On local attractivity of nonlinear functional integral equations via measures of noncompactness

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1 Malaya J. Mat On local attractivity of nonlinear functional integral equations via measures of noncompactness B. C. Dhage a,, S. B. Dhage a, S. K. Ntouyas b, and H. K. Pathak c, a Kasubai, Gurukul Colony, Ahmedpur , Dist: Latur, Maharashtra, India. b Department of Mathematics, University of Ioannina, Ioannina, Greece. c School of Studies in Mathematic, Pt. Ravishankar Shukla University, Raipur C.G. 4921, India. Abstract In this paper, we prove the local attractivity of solutions for a certain nonlinear Volterra type functional integral equations. We rely on a measure theoretic fixed point theorem of Dhage 28 for nonlinear D-setcontraction in Banach spaces. Finally, we furnish an example to validate all the hypotheses of our main result and to ensure the existence and ultimate attractivity of solutions for a numerical nonlinear functional integral equation. Keywords: solutions. Measure of noncompactness, fixed point theorem, functional integral equation, attractivity of 21 MSC: 45G1, 45G99. c 212 MJM. All rights reserved. 1 Introduction The last three decades witnessed the active area of research in the connotation of measure theoretic fixed point theory and its applications to the problems of nonlinear differential and integral equations. The novelty of this approach lies in the advantage that along with existence we obtain some additional information about some characterizations of the solutions automatically. Local and global stability of the solutions of certain functional integral equations is discussed via measures of noncompactness by many researchers see, for instance, Banas and Goebel [3], Banas and Rzepka [4], Dhage [8, 9], Dhage and Ntouyas [13] and the references therein. Very recently, Dhage [8] derived an abstract fixed point result more general than Darbo [5] fixed point theorem using the notion of measures of noncompactness and applied to stability problem of certain nonlinear functional integral equations. See Dhage and Lakshmikantham [12] and the references therein. Inspired or motivated by the idea of D-functions that given in the examples of Dhage [1, 11], we prove in this paper the local attractivity of solutions for a certain nonlinear Volterra type functional integral equations via Dhage s measure theoretic fixed point theorem. We now consider the following generalized nonlinear functional integral equation in short GNFIE γt xt = ut, xt p f t, s, xθsds, gt, s, xηsds 1.1 Corresponding author. Member of Nonlinear Analysis and Applied Mathematics NAAM-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia. This research was supported by the University Grant Commission, New Delhi, India MRP-MAJOR-MATH address: bcdhage@gmail.com B. C. Dhage, sbdhage4791@gmail.coms. B. Dhage, sntouyas@uoi.gr S. K. Ntouyas, hkpathak5@gmail.com, H. K. Pathak.

