On Hyers-Ulam-Rassias stability of a Volterra-Hammerstein functional integral equation
Sorina Anamaria Ciplea, Nicolaie Lungu, Daniela Marian, Themistocles M. Rassias
aa r X i v : . [ m a t h . G M ] J a n On Hyers-Ulam-Rassias stability of a Volterra-Hammersteinfunctional integral equation
Sorina Anamaria Ciplea, Nicolaie Lungu, Daniela Marian, Themistocles M. Rassias
Sorina Anamaria Ciplea, Technical University of Cluj-Napoca, Department of Management and Technology, 28Memorandumului Street, 400114, Cluj-Napoca, [email protected] Lungu, Technical University of Cluj-Napoca, Department of Mathematics, 28 Memorandumului Street,400114, Cluj-Napoca, [email protected] Marian, Technical University of Cluj-Napoca, Department of Mathematics, 28 Memorandumului Street,400114, Cluj-Napoca, [email protected] M. Rassias, National Technical University of Athens, Department of Mathematics, Zografou Campus,15780 Athens, [email protected]
Abstract
The aim of this paper is to study Hyers-Ulam-Rassias stability for a Volterra-Hammerstein func-tional integral equation in three variables via Picard operators.
Keywords:
Volterra-Hammerstein functional integral equation; Hyers-Ulam-Rassias stability : 26D10; 34A40; 39B82; 35B20
1. 1. Introduction
Ulam stability is an important concept in the theory of functional equations. The origin ofUlam stability theory was an open problem formulated by Ulam, in 1940, concerning the stabilityof homomorphism [46]. The first partial answer to Ulam’s question came within a year, whenHyers [8] proved a stability result, for additive Cauchy equation in Banach spaces. The firstresult on Hyers-Ulam stability of di ff erential equations was given by Obloza [36]. Alsina and Gerinvestigated the stability of di ff erential equations y ′ = y [2]. The result of Alsina and Ger wereextended by many authors [6], [13], [16], [17], [18], [38], [39], [40], [42], [44], [45] to the stabilityof the first order linear di ff erential equation and linear di ff erential equations of higher order. For Preprint submitted to Approximation and Computation in Science and Engineering , edited by N.J. Daras and Th.M. Rassias, SpringerJanuary 23, 2020 broader study of Hyers-Ulam stability for functional equations the reader is also referred to thefollowing books and papers: [1], [5], [9], [14], [11], [12], [19], [24], [25], [33], [34], [37], [46].The first result proved on the Hyers-Ulam stability of partial di ff erential equations is due toA. Prastaro and Th.M. Rassias [41]. Some results regarding Ulam-Hyers stability of partial dif-ferential equations were given by S.-M. Jung [14], S.-M. Jung and K.-S. Lee [15], N. Lungu andS.A Ciplea [26], N. Lungu and D. Popa [29], [30], [31], N. Lungu, C. Craciun [27], N. Lungu andD. Marian [28], D Marian, S.A. Ciplea and N. Lungu [32], I.A. Rus and N. Lungu [43]. In [4]Brzdek, Popa, Rasa and Xu presented a unified and systematic approach to the field. Some recentresults regarding stability analysis and their applications were established by H. Khan, A. Khan,T. Abdeljawad and A. Alkhazzan [20], A. Khan, J.F. Gmez-Aguilar, T.S. Khan and H. Khan [21],H. Khan, T. Abdeljawad, M. Aslam, R.A. Khan and A. Khan [22], H. Khan, J.F. Gmez-Aguilar,A. Khan and T.S. Khan [23]. Results regarding fixed point theory and the Ulam stability can befound in [3].In this paper we consider the following Volterra-Hammerstein functional integral equation inthree variables u ( x , y , z ) = g ( x , y , z , h ( u ) ( x , y , z )) (1) + Z x Z y Z z K ( x , y , z , r , s , t , f ( u ) ( r , s , t )) drdsdt + Z ∞ Z ∞ Z ∞ F ( x , y , z , r , s , t , f ( u ) ( r , s , t )) drdsdt via Picard operators.The present paper is motivated by a recent paper [35] of L.T.P. Ngoc, T.M. Thuyet and N.T.Long in which is studied the existence of asymptotically stable solution for a Volterra-Hammersteinintegral equation in three variables. Equation (1) is a generalization of equation (1.1) from [35].
