.afshari@yahoo.com Department of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran Full list of author information is available at the end of the article Abstract Using fixed point results of α ψ -Geraghty contractive type mappings, we examine the existence of solutions for some fractional differential equations in b-metric spaces. By some concrete examples we illustrate the obtained results. Keywords: Fractional differential equation; Normal cone; α ψ -Geraghty contractive type mapping Introduction In, Samet et al. [] presented the concepet of α-admissible mappings, which was expanded by several authors (see [5, 6, 9]). Baleanu, Rezapour, and Mohammadi [3] studied the existence of a solution for problem Dν w(ξ ) = h(ξ, w(ξ )) (ξ [, ], < ν ). Afshari, Aydi, and Karapinar [, ] considered generalized α ψ-geraghty contractive mappings in b-metric spaces. We investigate the existence of solutions for some fractional differential equations in b-metric spaces. We denote I = [, ]. Definition. ([7, ]) The Caputo derivative of order ν of a continuous function h : [, ) R is defined by c Dν h(ξ ) = (n ν) ξ (ξ ζ )n ν h(n) (ζ ) dζ, where n < ν < n, n = [ν], [ν] is the integer part of ν, and (z) = xz e x dx. () Definition. ([7, ]) The Riemann Liouville derivative of a continuous function h is defined by n ξ d h(ζ ) D h(ξ ) = dζ ν n (n ν) dξ (ξ ζ ) ν n = [ν], where the right-hand side is defined on (, ). The Author(s) 8. This article is distributed under the terms of the Creative Commons Attribution 4. International License ( 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.

2 Afshari et al. Advances in Difference Equations (8) 8:347 Page of Let be the set of all increasing continuous functions ψ :[, ) [, ) suchthat ψ(λx) λψ(x) λx for λ >,andletb be the family of nondecreasing functions γ : [, ) [, s )forsomes. Definition.3 ([]) Let (X, d)beab-metric space (with constant s). A function g : X X is a generalized α ψ-geraghty contraction if there exists α : X X [, )suchthat α(z, t)ψ ( s 3 d(gz, gt) ) γ ( ψ ( d(z, t) )) ψ ( d(z, t) ) () for all z, t X,whereγ B and ψ. Definition.4 ([]) Let g : X X and α : X X [, ) begiven.theng is called α-admissible if for z, t X, α(z, t) α(gz, gt). (3) Theorem.5 ([]) Let (X, d) be a complete b-metric space, and let f : X X be a generalized α ψ-geraghty contraction such that (i) f is α-admissible; (ii) there exists u X such that α(u, fu ) ; (iii) if {u n } X, u n u in X, and α(u n, u n ), then α(u n, u). Then f has a fixed point. Main result By X = C(I) we denote the set of continuous functions. Let d : X X [, )begivenby d(y, z)= ( ) (y z). = sup y(ξ) z(ξ) (4) ξ I Evidently, (X, d) isacompleteb-metric space with s =butisnotametricspace. Nowwestudytheproblem D ν Dξ w(ξ)=h( ξ, w(ξ) ), ξ I,3<ν 4, (5) under the conditions w() = w () = w() = w () =, (6) where D ν is the Riemann Liouville derivative, and h : I X R is continuous. Lemma. ([3]) Given h C(I X, R) and 3<ν 4, the unique solution of where D ν Dξ w(ξ)=h( ξ, w(ξ) ), ξ I,3<ν 4, (7) w() = w () = w() = w () =, (8)