2 192 B. C. Dhage et al. / On local attractivity of... for t R = [, R, where u : R R R, f, g : R R R R, p : R R R and γ, θ, σ, η : R R are continuous functions. Notice that the functional integral equation 1.1 is general in the sense that it includes several classes of known integral equations discussed in the literature see Banas and Rzepka [4], Dhage [8], Dhage and Ntouyas [13], O Regan and Meehan [15], Krasnoselskii [14], Väth [16], Dhage [7, 8] and the references therein. In this paper, we intend to obtain solution of GNFIE 1.1 in the space BCR, R of all bounded and continuous real-valued functions on R. We use a fixed point theorem of Dhage [8] involving general measures of noncompactness to prove the existence and ultimate attractivity of solutions of GNFIE 1.1 under certain new conditions. The results of this paper are new to the theory of nonlinear differential and integral equations. 2 Auxiliary Results This section is devoted to presenting a few auxiliary results needed in the sequel. Assume that E is a Banach space with the norm and the zero element θ. Denote by B[x, r] the closed ball centered at x and with radius r. If X, Y are arbitrary subsets of E then the symbols λx and X Y stand for the usual algebraic operations on those sets. Moreover, we write X, co X to denote the closure and the closed convex hull of X, respectively. Further, let P p E denote the class of all nonempty subsets of E with a property p. Here p may be p =closed cl, in short, p =bounded bd, in short, p =relatively compact rcp, in short etc. Thus, P cl E, P bd E, P cl,bd E and P rcp E denote respectively the classes of closed, bounded, closed and bounded and relatively compact subsets of E. The axiomatic way of defining the concept of the measure of noncompactness has been adopted in several papers in the literature. See Akhmerov et al. [2], Deimling [6], Väth [16] and Zeidler [17]. In this paper, we adopt the following axiomatic definition of the measure of noncompactness in a Banach space given in Banas and Goebel [3] and Dhage [7, 8]. Definition 2.1. A mapping µ : P bd E R is called the measure of noncompactness in E if it satisfies the following conditions: 1 o The family ker µ = {X P bd E : µx = } is nonempty and ker µ P rcp E. 2 o µx = µx. 3 o µco X = µx. 4 o X Y µx µy. 5 o If {X n } is a decreasing sequence of sets in P cl,bd E such that lim µx n =, then the intersection set X = n X n is nonempty. n=1 The family ker µ described in 1 o is said to be the kernel of the measure of noncompactness µ. We refer to [2, 3, 4, 6, 16, 17] for further facts concerning the measures of noncompactness and their properties. Let us only observe that the intersection set X is a member of the family ker µ. Indeed, since µx µx n for any n, we infer that µx =. In view of 1 o this yields that X ker µ. A measure µ of noncompactness is said to be sublinear if 6 o µx Y µa µb for all X, Y P bd E, and 7 o µλx λ µx for all λ R and X P bd E. Let E = BCR, R be the space of all continuous and bounded functions on R and define a norm in E by x = sup{ xt : t }.

3 B. C. Dhage et al. / On local attractivity of Clearly E is a Banach space with this supremum norm. Let us fix a bounded subset A of E and a positive real number T. For any x A and ɛ, denote by ω T x, ɛ, the modulus of continuity of x on the interval [, T] defined by ω T x, ɛ = sup{ xt xs : t, s [, T], t s ɛ}. Moreover, let ω T A, ɛ = sup{ω T x, ɛ : x A}, ω T A = lim ɛ ωt A, ɛ, ω A = lim T ωt A. By At we mean a set in R defined by At = {xt x A} for t R. We denote diam At = sup{ xt yt : x, y A}. Finally we define a function µ on P bd E by the formula µa = ω A lim sup diamat. 2.2 t It has been shown in Banas and Goebel [3] that µ is a sublinear measure of noncompactness in E. From the definition of the measure µ, it is clear that the thickness of the bundle of functions At tends to zero as t tends to. This particular characteristic of µ has been utilized in formulating the main existence and attractiivity result of this paper. Before going to the key tool used in this paper, we recall the following useful definition introduced by Dhage [8]. Definition 2.2. A mapping T : E E is called D-set-Lipschitz if there exists a upper semi-continuous nondecreasing function ϕ : R R such that µt A ϕµa for all A P bd E with T A P bd E, where ϕ =. The function ϕ is sometimes called a D-function of T on E. Especially when ϕr = kr, k >, U is called a k-set-lipschitz mapping and if k < 1, then T is called a k-set-contraction on E. Further, if ϕr < r for r >, then T is called a nonlinear D-set-contraction on E. Lemma 2.1 Dhage [8]. If ϕ is a D-function with ϕr < r for r >, then lim n ϕ n t = for all t [, and vice-versa. Using Lemma 2.1, Dhage [8] proved the following important result. Theorem 2.1. Let C be a closed, convex and bounded subset of a Banach space E and let T and nonlinear D-set-contraction. Then T has a fixed point. : C C be a continuous Remark 2.1. Let us denote by FixT the set of all fixed points of the operator T which belong to C. It can be shown that the set FixT existing in Theorem 2.1 belongs to the family ker µ. Indeed, if FixT ker µ, then µfixt > and T FixT = FixT. Now from nonlinear set-contractivity it follows that µt FixT φµfixt which is a contradiction since φr < r for r >. Hence Fix T ker µ. This particular characteristic has been utilized in our study of local attractivity of the solutions of nonlinear integral equations. 3 Local Attractivity Results In this section we prove our main existence and attractivity results for the GNFIE 1.1 under some suitable conditions. We need the following definition in what follows. Let us assume that E = BCR, R and let Ω be a subset of E. Let T : E E be an operator and consider the operator equation in E, T xt = xt for all t R. 3.3 Below we give an attractivity characterizations of the solutions for the operator equation 3.3 on R.

