Construction of a measure of noncompactness in Sobolev spaces with an application to functional integral-differential equations

In this paper, first we introduce a measure of noncompactness in the Sobolev space $$W^{k,1}(\Omega )$$Wk,1(Ω) and then, as an application, we study the existence of solutions for a class of the functional integral-differential equations using Darbo’s fixed point theorem associated with this new measure of noncompactness. Further, two examples are presented to verify the effectiveness and applicability of our main results.


Introduction
Sobolev spaces [11], i.e., the class of functions with derivatives in L p ; play an outstanding role in the modern analysis. In the last decades, there has been increasing attempts to study these spaces. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces. They also highlighted in approximation theory, calculus of variation, differential geometry, spectral theory etc.
On the other hand, integral-differential equations (IDE) have a great deal of applications in some branches of sciences. It arises especially in a variety of models from applied mathematics, biological science, physics and another phenomenon, such as the theory of electrodynamics, electromagnetic, fluid dynamics, heat and oscillating magnetic, etc. [9,12,18,21,24]. There have appeared recently a number of interesting papers [2,6,10,19,22,23,27] on the solvability of various integral equations with help of measures of noncompactness.
The first such measure was defined by Kuratowski [25]. Next, Banaś et al. [8] proposed a generalization of this notion which is more convenient in the applications. The technique of measures of noncompactness is frequently applicable in several branches of nonlinear analysis, in particular the technique turns out to be a very useful tool in the existence theory for several types of integral and integral-differential equations. Furthermore, they are often used in the functional equations, fractional partial differential equations, ordinary and partial differential equations, operator theory and optimal control theory [1, 3, 7, 13, 15-17, 26, 28, 29]. The most important application of measures of noncompactness in the fixed point theory is contained in the Darbo's fixed point theorem [4,5]. Now, in this paper, we introduce a new measure of noncompactness in the Sobolev space W k;1 ðXÞ as a more effective approach. Then, we study the problem of existence of solutions of the functional integral-differential equation We provide some notations, definitions and auxiliary facts which will be needed further on.
Throughout this paper, R þ indicates the interval ½0; þ1Þ and for the Lebesgue measurable subset D of R; m(D) denotes the Lebesgue measure of D. Moreover, let L 1 ðDÞ be the space of all Lebesgue integrable functions f on D equipped with the standard norm kf k L 1 ðDÞ ¼ R D jf ðxÞjdx. Let ðE; k Á kÞ be a real Banach space with zero element 0. The symbol Bðx; rÞ stands for the closed ball centered at x with radius r and put B r ¼ Bð0; rÞ. Denote by M E the family of nonempty and bounded subsets of E and by N E its subfamily consisting of all relatively compact sets of E. For a nonempty subset X of E, the symbols X and ConvX will denote the closure and the closed convex hull of X, respectively. Definition 1.1 [8] A mapping l : M E ! R þ is said to be a measure of noncompactness in E if it satisfies the following conditions: 3 lðXÞ ¼ lðXÞ. 4 lðConvXÞ ¼ lðXÞ. 5 lðkX þ ð1 À kÞYÞ klðXÞ þ ð1 À kÞlðYÞ for k 2 ½0; 1. 6 If fX n g is a sequence of closed sets from M E such that X nþ1 & X n for n ¼ 1; 2; . . . and if lim n!1 lðX n Þ ¼ 0 then the set X 1 ¼ \ 1 n¼1 X n is nonempty. A measure of noncompactness l is said to be regular if it additionally satisfies the following conditions: 7 lðX S YÞ ¼ maxflðXÞ; lðYÞg: 8 lðX þ YÞ lðXÞ þ lðYÞ: 9 lðkXÞ ¼ jkjlðXÞ for k 2 R: In what follows, we recall the well known Darbo's fixed point theorem.
Theorem 1.2 [13] Let X be a nonempty, bounded, closed and convex subset of a Banach space E and let F : X ! X be a continuous mapping such that there exists a constant k 2 ½0; 1Þ with the property for any nonempty subset X of X, where l is a measure of noncompactness defined in E. Then, F has a fixed point in the set X.