2. 2. Existence and uniqueness
In what follows we consider some conditions relative to equation (1).Let ( E , |·| ) be a Banach space and ∆ = n ( x , y , z , r , s , t ) ∈ R + : s ≤ x , r ≤ y , t ≤ z o . Let τ > X τ : = n u ∈ C (cid:16) R + , E (cid:17) | ∃ M ( u ) > | u ( x , y , z ) | e − τ ( x + y + z ) ≤ M ( u ) , ∀ ( x , y , z ) ∈ R + o . X τ we consider Bielecki’s norm k u k τ : = sup x , y , z ∈ R + (cid:16) | u ( x , y , z ) | e − τ ( x + y + z ) (cid:17) . It is clear that ( X τ , k·k τ ) is a Banch space. In what follows we assume, relative to (1), the conditions:(C1) g ∈ C (cid:16) R + × E , E (cid:17) , K ∈ C ( ∆ × E , E ) , F ∈ C ( ∆ × E , E ) , h ∈ C ( X τ , X τ ) , f ∈ C ( X τ , X τ ) , f ∈ C ( X τ , X τ ) ;(C2) there exists l h > | h ( u ) ( x , y , z ) − h ( v ) ( x , y , z ) | ≤ l h k u − v k τ e τ ( x + y + z ) , ∀ x , y , z ∈ R + , ∀ u , v ∈ X τ ;(C3) there exists l g > | g ( x , y , z , e ) − g ( x , y , z , e ) | ≤ l g | e − e | , ∀ x , y , z ∈ R + , ∀ e , e ∈ E ;(C4) there exists l K ∈ C ( ∆ , R + ) such that | K ( x , y , z , r , s , t , e ) − K ( x , y , z , r , s , t , e ) | ≤ l K ( x , y , z , r , s , t ) | e − e | , ∀ ( x , y , z , r , s , t ) ∈ ∆ , ∀ e , e ∈ E ;(C5) there exists l F ∈ C ( ∆ , R + ) such that | F ( x , y , z , r , s , t , e ) − F ( x , y , z , r , s , t , e ) | ≤ l F ( x , y , z , r , s , t ) | e − e | , ∀ ( x , y , z , r , s , t ) ∈ ∆ , ∀ e , e ∈ E ;(C6) there exists l f > l f > | f ( u ) ( r , s , t ) − f ( v ) ( r , s , t ) | ≤ l f | u ( r , s , t ) − v ( r , s , t ) | , ∀ ( r , s , t ) ∈ R + , ∀ u , v ∈ X τ , | f ( u ) ( r , s , t ) − f ( v ) ( r , s , t ) | ≤ l f | u ( r , s , t ) − v ( r , s , t ) | , ∀ ( r , s , t ) ∈ R + , ∀ u , v ∈ X τ ;(C7) there exists l > l > Z x Z y Z z l f l K ( x , y , z , r , s , t ) e τ ( r + s + t ) drdsdt ≤ l e τ ( x + y + z ) , ∀ ( x , y , z , r , s , t ) ∈ ∆ , Z ∞ Z ∞ Z ∞ l f l F ( x , y , z , r , s , t ) e τ ( r + s + t ) drdsdt ≤ l e τ ( x + y + z ) , ∀ ( x , y , z , r , s , t ) ∈ ∆ ;3C8) l g l h + l + l < | g ( x , y , z , h ( u ) ( x , y , z )) | + Z x Z y Z z | K ( x , y , z , r , s , t , f (0) ( r , s , t )) | drdsdt + Z ∞ Z ∞ Z ∞ | F ( x , y , z , r , s , t , f (0) ( r , s , t )) | drdsdt ≤ α exp ( τ ( x + y + z )) , ∀ ( x , y , z , r , s , t ) ∈ ∆ ;(C10) there exists m > Z ∞ Z ∞ Z ∞ h l f l K ( x , y , z , r , s , t ) + l f l F ( x , y , z , r , s , t ) i drdsdt ≤ m , ∀ ( x , y , z , r , s , t ) ∈ ∆ . Theorem 2.1.
