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AppliedMath
, Volume 2 (3) – Sep 19, 2022

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Article The Wiener–Hopf Equation with Probability Kernel and Submultiplicative Asymptotics of the Inhomogeneous Term Mikhail Sgibnev Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia; sgibnev@math.nsc.ru Abstract: We consider the inhomogeneous Wiener–Hopf equation whose kernel is a nonarithmetic probability distribution with positive mean. The inhomogeneous term behaves like a submultiplica- tive function. We establish asymptotic properties of the solution to which the successive approxima- tions converge. These properties depend on the asymptotics of the submultiplicative function. Keywords: Wiener–Hopf equation; inhomogeneous equation; nonarithmetic probability distribution; positive mean; submultiplicative function; asymptotic behavior MSC: 45E10; 60K05 1. Introduction The classical Wiener–Hopf equation has the form z(x) = k(x y)z(y) dy + g(x), x 0, or, equivalently, z(x) = z(x y)k(y) dy + g(x), x 0. Citation: Sgibnev, M. The Wiener–Hopf Equation with We shall consider the inhomogeneous generalized Wiener–Hopf equation Probability Kernel and Submultiplicative Asymptotics of the z(x) = z(x y) F(dy) + g(x), x 0, (1) Inhomogeneous Term. AppliedMath ¥ 2022, 2, 501–511. https://doi.org/ where z is the function sought, F is a given probability distribution on R, and the inho- 10.3390/appliedmath2030029 mogeneous term g is a known complex function. A probability distribution G on R is Received: 17 August 2022 called nonarithmetic if it is not concentrated on the set of points of the form 0, l, 2l, Accepted: 14 September 2022 . . . (see Section V.2, Deﬁnition 3 of [1]). Let R be the set of all nonnegative numbers and Published: 19 September 2022 R := RnR be the set of all negative numbers. For c 2 C, we assume that c/¥ is equal to zero. The relation a(x) cb(x) as x ! ¥ means that a(x)/b(x) ! c as x ! ¥; if c = 0, Publisher’s Note: MDPI stays neutral then a(x) = o(b(x)). with regard to jurisdictional claims in published maps and institutional afﬁl- iations. Deﬁnition 1. A positive function j(x), x 2 R, is called submultiplicative if it is ﬁnite, Borel measurable, and satisﬁes the conditions: j(0) = 1, j(x + y) j(x) j(y), x, y 2 R. The following properties are valid for submultiplicative functions deﬁned on the Copyright: © 2022 by the author. whole line (Theorem 7.6.2) of [2]: Licensee MDPI, Basel, Switzerland. This article is an open access article log j(x) log j(x) ¥ < r := lim = sup distributed under the terms and x!¥ x x x<0 conditions of the Creative Commons log j(x) log j(x) Attribution (CC BY) license (https:// inf = lim =: r < ¥. (2) x>0 x x!¥ x creativecommons.org/licenses/by/ 4.0/). AppliedMath 2022, 2, 501–511. https://doi.org/10.3390/appliedmath2030029 https://www.mdpi.com/journal/appliedmath AppliedMath 2022, 2 502 Here are some examples of submultiplicative function on R : (i) j(x) = (x + 1) , r > 0; (ii) j(x) = exp(cx ), where c > 0 and 0 < b < 1; and (iii) j(x) = exp(gx), where g 2 R. In (i) and (ii), r = 0, while in (iii), r = g. The product of a ﬁnite number of + + submultiplicative function is again a submultiplicative function. In the present paper, we investigate the asymptotic behavior of the solution to Equation (1), where F is a nonarithmetic probability distribution with ﬁnite positive mean m := x F(dx) and the function g(x) is asymptotically equivalent (up to a constant fac- tor) to a nondecreasing submultiplicative function j(x) tending to inﬁnity as x ! ¥: g(x) cj(x) as x ! ¥. In the main theorems (Theorems 2 and 3), j(x), x 2 R , is a nondecreasing submultiplicative function for which there exists lim j(x + y)/j(x) for x!