4 194 B. C. Dhage et al. / On local attractivity of... Definition 3.3. We say that solutions of the equation 3.3 are locally ultimately attractive if there exists a closed ball B[x, r ] in the space BCR, R for some x BCR, R such that, for arbitrary solutions x = xt and y = yt of equation 3.3 belonging to B[x, r ] Ω, we have lim xt yt =. 3.4 In case the limit 3.2 is uniform with respect to the set B[x, r ] Ω, i.e., for each ɛ > there exists T > such that xt yt ɛ 3.5 for all solutions x, y B[x, r ] Ω of 3.3 and for t T, we will then say that solutions of equation 3.3 are uniformly locally ultimately attractive on R. We consider the following set of hypotheses in the sequel. H The functions γ, θ, σ, η : R R are continuous and lim t γt =. H 1 The function u : R R R is continuous and there exists a continuous and nondecreasing function ϕ : R R such that ut, x ut, y ϕ x y for each t R and x, y R. Moreover, we assume ϕr < r for r >. H 2 The function U : R R defined by Ut = ut, is bounded with c 1 = sup t Ut. H 3 The function f : R R R R is continuous and there exist a constant k > and a function ϕ as appears in H 1 such that for t, s R and x, y R. H 4 The function F : R R defined by Ft = f t, s, x f t, s, y k ϕ x y γt f t, s, ds is bounded with c 2 = sup t Ft. H 5 The function g : R R R R is continuous and there exist functions a, b : R R satisfying for t, s R and x R. Moreover, lim t at gt, s, x atbs bsds =. H 6 The function p : R R R satisfies the following condition for all t 1, t 2, t 1, t 2 R. Moreover, p, =. pt 1, t 2 pt 1, t 2 t 1 t 1 t 2 t 2 Remark 3.2. Since the hypothesis H holds, there exists a constant c > such that c = sup t γt. Similarly, since H 5 holds, the function v : R R defined by vt = at bsds is continuous and the number c 3 = sup t vt exists. Theorem 3.2. Assume that the hypotheses H H 6 hold. Further if there exists a positive solution r of the inequality 1 c kϕr q r, 3.6 where q is the constant defined by q = 3 i=1 c i, then the GNFIE 1.1 has a solution and the solutions are uniformly locally ultimately attractive on R.

5 B. C. Dhage et al. / On local attractivity of Proof. Now consider the closed ball B[, r ] in E centered at origin of radius r. Define the mapping T on E by γt T xt = ut, xt p f t, s, xθsds, gt, s, xηsds for t R. We shall show that the map T satisfies all the conditions of Theorem 3.1 on E. Step I: First we show that T defines a mapping T : E E. Since p, q, γ, σ are continuous, T x is continuous and hence it is measurable on R for each x E. As θr R, we have max t xθt max t xt. On the other hand, hypotheses H H 3 and H 5 imply that γt T xt ut, xt p f t, s, xθsds, ut, xt ut, ut, ϕ xt ut, ϕ xt ut, γt γt γt f t, s, xθs ds ϕ x Ut kγtϕ x c 2 vt 1 c kϕ x q, for all t R. Taking supemum over t, we obtain, gt, s, xηsds f t, s, xθsds f t, s, xθs f t, s, ds p, gt, s, xηs ds γt gt, s, xηsds 3.7 f t, s, ds atbsds T x 1 c kϕ x q r. 3.8 From 3.7, we deduce that T x E and T defines a mapping T : B[, r ] B[, r ]. Step II: We show that T is continuous on B[, r ]. Let ɛ > be given and let x, y B[, r ] be such that x y ɛ. Then by hypotheses H 1 H 5 γt T xt T yt ut, xt ut, yt p f t, s, xθsds, gt, s, xηsds γt p f t, s, yθsds, ϕ xt yt ϕ xt yt γt gt, s, yηsds [ f t, s, xθs f t, s, yθs]ds [gt, s, xηs gt, s, yηs]ds γt f t, s, xθs f t, s, yθs ds gt, s, xηs gt, s, yηs ds ϕ x y kγtϕ x y 2 1 c kϕɛ 2vt atbsds 1 c kɛ 2vt. 3.9 Since vt as t, there exists T > such that vt ɛ, t > T. Thus if t > T, then from 3.8 we have that T xt T yt 3 c kɛ. If t < T, then define a function ω = ωɛ by the formula ωɛ = sup{ gt, s, x gt, s, y : t, s [, T], x, y [ r, r ], x y ɛ}. 3.1 Now gt, s, x is continuous and hence uniformly continuous on [, T] [, T] [ r, r ]. As a result we have ωɛ as ɛ. Therefore, from 3.1,