Under the conditions ( C − ( C the equation (1) has in X τ a unique solution u ∗ . Proof.
We consider the operator A : X τ → X τ , A ( u ) ( x , y , z ) : = second part of (1) . First we prove that A ( u ) maps X τ in X τ . For u ∈ X τ we have: | A ( u ) ( x , y , z ) | ≤ | g ( x , y , z , h ( u ) ( x , y , z )) | + Z x Z y Z z | K ( x , y , z , r , s , t , f ( u ) ( r , s , t )) | drdsdt + Z ∞ Z ∞ Z ∞ | F ( x , y , z , r , s , t , f ( u ) ( r , s , t )) | drdsdt ≤ | g ( x , y , z , h ( u ) ( x , y , z )) | + Z x Z y Z z | K ( x , y , z , r , s , t , f ( u ) ( r , s , t )) − K ( x , y , z , r , s , t , f (0) ( r , s , t )) | drdsdt + Z ∞ Z ∞ Z ∞ | F ( x , y , z , r , s , t , f ( u ) ( r , s , t )) − F ( x , y , z , r , s , t , f (0) ( r , s , t )) | drdsdt + Z x Z y Z z | K ( x , y , z , r , s , t , f (0) ( r , s , t )) | drdsdt + Z ∞ Z ∞ Z ∞ | F ( x , y , z , r , s , t , f (0) ( r , s , t )) | drdsdt .
4e obtain | A ( u ) ( x , y , z ) | ≤ | g ( x , y , z , h ( u ) ( x , y , z )) | + Z x Z y Z z | K ( x , y , z , r , s , t , f (0) ( r , s , t )) | drdsdt + Z ∞ Z ∞ Z ∞ | F ( x , y , z , r , s , t , f (0) ( r , s , t )) | drdsdt + Z x Z y Z z l K ( x , y , z , r , s , t ) | f ( u ) ( r , s , t ) − f (0) ( r , s , t ) | drdsdt + Z ∞ Z ∞ Z ∞ l F ( x , y , z , r , s , t ) | f ( u ) ( r , s , t ) − f (0) ( r , s , t ) | drdsdt ≤ α exp ( τ ( x + y + z )) + Z x Z y Z z l K ( x , y , z , r , s , t ) l f | u ( r , s , t ) | e τ ( r + s + t ) · e − τ ( r + s + t ) drdsdt + Z ∞ Z ∞ Z ∞ l F ( x , y , z , r , s , t ) l f | u ( r , s , t ) | e τ ( r + s + t ) · e − τ ( r + s + t ) drdsdt ≤ α exp ( τ ( x + y + z )) + k u k τ Z x Z y Z z l K ( x , y , z , r , s , t ) l f e τ ( r + s + t ) drdsdt + k u k τ Z ∞ Z ∞ Z ∞ l F ( x , y , z , r , s , t ) l f e τ ( r + s + t ) drdsdt We have | A ( u ) ( x , y , z ) | ≤ (cid:2) α + k u k τ ( l + l ) exp ( τ ( x + y + z )) (cid:3) , hence A ( u ) ∈ X τ . A is a contraction in X τ with respect to k·k τ . Indeed, for u , v ∈ X τ , we have: | A ( u ) ( x , y , z ) − A ( v ) ( x , y , z ) | ≤ | g ( x , y , z , h ( u ) ( x , y , z )) − g ( x , y , z , h ( v ) ( x , y , z )) | + Z x Z y Z z | K ( x , y , z , r , s , t , f ( u ) ( r , s , t )) − K ( x , y , z , r , s , t , f ( v ) ( r , s , t )) | drdsdt + Z ∞ Z ∞ Z ∞ | F ( x , y , z , r , s , t , f ( u ) ( r , s , t )) − F ( x , y , z , r , s , t , f ( v ) ( r , s , t )) | drdsdt ≤ l g | h ( u ) ( x , y , z ) − h ( v ) ( x , y , z ) | + Z x Z y Z z l K ( x , y , z , r , s , t ) l f | f ( u ) ( r , s , t ) − f ( v ) ( r , s , t ) | drdsdt + Z ∞ Z ∞ Z ∞ l F ( x , y , z , r , s , t ) l f | f ( u ) ( r , s , t ) − f ( v ) ( r , s , t ) | drdsdt ≤ l g l h k u − v k τ e τ ( x + y + z ) + Z x Z y Z z l K ( x , y , z , r , s , t ) l f k u − v k τ e τ ( r + s + t ) drdsdt + Z ∞ Z ∞ Z ∞ l F ( x , y , z , r , s , t ) l f k u − v k τ e τ ( r + s + t ) drdsdt ≤ l g l h k u − v k τ e τ ( x + y + z ) + l k u − v k τ e τ ( x + y + z ) + l k u − v k τ e τ ( x + y + z ) . Then we have: k A ( u ) − A ( v ) k τ ≤ (cid:16) l g l h + l + l (cid:17) k u − v k τ for all u , v ∈ X τ . From ( C
8) we have that A is a contraction. Hence A is a c − Picard operator, with c = − l g l h − l − l . Hence the equation (1) has a unique solution in X τ .
3. Hyers-Ulam-Rassias stability
In what follows we consider the equation u ( x , y , z ) = g ( x , y , z , h ( u ) ( x , y , z )) (2) + Z x Z y Z z K ( x , y , z , r , s , t , f ( u ) ( r , s , t )) drdsdt + Z ∞ Z ∞ Z ∞ F ( x , y , z , r , s , t , f ( u ) ( r , s , t )) drdsdt and the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ( x , y , z ) − g ( x , y , z , h ( u ) ( x , y , z )) − Z x Z y Z z K ( x , y , z , r , s , t , f ( u ) ( r , s , t )) drdsdt −− Z ∞ Z ∞ Z ∞ F ( x , y , z , r , s , t , f ( u ) ( r , s , t )) drdsdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ϕ ( x , y , z ) , (3)6here ( E , |·| ) is a Banach space and ϕ ∈ C (cid:16) [0 , a ) , R + (cid:17) is increasing , g ∈ C (cid:16) [0 , a ) × E , E (cid:17) , K ∈ C (cid:16) [0 , a ) × E , E (cid:17) , F ∈ C (cid:16) [0 , a ) × E , E (cid:17) , h ∈ C ( X τ , X τ ) , f ∈ C ( X τ , X τ ) , f ∈ C ( X τ , X τ ) . Theorem 3.1.