3 Afshari et al. Advances in Difference Equations (8) 8:347 Page 3 of is given by w(ξ)= G(ξ, ζ )h(s, w(s)) ds, where G(ξ, ζ )= (ξ ) ν ( ζ ) ν ξ ν [(ζ ξ)(ν )( ξ)ζ ], ζ ξ, ( ζ ) ν ξ ν [(ζ ξ)(ν )( ξ)ζ ], ξ ζ. (9) If h(ξ, w(ξ)) =, then the unique solution of (7) (8) is given by f (ξ)= G(ξ, ζ ) ds = Ɣ(ν ) ξ ν ( ξ). Lemma. ([3]) In Lemma., G(ξ, ζ ) given in (9) satisfies the following conditions: () G(ξ, ζ )>, and G(ξ, ζ ) is continuous for ξ, ζ I; (ν )σ (ξ)ρ(ζ ) () G(ξ, ζ ) r ρ(ζ ), where r = max { ν,(ν ) }, σ (ξ)=ξ ν ( ξ), and ρ(ζ )=ζ ( ζ ) ν. Theorem.3 Suppose (i) there exist θ : R R and ψ such that h(ξ, c) h(ξ, d) Ɣ(ν ) 4ν ψ( c d ) 4 (c d) for ξ I and c, d R with θ(c, d) ; (ii) there exists y C(I) such that θ(y (ξ), G(ξ, ζ )h(ζ, y (ξ)) dζ ), ξ I; (iii) for ξ I and y, z C(I), θ(y(ξ), z(ξ)) implies ( θ G(ξ, ζ )h ( ζ, y(ζ ) ) dζ, G(ξ, ζ )h ( ζ, z(ξ) ) ) dζ ; (iv) if {y n } C(I), y n y in C(I), and θ(y n, y n ), then θ(y n, y). Then problem (7) has at least one solution. Proof By Lemma. y C(I) is a solution of (7) if and only if it is a solution of y(ξ) = G(ξ, ζ )h(ζ, y(ζ )) dζ,andwedefinea: C(I) C(I)byAy(ξ)= G(ξ, ζ )h(ζ, y(ζ )) dζ for ξ I.Forthispurpose,wefindafixedpointofA.Lety, z C(I)besuchthatθ(y(ξ), z(ξ)) forξ I. Using (i), we get Ay(ξ) Az(ξ) = G(ξ, ζ ) ( h ( ζ, y(ζ ) ) h ( ζ, z(ζ ) )) dζ [ G(ξ, ζ ) ] ( ) ( ) h ζ, y(ζ ) h ζ, z(ζ ) dζ [ G(ξ, ζ ) (ψ( (y z) )) 8 4 (y z). Ɣ(ν ) 4ν ψ( y(ζ ) z(ζ) ] ) 4 (y z) dζ

4 Afshari et al. Advances in Difference Equations (8) 8:347 Page 4 of Hence, for y, z C(I)andξ I with θ(y(ξ), z(ξ)), we have (Ay Az) (ψ( (y z) )) 8 4 (y z). Let α : C(I) C(I) [, )bedefinedby, θ(y(ξ), z(ξ)), ξ I, α(y, z)= otherwise. Define γ :[, ) [, )byγ(q)= q and s =. 4 4q So α(y, z)ψ ( 8d(Ay, Az) ) 8α(y, z)ψ ( d(ay, Az) ) = (ψ(d(y, z))) 4ψ(d(y, z)) (ψ(d(y, z))) 4d(y, z) γ (ψ(d(y, z))) γ ( ψ ( d(y, z) )) (ψ(d(y, z))) 4ψ(d(y, z)) γ ( ψ ( d(y, z) )) ψ ( d(y, z) ), γ B. Then A is an α ψ contractive mapping. From (iii) and the definition of α we have α(y, z) θ ( y(ξ), z(ξ) ) θ ( A(y), A(z) ) α ( A(y), A(z) ), for y, z C(I). Thus, A is α-admissible. By (ii) there exists y C(I)suchthatα(y, Ay ). By (iv) and Theorem.5 there is y C(I)suchthaty = Ay.Hencey is a solution of the problem. Corollary.4 Suppose that there exist θ : R R and ψ such that h(ξ, c) h(ξ, d) ψ( c d ) 4 (c d) () for ξ Iandc, d R with θ(c, d). Also, suppose that conditions (ii) (iv) from Theorem.3 hold for h, where G(ξ, ζ ) is given in (9). Then the problem where D 7 Dξ w(ξ)=h( ξ, w(ξ) ), ξ I, () w() = w () = w() = w () =, has at least one solution.