6 196 B. C. Dhage et al. / On local attractivity of... T xt T yt 1 c kɛ σ ωɛ for all t R, where σ = max{σt : t [, T]}. Hence, it follows that T x T y max{3 c kɛ, 1 c kɛ σ ωɛ} as ɛ. Hence T is a continuous mapping from B[, r ] into itself. Step III: Here we show that T is a nonlinear set-contraction on B[, r ]. This will be done in the following two cases: Case I : Let A B[, r ] be non-empty. Further fix the number T > and ɛ >. Since the functions f and g are continuous on compact [, T] [, T] [ r, r ], there are constants c 4 > and c 5 > such that f t, s, x c 4 and gt, s, x c 5 for all t, s [, T] and x [ r, r ]. Then choosing t, τ [, T] such that t τ ɛ and taking into account our hypotheses, we obtain where γt T xt T xτ ut, xt uτ, xτ p στ γτ p f τ, s, xθsds, gτ, s, xηsds ut, xt ut, xτ ut, xτ uτ, xτ γt f t, s, xθsds γτ στ gt, s, xηsds f t, s, xθsds, gt, s, xηsds f τ, s, xθsds gτ, s, xηsds ϕ xt xτ ut, xτ uτ, xτ γt γt f t, s, xθsds f τ, s, xθsds γt γτ f τ, s, xθsds f τ, s, xθsds στ gt, s, xηsds σt στ [gt, s, xηsds gτ, s, xηs]ds ϕ xt xτ ut, xτ uτ, xτ γt γt γτ στ f t, s, xθs f τ, s, xθs ds στ f τ, s, xθs ds gt, s, xηs ds σt gt, s, xηsds gτ, s, xηs ds ϕ xt xτ ω T u, ɛ kω T f, ɛ c 4 ω T γ, ɛ Tω T g, ɛ c 5 ω T σ, ɛ, ω T γ, ɛ = sup{ γt γτ : t, τ [, T], t τ ɛ}, ω T σ, ɛ = sup{ σt στ : t, τ [, T], t τ ɛ},

7 B. C. Dhage et al. / On local attractivity of ω T u, ɛ = sup{ ut, x uτ, x : t, τ [, T], t τ ɛ, x r }, ω T f, ɛ = sup{ f t, s, x f τ, s, x : t, τ [, T], t τ ɛ, x r }, ω T g, ɛ = sup{ gt, s, x gτ, s, x : t, τ [, T], t τ ɛ, x r }. The above inequality further implies that ω T T x, ɛ ϕω T x, ɛ ω T u, ɛ c kω T f, ɛ c 4 ω T γ, ɛ Tω T g, ɛ c 5 ω T σ, ɛ Since by hypotheses, the functions u, ϕ, γ, σ and f, g are continuous respectively on [, T] and [, T] [, T] [ r, r ], we infer that they are uniformly continuous there. Hence we deduce that ϕω T x, ɛ, ω T u, ɛ, ω T γ, ɛ, ω T f, ɛ, ω T g, ɛ as ɛ. Hence from the above estimate 3.11, we obtain and consequently Case II: Now for any x, y A one has: ω T T A =, γt T xt T yt ut, xt ut, yt p γt p f t, s, yθsds, ϕ xt yt ϕdiam At ω T A = γt f t, s, xθsds, gt, s, xηsds gt, s, yηsds f t, s, xθs f t, s, yθsds gt, s, xηsds gt, s, yηsds γt f t, s, xθs f t, s, yθs ds gt, s, xηsds gt, s, yηs ds ϕdiam At k ϕdiam At k ϕdiam At k As a result of the above inequality we obtain γt γt γt xθs yθs ds 2vt diam Aθsds 2vt diam A ds 2vt ϕdiam At kγtdiam A ds 2vt. diam T At ϕdiam At kγtdiam A 2vt. Taking the limit superior as t in the above inequality yields lim sup diam T At ϕ lim sup diam At k lim sup γt diam A 2 lim sup vt. Since both the limits, namely lim vt and lim γt exist and each one is equal to, it follows that lim sup vt = and lim sup γt =. Hence, from the above inequality, we have lim sup diam T At ϕ lim sup diam At. 3.13