Construction of a measure of noncompactness in Sobolev spaces
In this section, we introduce a measure of noncompactness in the Sobolev space W k;1 ðXÞ.
Let X be a subset of R n and k 2 N, we denote by W k;1 ðXÞ the space of functions f which, together with all their distributional derivatives D a f of order jaj k, belong to L 1 ðXÞ. Here a ¼ ða 1 ; . . .; a n Þ is a multi-index, i.e., each a j is a nonnegative integer, jaj ¼ a 1 þ Á Á Á þ a n , and D a ¼ o jaj =ox a 1 1 . . .ox a n n : Then, W k;1 ðXÞ is equipped with the complete norm We present the following theorem which characterizes the compact subsets of the Sobolev spaces. (i) F is bounded, i.e., there is some M so that Z jD a f ðxÞjdx\M; f 2 F ; jaj k: Now, we are going to describe a measure of noncompactness in W k;1 ðXÞ. Theorem 2.2 Suppose 1 k\1 and U is a bounded subset of W k;1 ðXÞ. For u 2 U, e [ 0 and 0 jaj k, let x T ðu; eÞ ¼ supfkT h D a u À D a uk L 1 ðB T Þ : h 2 X; khk R n \e; 0 jaj kg; x T ðU; eÞ ¼ supfx T ðu; eÞ : u 2 Ug; x T ðUÞ ¼lim e!0 x T ðU; eÞ; where B T ¼ fa 2 X : kak R n Tg and T h uðtÞ ¼ uðt þ hÞ. Then x 0 : M W k; ðXÞ ! R given by defines a measure of noncompactness in W k;1 ðXÞ.
Proof Take U 2 M W k; ðXÞ such that x 0 ðUÞ ¼ 0. Fix arbitrary a such that 0 jaj k. Let g [ 0 be arbitrary, since x T ðU; eÞ ¼ 0: Thus, there exists small enough d [ 0 and large enough T [ 0 such that x T ðU; dÞ\g. This implies that for all u 2 U and h 2 X such that khk R n \d. Since g [ 0 was arbitrary, we obtain Using again the fact that and so for e [ 0 there exists large enough T [ 0 such that kD a uk L 1 ðXnB T Þ \e for all u 2 U: It follows then from Theorem 2.1 that U is totally bounded. Thus, 1 holds. 2 is obvious by the definition of x 0 . Now, we check that condition 3 holds. For this purpose, suppose that U 2 M W k; ðXÞ and fu n g & U such that u n ! u 2 U in W k;1 ðXÞ. From the definition of x T ðU; eÞ; we have kT h D a u n À D a u n k L 1 ðB T Þ x T ðU; eÞ; for any n 2 N, T [ 0 and h 2 X with khk R n \e. Letting n ! 1; we get for any T [ 0 and h 2 X with khk R n \e. Hence and thus From (3) and 2 we obtain x 0 ðUÞ ¼ x 0 ðUÞ. 4 follows directly from D a ½ConvðUÞ ¼ ConvðD a UÞ and hence is omitted.
The proof of condition 5 can be obtained by using the equality for all k 2 ½0; 1; u 1 2 X and u 2 2 Y.
It remains only to verify 6 , suppose that fU n g is a sequence of closed and nonempty sets of M W k; ðXÞ such that U nþ1 & U n for n ¼ 1; 2; . . ., and lim n!1 x 0 ðU n Þ ¼ 0. Now, for any n 2 N, take u n 2 U n and set G ¼ fu n g.
Claim: G is a compact set in W k;1 ðXÞ. Let e [ 0 be fixed, since lim n!1 x 0 ðU n Þ ¼ 0, there exists sufficiently large m 1 2 N such that x 0 ðU m 1 Þ\e. Hence, we can find small enough d 1 [ 0 and large enough On the other hand, we know that the set fu 1 ; for all n ¼ 1; 2; . . .; m 1 , 0 jaj k and h 2 X with khk R n \d 2 . Furthermore, which implies that kT h D a u n À D a u n k L 1 ðXÞ \3e; for all n ¼ 1; 2; . . .; m 1 . Thus, kT h D a u n À D a u n k L 1 ðXÞ \3e; and for all n 2 N, khk R n \ minfd 1 ; d 2 g and T ¼ maxfT 1 ; T 2 g: Therefore, all the hypotheses of Theorem 2.1 are satisfied, that completes the proof of the claim.
Using the above claim, there exists a subsequence fu n j g and u 0 2 W k;1 ðXÞ such that u n j ! u 0 . Since u n 2 U n , U nþ1 & U n and U n is closed for all n 2 N, we yield that finishes the proof of 6 . h We now investigate the regularity of x 0 .
x T ðX þ Y; eÞ x T ðX; eÞ þ x T ðY; eÞ; x T ðkX; eÞ ¼jkjx T ðX; eÞ; Then, the hypotheses 7 -9 hold. Next, we show that 10 holds. Take U 2 N W k; ðXÞ . Thus, the closure of U in W k;1 ðXÞ is compact. By Theorem 2.1, for all jaj k and for all e [ 0, there exists T [ 0 such that kD a uk L 1 ðXnB T Þ \e for all u 2 U; and there exists d [ 0 such that kT h D a u À D a uk L 1 ðB T Þ \e for all h 2 X with khk R n \d. Then, for all u 2 U we have khk R n \dg e: Therefore, x T ðU; dÞ ¼ supfkx T ðu; dÞk : u 2 Ug ¼ 0: It yields that Proof Applying the same strategy as ( [4], Theorem [14]), we observe that x 0 ðQÞ 3: It remains to verify x 0 ðQÞ ! 3: For any k 2 N, there exists E k & R n such that mðE k Þ ¼ 1 10k ,

Application
In this section, we study the existence of solutions for some functional integral-differential equations. We also provide some illustrative examples to verify effectiveness and applicability of our results. We start with some preliminaries which we need in subsequence.