5 Afshari et al. Advances in Difference Equations (8) 8:347 Page 5 of Proof By Lemma. min G(ξ, ζ ) dζ = 5 and max Using ()and(), by Theorem.3we obtain Ay(ξ) Az(ξ) (ψ( y z )) 8 4 (y z). G(ξ, ζ ) dζ =4 3. () TherestoftheproofisaccordingtoTheorem.3. Lemma.5 ([8]) If h C(I X, R) and h(ξ, w(ξ)), then the problem D ν w(ξ)=h( ξ, w(ξ) ), (<ξ <,3<ν 4), w() = w () = w () = w () = (3) has a unique positive solution w(ξ)= where G(ξ, ζ ) is given by G(ξ, ζ )h ( ζ, w(ζ ) ) dζ, G(ξ, ζ )= ξ ( ζ ) ν 3 (ξ ζ ) ν, ζ ξ, ξ ( ζ ) ν 3, ξ ζ. (4) Lemma.6 ([]) The function G(ξ, ζ ) in Lemma.5 has the following property: ζ ( ζ )( ζ )ν 3 ξ ν G(ξ, ζ ) ( ζ )ν 3 ξ ν, where ξ, ζ Iand3<ν 4. Based on Theorem.3, we get the following result. Corollary.7 Assume that there exist θ : R R and ψ such that h(ξ, c) h(ξ, d) ψ( c d ) M 4 (c d), where M = sup ξ I G(ξ, ζ ) dζ. Also, suppose that conditions (ii) (iv) from Theorem.3 are satisfied, where G(ξ, ζ ) is given in (4). Then problem (3) has at least one solution. Proof By Lemma.5 y C(I) is a solution of (3) if and only if a solution of y(ξ) = G(ξ, ζ )h(ζ, y(ζ )) dζ.definea: C(I) C(I) byay(ξ)= G(ξ, ζ )h(ζ, y(ζ )) dζ for ξ I. We find a fixed point of A.Lety, z C(I)besuchthatθ(y(ξ), z(ξ)) forξ I.By(i)and

6 Afshari et al. Advances in Difference Equations (8) 8:347 Page 6 of Lemma.6 we get Ay(ξ) Az(ξ) = G(ξ, ζ ) ( h ( ζ, y(ζ ) ) h ( ζ, z(ζ ) )) dζ [ G(ξ, ζ ) ] ( ) ( ) h ζ, y(ζ ) h ζ, z(ζ ) dζ [ G(ξ, ζ ) [ [ [ M ψ( y(ζ ) z(ζ) ] ) 4 (y z) dζ ] G(ξ, ζ ) ψ( y(ζ ) z(ζ ) ) (sup ξ I G(ξ, ζ ) dζ ) 4 (y z) dζ ] G(ξ, ζ ) ( ψ( y(ζ ) z(ζ ) ) G(ξ, ζ ) dζ ) 4 (y z) dζ ( ζ )ν 3 ξ ν ( ζ ( ζ )( ζ )ν 3 ξ ν dζ ) ψ( y(ζ ) z(ζ ] ) ) 4 (y z) dζ (ψ( (y z) )) 8 4 (y z). Suppose that conditions (ii) (iv) from Theorem.3 are satisfied, where G(ξ, ζ ) is given in (4). By Theorem.3 problem (3) has at least one solution. Let (X, d) be given in (4). For the equation c D ν y(ξ)=h ( ξ, y(ξ) ), (ξ I,<ν ), (5) via y() =, y() = y(ζ ) dζ ( < η <), where h : I X R is continuous, we have the following result. Theorem.8 Assume that there exist θ : R R, γ B, and ψ such that h(ξ, c) h(ξ, d) Ɣ(ν ) 5 8 γ ( ψ ( c d )) ψ ( c d ). Suppose conditions (ii) (iv) from Theorem.3 hold, where A : C(I) C(I) is defined by Ay(ξ):= (ξ ζ ) ν h ( ζ, y(ζ ) ) ξ dζ ( ζ ) ν h ( ζ, y(ζ ) ) dζ ( η ) ( ζ (ζ n) ν h ( n, y(n) ) ) dn dζ (ξ I); ξ ( η ) Then (5) has at least one solution.