¥ each y 2 R. If such a limit exists, then it is equal to exp(r y). Earlier [3], the asymptotic behavior of z was studied in detail under the following assumptions: (i) m 2 (0, +¥] and (ii) g belong to either g 2 L (0, ¥) or g 2 L (0, ¥). 1 ¥ Roughly speaking, if g 2 L (0, ¥), then z(x) tends to a speciﬁc ﬁnite limit as x ! ¥. Moreover, under appropriate conditions, a submultiplicative rate of convergence was given in the form o(1/j(x)). If g 2 L (0, ¥), then z(x) = O(x) or even z(x) = f (¥)x/m as x ! ¥, provided f (¥) := lim f (x) exists. x!¥ The existence of the solution to Equation (1) and its explicit form (5) were established in [4] for g 2 L (0, ¥) and arbitrary probability distributions F, regardless of whether F is of oscillating or drifting type. If m = 0 and if some other hypotheses are fulﬁlled, then z(x) tends to a speciﬁc ﬁnite limit as x ! ¥ (Theorem 4 of [4]). The stability of an integro-differential equation with a convolution type kernel was studied in [5,6]. 2. Preliminaries Consider the collection S(j) of all complex-valued measures {, such that k{k := j(x)j{j(dx) < ¥; here, j{j stands for the total variation of {. The collection S(j) is a Banach algebra with norm kk by the usual operations of addition and scalar multiplication of measures; the product of two elements n and { of S(j) is deﬁned as their convolution n{ (Section 4.16) of [2]. The unit element of S(j) is the measure d of unit mass concentrated at zero. Deﬁne the Laplace transform of a measure { as {b(s) := exp(sx){(dx). It follows from (2) that the Laplace transform of any { 2 S(j) converges absolutely with respect to j{j for all s in the strip P(r , r ) := fs 2 C : r <s r g. Let n and { be two complex-valued + + measures on the s-algebra B of Borel sets in R. Their convolution is the measure ZZ Z n{( A) := n(dx){(dy) = n( A x){(dx), A 2 B, fx+y2 Ag provided the integrals make sense; here, A x := fy 2 R : x + y 2 Ag. Denote by F the n-th convolution power of F: 0 1 (n+1) n F := d , F := F, F := F F, n 1. Let U be the renewal measure generated by F: U := F . n=0 Let X , k 1, be independent random variables with the same distribution F not concentrated at zero. These variables generate the random walk S = 0, S = X + . . . + X , 0 n 1 n n 1. Put T := min n 1 : S 0 . The random variable H := S is called the + n + ﬁrst weak ascending ladder height. Similarly, T := min n 1 : S < 0 and H := S AppliedMath 2022, 2 503 is the ﬁrst strong descending ladder height. We have the factorization identity (the symbol E stands for “expectation”). sX T sH T sH + + 1 xE(e ) = 1 E x e 1 E x e , jxj 1, <s = 0. (3) This can easily be deduced from an analogous identity in Section XVIII.3 of [1] for another collection of ladder variables. Denote by F the distributions of the random variables H , respectively. It follows from the identity (3) that d F = (d F ) (d F ). (4) 0 0 0 + ¥ k Let U := F be the renewal measures generated by the distributions F , k=0 respectively. Denote by 