Lemma 3.1 [14]
Let X be a Lebesgue measurable subset of R n and 1 p 1: If ff n g is convergent to f in the L pnorm, then there is a subsequence ff n k g which converges to f a.e., and there is g 2 L p ðXÞ, g ! 0; such that jf n k ðxÞj gðxÞ for a:e: x 2 X: Let X be a subset of R n and k 2 N, we denote by BC k ðXÞ the space of functions f which are bounded and ktimes continuously differentiable on X with the standard norm kf k BC k ðXÞ ¼ max where kD a f k u ¼ supfjD a f ðxÞj : x 2 Xg. Theorem 3.3 Let X be a subset of R n with mðXÞ\1. Assume that the following conditions are satisfied: (i) p 2 W 1;1 ðXÞ; q 2 BC 1 ðXÞ and (ii) g : X Â R nþ2 ! R satisfies the Carath eodory conditions and there exist a bounded continuous function a : X ! R þ with jaðxÞj M for all x 2 X and some M [ 0 and a concave, lower semi-continuous and nondecreasing function f : R þ ! R þ such that jgðx; u 0 ; u 1 ; . . .; u nþ1 Þj aðxÞfð max 0 i nþ1 ju i jÞ: ð8Þ (iii) k : X Â X ! R satisfies the Carath eodory conditions and has a derivative of order 1 with respect to the first argument. Moreover, there exist g 1 ; g 3 2 W 1;1 ðXÞ and g 2 2 L 1 ðXÞ such that jkðx; yÞj g 1 ðxÞg 2 ðyÞ; jkðx 1 ; yÞ À kðx 2 ; yÞj g 2 ðyÞjg 3 ðx 1 Þ À g 3 ðx 2 Þj; and ok ox i ðx; yÞ g 1 ðxÞg 2 ðyÞ; for almost x; y; x 1 ; x 2 2 X and 1 i n. (iv) There exists a positive solution r 0 of the inequality (v) T : W 1;1 ðXÞ ! L 1 ðXÞ is a continuous operator such that for any x 2 W 1;1 ðXÞ we have kTðxÞk L 1 ðXÞ kxk 1;1 : Then, the functional integral-differential equation Due to (11) and using condition (iv), we derive that F is a mapping from B r 0 into B r 0 . Now, we show that the map F is continuous. Let fu m g be an arbitrary sequence in W 1;1 ðXÞ which converges to u 2 W 1;1 ðXÞ. By Lemma 3.1 there is a subsequence fu m k g which converges to u a.e., f ou m k ox i g converges to f ou ox i g a.e., fTu m k g converges to Tu a.e. and there is h 2 L 1 ðXÞ, h ! 0; such that maxfju m k ðyÞj; j ou m k ox 1 ðyÞj; j ou m k ox 2 ðyÞj; . . .; jTu m k ðyÞjg hðyÞ for a.e. y 2 X: Since u m k ! u almost everywhere and g satisfies the Carath eodory conditions, it follows that for almost all y 2 X: Inequality (12) and condition (iii) imply that kFu m k À Fuk 1;1 ! 0 and oFu m k ox i À oFu ox i 1;1 ! 0 as k ! 1 ð1 i nÞ: Therefore, F : W 1;1 ðXÞ À! W 1;1 ðXÞ is continuous.
To finish, the proof we have to verify that condition (1) is satisfied. We fix arbitrary T [ 0 and e [ 0. Let U be a nonempty and bounded subset of B r 0 . Choose u 2 U and x; h 2 B T with khk R n e, then we have jkðx; yÞ À kðx þ h; yÞj g À y; uðyÞ; ou ox 1 ðyÞ; . . .; ou ox n ðyÞ; TuðyÞ Á dydx Obviously, x T ðp; eÞ ! 0, x T ðg 3 ; eÞ ! 0 and by continuity of q, Z B T jqðxÞ À qðx þ hÞjjuðxÞjdx ! 0; as e ! 0. Then the right hand side of (13) tends to kx T ðUÞ as e ! 0.
Example 3.5 Consider the following functional integraldifferential equation , then condition (ii) of Theorem 3.3 holds. Moreover, k is continuous and has a continuous derivative of order 1 with respect to the first argument. On the other hand, g 1 ðxÞ ¼ g 3 ðxÞ ¼ e x and g 2 ðxÞ ¼ e Àx . It can be easily shown that each number r ! 10 satisfies the inequality in condition (iv), i.e., Hence, as the number r 0 we can take r 0 ¼ 10. Consequently, all the conditions of Theorem 3.3 are satisfied. It implies that the functional integral-differential equation (18) has at least one solution in the space W 1;1 ðXÞ.
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