7 Afshari et al. Advances in Difference Equations (8) 8:347 Page 7 of Proof Afunctiony C(I) is a solution of (5) if and only if it is a solution of y(ξ)= (ξ ζ ) ν h ( ζ, y(ζ ) ) ξ dζ ( ζ ) ν h ( ζ, y(ζ ) ) dζ ( η ) ( ζ (ζ n) ν h ( n, y(n) ) ) dn dζ (ξ I). ξ ( η ) Then (5) is equivalent to finding y C(I)thatisafixedpointofA.Lety, z C(I)with θ(y(ξ), z(ξ)), ξ I.By(i)wehave Ay(ξ) Az(ξ) = ξ ( η ) ξ ( η ) Ɣ(α) (ξ ζ ) ν h ( ζ, y(ζ ) ) dζ ( ζ ) ν h ( ζ, y(ζ ) ) dζ ( ζ (ξ ζ ) ν h ( ζ, z(ζ ) ) dζ (ζ n) ν h ( n, y(n) ) ) dn dζ ξ ( ζ ) ν h ( ζ, z(ζ ) ) dζ ( η ) ξ η ( ζ (ζ n) ν h ( n, z(n) ) ) dn dζ ( η ) ξ ζ ν ( ) ( ) h ζ, y(ζ ) h ζ, z(ζ ) dζ ξ ( η ) ζ ζ ν h ( ζ, y(ζ ) ) h ( ζ, z(ζ ) ) dζ ξ ( η ) ζ n ν ( ) ( ) h n, y(n) h n, z(n) dn dζ ν Ɣ(ν ) ξ ζ 5 8 γ ( ψ ( y(ζ ) z(ζ) )) ( ψ y(ζ ) z(ζ) ) dζ ξ ν Ɣ(ν ) ζ ( η ) 5 8 γ ( ψ ( y(ζ ) z(ζ) )) ( ψ y(ζ ) z(ζ) ) dζ ζ ξ ζ n ν ( η ) 8 γ ( ψ ( y(n) z(n) )) ( ψ y(n) z(n) ) dζ ( ) Ɣ(ν ) 5 8 γ ( ψ ( y z )) ( ) [ ψ y z sup( ξ ζ ν dζ ξ ( η ) ζ ν dζ

8 Afshari et al. Advances in Difference Equations (8) 8:347 Page 8 of ξ ( η ) ( ζ 8 γ ( ψ ( y z )) ( ) ψ y z ) )] ζ n ν dn dζ for all y, z C(I)withθ(y(ξ), z(ξ)), ξ I,sothat (Ay Az) 8 γ ( ψ ( y z )) ψ ( y z ). Let α : C(I) C(I) [, )bedefinedby θ(y(ξ), z(ξ)), ξ I, α(y, z)= otherwise. Then α(y, z)ψ ( 8d(Ay, Az) ) 8α(y, z)ψ ( d(ay, Az) ) α(y, z)ψ ( γ ( ψ ( d(y, z) )) ψ ( d(y, z) )) γ ( ψ ( d(y, z) )) ψ ( d(y, z) ) for all y, z C(I), and thus A is an α ψ contractive mapping. From Theorem.5,based on the proof of Theorem.3, we can deduce the proof of Theorem.8. Here we find a positive solution for where c D ν Dξ w(ξ)=h( ξ, w(ξ) ), <ν, ξ I, (6) w() w(ζ ) dζ = w(). Note that c D ν is the Caputo derivative of order ν. We consider the Banach space of continuous functions on I endowed with the sup norm. We have the following lemma. Lemma.9 ([4]) Let <ν and h C([, T] X, R) be given. Then the equation c D ν w(ξ)=h ( ξ, w(ξ) ) ( ξ [, T], T ) with w() T w(ζ ) dζ = w(t) has a unique solution given by w(ξ)= T G(ξ, ζ )h ( ζ, w(ζ ) ) dζ,