1 the indicator of the subset R in R: 1 (x) = 1 for x 2 R and R + R + + + 1 (x) = 0 for x 2 R . Extend the function g onto the whole line: g(x) := 0, x < 0. This convention will be valid throughout. Let n be a measure deﬁned on B, and a(x), x 2 R, a function. Deﬁne the convolution n a(x) as the function a(x y) n(dy), x 2 R. The following theorem has been proven in [4]. Theorem 1. Let F be a probability distribution and g 2 L (R ). Then, the function 1 + z(x) = U (U g)1 (x), x 2 R , (5) + R + is the solution to Equation (1), which coincides with the solution obtained by successive approxima- tions. If m is ﬁnite and positive, then m := x F (dx) is also ﬁnite and positive (Sec- + + tion XII.2, Theorem 2 of [1]). We have m = m (1 F (R )), U (R [f0g) = . (6) 1 F (R ) In fact, pass in (4) to Laplace transforms and divide both sides by s. We get b b 1 F(s) 1 F (s) = 1 F (s) , s 6= 0, <s = 0. s s Let s tend to zero. Then, the fractions on both sides will tend to m and m , respectively. The second equality in (6) is a consequence of the fact that the distribution F is defective, i.e., F (R ) < 1. Lemma 1. Let F be a nonarithmetic probability distribution, such that m = x F(dx) 2 (0, ¥) and let j(x), x 2 R, be a submultiplicative function with r 0 r . Assume that j(x)F((¥, x]) dx < ¥. Suppose additionally that F(r ) < 1 if r < 0. Then U 2 S(j). Proof. By Theorem 4 in [7] with n = 1 and Remark 5 therein, we have j(x) F (dx) < ¥, ¥ AppliedMath 2022, 2 504 i.e., F 2 S(j). Let us prove that the element n := d F is invertible in S(j). Let n = n + n + n be the decomposition of n into absolutely continuous, discrete, and c d s singular components. By Theorem 1 of [8], the element n 2 S(j) has an inverse if n(s) 6= 0 for all s 2 P(r , r ), and if 1 2 d d inf nb (s) > max jn j(r ), jn j(r ) . (7) d s s + s2P(r ,r ) c d s Let F = F + F + F be the decomposition of F 2 S(j) into absolutely continuous, d d s s discrete, and singular components. Then, n = d F and n = F . We have c d c d inf nb (s) 1 sup F (s) = 1 F (r ). s2P(r ,r ) s2P(r ,r ) d d c On the other hand, max jn j(r ), jn j(r ) = F (r ). Hence, in order to prove (7), it s s + sufﬁces to show that c d s c c 1 F (r ) F (r ) 1 F (r ) > 0. If r = 0, this follows from the fact that the distribution F is defective. Let r < 0. b b By assumption, F(r ) < 1 and, obviously, F (r ) < 1. Relation (4) implies b c c 1 F(s) = 1 F (s) 1 F (s) , s 2 P(r , r ), (8) + + whence 1 F (r ) > 0 and (7) follows. Finally, c c c jnb(s)j 1jF (s)j 1 F (jsj) 1 F (r ) > 0, s 2 P(r , r ). Therefore, by Theorem 1 in [8], the measure d F is invertible in the Banach algebra S(j) and U = (d F ) 2 S(j). The proof of the lemma is complete. Lemma 2. Let a(x), x 2 R , be a monotone nondecreasing positive function. Suppose that lim a(x + y)/a(x) = 1 for each y 2 R. Then, x!¥ a(x) = o a(y) dy as x ! ¥. Proof. Let M > 0 be arbitrary. We have Z Z Z x x x a(y) a(y) a(x M) a(x M) dy dy dy = M . a(x) a(x) a(x) a(x) 0 x M x M It follows that lim inf a(y) dy/a(x) = ¥. The proof of the lemma is com- x!¥ plete. Lemma 3. Let G be a nonarithmetic probability distribution on R , such that m := x G(dx) 2 (0, ¥) and let U be the corresponding renewal measure: U := G . Suppose that a(x) and b(x), G G n=0 x 2 R , are nonnegative functions such that a(x) b(x) as x ! ¥.Then, I(x) := U a(x) U b(x) =: J(x) as x ! ¥. G G Proof. Given # > 0, choose A > 0, such that (1 #)b(x) a(x) (1 + #)b(x), x A. AppliedMath 2022, 2 505 Let Z Z x A x I(x) = + a(x y) U (dy) =: I (x) + I (x). 