Under the conditions ( C − ( C and(i) there exists N > such that | h ( u ) ( x , y , z ) − h ( v ) ( x , y , z ) | ≤ N | u ( x , y , z ) − v ( x , y , z ) | , ∀ x , y , z ∈ [0 , a ) , ∀ u , v ∈ X τ ; (ii) l g N < ,if u is a solution of (3) and u ∗ is the unique solution of (2) , we have | u ( x , y , z ) − u ∗ ( x , y , z ) | ≤ C KFgh f f ϕ ( x , y , z ) where C KFgh f f ϕ ( x , y , z ) = − l g N exp m − l g N ! , i.e the equation (2) is Hyers-Ulam-Rassias stable.Proof. We have | u ( x , y , z ) − u ∗ ( x , y , z ) |≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ( x , y , z ) − g ( x , y , z , h ( u ) ( x , y , z )) − Z x Z y Z z K ( x , y , z , r , s , t , f ( u ) ( r , s , t )) drdsdt − Z ∞ Z ∞ Z ∞ F ( x , y , z , r , s , t , f ( u ) ( r , s , t )) drdsdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | g ( x , y , z , h ( u ) ( x , y , z )) − g ( x , y , z , h ( u ∗ ) ( x , y , z )) | + Z x Z y Z z | K ( x , y , z , r , s , t , f ( u ) ( r , s , t )) − K ( x , y , z , r , s , t , f ( u ∗ ) ( r , s , t )) | drdsdt + Z ∞ Z ∞ Z ∞ | F ( x , y , z , r , s , t , f ( u ) ( r , s , t )) − F ( x , y , z , r , s , t , f ( u ∗ ) ( r , s , t )) | drdsdt ≤ ϕ ( x , y , z ) + l g | h ( u ) ( x , y , z ) − h ( u ∗ ) ( x , y , z ) | + Z x Z y Z z l K ( x , y , z , r , s , t ) l f | f ( u ) ( r , s , t ) − f ( u ∗ ) ( r , s , t ) | drdsdt + Z ∞ Z ∞ Z ∞ l F ( x , y , z , r , s , t ) l f | f ( u ) ( r , s , t ) − f ( u ∗ ) ( r , s , t ) | drdsdt . i ) , ( ii ) we have: | u ( x , y , z ) − u ∗ ( x , y , z ) | ≤ ϕ ( x , y , z ) + l g N | u ( x , y , z ) − u ∗ ( x , y , z ) | + Z x Z y Z z l K ( x , y , z , r , s , t ) l f | u ( r , s , t ) − u ∗ ( r , s , t ) | drdsdt + Z ∞ Z ∞ Z ∞ l F ( x , y , z , r , s , t ) l f | u ( r , s , t ) − u ∗ ( r , s , t ) | drdsdt ≤ ϕ ( x , y , z ) + l g N | u ( x , y , z ) − u ∗ ( x , y , z ) | + Z ∞ Z ∞ Z ∞ (cid:16) l K ( x , y , z , r , s , t ) l f + l F ( x , y , z , r , s , t ) l f (cid:17) | u ( r , s , t ) − u ∗ ( r , s , t ) | drdsdt . Then (cid:16) − l g N (cid:17) | u ( x , y , z ) − u ∗ ( x , y , z ) |≤ ϕ ( x , y , z ) + Z ∞ Z ∞ Z ∞ (cid:16) l K ( x , y , z , r , s , t ) l f + l F ( x , y , z , r , s , t ) l f (cid:17) | u ( r , s , t ) − u ∗ ( r , s , t ) | drdsdt and we have | u ( x , y , z ) − u ∗ ( x , y , z ) | ≤ ϕ ( x , y , z )1 − l g N + − l g N Z ∞ Z ∞ Z ∞ (cid:16) l K ( x , y , z , r , s , t ) l f + l F ( x , y , z , r , s , t ) l f (cid:17) | u ( r , s , t ) − u ∗ ( r , s , t ) | drdsdt . From Wendorf Lemma [7] for unbounded domain it follows that | u ( x , y , z ) − u ∗ ( x , y , z ) |≤ ϕ ( x , y , z )1 − l g N exp " − l g N Z ∞ Z ∞ Z ∞ (cid:16) l K ( x , y , z , r , s , t ) l f + l F ( x , y , z , r , s , t ) l f (cid:17) drdsdt and we have | u ( x , y , z ) − u ∗ ( x , y , z ) | ≤ − l g N exp " m − l g N · ϕ ( x , y , z )and | u ( x , y , z ) − u ∗ ( x , y , z ) | ≤ C KFgh f f · ϕ ( x , y , z )where C KFgh f f ϕ ( x , y , z ) = − l g N exp m − l g N ! , and the equation (2) is Hyers-Ulam-Rassias stable.8 eferences [1] M. R. Abdollahpour, R. Aghayaria, and M. Th. Rassias, Hyers-Ulam stability of associated Laguerre di ff erentialequations in a subclass of analytic functions, Journal of Mathematical Analysis and Applications, 437(2016),605- 612.[2] C. Alsina and R. Ger, On some inequalities and stability results related to exponential function, J. Inequal. 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