9 Afshari et al. Advances in Difference Equations (8) 8:347 Page 9 of where G(ξ, ζ ) is defined by G(ξ, ζ )= (T ζ ) ν νt(ξ ζ ) ν TƔ(ν) (T ζ ) ν TƔ(ν) (T ζ )ν, T ζ < ξ, (T ζ )ν, T ξ ζ < T. (7) By Lemma.9 and Theorem.4 we get the following conclusion. Corollary. Assume that there exist θ : R R and ψ such that h(ξ, c) h(ξ, d) ψ( c d ) 4 (c d) for ξ Iandc, d R with θ(c, d). Suppose conditions (ii) (iv) from Theorem.3 are satisfied, where G(ξ, ζ ) is given in (7). Then the following problem has at least one solution: c D w(ξ)=h ( ξ, w(ξ) ), ( ) ξ [, ], w() w(ζ ) dζ = w(). Proof It is easily that min t [,] G(t, s) ds = 3 and max 8 t [,] G(t, s) ds = 5.ByTheorem.3 we conclude the desired result. Example. Let ψ(r) =r, θ(x, z) =xz, andy n (ξ) = ξ. We consider h : I [, ] n [, ] and the periodic boundary value problem with D 7 Dξ w(ξ)=h( ξ, w(ξ) ) = w(ξ), ξ I, (8) w() = w () = w() = w () =. Then h(ξ, c) h(ξ, d) 3 = c d 4 ψ( c d ) 8 4 (c d) for ξ I and c, d [, ] with θ(c, d). Because y (ξ)=ξ,thus ( θ y (ξ), G(ξ, ζ )h ( ζ, y (ζ ) ) ) dζ for all ξ I.Also,θ(y(ξ), z(ξ)) = y(ξ)z(ξ) impliesthat ( θ G(ξ, ζ )h ( ζ, y ( ζ ) ) dζ, G(ξ, ζ )h ( ζ, z(ζ ) ) ) dζ. It is obvious that condition (iv) in Corollary (.4) holds. Hence by Corollary.4 problem (8) has at least one solution.

10 Afshari et al. Advances in Difference Equations (8) 8:347 Page of Acknowledgements Not applicable. Funding Not applicable. Competing interests The authors declare that they have no competing interests. Authors contributions All authors read and approved the final manuscript. Author details Department of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran. Department of Mathematics, Cankaya University, Ankara, Turkey. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 5 June 8 Accepted: 8 September 8 References. Afshari, H., Aydi, H., Karapinar, E.: Existence of fixed points of set-valued mappings in b-metric spaces. East Asian Math. J. 3(3), (6). Afshari, H., Aydi, H., Karapinar, E.: On generalized α ψ-geraghty contractions on b-metric spaces. Georgian Mathematical Journal, (8) In press 3. Baleanu, D., Rezapour, S., Mohammadi, H.: Some existence results on nonlinear FDEs. rsta.royalsocietypublishing.org, on May 8, 7 4. Benchohra, M., Ouaar, F.: Existence results for nonlinear fractional differential equations with integral boundary conditions. Bull. Math. Anal. Appl. 4,7 5 () 5. Bota, M.-F., Chifu, C., Karapinar, E.: Fixed point theorems for generalized (α ψ)-ciric-type contractive multivalued operators in b-metric spaces. J. Nonlinear Sci. Appl. 9(3), (6) 6. Bota, M.-F., Karapinar, E., Mlesnite, O.: Ulam Hyers stability results for fixed point problems via alpha-psi-contractive mapping in b-metric space. Abstr. Appl. Anal. 3,Article ID8593 (3) 7. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (6) 8. Liang, S., Zhang, J.: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 7, (9) 9. Karapınar, E., Samet, B.: Generalized α ψ-contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal.,Article ID (). Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (999). Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for α ψ-contractive type mappings. Nonlinear Anal. 75, (). Wang, H., Zhang, L.: The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems. Bound. Value Probl. 5,3 (5) 3. Xu, X., Jiang, D., Yuan, C.: Multiple positive solutions for the BVPs of a nonlinear fractional differential equation. Nonlinear Anal. TMA 7, (9)