1 2 0 x A Similarly, let J(x) = J (x) + J (x). Obviously, I (x) I (x) 1 1 1 # lim inf lim sup 1 + #. x!¥ J (x) J (x) 1 x!¥ 1 Since # is arbitrary, lim I (x)/ J (x) = 1, i.e., I (x) J (x) as x ! ¥. Moreover, x!¥ 1 1 1 1 I (x) a(x A)U ([0, x A]) ! ¥ as x ! ¥ by the elementary renewal theorem for the 1 G measure U : U ([0, x]) x/m as x ! ¥ (see Section 1.2 of [9]). According to Blackwell’s G G G theorem (Section XI.1, Theorem 1 of [1]), I (x) a( A)U ((x A, x]) ! a( A) A/m as x ! ¥. 2 G G Hence, I(x) I (x) as x ! ¥. A similar relation also holds for J(x), which completes the proof of the lemma. Lemma 4. Let j(x), x 2 R , be a submultiplicative function, such that there exists n(y) := lim j(x + y)/j(x) for each y 2 R. Then n(y) = exp(r y), y 2 R. x!¥ + Proof. By the Corollary of Theorem 4.17.3 in Section 4.17 of [2], n(y) = exp(ay) for some j(n + 1) a 2 R. Given # > 0, there exists n = n (#), such that log a + # for n n . 0 0 0 j(n) m(a+#) Hence, j(n + m) j(n )e and 0 0 log j(n + m) log j(n ) m(a + #) 0 0 r = lim lim + lim = a + #. m!¥ m m!¥ m m!¥ m Similarly, r a #. Since # > 0 is arbitrary, a = r . The proof of the lemma is + + complete. 3. Main Results Theorem 2. Let F be a nonarithmetic probability distribution, such that m = x F(dx) 2 (0, ¥) and let j(x), x 2 R , be a nondecreasing continuous submultiplicative function tending to inﬁnity as x ! ¥, such that r = 0 and there exists lim j(x + y)/j(x) for each y 2 R. Suppose + x!¥ that the inhomogeneous term g(x), x 2 R , is bounded on ﬁnite intervals and satisﬁes the relation g(x) cj(x) as x ! ¥, where c 2 C. Assume that j(jxj)F((¥, x]) dx < ¥. Then, the function z(x), x 2 R , deﬁned by (5) is a solution to Equation (1) and satisﬁes the asymptotic relation z(x) j(y) dy as x ! ¥. Proof. Put M(x) = j(y) dy. By Lemma 4, lim j(x + y)/j(x) = 1 for each y 2 R. x!¥ Extend the function j(x) onto the whole line R by setting j(x) = j(jxj) for x 2 R . The extended function retains the submultiplicative property and r = 0. To prove the AppliedMath 2022, 2 506 ﬁrst statement of the theorem, it sufﬁces to assume g 0. Choose C > 0, such that g(x) Cj(x), x 2 R . The function z(x) deﬁned by (5) is ﬁnite, since 0+ U g(x) CU j(x) = C j(x y) U (dy) Cj(x)kU k , z(x) CkU k j(x y) U (dy) CkU k j(x)U ([0, x]) < ¥ j + j + for all x 2 R . Let n be a natural number. Denote by 1 the indicator of [0, n]. Consider [0,n] Equation (1) with the inhomogeneous term g (x) = g(x)1 (x). Let z be the solution to n n [0,n] the equation z (x) = z (x y) F(dy) + g (x), x 2 R , (9) n n n + deﬁned by formula (5): z (x) = U (U g )1 (x), x 2 R . (10) n + n + The integral in (9) can be written as z (x y)1 (y) F(dy) z (x) z(x) < ¥. n n [0,x] The last two inequalities are consequences of (5). Obviously, z (x) " as n ". By Section 27, Theorem B of [10], the integral tends to z(x y) F(dy) as n " ¥. Letting n " ¥ in (9) and (10), we get that z is a solution to (1). Let us prove the assertion of the theorem for the solution z to (1) for g = j. Let us show that U j(x) ! U (R [f0g) as x ! ¥. (11) j(x) We have U j(x) j(x y) = U (dy). (12) j(x) j(x) By Lemma 4, the integrand tends to 1 as x ! ¥ and it is majorized by the U - integrable function j(y), since j(x y) j(y) = j(y) j(x) and U 2 