8 198 B. C. Dhage et al. / On local attractivity of... Now from the inequalities 3.12, 3.13 and the definition of µ it follows that µt A = ω T A lim sup diam T At ϕ lim sup diam At ϕ ω A lim sup diam At, or, equivalently, µt A ϕµa, 3.14 where µ is the measure of noncompactness defined in the space BCR, R. This shows that T is a nonlinear D-set-contraction on B[, r ]. Thus, the map T satisfies all the conditions of Theorem 2.2 with C = B[, r ] and an application of it yields that T has a fixed point in B[, r ]. This further by definition of T implies that the GNFIE 1.1 has a solution in B[, r ]. Moreover, taking into account that the image of B[, r ] under the operator T is again contained in the ball B[, r ] we infer that the set FT of all fixed points of T is contained in B[, r ]. If the set FT contains all solutions of the equation 1.1, then we conclude from Remark 2.1 that the set FT belongs to the family ker µ. Now, taking into account the description of sets belonging to ker µ given in Section 2 we deduce that all solutions of the equation 1.1 are uniformly locally ultimately attractive on R. This completes the proof. 4 An Example As an application, we consider the following nonlinear functional integral equation for all t R. xt = t ln 12 xt t 1t t 2 t t 1 t t 2 ln 1 12 x ds exp t 2 s 2 cos xs ds, sin xs Let pt, t = t t, ϕt = ln 1 12 t, θt = t 2 1, σt = t, ηt = t, 1 t γt = t2 1 t 3, ut, x = t ln 12 xt, at = 1 t 3 exp t, bs = s 2, for all t, t, s R, and f t, s, x = 1 t 1 t t 2 ln 1 12 x, for all t, s R and x R. Notice that: gt, s, x = 1 t 3 s 2 cos xs exp t 1 sin xs t 2 i The functions γ, θ, σ, η are obviously continuous and lim γt = lim 1 t 3 =. Also c = sup γt = sup t t 2 1 t t

9 B. C. Dhage et al. / On local attractivity of ii For arbitrary fixed x, y R we have ut, x ut, y = 1 1 t ln x ln ln x 1 1 ln 2 y < ln 1 12 x y = ϕ x y y x y y Therefore, hypothesis H 1 is satisfied with ϕr = ln 1 12 r < r, for r >. iii H 2 is satisfied since Ut = ut, = and c 1 = sup t ut, =. iv For arbitrary fixed x, y R such that x y and for t > we obtain 1 t f t, s, x f t, s, y = 1 t t 2 ln x 1 1 ϕ x y, 2 y as in ii. The case is similar when y x. Thus H 3 is satisfied with k = 1 and ϕr = ln 1 12 r < r, for r >. v Next, hypothesis H 4 is satisfied, since the function F : R R defined by Ft = is bounded with c 2 = sup t Ft =. γt f t, s, ds = t 2 t 3 1 ds = vi The function g acts continuously from the set R R R into R. Moreover, we have gt, s, x 1 t 3 exp t s 2 = atbs, for all t, s R and x R, then we can see that hypothesis H 5 is satisfied. Indeed, we have Also we have c 3 = sup t lim at t 1t 1 3 t3 exp t.37. vii Obviously, hypothesis H 6 is satisfied. bsds = lim 1 t 3 exp t = 1 3 lim t3 exp t =. t 1t s 2 ds The inequality reduces to the form 1 c kϕr q r ln 1 12 r.37 r. It is easily seen that each number r.6 satisfies the above inequality. Thus, as the number r we can take r =.6. Note that this estimate of r can be improved. Keeping in view the above observations, we find that the functions γ, ϕ, θ, σ, η, u, f, g,a and b satisfy all the conditions of Theorem 3.2 and hence the GNFIE 4.1 has at least one solution in the space BCR, R and the solutions of the equation 4.1 are uniformly locally ultimately attractive on R located in the ball B[,.6].