S(j) by Lemma 1. Applying Lebesgue’s bounded convergence theorem (Sec- tion 26, Theorem D of [10]), we can pass to the limit under the integral sign in (12), which proves (11). Apply Lemma 3 with the following choice of G, a(x) and b(x): G := F , a(x) := 1 (x)U j(x), b(x) := U (R [f0g)1 (x)j(x). R R + + We get Z Z x x z (x) = U j(x y) U (dy) U (R [f0g) j(x y) U (dy) as x ! ¥. j + + 0 0 Recalling (6), we see that in order to prove the theorem for z , it sufﬁces to establish Z Z x x 1 1 U (1 j)(x) = j(x y) U (dy) j(y) dy = M(x) as x ! ¥. (13) + + m m 0 + 0 + AppliedMath 2022, 2 507 Integrating by parts, we get Z Z x x j(x y) U (dy) = j(x y)U ([0, y]) U ([0, y]) d j(x y) + + + y y=0 0 0 = U ([0, x]) j(x) U ([0, y]) d j(x y). (14) + + y The following three estimates hold: j(x), x, U ([0, x]) = o( M(x)) as x ! ¥. (15) The ﬁrst estimate follows from Lemma 2 with a(x) = j(x). The second one follows from the assumption j(y) ! ¥ as y ! ¥. The third estimate follows from the second one and the elementary renewal theorem for the measure U : U ([0, x]) x/m as x ! ¥. + + + Show that Z Z x x U ([0, y]) d j(x y) y d j(x y) as x ! ¥, (16) + y y 0 + 0 1 1 y d j(x y) M(x) as x ! ¥. (17) m m + 0 + We prove ﬁrst (17). This follows from the second estimate in (15) and the equality Z Z x x y d j(x y) = yj(x y) + j(x y) dy = x + M(x). y=0 0 0 Let # > 0 be arbitrary. Use the elementary renewal theorem and choose y = y (#), 0 0 such that (1 #)U ([0, y]) (1 + #)U ([0, y]), y y . + + 0 Write the left-hand side of (16) in the form Z Z y x + U ([0, y]) d j(x y) =: K (x) + K (x), + y 1 2 0 y and let M (x) + M (x) be a similar decomposition for the right-hand side. Obviously, 1 2 (1 #) M (x) K (x) (1 + #) M (x). (18) 2 2 2 Let us prove that, as x ! ¥, both sides in (16) are asymptotically equivalent to K (x) and M (x), respectively. We have Z Z x x x 1 y 1 M (x) = y d j(x y) = j(x y) + j(x y) dy 2 y m m m y y=y y + + 0 + 0 0 xy x y 1 = + j(x y ) + j(y) dy. m m m + + + Let us show that xy M (x) := j(y) dy M(x) as x ! ¥. Using the ﬁrst estimate in (15), we get Z Z x x j(y) dy j(x) j(y x) dy xy xy 0 0 = j(x) j(y) dy j(x)j(y )y = o( M(x)) as x ! ¥. 0 0 0 AppliedMath 2022, 2 508 Finally, Z Z x x M (x) 1 = j(y) dy j(y) dy M(X) M(x) 0 xy = 1 j(y) dy = 1 o(1) ! 1 as x ! ¥, M(x) xy which establishes the desired equivalence M (x) M(x) as x ! ¥. Taking into account the estimates in (15), we see that M (x) M(x)/m as x ! ¥. Moreover, 2 + y j(x y ) 1 0 0 M (x) = + j(u) du. m m + + xy The integral is estimated by y j(x)/m . Thus, M (x) = o( M(x)) as x ! ¥ (see (15)). 0 1 Relation (17) is proven. Now, divide all parts of (18) by M (x) and let x tend to inﬁnity. We obtain K (x) K (x) 2 2 1 # lim inf lim sup 1 + #. x!¥ M (x) M (x) 2 2 x!¥ Hence, K (x) M (x) M(x) as x ! ¥. Relation (16) is proven, since, as x ! ¥, 2 2 K (x) U ([0, y ]) d j(x y) 1 + 0 y = U ([0, y ])[j(x) j(x y )] U ([0, y ])j(x) = o( M(x)). + + 0 0 0 The equivalence (13) now follows from (14)–(17), which proves the theorem in the particular case g = j. Let g satisfy the hypotheses of the theorem. If, for some C > 0, jg(x)j Cj(x), x 2 R , then lim supjz(x)j j(y) dy . x!¥ It follows that if c = 0, then z(x) = o(z (x)) as x ! ¥. To see this, choose a small # > 0 and a natural number n, such that jg(x)j #j(x), x n. Write g = 1 g + (g 1 g) =: g + g . [0,n] [0,n] 1 Let z and z be the solutions to (1) corresponding to g and g , respectively. Then, 2 2 1 1 z = z + z and jz (x)j #z (x), x 2 R . By Theorem 6.2 in [3], z (x) = o(x) as x ! ¥. 1 2 2 j + 1 Since j(x) 1, x 2 R , it follows that z (x) = o j(y) dy as x ! ¥. Therefore, lim supjz(x)j j(y) dy . x!¥ Since # > 0 is arbitrary, the assertion of the theorem is true for c = 0. Let c 6= 0. Write g in the form g = cj + g . Then, g (x) = o(j(x)) as x ! ¥, and we have z = cz + z , 1 1 j 1 where z is the solution to Equation (1) with the inhomogeneous term g . The proof of the 1 1 theorem is complete. Theorem 3. Let F be a nonarithmetic probability distribution, such that m = x F(dx) 2 (0, ¥), and let j(x), x 2 R , be a nondecreasing submultiplicative function, such that r > 0, and + + there exists lim j(x + y)/j(x) for each y 2 R. Suppose that the inhomogeneous term g(x), x!¥ AppliedMath 2022, 2 509 x 2 R , is bounded on ﬁnite intervals and satisﬁes the relation g(x) cj(x) as x ! ¥, where c 2 C. Assume that j(jxj)F((¥, x]) dx < ¥ and F(r ) < 1. Then, the function z(x), x 2 R , deﬁned by (5) is a solution to Equation (1) + + and satisﬁes the asymptotic relation z(x) j(x) as x ! ¥. 1 F(r ) Proof. As in the proof of the preceding theorem, we verify that z(x) is a solution to (1). First, let us prove the assertion of the theorem for the solution z to (1) corresponding to g = j, i.e., let us prove that, as x ! ¥, z (x) U j(x y) 1 b b = U (dy) ! U (r )U (r ) = . (19) + + + + j(x) j(x) 0 1 F(r ) Write the integrand in the form U j(x y) j(x y) I(x, y) := 1 (y) , y 2 R . [0,x] j(x y) j(x) Notice that U j(x) j(x y) = U (dy) ! U (r ) as x ! ¥. (20) j(x) j(x) r y In fact, j(x y)/j(x) ! e as x ! ¥ by Lemma 4 and, according to Lemma 1, this ratio is majorized by the U -integrable function j(y), y 2 R : Z Z 0 0 U j(x) j(x y) = U (dy) j(jyj) U (dy) = kU k < ¥. j(x) j(x) ¥ ¥ Relation (20) now follows from Lebesgue’s bounded convergence theorem. Our further actions are as follows. We will pick out a majorant for the function I(x, y), y 2 R , in the by form Me with b 2 (r , 0). Then, by Lebesgue’s theorem, we pass to the limit under the integral sign in the left-side integral in (19) as x ! ¥, and thus prove relation (19). Put f (x) = log j(x) r x. By hypothesis, we have f (x y) f (x) = log j(x y) log j(x) + r x ! 0 as x ! ¥ (21) for each y 2 R. According to Lemma 1.1 in [11], relation (21) is fulﬁlled uniformly in y 2 [0, 1]. Hence, j(x y) exp(r y) ! 1 as x ! ¥ j(x) uniformly in y 2 [0, 1]. Choose a small # > 0 such that b := log(1 + #) r < 0. Let N = N(#) > 0 be an integer such that j(x y) exp(r y) 1 + #, x N, y 2 [0, 1]. j(x) AppliedMath 2022, 2 510 Denote by [x] the integral part of a real number x; i.e., [x] is the maximal integer not exceeding x: x = [x] + J, J 2 [0, 1). For y 2 [l, l + 1], l = 0, . . . , [x] N 1, we have j(x y) j(x l (y l)) j(x l) = , j(x) j(x l) j(x) j(x l (y l)) (1 + #) exp(r (y l)), j(x l) j(x l) j(x l) j(x l + 1) j(x 1) = . . . (1 + #) exp(lr ). j(x) j(x l + 1) j(x l + 2) j(x) Ultimately, j(x y) l+1 l+1 (1 + #) exp(r (y l)) exp(lr ) = (1 + #) exp(r y) + + + j(x) (1 + #) exp(by), y 2 [l, l + 1], l = 0, . . . , [x] N 1. Now, let y 2 ([x] N 1, x]. We have j(x y) j(N + 2) j(N + 2) j(N + 2) j(N + 2) exp(by). j(x) j(x) exp(r x) exp(r y) + + Thus, the U -integrable majorant sought for the function I(x, y), y 2 R , which does + + not depend on x, is of the form kU k maxf(1 + #), j(N + 2)g exp(by), y 2 R . j + Now, in order to prove relation (19), it sufﬁces, by Lebesgue’s theorem, to pass to the limit under the integral sign in (19). The last equality in (19) is a consequence of (8) for <s = r : 1 1 1 b b b U(s) = = = U (s)U (s), b b b 1 F(s) 1 F (s) 1 F (s) which is admissible, since b b b b jF(s)j F(r ) < 1, jF (s)j F (r ) < 1, <s = r . + + + In the general case, it sufﬁces to repeat the concluding reasoning of the previous proof using the estimate jz(x)j C lim sup j(x) 1 F(r ) x!¥ for jg(x)j Cj(x), x 2 R , and, considering the case c = 0, take into account the relation r x z (x) = o(x) as x ! ¥ and all the more z (x) = o(j(x)) as x ! ¥, since x e j(x), 1 1 x 2 R . 4. Conclusions We have established the asymptotic behavior of the solution z of the generalized Wiener–Hopf Equation (1), where the inhomogeneous term g behaves like an unbounded submultiplicative function, up to a constant factor, i.e., g(x) cj(x) as x ! ¥. Depending on whether r = 0 or r > 0, there are two different types of asymptotics for z (Theorems 2 + + and 3): either z(x) c j(y) dy or z(x) c j(x) as x ! ¥, where c and c are speciﬁc 1 2 1 2 constants. Here are two simple examples (c = 1): (i) If j(x) = (x + 1) , r > 0, then r+1 z(x) as x ! ¥; m(r + 1) AppliedMath 2022, 2 511 (ii) If j(x) = exp(gx), g > 0, then gx z(x) as x ! ¥. 1 F(r ) Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The work was carried out within the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF-2022-0004). Conﬂicts of Interest: The author declares no conﬂict of interest. References 1. Feller, W. An Introduction to Probability Theory and Its Applications; Wiley: New York, NY, USA, 1966; Volume 2. 2. Hille, E.; Phillips, R.S. Functional Analysis and Semi-Groups; American Mathematical Society Colloquium Publications; American Mathematical Society: Providence, RI, USA, 1957; Volume 31. 3. Sgibnev, M.S. Wiener–Hopf equation whose kernel is a probability distribution. Differ. Equ. 2017, 53, 1174–1196. [CrossRef] 4. Sgibnev, M.S. The Wiener–Hopf equation with probability kernel of oscillating type. Sib. Èlektron. Mat. Izv. 2020, 17, 1288–1298. [CrossRef] 5. Inoan, D.; Marian, D. Semi-Hyers–Ulam–Rassias Stability of a Volterra Integro-Differential Equation of Order I with a Convolution Type Kernel via Laplace Transform. Symmetry 2021, 13, 2181. [CrossRef] 6. Inoan, D.; Marian, D. Semi-Hyers–Ulam–Rassias Stability via Laplace Transform, for an Integro-Differential Equation of the Second Order. Mathematics 2022, 10, 1893. [CrossRef] 7. Sgibnev, M.S. Semimultiplicative moments of factors in Wiener—Hopf matrix factorization. Sb. Math. 2008, 199, 277–290. [CrossRef] 8. Sgibnev, M.S. On invertibilty conditions for elements of Banach algebras of measures. Math. Notes 2013, 93, 763–765. [CrossRef] 9. Smith, W.L. Renewal theory and its ramiﬁcations. J. R. Stat. Soc. Ser. B 1958, 20, 243–302. [CrossRef] 10. Halmos, P.R. Measure Theory; Springer: New York, NY, USA, 1974. 11. Seneta, E. Regularly Varying Functions; Springer: Berlin, Germany, 1976.

AppliedMath – Multidisciplinary Digital Publishing Institute

**Published: ** Sep 19, 2022

**Keywords: **Wiener–Hopf equation; inhomogeneous equation; nonarithmetic probability distribution; positive mean; submultiplicative function; asymptotic behavior

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