10 2 B. C. Dhage et al. / On local attractivity of... Remark 4.3. We remark that: i Taking ut, xt = qt, pt, t = t t for all t, t R and for any x R the generalized nonlinear functional integral equation 4.1 reduces to the nonlinear functional integral equation considered by Dhage [8] which, in turn, includes several classes of known integral equations discussed in the literature. ii Taking pt, t = t, γt = t and θs = s for all t, s R, we retrieve the functional integral equation studied by Aghajani, Banas and Sabzali [1]. iii The authors in [1] generalized Theorem 2.2 under the weaker upper semi-continuity of the D-function ψ and the requirement of the condition that lim n ψ n r = for all t >, however to hold this condition, they needed an additional condition on the function ψ that ψr < r for r >. But in actual practice, it is very difficult to verify this condition and the authors in [1] did not provide any example of the function ψ illustrating the comparison between two conditions in applications. Moreover, for applications to the existence result, they assumed an additional condition on the function ψ, namely, supperadditivity which automatically yields the upper semi-continuity together with the monotone characterization of the function ψ and so, the existence theorem for the nonlinear integral equation considered in Aghajani et.al. [1] follows by a direct application of Theorem 2.2 of Dhage [8]. References [1] A. Aghajani, J. Banas and N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin 2 213, [2] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodhina and B. N. Sadovskii, Measures of noncompactness and Condensing Operators, Birkhauser Verlag, [3] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, LNPAM Vol. 6, Marcel Dekker, New York 198. [4] J. Banas and B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability, Appl. Math. Lett , 1-6. [5] G. Darbo, Punti uniti i transformazion a condominio non compatto, Rend. Sem. Math. Univ. Padova , [6] K. Deimling, Ordinary Differential Equations in Banach Spaces, LNM Springer Verlag, [7] B.C. Dhage, A nonlinear alternative with applications to nonlinear perturbed differential equations,nonlinear Studies , [8] B.C. Dhage, Asymptotic stability of the solution of certain nonlinear functional integral equations via measures of noncompactness, Comm. Appl. Nonlinear Anal , no. 2, [9] B.C. Dhage, Attractivity and positivity results for nonlinear functional integral equations via measures of noncompactness, Differ. Equ. Appl , [1] B.C. Dhage, Partially condensing mappings in ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math , [11] B.C. Dhage, Global attractivity results for comparable solutions of nonlinear hybrid fractional integral equations, Differ. Equ. Appl , [12] B.C. Dhage, V. Lakshmikatham, On global existence and attrctivity results for nonlinear functional integral equations, Nonlinear Analysis 72 21, [13] B. C. Dhage and S. K.Ntouyas, Existence results for nonlinear functional integral equations via a fixed point theorem of Krasnoselskii-Schaefer type, Nonlinear Studies ,

11 B. C. Dhage et al. / On local attractivity of [14] M.A. Krasnoselskii, Topological Methods in The Theory of Nonlinear Integral Equations, Pergamon Press, New York, [15] D. O Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integro-differential Equations, Kluwer Academic Dordrechet [16] M. Väth, Voltera and Integral Equations of Vector Functions, PAM, Marcel Dekker, New York 2. [17] E. Zeidler, Nonlinear Functional Analysis and Its Applications: Part I, Fixed Point Theory, Springer Verlag, Received: December 11, 214; Accepted: March 1, 215 UNIVERSITY PRESS Website: